Rovelli on EPR
tttito
correlations do not entail any form of "non-locality", when viewed
in the context of a relational interpretation of quantum mechanics.'
This sounds somewhat familiar (cf. [2]).
IV
[1] http://arxiv.org/abs/quant-ph/0604064
[2] in
http://groups.google.com/group/sci.physics.research/msg/9bebf67819f08315
I wrote "Entanglement will then appear as a property of the
interaction/information-exchange between superposed D1 and D2 , when
measurement outcomes are matched/compared. In this setting nonlocality
disappears, together with the hidden assumptions that spawned it."
Oh No
>In barely one-month-old [1], Rovelli argues that 'EPR-type
>correlations do not entail any form of "non-locality", when viewed
>in the context of a relational interpretation of quantum mechanics.'
>
In so far as I can see, all this is saying is what we already know, that
there is nothing we can say within the formal structure of quantum
theory which yields a contradiction.
I think that is a slightly different, and somewhat weaker, statement
than the one which I, and I think also rof, would like to see, namely an
answer to the question "what is really going on?".
The program outlined by Rovelli (1996, Relational Quantum Mechanics,
Int. J. Th. Phys., 35 1637) is this:
"The program outlined is thus to do for the formalism of quantum
mechanics what Einstein did for the Lorentz transformations: i. Find a
set of simple assertions about the world, with clear physical meaning,
that we know are experimentally true (postulates); ii. Analyze these
postulates, and show that from their conjunction it follows that certain
common assumptions about the world are incorrect; iii. Derive the full
formalism of quantum mechanics from these postulates. I expect that if
this program could be completed, we would at long last begin to agree
that we have understood quantum mechanics".
I believe that that is what I have done in gr-qc/0508077. But it only
answers the question "what is quantum mechanics saying", and I am not
convinced that it is really a huge step forward from what Von Neumann
was saying when he identified Hilbert space with a formal language, vis
quantum logic. In the context of EPR I don't think it answers the
question, "what is really going on?".
To answer that question I think we have to go much deeper. I think we
have to first use the formalism of quantum mechanics to construct
quantum electrodynamics - itself regarded as an unsolved question which
I tackle in Discrete Quantum Electrodynamics (physics/0101062). Then we
have to introduce a physical metric by reworking Einstein's development
of special relativity in terms of photon interactions as defined in qed.
This will show us that the metric is a product of particle interactions,
and not a prior physical property of space. A satisfactory treatment
will produce gtr, rather than sr, which I seek to show in gr-qc/0508077.
Such a treatment shows that spin is not a property of a particle in
isolation, but a part of a relationship between a particle and space-
time. At the time when the entangled pair is produced, as it seems to
me, the relationship which they have with spacetime is not fully
defined, and in particular their spin properties are not defined. Their
spin properties only become defined when they interact with A's or B's
measurement apparatus. Spin is conserved, so that the spin property does
become defined, it becomes defined for the past as well as the present.
Thus A's measurement determines the spin relationship between the
particles and space-time at the time of the original production of the
particles, and hence it also determines it in B's measurement.
As an explanation that does not violate locality. It does violate a
traditional notion of causality, which is one of the possibilities
mentioned by Bell. I don't have much of a problem with that. If space-
time only exists as a consequence of particle interactions, then
traditional causality is out of the window anyway. Also it ties in with
time reversibility and the Feynman-Stuckelberg interpretation that an
antiparticle is a time reversed particle. As has been discussed
elsewhere, the "arrow of time" which we perceive is a result of entropy,
a statistical effect based on many particle interactions. I see no
reason to think that the arrow of time should exist within the quantum
domain.
Regards
--
Charles Francis
substitute charles for NotI to email
Eugene Stefanovich
> The program outlined by Rovelli (1996, Relational Quantum Mechanics,
> Int. J. Th. Phys., 35 1637) is this:
>
> "The program outlined is thus to do for the formalism of quantum
> mechanics what Einstein did for the Lorentz transformations: i. Find a
> set of simple assertions about the world, with clear physical meaning,
> that we know are experimentally true (postulates); ii. Analyze these
> postulates, and show that from their conjunction it follows that certain
> common assumptions about the world are incorrect; iii. Derive the full
> formalism of quantum mechanics from these postulates. I expect that if
> this program could be completed, we would at long last begin to agree
> that we have understood quantum mechanics".
I believe this program was successfully completed a while ago:
G. Birkhoff and J. von Neumann,
The logic of quantum mechanics, Ann. Math. 37 (1936), 823.
G. W. Mackey, The mathematical foundations of
quantum mechanics (W. A. Benjamin, New York, 1963), see
esp. Section 2-2.
C. Piron, Foundations of Quantum Physics,
(W. A. Benjamin, Reading, 1976)
Eugene Stefanovich.
Cl.Massé
1147284408.8...@y43g2000cwc.googlegroups.com
> In barely one-month-old [1], Rovelli argues that 'EPR-type
> correlations do not entail any form of "non-locality", when viewed
> in the context of a relational interpretation of quantum mechanics.'
I would rather say: "when not looked at" As soon as you compare each side,
locality doesn't intervene since it can only be made at a same point. The
relational interpretation entails some kind of globality, which when set
aside as belonging to the theory and not to the system, leave
nothing non-local. Indeed, this interpretation only considers relations
between systems at a given point, similarly as a circle is seen locally as a
line.
> This sounds somewhat familiar (cf. [2]).
>
> IV
>
> [1] http://arxiv.org/abs/quant-ph/0604064
> [2] in
> http://groups.google.com/group/sci.physics.research/msg/9bebf67819f08315
> I wrote "Entanglement will then appear as a property of the
> interaction/information-exchange between superposed D1 and D2 , when
> measurement outcomes are matched/compared. In this setting nonlocality
> disappears, together with the hidden assumptions that spawned it."
The relational interpretation isn't necessary to see that, the Copenhagen
interpretation suffices if used appropriately, that is, if the collapse
occurs only at the time of the comparison. Quantum mechanics give exactly
the same result whenever is introduced the projection onto the eigenstate,
be it at the time of the polarization measurement or even just before
conscious perception.
Of course, the advertisement of the relational interpretation is quite
another stuff.
--
~~~~ clmasse on free F-country
Liberty, Equality, Profitability.
Oh No
foundations which are not nearly so sure. Neither are in print. I am not
convinced that Birkhoff and Von Neumann gave a set of simple assertions
about the world with clear physical meaning. They simply said that qm
has the structure of a formal language which tells us everything we can
find out from experiment, and that, pretty much is what is said of the
orthodox interpretation. To carry out the program the language must also
be shown to make sense in translation into English, imv. And it should
not simply enable us to predict the results of experiments, it should
work from precepts which, like Einstein's, are obviously true.
If you think the programme has been completed, perhaps you could tell
me, for example, why the Schrodinger equation is obeyed.
Cl.Massé
news: 4463CBC...@synopsys.com
> I believe this program was successfully completed a while ago:
>
> G. Birkhoff and J. von Neumann,
> The logic of quantum mechanics, Ann. Math. 37 (1936), 823.
>
> G. W. Mackey, The mathematical foundations of
> quantum mechanics (W. A. Benjamin, New York, 1963), see
> esp. Section 2-2.
>
> C. Piron, Foundations of Quantum Physics,
> (W. A. Benjamin, Reading, 1976)
They aren't, by far, the only people who claim to have "understood" quantum
mechanics. Alas, all these works either muddy the water, claim to "shed
some light" on an anyway unessential part of QM, or merely reformulate or
renomenclaturize it without solving the interpretation problem.
But we don't yet know what is the incorrect assumption, that is, something
contradictory with, and replaced by the logical consequences of the
postulates. Indeed, many assumptions may be removed giving back
consistency, but at the expense of completeness.
I think the incorrect assumption is linearity and locality, for a good and
simple reason, without them it is virtually impossible to make
calculations. Together, they reduce the set of available tractable concepts
to a doubleton: corpuscle and wave. For long the question has been "which?"
Now it should be "What else?"
tttito
...
> In so far as I can see, all this is saying is what we already know, that there is
>nothing we can say within the formal structure of quantum theory which yields a
>contradiction.
The "novelty" in Smerlak-Rovelli ([1]) is that they introduce the
notion of local information exchange between superposed observers as
the key to EPR locality. Here are a few pointers.
"What changes instantaneously at time t 0, for A , is not the objective
state of ß , but only its (subjective) relative state, that codes the
information that A has about ß .... if I see an elephant and I ask you
what you see, I expect you to tell me that you too you see an elephant.
If not, something is wrong. ... everybody hears everybody else stating
that they see the same elephant he sees. This, after all, is the best
definition of objectivity.
[i.e. in RQM regards "objective" reality as a locus of intersubjective
agreement and ...]
"any such conversation about elephants is ultimately an interaction
between quantum systems"
[i.e. between superposed observers, see below]
"This fact may be irrelevant in everyday life, but disregarding it may
give rises to subtle confusions, such as the one leading to the
conclusion of nonlocal EPR influences. ... .In the EPR situation, A and
B can be considered two distinct observers, both making measurements on
á and â. The comparison [!!!] of the results of their measurements,
we have argued, cannot be instantaneous, that is, it requires A and B
to be in causal contact."
[i.e. since information exchange is a local process, EPR is local]
"More importantly, with respect to A, B is to be considered as a normal
quantum system (and, of course, with respect to B, A is a normal
quantum system)"
[i.e A is superposed in B's perspective and viceversa]
As far as superpositions can be detected, all the above is testable.
Cheers,
IV
PS Smerlak and Rovelli also realise that "From this perspective,
probability needs clearly to be interpreted subjectively", i.e.
according to the DeFinetti interpretation which I have been ranting
about ([2]).
[1] http://arxiv.org/abs/quant-ph/0604064
[2]
http://groups.google.com/group/sci.physics.research/msg/fc81d7a091622078
J. Horta
> ...................................... They simply said that qm
> has the structure of a formal language which tells us everything we can
> find out from experiment, and that, pretty much is what is said of the
> orthodox interpretation. To carry out the program the language must also
> be shown to make sense in translation into English, imv. And it should
> not simply enable us to predict the results of experiments, it should
> work from precepts which, like Einstein's, are obviously true.
>
IMV nothing Einstein said was obvious in the above sense. That the speed
of light is (or is not for that mater) a constant independent of
inertial frame simply fits the facts better. I for one can't say
even in retrospect that this is or is not the more obvious fact.
> If you think the programme has been completed, perhaps you could tell
> me, for example, why the Schrodinger equation is obeyed.
>
No more than you could tell me why local geometry is Lorentzian or
why geometry exists at all. To me these things are not obvious they
simply fit the data and facts.
> Regards
sigol...@gmail.com
replace non-locality by the axiom that "reality is local" to the
extent that a distant measurement event does not become "real" for an
observer until it enters that observer's past light cone.
So e.g., there is no "element of reality" (for observer Alice) to be
associated with the (for her as yet unmeasured) spin state of the
distant space-like separated member of a pair of spin anti-correlated
particles, at the time of her selection of the earlier (for her)
orientation of the axis of the measurement of the near particle's
spin. Such an "element of reality" need only arise for her when the
distant (Bob's) measurement enters into her past. Symmetrically
similar for Bob, or any other observer.
There is (to me) an oblique reference to this loophole in the original
EPR paper and other writings of Einstein, but this localization of
"elements of reality" was apparently distasteful to him.
So the question becomes, what, if anything to we actually give up by
localizing "reality" in this way?
Eugene Stefanovich
Cl.Massé wrote:
>>I believe this program was successfully completed a while ago:
>>
>>G. Birkhoff and J. von Neumann,
>>The logic of quantum mechanics, Ann. Math. 37 (1936), 823.
>>
>>G. W. Mackey, The mathematical foundations of
>>quantum mechanics (W. A. Benjamin, New York, 1963), see
>>esp. Section 2-2.
>>
>>C. Piron, Foundations of Quantum Physics,
>>(W. A. Benjamin, Reading, 1976)
>
>
> They aren't, by far, the only people who claim to have "understood" quantum
> mechanics. Alas, all these works either muddy the water, claim to "shed
> some light" on an anyway unessential part of QM, or merely reformulate or
> renomenclaturize it without solving the interpretation problem.
>
> But we don't yet know what is the incorrect assumption, that is, something
> contradictory with, and replaced by the logical consequences of the
> postulates. Indeed, many assumptions may be removed giving back
> consistency, but at the expense of completeness.
>
> I think the incorrect assumption is linearity and locality, for a good and
> simple reason, without them it is virtually impossible to make
> calculations. Together, they reduce the set of available tractable concepts
> to a doubleton: corpuscle and wave. For long the question has been "which?"
> Now it should be "What else?"
Are you saying that quantum mechanics is logically inconsistent?
Do I understant you right? Where do you see the inconsistency?
Eugene.
Arkadiusz Jadczyk
<No...@charlesfrancis.wanadoo.co.uk> wrote:
>If you think the programme has been completed, perhaps you could tell
>me, for example, why the Schrodinger equation is obeyed.
In fact, it does not have to be obeyed. In theories that are based on
"flash ontology" or "event ontology"
(See e.g. "Some Comments on the Formal Structure of Spontaneous
Localization Theories" http://arxiv.org/abs/quant-ph/0603046 )
the Schrodinger equation is valid only when "no measurement is being
made". But measurements are being made all the time!
Von Neumann and Jauch did not define what a measurement is. John Bell
was angry with that lack of a dynamical definition. That is why
physicists (de Broglie, Bohm, Ghirardi-Rimini-Weber, Bell etc. ) were
looking for alternatives which better correspond to the elementary
observations)
ark
--
Arkadiusz Jadczyk
http://quantumfuture.net/quantum_future/jadpub.htm
--
Eugene Stefanovich
Briefly, what these people did is the following:
1) they recognized that classical Boolean logic has its realization
in terms of unions and intersections of subsets of a given set. In
classical physics this set can be identified with the phase space of
the physical system.
2) they noticed that the distributive law of classical logic is far
from obvious, and they substituted it with a weaker postulate
(orthomodularity). This gives rise to the so-called "quantum logic".
This logic has a mathematical realization in terms of intersections and
spans of closed subspaces in the Hilbert space. So, the phase space
of classical mechanics should be generalized to the Hilbert space
of quantum mechanics.
I think, this is a powerful result as it shows that classical theories
form a subset of quantum theories: classical distributivity is a
particular case of quantum othomodularity.
The interpretation of quantum mechanics that is most consistent with
the Bikhoff-von Neumann logical approach is the "ensemble" or
"statistical" interpretation presented in
L.E. Ballentine Quantum Mechanics: A Modern
Development (World Scientific, Singapore, 1998)
I highly recommend this book.
>
> If you think the programme has been completed, perhaps you could tell
> me, for example, why the Schrodinger equation is obeyed.
Yes, quantum logic says nothing about dynamics. In order to get the
Schrodinger equation you need to add the principle of relativity to
your postulates. As demonstrated by
E. P. Wigner, On unitary representations of
the inhomogeneous Lorentz group, Ann. Math. 40 (1939), 149.
and
P. A. M. Dirac, Forms of relativistic
dynamics, Rev. Mod. Phys. 21
(1949), 392.
this requires a definition of the unitary representation of the
Poincare group in the Hilbert space of the system. The Hamiltonian
is a representative of the generator of time translations of the
Poincare group, and the Schrodinger equation
-ih d/dt |Psi(t)> = H |Psi(t)>
is just a compact form of writing how the state vector changes
under time translations.
Eugene.
r...@maths.tcd.ie
>Cl.Massi wrote:
>>>I believe this program was successfully completed a while ago:
>>>
>>>G. Birkhoff and J. von Neumann,
>>>The logic of quantum mechanics, Ann. Math. 37 (1936), 823.
>>>
>>>G. W. Mackey, The mathematical foundations of
>>>quantum mechanics (W. A. Benjamin, New York, 1963), see
>>>esp. Section 2-2.
>>>
>>>C. Piron, Foundations of Quantum Physics,
>>>(W. A. Benjamin, Reading, 1976)
>>
>> They aren't, by far, the only people who claim to have "understood" quantum
>> mechanics. Alas, all these works either muddy the water, claim to "shed
>> some light" on an anyway unessential part of QM, or merely reformulate or
>> renomenclaturize it without solving the interpretation problem.
On page 71, Mackey introduces "Axiom VII", which says:
The partially ordered set of all questions in quantum mechanics is
isomorphic to the partially ordered set of all closed subspaces
of a separable, infinite dimensional Hilbert space.
That this is an axiom, and not a theorem, is the sense in which
we do not know why quantum mechanics works. Mackey is refreshingly
honest about this point:
This axiom has rather a different character from Axioms I through
VI. These all had some degree of physical naturalness and
plausibility. Axiom VII seems entirely ad hoc. Why do we make it?
Can we justify making it? What else might we assume? We shall
discuss these questions in turn. The first is the easiest to
answer. We make it beacuse it "works," that is, it leads to
a theory which explains physical phenomena and successfully
predicts the results of experiments. It is conceivable that a
quite different assumption would do likewise but this is a
possibility that no one seems to have explored. Indeed, one
would like to have a list of physically plausible assumptions
from which one could deduce Axiom VII. Short of this one
would like a list from which one could deduce a set of
possibilities for the structure of Q, all but one of which
could be shown to be inconsistent with suitably planned
experiments. At the moment such lists are not available
and we are far from being forced to accept axiom VII as
logically inevitable. ...
Mackey's description of the situation is fairly accurate, and he
thankfully does not attempt (as, for example, Gottfried does,
with his "algebra of filters") to dupe the reader into thinking
that we know why the probabilities assigned by quantum mechanics
coincide with experimentally observed frequencies of individual
experimental results. Birkhoff and Von Neumann's argument, which
Mackey refers to, is the closest that anybody has come, as far
as I can determine, to providing some justification for the
use of quantum mechanics, although at best they have shown
that quantum mechanics is one of a number of procedures which
could conceivably be used to assign probabilities.
Charles has presented a paper which he claims provides the
explanation required. I have examined this paper but cannot
find a satisfactory explanation of why quantum mechanics,
rather than some other procedure, is the correct procedure
to use for assigning probabilities to the results of
measurements. There are some things in the paper with which
I agree, but two of the principal foundations, namely
relationalism and quantum logic, seem to me to be unlikely
to be fruitful.
I have explained before why I find relationalism to be incoherent.
Either it is a founding principle on which his argument is based,
which I hope it is not, because it would be difficult to base
a coherent argument on an incoherent principle, or the argument
can be presented without any appeal to relationalism (or any
other "ism"), which would greatly improve its clarity.
The use of quantum logic also seems to me to present no improvement
in understanding, since it merely replaces one unexplained procedure
with another. To say that the procedures used in quantum mechanics
consitute logic is at best a poetical metaphor, because quantum
mechanics is literally not logic. When somebody says that statements
have "truth values" which can be complex, I do not know what they
mean. I can consider maps from statements to sets other than
{true,false}, but if somebody wants to say that these are "truth
values", then there are two possible cases.
One possibility is that this is just the introduction of a new term,
"truth value", and that any other term could have been used just
as well, for example "fribble". In that case, one would have to
conclude that quantum logic is not a more appropriate name than
quantum fribbology, and that those who opt to study quantum logic
should be warned that it is not a system used to model inference,
as actual logic is, but that it is a particular mathematical system
to which actual normal true-and-false logic applies, and that calling
it logic is just poetry.
(In mathematics, we often speak of a distance between numbers, and
this is an example of perhaps similar poetry, because numbers do
not have locations in space, and the map from (say) R times R to R
could have been given any other name apart from distance. It could
have been called the alpha function, for example, and mathematics
would be just the same. The usage of terms like "distance" is
whimsical and metaphorical.)
The other possible position that an advocate of quantum logic
could adopt is that the choice of the term "truth value" does
indeed carry an important message of some kind, and then
they have the obligation to tell us what that message is. Is
the message that normal logic has to be abandoned and replaced
by this new system? Why, and how could anybody possibly ever
know such a thing?
In Charles' paper, and in my correspondence with him, he has
been keen to insist that his introduction of the entire Hilbert
space formalism, along with the identification of kets as
propositions about measurement results in a formal language,
is little more than a choice of notation. However, the set
of things which can be proven about the results of measurements
if one merely accepts the Hilbert space formalism and the
rules of quantum mechanics for assigning probabilities to
the results of measurements is non-trivial.
For example, if there are 10 mutually exclusive complete
measurements than I can perform on a system, each
of which can give 10 possible results, then a complete
specification of the probabilities of obtaining each
of the 100 results requires 90 parameters to specify.
That's 100 possible results, grouped into sets of 10.
In each set of 10, I assign 9 probabilities, and the
10th is fixed by the fact that the probabilities must
sum to one.
If I suppose that measuring the same thing twice always
gives the same result, then I can prepare a system by
making a measurement. Then 10 probabilities are fixed
(one probability is 1 - if I measure the same thing
again, I will get the same result, nine are zero - if
I measure the same thing again, the chance of me getting
any of the other nine results is zero). That leaves 81
parameters which I need in order to specify the probabilities
of obtaining the various results of the different measurements
I could perform.
Quantum mechanics, however, tells us that we can
specify all of the probabilities with only 18 parameters.
All that we need to do is specify a ray in a ten-dimensional
Hilbert space, so that's 9 complex numbers or 18 parameters.
So if we have accepted the Hilbert space formalism, and if
we have accepted that the square of the inner product
gives us probabilities of finding particular measurement
results given particular preparations, then we have
already agreed to some very non-trivial statements
about the results of experiments and how many parameters
are needed to describe them. A mere choice of notation
can't change an 81-dimensional space into an 18-dimensional
space. It is noteworthy that this is all before any mention
has been made of time evolution, the Hamiltonian, or any dynamics.
The question is: "Why are there 18 and not 81 parameters needed to
specify the probabilities of obtaining the various possible measurement
results?" Right now, the only answer we have is that the symbols
tell us so, and I do not believe that Charles, Mackey, Rovelli,
Birkhoff & Von Neumann, or anybody else has come even remotely
close to addressing this question.
R.
Oh No
>On Sun, 14 May 2006 20:26:27 +0000, Oh No wrote:
>
>> has the structure of a formal language which tells us everything we can
>> find out from experiment, and that, pretty much is what is said of the
>> orthodox interpretation. To carry out the program the language must also
>> be shown to make sense in translation into English, imv. And it should
>> not simply enable us to predict the results of experiments, it should
>> work from precepts which, like Einstein's, are obviously true.
>>
>
>of light is (or is not for that mater) a constant independent of
>inertial frame simply fits the facts better. I for one can't say
>even in retrospect that this is or is not the more obvious fact.
One makes the argument more rigorous if one replaces speed of light with
maximum speed of information transfer - that way one finds that sr does
not actually depend on the photon having zero mass, for example.
>
>> If you think the programme has been completed, perhaps you could tell
>> me, for example, why the Schrodinger equation is obeyed.
>>
>
>No more than you could tell me why local geometry is Lorentzian or
>why geometry exists at all. To me these things are not obvious they
>simply fit the data and facts.
Certainly it is not obvious, but it is possible to show that a
Schrodinger equation is required by local Lorentz invariance. In essence
when this condition is combined with quantum logic one is restricted to
qed and certain generalisations of it.
As for why geometry is locally Minkowski, there are only two
possibilities. Either there is a maximum speed of information transfer,
in which case we have sr locally, or there is no maximum speed of
information transfer. In the latter case there is a concept of absolute
time and instantaneous action at a distance is permitted. There will be
other conclusions. Personally I don't like the concept of instantaneous
action at a distance, but I do not know if a theory formulated like this
can work as a purely theoretical construction. What I do know is that we
can reject it empirically, so I don't see much point in attempting to
bring such a model up to a level of theoretical development where it can
be properly examined for theoretical consistency.
Ilja Schmelzer
> So the question becomes, what, if anything to we actually give up by
> localizing "reality" in this way?
The answer has been found by Bell.
If we want to preserve Einstein causality, we have
to give up realism in the form used by Bell in his proof.
That means, we can talk only about statistical
experiments, with probability distributions
rho(m,c) influenced by macroscopic decisions
of observers c in C and resulting expectation values
for functions f on the measurement results M
E(f|c) = int f(m) rho(m,c) dm.
We can no longer talk about some state of
reality x in X in each particular experiment,
which, together with the decisions c,
defines to measurement results m so that
E(f|c)= int f(m(x,c)) rho(x) dx.
and talk about causality in the form that
the decision c influences the measurement
result m if m(x,c) depends on c.
This essentially weakens the notion of
causality - it remains possible to talk
only about statistical influence of c on
rho(m|c).
IMHO, we have to give up far too much,
in comparison with the realistic alternative
(Bohmian mechanics etc.).
Ilja
Oh No
>Oh No wrote:
>
If so, then I do not accept this part of it. What we can say about
reality is subjective. That does not mean there is no objective reality.
I do not think Rovelli thinks there is no objective reality either.
As for the rest of what you say, it does not alter what I said. In
discussions here on spr a few years back Matt McIrvin proved very good
at expressing this point of view. Nonetheless it does not alter the
fundamental position of the realist, that reality exists whether or not
it is outside the light cone.
>
>
>As far as superpositions can be detected, all the above is testable.
>
>Cheers,
>
>IV
>
>PS Smerlak and Rovelli also realise that "From this perspective,
>probability needs clearly to be interpreted subjectively",
Certainly, in the sense that probability depends on the information
available to an observer.
Oh No
Also to me.
>
>So the question becomes, what, if anything to we actually give up by
>localizing "reality" in this way?
>
Yes, I think there is. We have to give up realism, that some sort of
material reality exists independent of observation, and adopt some sort
of sophisticated form of idealism in which reality only exists for
observers inside a light cone. If we are doing that then I don't think
we have really solved the question of interpretation at all.
mark...@yahoo.com
> "The program outlined is thus to do for the formalism of quantum
> mechanics what Einstein did for the Lorentz transformations: i. Find a
> set of simple assertions about the world, with clear physical meaning,
> that we know are experimentally true (postulates); ii. Analyze these
> postulates, and show that from their conjunction it follows that certain
> common assumptions about the world are incorrect; iii. Derive the full
> formalism of quantum mechanics from these postulates. I expect that if
> this program could be completed, we would at long last begin to agree
> that we have understood quantum mechanics".
>
> answers the question "what is quantum mechanics saying", and I am not
> convinced that it is really a huge step forward from what Von Neumann
> was saying when he identified Hilbert space with a formal language, vis
> quantum logic. In the context of EPR I don't think it answers the
> question, "what is really going on?".
What Rovelli is asking for can be done more simply still. I show this
in the suggestively titled "On The Quantum Dynamics of Moving Bodies",
which can be found under
http://federation.g3z.com/Physics/Index.htm
I'll summarize, here, what's done there.
In effect, the postulates are
(1) A system is described by a set (q1,q2,...,qN) of configuration
space coordinates that satisfy a 2nd order law of motion, q''(t) =
A(q(t),q'(t))
(2) At each time [q(t),q(t)] = 0
(3) At each time [q(t),[q(t),q'(t)]] = 0.
A stronger version of (3) is taken: particularly, that the [q,q']
commutators are c-numbers. There is a brief mention of weakening this
condition to only requiring that [q,q'] be a function of q's only,
independent of q'.
For the following, define the "quantum" Poisson brackets by
{A,B} = [A,B]/(i h-bar).
What Hojman and Shepley showed in 1990 is that if one assumes that the
classical limit of the {q,q'} commutators forms a non-singular matrix,
W, then W will be the inverse mass matrix of a non-singular Lagrangian
system. Thus, the quantum theory must have a classical Hamiltonian
theory as its classical limit, and must therefore be the quantization
of a classical Hamiltonian theory.
As I point out initially, this generalizes. If one starts out with
general phase space coordinates {s1,s2,...,sp} without making any
assumptions about the commutators; and one assumes they satisfy a first
order law s'(t) = V(s(t)), then in the classical limit one gets what is
known as a Poisson manifold.
Locally, this layers into what are known as "symplectic leaves". This
layering defines what eventually becomes the superselection structure
of the quantum theory. On each layer, the Poisson tensor Omega = {s,s'}
is invertible, with Omega^{-1} = omega, which yields the symplectic
form
omega_{ij} ds^i ^ ds^j.
The requirement that the commutators cohabit with the equations of
motion is a very strong restriction and, here, implies that the system
comes from a Lagrangian that is the first order in ds/dt -- as
expected.
Locally, the coordinates can be divided into coordinates and velocities
s = (q, v = dq/dt) such that the symplectic form takes on the form
m_{ij} dq^i dv^j,
which yields the mass matrix m = d^L/dv^2 of a Lagrangian. The inverse
W = m^{-1} gives you the [q,v] commutator matrix (in the classical
limit),
{q, dq/dt} -> W, as h-bar -> 0.
The rest of the writeup focuses on the special case where the total
system has a decomposition into (q, v), with a 1st order law (dq/dt =
v, dv/dt = A(q,v)), such that the (q) part is a classical algebra,
[q,q] = 0, and [q,v] is restricted to c-number.
There is, at this point, no restriction on the number N of degrees of
freedom (N may be infinite). This case, therefore, covers much of field
theory and the quantum mechanics. What's not included in here is a
prospective quantum mechanics of a test particle in a gravitational
field, since then the [q,v] commutators will yield -- up to proportion
-- the dual spacetime metric, which is dependent on q.
Again, here, the requirement of having both the commutators and the
equations of motion is a severe restriction. It shows up by
differentiating the [q,q] = 0 equation and by taking various Jacobi
identities (the same strategy Hojman and Shepley used).
Definining W = {q,v}, and S = {v,v}, the constraints amout to the
following:
d/dt {q,q} = 0 -> W^{ab} = W^{ba}
dW/dt = d/dt {q,v} -> dW^{ab}/dt = 1/2 ({q^a, A^b} + {q^b,
A^a})
Also one gets from this equation
S^{ab} = 1/2 ({q^b,A^a} - {q^a,A^b}).
Finally
dS/dt = d/dt {v,v} -> dS^{ab}/dt = 1/2 ({v^a,A^b} -
{v^b,A^a}).
The Jacobi identities yield
On (q,q,q) -> nothing
On (q,q,v) -> {q^a,W^{bc}} = {q^b,W^{ac}}
On (q,v,v) -> {q^a,S^{bc}} = {v^b,W^{ac}} - {v^c,W^{ab}}
On (v,v,v) -> {v^a,S^{bc}} + {v^b,S^{ca}} + {v^c,S^{ab}} = 0.
It's at this point that the punchline is dropped. Since the commutators
are already c-numbers, there's no need to take any classical limit, as
Hojman and Shepley did. The adjoint action of the q coordinates on
functions F(q,v) is that of a derivative operator
{q^a,F(q,v)} = W^{ab} dF/dv^b
and the adjoint action of the velocities v on functions F(q) alone is
{v^a,F(q)} = -W^{ab} dF/dq^b.
This is the case, when the functions are restricted to polynomials. One
has to make technical assumptions (that are not yet spelled out clearly
in the writeup) on the underlying topology of the algebra in order to
pass over to a limit and incorporate a larger class of operators within
this.
I assume this can be taken care of appropriately (e.g. one method may
be to restrict one's focus to C*-algebras and use the method of induced
representations to arrive at operator forms for the respective adjoint
actions -- this is where the number N of degrees of freedom may be
required to be finite).
The result of this is that one arrives at what are in essence the
Helmholtz conditions. The Helmholtz conditions determine when a 2nd
order system has a Lagrangian.
More precisely, one has to factor out the 0-subspace of the W matrix.
This is the primary reason I required the W matrix to be c-numbers. The
0-subspace gives you the coordinates (q_C, v_C) which form a classical
subsystem. The remainder of the system (q_Q, v_Q) is the quantum part
of the system.
The classical subsystem evolves independently of the quantum
coordinates, and must therefore be regarded as external.
So, in the following I'll let (q,v) refer to just the quantum
coordinates (q_Q,v_Q).
In the quantum subsystem, the W matrix is invertible, giving you the
mass matrix m. Defining the KINETIC MOMENTUM P = m v, one arrives at
commutators
{q,q} = 0, {q,P} = I, {P,P} = s,
which almost gets you there. The 2nd order system takes on the form
m dq/dt = P; dP/dt = F(q,p).
The Jacobi relations on (q,p,p) establish that s is a function of q
only. The relations on (p,p,p) show that the 2-form s = s_{ij} dq^i ^
dq^j is exact and that, locally, s = dA, with A a function of q only.
This allows one to define the CANONICAL MOMEMTNUM, p = mv + A,
resulting in commutators
{q,q} = 0, {q,p} = I, {p,p} = 0.
The equations of motion now become
m dq/dt + A = p; dp/dt = f.
Differentiating the commutator relations and applying these equations
yields
df^j/dp_i = W^{ik} dA_k/dq^j; df_i/dq^j =
df_j/dq^i,
from which one arrives at (locally):
f_i = -dU/dq^i, A_i = -m_{ij} dU/dp_j.
But A is a function of q only, therefore U is linear in the momenta.
The result is that the equations of motion become
dp_i/dt = -dU/dq^i = -d/dq^i (U + 1/2 W^{ij} p_i
p_j)
dq^i/dt = d/dp_i (U + 1/2 W^{ij} p_i p_j)
showing that the quantum part of the system is canonically quantized
with a Hamiltonian that is quadratic in the momenta.
Competing the square, one arrives at the final form for the
Hamiltonian,
H = 1/2 W^{ij} (p_i - A_i(q)) (p_j - A_j(q))
+ phi(q).
The final result is that the system in question is proven to be -- in
the most general case -- a hybrid classico-quantum system, in which the
quantum subsystem is canonically quantized with respect to a
Hamiltonian quadratic in the momenta evolving in superselection sectors
parametrized by the (external) classical subsystem.
A good example of how this works -- and an illustration of how the
constraint of having both equal-time commutators and equations of
motion -- is to consider the case of a 2-body system in ordinary
3-space, with
{q,v} = I/m; {Q,V} = I/M; {q,q} = {v,v} = {Q,Q} =
{V,V} = 0
and with (q,v)'s commuting with (Q,V)'s. If one assumes a force law of
the form
dq/dt = v; m dv/dt = F(q,v,Q,V); dQ/dt = V; M dV/dt
= -F(q,v,Q,V),
then the requirement of consistency severely restricts the form the
interaction force F may have.
Differentiating the {q,v} relation, one gets {q,F} = 0. Similarly,
{Q,F} = 0. Therefore F is independent of v and V. Differentiating the
{v,v} relation, one gets -- in component form --
{v^a, F^b/m} - {F^a/m, v^b} = 0 -> dF^a/dq^b =
dF^b/dq^a.
Similarly for the Q coordinates
dF^a/dQ^b = dF^b/dQ^a.
Differentiating the {v,V} relations yields
(1/M) dF^a/dq^b + (1/m) dF^b/dq^a = 0.
This shows that the force is independent of the center of mass
coordinates (mq + MQ) and depends only on the relative coordinates
(q-Q) with the other two relations showing that the dependency is given
by
F = -dU(q-Q)/dq = dU(q-Q)/dQ.
Therefore, the quantum system must be Hamiltonian, with a Hamiltonian
given by
H = 1/2 m v^2 + 1/2 M V^2 + U(q-Q).
The interaction depends only on the relative difference of the
coordinates.
Oh No
>Oh No wrote:
>> Thus spake Eugene Stefanovich <eug...@synopsys.com>
>>
>>
do more, that is to translate the fundamental statements of quantum
logic into simple, and preferably obvious, statements about measurement
in the English language.
>>
>> If you think the programme has been completed, perhaps you could tell
>> me, for example, why the Schrodinger equation is obeyed.
>
>Yes, quantum logic says nothing about dynamics. In order to get the
>Schrodinger equation you need to add the principle of relativity to
>your postulates. As demonstrated by
>
I also do.
Cl.Massé
> Are you saying that quantum mechanics is logically inconsistent?
> Do I understant you right? Where do you see the inconsistency?
The mathematical formulation is logically sound, but every interpretation
entails some conceptual inconsistency. For example, we have no mechanism
for correlating entangled pair faster than light. Of course, in mathematics
this is described by the projection of a vector in a Hilbert space.
However, it's
but a formal description, and there is no covariant one in terms of a wave
function in space. Of course there is no violation of the causality, but we
haven't the explanation why, it is still a mystery. This precisely points
to the possibility of a covariant description, that wouldn't be orthodox QM.
tttito
> "tttito" <vec...@weirdtech.com> a écrit dans le message de news:
> 1147284408.8...@y43g2000cwc.googlegroups.com
...
> > in
> > http://groups.google.com/group/sci.physics.research/msg/9bebf67819f08315
> > I wrote "Entanglement will then appear as a property of the
> > interaction/information-exchange between superposed D1 and D2 , when
> > measurement outcomes are matched/compared. In this setting nonlocality
> > disappears, together with the hidden assumptions that spawned it."
>
> The relational interpretation isn't necessary to see that, the Copenhagen
> interpretation suffices if used appropriately, that is, if the collapse
> occurs only at the time of the comparison.
Exactly, but then you need superposed observers,
> Quantum mechanics give exactly
> the same result whenever is introduced the projection onto the eigenstate,
> be it at the time of the polarization measurement or even just before
> conscious perception.
No. That depends on what you measure upon information exchange (i.e. on
the question that observers ask each other and hence on the type of
information that is exchanged). Experimental verification of RQM hinges
on detecting (i.e. measuring interference patterns of) superpositions
of measurement devices/observers (cf. [1]).
Detecting macroscopic superpositions is the key step to move RQM from
speculative to experimental relevance.
IV
[1]
http://groups.google.com/group/sci.physics.research/msg/9d998d3a03bd2301
"When Alice observes an electron in a 1/sqrt(2)(|spin up>+|spin down>)
superposition she enters a superposition 1/sqrt(2)|Alice>|spin
up>+|Alice>|spin down>" . Alice may be the name of any measurement
device.
Oh No
>
>Charles has presented a paper which he claims provides the
>explanation required. I have examined this paper but cannot
>find a satisfactory explanation of why quantum mechanics,
>rather than some other procedure, is the correct procedure
>to use for assigning probabilities to the results of
>measurements. There are some things in the paper with which
>I agree, but two of the principal foundations, namely
>relationalism and quantum logic, seem to me to be unlikely
>to be fruitful.
>
>I have explained before why I find relationalism to be incoherent.
>Either it is a founding principle on which his argument is based,
>which I hope it is not, because it would be difficult to base
>a coherent argument on an incoherent principle, or the argument
>can be presented without any appeal to relationalism (or any
>other "ism"), which would greatly improve its clarity.
The fundamental principle is that we can only say where something is if
we say where it is relative to other matter. That does not strike me as
being incoherent. In fact I find it simple, and even empirically
obvious. What is perhaps a lot less simple is building it into a
mathematical structure. Doing so certainly defeated Descartes and
Leibniz, but then we know a lot more now than they did then.
>The use of quantum logic also seems to me to present no improvement
>in understanding, since it merely replaces one unexplained procedure
>with another. To say that the procedures used in quantum mechanics
>consitute logic is at best a poetical metaphor, because quantum
>mechanics is literally not logic. When somebody says that statements
>have "truth values" which can be complex, I do not know what they
>mean. I can consider maps from statements to sets other than
>{true,false}, but if somebody wants to say that these are "truth
>values", then there are two possible cases.
>
>One possibility is that this is just the introduction of a new term,
>"truth value", and that any other term could have been used just
>as well, for example "fribble". In that case, one would have to
>conclude that quantum logic is not a more appropriate name than
>quantum fribbology, and that those who opt to study quantum logic
>should be warned that it is not a system used to model inference,
>as actual logic is, but that it is a particular mathematical system
>to which actual normal true-and-false logic applies, and that calling
>it logic is just poetry.
I would be happy enough with that. Actually as a result of the
correspondence we have had I have altered my terminology, so that "truth
value" now refers to the modulus |<f|g>|, so that it is real. This also
remains consistent with the formal definition of a many valued logic. I
have also demoted the notion of a truth value in the account, and it now
only serves to complete that formal definition, and is not used in any
other way. In practice quantum logic gives a formal structure for the
calculation of probabilities (which may also be regarded as truth
values, as they are in Bayesian reasoning).
>The other possible position that an advocate of quantum logic
>could adopt is that the choice of the term "truth value" does
>indeed carry an important message of some kind, and then
>they have the obligation to tell us what that message is. Is
>the message that normal logic has to be abandoned and replaced
>by this new system? Why, and how could anybody possibly ever
>know such a thing?
In general one should not treat truth values as carrying much in the way
of an important message. They are really just a measure of how much we
think we ought to believe in a proposition, which is itself a pretty
woolly, and frequently subjective concept (as in fuzzy logic) unless one
can find objective reasons which may, for example, relate truth values
to probabilities. In my paper I start with the idea of probabilistic
results to experiments, and the structure of Hilbert space as a logic is
determined from that, and not the other way around.
>
>In Charles' paper, and in my correspondence with him, he has
>been keen to insist that his introduction of the entire Hilbert
>space formalism, along with the identification of kets as
>propositions about measurement results in a formal language,
>is little more than a choice of notation. However, the set
>of things which can be proven about the results of measurements
>if one merely accepts the Hilbert space formalism and the
>rules of quantum mechanics for assigning probabilities to
>the results of measurements is non-trivial.
Yes. But quantum logic is not the whole of quantum mechanics. It only
applies to experimental results at a given time. We cannot usefully
determine anything much from that unless we also have a time evolution
equation. As Eugene has just remarked time evolution comes from adding
the principle of relativity. The principle of relativity is contained
within the fundamental principle of relationism, so introducing it is
not a problem.
>For example, if there are 10 mutually exclusive complete
>measurements than I can perform on a system, each
>of which can give 10 possible results, then a complete
>specification of the probabilities of obtaining each
>of the 100 results requires 90 parameters to specify.
>That's 100 possible results, grouped into sets of 10.
>In each set of 10, I assign 9 probabilities, and the
>10th is fixed by the fact that the probabilities must
>sum to one.
>
>If I suppose that measuring the same thing twice always
>gives the same result, then I can prepare a system by
>making a measurement. Then 10 probabilities are fixed
>(one probability is 1 - if I measure the same thing
>again, I will get the same result, nine are zero - if
>I measure the same thing again, the chance of me getting
>any of the other nine results is zero). That leaves 81
>parameters which I need in order to specify the probabilities
>of obtaining the various results of the different measurements
>I could perform.
>
>Quantum mechanics, however, tells us that we can
>specify all of the probabilities with only 18 parameters.
>All that we need to do is specify a ray in a ten-dimensional
>Hilbert space, so that's 9 complex numbers or 18 parameters.
You can lose some of those. There are only nine parameters; so long as
we stick strictly within a Hilbert space defined at given time, phase is
meaningless. It becomes important when dynamics are introduced.
>
>So if we have accepted the Hilbert space formalism, and if
>we have accepted that the square of the inner product
>gives us probabilities of finding particular measurement
>results given particular preparations, then we have
>already agreed to some very non-trivial statements
>about the results of experiments and how many parameters
>are needed to describe them. A mere choice of notation
>can't change an 81-dimensional space into an 18-dimensional
>space. It is noteworthy that this is all before any mention
>has been made of time evolution, the Hamiltonian, or any dynamics.
>
>The question is: "Why are there 18 and not 81 parameters needed to
>specify the probabilities of obtaining the various possible measurement
>results?" Right now, the only answer we have is that the symbols
>tell us so, and I do not believe that Charles, Mackey, Rovelli,
>Birkhoff & Von Neumann, or anybody else has come even remotely
>close to addressing this question.
In practice your fundamental assumption appears to be wrong, that you
can have 10 independent mutually exclusive complete measurements of a
system, each giving 10 possible results. All the measurements described
in quantum theory are represented as operators, and as such are linear
combinations of each other. I think you are overlooking a fundamental
point which I make in my paper, that all measurements can be reduced to
measurements of position.
If we stick to measurement at one particular time, which is the way
Hilbert space is formulated, then strictly we can only measure position,
and your other nine possible measurements are a fiction. If you want to
measure something else, momentum say, then you have to take measurements
over a period of time, and therefore you do have to invoke dynamics.
That means a determination of changes in position, so it is not
unreasonable to expect the momentum operator to be a linear combination
of position operators.
To make your case stick you would have to find a genuinely independent
measurement. That's not impossible. Spin, for example. But if there is
such a genuinely independent measurement, then you were wrong in your
original claim that your first measurement was complete, and in that
case you can't describe the system with a 10 dimensional Hilbert space.
Cl.Massé
news: lwA0H7KA...@charlesfrancis.wanadoo.co.uk
> If you think the programme has been completed, perhaps you could tell
> me, for example, why the Schrodinger equation is obeyed.
Because it is not. Considering Dirac equation instead, it's but a wave
equation with a peculiar geometry, the one of a spinor. The Maxwell
equation is analogous with another geometry. Even them aren't obeyed,
especially at measurement, and we need the complementary notion of
corpuscle.
Looking more closely to the propagation process of a wave and of an ensemble
of Brownian corpuscles, very strong similarities pop up. The only
difference is in the time derivative term, that may be @^2 (classical), i@
(Schrödinger), or @ (diffusion), and which corresponds to how amplitudes
add.
The similarities can be named by a single term: superposition principle,
meaning that the amplitudes add linearly. In few words, the Schrödinger
equation is obeyed because it's the part describing the linear behaviour of
Nature, that is, when nothing happends.
Eugene Stefanovich
You are right, the Hilbert space is just postulated in Mackey's axioms.
However, there were quite a few developments in this field since
Mackey's book was published (1963). The most important step is in
Piron's book
C. Piron, Foundations of Quantum Physics,
(W. A. Benjamin, Reading, 1976)
Instead of Mackey's axiom VII and instead of the classical distributive
law of logic, and instead of Birkhoff-von Neumann "modular law",
Piron introduces the "orthomodular law". This law can be formulated in a
number of different ways. One of the most transparent formulations is
"if proposition x implies proposition y, then x and y are compatible".
Then Piron goes on to prove a theorem which says that above axioms can
be realized if logical propositions are identified with closed subspaces
in a Hilbert space over R, C, or quaternions. Quantum theories with
real or quaternionic scalars have been studied, but, as far as I know,
nothing interesting came out of this. So, we are left with the usual
C-number quantum mechanics whose mathematical apparatus directly
follows from Birkhoff-von Neumann-Mackey-Piron axioms via Piron's theorem.
There are quite a few reviews and book where you can find more
details. You can check, for example,
E.G. Beltrametti and G. Casinelli "The logic of quantum
mechanics" (Addison-Wesley, Reading, 1981)
or search the web for "quantum logic", "orthomodular lattice",
etc.
> When somebody says that statements
> have "truth values" which can be complex, I do not know what they
> mean.
In quantum logic the "truth values" are not complex. They are
real numbers from the interval [0,1], i.e., the probabilities that the
result of the "yes-no experiment" is "yes". Complex numbers arise
only after the propositions of quantum logics are mapped into the
set of subspaces of the complex Hilbert space via Piron's theorem.
This mapping identifies the "truth value" as the square of the modulus
of the projection of the state vector on the subspace, i.e., still
a real number from [0,1].
> ... those who opt to study quantum logic
> should be warned that it is not a system used to model inference,
> as actual logic is, but that it is a particular mathematical system
> to which actual normal true-and-false logic applies, and that calling
> it logic is just poetry.
I don't think so. In my opinion, classical logic developed by Aristotle
and Boole refers only to propositions about classical objects.
For quantum objects we need to take into account the statistical nature
of measurements and indeterminism. This requires a change in the rules
of logic. Quantum logic says that all classical axioms are still OK,
except the axiom of distributivity. This axiom wasn't very intuitive
in the classical system anyway. Quantum logic uses the "orthomodular
law" instead. The distributivity axiom is a particular case of the
"orthomodular law".
Eugene.
Cl.Massé
news: 4468B72E...@synopsys.com
> Yes, quantum logic says nothing about dynamics. In order to get the
> Schrodinger equation you need to add the principle of relativity to
> your postulates. As demonstrated by
>
> E. P. Wigner, On unitary representations of
> the inhomogeneous Lorentz group, Ann. Math. 40 (1939), 149.
>
> and
>
> P. A. M. Dirac, Forms of relativistic
> dynamics, Rev. Mod. Phys. 21
> (1949), 392.
>
> this requires a definition of the unitary representation of the
> Poincare group in the Hilbert space of the system. The Hamiltonian
> is a representative of the generator of time translations of the
> Poincare group, and the Schrodinger equation
>
> -ih d/dt |Psi(t)> = H |Psi(t)>
>
> is just a compact form of writing how the state vector changes
> under time translations.
Isn't it the whole idea of dynamics? Here, we would call it a
"Lapalissade" (obvious statement), but it is said so seriously and so
formally that it seems a discovery.
The evolution of any system, including classical ones, can be written
under this form. But that doesn't say what concretely is H, and why
this one and not another.
Actually, we have the equivalence -ih d/dt = H, and as d/dt is a
generator of time translation, H is too. The representation space may
be the Hilbert space for a quantum system, or the phase space for a
classical system.
scerir
> The mathematical formulation is logically sound,
> but every interpretation entails some conceptual
> inconsistency.
Inconsistency or incompleteness? There are simple
cases, such as the Einstein-de Broglie paradox
http://www.arxiv.org/abs/quant-ph/0404016 ,
in which there is no physical solution, there
is just a meta-physical (actually merely a logical)
solution. (In that paradox if P1 is the projection
operator representing the physical proposition
that the particle is in box1, and P2 is the
projection operator representing the physical
proposition that the particle is in box2,
then the _logical structure_ of the lattice of
projections _must_ ensure that P1 and P2
are orthogonal. Hence any state which is an
eigenstate of P1 with eigenvalue 1/0 will also
be an eigenstate of P2 with eigenvalue 0/1.)
> For example, we have no mechanism
> for correlating entangled pair faster
> than light.
Yes. But Cerf and Adami pointed out
http://www.arxiv.org/abs/quant-ph/9512022
that conditional entropies can be negative when
considering quantum entangled systems, and these
negative conditional entropies _might_ simulate
_time reversed_ actions between each analyzer
and the common source of the entangled pair.
This _might_ be a mechanism for correlation,
since at the emission time t=0 the source of
the entangled pair would know the parameters
(if not the outcomes) of both analyzers.
Essentially this is also what Olivier Costa
de Beauregard answered to de Broglie, in 1947.
(I'm not saying it is an easy mechanism though!).
Regards,
s.
Eugene Stefanovich
>>Yes, quantum logic says nothing about dynamics. In order to get the
>>Schrodinger equation you need to add the principle of relativity to
>>your postulates. As demonstrated by
>>
>>E. P. Wigner, On unitary representations of
>>the inhomogeneous Lorentz group, Ann. Math. 40 (1939), 149.
>>
>>and
>>
>>P. A. M. Dirac, Forms of relativistic
>>dynamics, Rev. Mod. Phys. 21
>>(1949), 392.
>>
>>this requires a definition of the unitary representation of the
>>Poincare group in the Hilbert space of the system. The Hamiltonian
>>is a representative of the generator of time translations of the
>>Poincare group, and the Schrodinger equation
>>
>>-ih d/dt |Psi(t)> = H |Psi(t)>
>>
>>is just a compact form of writing how the state vector changes
>>under time translations.
>
>
> Isn't it the whole idea of dynamics? Here, we would call it a
> "Lapalissade" (obvious statement), but it is said so seriously and so
> formally that it seems a discovery.
>
> The evolution of any system, including classical ones, can be written
> under this form. But that doesn't say what concretely is H, and why
> this one and not another.
Yes, Schroedinger equation is useless until you specified H, and this
is the most challenging task of the theory. As a minimum, the
Hamiltonian should satisfy two requirements:
1. relativistic invariance - i.e., correct Poincare Lie algebra
commutators with other generators - of space translations P, of
rotations J, and of boosts K
2. cluster separability - i.e., the dynamics of spatially separated
groups of particles should be independent.
These conditions are not so easy to satisfy. Weinberg in his vol. 1
"The quantum theory of fields" claims that the only way to do that
is local quantum field theory. However, he misses (at least) three
other possibilities that were proven to work:
1. "direct interaction" theories initiated in
Bakamjian, B.; Thomas, L. H., Relativistic particle dynamics. II, Phys.
Rev. 92, 1300 (1953).
When there are more than 2 particles, the cluster separability requires
a complicated combinatorial construction of interactions, but it can be
done (see works by W. N. Polyzou)
2. Non-local quantum field theory:
M. I. Shirokov, Relativistic
nonlocal quantum field theory, Int. J. Theor. Phys., 41 (2002), 1027.
3. Direct solution of the Poincare Lie algebra equations for
interactions built as polynomials in particle creation and annihilation
operators. This approach is presented in a
beautiful series of papers by H. Kita in 1966-1973. The last paper in
this series is `A realistic model of convergent quantum mechanics of
interacting particles' Prog. Theor. Phys. 49 (1973), 1704 where you can
pick up the other references if interested.
Eugene.
Eugene Stefanovich
mark...@yahoo.com wrote:
>
> What Rovelli is asking for can be done more simply still. I show this
> in the suggestively titled "On The Quantum Dynamics of Moving Bodies",
> which can be found under
>
> http://federation.g3z.com/Physics/Index.htm
[...]
> This shows that the force is independent of the center of mass
> coordinates (mq + MQ) and depends only on the relative coordinates
> (q-Q) with the other two relations showing that the dependency is given
> by
> F = -dU(q-Q)/dq = dU(q-Q)/dQ.
>
> Therefore, the quantum system must be Hamiltonian, with a Hamiltonian
> given by
> H = 1/2 m v^2 + 1/2 M V^2 + U(q-Q).
> The interaction depends only on the relative difference of the
> coordinates.
There is (in my opinion) even more elegant demonstration of this
property for non-relativistic quantum 2-particle systems. It uses
the unitary representation of the Galilei group in the Hilbert space
of the system:
T. F. Jordan, "Linear operators for quantum mechanics"
(published by T.F. Jordan, Duluth, 1969)
This demonstration can be easily generalized to relativistic systems
of particles: just change the Galilei group to the Poincare group:
Bakamjian, B.; Thomas, L. H., Relativistic particle dynamics. II, Phys.
Rev. 92 (1953), 1300.
Eugene.
Oh No
>"Oh No" <No...@charlesfrancis.wanadoo.co.uk> a écrit dans le message de
>news: lwA0H7KA...@charlesfrancis.wanadoo.co.uk
>
>> If you think the programme has been completed, perhaps you could tell
>> me, for example, why the Schrodinger equation is obeyed.
>
>Because it is not. Considering Dirac equation instead, it's but a wave
>equation with a peculiar geometry, the one of a spinor.
I agree with Eugene. Dirac himself described the Dirac equation as "a
Schrodinger equation
Cl.Massé
BZ4bg.3399$cX1....@twister2.libero.it
> Inconsistency or incompleteness? There are simple
> cases, such as the Einstein-de Broglie paradox
> http://www.arxiv.org/abs/quant-ph/0404016 ,
> in which there is no physical solution, there
> is just a meta-physical (actually merely a logical)
> solution. (In that paradox if P1 is the projection
> operator representing the physical proposition
> that the particle is in box1, and P2 is the
> projection operator representing the physical
> proposition that the particle is in box2,
> then the _logical structure_ of the lattice of
> projections _must_ ensure that P1 and P2
> are orthogonal. Hence any state which is an
> eigenstate of P1 with eigenvalue 1/0 will also
> be an eigenstate of P2 with eigenvalue 0/1.)
There is no covariant description either. I don't know if that should be
classified as incompleteness (no description) or inconsistency (non
covariant.)
> Yes. But Cerf and Adami pointed out
> http://www.arxiv.org/abs/quant-ph/9512022
> that conditional entropies can be negative when
> considering quantum entangled systems, and these
> negative conditional entropies _might_ simulate
> _time reversed_ actions between each analyzer
> and the common source of the entangled pair.
> This _might_ be a mechanism for correlation,
> since at the emission time t=0 the source of
> the entangled pair would know the parameters
> (if not the outcomes) of both analyzers.
> Essentially this is also what Olivier Costa
> de Beauregard answered to de Broglie, in 1947.
> (I'm not saying it is an easy mechanism though!).
I think it is called the transactional interpretation. It seems elegant in
principle, but in practice it is difficult to implement in more than 1
dimension.
Cl.Massé
>> Quantum mechanics give exactly
>> the same result whenever is introduced the projection onto the
>> eigenstate, be it at the time of the polarization measurement or even
>> just before conscious perception.
"tttito" <vec...@weirdtech.com> a écrit dans le message de news:
1147865489....@j73g2000cwa.googlegroups.com
> No. That depends on what you measure upon information exchange (i.e. on
> the question that observers ask each other and hence on the type of
> information that is exchanged). Experimental verification of RQM hinges
> on detecting (i.e. measuring interference patterns of) superpositions
> of measurement devices/observers (cf. [1]).
>
> Detecting macroscopic superpositions is the key step to move RQM from
> speculative to experimental relevance.
But you're taking about the case where there is no projection, then not of
(orthodox) quantum mechanics, or the Copenhagen interpretation. The
multi-world interpretation doesn't use the projection either. In what is it
different from RQM?
I doubt you can observe the superposition of two observers, since an
observer has a sole, non superposed, consciousness. Same thing for the
apparatus since it is correlated to the observer that should precisely
observe it. It's the whole idea of an interpretation, it yields no
prediction.
tttito
...
>
> ...you're taking about the case where there is no projection, then not of
> (orthodox) quantum mechanics, or the Copenhagen interpretation.
If unitarity holds, projection is a subjective process, as in RQM,
which I regard as a logical completion of Copenhagen. If "orthodox QM"
(whatever that means) is silent about Wigner's friend , still the
issues he raises have to be tackled .
> The
> multi-world interpretation doesn't use the projection either. In what is it
> different from RQM?
That's indeed an interpretational issue, which I will not waste time
discussing (see [1] for my take on it).
> I doubt you can observe the superposition of two observers, since an observer has a
> sole, non superposed, consciousness.
> Same thing for the apparatus since it is correlated to the observer that should precisely
> observe it.
Consciousness ia slippery concept, let's deal first with pointer
states, which we can measure/observe. Bob does not measure Alice's
consciousness (if Alice is just a measurement device, she may not need
one). He observes only her pointer states, whose superpositions he may
detect (although that may be hard to implement experimentally cf. [3]).
So Alice's state may be detected as
1/sqrt(2)(|Alice_pointer_A>+|Alice_pointer_B>) by Bob, while Alice (if
we want to assign her a consciousness ) may perceive it as
|Alice_pointer_A> or |Alice_pointer_B>, but not both (unless she is
schizophrenic ). The key point here is that the state vector of a
system is observer-dependent, i.e. it encodes the information relative
to a given observer's perpective. Different observers -> different
state vectors. Bob's perpective of Alice is not necessarily the same
that any instance of Alice has of Alice ([2]).
> It's the whole idea of an interpretation, it yields no
> prediction.
If I thought it were so I would not waste my time. Detecting
macroscopic superpositions is a concrete challenge.
Thanks for the stimulating feedback.
IV
[1]
http://groups.google.com/group/sci.physics.research/msg/6a73a0f2cbdfa344
[2] As Rovelli writes in http://arxiv.org/abs/quant-ph/9609002, "the
notion rejected here is the notion of absolute, or
observer-independent, state of a system; equivalently,the notion of
observer-independent values of physical quantities."
[3] http://physicsweb.org/articles/world/13/8/3
--------------------------------
" Aquellos que allí se parecen no son gigantes, sino molinos de
viento"
"What we see there are not giants but windmills"
Miguel de Cervantes , Don Quixote de la Mancha
scerir
> [...] So Alice's state may be detected as
> 1/sqrt2(|Alice_pointer_A>+|Alice_pointer_B>)
> by Bob, while Alice (if we want to assign her
> a consciousness) may perceive it as
> |Alice_pointer_A> or |Alice_pointer_B>,
> but not both.
Let us imagine that Alice may 'perceive' it
(but is that a measurement, or a pre-measurement?)
as |Alice_pointer_A>.
And let us suppose that the observable
(|Alice_pointer_A>+|Alice_pointer_B>)
may be detected by Bob.
The question seems (to me) to be: do the
observable |Alice_pointer_A> and the observable
(|Alice_pointer_A>+|Alice_pointer_B>)
commute?
s.
'The German language allows to make the distinction
between "Realitaet" and "Wirklichkeit".
Realitaet relates to the Latin word for "thing" ("res").
Wirklichkeit contains the notion of effecting something
("Wirkung").
I prefer "Wirklichkeit" because I think that no object
can be defined without reference to its external world,
which is in perpetual change.'
-Marcus Arndt
John Bell
> Thus spake r...@maths.tcd.ie
> >
> >Charles has presented a paper which he claims provides the
> >explanation required. I have examined this paper but cannot
> >find a satisfactory explanation of why quantum mechanics,
> >rather than some other procedure, is the correct procedure
> >to use for assigning probabilities to the results of
> >measurements. There are some things in the paper with which
> >I agree, but two of the principal foundations, namely
> >relationalism and quantum logic, seem to me to be unlikely
> >to be fruitful.
> >
> >I have explained before why I find relationalism to be incoherent.
I missed that so would appreciate a reminder/link as I have usually
found your comments to be worth reading.
> >Either it is a founding principle on which his argument is based,
> >which I hope it is not, because it would be difficult to base
> >a coherent argument on an incoherent principle, or the argument
> >can be presented without any appeal to relationalism (or any
> >other "ism"), which would greatly improve its clarity.
>
> The fundamental principle is that we can only say where something is if
> we say where it is relative to other matter.
That is a very 3D argument. It would be more relativistically logical
to argue that we can only say where a 4D event is relative to other 4D
events. However, neither statement is unconditionally true. The second
statement can be untrue because of the relativity of simultaneity. The
first statement is generally untrue because reality is dynamic.
I would argue that the only way one can logically and relativistically
consistently say where (and when) something is, is relative to oneself.
One can thus also only consistently then say where and when other
events are in relation to this given event, yet again relative to
oneself. Yes, one can then generalise to the extent of potentially
confirming that all observers can agree on certain things. However, if
you do this by making further theoretical assumptions as opposed to
direct comparison of empirical observations of various researchers,
then one is merely reinforcing ad hoc theoretical assumptions as
opposed to performing genuine research. This appears to be one
fundamental way in which the foundations of our respective conclusions
differ.
Although I find some of your comments to be reasonable, if your axioms
are valid, your conclusions should be valid (assuming you have made no
logical blunders en route). However, we already appear to have
established via the discussion "is temporal sign ambiguity...." that
your final conclusions are logically and mathematically self
contradictory, and directly refutable by empirical observation.
> That does not strike me as
> being incoherent. In fact I find it simple,
At the risk of being accused of rubbing salt into the wound, I am
tempted to say "overly simplistic".
John Bell
http://global.accelerators.co.uk
(Change John to Liberty to respond)
r...@maths.tcd.ie
> Thus spake r...@maths.tcd.ie
>>
>>Charles has presented a paper which he claims provides the
>>explanation required. I have examined this paper but cannot
>>find a satisfactory explanation of why quantum mechanics,
>>rather than some other procedure, is the correct procedure
>>to use for assigning probabilities to the results of
>>measurements. There are some things in the paper with which
>>I agree, but two of the principal foundations, namely
>>relationalism and quantum logic, seem to me to be unlikely
>>to be fruitful.
>>
>>I have explained before why I find relationalism to be incoherent.
>>Either it is a founding principle on which his argument is based,
>>which I hope it is not, because it would be difficult to base
>>a coherent argument on an incoherent principle, or the argument
>>can be presented without any appeal to relationalism (or any
>>other "ism"), which would greatly improve its clarity.
>The fundamental principle is that we can only say where something is if
>we say where it is relative to other matter. That does not strike me as
>being incoherent. In fact I find it simple, and even empirically
>obvious. What is perhaps a lot less simple is building it into a
>mathematical structure. Doing so certainly defeated Descartes and
>Leibniz, but then we know a lot more now than they did then.
If the principle is that "we can only say where something is if
we say where it is relative to other matter", then that on its
own is not incoherent, but Rovelli's claim that this is the
secret to understanding quantum mechanics is. It will not be easy
to build it into a mathematical stucture. It is a statement about
the absence of something (an absolute background). Mathematically
representing the absence of something is a daunting task, especially
if you want it to be your starting point. If this principle has
entered into the formalism you present in your paper, I have not
understood where.
I would agree with the truth of the so-called fundamental principle
if it is understood to assert that all location is relative location,
or, equivalently, that absolute motion is only an idea, not a feature
of the physical world. I must regard a principle which attempts
to place restrictions on what I can say as belonging to the field
of linguistics, and not physics.
But even in the form in which I agree with it, the principle
does not provide us with any "positive knowledge". It only
tells us what is not the case, without giving us any clearer
idea of what might actually be the case. As such, it can
perhaps serve to prevent error, but it cannot be the foundation
of any body of knowledge about actual facts.
> ... In my paper I start with the idea of probabilistic
>results to experiments, and the structure of Hilbert space as a logic is
>determined from that, and not the other way around.
>>
>>In Charles' paper, and in my correspondence with him, he has
>>been keen to insist that his introduction of the entire Hilbert
>>space formalism, along with the identification of kets as
>>propositions about measurement results in a formal language,
>>is little more than a choice of notation. However, the set
>>of things which can be proven about the results of measurements
>>if one merely accepts the Hilbert space formalism and the
>>rules of quantum mechanics for assigning probabilities to
>>the results of measurements is non-trivial.
>Yes. But quantum logic is not the whole of quantum mechanics. It only
>applies to experimental results at a given time. We cannot usefully
>determine anything much from that unless we also have a time evolution
>equation. As Eugene has just remarked time evolution comes from adding
>the principle of relativity. The principle of relativity is contained
>within the fundamental principle of relationism, so introducing it is
>not a problem.
The principle of relativity which Eugene was referring to was the
principle of the constant speed of propagation of information.
The principle of relativity which is contained (let us say,
for the sake of the argument) in the fundamental principle
of relationism is the principle that one cannot say how
fast something is moving unless we specify an object
to which that motion is relative.
Those are two different principles of relativity.
In any case, your argument seems to be that quantum
logic on its own places no restrictions on the results
of measurements. You claim that my demonstration of the
contrary doesn't apply. Your reasons are obscure to me.
Quantum logic (meaning quantum mechanics) applies to circumstances
when a system is prepared (by filtering, namely by selecting those
which give a particular result to a measurement), and then a
measurement is performed on it. There is no restriction placed on
the time at which the physical activities involved in the measurement
need to take place. Waiting five seconds and measuring the position
is simply a different experiment from waiting four seconds and
measuring the position. Each of those two experiments has
an associated operator.
Indeed, that there is no restriction on the time at which the
measurement takes places means that the measurement can occur
before the preparation, and the statistical predictions of
quantum mechanics will still hold. This is true because
preparation and measurement are the same procedure.
You say: "We cannot usefully determine anything much from that
unless we also have a time evolution equation." I don't know
what you mean by "determine anything much from that". What
we can do, without a time evolution operator, is observe,
for each preparation, the probability of observing each
result to each measurement. Measurements performed at
different times after the preparation of the system simply
count as different measurements. After we have collected
the statistics of the measurement results, we then
go about constructing a time evolution operator to describe
the statistical relationships we have empirically discovered.
As I explained above, measurements performed at different
times are different measurements and get represented by
different linear operators. We do not have to throw
away the Hilbert space and get a new one if we want to
mathematically represent measurements performed at different
times. If a Hilbert space is "defined at a given time" then
it is defined for all time.
So, imagine I had said: Consider a set of three mutually exclusive
complete measurements, each of which can give 2 possible results.
You could complain that this is unfair, because I'm assuming
dynamics and the Hilbert space is only defined at a given time and
I'm overlooking points in your paper and so on. By these means
and with these arguments you could try to convince me that it's
absolutely unacceptable for me to suppose that there can be
a set of three mutually exclusive complete measurements, each
of which has two possible results.
But we have a clear example of such a system. The spin of
a spin-half particle can be measured along three orthogonal
axes, and each of those measurements has two possible
results.
Would you tell me that I can only measure position and that therefore
these spin measurements are a fiction? Or perhaps that, "if there
is such a genuinely independent measurement, then you were wrong
in your original claim that your first measurement was complete,
and in that case you can't describe the system with a 2 dimensional
Hilbert space." Would you expect me, upon reading this expression,
to become enlightened and regard spin measurements as impossible?
Are you saying that the simple story of the
spin-half particle and the Pauli matrices and measurements
with Stern-Gerlach magnets and so on is all a pack of
lies, because:
>[the] fundamental assumption appears to be wrong, that you
>can have [3] independent mutually exclusive complete measurements of a
>system, each giving [2] possible results. All the measurements described
>in quantum theory are represented as operators, and as such are linear
>combinations of each other. ?
Or would you say that when 10 and 10 are substituted for 3 and 2
in the above paragraph it makes a difference to the truth of the
claim? It does not, because it is possible to have 3 distinct
Hermitian operators on a 2 dimensional Hilbert space and it is
possible to have 10 distinct Hermitian operators on a 10
dimensional Hilbert space. Whether they are linearly independent
or not is irrelevant, and in any case there are certainly enough
dimensions in a ten-dimensional space to accommodate 10 linearly
independent Hermitian operators.
But I gather from this that I have failed to explain properly what
I was saying in the first place. Let me try again.
Suppose you know no quantum mechanics. There are 10 different
measurements you perform, each of which can give 10 different
results. (Despite your protestations to the contrary, this is
quite possible. Consider a spin-9/2 particle and measurements
along any ten distinct axes in three-dimensional space as an
example.)
Call the measurements M0, M1, ... , M9.
Let r_ij denote the jth result to the ith experiment. Let
P(r_ij|r_kl) denote the probability of observing result
r_ij when performing measurement Mi on a system which
has just given result r_kl upon measurement of Mk.
The numbers P(r_ij|r_kl) can be measured.
Suppose you have already measured the numbers P(r_ij|r_kl)
for all values of i,j,k, and l except for i=0 or k=0. That is,
for any preparation which consists of selecting the system
which gives a specific result to any of the nine experiments
M1 to M9, the probability of obtaining any given result to any of the
same nine experiments in known.
Now suppose you introduce the 0th measurement. The systems are
prepared by selecting those which give the result r_00 to
measurement M0. You are about to measure the numbers
P(r_ij|r_00) except when i=0.
It is these numbers which one needs 81 independent parameters
to specify. If you do not know quantum mechanics, you
will have to start the laborious task of measuring 81 numbers.
However, if you suddenly learn quantum mechanics, then you can save
yourself some experimental work. You can use the numbers P(r_ij|r_kl)
which you have already measured to construct the relevant Hermitian
operators on a ten-dimensional Hilbert space. After this, you will
have nine ten-by-ten Hermitian matrices written down, representing
the nine measurements which you have collected statistics about.
You also have a state, |r_00>, which you can prepare by performing
M0 and selecting the systems which give the result r_00. By measuring
18 parameters, that is, by measuring 18 of the numbers P(r_ij|r_00),
you can write a ten-dimensional complex vector representing |r_00>.
After this, you do not have to measure any of the remaining 63
P(r_ij|r_00). You can calculate them using the ten-by-ten Hermitian
matrices and the ten-dimensional complex vector.
This is why I am saying that "quantum logic" places nontrivial
constraints on the statistics of the results of measurements. I
have used nothing other than the normal rules of quantum
mechanics above. Whether there is some hidden assumption about
time passing somewhere is irrelevant. If such a hidden assumption
does not invalidate quantum mechanics then it does not invalidate
this argument.
R.
John F
<<snip>>
: Suppose you know no quantum mechanics. There are 10 different
: measurements you perform, each of which can give 10 different
: Let r_ij denote the jth result to the ith experiment. Let
: P(r_ij|r_kl) denote the probability of observing result
: r_ij when performing measurement Mi on a system which
: has just given result r_kl upon measurement of Mk.
This sounds a lot like the measurement symbols M(a',a")
introduced in Chapter One, The Algebra of Measurement,
of Quantum Kinematics and Dynamics, Julian Schwinger,
W.A.Benjamin 1970, SBN 8053-8510-X. Much later (in fact,
after his death), in Quantum Mechanics - Symbolism of
Atomic Measurements, Springer 2001, ISBN 3-540-41408-8,
he introduces new notation |a',a"| for his M-symbols,
and an updated discussion. A very interesting, in my
opinion, ab initio operational-style development of
the subject.
--
John Forkosh ( mailto: j...@f.com where j=john and f=forkosh )
r...@maths.tcd.ie
>Oh No wrote:
>> Thus spake r...@maths.tcd.ie
>> >I have explained before why I find relationalism to be incoherent.
>I missed that so would appreciate a reminder/link as I have usually
>found your comments to be worth reading.
Thanks. An example me criticizing relationalism is at:
http://groups.google.com/group/sci.physics.research/msg/4929dfa2395fb9aa?dmode=source
My criticism is simple. Rovelli claims that quantum mechanics suddenly
all makes perfect sense if we say things like "The cat is alive
relative to Bob", instead of things like "The cat is alive". He
says that the expression "The cat is alive" is meaningless, unless
we specify somebody who the cat is alive relative to.
Apart from the fact that this is just a suggestion about the way
that we should speak, or the way that Rovelli thinks we should
speak, there is another problem. If the cat is alive relative
to Bob, can it be dead relative to anybody else? Rovelli doesn't
answer this question. If the answer is yes, then that means that
the interpretation is just the many-worlds interpretation, because
Rovelli is saying that one person can see a dead cat while another
person can see the same cat alive at the same time. On the other
hand, if the answer is no, then that means that if the cat is
dead for one person then it is dead for everybody. That means
that it is absolutely dead and its being dead isn't relative
at all. In that case, Rovelli's claim that we shouldn't say
"The cat is alive" is wrong. If the cat is alive relative to
anybody then it is absolutely alive and not dead.
Best,
R.
r...@maths.tcd.ie
>r...@maths.tcd.ie wrote:
>> On page 71, Mackey introduces "Axiom VII", which says:
>> The partially ordered set of all questions in quantum mechanics is
>> isomorphic to the partially ordered set of all closed subspaces
>> of a separable, infinite dimensional Hilbert space.
>> ...
>You are right, the Hilbert space is just postulated in Mackey's axioms.
>However, there were quite a few developments in this field since
>Mackey's book was published (1963). The most important step is in
>Piron's book
>C. Piron, Foundations of Quantum Physics,
>(W. A. Benjamin, Reading, 1976)
>Instead of Mackey's axiom VII and instead of the classical distributive
>law of logic, and instead of Birkhoff-von Neumann "modular law",
>Piron introduces the "orthomodular law". This law can be formulated in a
>number of different ways. One of the most transparent formulations is
>"if proposition x implies proposition y, then x and y are compatible".
Thanks for bringing this to my attention. I had already realised
that one could, in certain cases, reduce the number of parameters
needed to specify probabilities by using observations like these.
If it has been empirically established that P(y|x)=1, then one can
often infer, instead of measuring, P(y|z), after P(x|z) has been
measured.
>Then Piron goes on to prove a theorem which says that above axioms can
>be realized if logical propositions are identified with closed subspaces
>in a Hilbert space over R, C, or quaternions.
Unfortunately I have not been able to find a copy of Piron's book,
either for sale on the web or at the libraries of two major
American universities. From what you say above, though, it sounds
like Piron does more or less what the others do, namely introduce
a set of reasonable axioms, and then say that quantum mechanics is
one of a set of systems which satisfy those axioms.
Using empirically established implications allows us to reduce the
number of parameters we need to measure to specify all of the
probabilities involved, but not nearly as much as quantum mechanics
does, so other principles are needed. In most circumstances, there
will be no implications at all, and yet quantum mechanics still
places nontrivial constraints on the results of measurements.
For example, if we consider measurements of properties of a given
system, where each which measurement is a complete measurement
and has only two possible results, then we can ask the question:
What is the maximum number of measurements that we can have
in a set of measurements such that knowing the result of one
of the measurements requires maximum ignorance of what
the results of the other measurements would be? Maximum
ignorance in this case means that the probabilities assigned
to the results of all measurements (apart from the one
measurement whose result we know) is 0.5.
Quantum mechanics tells us that the answer is 3, because
there are three Pauli matrices. Without the identification
of the lattice of propositions with the subspace lattice of
a complex Hilbert space, no chain of reasoning that we know
of could lead somebody who didn't know quantum mechanics
to the number 3 in this case.
>Quantum theories with
>real or quaternionic scalars have been studied, but, as far as I know,
>nothing interesting came out of this. So, we are left with the usual
>C-number quantum mechanics whose mathematical apparatus directly
>follows from Birkhoff-von Neumann-Mackey-Piron axioms via Piron's theorem.
>There are quite a few reviews and book where you can find more
>details. You can check, for example,
>E.G. Beltrametti and G. Casinelli "The logic of quantum
>mechanics" (Addison-Wesley, Reading, 1981)
I will examine this book more carefully, but it appears
to provide a tortuous explanation of why quantum mechanics
is not unreasonable, rather than a demonstration that
reasonable starting points lead to the conclusion that
quantum mechanics is likely to work.
>or search the web for "quantum logic", "orthomodular lattice",
>etc.
I have already; I don't find anything enlightening.
>> When somebody says that statements
>> have "truth values" which can be complex, I do not know what they
>> mean.
>In quantum logic the "truth values" are not complex.
I was talking more generally about the assignment of truth
values which are neither probabilities nor true/false. Charles
introduces a bizarre truth value in his paper, which isn't
a probability and isn't true or false.
>> ... those who opt to study quantum logic
>> should be warned that it is not a system used to model inference,
>> as actual logic is, but that it is a particular mathematical system
>> to which actual normal true-and-false logic applies, and that calling
>> it logic is just poetry.
>I don't think so. In my opinion, classical logic developed by Aristotle
>and Boole refers only to propositions about classical objects.
>For quantum objects we need to take into account the statistical nature
>of measurements and indeterminism. This requires a change in the rules
>of logic.
When we find that we need to use probability, then we have found
that we need to use probability. It doesn't mean that logic
needs to be revised. The statisticians would be surprised to
hear that one has to discard logic itself in order to do
statistics.
>Quantum logic says that all classical axioms are still OK,
>except the axiom of distributivity. This axiom wasn't very intuitive
>in the classical system anyway. Quantum logic uses the "orthomodular
>law" instead. The distributivity axiom is a particular case of the
>"orthomodular law".
Well, the axiom which you claim isn't intuitive is the one which
says that if two statements are not both true, then at least one
of them is false. I suppose one could be sceptical enough to doubt
that axiom, but I do not think that "if proposition x implies
proposition y, then x and y are compatible" is more intuitive.
In fact, I would go so far as to say that "If two statements
are not both true then at least one of them is false" is
so intuitive that it is impossible to understand how it
could be false.
Moreover, normal logic has not failed at all. The distributive
law always has and always will work just fine. So-called
quantum logic has a non-distributive lattice because the
operations meet and join are not logical operations.
In quantum logic, the practice is to restrict the set of propositions
in the system to experimentally verifiable propositions. Then operations
of meet and join are introduced to identify the weakest proposition
implying two given propositions and the strongest proposition implied by
two propositions, respectively. But these operations are not the
logical AND and OR operations with which we are familiar. In fact,
they are not even logical operations at all, because what the
meet and join of two given propositions are is an empirical
question. One cannot know in advance whether measuring one
quantity will disturb the value of another quantity; one must
find out by experiment which measurements commute. Only after
sufficient experimentation with the system will we know the
structure of the lattice of experimentally verifiable
propositions.
Meanwhile, the origin of the non-distributivity in the lattice
of experimentally verifiable propositions come from cases
when we take the meet of two statements like:
"If I measure A I will get the result a1"
"If I measure B I will get the result b1"
If A and B are incompatible measurements, then there is no
way one could know the truth of both of these propositions,
hence there is no experimentally verifiable proposition
implying both of these propositions, so the meet of the
two propositions is the absurd proposition, which is the
equivalent of "false" in this system.
However, the distributive law of actual logic still works
fine. The proposition that we get by taking the logical
AND of the two propositions above is a perfectly valid
proposition, which may even be true, even though it is
not experimentally verifiable. You can even introduce
a predicate called "experimentally verifiable" into
normal logic, and deal with it in just the same way
as you deal with any other predicate.
If you follow the rules of normal logic you will not
be led to any contradictions. If the reasoning which
leads you to manipulate the symbols, in the way that
quantum mechanics prescribes, is good reasoning then
it can be expressed in terms of actual logic. If
somebody is explaining why a particular procedure
is followed then if they say "This procedure must be
followed because there is a class of objects to which
logic doesn't apply and ...", then they might just as well
say "There is no reason."
So what has happened is that the term "weakest experimentally
verifiable proposition implying two given propositions" was renamed
"meet", and something similar happened with "join". Then it was
observed that "meet" and "join" have algebraic properties similar
in some respects to the logical AND and OR. From this observation
the whimsical and metaphorical name "quantum logic" was chosen.
This name was heard by the masses, who interpreted it as a failure
of "classical" logic. Now we have a population who think that logic
itself is wrong and needs to be modified.
R.
tttito
..
> Let us imagine that Alice may 'perceive' it
> (but is that a measurement, or a pre-measurement?)
> as |Alice_pointer_A>.
I consider this one a measurement.
> And let us suppose that the observable
> (|Alice_pointer_A>+|Alice_pointer_B>)
> may be detected by Bob.
> The question seems (to me) to be: do the
> observable |Alice_pointer_A> and the observable
> (|Alice_pointer_A>+|Alice_pointer_B>)
> commute?
They do not commute in the sense that no observer can view Alice as
being in 1/sqrt(2)(|Alice_pointer_A>+|Alice_pointer_B>)(*) and in
|Alice_pointer_A> at the same time.
However my notation may be misleading. The shorthand (*) however
actually denotes a collection of similar states
1/sqrt(2)(a|Alice_pointer_A>+b |Alice_pointer_B>) with different
relative phases. Note also that the detection of macroscopic
superpositions may require a long run of independent 'weak'
measurements (see [1],[2]).
..
> 'The German language allows to make the distinction
> between "Realitaet" and "Wirklichkeit".
> Realitaet relates to the Latin word for "thing" ("res").
Res comes from reor, ratus (I think, I estimate, cf. Ding<->denken,
thing<->think in German and English) arguably from an Indoeuropean root
rt- (rectus, right, recht). The meme of objects as mental constructs is
not exactly new.
..
Cura ut valeas,
IV
PS Here is a little add-on to my previous post, where I wrote
>> Alice (if we want to assign her a consciousness ) may perceive [her state] as
>> |Alice_pointer_A> or |Alice_pointer_B>, but not both (unless she is schizophrenic ).
The remark about Alice mental health is very speculative, to put it
mildly, and it has no bearing on the main argument. However, since in
RQM collapse is subjective, one may conjecture that non-standard or
"defective" state-vector reduction corresponds to the breakdown of
intersubjective agreement (i.e. loss of contact with "reality") that
characterises some mental illnesses. Such an argument exemplifies my
take on quantum relationalism.
[1] "At first sight the above method of detection of QIMDS seems to
violate a fundamental
principle of quantum measurement theory: should not continuous
measurement of the flux
value automatically destroy the possibility of superposition of
different values? As emphasized
by the authors, the reason that it does not is that the measurement is
very 'weak' and the
data are obtained only by statistical averaging over a large sample."
in T.Leggett , "Testing the limits of quantum mechanics: motivation,
state of play, prospects" at
"http://www.iop.org/EJ/abstract/0953-8984/14/15/201/
[2]
http://groups.google.com/group/sci.physics.research/msg/1f18a6ef45054071
Cl.Massé
> Consciousness ia slippery concept,
Agreed, but that is irrelevant. See below.
> let's deal first with pointer
> states, which we can measure/observe. Bob does not measure Alice's
> consciousness (if Alice is just a measurement device, she may not need
> one). He observes only her pointer states, whose superpositions he may
> detect (although that may be hard to implement experimentally cf. [3]).
> So Alice's state may be detected as
> 1/sqrt(2)(|Alice_pointer_A>+|Alice_pointer_B>) by Bob, while Alice (if
> we want to assign her a consciousness ) may perceive it as
>> Alice_pointer_A> or |Alice_pointer_B>, but not both (unless she is
> schizophrenic ). The key point here is that the state vector of a
> system is observer-dependent, i.e. it encodes the information relative
> to a given observer's perpective. Different observers -> different
> state vectors. Bob's perpective of Alice is not necessarily the same
> that any instance of Alice has of Alice ([2]).
All theory, all experiment need ultimately to get into *my* consciousness in
order to make sense (for me, I don't care for others). Even if conscious
physicists are developing theories and observing. Even if their
consciousness is a superposition, I don't sense mine as such. Therefore, my
consciousness can't observe an observer observing a superposition of
apparatuses. In my world, therefore in yours as far as I can perceive it,
nobody can say me he is experiencing a superposition of consciousness.
> If I thought it were so I would not waste my time. Detecting
> macroscopic superpositions is a concrete challenge.
Macroscopic superposition can be measured by an experiment that would give
different outcomes for different systems in the same superposed state, even
in the Copenhagen interpretation.
Eugene Stefanovich
r...@maths.tcd.ie wrote:
>
> My criticism is simple. Rovelli claims that quantum mechanics suddenly
> all makes perfect sense if we say things like "The cat is alive
> relative to Bob", instead of things like "The cat is alive". He
> says that the expression "The cat is alive" is meaningless, unless
> we specify somebody who the cat is alive relative to.
>
> Apart from the fact that this is just a suggestion about the way
> that we should speak, or the way that Rovelli thinks we should
> speak, there is another problem. If the cat is alive relative
> to Bob, can it be dead relative to anybody else? Rovelli doesn't
> answer this question. If the answer is yes, then that means that
> the interpretation is just the many-worlds interpretation, because
> Rovelli is saying that one person can see a dead cat while another
> person can see the same cat alive at the same time. On the other
> hand, if the answer is no, then that means that if the cat is
> dead for one person then it is dead for everybody. That means
> that it is absolutely dead and its being dead isn't relative
> at all. In that case, Rovelli's claim that we shouldn't say
> "The cat is alive" is wrong. If the cat is alive relative to
> anybody then it is absolutely alive and not dead.
We know from special relativity that the length of a stick can be 1m
from the point of view of observer A, and the length of the same stick
can be 0.5m from the point of view of another (moving) observer B.
How different it is from saying: the cat is alive from the point of view
of A while the cat is dead from the point of view of B?
Eugene.
sigol...@gmail.com
precisely Rovelli's point, namely that the state of the cat might be
described by sheaf theory. This would mean that although the correct
statement is "the cat is alive to Alice", whenever two observers are
able to check on the state of the cat, they always will agree according
to the sheaf gluing property. This is only important here because it
means that there is no need to expect the quantum mechanical formalism
to reflect "elements of reality" which are not present in the sheaf
interpretation, so EPR's argument for the incompleteness of the QM
formalism fails. There are preprints on this e.g. by Zafiris (math-ph
0306045) and Kafotos, kato, and Roy (Sheaf cohomology and geometrical
approach to EPR) as well as others by Isham et. al. which I'm still
reading.
Eugene Stefanovich
> Unfortunately I have not been able to find a copy of Piron's book,
> either for sale on the web or at the libraries of two major
> American universities.
You can ask your library to get this book through interlibrary loan.
This is a small book with powerful ideas. Definitely worth the effort.
Otherwise, you can look for Piron's journal articles in 1960's and
1970's, e.g.,
C. Piron, Helv. Phys. Acta 37 (1964), 439.
> From what you say above, though, it sounds
> like Piron does more or less what the others do, namely introduce
> a set of reasonable axioms, and then say that quantum mechanics is
> one of a set of systems which satisfy those axioms.
He says that QM is, basically, a unique system that satisfies those
axioms.
>> In my opinion, classical logic developed by Aristotle
>>and Boole refers only to propositions about classical objects.
>>For quantum objects we need to take into account the statistical nature
>>of measurements and indeterminism. This requires a change in the rules
>>of logic.
>
>
> When we find that we need to use probability, then we have found
> that we need to use probability. It doesn't mean that logic
> needs to be revised. The statisticians would be surprised to
> hear that one has to discard logic itself in order to do
> statistics.
Note that quantum probabilities are quite different from classical
probabilities. Classical probabilities arise in a classical mixed
state. Quantum probabilities are present in both pure quantum state
(a ray in the Hilbert space) and in the mixed quantum state
(the density operator).
Doesn't Feynman's two-slit experiment defies the rules of classical
logic? According to these rules we should admit that the electron
passes through both slits, which is nonsense. Of course, this experiment
can be described by invoking the formalism of quantum mechanics, i.e.,
wave functions and the Hilbert space and al that.
The contribution of Birkhoff,
von Neumann and others was to recognize that at a deeper level this
formalism amounts to the change of the rules of logic.
>>Quantum logic says that all classical axioms are still OK,
>>except the axiom of distributivity. This axiom wasn't very intuitive
>>in the classical system anyway. Quantum logic uses the "orthomodular
>>law" instead. The distributivity axiom is a particular case of the
>>"orthomodular law".
>
>
> Well, the axiom which you claim isn't intuitive is the one which
> says that if two statements are not both true, then at least one
> of them is false.
Are we talking about the same distributive law? The law I know about
involves three propositions A, B, and C, and it says:
A or (B and C) = (A or B) and (B or C)
> I suppose one could be sceptical enough to doubt
> that axiom, but I do not think that "if proposition x implies
> proposition y, then x and y are compatible" is more intuitive.
I may agree that this axiom is not very intuitive, but it
ought to be true, because it leads directly to the formalism of
quantum mechanics which has been verified by experiment an uncountable
number of times.
> So what has happened is that the term "weakest experimentally
> verifiable proposition implying two given propositions" was renamed
> "meet", and something similar happened with "join". Then it was
> observed that "meet" and "join" have algebraic properties similar
> in some respects to the logical AND and OR. From this observation
> the whimsical and metaphorical name "quantum logic" was chosen.
> This name was heard by the masses, who interpreted it as a failure
> of "classical" logic. Now we have a population who think that logic
> itself is wrong and needs to be modified.
You seem to suggest that logic is independent on physical experience.
I don't think so. I think that Aristotle's and Boole's postulates
look so obvious to us simply because we never meet quantum objects in
our everyday life. For quantum objects, two properties may not be
measurable simultaneously and measurements performed in an ensemble
of identically prepared systems may not be reproducible. From classical
standpoint these are pretty unusual features. So, I wouldn't be
surprised to discover that the rules of logic itself should be changed
in order to accomodate these features.
Eugene.
J. Horta
Thank you for a very stimulating and interesting discussion but...
Perhaps I'm simply not good enough to understand the deeper
issues being discussed in this thread (and the cited papers).
But, I can't help but ask what is wrong with the usual
interpretation that the wave function for (say) a cat + atom
system referring to an ensemble of similarly prepared cats
and atoms? If so determining the exact time of death for a given
cat can then only be address as a statistical statement about
an ensemble of cats and atoms. From my perspective the confusion
creeps in from people desperately wanting the wave function to address the
individual member and not the full ensemble. BTW if the usual bit about
"addressing the statistics of an ensemble" is indeed truly the way of
things then the cat is surely dead when observed to be so. Once death has
been determined each observer places the dead cat in a new ensemble
described by a new and improved state vector of known dead cats and then
moves on.
I recognize I could truly be one of those people who just
won't (or can't) get it so trying to convince me may well
be futile.
thomas_l...@hotmail.com
>We know from special relativity that the length of a stick can be 1m
>from the point of view of observer A, and the length of the same stick
>can be 0.5m from the point of view of another (moving) observer B.
>How different it is from saying: the cat is alive from the point of view
>of A while the cat is dead from the point of view of B?
I believe that what Rovelli is pointing at is not just a problem with
the interpretation of QM, but a problem with QM itself. This is of
course a drastic conclusion, since QM has agreed with every experiment
for a century. OTOH, Newtonian mechanics agreed with every experiment
for over 200 years, and it still turned out to be wrong.
Specifically, I claim that QM, or at least QFT, needs to be modified
to allow for an explicit representation of the observer. A crucial
assumption in Rovelli's paper is that the observer has a well-defined
position x. However, in the Hamiltonian formalism, we relate
everything to a particular Lorentz frame, which can be interpreted as
the observer's rest frame. Thus, we implicitly introduce an observer
which moves at 3-velocity v = 0. But assuming that both x and v have
sharp values runs into trouble with the uncertainty principle, since
we would expect something like
[x, v] = i hbar / M,
where M is the observer's mass. Only when M = infinity can both x and
v be measured sharply. But this is the limit where the observer
becomes classical, which is manifest in the Copenhagen interpretation.
Hence, one may expect that there exists a manifestly observer-
dependent generalization of QM, which reduces to conventional,
observer-independent QM in the limit that the observer is infinitely
massive. This idea is similar to how SR and QM reduce to Newtonian
mechanics in the limits c -> infinity and hbar -> 0. It is also
notable how reasoning about observers led to both SR (time and length
depend on the observer) and QM (observation affects the canonical
conjugate variable).
It is clear that an implicit assumption that the observer is
infinitely massive will cause trouble for quantum gravity, because an
infinite mass couples to gravity and collapses into a black hole.
John Bell
> r...@maths.tcd.ie wrote:
>
> >
> > My criticism is simple. Rovelli claims that quantum mechanics suddenly
> > all makes perfect sense if we say things like "The cat is alive
> > relative to Bob", instead of things like "The cat is alive". He
> > says that the expression "The cat is alive" is meaningless, unless
> > we specify somebody who the cat is alive relative to.
> >
> > Apart from the fact that this is just a suggestion about the way
> > that we should speak, or the way that Rovelli thinks we should
> > speak, there is another problem. If the cat is alive relative
> > to Bob, can it be dead relative to anybody else? Rovelli doesn't
> > answer this question. If the answer is yes, then that means that
> > the interpretation is just the many-worlds interpretation, because
> > Rovelli is saying that one person can see a dead cat while another
> > person can see the same cat alive at the same time. On the other
> > hand, if the answer is no, then that means that if the cat is
> > dead for one person then it is dead for everybody. That means
> > that it is absolutely dead and its being dead isn't relative
> > at all. In that case, Rovelli's claim that we shouldn't say
> > "The cat is alive" is wrong. If the cat is alive relative to
> > anybody then it is absolutely alive and not dead.
>
>
> from the point of view of observer A, and the length of the same stick
> can be 0.5m from the point of view of another (moving) observer B.
> How different it is from saying: the cat is alive from the point of view
> of A while the cat is dead from the point of view of B?
>
unsatisfactory if for no better reason than that it implies that
nothing is real without the intervention of human beings. In this
experiment the possibility of a radioactive decay is arranged to poison
the cat (or not). One can argue more realistically that here it is the
cat actually performing the experiment (albeit unwillingly), and the
cat's observation that seals its fate. There is then no real paradox,
irrespective of whether humans are subsequently involved or not.
I am inclined to agree from the above that Rovelli's position does
appear empirically untenable
Oh No
>Oh No <No...@charlesfrancis.wanadoo.co.uk> writes:
>
>
>>The fundamental principle is that we can only say where something is if
>>we say where it is relative to other matter.
>If the principle is that "we can only say where something is if
>we say where it is relative to other matter", then that on its
>own is not incoherent, but Rovelli's claim that this is the
>secret to understanding quantum mechanics is.
I am not certain that Rovelli makes that claim in a clear way. He does
discuss Descartes who I think gave the first clear expression of the
principle, but then, as it seems to me Rovelli gets diverted by too much
abstract discussion of observers. However, I certainly make that claim.
>It will not be easy
>to build it into a mathematical stucture.
I don't think it is.
>It is a statement about
>the absence of something (an absolute background). Mathematically
>representing the absence of something is a daunting task, especially
>if you want it to be your starting point.
Fortunately most of the work has already been done by others. The task
is much more mathematically daunting than could be done by any one
person.
>If this principle has
>entered into the formalism you present in your paper, I have not
>understood where.
That is probably a central reason that you haven't understood what I am
trying to say about the interpretation of quantum mechanics. And yet to
me, the principle enters into every part of the paper. Certainly I
discuss it in the introduction, and the whole purpose of introducing
quantum logic is to have a formal language in which we can discuss
matter and measurement in the absence of background.
>I would agree with the truth of the so-called fundamental principle
>if it is understood to assert that all location is relative location,
>or, equivalently, that absolute motion is only an idea, not a feature
>of the physical world.
Distinguish two principles of relationism, relativity of position
expressed above, and essentially due to Descartes, and relativity of
motion, on which foundation Einstein based special relativity. They are
closely related, and indeed the second is contained in the former. Both
are needed to get very far with quantum theory.
> I must regard a principle which attempts
>to place restrictions on what I can say as belonging to the field
>of linguistics, and not physics.
I don't think you would really. Actually the opposite. The only
restriction I wish to place on language is to ensure that it is used to
discuss physics rather than fantasy. Language can describe a land of
wizards and dragons, but such a land does not exist. Likewise language
can discuss absolute space, and yet with a careful analysis of
measurement, we may recognise that absolute space has no empirical
justification.
>But even in the form in which I agree with it, the principle
>does not provide us with any "positive knowledge". It only
>tells us what is not the case, without giving us any clearer
>idea of what might actually be the case.
True, but that changes the question. Having answered the question "what
can we not say?" the question becomes "what can we say?". The objective
now is to write down postulates for what we can say, in accordance with
observation and without contradicting the fundamental principle. The
development of a formal language is just a tool for doing that.
>As such, it can
>perhaps serve to prevent error, but it cannot be the foundation
>of any body of knowledge about actual facts.
Preventing error may be sufficient. If one can eliminate all
possibilities but one, then that which remains must be the truth. If we
can write down a set of postulates, based on observation, and of
sufficient strength to constrain physics, then we have something
worthwhile. If we can show that these postulates describe what is
essentially a unique structure, then we will have a proven theory of
physics which does not rely on induction. My paper, gr-qc/0508077, has
been updated a couple of times since you read it. In the latest
revision, which I am replacing today, I have split the definitions which
determine the mathematical structure into definitions and postulates.
The distinction is that postulates contain empirical assertions about
the world, whereas definition are purely semantic and determine only
mathematical structure. I have had some trepidation about doing this,
because I am not sure that it is always completely obvious which is
which. Overall I think it is as step forward, but I would appreciate if
you felt strong enough to subject it again to your logical mind.
The main point is that both principles are necessary. I would argue,
however, that, according to the fundamental principle of relationism, to
talk of speed or even of space-time coordinates we have first to
propagate information. I only see two options. Either there is
instantaneous propagation which is empirically false, or there exists a
maximal speed for the propagation of information. I would argue further,
that the principle requires that the properties of matter have no
dependency on time or position, and that this is expressed in the
cosmological principle, from which we may infer the principle of general
relativity.
>In any case, your argument seems to be that quantum
>logic on its own places no restrictions on the results
>of measurements.
I don't say no restriction; just that on its own it doesn't get us very
far.
> You say: "We cannot usefully determine anything much from that
> unless we also have a time evolution equation." I don't know
> what you mean by "determine anything much from that". What
> we can do, without a time evolution operator, is observe,
> for each preparation, the probability of observing each
> result to each measurement. Measurements performed at
> different times after the preparation of the system simply
> count as different measurements. After we have collected
> the statistics of the measurement results, we then
> go about constructing a time evolution operator to describe
> the statistical relationships we have empirically discovered.
I don't do things in that order at all. I create a mathematical
structure then define time evolution as dictated by covariance
considerations. That leads me to qed, plus some variants which appear to
include theories of weak and strong interactions. The time evolution for
any given situation must be an application of these fundamental
theories. I have just put a paper on this on arxiv, gr-qc/0605127
>As I explained above, measurements performed at different
>times are different measurements and get represented by
>different linear operators. We do not have to throw
>away the Hilbert space and get a new one if we want to
>mathematically represent measurements performed at different
>times. If a Hilbert space is "defined at a given time" then
>it is defined for all time.
That is probably mathematically true, but the application of mathematics
to physics does, in this instance, require some delicacy. As I have
formulated it Hilbert space is defined at given time, or at any rate the
equivalent quantum logic discusses measurement results at given time. It
then becomes necessary to find an operator to describe time evolution as
a map from Hilbert space at time t to Hilbert space at time t+dt. We
need to forget that the two Hilbert spaces can be isomorphically
identified, because that loses some of the physical information required
to describe dynamics (it also leads to indefinability problems in qed,
but that is another story).
>But we have a clear example of such a system. The spin of
>a spin-half particle can be measured along three orthogonal
>axes, and each of those measurements has two possible
>results.
>Are you saying that the simple story of the
>spin-half particle and the Pauli matrices and measurements
>with Stern-Gerlach magnets and so on is all a pack of
>lies, because:
>
Of course not.
>But I gather from this that I have failed to explain properly what
>I was saying in the first place.
I prefer that theory, but the theory I actually subscribe to is that
that I have failed to explain coherently what my objection actually is.
>Let me try again.
Ok
>Suppose you know no quantum mechanics. There are 10 different
>measurements you perform, each of which can give 10 different
>results. (Despite your protestations to the contrary, this is
>quite possible. Consider a spin-9/2 particle and measurements
>along any ten distinct axes in three-dimensional space as an
>example.)
Trouble is, when I want to determine spin, I can't think of a way of
doing it which does not require an analysis of dynamics, like bending a
path in a Stern Gerlach experiment. In fact, the formulation of quantum
logic in A Relational Quantum Theory incorporating Gravity (RQG), gr-
qc/0508077 (revised from the version you read) makes no mention of spin.
It is introduced in a follow up paper "A Treatment of Quantum
Electrodynamics as a Model of Interactions between Sizeless Particles in
Relational Quantum Gravity" because it turns out that there is no
covariant formulation which does not require it.
>
> Call the measurements M0, M1, ... , M9.
>
> Let r_ij denote the jth result to the ith experiment. Let
> P(r_ij|r_kl) denote the probability of observing result
> r_ij when performing measurement Mi on a system which
> has just given result r_kl upon measurement of Mk.
>
As you have formulated it here, I think the conclusion is that quantum
mechanics does contain an additional assumption over and above the quite
artificial construction of states from a basis at given time. I think,
in the context of a post on spr, I can only point to what I think this
is, because for a complete, full, and rigorous analysis of measurement,
I think we need a full, consistent, and working theory of qed, which has
been one of the main objects of my research over the years. The paper I
have just put on arxiv represents the culmination of this research so
far. In short, I think the complete answer to the issue you raise is not
simple.
As I formulate quantum theory I start with only one type of measurement,
specifically measurement of position. To put this in the context of your
example, let us call this measurement M9. I then construct labels for
other states artificially, by creating a Hilbert space which is, in
essence, determined by the probabilities for getting each of the results
r_9j. The claim that M9 is complete means to me that all physical states
can be represented by states of this Hilbert space. (in fact this
assumption has to be relaxed in qed, but I retain the Hilbert space even
though not all states in it necessarily correspond to real measurement
results - there will be so called "virtual" photons, which I treat as
real photons which cannot be directly measured).
Now, if I am to do another measurement, M8, say, then I have to carry
out a physical interaction with the system. The problem now is that so
far I have only developed a static labelling system for states, at the
time of the measurement M9. In order to describe what happens in the
measurement M8, I need to be able to discuss what happens dynamically,
not only for a non-interacting particle but also in the interacting
theory. Strictly speaking it is quite premature to discuss any other
measurement at all. The fact that I do so requires the introduction of a
new postulate, vis luder's projection postulate.
Introducing Luder's projection postulate at this stage does require an
additional assumption about the behaviour of matter. I don't think it is
quite fair to say it is a hidden assumption, but what is true is that it
constrains the theory in a non-trivial way, just as you suggest. As far
as the logical development of the model as a physical theory is
concerned, it is perhaps premature. Whether this is a legitimate
constraint, or even a necessary one, is an issue which I don't think can
be answered properly without first developing a complete account of time
evolution, including an account of the interactions of elementary
particles, namely a full and consistent qed. I have included it where I
have for the sake of reproducing standard quantum theory, but the
justification for doing so requires looking ahead to the rest of the
development in the new paper. Ultimately, one wants to show that it is a
redundant assumption, in other words, that by applying the fundamental
principle one can show that the time evolution of the complete
interacting theory, i.e. qed, is such that measurement is already
constrained such that the projection postulate is obeyed.
What I have actually done, in introducing the projection postulate in
the manner in which I have, is to reduce all measurement to measurement
of position, but to a measurement of position in a larger system in Fock
space. If M9 corresponds to measurement of position of an elementary
particle, then M8 corresponds to measurement of position of some
particle in a larger quantum system containing the measurement
apparatus. A discussion of the behaviour of such larger systems requires
the full development of qed in the second paper, however in so far as I
can see, that development does not specifically depend on the projection
postulate, only on the fundamental principle contained in relationism.
So, what I would say is, have I given a consistent interpretation of
quantum theory, I think I have: It is a model of interactions of between
sizeless particles in the absence of space-time background. Have I given
a full and complete account of the measurement problem, demonstrating
that Luder's projection postulate is a consequence of the theory? No, I
haven't, but I think I have made it reasonable. Do I believe that such
an account is possible in principle within the theory? Yes I do. Do I
think that such an account would be publishable, or indeed that if it
were given anyone would understand a word of it? No, at the present
time, I'm afraid I don't believe that.
r...@maths.tcd.ie
>Eugene Stefanovich <eug...@synopsys.com> writes:
>>Quantum logic says that all classical axioms are still OK,
>>except the axiom of distributivity. This axiom wasn't very intuitive
>>in the classical system anyway. Quantum logic uses the "orthomodular
>>law" instead. The distributivity axiom is a particular case of the
>>"orthomodular law".
>Well, the axiom which you claim isn't intuitive is the one which
>says that if two statements are not both true, then at least one
>of them is false.
My mistake; I confused distributive with de Morgan. The distributive
laws say that x AND (y OR z) = (x AND y) OR (x AND z), and the
same thing with AND and OR swapped. The distributive law is, of
course, absolutely true, and has not been found to be deficient
in any way. If one takes the time to understand what it asserts,
one can clearly see its truth. There can be no counterexample
to it because any counterexample would be self-contradictory.
R.
tttito
..
>
> We know from special relativity that the length of a stick can be 1m
> from the point of view of observer A, and the length of the same stick
> can be 0.5m from the point of view of another (moving) observer B.
> How different it is from saying: the cat is alive from the point of view
> of A while the cat is dead from the point of view of B?
..
Good point.
The question is whether A and B have an information exchange protocol
that allows them to compare their measurements in a testable and
reproducible way. In the stick's case they do. In the cat's case the
jury is still out.
I think some day we will be able to detect interference patterns of
dead and alive cats (or people for that matter). The recent results on
macroscopic superpositions are encouraging and there may be more in
the pipeline. Detecting superpositions however does not entail
information exchange on distinct measurement outcomes.
Direct communication between observers perceiving mutually exclusive
measurement outcomes is another matter, not just for cats but for
electrons too. Once you've measured the position of an electron it's
hard to exchange information with someone who got a different reading
of the same measurement. How that could be done exceeds even my rather
fertile fantasy.
IV
r...@maths.tcd.ie
>r...@maths.tcd.ie wrote:
>> My criticism is simple. Rovelli claims that quantum mechanics suddenly
>> all makes perfect sense if we say things like "The cat is alive
>> relative to Bob", instead of things like "The cat is alive". He
>> says that the expression "The cat is alive" is meaningless, unless
>> we specify somebody who the cat is alive relative to.
>>
>> Apart from the fact that this is just a suggestion about the way
>> that we should speak, or the way that Rovelli thinks we should
>> speak, there is another problem. If the cat is alive relative
>> to Bob, can it be dead relative to anybody else? Rovelli doesn't
>> answer this question. If the answer is yes, then that means that
>> the interpretation is just the many-worlds interpretation, because
>> Rovelli is saying that one person can see a dead cat while another
>> person can see the same cat alive at the same time. On the other
>> hand, if the answer is no, then that means that if the cat is
>> dead for one person then it is dead for everybody. ...
>We know from special relativity that the length of a stick can be 1m
>from the point of view of observer A, and the length of the same stick
>can be 0.5m from the point of view of another (moving) observer B.
>How different it is from saying: the cat is alive from the point of view
>of A while the cat is dead from the point of view of B?
Indeed it is very different. In special relativity we can accept the
fact that different people can measure the length of a stick and
get different answers. The reason we can accept it is that each
individual's account of the events that they saw is consistent
with each other individual's account, in specific regards. Those
things which two observers will agree on are the relativistic
invariants. A good example of a relativistic invariant is whether
a cat is alive or dead. Everybody should agree about that, regardless
of their direction or speed of travel.
Indeed, everything measurable is a relativistic invariant. The
distance from here to the moon right now is not a relativistic
invariant, but it isn't measurable. The amount of time it takes
a light beam to get there and back is measurable, and it's a
relativistic invariant.
It is because such things are the same for everybody that we
consider them to be features of the objective world. So while
it's okay for a theory to suppose that things will look one
way for one person and a different way for a different person,
the two different people must agree on some facts if they are
to be considered to be living in the same world. If you see
a dead cat in front of you, while I see the same cat alive, then
you will go on to live in a world in which the cat has died,
while I will go on to live in a world in which the cat is still
alive. These are different worlds.
So I'm not saying that it's completely unacceptable to suppose,
when interpreting quantum mechanics, that a cat is alive relative
one person and dead relative to somebody else. What I'm saying
is that this expression ("alive relative to one person and dead
relative to somebody else") only makes sense if there are parallel
worlds. Consequently, if Rovelli's interpretation is not just the
same as the many worlds interpretation, and if it isn't incoherent,
then Rovelli's answer to the question: Can a cat be alive relative
to one person and dead relative to another? would have to be no.
Rovelli himself says (*), "In Everett, there is an ontological
multiplicity of realities, which is absent in the relational point
of view ... " Without an ontological multiplicity of realities, one
couldn't have a cat which was alive relative to one person and dead
relative to another. So Rovelli must be saying that if a cat is
alive relative to one person then it is alive for everybody. That
is what it means to say that there is no "multiplicity of realities".
So then whether the cat is alive or dead, or whether a system is
in a particular state, or whether a (measurable) quantity has a
particular value must be the same for everybody. But in that case
there is nothing relational at all about anything, and putting
"relative to Bob" at the end of statements is completely unnecessary.
Best,
R.
Oh No
>
>
>
>We know from special relativity that the length of a stick can be 1m
>from the point of view of observer A, and the length of the same stick
>can be 0.5m from the point of view of another (moving) observer B.
>How different it is from saying: the cat is alive from the point of
>view of A while the cat is dead from the point of view of B?
>
I would have said quite a lot. The stick has a proper length, which is
the same for all observers. It is only the niceties of determining
coordinate systems which makes it measured length different for
different observers. Now we have the situation that every observer who
observes the cat can agree as to whether the cat is alive or dead. Only
those observers who have not observed the cat cannot say which.
There is a lot of difference between saying "I don't know whether the
cat is alive or dead" and saying "The cat is some peculiar mixed state
of being alive or dead". I maintain that quantum mechanics is simply
saying the former. One can simply accept that it is the case, and that
since one gets no contradictions it must be right. That possibly sums up
the orthodox view. But actually I don't think that is enough. Lack of
knowledge is described perfectly by probability theory, and probability
theory is not quantum logic. One needs to be able to explain precisely
what is going on in the quantum world, different from the classical,
that leads to the precise equations which describe quantum logic.
Cl.Massé
news: 446E080F...@synopsys.com
> Yes, Schroedinger equation is useless until you specified H, and this
> is the most challenging task of the theory. As a minimum, the
> Hamiltonian should satisfy two requirements:
>
> 1. relativistic invariance - i.e., correct Poincare Lie algebra
> commutators with other generators - of space translations P, of
> rotations J, and of boosts K
Isn't it automatically satisfied if H is a generator of time translation?
> 2. cluster separability - i.e., the dynamics of spatially separated
> groups of particles should be independent.
>
> These conditions are not so easy to satisfy. Weinberg in his vol. 1
> "The quantum theory of fields" claims that the only way to do that
> is local quantum field theory. However, he misses (at least) three
> other possibilities that were proven to work:
4. Non linear Schrödinger equation, with H depending on Psi.
5. Trivial representation, i.e. non propagating field.
and...
6. Classical mechanics.
Eugene Stefanovich
r...@maths.tcd.ie wrote:
>>We know from special relativity that the length of a stick can be 1m
>>from the point of view of observer A, and the length of the same stick
>>can be 0.5m from the point of view of another (moving) observer B.
>>How different it is from saying: the cat is alive from the point of view
>>of A while the cat is dead from the point of view of B?
>
>
> Indeed it is very different. In special relativity we can accept the
> fact that different people can measure the length of a stick and
> get different answers. The reason we can accept it is that each
> individual's account of the events that they saw is consistent
> with each other individual's account, in specific regards. Those
> things which two observers will agree on are the relativistic
> invariants. A good example of a relativistic invariant is whether
> a cat is alive or dead. Everybody should agree about that, regardless
> of their direction or speed of travel.
I am wondering if you have any hard evidence (besides your intuition) to
claim that being dead or alive is a relativistic invariant.
Take a look from a different perspective. Let A and B be two observers
that are shifted in time with respect to each other. Let's say that
the time shift is 10 years. Then, you wouldn't find it surprising to
discover that A and B disagree on whether the (same) cat is dead or
alive.
Now assume that instead of being time separated the observers
A and B move with respect to each other. Then, according to you,
they should see no difference in the health of the observed cat, no
matter how high their relative speed is. Why shifting the observer in
time is so profoundly different from changing the velocity of the
observer? Or translating the observer in space? Isn't it true that
relativistic theories require consideration
of the Poincare group in which all inertial transformations
(time shifts, space translations, rotations, and boosts) have equal
rights? Then why the time shifts are so different from other
transformations?
All these questions have nothing to do with quantum mechanics per se.
They remain relevant in the classical world as well?
Eugene.
Eugene Stefanovich
Oh No wrote:
> There is a lot of difference between saying "I don't know whether the
> cat is alive or dead" and saying "The cat is some peculiar mixed state
> of being alive or dead". I maintain that quantum mechanics is simply
> saying the former.
I think it is a wrong way to think about quantum mechanics.
First, when you perform a measurement on the cat you can say with
full certainty whether it is dead or alive. The question is only
about the theoretical description. Can the theory (=quantum mechanics)
predict the result of the measurement before it is done? The answer is
'no'. In the "Schroedinger cat" situation, quantum mechanics does not
give you a definite answer. It only gives you the probabilities of
possible outcomes. You can say (together with Einstein) that this
means that quantum mechanics is not a complete theory. I can agree with
that. But this is the best theory we have.
The peculiar superposition "dead + alive" does not refer to the
individual cat,
but to the ensemble of identically prepared cats in identical boxes.
If you keep in mind that the wavefunction provides you a description
of the ensemble rather that individual system, you can avoid all
kinds of paradoxes with the "wavefunction collapse" and with the role
of "mind" in this collapse.
Eugene.
Eugene Stefanovich
r...@maths.tcd.ie wrote:
> My mistake; I confused distributive with de Morgan. The distributive
> laws say that x AND (y OR z) = (x AND y) OR (x AND z), and the
> same thing with AND and OR swapped. The distributive law is, of
> course, absolutely true, and has not been found to be deficient
> in any way. If one takes the time to understand what it asserts,
> one can clearly see its truth. There can be no counterexample
> to it because any counterexample would be self-contradictory.
If x, y, and z are propositions abot classical objects, like
books or cars, then I fully agree with you - the distributive
law is valid. However, I am not sure if it is still true
for propositions about the quantum electron. Of course, in the quantum
case, one needs to first define the meaning of operations 'and'
and 'or'.
Eugene.
Eugene Stefanovich
>
> Perhaps I'm simply not good enough to understand the deeper
> issues being discussed in this thread (and the cited papers).
> But, I can't help but ask what is wrong with the usual
> interpretation that the wave function for (say) a cat + atom
> system referring to an ensemble of similarly prepared cats
> and atoms? If so determining the exact time of death for a given
> cat can then only be address as a statistical statement about
> an ensemble of cats and atoms. From my perspective the confusion
> creeps in from people desperately wanting the wave function to address the
> individual member and not the full ensemble. BTW if the usual bit about
> "addressing the statistics of an ensemble" is indeed truly the way of
> things then the cat is surely dead when observed to be so. Once death has
> been determined each observer places the dead cat in a new ensemble
> described by a new and improved state vector of known dead cats and then
> moves on.
>
> I recognize I could truly be one of those people who just
> won't (or can't) get it so trying to convince me may well
> be futile.
If I understand you correctly, then your approach to quantum
mechanics is exactly the same as the "statistical" or "ensemble"
intepretation in
L.E. Ballentine Quantum Mechanics: A Modern
Development (World Scientific, Singapore, 1998)
In my view, this is the only interpretation that makes perfect sense.
Eugene.
Oh No
>
>> There is a lot of difference between saying "I don't know whether the
>> cat is alive or dead" and saying "The cat is some peculiar mixed state
>> of being alive or dead". I maintain that quantum mechanics is simply
>> saying the former.
>
>I think it is a wrong way to think about quantum mechanics.
>First, when you perform a measurement on the cat you can say with
>full certainty whether it is dead or alive. The question is only
>about the theoretical description. Can the theory (=quantum mechanics)
>predict the result of the measurement before it is done? The answer is
>'no'. In the "Schroedinger cat" situation, quantum mechanics does not
>give you a definite answer. It only gives you the probabilities of
>possible outcomes. You can say (together with Einstein) that this
>means that quantum mechanics is not a complete theory. I can agree with
>that. But this is the best theory we have.
That is not why Einstein said quantum mechanics is not a complete
theory. He was talking of the impossibility of describing the situation
in EPR.
>
>The peculiar superposition "dead + alive" does not refer to the
>individual cat,
>but to the ensemble of identically prepared cats in identical boxes.
>If you keep in mind that the wavefunction provides you a description
>of the ensemble rather that individual system, you can avoid all
>kinds of paradoxes with the "wavefunction collapse" and with the role
>of "mind" in this collapse.
There are several problems with that. First of all it assumes a
frequentist interpretation, and yet a frequentist interpretation of
probability theory is now considered passe and inadequate. You do not
need an ensemble of cats even to discuss the probability that the cat
will be alive or dead, and nor is it even empirically valid to do so.
Most people think it would be bad enough to do the experiment at all,
let alone repeat it!
The cat is a bad example, because actually the predictions of the model
are identical to those of probability theory. It is actually an instance
where a frequentist interpretation of probability theory can be made to
work, but that only confuses the issue. What is more to the point is
that the superposition of cat states is not different in principle from
the superposition of "left slit" "right slit" states which gives you
interference patterns in a Young's slits experiment.
Now you cannot claim that the result of a Young's slits experiment is
the same as that given by probability theory for an ensemble of
particles each of which is either "left slit" or "right slit". So if you
want to give a frequentist interpretation of the Young's slits
experiment, every member of your ensemble has to be a superposed state.
Now if you apply that analysis to the cat, introducing an ensemble has
not solved any problem at all, it has if anything multiplied it because
now you have an ensemble of half live half dead cats.
But the worse problem is that whatever description is given to the cat
should also apply
Eugene Stefanovich
Oh No wrote:
> The cat is a bad example, because actually the predictions of the model
> are identical to those of probability theory. It is actually an instance
> where a frequentist interpretation of probability theory can be made to
> work, but that only confuses the issue. What is more to the point is
> that the superposition of cat states is not different in principle from
> the superposition of "left slit" "right slit" states which gives you
> interference patterns in a Young's slits experiment.
>
> Now you cannot claim that the result of a Young's slits experiment is
> the same as that given by probability theory for an ensemble of
> particles each of which is either "left slit" or "right slit". So if you
> want to give a frequentist interpretation of the Young's slits
> experiment, every member of your ensemble has to be a superposed state.
>
> Now if you apply that analysis to the cat, introducing an ensemble has
> not solved any problem at all, it has if anything multiplied it because
> now you have an ensemble of half live half dead cats.
OK, I also feel more comfortable not discussing those poor cats.
Particles are much better, because nobody cares whether they are dead or
alive.
I fully agree with you that if we divide the particles in the ensemble
into "left slit" particles and "right slit" particles, then we will lose the
interference picture. This is exactly why I refuse to talk about the
state of the individual particle. Quantum mechanics tells us nothing about
the state of the individual particle. Quantum mechanics with all its
apparatus cannot say which point on the screen will be hit by the
particle passed through the hole(s).
Quantum mechanics has no clue when exactly a given radioactive
nucleus will decay. So, there are certain observable things in
nature (like the "particle hits the screen at point A" or the "nucleus
decays at time t") which are
not covered by our theory at all. All the theory can do is to tell
us the probabilities. This is why quantum mechanics may
be called "incomplete". You may be right that EPR paper did not discuss
the incompleteness in this sense. But there are other Einstein's
writings which make me believe that this "incompleteness" is exactly
what he had in mind. Consider, for example, the following passage:
"I now imagine a quantum theoretician who may even admit that
the quantum-theoretical description refers to ensembles of systems
and not to individual systems, but who, nevertheless, clings to the
idea that the type of description of the statistical quantum theory
will, in its essential features, be retained in the future. He may
argue as follows: True, I admit that the quantum-theoretical
description is an incomplete description of the individual system. I
even admit that a complete theoretical description is, in principle,
thinkable. But I consider it proven that the search for such a
complete description would be aimless. For the lawfulness of nature
is thus constructed that the laws can be completely and suitably
formulated within the framework of our incomplete description. To
this I can only reply as follows: Your point of view - taken as
theoretical possibility - is incontestable." A. Einstein, in
Albert Einstein: Philosopher-Scientist, (Open Court, Peru, 1949)
This pretty much summarizes my position regarding quantum mechanics:
QM does not give us any knowledge about the state of individual
system. The word "state" refers to the ensemble of systems, not to
its individual members. The state of the ensemble is described by
the wave function. In order to compare this theoretical description
with actual observations, one needs to arrange experimentally
an ensemble of identically prepared systems. Then the frequencies
of measurements of different values of observables in this ensemble
will exactly coincide with numbers calculated as (integrals of)
the squares of the wave function. The results of individual measurements
have no theoretical explanation at all.
Eugene.
Dirk Bruere
> Thank you for a very stimulating and interesting discussion but...
>
> Perhaps I'm simply not good enough to understand the deeper
> issues being discussed in this thread (and the cited papers).
> But, I can't help but ask what is wrong with the usual
> interpretation that the wave function for (say) a cat + atom
> system referring to an ensemble of similarly prepared cats
> and atoms? If so determining the exact time of death for a given
> cat can then only be address as a statistical statement about
> an ensemble of cats and atoms. From my perspective the confusion
> creeps in from people desperately wanting the wave function to address the
> individual member and not the full ensemble. BTW if the usual bit about
> "addressing the statistics of an ensemble" is indeed truly the way of
> things then the cat is surely dead when observed to be so. Once death has
> been determined each observer places the dead cat in a new ensemble
> described by a new and improved state vector of known dead cats and then
> moves on.
And if there is only one experiment, how can QM refer to an ensemble
except as a work of fiction? And who said that QM had to be meaningless
except in terms of ensembles?
Dirk
J. Horta
it up or take the discussion to private mail. -ik ]
Probability theory says the chance of rolling a 6, even
if you roll the die only once. Probability theory says nothing
about a single roll of the die. The part people find really
irritating is the notion that QM makes probability fundamental
and so, from a classical perspective, our descriptions of
nature incomplete. This was ordained from the day momentum and
position values were replaced with non-commuting operators
acting on a Hilbert space.
> And who said that QM had to be meaningless
> except in terms of ensembles?
>
70+ years of experimental observation. We should go on empirical
observation and not what we want to be true. What rules out
nature dictating what is knowable? Some instinctive expectation
of a "reality" behind the observations? What if the statistics of the
ensemble is all that is knowable?? I tend to believe this is what
experiments show us.
> Dirk
Cl.Massé
news: 4HRpPjAP...@charlesfrancis.wanadoo.co.uk
> The fundamental principle is that we can only say where something is if
> we say where it is relative to other matter. That does not strike me as
> being incoherent. In fact I find it simple, and even empirically
> obvious. What is perhaps a lot less simple is building it into a
> mathematical structure. Doing so certainly defeated Descartes and
> Leibniz, but then we know a lot more now than they did then.
The metaphysical system of Leibniz doesn't postulate any space and time.
Actually, he was the first relationist.
On the other hand, the Einstein's relativity open the task of describing
matter without a time coordinate. There is no way to geometrically define
simultaneity. For doing that, postulates on light have first to be laid
down, that is, on the object that will populate space-time.
Looking at the least action principle, we can first define a geometrical
framework, and then formulate a model with this principle. We can also
observe that the least action between any too points has the properties of a
distance, and use it to build the space.
Even though relativity rests on the synchronisation between two points, it
relies on the use of a loop for doing it. The historical Michelson-Morley
experiment also relied on a loop setup to see the relative motion of ether.
And when relativity leads to a paradox (twins), it is anew in a loop
configuration.
More generally, physics is based on comparisons, that's the only way to
make a measurement, which is a comparison with a standard. Now it is
impossible to make a comparison between too different points. The false
hidden assumption, the same one that Poincaré didn't see, is that a physical
property is assigned to points. Physical properties should be assigned to
loops instead. That is more obvious for the gauges theories, where the
gauge is a useless value that disappears along a loop.
--
~~~~ clmasse on free F-country
Wanted: Schrödinger's cat, dead and alive.
r...@maths.tcd.ie
>r...@maths.tcd.ie wrote:
>> Unfortunately I have not been able to find a copy of Piron's book,
>> either for sale on the web or at the libraries of two major
>> American universities.
>You can ask your library to get this book through interlibrary loan.
>This is a small book with powerful ideas. Definitely worth the effort.
>Otherwise, you can look for Piron's journal articles in 1960's and
>1970's, e.g.,
>C. Piron, Helv. Phys. Acta 37 (1964), 439.
Thanks; I'll get hold of a copy soon.
>> From what you say above, though, it sounds
>> like Piron does more or less what the others do, namely introduce
>> a set of reasonable axioms, and then say that quantum mechanics is
>> one of a set of systems which satisfy those axioms.
>He says that QM is, basically, a unique system that satisfies those
>axioms.
Well, in that case, he and I are likely to disagree about what
constitutes a reasonable axiom. If the introduction of an
axiom is justified by an appeal to a failure of what you
call classical logic, then I have to regard that as an
indication that the axioms are chosen merely because
they give the desired result.
After all, if I send a mechanic to find out why a mechanical
failure occurred, and he returns to tell me that the failure
happened because the laws of logic failed, I would not find
his explanation plausible. If a physicist gives me the same
excuse, then I do not find it any more plausible than when
the mechanic says it.
>>> In my opinion, classical logic developed by Aristotle
>>>and Boole refers only to propositions about classical objects.
>>>For quantum objects we need to take into account the statistical nature
>>>of measurements and indeterminism. This requires a change in the rules
>>>of logic.
>>
>>
>> When we find that we need to use probability, then we have found
>> that we need to use probability. It doesn't mean that logic
>> needs to be revised. The statisticians would be surprised to
>> hear that one has to discard logic itself in order to do
>> statistics.
>Note that quantum probabilities are quite different from classical
>probabilities. Classical probabilities arise in a classical mixed
>state. Quantum probabilities are present in both pure quantum state
>(a ray in the Hilbert space) and in the mixed quantum state
>(the density operator).
There are two sides from which probability can be viewed. One can
consider only the mathematical representation of probability,
which is a field of pure mathematics, or one can consider the
physics involved in unpredictable events. If there is any
non-circular distinction between "classical" probability
and "quantum" probability, then it is a purely mathematical
distinction, and it concerns the symbols which we use when
we calculate the probabilities.
>Doesn't Feynman's two-slit experiment defies the rules of classical
>logic?
This is not the case, but before examining why it isn't, I think
it is important to understand why no experimental result can
ever "defy the rules of classical logic".
Logic deals with propositions, implication, the relations of
and, or, and not, and objects and predicates. We start with
a set of propositions, which are given (the data), and then
we proceed to deduce new propositions. In this way, we use
logic to increase the set of propositions which we know to
be true. That is all that logic does, and it does not make
predictions about the results of experiments, because
the notion of a prediction, or an experiment, is foreign
to logic.
Pure mathematics is built upon logic. The theorems of
mathematics are only acceptable to us because all of
the inferences which are used in the proofs come
from the rules of inference of logic. Each line of
a valid mathematical theorem is a proposition, and,
in annotated proofs, the rule of logic which allows
that proposition to be asserted should be listed
beside the proposition. That way we know that the
theorems of mathematics are acceptable. If somebody
tells us that logic is wrong, then he is saying
that the theorems of mathematics are no longer valid.
The statement that logic is wrong is, therefore,
absurd.
Logic makes no predictions about the results of experiments.
The discipline of logic ignores the content of the
propositions with which it deals. Propositions about
experiments or what will or did happen are no different
from those about set theory or legal procedure, as far
as logic is concerned. If you want to specialise in
propositions about experiments and make predictions
about the results of experiments, you need a different
discipline, namely physics.
For this reason, if a prediction about the result of
an experiment is incorrect, it is not logic which has
failed; the person who made the prediction has failed.
If you make a prediction, you must have some procedure
for making predictions. Logic does not give you that
procedure; you must find it elsewhere. If that procedure
doesn't work, don't blame logic.
With regard to the two-slit experiment, here is what
you can do with logic. You can write down a set of
propositions which say things like "A spot was observed
at point X", and so on, and deduce what you can from
those propositions. You will not be able to derive
a contradiction if you merely describe what was observed.
>According to these rules we should admit that the electron
>passes through both slits, which is nonsense.
The statement that the electron passes through both slits
is not a statement about what was observed. It is a story
that people like to tell when talking about the two-slit
experiment, but it has no relationship to either the
experimental facts or the mathematical formalism. In quantum
mechanics, the wavefunction is not something which waves
around in physical space; it is a function on configuration
space, so it lives in an abstract mathematical space which
looks nothing like the physical world.
A person might mathematically examine the propagation of the
wavefunction in configuration space. Such a person is qualified to
tell us about the solution to a differential equation, and we respect
him as a mathematician. However, he is not qualified to tell us
what transpires in physical space, and if he says that the electron
passed through both slits then he is talking about matters which
he knows nothing about. At best he is telling us a fictional
story inspired by true events (where the true events constitute
the interference pattern). I suspect this story is repeated so
frequently merely because people think that it's a cool thing
to say.
>Of course, this experiment
>can be described by invoking the formalism of quantum mechanics, i.e.,
>wave functions and the Hilbert space and al that.
>The contribution of Birkhoff,
>von Neumann and others was to recognize that at a deeper level this
>formalism amounts to the change of the rules of logic.
I'm afraid that for the reasons that I describe above the rules
of logic rest on a secured foundation, established forever. Birkhoff
and von Neumann may have established that a particular symbolic
system has something to do with incompatible measurements, but
the rules of logic remain in force when we want to prove theorems
about orthomodular lattices. Logic cannot be declared
invalid without removing the foundation upon which the theory of
orthomodular lattices is built. Logic has not changed; it is
merely being applied to a particular system.
>>>>Quantum logic says that all classical axioms are still OK,
>>>>except the axiom of distributivity. This axiom wasn't very intuitive
>>>>in the classical system anyway. Quantum logic uses the "orthomodular
>>>>law" instead. The distributivity axiom is a particular case of the
>>>>"orthomodular law".
>>>
>>> Well, the axiom which you claim isn't intuitive is the one which
>>> says that if two statements are not both true, then at least one
>>> of them is false.
>
>>Are we talking about the same distributive law? ...
It is absolutely and completely true for all propositions about
all objects. Suppose that there were three propositions, x, y,
and z, such that the distributive law is false. That is, suppose
x AND (y OR z) is true, while (x AND y) OR (x AND z) is false.
This is a contradiction. If we accept the first supposition,
then we really accept that x is true and that either y or z is
true. If the first option (y) is right then x AND y is true.
If the section option is right then x AND z is true. This
contradicts the second supposition, namely that (x AND y) OR (x AND z)
is false.
Now, a person might respond to me by saying that I have assumed
above that the laws of "classical" logic apply, and that this is
precisely what is in question so I may not assume it. However, one
must understand that I am not saying "If we follow these rules then
we get this conclusion". I am saying that these rules must inevitably
be followed and we must inevitably be led to the conclusion that
the distributive law always holds. It is as simple as saying that
if x is true and x implies y then y is true. It follows from the
truth tables. That is, it follows from the definitions of AND, OR,
NOT, and so on.
You cannot keep the same definitions of AND, OR and NOT and
have a counterexample to the distributive law. This is why
you say:
>Of course, in the quantum
> case, one needs to first define the meaning of operations 'and'
> and 'or'.
But if we put something different in place of "and", then we should
not call it "and" unless we are being poetic or want to deliberately
cause confusion. If we replace both "and" and "or" with some
completely different operations, whose actions must be discovered
by experiment, then we are not talking about logic any more,
and we should not call it logic because it isn't.
>> I suppose one could be sceptical enough to doubt
>> that axiom, but I do not think that "if proposition x implies
>> proposition y, then x and y are compatible" is more intuitive.
>I may agree that this axiom is not very intuitive, but it
>ought to be true, because it leads directly to the formalism of
>quantum mechanics which has been verified by experiment an uncountable
>number of times.
When you say "leads directly", I suspect you are assuming that
throwing away logic is acceptable.
>> So what has happened is that the term "weakest experimentally
>> verifiable proposition implying two given propositions" was renamed
>> "meet", and something similar happened with "join". Then it was
>> observed that "meet" and "join" have algebraic properties similar
>> in some respects to the logical AND and OR. From this observation
>> the whimsical and metaphorical name "quantum logic" was chosen.
>> This name was heard by the masses, who interpreted it as a failure
>> of "classical" logic. Now we have a population who think that logic
>> itself is wrong and needs to be modified.
>You seem to suggest that logic is independent on physical experience.
>I don't think so. I think that Aristotle's and Boole's postulates
>look so obvious to us simply because we never meet quantum objects in
>our everyday life.
No; the rules of propositional logic follow from the definitions.
You can only object to the rules of logic if you object to the
procedure of considering the consequences of definitions to be
true.
This principle - that the consequences of definitions are true,
is a fundamental principle which mathematics and physics both
assume. You cannot reject logic without rejecting mathematics
and physics too. A mathematician is not qualified to discover
that logic is false. A physicist is not qualified to discover
that mathematics is false, or that logic is false.
With regard to experience, logic does not say anything about
what we will or won't experience. If you write down a set
of propositions describing any experience you have had,
regardless of what type of objects you interact with in
everyday life, you will not be able to derive a contradiction.
If you start from a set of propositions and derive a contradiction,
then one or more of those propositions is not true. Hence,
if you write down a set of propositions describing your
experiences and derive a contradiction from them, then some
of those propositions were false and you shouldn't have written
them down.
>For quantum objects, two properties may not be
>measurable simultaneously and measurements performed in an ensemble
>of identically prepared systems may not be reproducible. From classical
>standpoint these are pretty unusual features.
No; there is no such concept as "unusual" in logic. Logic does
not state that all properties are simultaneously measurable.
It doesn't say anything about what is or isn't measurable. Hence,
the discovery that something is or isn't measurable is not
pretty unusual for logic.
R.
Oh No
>But, I can't help but ask what is wrong with the usual interpretation
>that the wave function for (say) a cat + atom system referring to an
>ensemble of similarly prepared cats and atoms? If so determining the
>exact time of death for a given cat can then only be address as a
>statistical statement about an ensemble of cats and atoms. From my
>perspective the confusion creeps in from people desperately wanting the
>wave function to address the individual member and not the full
>ensemble. BTW if the usual bit about "addressing the statistics of an
>ensemble" is indeed truly the way of things then the cat is surely dead
>when observed to be so. Once death has been determined each observer
>places the dead cat in a new ensemble described by a new and improved
>state vector of known dead cats and then moves on.
So long as it is understood that a cat is a cat, and an ensemble is an
idea, not evidence of the physical existence of many worlds and many
cats, not a great deal is wrong, because in this instance quantum theory
just yields the ordinary results of probability theory. The point is one
of principle, because in other instances, such as EPR, or Young's slits,
quantum theory yields quite different results from classical probability
theory.
Eugene Stefanovich
I see your point. If observers are not classical, then we cannot label
them by their time, position, orientation and velocity. Then we lose
the Poincare group properties of the transformations between inertial
observers. Without the Poincare group, there is no way to have
relativistic quantum mechanics, or QFT. Everything falls apart...
Eugene.
John Bell
> The peculiar superposition "dead + alive" does not refer to the
> individual cat,
> but to the ensemble of identically prepared cats in identical boxes.
> If you keep in mind that the wavefunction provides you a description
> of the ensemble rather that individual system, you can avoid all
> kinds of paradoxes with the "wavefunction collapse" and with the role
> of "mind" in this collapse.
This doesn't sound like much of an improvement to me since you appear
to be replacing QM wavefunction collapse with the collapse (or
perpetual branching) of macroscopic parallel universes. That strikes me
as a far more desperate measure than simply admitting that you do not
yet have an adequately comprehensive explanation of quantum
wavefunction collapse.
When you have something like a potentially dead cat in a box, you can't
pass it simultaneously through a pair of slits and obtain an
interference pattern. It is therefore nonsensical to pretend that it is
the same thing as a QM wavefunction. It obviously just represents a
statistical probability, like the result of turning over an already
dealt card. You can confirm this on opening the box which will reveal
not only dead or alive, but also when the original QM wavefunction
collapsed (by the temperature and smell of the corpse).
Ilja Schmelzer
> ... it
> means that there is no need to expect the quantum mechanical formalism
> to reflect "elements of reality" which are not present in the sheaf
> interpretation, so EPR's argument for the incompleteness of the QM
> formalism fails.
There is no need to do physics at all, but that does not mean that
EPR's argument fails.
According to Popper, we have to make nontrivial assumptions.
The question is which nontrivial assumptions better describe reality.
Ilja
Blackbird
If there is only one experiment, performed once, QM is no longer a
falsifiable theory. We belive in QM because we have performed the
experiments over and over again. And the results, while generally different
each time, has a distribution in agreement with the theory.
> And who said that QM had to be
> meaningless except in terms of ensembles?
It's not meaningless; but I believe that QM then becomes a theory of
philosophy rather than a theory of science.
Cl.Massé
news: 447662C8...@synopsys.com
> The peculiar superposition "dead + alive" does not refer to the
> individual cat,
> but to the ensemble of identically prepared cats in identical boxes.
> If you keep in mind that the wavefunction provides you a description
> of the ensemble rather that individual system, you can avoid all
> kinds of paradoxes with the "wavefunction collapse" and with the role
> of "mind" in this collapse.
I agree with this description, but other paradoxes emerge. We have no
description of a single particle, like with the Young holes and a one by one
beam, while the premise implies there must be one. Indeed, QM postulates
particular relations between the probabilities of different observables.
These relations are different from those of classical statistics, that's why
probability amplitudes are used.
r...@maths.tcd.ie
>r...@maths.tcd.ie wrote:
>>>We know from special relativity that the length of a stick can be 1m
>>>from the point of view of observer A, and the length of the same stick
>>>can be 0.5m from the point of view of another (moving) observer B.
>>>How different it is from saying: the cat is alive from the point of view
>>>of A while the cat is dead from the point of view of B?
>>
>>
>> Indeed it is very different. In special relativity we can accept the
>> fact that different people can measure the length of a stick and
>> get different answers. The reason we can accept it is that each
>> individual's account of the events that they saw is consistent
>> with each other individual's account, in specific regards. Those
>> things which two observers will agree on are the relativistic
>> invariants. A good example of a relativistic invariant is whether
>> a cat is alive or dead. Everybody should agree about that, regardless
>> of their direction or speed of travel.
>I am wondering if you have any hard evidence (besides your intuition) to
>claim that being dead or alive is a relativistic invariant.
Yes, I do. The simplest example of a relativistic invariant is
x^2 + y^2 + z^2 - t^2, where x, y, and z are the distances
between two events along three spatial axes while t is the
time interval between the two events. From this one can
see that the round-trip time for a light beam to travel
from one side of a chamber to the other and back again is
a relativistic invariant. Each observer will agree about
how many seconds this is. Dividing this time by two and
multiplying it by the speed of light gives a measure
of the diameter of the chamber, so every observer will
agree about the diameter of a chamber defined this
way, regardless of his motion.
The left ventricle in the cat's heart is a chamber
which is rhythmically changing shape for a live cat
but not changing shape for a dead cat. Every observer
will agree about whether the cat's heart is beating
because everybody will agree about the diameter of
the left ventricle as defined above, and whether
it changes rhythmically in time.
>Take a look from a different perspective. Let A and B be two observers
>that are shifted in time with respect to each other. Let's say that
>the time shift is 10 years. Then, you wouldn't find it surprising to
>discover that A and B disagree on whether the (same) cat is dead or
>alive.
Right.
>Now assume that instead of being time separated the observers
>A and B move with respect to each other. Then, according to you,
>they should see no difference in the health of the observed cat, no
>matter how high their relative speed is. Why shifting the observer in
>time is so profoundly different from changing the velocity of the
>observer? Or translating the observer in space? Isn't it true that
>relativistic theories require consideration
>of the Poincare group in which all inertial transformations
>(time shifts, space translations, rotations, and boosts) have equal
>rights? Then why the time shifts are so different from other
>transformations?
>All these questions have nothing to do with quantum mechanics per se.
>They remain relevant in the classical world as well?
Yes; your questions apply just as well to Newtonian physics
as they do to relativistic physics.
I would explain it this way: We have to consider when and where
A and B will meet up and express their agreement or disagreement.
If they were initially separated in time, by 10 years, then,
after they make the measurements on their cats, A will have to
wait ten years to meet B. When they meet, A will say "Ten years
ago I saw a live cat", and B will say "ten seconds ago I
saw a dead cat". They will then understand that they do not
disagree about the history of the cat.
Similarly, with spatial translations, A will have to
travel some distance to meet B, and he will remember
that he travelled that distance, and when he takes that
into account, A and B will agree about what happened.
If A hadn't taken the distance he travelled into
account, then A and B would have disagreed about where
the cat was.
With regard to the Poincare group, one must distinguish
between shifting or boosting the entire system on one hand,
and changing the relative location or motion of two parts of the
system, on the other. If we are changing from one observer's
point of view to the other (like in the case of two moving
observers who agree about the health of a cat), then it
is like shifting or boosting the entire system, so everything
which is invariant remains the same. If we change the location
of one of the observers relative to the other, then we are
changing the system and the invariants will change too.
R.
Eugene Stefanovich
[...]
> Pure mathematics is built upon logic. The theorems of
> mathematics are only acceptable to us because all of
> the inferences which are used in the proofs come
> from the rules of inference of logic. Each line of
> a valid mathematical theorem is a proposition, and,
> in annotated proofs, the rule of logic which allows
> that proposition to be asserted should be listed
> beside the proposition. That way we know that the
> theorems of mathematics are acceptable. If somebody
> tells us that logic is wrong, then he is saying
> that the theorems of mathematics are no longer valid.
> The statement that logic is wrong is, therefore,
> absurd.
[...]
> It follows from the
> truth tables.
[...]
> That is, it follows from the definitions of AND, OR,
> NOT, and so on.
>
> You cannot keep the same definitions of AND, OR and NOT and
> have a counterexample to the distributive law.
[...]
In all mathematical theorems, we deal with propositions that are
certain: they are either certainly true or certainly false:
The line L either passes through the point P or it doesn't. There is
no uncertainty.
In such situations the rules of Boolean logic are perfectly valid.
You can use truth tables and rigorously derive the distributive
law. The meaning of AND and OR is exactly as you described.
Now, imagine the world in which all measurements are inherently
probabilistic. You select a point P and draw a line L many times.
Sometimes you find that L passes through P, sometimes you don't.
All you can say is that there is a probability q that the line L
passes through the point P. Now, how in this world you are going
to formulate Euclid axioms and prove all the theorems? You don't
have certain yes-no statements anymore. You cannot apply the rules
of Boolean logic to the statements like "the line L passes through
the point P". The truth tables are now populated not by clean
"yes" and "no", but by probabilities. The meaning of AND and OR
operations is not that clear anymore.
This is the world of quantum mechanics. This new world requires new
quantum logic.
Eugene.
r...@maths.tcd.ie
>... The simplest example of a relativistic invariant is
>x^2 + y^2 + z^2 - t^2, where x, y, and z are the distances
>between two events along three spatial axes while t is the
>time interval between the two events. From this one can
>see that the round-trip time for a light beam to travel
>from one side of a chamber to the other and back again is
>a relativistic invariant.
Sorry; the expression "the round-trip time for a light beam to
travel from one side of a chamber to the other and back again" is
ambiguous. What I'm referring to is the proper time between the
event when the light is first emitted and the event when the
light arrives back at the side of the chamber from which it
was emitted. Being a proper time, this is an expression of
the form x^2 + y^2 + z^2 - t^2, and hence a relativistic
invariant.
Best,
R.
scerir
> And if there is only one experiment, how can QM refer
> to an ensemble except as a work of fiction?
> And who said that QM had to be meaningless
> except in terms of ensembles?
Interesting point, Dirk.
According to Einstein (and his "ignorance interpretation")
each physical variable has a value, in every state
of the system, and the indeterminacy is only due to
our knowledge of the physical state. So the wave function
provides a description of certain statistical properties
of an ensemble of similarly prepared systems, and it is not
a complete description of individual systems.
But - in principle - such a complete description
should be possible, according to Einstein.
On the contrary, according to Ballentine (and his "statistical
interpretation") quantum states do not represent entities.
At least the above is what I remember of Ballentine's papers
about Einstein's interpretation and his own interpretation.
-serafino
"To say that psi describes the 'state'
of one single system is just a figure
os speech, just as one might say in
every day life: 'My life expectancy
(at 67) is 4.3 years!' ... what it
really means, of course, that you
take all individuals of 67 and count
the percentage of those who live for
a certain lenght of time. This has
always been my own concept of how to
interpret |psi|^2."
-Max Born (4 September 1950)
r...@maths.tcd.ie
>Thus spake r...@maths.tcd.ie
>>Oh No <No...@charlesfrancis.wanadoo.co.uk> writes:
>>
>>>The fundamental principle is that we can only say where something is if
>>>we say where it is relative to other matter.
>>If the principle is that "we can only say where something is if
>>we say where it is relative to other matter", then that on its
>>own is not incoherent, but Rovelli's claim that this is the
>>secret to understanding quantum mechanics is.
>I am not certain that Rovelli makes that claim in a clear way.
He describes it as providing "The way out from this dilemma", and
says that "The apparent contradiction between the two statements
that a variable has or hasn't a value is resolved by indexing the
statements with the different systems with which the system in
question interacts." That sounds to me like a claim that it's a
great breakthrough in understanding, but perhaps I'm misinterpreting
him. He does say "In fact, one may conjecture that this peculiar
consistency between the observations of different observers is the
missing ingredient for a reconstruction theorem of the Hilbert space
formalism of quantum theory." So he's made the great conceptual
breakthrough and just needs to tidy up the technical details, I
gather.
>He does
>discuss Descartes who I think gave the first clear expression of the
>principle, but then, as it seems to me Rovelli gets diverted by too much
>abstract discussion of observers. However, I certainly make that claim.
>>It will not be easy
>>to build it into a mathematical stucture.
>I don't think it is.
Perhaps I was being facetious; in fact I believe it will be
impossible.
>>If this principle has
>>entered into the formalism you present in your paper, I have not
>>understood where.
>That is probably a central reason that you haven't understood what I am
>trying to say about the interpretation of quantum mechanics.
I think that's very likely to be true.
>And yet to
>me, the principle enters into every part of the paper. Certainly I
>discuss it in the introduction, and the whole purpose of introducing
>quantum logic is to have a formal language in which we can discuss
>matter and measurement in the absence of background.
You mention it in various places, but it seems to me to
have no relation to quantum logic or any of the other
formalism.
>>I would agree with the truth of the so-called fundamental principle
>>if it is understood to assert that all location is relative location,
>>or, equivalently, that absolute motion is only an idea, not a feature
>>of the physical world.
>Distinguish two principles of relationism, relativity of position
>expressed above, and essentially due to Descartes, and relativity of
>motion, on which foundation Einstein based special relativity. They are
>closely related, and indeed the second is contained in the former. Both
>are needed to get very far with quantum theory.
My apologies; I meant to say "absolute location is only an idea".
>> I must regard a principle which attempts
>>to place restrictions on what I can say as belonging to the field
>>of linguistics, and not physics.
>I don't think you would really. Actually the opposite. The only
>restriction I wish to place on language is to ensure that it is used to
>discuss physics rather than fantasy. Language can describe a land of
>wizards and dragons, but such a land does not exist. Likewise language
>can discuss absolute space, and yet with a careful analysis of
>measurement, we may recognise that absolute space has no empirical
>justification.
If I were to put a restriction on language, I would insist that
when talking about physics we should speak literally. In my ideal
world, people would not say that quantum mechanics involves a change
in the rules of logic, because that is a metaphor.
You claim that the introduction of the Hilbert space does
nothing more than provide a "formal language in which we can discuss
matter and measurement in the absence of background." But
my point is that the Hilbert space structure puts many
nontrivial constraints on the statistics of the results
of measurements. So you are not merely introducing a
language, but you are also introducing assumptions about
the statistics of the results of experiments. The reason
that the steps which you use to justify the Hilbert space
formalism don't seem adequate to me is that you never
say "Here I'm assuming in advance that the statistics
of the results of experiments will satisfy certain constraints
and these are the constraints and these are the reasons
why it's reasonable for me to suppose that actual
experimental results will satisfy those constraints."
If you said that, and gave good reasons, I'd be delighted.
But instead you make a brief mention that phase has something
to do with motion and say it will all be explained in
section 3, but in section 3 you assume that no more
justification is needed for the use of complex Hilbert
spaces. You seem to assume that it was all settled
earlier.
>>But even in the form in which I agree with it, the principle
>>does not provide us with any "positive knowledge". It only
>>tells us what is not the case, without giving us any clearer
>>idea of what might actually be the case.
>True, but that changes the question. Having answered the question "what
>can we not say?" the question becomes "what can we say?". The objective
>now is to write down postulates for what we can say, in accordance with
>observation and without contradicting the fundamental principle. The
>development of a formal language is just a tool for doing that.
But it places nontrivial constraints on the statistics of the
results of experiments, so it's not just a choice of notation.
>>As such, it can
>>perhaps serve to prevent error, but it cannot be the foundation
>>of any body of knowledge about actual facts.
>Preventing error may be sufficient. If one can eliminate all
>possibilities but one, then that which remains must be the truth. If we
>can write down a set of postulates, based on observation, and of
>sufficient strength to constrain physics, then we have something
>worthwhile. If we can show that these postulates describe what is
>essentially a unique structure, then we will have a proven theory of
>physics which does not rely on induction. My paper, gr-qc/0508077, has
>been updated a couple of times since you read it. In the latest
>revision, which I am replacing today, I have split the definitions which
>determine the mathematical structure into definitions and postulates.
>The distinction is that postulates contain empirical assertions about
>the world, whereas definition are purely semantic and determine only
>mathematical structure. I have had some trepidation about doing this,
>because I am not sure that it is always completely obvious which is
>which. Overall I think it is as step forward, but I would appreciate if
>you felt strong enough to subject it again to your logical mind.
Why thanks. I've got a version of it here and I'll look through
it for differences, but my central problems seems to be that
the use of quantum logic seems to be unjustified and the
relationalism doesn't seem to have anything to do with
quantum logic.
Also, the idea of assigning weird "truth values" to statements
about counterfactuals seems to me to be unnecessary and
confusing. Why can't everything be expressed with statements which
are actually true?
>>The principle of relativity which Eugene was referring to was the
>>principle of the constant speed of propagation of information.
>>
>>The principle of relativity which is contained (let us say,
>>for the sake of the argument) in the fundamental principle
>>of relationism is the principle that one cannot say how
>>fast something is moving unless we specify an object
>>to which that motion is relative.
>>
>>Those are two different principles of relativity.
>The main point is that both principles are necessary. I would argue,
>however, that, according to the fundamental principle of relationism, to
>talk of speed or even of space-time coordinates we have first to
>propagate information.
It seems that you would have to appeal to some principles other
than "absolute space does not exist" to deduce that "to talk of
speed or even of space-time coordinates we have first to propagate
information". The first doesn't say anything about information,
so some principle about information would need to be used as well.
>I only see two options. Either there is
>instantaneous propagation which is empirically false, or there exists a
>maximal speed for the propagation of information.
Why not a variable maximum speed, or a probability distribution
of speeds which is vanishingly small at very high speeds?
>I would argue further,
>that the principle requires that the properties of matter have no
>dependency on time or position, and that this is expressed in the
>cosmological principle, from which we may infer the principle of general
>relativity.
Have these arguments been expressed rigorously?
>>In any case, your argument seems to be that quantum
>>logic on its own places no restrictions on the results
>>of measurements.
>I don't say no restriction; just that on its own it doesn't get us very
>far.
It can reduce the number of measurements we need to make from
81 to 18. That seems to me to get us quite far. In fact, this
seems to me to be the principal issue in need of explanation.
>> You say: "We cannot usefully determine anything much from that
>> unless we also have a time evolution equation." I don't know
>> what you mean by "determine anything much from that". What
>> we can do, without a time evolution operator, is observe,
>> for each preparation, the probability of observing each
>> result to each measurement. Measurements performed at
>> different times after the preparation of the system simply
>> count as different measurements. After we have collected
>> the statistics of the measurement results, we then
>> go about constructing a time evolution operator to describe
>> the statistical relationships we have empirically discovered.
>I don't do things in that order at all. I create a mathematical
>structure then define time evolution as dictated by covariance
>considerations. That leads me to qed, plus some variants which appear to
>include theories of weak and strong interactions. The time evolution for
>any given situation must be an application of these fundamental
>theories. I have just put a paper on this on arxiv, gr-qc/0605127
I think I won't be able to understand anything later in
the paper unless the foundations make sense to me.
>>Suppose you know no quantum mechanics. There are 10 different
>>measurements you perform, each of which can give 10 different
>>results. (Despite your protestations to the contrary, this is
>>quite possible. Consider a spin-9/2 particle and measurements
>>along any ten distinct axes in three-dimensional space as an
>>example.)
>Trouble is, when I want to determine spin, I can't think of a way of
>doing it which does not require an analysis of dynamics, like bending a
>path in a Stern Gerlach experiment. In fact, the formulation of quantum
>logic in A Relational Quantum Theory incorporating Gravity (RQG), gr-
>qc/0508077 (revised from the version you read) makes no mention of spin.
>It is introduced in a follow up paper "A Treatment of Quantum
>Electrodynamics as a Model of Interactions between Sizeless Particles in
>Relational Quantum Gravity" because it turns out that there is no
>covariant formulation which does not require it.
These considerations are unnecessary. We can abstract from the
procedure involved in the measurement and from the
considerations involved in determining the final result.
All that you need to suppose is the case, in order for
my argument to work, is that the predictions of quantum
mechanics are actually correct.
>>This is why I am saying that "quantum logic" places nontrivial
>>constraints on the statistics of the results of measurements. I
>>have used nothing other than the normal rules of quantum
>>mechanics above.
>> ...
>As you have formulated it here, I think the conclusion is that quantum
>mechanics does contain an additional assumption over and above the quite
>artificial construction of states from a basis at given time. I think,
>in the context of a post on spr, I can only point to what I think this
>is, because for a complete, full, and rigorous analysis of measurement,
>I think we need a full, consistent, and working theory of qed, which has
>been one of the main objects of my research over the years. The paper I
>have just put on arxiv represents the culmination of this research so
>far. In short, I think the complete answer to the issue you raise is not
>simple.
I also think that it's not simple.
>As I formulate quantum theory I start with only one type of measurement,
>specifically measurement of position. To put this in the context of your
>example, let us call this measurement M9. I then construct labels for
>other states artificially, by creating a Hilbert space which is, in
>essence, determined by the probabilities for getting each of the results
>r_9j. The claim that M9 is complete means to me that all physical states
>can be represented by states of this Hilbert space. (in fact this
>assumption has to be relaxed in qed, but I retain the Hilbert space even
>though not all states in it necessarily correspond to real measurement
>results - there will be so called "virtual" photons, which I treat as
>real photons which cannot be directly measured).
But the Hilbert space is not "determined by the probabilities ...".
It is a set of linear sums of symbols like |x>, with complex
coefficients. The symbols |x> are associated with the possible
results of measurements of position. This introduces a
vector space whose dimensionality is twice the number of
possible measurement results. How do you know that this
is the right number of dimensions to encode the statistics
of the results of all other possible measurements?
>Introducing Luder's projection postulate at this stage does require an
>additional assumption about the behaviour of matter. I don't think it is
>quite fair to say it is a hidden assumption, but what is true is that it
>constrains the theory in a non-trivial way, just as you suggest. As far
>as the logical development of the model as a physical theory is
>concerned, it is perhaps premature. Whether this is a legitimate
>constraint, or even a necessary one, is an issue which I don't think can
>be answered properly without first developing a complete account of time
>evolution, including an account of the interactions of elementary
>particles, namely a full and consistent qed.
Qed seems to me to be just a particular application of quantum
mechanics. Quantum mechanics seems to be a procedure for
assigning probabilities to the results of experiments, regardless
of what fields or particles we might suppose to be responsible
for the results.
>So, what I would say is, have I given a consistent interpretation of
>quantum theory, I think I have: It is a model of interactions of between
>sizeless particles in the absence of space-time background.
Well, I'm not sure what you mean by a consistent interpretation.
It is merely a conjecture of yours? Do you consider your own
arguments to be completely rigorous or merely plausible?
>Have I given
>a full and complete account of the measurement problem, demonstrating
>that Luder's projection postulate is a consequence of the theory? No, I
>haven't, but I think I have made it reasonable. Do I believe that such
>an account is possible in principle within the theory? Yes I do. Do I
>think that such an account would be publishable, or indeed that if it
>were given anyone would understand a word of it? No, at the present
>time, I'm afraid I don't believe that.
Probably not. Anything which requires a lot to thought to
understand is unlikely to be widely understood.
R.
Ilja Schmelzer
> Specifically, I claim that QM, or at least QFT, needs to be modified
> to allow for an explicit representation of the observer.
Such a modification already exists, and is known
as Bohmian mechanics. In BM, you can have an observer
q_obs(t) together with the observed object q_obj(t) and
the wave function Psi(q_obs,q_obj,t).
> Only when M = infinity can both x and
> v be measured sharply.
Not a problem, because dq/dt cannot be measured sharply
in BM.
> It is also
> notable how reasoning about observers led to both SR (time and length
> depend on the observer) and QM (observation affects the canonical
> conjugate variable).
I disagree that reasoning about observers leads to SR, GR or QM.
The (unjustified) positivistic rejection was popular in this time
(the time of logical positivism, before Popper).
The rejection of additional unobservable structures of scientific
theories (preferred frames, equations for gauge potentials, for
trajectories in BM, wave function instead of density matrix
and so on) never increases the predictive power of the theory
(Popper's criterion), but sometimes decreases it.
The point is that these additional hidden structures sometimes
lead to additional compatibility conditions. For example, for
a preferred frame we the homogeneity assumption of the universe
immediately gives curvature 0, while GR allows different
values of curvature for a homogeneous universe.
But I find it also remarkable that different natural types of
hidden variables require each other: BM needs a preferred
frame. Local energy and momentum densities for gravity
need preferred coordinates (Noether). Observable gauge
potentials also need a preferred frame.
Ilja
Eugene Stefanovich
> "Eugene Stefanovich" <eug...@synopsys.com> a écrit dans le message de
> news: 447662C8...@synopsys.com
>
>
>>The peculiar superposition "dead + alive" does not refer to the
>>individual cat,
>>but to the ensemble of identically prepared cats in identical boxes.
>>If you keep in mind that the wavefunction provides you a description
>>of the ensemble rather that individual system, you can avoid all
>>kinds of paradoxes with the "wavefunction collapse" and with the role
>>of "mind" in this collapse.
>
>
> I agree with this description, but other paradoxes emerge. We have no
> description of a single particle, like with the Young holes and a one by one
> beam, while the premise implies there must be one.
This is exactly the point: quantum mechanics does not have description
of one particle (or one cat, or one whatever). QM cannot say what will be
the result of measurement performed on an individual system. QM can make
meaningful statements only about ensembles of identically prepared systems.
Eugene.
Eugene Stefanovich
> Eugene Stefanovich <eug...@synopsys.com> writes:
>>I am wondering if you have any hard evidence (besides your intuition) to
>>claim that being dead or alive is a relativistic invariant.
>
>
> Yes, I do. The simplest example of a relativistic invariant is
> x^2 + y^2 + z^2 - t^2, where x, y, and z are the distances
> between two events along three spatial axes while t is the
> time interval between the two events. From this one can
> see that the round-trip time for a light beam to travel
> from one side of a chamber to the other and back again is
> a relativistic invariant. Each observer will agree about
> how many seconds this is.
I respectfully disagree. Remember the famous Einstein's "light clock"
in which a pulse of light is bouncing between two parallel
mirrors? The round-trip time is different for different observers.
This example is used to demonstrate the time dilation effect.
Eugene.
r...@maths.tcd.ie
>Eugene Stefanovich skrev:
>>We know from special relativity that the length of a stick can be 1m
>>from the point of view of observer A, and the length of the same stick
>>can be 0.5m from the point of view of another (moving) observer B.
>>How different it is from saying: the cat is alive from the point of view
>>of A while the cat is dead from the point of view of B?
>I believe that what Rovelli is pointing at is not just a problem with
>the interpretation of QM, but a problem with QM itself. This is of
>course a drastic conclusion, since QM has agreed with every experiment
>for a century. OTOH, Newtonian mechanics agreed with every experiment
>for over 200 years, and it still turned out to be wrong.
>Specifically, I claim that QM, or at least QFT, needs to be modified
>to allow for an explicit representation of the observer. A crucial
>assumption in Rovelli's paper is that the observer has a well-defined
>position x. ...
I am by no means a supporter of Rovelli's ideas, but it seems to
me that his entire project is to assert that the observer doesn't
have a well-defined position, and to claim that this has something
to do with quantum mechanics.
R.
Oh No
>Oh No <No...@charlesfrancis.wanadoo.co.uk> writes:
>
>>Thus spake r...@maths.tcd.ie
>>>Oh No <No...@charlesfrancis.wanadoo.co.uk> writes:
>>>
>>>>The fundamental principle is that we can only say where something is if
>>>>we say where it is relative to other matter.
>>>It will not be easy
>>>to build it into a mathematical stucture.
>
>>I don't think it is.
>
>Perhaps I was being facetious; in fact I believe it will be
>impossible.
I find that unnecessarily defeatist. Einstein already built a fair bit
of the principle into special and general relativity. People other than
Rovelli have thought that quantum uncertainty had something to do with
the principle (starting with Heisenberg). Clearly the principle is
restrictive. The problem is with formalising it. I don't believe the
universe can be inconsistent with itself, so I think it natural that it
must be describable by a formal mathematical structure.
>>And yet to
>>me, the principle enters into every part of the paper. Certainly I
>>discuss it in the introduction, and the whole purpose of introducing
>>quantum logic is to have a formal language in which we can discuss
>>matter and measurement in the absence of background.
>
>You mention it in various places, but it seems to me to
>have no relation to quantum logic or any of the other
>formalism.
I think the problem is with measurement in general. Quantum logic, as
you understand it (probably correctly) incorporates the projection
postulate, and with that the measurement problem, as you have outlined.
The resolution of the problem would be to drop the projection postulate
and to prove it (ultimately) as a theorem.
>>> I must regard a principle which attempts
>>>to place restrictions on what I can say as belonging to the field
>>>of linguistics, and not physics.
>
>>I don't think you would really. Actually the opposite. The only
>>restriction I wish to place on language is to ensure that it is used to
>>discuss physics rather than fantasy. Language can describe a land of
>>wizards and dragons, but such a land does not exist. Likewise language
>>can discuss absolute space, and yet with a careful analysis of
>>measurement, we may recognise that absolute space has no empirical
>>justification.
>
>If I were to put a restriction on language, I would insist that
>when talking about physics we should speak literally. In my ideal
>world, people would not say that quantum mechanics involves a change
>in the rules of logic,
I agree with you completely on that. I think this is this misconception
about quantum logic which has held up its development as a tool for
understanding nature.
>You claim that the introduction of the Hilbert space does
>nothing more than provide a "formal language in which we can discuss
>matter and measurement in the absence of background." But
>my point is that the Hilbert space structure puts many
>nontrivial constraints on the statistics of the results
>of measurements. So you are not merely introducing a
>language, but you are also introducing assumptions about
>the statistics of the results of experiments. The reason
>that the steps which you use to justify the Hilbert space
>formalism don't seem adequate to me is that you never
>say "Here I'm assuming in advance that the statistics
>of the results of experiments will satisfy certain constraints
>and these are the constraints and these are the reasons
>why it's reasonable for me to suppose that actual
>experimental results will satisfy those constraints."
Again, I think the issue is the measurement problem. If the projection
postulate were dropped, would you still have a problem?
>If you said that, and gave good reasons, I'd be delighted.
>But instead you make a brief mention that phase has something
>to do with motion and say it will all be explained in
>section 3, but in section 3 you assume that no more
>justification is needed for the use of complex Hilbert
>spaces. You seem to assume that it was all settled
>earlier.
At that stage of the paper, all I have considered is measurement results
of position for a single particle at time t. At that point phase is an
irrelevance. I don't see need for justification of something which has
no effect. Later I want to be able to do Lorentz transforms, since I
also know that they are part and parcel of the fundamental principle.
When covariance is brought in, phase starts to play a role, and with it
equations of motion appear for the phase relationships. This places
restrictions on phase, but I see no problem with that because in the
first instance phase was completely arbitrary.
>
>>>But even in the form in which I agree with it, the principle
>>>does not provide us with any "positive knowledge". It only
>>>tells us what is not the case, without giving us any clearer
>>>idea of what might actually be the case.
>
>>True, but that changes the question. Having answered the question "what
>>can we not say?" the question becomes "what can we say?". The objective
>>now is to write down postulates for what we can say, in accordance with
>>observation and without contradicting the fundamental principle. The
>>development of a formal language is just a tool for doing that.
>
>But it places nontrivial constraints on the statistics of the
>results of experiments, so it's not just a choice of notation.
Will allow that these constraint come from two sources, 1) covariance,
which is legitimate, and 2) the projection postulate, which I have
suggested we drop (along with its implications).
> my central problems seems to be that
>the use of quantum logic seems to be unjustified and the
>relationalism doesn't seem to have anything to do with
>quantum logic.
Do you accept that without the projection postulate, and without time
evolution, all I have done is make simple (almost trivial) statements
about probabilistic results of measurement of position at given time
without an assumption of background space, and created a slightly
elaborate mathematical structure with which to discuss them?
>Also, the idea of assigning weird "truth values" to statements
>about counterfactuals seems to me to be unnecessary and
>confusing. Why can't everything be expressed with statements which
>are actually true?
If you don't feel comfortable with the notion of "truth value", I am
happy to ignore it. However I do not think the situation is
fundamentally different from probability theory. The statement "Next
time I throw a die, it will be a 6" cannot be actually true. But
presumably you accept the truth of the statement "The probability that I
throw a six is 1/6"? and that if one creates a formal language in which
1/6 is said to be the truth value for the former statement, then the
statements of that language can be true? In fact, I believe you already
reduced quantum logic to Boolean logic much in this manner in another
post.
>
>>>The principle of relativity which Eugene was referring to was the
>>>principle of the constant speed of propagation of information.
>>>
>>>The principle of relativity which is contained (let us say,
>>>for the sake of the argument) in the fundamental principle
>>>of relationism is the principle that one cannot say how
>>>fast something is moving unless we specify an object
>>>to which that motion is relative.
>>>
>>>Those are two different principles of relativity.
>
>>The main point is that both principles are necessary. I would argue,
>>however, that, according to the fundamental principle of relationism, to
>>talk of speed or even of space-time coordinates we have first to
>>propagate information.
>
>It seems that you would have to appeal to some principles other
>than "absolute space does not exist" to deduce that "to talk of
>speed or even of space-time coordinates we have first to propagate
>information". The first doesn't say anything about information,
>so some principle about information would need to be used as well.
>
>>I only see two options. Either there is
>>instantaneous propagation which is empirically false, or there exists a
>>maximal speed for the propagation of information.
>
>Why not a variable maximum speed, or a probability distribution
>of speeds which is vanishingly small at very high speeds?
I don't know that a variable maximum speed makes sense. Since all speeds
are determined relative to the maximum (or we have no empirical
definition of speed) the maximum speed can only be 1 relative to itself.
Strictly, since relativity is a classical theory, I think we should
assume that there may be a probability distribution of speeds in the
quantum domain, and that the maximum speed is either a theoretical
maximum or a mean. I don't think either of those suppositions changes
the validity of special relativity as a theory, however.
>>I would argue further,
>>that the principle requires that the properties of matter have no
>>dependency on time or position, and that this is expressed in the
>>cosmological principle, from which we may infer the principle of general
>>relativity.
>
>Have these arguments been expressed rigorously?
I do not see how it is possible to argue rigorously prior to creating a
formal mathematical structure. What one can do is examine the arguments
one has for choosing axioms, and make them more and more solid, or
alternatively find fault with them and discard or modify them as
appropriate. In this way one seeks to find more fundamental axioms with
which to define a mathematical structure.
>>>In any case, your argument seems to be that quantum
>>>logic on its own places no restrictions on the results
>>>of measurements.
>
>>I don't say no restriction; just that on its own it doesn't get us very
>>far.
>It can reduce the number of measurements we need to make from
>81 to 18. That seems to me to get us quite far. In fact, this
>seems to me to be the principal issue in need of explanation.
Yes. But in the interests of proceeding formally, I am suggesting we
drop the projection postulate, and lose all these other measurements in
the process, at least for the time being. I have said that other
measurement must be reduced to measurement of position, or a combination
of measurements of position of particles in a more complex system. We
have a lot more material to cover, concerning what more complex systems
are possible in principle, before we can start to think about a proper
treatment of measurement in general.
>>> You say: "We cannot usefully determine anything much from that
>>> unless we also have a time evolution equation." I don't know
>>> what you mean by "determine anything much from that". What
>>> we can do, without a time evolution operator, is observe,
>>> for each preparation, the probability of observing each
>>> result to each measurement. Measurements performed at
>>> different times after the preparation of the system simply
>>> count as different measurements. After we have collected
>>> the statistics of the measurement results, we then
>>> go about constructing a time evolution operator to describe
>>> the statistical relationships we have empirically discovered.
>
>>I don't do things in that order at all. I create a mathematical
>>structure then define time evolution as dictated by covariance
>>considerations. That leads me to qed, plus some variants which appear to
>>include theories of weak and strong interactions. The time evolution for
>>any given situation must be an application of these fundamental
>>theories. I have just put a paper on this on arxiv, gr-qc/0605127
>
>I think I won't be able to understand anything later in
>the paper unless the foundations make sense to me.
Is there anything that does not make sense about the treatment of
Hilbert space built on the discrete measurement results for measurement
of position of a single particle at given time, ignoring both the
possibility of another type of measurement, and that of a second
measurement at later time?
>>>Suppose you know no quantum mechanics. There are 10 different
>>>measurements you perform, each of which can give 10 different
>>>results. (Despite your protestations to the contrary, this is
>>>quite possible. Consider a spin-9/2 particle and measurements
>>>along any ten distinct axes in three-dimensional space as an
>>>example.)
>
>>Trouble is, when I want to determine spin, I can't think of a way of
>>doing it which does not require an analysis of dynamics, like bending a
>>path in a Stern Gerlach experiment. In fact, the formulation of quantum
>>logic in A Relational Quantum Theory incorporating Gravity (RQG), gr-
>>qc/0508077 (revised from the version you read) makes no mention of spin.
>>It is introduced in a follow up paper "A Treatment of Quantum
>>Electrodynamics as a Model of Interactions between Sizeless Particles in
>>Relational Quantum Gravity" because it turns out that there is no
>>covariant formulation which does not require it.
>
>These considerations are unnecessary. We can abstract from the
>procedure involved in the measurement and from the
>considerations involved in determining the final result.
>All that you need to suppose is the case, in order for
>my argument to work, is that the predictions of quantum
>mechanics are actually correct.
Ok, so I am not going to suppose that. As I say, it is something which
should be proven, not assumed. In fact, for the purpose of
interpretation it does not even have to be proven, merely made sensible
and reasonable within the context of a physical model.
>> I think the complete answer to the issue you raise is not
>>simple.
>
>I also think that it's not simple.
Then, will you accept, that for me to make my case, I have somehow to
lead you through to later parts of the papers?
>>As I formulate quantum theory I start with only one type of measurement,
>>specifically measurement of position. To put this in the context of your
>>example, let us call this measurement M9. I then construct labels for
>>other states artificially, by creating a Hilbert space which is, in
>>essence, determined by the probabilities for getting each of the results
>>r_9j. The claim that M9 is complete means to me that all physical states
>>can be represented by states of this Hilbert space. (in fact this
>>assumption has to be relaxed in qed, but I retain the Hilbert space even
>>though not all states in it necessarily correspond to real measurement
>>results - there will be so called "virtual" photons, which I treat as
>>real photons which cannot be directly measured).
>
>But the Hilbert space is not "determined by the probabilities ...".
>It is a set of linear sums of symbols like |x>, with complex
>coefficients. The symbols |x> are associated with the possible
>results of measurements of position. This introduces a
>vector space whose dimensionality is twice the number of
>possible measurement results. How do you know that this
>is the right number of dimensions to encode the statistics
>of the results of all other possible measurements?
At the present stage of the development, all we have actually done is
introduce free parameters. The next stage of the argument involves
bringing in covariance, justified from the fundamental principle, not
from statistics. We have to show that this restricts the free
parameters.
>>Introducing Luder's projection postulate at this stage does require an
>>additional assumption about the behaviour of matter. I don't think it is
>>quite fair to say it is a hidden assumption, but what is true is that it
>>constrains the theory in a non-trivial way, just as you suggest. As far
>>as the logical development of the model as a physical theory is
>>concerned, it is perhaps premature. Whether this is a legitimate
>>constraint, or even a necessary one, is an issue which I don't think can
>>be answered properly without first developing a complete account of time
>>evolution, including an account of the interactions of elementary
>>particles, namely a full and consistent qed.
>
>Qed seems to me to be just a particular application of quantum
>mechanics.
I agree with you in principle, but I assume you also know that this is
regarded by those most qualified to judge as an unsolved mathematical
problem. We have both agreed that resolving the issue of interpretation
will be difficult. The fact that we have first to construct qed is a
measure of how difficult. All the more so, because we will have to make
it compatible with general relativity (in appropriate approximations and
limits) at the same time. I say this, not to arbitrarily introduce new
fields, but because I hold that these apparently disparate problems are
all part and parcel of the same problem, and that there is no solution
to one which is not also a solution to the others.
>Quantum mechanics seems to be a procedure for
>assigning probabilities to the results of experiments, regardless
>of what fields or particles we might suppose to be responsible
>for the results.
Yes, but interpretation means that we also understand the physical
principles responsible for the results.
>>So, what I would say is, have I given a consistent interpretation of
>>quantum theory, I think I have: It is a model of interactions of between
>>sizeless particles in the absence of space-time background.
>
>Well, I'm not sure what you mean by a consistent interpretation.
>It is merely a conjecture of yours? Do you consider your own
>arguments to be completely rigorous or merely plausible?
Many of them I consider rigorous. Certainly enough for me to be
convinced that there is essentially no other interpretation. I am not
going to claim to have made every aspect rigorous.
Cl.Massé
> This is exactly the point: quantum mechanics does not have description
> of one particle (or one cat, or one whatever). QM cannot say what will be
> the result of measurement performed on an individual system. QM can make
> meaningful statements only about ensembles of identically prepared
> systems.
That summarizes the very idea of probability, but the ensemble device can't
be applied when measuring a single particle. For example, when the
impulsion of a particle is measured just after its position is, which
initial ensemble would describe the resulting uniform probability law?
--
~~~~ clmasse on free F-country
Liberty, Equality, Profitability.
Eugene Stefanovich
That's a good question and, frankly, I don't have an answer.
Earlier in this thresd we were talking about a single measurement of
a physical observable or about an ensemble of such measurements in
identically
prepared systems. In such measurements, the particle is prepared, then
its observable is measure, then the particle is discarded. Now you added
another level of complexity: you are considering double measurements:
i.e., the particle is prepared,
then its position is measured, then, after some time, the momentum is
measured on the same particle.
I am not sure if this more complicated setup is realizable in practice,
and I am even not sure that quantum mechanics is the right way to
describe this setup. I think we first need to understand fully the
simplest case, i.e., single measurements, rather that throw additional
complications into the mix.
Eugene.
Guy Neapig
> QM cannot say what will be
> the result of measurement performed on an individual system.
this is not always true: if we know that the system is in an eigenstate
of some observable, then the we know for sure that a measurement of
that observable will give as result the corresponding eigenvalue.
Theoretically, the projector |psi><psi| is such an observable for any
given state |psi>, although in practice it will generally be very
difficult to set-up an apparatus measuring that observable.
Guy.
scerir
> For example, when the impulsion of a particle
> is measured just after its position is,
> which initial ensemble would describe
> the resulting uniform probability law?
Can we say that every measurement 'reset'
the (supposed) ensemble?
-serafino
'In the case of undefiniteness of a property
of a system for a certain arrangement
(with certain state of the system) any attempt
to measure that specific property destroys
(at least partially) the influence of earlier
knowledge of the system on (possibly statistical)
statements about later possible measurement
results.'
-W.Pauli, in Handbuch der Physik,
edited by S.Flugge, Springer-Verlag,
Berlin, 1958, Vol. 1, p. 7.
mark...@yahoo.com
> There is (in my opinion) even more elegant demonstration of this
Unfortunately, since you didn't exactly indicate what "this" is
referring to, it's difficult to address or even understand you're
actually stating.
> property for non-relativistic quantum 2-particle systems. It uses
> the unitary representation of the Galilei group in the Hilbert space
> of the system:
There is nothing specifically pertaining to quantum theory in this mode
of representation. It's a generic theory-indepedent formalization of
the principle of relativity (Galilean relativity, Poincare' relativity,
Euclidean relativity, etc.) It describes classical physics equally well
-- in the Hilbert space representation of classical physics.
Those kinds of arguments -- borne ultimately of a Wigner analysis or
something similar, would not replace the axiomatic framework outlined
here, but SUPPLEMENT them, essentially as an application of the general
framework.
First you classify the theories, then you show the instances where such
and such symmetry space representation(s) hold.
> This demonstration can be easily generalized to relativistic systems
> of particles: just change the Galilei group to the Poincare group:
"Wigner Classification for Galilei/Poincare/Euclid"
http://federation.g3z.com/Physics/Index.htm
mark...@yahoo.com
> In barely one-month-old [1], Rovelli argues that 'EPR-type
> correlations do not entail any form of "non-locality", when viewed
> in the context of a relational interpretation of quantum mechanics.'
>
> This sounds somewhat familiar (cf. [2]).
Rovelli's concerns were laid out in more detail in Chapter 5 of his
latest "treatise" on Quantum Gravity.
Despite advocating and detailing the "timeless approach" to mechanics
in Chapter 3, he apparently forgot his entire program when deliberating
over the apparent problems associated with the "changes in state" and
the dilemma of "does anything ever get measured" issues that are really
little more than the Wigner Friend issue in disguise.
First of all, since the "t" variable was completely dropped by Chapter
3, there's no more Schroedinger Pictures. It's all Heisenberg.
So, the question of what a "state becomes" is irrelevant. A state is
timeless in Heisenberg. Nothing becomes anything.
So, it boils down to how to write the Lueder rule (the formalization of
"collapse") in timeless form. It's also necessary, since you're in the
Heisenberg picture.
Keep in mind, by the way, none of this has anything specifically to do
with quantum theory. Wave function "collapse", the Born rule, the
Lueder rule all apply in classical physics -- exactly in the form
stated for quantum theory -- when the latter is written in the Hilbert
state representation of classical physics.
(Indeed, this classical "correspondence" could be used as an argument,
via a "correspondence principle" (what's true of Classical physics in
its Hilbert space representation is true of quantum and
classico-quantum systems), to assert the rule's generalization to
classico-quantum and pure quantum systems.
To get rid of time, you have to generalize the rule so that it applies
to arbitrary measurement sequences, rather than to single measurements.
Then, having done that, the expedient of knowing "what the state is
'after measurement'" will be rendered superfluous and can be dropped
from consideration.
If W is the state of the system, then associated with a set of
measurements of observables (A1,A2,...,An) is a probability
distribution over the product space
spec(A1) x spec(A2) x ... x spec(An)
given explicitly by
p(A1 = a1, A2 = a2, ..., An = an) = W[T'[P1...Pn]
T[P1...Pn]]
where T' and T are the reverse and forward time-ordered operator
orderings and W[] the linear functional associated with the state W,
and P1, ..., Pn the projection associated with (A1 = a1), ..., (An =
an), respectively.
This is the Heisenberg Lueder Rule.
It replaces both the "projection postulate" and the "evolution
postulate", both of which are rendered superfluous. Dynamics is in the
observables where it belongs.
Again, note: this rule applies across the board -- classical physics,
quantum physics and all the instances of classico-quantum physics (i.e.
quantum theories with superselection).
Which gets to the other part of the criticism of Rovelli's comments.
The hard edge of his paradox is significantly blunted if you simply
reject the axiom that everything under the sun gets rendered as a
quantum mode and simply accept the manifest fact that classical modes
to coexist alongside quantum modes.
Examples are not hard to come by, they're simply disguised. The "t"
variable (and the "x", "y", "z" and 't") variables are, in fact,
classical observables that exist within the quantum theory. Extra
degrees of freedom (like in electromagnetism) where the commutators are
constrained to zero represent de facto classical modes.
As another example, one can conceive of the vacuum as a classical heat
bath in which the quantum systems are immersed. The classical modes
associated with this heat bath become inextricably intermingled with
the microscopic degrees of freedom, producing an effective "mean time
to decoherence". This extra element closes up the remaining 10% of the
gap that decoherence spans 90% of to turn its quasi-superselections
into the bona fide superselections we all, in fact, see. Treating the
universe as an de facto open system also gets around the obstacle that
prevents quantum modes undergoing automorphic evolution from
commuicating with classical modes.
The formalism I just outlined in a previous reply under this header
essentially pushs one in this general direction already, showing that
the most general solution to (equal time commutators + 2nd order
equations of motion) is a hybrid classico-quantum system.
J. Horta
> Eugene Stefanovich wrote:
>
>> QM cannot say what will be
>> the result of measurement performed on an individual system.
>
> this is not always true: if we know that the system is in an eigenstate
> of some observable, then the we know for sure that a measurement of
> that observable will give as result the corresponding eigenvalue.
>
Well, be careful making this assertion. QM may say that a probability
is certain or 1, however, to verify this one would need to
form an ensemble and do the required statistical test to show the result
is certain to within some confidence level. Thinking along these lines
measurement on an eigenstate is no different than any other measurement.
In all cases QM provides a purely statistical discription of nature that
cannot be verified by a single measurement on a single system.
Eugene Stefanovich
> Eugene Stefanovich wrote:
>
>
>>QM cannot say what will be
>>the result of measurement performed on an individual system.
>
>
> this is not always true: if we know that the system is in an eigenstate
> of some observable, then the we know for sure that a measurement of
> that observable will give as result the corresponding eigenvalue.
You are right. My statement doesn't apply to ensembles prepared
in an eigenstates of the measured observable. I should better say:
"Quite often QM cannot say..."
Eugene.
Cl.Massé
8Zkhg.26828$jP5.6...@twister1.libero.it
> Can we say that every measurement 'reset'
> the (supposed) ensemble?
"resetting" an ensemble is inconsistent with the very idea of an ensemble.
We don't remove the elements that aren't the case in the ensemble, we
redefine it completely, and that is impossible since the actual system is
only one element of the ensemble.