Many worlds, Copenhagen and the universal wave function
SIGFPE
interpretation of quantum mechanics. I dont't like to waste too much
effort on such things but as I share a universe with people I do it
makes sense to try to understand what the people around me are talking
about.
In the Copenhagen interpretation there are observers and the observed
universe. It seems that the state of the universe at any time is a
tuple that looks like this:
(O_1,...,O_n,psi)
where the O_i are quasi-mysterious observers and the wavefunction psi
describes the rest of the universe. psi is governed by the Schrodinger
wave equation and from time to time (at times unpredictable, nay,
*theoretically* unpredictable, by theory) an interaction between an
observer O_i and psi takes place whereby psi is projected onto some
subspace.
In the many worlds approach we merely have psi. We can think of the
entire universe in the 'many worlds' approach as a tensor product of
various subsystems so in fact psi lives in some tensor product space
S_1 \otimes ... \otimes S_n
If you have a basis for each S_i then you may expand these out to form a
tensor product basis and any psi will be a linear combination of these.
Each of these basis elements describes an entire universe and any psi is
a linear combination of these. Hence the name 'many worlds'.
The problem I have is this: in the Copenhagen interpretation we also have
a wave function psi but this time it lives in a slightly reduced space
S_1 \otimes ... \otimes S_n
where the product no longer contains terms describing observers (because
in the Copenhagen interpretation these don't have wavefunctions).
Nonetheless we still have a basis of states but in this case they are
states not of the entire universe but of the very large part of the
universe that doesn't contain observers. In other words the Copenhagen
interpretation is just as much a 'many worlds' theory except that there
are no superpositions of observers.
So in what sense is the Copenhagen interpretation not a 'many worlds'
interpretation?
It seems to me that the 'many worlds interpretation' should be called
the 'many observers interpretation'.
And of course there are physical differences between these
interpretations because in the 'many worlds' approach it should be
possible for observers to observe interference between the states of
other observers but the Copenhagen disallows such a thing. This means
that I'm also confused about why the 'many world interpretation' is
called an 'interpretation'.
Any alleviation of my confusion would be appreciated!
Alfred Einstead
> In the Copenhagen interpretation there are observers and the observed
> universe. It seems that the state of the universe at any time is a
> tuple that looks like this:
>
> (O_1,...,O_n,psi)
>
> where the O_i are quasi-mysterious observers and the wavefunction psi
> describes the rest of the universe.
Ah, a perfect sequey into part of what I wanted to talk about anyhow:
* Copenhagen = Everett + (___)?
* why Schroedinger's Cat is impossible
* where the macroscopic world, which Copenhagen posits at
the outset, comes from
A more appropriate description is that Copenhagen (i.e. Orthodox
QM a' la von Neumann) provides you with a means for determining
the probability distribution associated with the following data:
(a) An initial state psi(t0)
(b) A set M of "measurements"
where corresponding to each m in M, there are the data:
(b1) A time of measurement Tm
(b2) An operator Am
such that
Complimentarity: If Tm = Tm' then [Am, Am'] = 0
Each operator corresponds to a decomposition of the underlying
space into a family of projections:
Am = sum (a Pm_a: a in spectrum of Am)
sum (Pm_a) = 1, Pm_a a projection
Pm_a Pm_b = 0, for distinct a, b
Basically, that's all the information that's relevant. For,
if A = f(Am) for any one-to-one function f(x), the information
conveyed by A is exactly the same as that conveyed by Am. The
transformation Am -> f(Am) is just a recalibration of the dial.
So, a "measurement" is actually associated with a generalized
basis -- an orthogonal system of projections that add up to 1.
Without loss of generality, you can combine the Am, Am' for
when Tm = Tm'. As most, this refines the projection bases
corresponding to Am and Am'.
The elements of M are ordered by their times Tm:
m < m' if Tm < Tm'
So, you're actually talking about measurement "networks".
So, given a psi(t0), and network M, Copenhagen assigns a
stochastic evolution as follows.
For a minimal element m of M, you have:
psi(Tm-0) = U(Tm,t0) psi(t0)
where
U(s,t) = exp(-iH(s-t)/h-bar)
is the Evolution operator. The transition induced by m is:
psi(Tm+0) = Pm_a psi(Tm-0) / pa
with probability pa = |Pm_a psi(Tm-0)|
For an element m in M with an immediate predecessor m',
psi(Tm-0) = U(Tm,Tm') psi(Tm'+0)
and
psi(Tm+0) is determined as above.
> So in what sense is the Copenhagen interpretation not a 'many worlds'
> interpretation?
Obviously, it is. For each assignment of (psi(t0), M) it gives you
a family of outcomes, each with a probability measure assigned it.
Each member of that family is a world.
But this is NOT the same thing as Everett -- which assumes no M
whatsoever.
To get from Everett to Copenhagen you have to explain two things:
(a) Where a preferred basis comes from.
Every observation is a generalized basis. So, if there's
even one observation, then exactly what intrinsically
defines its basis?
(b) How to get all macroscopic entities to work off the
same basis. That is: how to get everyone to be on
the "same wavelength".
The 2nd problem is the Schroedinger Cat problem. Another way
of stating situation (b) is that macroscopic systems should
not be in superpositions with each other. Some kind of
superselection rule is preventing macroscopic superpositions.
Conversely, the problem commonly attributed to Copenhagen is
its seemingly ad hoc imposition of the macroscopic world on
the microscopic world via the definition of the (psi_0, M)'s.
Probabilities in Copenhagen can only be defined in reference
to a given (psi_0, M). So these elements are logically prior.
So ... what prevents macroscopic superpositions, and why do
they all have to be on the "same wavelength"? The reason is
quite simple:
No 2 macroscopic systems can be shielded
from one another.
They are in continual interaction with one another, unavoidably
so. In particular, the very existence of this universal
entanglement of macroscopic systems is brought about by the
following:
There exists a universal long-range interaction
which cannot be shielded.
That, of course, being gravity.
So, going back to Schroedinger's cat, imagine the situation
where the cat in the box is subjected to the poison vial
which, itself, is tripped by a radioactive counter device.
Gravity does not interact as a continuously operative classical
force, but interacts in quanta. So, for microscopic events,
the gravity is weak enough, we'd expect large times (in the
mean) between interaction events via gravity. So one can
have a microscopic superposition which lasts a long time.
But as soon as the event (the decay of an atom) is registered
by a large scale device, such as the mechanism which trips
the poison gas, this registers a HUGE difference (relatively
speaking) in terms of the respective gravity fields which
(for macroscopic systems) is in continuous operation.
This difference leads to a natural superselection between the
different outcomes. It cannot be shielded, therefore all
other macroscopic systems in the vicinity will line up with
these two outcomes as soon as the difference in the gravity
fields reaches the respective systems outside the box.
The box the cat was in didn't do anything at all. It certainly
didn't shield the cat from observation! The cat might as well
have been out in the open, for all the difference it made.
So, because of the existence of a universal, unshieldable
long-range force, you have a situation in which all
macroscopic systems will be lined up on a common basis.
And this is what creates your macroscopic world out of the
Quantum World.
Penrose was right (for a different reason than he said).
Gravity is the culprit.
Mark
>From whop...@csd.uwm.edu (Alfred Einstead):
>To get from Everett to Copenhagen you have to explain two things:
>(a) Where a preferred basis comes from.
>(b) How to get all macroscopic entities to work off the
> same basis. That is: how to get everyone to be on
> the "same wavelength".
item (b) is really referring specifically to the condition called
"Complimentarity" below:
>[Orthodox QM a' la von Neumann] provides you with a means for determining
>the probability distribution associated with [...]
> (a) An initial state psi(t0)
> (b) A set M of "measurements"
>where corresponding to each m in M, there are the data:
> (b1) A time of measurement Tm
> (b2) An operator Am
>such that
> Complimentarity: If Tm = Tm' then [Am, Am'] = 0
The name is somewhat misleading, even though it's just the converse
of the usual statement of complimentarity:
Non-commuting operators cannot be applied simultaneously
In order to get a consistent macroscopic picture, it's an absolute
necessity that any two simultaneous observers be on the same basis or a
compatible system of bases. And that's what's being referred to by
"getting everyone to be on the same wavelength".
But this principle makes implicit use of the concept of simultaneity.
That's not a suprise, though, because historically the theory (Quantum
Mechanics) is formulated in a non-Relativistic setting, and makes
essential use of one of the crucial properties of Newtonian space,
which does not hold in Relativity; namely that:
If a || b and b || c then a || c
where a || b means "a is neither before nor after b"
So, if you try to apply the principle as is you run immediately into
problems whether you have any 3 observations located at a, b and c
such that
a || b, b || c, but a is before c
where the corresponding operators are such that:
[A(a), A(c)] != 0
Then you're immediately confronted with the issue: which of the operators
does A(b) commute with?
This is the Paradox of Simultaneous Measurement.
The resolution is simple, but inevitable. It is possible for A(b) to
commute with both A(a) and A(c) if you have a situation like so:
A(b) = I_H x AK
AK: K --> K
A(a) = AH(a) x I_K, A(c) = AH(c) x I_K
AH(a), AH(c): H -> H
The underlying state space splits up into separate, mutually commuting,
components for each spacelike separated observer.
Hence, Complimentaity is really just the Newtonian form of Microcausality.
Thus, also, it follows that LOCALITY is a necessary concept in a Relativistic
form of Quantum Theory.
The significance of this is that it makes the process of translating the
standard orthodox axiomatization of QM into relativistic form non-trivial
and it is, in fact, STILL an open problem of sorts!
In particular, the question is how to generalize von Neumann's Projection
Axiom into a purely local form suitable for relativistic spaces. Nor is
the issue lost on any of the measurement theories, such as Everett.
There, too, you STILL have the (still basically open) question of how to
render the relative state formalism into a form that respects locality
and is suitable for a relativistic space.
Arkadiusz Jadczyk
(Mark) wrote:
> Non-commuting operators cannot be applied simultaneously
Who says so and why?
Check papers available on the Cassiopaea
web site:
HOW EVENTS COME INTO BEING: EEQT, PARTICLE TRACKS, QUANTUM CHAOS, AND
TUNNELING TIME
http://www.cassiopaea.org/quantum_future/papers/garda.htm
COMPLETELY MIXING QUANTUM OPEN SYSTEMS AND QUANTUM FRACTALS
http://www.cassiopaea.org/quantum_future/chaos.htm
EEQT A WAY OUT OF THE QUANTUM TRAP
http://www.cassiopaea.org/quantum_future/papers/petruc/petruc.html
TOPICS IN QUANTUM DYNAMICS.
http://www.cassiopaea.org/cgi-bin/getit.cgi?url=cassiopaea.org/quantum_future/papers/9506017.pdf
RELATIVISTIC QUANTUM EVENTS
http://www.cassiopaea.org/cgi-bin/getit.cgi?url=cassiopaea.org/quantum_future/papers/9610028.pdf
Very soon (a week or two from now) online simulations of chaos caused by
"simultaneous mesurement" of noncommuting quantities (in java) will be
available.
I say "simulatenous" in quotation marks because the theory and
simulation deals with "cotinuous" measurement - that is extended
in time - as any real measurement is.
ark
Charles Francis
SIGFPE <usen...@sigfpe.com> writes
>In the Copenhagen interpretation there are observers and the observed
>universe. It seems that the state of the universe at any time is a
>tuple that looks like this:
>
>(O_1,...,O_n,psi)
I think you would do better to split this differently, into a set of
ordered triples
{(observer, apparatus, observed system)}
then qm describes the observations made in this system. You should allow
that an observer may be a part of the observed system of another
observer, as in Wigner's friend.
>The problem I have is this: in the Copenhagen interpretation we also have
>a wave function psi but this time it lives in a slightly reduced space
>
>S_1 \otimes ... \otimes S_n
>
>where the product no longer contains terms describing observers (because
>in the Copenhagen interpretation these don't have wavefunctions).
The wave function exists within the triple pertaining to one observer.
It is not in general compatible with the wave function pertaining to
another observer because for one observer collapse takes place
differently from another. This is understandable in Copenhagen if the
wave function describes the information an observer has about nature,
not nature itself.
>So in what sense is the Copenhagen interpretation not a 'many worlds'
>interpretation?
The many observers described above is rather like it.
>It seems to me that the 'many worlds interpretation' should be called
>the 'many observers interpretation'.
There are many worlders who describe a many minds interpretation. But I
have tried pointing to the idea that since qm describes only information
from measurement it produces a different description for each observer
with different information, but the response from many worlders has
generally been negative, and despite denials, it seems to me that they
really do believe in the existence of some sort of naive many worlds.
>And of course there are physical differences between these
>interpretations because in the 'many worlds' approach it should be
>possible for observers to observe interference between the states of
>other observers but the Copenhagen disallows such a thing. This means
>that I'm also confused about why the 'many world interpretation' is
>called an 'interpretation'.
>Any alleviation of my confusion would be appreciated!
The trouble I find is that when something does not make sense it is just
confusing. I don't think many worlds makes sense, hence I cannot help to
explain it, sorry. Copenhagen, OTOH, not in its original form but in the
orthodox or Dirac-Von Neumann form does, to me make very good sense. But
while it can be used to understand qm as a quantum logic or as an exotic
probability theory, it does not explain the Schrodinger equation and
hence as interpretation it is not complete.
Regards
- --
Charles Francis
Ralph E. Frost
news:ImkLYaW$Oxl8...@clef.demon.co.uk...
> In article <usenet123-C042A...@nnrp4-w.snfc21.pbi.net>,
> SIGFPE <usen...@sigfpe.com> writes
>
> >In the Copenhagen interpretation there are observers and the observed
> >universe. It seems that the state of the universe at any time is a
> >tuple that looks like this:
> >
> >(O_1,...,O_n,psi)
>
> I think you would do better to split this differently, into a set of
> ordered triples
>
> {(observer, apparatus, observed system)}
>
> then qm describes the observations made in this system. You should allow
> that an observer may be a part of the observed system of another
> observer, as in Wigner's friend.
...
> The trouble I find is that when something does not make sense it is just
> confusing. I don't think many worlds makes sense, hence I cannot help to
> explain it, sorry. Copenhagen, OTOH, not in its original form but in the
> orthodox or Dirac-Von Neumann form does, to me make very good sense. But
> while it can be used to understand qm as a quantum logic or as an exotic
> probability theory, it does not explain the Schrodinger equation and
> hence as interpretation it is not complete.
Could you please outline, summarized, illuminate what it is that you mean
by this last phrase:
"it [D-VN QM] does not explain the Schrodinger equation and
hence as interpretation it is not complete"
What's the real difficulty?
Thanks.
--
Best regards,
Ralph Frost
http://flep.refrost.com
"The essential nature of external reality, Comenius thought,
could be conveyed by education to the simplest intelligence
if all knowledge could be reduced to a basic principle."
- notions ascribed to John Amos Comenius (1592-1670), circa 1640
[Dobbs, Betty Jo Teeter, THE FOUNDATIONS OF NEWTON'S ALCHEMY, Cambridge
University Press, Cambridge 1975 p. 60]
Charles Francis
<ref...@dcwi.com> writes
>>Copenhagen, OTOH, not in its original form but in the
>> orthodox or Dirac-Von Neumann form does, to me make very good sense. But
>> while it can be used to understand qm as a quantum logic or as an exotic
>> probability theory, it does not explain the Schrodinger equation and
>> hence as interpretation it is not complete.
>
>Could you please outline, summarized, illuminate what it is that you mean
>by this last phrase:
>
> "it [D-VN QM] does not explain the Schrodinger equation and
> hence as interpretation it is not complete"
>
>What's the real difficulty?
Interference effects are real, and it is much thought that this is
indication that there must be physical matter which in some sense obeys
the equation, and that this is fundamental physical law. If the wave
function only describes information then a reason must be given as to
why the information should obey a wave equation.
In fact I believe we can now explain this, since the wave equation can
be drawn out of local U(1) symmetry (known also as gauge symmetry, or
phase invariance) together with covariance. However I would still not
regard this explanation as complete in the absence of a fully rigorous
and convincing form of qft.
Regards
--
Charles Francis