The graviton, and the fallacy of the stolen concept.
Starblade Darksquall
usage of the idea of a graviton in their theories. That's just plain
wrong. And let me tell you why.
First off, most particles are things that exist IN time and space. Or
at least that's how we understand them now. It's pretty hard to try to
consider particles that are made of time and space, and for obvious
reasons which I will explain later. Now, because of this, we can
easily conceptualize the particle's properties as being
indeterministic, even though we have to use a fixxed metric, or at
most, a fixxed timespace with metric varying from point to point.
Gravity is different, in that we have to conceptualize time and space
being different relative to itself, so that theory is going to be very
difficult even in the deterministic realm, and pretty much impossible
in the indeterministic realm, which pretty much makes quantization a
fantasy, at least for those unwilling to let go of current physical
paradigm.
However, their solution to these problems, to make the graviton itself
a particle, when gravity is itself space and time, is a contradiction.
A particle, as such, is what exists IN space and time. Gravity IS
space and time. Space and time may neighbor other space and time, but
other than this, it makes no sense to consider it ON space and time.
The idea of a graviton uses the idea of a particle of gravity, and
because you ignore what a particle is, and what gravity is, and simply
use them at your convenience, without paying attention to what makes
them what they are, and how one understands them conceptually, you are
guilty of the fallacy of the stolen concept.
It takes a rational mind to help destroy the irrationalities in
physics.
(...Starblade Riven Darksquall...)
Lubos Motl
> First off, most particles are things that exist IN time and space. Or
> at least that's how we understand them now. It's pretty hard to try to
> consider particles that are made of time and space, and for obvious
> reasons which I will explain later.
Any idea in physics or science - or anywhere for that matter - can be
difficult for someone, but a much more important question is whether it
makes sense. Gravitons certainly make sense, and most likely they are
reality: their existence is a direct consequence of the simplest
principles of quantum mechanics applied to a theory that contains general
relativity.
Imagine simply gravitational waves - the same waves that are generated by
rotating double stars and similar objects and that are expected to be
confirmed by direct observations in the next 10 years. According to
quantum mechanics, the energy carried away by these waves must be
quantized in units of E=hf, just like for the photons, and the quanta are
called "gravitons".
They can still be viewed as objects living on a fixed classical
background.
> Gravity is different, in that we have to conceptualize time and space
> being different relative to itself, so that theory is going to be very
> difficult even in the deterministic realm, and pretty much impossible ...
Well, GR is a beautiful theory, but it is also a simple theory in a sense
- Einstein could have found it and investigated himself by 1915, for
example. Today we study much more complex theories than just ordinary GR.
If the physicists were lazy and stopped every time when things start to
look a bit difficult, they would have made no progress. Ever.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Only two things are infinite, the Universe and human stupidity,
and I'm not sure about the former. - Albert Einstein
Doug Sweetser
I think it is the non-linearity of general relativity that keeps me
confused, that gravity fields gravitate. For peace of mind, I work in
a linear approximation to general relativity. There, mass will work
just like electric charge. Photons do not exchange photons with other
phonts, thank goodness! Particles with electric charge may exchange
virtual photons so they fly along the right path. A linear field
approach to gravity would behave the same way. Gravitons would not
exchange virtual gravitons with other gravitons, getting rid of a nasty
game of cat chasing its own tail. For particles with mass charge (read
a rest mass), they would exchange virtual spin-2 gravitons and go their
merry way along geodesics. I can live with that.
doug
quaternions.com
Doug Sweetser
Hello Lubos:
There is something wonderfully odd about this standard explanation:
> Imagine simply gravitational waves - the same waves that are generated
> by rotating double stars and similar objects and that are expected to
> be confirmed by direct observations in the next 10 years. According
> to quantum mechanics, the energy carried away by these waves must be
> quantized in units of E=hf, just like for the photons, and the quanta
> are called "gravitons".
When I think about objects that make quanta, I always imagine really
little things, the stuff happening on atomic scale. To make a few
gravitational quanta takes two enormous stars rotating around each
other :-) The math is the same, E=hf, but the objects creating the
events are anomalously large!
doug
quaternions.com
Starblade Darksquall
> On Fri, 6 Feb 2004, Starblade Darksquall wrote:
>
> > First off, most particles are things that exist IN time and space. Or
> > at least that's how we understand them now. It's pretty hard to try to
> > consider particles that are made of time and space, and for obvious
> > reasons which I will explain later.
>
> Any idea in physics or science - or anywhere for that matter - can be
> difficult for someone, but a much more important question is whether it
> makes sense.
Whether or not it 'makes sense' is a matter of perspective. Isn't
science based on the idea that we need to make decisions based on
logic and fact rather than simply 'good argument'? Good argument is
the basis for philosophy, and philosophy help guide scientific
principles, which in turn give rise to scientific methods, including,
of course, as the best example, is THE scientific method.
The best way we can utilize this principle is, of course, by
recognizing exactly what the tenets of GR and QFT are, then deciding
for ourselves if the idea of a graviton makes sense. A graviton is in
fact a representation of a quantized gravitational field. However, if
there is some fundamental conflict between the concept of a quantized
X field and the gravitational field to which the quantization applies,
then there cannot be a graviton.
A simple argument would proceed as follows:
Quantum Field Theory works on fields which are a part of, and in fact
exist on a fixxed background metric, with some indeterminite,
noncausal activity at the really small scale. This, quantum field
theory is what we use to determine the concept of quantization.
Quantization means to make a field work with quantum field theory.
General Relativity is itself a generalization of relativity theory to
gravitational fields, and treats gravitational fields as deterministic
entities which, in fact, comprise a nonfixxed metric. Gravitation is
due, then, to the bending of space and time, which is quite distinct
from being a field IN space and time.
As you can see, the two ideas clash. One consists of a constant metric
with a indeterminate field, whereas the other consists of a variable
metric with a determinate field. A GOOD idea would, in fact, encompass
both a variable metric and an indeterminite field. It would seek to
reverse engineer the universe at the quantum gravity scale rather than
naively attempting to mix the principles together in a messy
compromise.
Do you see the logic in this? That we should reverse engineer the
universe rather than simply applying the principles we know to either
different fields in an attempt to mix together QFT and GR?
Gravitons certainly make sense, and most likely they are
> reality: their existence is a direct consequence of the simplest
> principles of quantum mechanics applied to a theory that contains general
> relativity.
>
And this is exactly what I am arguing against. We need to find
principles that are neither QFT nor GR, but rather, of which they are
both a subset. Loop Quantum Gravity is a disorganized mess, and I
doubt it will produce anything even remotely useful, but it's still a
good exercize. String Theory and its children, on the other hand,
while not very well grounded, nevertheless are the leading theory, and
for lack of a more competent theory, it's really the closest thing
we've got. Furthermore, we have numerous other principles to work
with, such as that of Blackhole Thermodynamics, but that can't really
be a starting point itself. What we need is a bold new theory... we
need to go where, quite literally, No Man Has Gone Before. That's what
I believe, and I consider myself quite well grounded in said belief.
> Imagine simply gravitational waves - the same waves that are generated by
> rotating double stars and similar objects and that are expected to be
> confirmed by direct observations in the next 10 years. According to
> quantum mechanics, the energy carried away by these waves must be
> quantized in units of E=hf, just like for the photons, and the quanta are
> called "gravitons".
>
While I don't doubt that there is some rule for determining how much
energy a gravitational wave can carry, I doubt that any of our current
rules for determining the energy will work on gravity.
> They can still be viewed as objects living on a fixed classical
> background.
>
Which is wrong. You're subjugating GR to QFT. That would be just as
wrong as subjugating QFT to GR. The very fact is, we may have to look
at situations which consider both QFT and GR too primitive to work on
their own, but which attempt to create a theory of which both QFT and
GR are a subset.
> > Gravity is different, in that we have to conceptualize time and space
> > being different relative to itself, so that theory is going to be very
> > difficult even in the deterministic realm, and pretty much impossible ...
>
> Well, GR is a beautiful theory, but it is also a simple theory in a sense
> - Einstein could have found it and investigated himself by 1915, for
> example. Today we study much more complex theories than just ordinary GR.
> If the physicists were lazy and stopped every time when things start to
> look a bit difficult, they would have made no progress. Ever.
>
That's true. And we need a good theory upon which to exercize the
principles of science and of scientific progress. So I can't hold too
much of a grudge against their current attempts to solve the riddle of
finding a truly unifying theory of modern physics.
______________________________________________________________________________
> E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
> eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> Only two things are infinite, the Universe and human stupidity,
> and I'm not sure about the former. - Albert Einstein
(...Starblade Riven Darksquall...)
mike.james
"Doug Sweetser" <swee...@alum.mit.edu> wrote in message
news:c0ajpt$89k$1...@pcls4.std.com...
Also why does gravity have to obey E=hf?
What logic or physics rules out the existence of a non-quantised field?
Isn't this a matter for experiment?
mikej
Buzurg Shagird
>
> There is something wonderfully odd about this standard explanation:
> When I think about objects that make quanta, I always imagine really
> little things, the stuff happening on atomic scale. To make a few
> gravitational quanta takes two enormous stars rotating around each
> other :-) The math is the same, E=hf, but the objects creating the
> events are anomalously large!
Although I am not Lubos ;-) may I point out that there is nothing odd
about this explanation -- the scale of the source will determine only
the scale of the typical wavelength of the radiation, which will be
very large for stars, and the frequency will be correspondingly small,
and the energy of each graviton will be absolutely tiny! But at the
energy these objects radiate, there will be lots and lots of gravitons,
which will add up to look like classical gravitational waves.
-S.
Doug Sweetser
> Also why does gravity have to obey E=hf?
> What logic or physics rules out the existence of a non-quantized
> field? Isn't this a matter for experiment?
Everything should be a matter of experiment :-) I think for gravity we
will end up with the usual mathematical machinery of quantum mechanics:
Hilbert spaces, operators, commutators, ... Why? I am typing in a
room where gravity, EM, the weak and strong forces are doing their own
things simultaneously. Nature would have to be inconsistent to treat
the simplest, most symmetric force differently from the others.
Millikan showed experimentally that there was one universal value for
electric charge. The numerical value of charge depends on how charge
is defined. Playing around with units one day I noticed that
(h c)^(1/2) has the same units as electric charge. I am not at all
sure if that is related to shared value of one unit of electric charge
used by all particles.
It is my hope to treat mass like electric charge, respecting relevant
differences. One difference is that the oil drop experiment for mass
applies only to the same types of particles. One electron has exactly
the same amount of mass charge (G^(1/2) 511 MeV) as another electron or
positron, and three electrons have three electron mass charges (G^(1/2)
1533 MeV). There is no universal mass charge like there is a universal
electrical charge however since no other particle starts out with the
same rest mass. Maybe someday a person will find a pattern in the
masses of particles, but that has yet to be seen.
doug
quaternions.com
mike.james
> Hello Mike:
>
> > Also why does gravity have to obey E=hf?
> > What logic or physics rules out the existence of a non-quantized
> > field? Isn't this a matter for experiment?
>
> Everything should be a matter of experiment :-) I think for gravity we
> will end up with the usual mathematical machinery of quantum mechanics:
> Hilbert spaces, operators, commutators, ... Why? I am typing in a
> room where gravity, EM, the weak and strong forces are doing their own
> things simultaneously. Nature would have to be inconsistent to treat
> the simplest, most symmetric force differently from the others.
I follow the "nice" idea that everything should be treated in the same way.
I can also see how gravity could be a QFT, if only it worked. What is
bothering me is that gravity might be different in the sense that it could
be the background for the other theories. There's a lot if talk of
background free theories and the desirability of such a theory but what if
(yes I know its not a popular view point) GR provides the background for the
other quantum theories without being a quantum theory itself. Perhaps the
manifold and its metric are different in nature from the fields, (other
fields?) it carries.
What I want to know is - is there anything that forces gravity to be a
quantum theory?
What is the paradox that would arise is say gravity were not part of the
same system as the other fields.
The answer might be some piece of very obvious theory that I'm missing but
just to say that it has to be quantised because the machinery to do so
exists and everything else is doesn't really seem to be enough.
I can well imagine experiments that would show that gravity is or is not
quantised but are they necessary?
mikej
eb...@lfa221051.richmond.edu
In article <c0gdpg$ltu$1$8302...@news.demon.co.uk>,
mike.james <mike....@infomax.demon.co.uk> wrote:
>What I want to know is - is there anything that forces gravity to be a
>quantum theory?
The problem is that gravity couples to matter. If gravity is described
by classical general relativity, while the rest of physics
is governed by quantum field theories, then how are we to interpret
the Einstein equation
G = T
which is of the form
(classical thing) = (quantum thing).
For instance, if you have particles with large uncertainties in their
positions, so that it's unknown whether there's a bunch of mass over
here or over there, then what does the (classical) geometry of
spacetime do?
Wald's book "General Relativity" goes into some detail about the
problems that arise if you try to treat gravity classically while
coupling it to quantum matter.
-Ted
--
[E-mail me at na...@domain.edu, as opposed to na...@machine.domain.edu.]
Arnold Neumaier
eb...@lfa221051.richmond.edu wrote:
> In article <c0gdpg$ltu$1$8302...@news.demon.co.uk>,
> mike.james <mike....@infomax.demon.co.uk> wrote:
>
>
>>What I want to know is - is there anything that forces gravity to be a
>>quantum theory?
>
>
> The problem is that gravity couples to matter. If gravity is described
> by classical general relativity, while the rest of physics
> is governed by quantum field theories, then how are we to interpret
> the Einstein equation
>
> G = T
>
> which is of the form
>
> (classical thing) = (quantum thing).
>
> For instance, if you have particles with large uncertainties in their
> positions, so that it's unknown whether there's a bunch of mass over
> here or over there, then what does the (classical) geometry of
> spacetime do?
This argument is far too simplistic. One could consider a unified theory
based on the equation
G = psi^* T psi,
where psi satisfies a Schroedinger equation in the so-called
functional Schroedinger picture of QFT. Then both sides have the
same status.
So the question that needs to be answered is
why this leads to contradictions with logic or experiment.
Arnold Neumaier
Lubos Motl
> > When I think about objects that make quanta, I always imagine really
> > little things, the stuff happening on atomic scale. To make a few
> > gravitational quanta takes two enormous stars rotating around each
> > other :-) The math is the same, E=hf, but the objects creating the
> > events are anomalously large!
You exaggerate ;-). The number of gravitons created per second by a binary
star is huge. Yes, the gravitational waves are very weak, and it is
difficult to detect them even as the classical waves. But at the same
moment one must realize that the smaller energy a wave of a fixed energy
carries, the more important the energy quantization becomes.
On Wed, 11 Feb 2004, mike.james wrote:
> Also why does gravity have to obey E=hf?
Any system in quantum mechanics is described by a wavefunction whose phase
is rotating in time with a frequency f=E/h where E is the energy of the
system (this is called the Schrodinger equation). Because the strength of
a gravitational wave is variable, there are many quantum microstates that
compose the object that we call "classical gravitational wave" (it is a
coherent state of gravitons, in fact). The energy difference between any
pair of states that contribute to the "classical wave" must be a multiplie
of E=hf, because we know that the ultimate classical wave has the
periodicity in time (with frequency f).
> What logic or physics rules out the existence of a non-quantised field?
The logic is that there is no logically consistent way to construct a
theory that treats some objects in a classical way, and other objects
using quantum mechanics. Because we know that the world is quantum
mechanical, its logic (e.g. the fact that only probabilities can be
predicted, and they are computed as squared absolute values of various
matrix elements of some operators on the Hilbert space) must be applied to
the whole world, and when we apply it to the gravitational field, we
obtain the same rule for energy quantization as in the case of the
electromagnetic field.
> Isn't this a matter for experiment?
It is very difficult to detect the individual gravitons - although it is
not completely impossible. But certainly, the assumption is that the
derivation of the existence of gravitons is correct, and unless you
propose a working alternative theory (which is very unlikely), there is
nothing to talk about. At this moment, the alternative - the attempt to
claim that gravity does not have to be quantum at all - is ruled out by
pure thought, and we don't even have to make an experiment.
Arkadiusz Jadczyk
<Arnold....@univie.ac.at> wrote:
>This argument is far too simplistic. One could consider a unified theory
>based on the equation
> G = psi^* T psi,
>where psi satisfies a Schroedinger equation in the so-called
>functional Schroedinger picture of QFT. Then both sides have the
>same status.
>
>So the question that needs to be answered is
>why this leads to contradictions with logic or experiment.
A nontrivial coupling of a classical and a quantum system needs to
satisfy certain consistency requirements. These have been studied
and described in the literature.
The equation of the type as above is not consistent, though it can serve
as a first order approximation to a consistent scheme.
Although the general principles of such a consistent scheme are clear,
and toy models have been studied (like quantum SQUID coupled to a
classical radio frequency oscillator), the particular case of classical
gravity coupled to a quantum field is still on the drawing board.
Additionally, gravity represented by a metric tensor in 4d is not an
"irreducible" geometrical object. It naturally splits into 9+1 fields,
where 9 stands for the causal structure and 1 is the volume (or length).
It may well happen that 1 can be quantized while 9 can be kept
classical.
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
Thomas Larsson
> This argument is far too simplistic. One could consider a unified theory
> based on the equation
> G = psi^* T psi,
> where psi satisfies a Schroedinger equation in the so-called
> functional Schroedinger picture of QFT. Then both sides have the
> same status.
>
> So the question that needs to be answered is
> why this leads to contradictions with logic or experiment.
>
Keeping gravity classical is discussed in section 2 of Carlip's review,
http://www.arxiv.org/abs/hep-th/9108028 . Consensus seems to be that this
brings in a host of new problems without really solving any of the old ones.
mike.james
"Thomas Larsson" <thomas_l...@hotmail.com> wrote in message
news:24a23f36.0402...@posting.google.com...
>
> Keeping gravity classical is discussed in section 2 of Carlip's review,
> http://www.arxiv.org/abs/hep-th/9108028 . Consensus seems to be that this
> brings in a host of new problems without really solving any of the old
ones.
>
Again it doesn't really 100% rule out the prospect of a classical theory of
gravity coupling to quantum everything else. It just makes it seem messy.
In this case I'm not even sure its that messy.
After all gravity as described by classical GR is non-linear so why the
surprise when it results in a non-linear wave equation for matter described
by the field.
It might be surprising that even the Newtonian form leads to a
non-linearlity but this is for a particle moving in its own field. I don't
see why the non-linearity in this case has to spread to the situations where
the wave equation is contrived to be linear by making the particle move in
someone elses field.
mikej
mike.james
"Thomas Larsson" <thomas_l...@hotmail.com> wrote in message
news:24a23f36.0402...@posting.google.com...
> Keeping gravity classical is discussed in section 2 of Carlip's review,
> http://www.arxiv.org/abs/hep-th/9108028 . Consensus seems to be that this
> brings in a host of new problems without really solving any of the old
ones.
>
Did you mean
http://www.arxiv.org/abs/gr-qc/0108040
mikej
Arnold Neumaier
Arkadiusz Jadczyk wrote:
> On 13 Feb 2004 15:42:56 -0500, Arnold Neumaier
> <Arnold....@univie.ac.at> wrote:
>
>>This argument is far too simplistic. One could consider a unified theory
>>based on the equation
>> G = psi^* T psi, (*)
>>where psi satisfies a Schroedinger equation in the so-called
>>functional Schroedinger picture of QFT. Then both sides have the
>>same status.
>>
>>So the question that needs to be answered is
>>why this leads to contradictions with logic or experiment.
>
>
> A nontrivial coupling of a classical and a quantum system needs to
> satisfy certain consistency requirements. These have been studied
> and described in the literature.
Please give a reference which states and proves these
consistency requirements.
> The equation of the type as above is not consistent,
Please explain why.
> Additionally, gravity represented by a metric tensor in 4d is not an
> "irreducible" geometrical object. It naturally splits into 9+1 fields,
> where 9 stands for the causal structure and 1 is the volume (or length).
> It may well happen that 1 can be quantized while 9 can be kept
> classical.
Should such a thing happen, I'd rather bet on the opposite -
classical volume and quantized conformal metric, as suggested
by the trace anomaly.
Arnold Neumaier
Arnold Neumaier
Lubos Motl wrote:
> Doug Sweeter wrote:
>
>>>When I think about objects that make quanta, I always imagine really
>>>little things, the stuff happening on atomic scale. To make a few
>>>gravitational quanta takes two enormous stars rotating around each
>>>other :-) The math is the same, E=hf, but the objects creating the
>>>events are anomalously large!
>
> You exaggerate ;-). The number of gravitons created per second by a binary
> star is huge.
Unlike the baryon or lepton number, the number of gravitons is a meaningless
concept, just as the number of photons in QED. Interactions produce infinitely
many of them, or rather superpositions of arbitrarily many...
Arnold Neumaier
mike.james
<eb...@lfa221051.richmond.edu> wrote in message
news:c0j90m$k88$1...@lfa222122.richmond.edu...
>
> The problem is that gravity couples to matter. If gravity is described
> by classical general relativity, while the rest of physics
> is governed by quantum field theories, then how are we to interpret
> the Einstein equation
>
> G = T
>
> which is of the form
>
> (classical thing) = (quantum thing).
>
> For instance, if you have particles with large uncertainties in their
> positions, so that it's unknown whether there's a bunch of mass over
> here or over there, then what does the (classical) geometry of
> spacetime do?
I can see that there might be difficulties in treating a quantum mass/energy
producing a non-quantised curvature but I still don't see that this rules it
out as a theory. See Arnold Neumaier's post a bit later in the thread for
example.
> Wald's book "General Relativity" goes into some detail about the
> problems that arise if you try to treat gravity classically while
> coupling it to quantum matter.
>
> -Ted
Thanks - I've got a copy of Wald's book some where I'll look it up.
I've already got a feeling that what I will find is a "weight of evidence"
type argument rather than one single paradox or counter argument.
mikej
Thomas Larsson
thomas_l...@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0402...@posting.google.com>...
> Keeping gravity classical is discussed in section 2 of Carlip's review,
> http://www.arxiv.org/abs/hep-th/9108028 . Consensus seems to be that this
> brings in a host of new problems without really solving any of the old ones.
Oops, that pointer belonged to another post. I meant of course
http://www.arxiv.org/abs/gr-qc/0108040
Arkadiusz Jadczyk
On 15 Feb 2004 09:51:33 -0500, Arnold Neumaier
<Arnold....@univie.ac.at> wrote:
>> A nontrivial coupling of a classical and a quantum system needs to
>> satisfy certain consistency requirements. These have been studied
>> and described in the literature.
>
>Please give a reference which states and proves these
>consistency requirements.
Algebraic formalism describes both classical and quantum theory.
Therefore to unify both we need algebraic formalism with classical
observables as the center of the algebra. Nontrivial coupling between
classical and quantum degrees of freedom is impossible when we describe
the dynamics via *-automorphisms. The simplest way out is by using
dynamical semigroups (such a semigroup mixes central and non-central
observables). This is the short proof. Literature gives just examples of
how this works in practice.
>> The equation of the type as above is not consistent,
>
>Please explain why.
>
>
>> Additionally, gravity represented by a metric tensor in 4d is not an
>> "irreducible" geometrical object. It naturally splits into 9+1 fields,
>> where 9 stands for the causal structure and 1 is the volume (or length).
>> It may well happen that 1 can be quantized while 9 can be kept
>> classical.
>
>Should such a thing happen, I'd rather bet on the opposite -
>classical volume and quantized conformal metric, as suggested
>by the trace anomaly.
Perhaps. Note however that we do have "Planck length", while there is no
"Planck conformal structure". There is only a "degenerated conformal
structure", when signature fluctuation starts, but there the Planck
constant is not involved there.
car...@no-physics-spam.ucdavis.edu
Arnold Neumaier <Arnold....@univie.ac.at> wrote:
> This argument is far too simplistic. One could consider a unified theory
> based on the equation
> G = psi^* T psi,
> where psi satisfies a Schroedinger equation in the so-called
> functional Schroedinger picture of QFT. Then both sides have the
> same status.
> So the question that needs to be answered is
> why this leads to contradictions with logic or experiment.
This idea was suggested by Moller and Rosenfeld about 40
years ago (and possibly by Potier a decade earlier, but I
haven't yet checked the reference). The first thing to
note is that it makes the Schrodinger equation nonlinear
-- the Schrodinger equation in a curved spacetime depends
on the metric, which, in this formulation of ``semiclassical
gravity,'' depends in turn on the state. It's not clear
whether this nonlinearity is large enough to be ruled out
by experiment; one of my students is working on this now.
But it would at least require a drastic revision of quantum
mechanics, since the principle of superposition would fail
(and with it much of the rationale for describing observables
as linear operators).
A second problem was analyzed by Eppley and Hannah, Found.
Phys. 7 (1977) 51. If gravity is ``classical,'' it seems
likely that one can use gravitational measurements to
simultaneously determine an object's position and momentum,
violating the uncertainty principle. Eppley and Hannah
show that if a gravitational measurement causes wave
function collapse, one must either give up the uncertainty
principle or momentum conservation, while if a gravitational
measurement does not wave function collapse, gravitational
interactions can be used to transmit information faster
than light.
None of this is conclusive, of course (unless my student
or someone else can rule out the required nonlinearity
experimentally). But it does mean that treating gravity
classically is not simple, and would require a serious
revision of quantum mechanics itself. This might not be
horrible -- Penrose, for example, would like to have
gravitational interactions responsible for wave function
collapse -- but it means that it's not a ``cheap'' solution.
Steve Carlip
Steve McGrew
On 16 Feb 2004 04:25:59 -0500, car...@no-physics-spam.ucdavis.edu
wrote:
>
>
>
>Arnold Neumaier <Arnold....@univie.ac.at> wrote:
>
>> This argument is far too simplistic. One could consider a unified theory
>> based on the equation
>> G = psi^* T psi,
>> where psi satisfies a Schroedinger equation in the so-called
>> functional Schroedinger picture of QFT. Then both sides have the
>> same status.
>
>> So the question that needs to be answered is
>> why this leads to contradictions with logic or experiment.
>
>This idea was suggested by Moller and Rosenfeld about 40
>years ago (and possibly by Potier a decade earlier, but I
>haven't yet checked the reference). The first thing to
>note is that it makes the Schrodinger equation nonlinear
>-- the Schrodinger equation in a curved spacetime depends
>on the metric, which, in this formulation of ``semiclassical
>gravity,'' depends in turn on the state. It's not clear
>whether this nonlinearity is large enough to be ruled out
>by experiment; one of my students is working on this now.
>But it would at least require a drastic revision of quantum
>mechanics, since the principle of superposition would fail
>(and with it much of the rationale for describing observables
>as linear operators).
>
>[snip]
If we are willing to contemplate the possibility of a "small"
nonlinearity in the Schroedinger equation, are there good reasons
not to consider a "small" nonconservation of charge?
Steve
Demian H.J. Cho
<unneccesary stuff>
> Additionally, gravity represented by a metric tensor in 4d is not an
> "irreducible" geometrical object. It naturally splits into 9+1 fields,
> where 9 stands for the causal structure and 1 is the volume (or length).
> It may well happen that 1 can be quantized while 9 can be kept
> classical.
>
>
> ark
Dear Ark,
Could you please elaborate more on the above statement?
especially the very last one.
Demian
eb...@lfa221051.richmond.edu
In article <402D33C8...@univie.ac.at>,
Arnold Neumaier <Arnold....@univie.ac.at> wrote:
>This argument is far too simplistic. One could consider a unified theory
>based on the equation
> G = psi^* T psi,
>where psi satisfies a Schroedinger equation in the so-called
>functional Schroedinger picture of QFT. Then both sides have the
>same status.
>
>So the question that needs to be answered is
>why this leads to contradictions with logic or experiment.
I completely agree with all this. If what I wrote suggests otherwise,
then I'm sorry to have been unclear.
I was trying to point out the sorts of problems that arise if
you try to couple classical spacetime to quantum matter. I didn't
mean to imply that such a coupling was known to be impossible,
but merely that it appeared to be problematic.
The proposal you give, G = <T>, with <> standing for the usual
quantum-mechanical expectation value, is discussed in Wald's book, by
the way.
mike.james
----- Original Message -----
From: "Arkadiusz Jadczyk" <arkREM...@ANDTHIScassiopaea.org>
> A nontrivial coupling of a classical and a quantum system needs to
> satisfy certain consistency requirements. These have been studied
> and described in the literature.
>
> The equation of the type as above is not consistent, though it can serve
> as a first order approximation to a consistent scheme.
>
> Although the general principles of such a consistent scheme are clear,
> and toy models have been studied (like quantum SQUID coupled to a
> classical radio frequency oscillator), the particular case of classical
> gravity coupled to a quantum field is still on the drawing board.
So the issue isn't as settled as it seems - and its a matter for either
experiment or theoretical study to obtain a paradox or contradiction.
Something like classical coupling to quantum or vice versa produces non
conservation of something or other.
> Additionally, gravity represented by a metric tensor in 4d is not an
> "irreducible" geometrical object. It naturally splits into 9+1 fields,
> where 9 stands for the causal structure and 1 is the volume (or length).
> It may well happen that 1 can be quantized while 9 can be kept
> classical.
I don't follow this but you seem to be saying that there might be a "core"
part of the metric field which has to be quantised to avoid the sort of
paradox mentioned above.
Any reference to the splitting into 9+1 fields?
mikej
mike.james
> > What logic or physics rules out the existence of a non-quantised field?
>
> The logic is that there is no logically consistent way to construct a
> theory that treats some objects in a classical way, and other objects
> using quantum mechanics.
You don't really give me any hard evidence.
While I'm sympathetic to the view that if one thing is quantum everything is
quantum this isn't a hard and fast proof ruling out something more
complicated.
When you say "there is no logically consistent way" do you really mean that
we have a proof that says that classical and quantum cannot exist and it
extends to the case of gravity coupling to other fields?
mikej
Lubos Motl
> Unlike the baryon or lepton number, the number of gravitons is a
> meaningless concept, just as the number of photons in QED.
> Interactions produce infinitely many of them, or rather superpositions
> of arbitrarily many...
My goal was not to create an exact concept of the "number of gravitons" -
even though this operator ("number of gravitons") certainly exists
perturbatively. My goal was to say that the number of gravitons that are
created (from a binary star) is huge, and this qualitative statement does
not require any specific exact definition. Take the total energy that has
been emitted and divide it by the frequency where the distribution has a
peak, for example.
Alfred Einstead
>This argument is far too simplistic. One could consider a unified theory
>based on the equation
> G = psi^* T psi, (*)
>where psi satisfies a Schroedinger equation in the so-called
>functional Schroedinger picture of QFT. Then both sides have the
>same status.
But then what constrains the underlying manifold to be globally
hyperbolic? There's nothing in the field equations, themselves,
that says this has to be so.
The formalism is, strictly speaking, weaker than the most general
situation desired, because it's now got global hyperbolicity as a
prerequisite to merely writing down (*).
This also seems to identify a distinguished state, or is there one
such equation for each possible psi? (*) is also restricting the
state to being a pure state. Maybe, for instance, it's more
appropriate for the state to be mixed (e.g., a thermal state).
How is T defined? The approach I'm familiar with (Wald et al.)
involves going BACK to a purely algebraic (i.e. state-independent)
setting.
More generally, the question of how to represent
T = G
may not be as vexing as it seems. There's nothing, for instance,
at the outset that says that no c-number T can be derived from
purely q-number quantities. For instance, in ordinary QM, even
though the p's and q's are q-numbers, their commutators [q,p]'s
are c-numbers. Likewise, certain combinations of commutators
involving (free) field quantities yield c-numbers. Perhaps it's
possible to synthesize c-number values in the more general
situation out of the field q-number values and get, from that,
a c-number quantitity for T that is purely algebraic and therefore
does NOT require making any reference to any underlying state
space representaiton or distinguished state.
Lubos Motl
> Algebraic formalism describes both classical and quantum theory.
> Therefore to unify both we need algebraic formalism with classical
> observables as the center of the algebra. Nontrivial coupling between
> classical and quantum degrees of freedom is impossible when we describe
> the dynamics via *-automorphisms. The simplest way out is by using
> dynamical semigroups (such a semigroup mixes central and non-central
> observables). This is the short proof. Literature gives just examples of
> how this works in practice.
The algebraic formalism does not and cannot describe quantum gravity, and
in fact it would even have problems classically. There are no local
gauge-invariant observables in GR, because general covariance eliminates
the link between the notion of a "point" and its coordinates. The
dynamical character of the metric prevents us from defining what the
"vanishing commutators at spacelike separation mean": whether or not two
points are spacelike-separated depends on the dynamical metric.
More generally, holography seems to be a general feature of quantum
gravity, and it simply means that the bulk off-shell observables can't be
fundamental.
Moreover, if we deal with a quantum theory, there can't be any true
"classical observables". The closest thing that you can have is an
observable that is relevant in a classical limit. Also, physics in general
simply does not allow a complete decoupling of two sets of degrees of
freedom; if some "degrees of freedom" are completely decoupled from the
physical ones, then they must either be unphysical, or they must describe
a different Universe. Finally, a collection of five very controversial and
unjustified new statements is not called a "short proof", and literature
cannot show any examples of the workings of something if it actually does
not work.
As you can see, I did not succeed in my attempt to find a true sentence in
your first paragraph.
> Perhaps. Note however that we do have "Planck length", while there is no
> "Planck conformal structure". There is only a "degenerated conformal
> structure", when signature fluctuation starts, but there the Planck
> constant is not involved there.
This paragraph would do better.
Danny Ross Lunsford
> Additionally, gravity represented by a metric tensor in 4d is not an
> "irreducible" geometrical object. It naturally splits into 9+1 fields,
> where 9 stands for the causal structure and 1 is the volume (or length).
> It may well happen that 1 can be quantized while 9 can be kept
> classical.
Well although I applaued the insight here, it has to be mentioned that
the metric and the calibration are *jointly* irreducible in Weyl
geometry. Even in Riemannian geometry, the metric is algebraically
irreducible, if not, as you point out - what to call it? -
"geometrodynamically" irreducible. It is fascinating that only the
combination of metric and calibration taken together are unquestionably
irreducible as such, and as you point out this has to be taken as a
fundamental clue about the nature of the underlying structure of a
correct physics.
-drl
Italo Vecchi
> Arnold Neumaier <Arnold....@univie.ac.at> wrote:
>
> > This argument is far too simplistic. One could consider a unified theory
> > based on the equation
> > G = psi^* T psi,
> > where psi satisfies a Schroedinger equation in the so-called
> > functional Schroedinger picture of QFT. Then both sides have the
> > same status.
You get an expectation value on the left side. How is the "classical"
Einstein tensor to be interpreted as an expectation value?
As long as this question is not answered the above equates apples and
pears, conceptually if not formally.
QM is a theory of measurement , so coupling GR to QM requires to
implement space and time as observables, i.e. it requires "quantising"
them. That's obviously hard to do , but i do not see how it can be
avoided.
I am sure that a clarification on this basic issue would be welcome by
others too.
Regards,
IV
Arkadiusz Jadczyk
On Tue, 17 Feb 2004 19:29:38 +0000 (UTC), Lubos Motl
<mo...@feynman.harvard.edu> wrote:
>The algebraic formalism does not and cannot describe quantum gravity, and
>in fact it would even have problems classically. There are no local
>gauge-invariant observables in GR, because general covariance eliminates
>the link between the notion of a "point" and its coordinates.
Algebraic formalism does not require local gauge-invariant observables.
It requires an algebra of operators.
The
>dynamical character of the metric prevents us from defining what the
>"vanishing commutators at spacelike separation mean": whether or not two
>points are spacelike-separated depends on the dynamical metric.
Algberaic formalism does not require ""vanishing commutators at
spacelike separation". It requires algebra of operators and states on
this algbebra.
>More generally, holography seems to be a general feature of quantum
>gravity, and it simply means that the bulk off-shell observables can't be
>fundamental.
Algbebraic formalism is simply more general than Hilbert space
formalism. If you tell me that you want to have quantum theory
without Hilbert space - then please tell me what YOU mean by a "quantum
theory" ?
>Moreover, if we deal with a quantum theory, there can't be any true
>"classical observables".
And who says so? And why?
> The closest thing that you can have is an
>observable that is relevant in a classical limit.
In fact I can have a lot of classical observables. For instant "number
2". And also number Pi. And also the sentence that tells me what the
theory formulation is.
> Also, physics in general
>simply does not allow a complete decoupling of two sets of degrees of
>freedom; if some "degrees of freedom" are completely decoupled from the
>physical ones, then they must either be unphysical, or they must describe
>a different Universe.
It depends on the definition of the universe. My definition may be
broader than yours.
> Finally, a collection of five very controversial and
>unjustified new statements is not called a "short proof", and literature
>cannot show any examples of the workings of something if it actually does
>not work.
The literature shows how using the coupling to classical degrees of
freedom using the algebraic formalism you can simulate, in real time,
forming a particle track in a cloud chamber. I did not not show any
other theoretical model that is able to do this (including the random
timings of events). If you know one, and can rund a simulation on your
computer - please let me know.
Perhaps, indeed, I should not call my idea of a proof a proof. It is
indeed only an idea. A proof, even a "short one" would take
more space.
Concerning "controversial and unjustified statements" - this can apply
as well to what you wrote. Therefore, I believe, it is better to avoid
"unjustified judgements.
ark
>
>As you can see, I did not succeed in my attempt to find a true sentence in
>your first paragraph.
Perhaps because of your prejudices, and also because you do not know the
relevant stuff that is not in your domain in expertise?
Arnold Neumaier
Arkadiusz Jadczyk wrote:
>>>Additionally, gravity represented by a metric tensor in 4d is not an
>>>"irreducible" geometrical object. It naturally splits into 9+1 fields,
>>>where 9 stands for the causal structure and 1 is the volume (or length).
>>>It may well happen that 1 can be quantized while 9 can be kept
>>>classical.
>>
>>Should such a thing happen, I'd rather bet on the opposite -
>>classical volume and quantized conformal metric, as suggested
>>by the trace anomaly.
>
> Perhaps. Note however that we do have "Planck length", while there is no
> "Planck conformal structure".
There is no need for the latter if the conformal metric's spectrum
(suitably defined) is continuous. After all, momentum also remains
continuous after quantization.
And the former is probably related to the total mass
of the universe (by Dirac-type arguments), which already sets
a mass scale in nonrelativistic QM. Thus there is no need for
another scale-invariance breaking mechanism.
Arnold Neumaier
Lubos Motl
On 18 Feb 2004, Arkadiusz Jadczyk wrote:
> Algebraic formalism does not require local gauge-invariant observables.
> It requires an algebra of operators.
Let me just recall that only gauge-invariant observables have a physical
meaning (that is independent on your calculational tricks) and can be
measured. A formalism without gauge-invariant observables is not a
physical theory. Gravity is associated with the general covariance, and
this symmetry implies that the gauge-invariant operators can't be local.
> Algberaic formalism does not require ""vanishing commutators at
> spacelike separation". It requires algebra of operators and states on
> this algbebra.
I can explicitly quote the causality axiom E, see pages 59-60 of Haag's
"Local quantum physics". This axiom says exactly what you think that it
does not say; spacelike-separated operators must commute.
If your "very tolerant" description of algebraic quantum field theory were
correct, AQFT would be a harmless concept without any physical meaning
because "everything goes". However, it is not harmless, and quantum
gravity cannot fit its axioms.
If you like the words "algebraic quantum field theory", try to learn how
to spell the word "algebra" correctly because you have misspelled it at
least 3 times.
> >More generally, holography seems to be a general feature of quantum
> >gravity, and it simply means that the bulk off-shell observables can't be
> >fundamental.
>
> Algbebraic formalism is simply more general than Hilbert space
> formalism. If you tell me that you want to have quantum theory
> without Hilbert space - then please tell me what YOU mean by a "quantum
> theory" ?
I did not say that a quantum theory does not have a Hilbert space.
Conventional quantum theories always have a Hilbert space. I said that
quantum gravity can't be fully formulated in terms of operators associated
with particular points in the classical geometry.
Let me answer your question: A quantum theory is a set of rules and
equations that allow one to calculate the probabilities of various
alternative outcomes of the experiments (or histories, more generally) in
terms of the expectation values of some linear operators and their
products.
In the textbook examples, a quantum theory is determined by a Hilbert
space - usually generated as a representation of a suitable set of
observables - together with a Hamiltonian that defines the time evolution,
or equivalently in terms of some classical action whose quantized path
integral can be used to calculate various correlators and amplitudes.
Quantum gravity requires one to fix the gauge (e.g. by light-cone gauge)
if we want to talk about the finite time evolution. Covariantly, no such
full gauge-fixing is possible, and the S-matrix - the evolution operator
between the asymptotic past and the asymptotic future - is the only known
exactly definable operator that replaces the Hamiltonian.
> >Moreover, if we deal with a quantum theory, there can't be any true
> >"classical observables".
>
> And who says so? And why?
Everyone who knows the principles of quantum mechanics also knows that
once we adopt quantum mechanical principles, they can't be reconciled with
non-quantum principles in a single coherent theory. Quantum mechanics
means that various possibilities are not uniquely predictable; only the
probabilities are predicted, and the total probability must be conserved.
Quantum evolution implies that essentially every possibility gets a
nonzero probability and every attempt to suddenly introduce classical
physics - where the evolution is deterministic, at least in principle -
would destroy the picture. The probabilities would not sum to one, for
example.
It is not possible to give a rigorous proof of the no-go theorem about
this fictitious theory that mixes quantum mechanics with something else
because what this theory should exactly mean has not been defined by
anyone. Let me therefore summarize that with the same accuracy as you
defined the problem, it is obvious that no such mixture is possible. You
would have to say more concretely what sort of modification of QM you have
in mind, and then we could see whether one can give you a more rigorous
proof why it is impossible.
> In fact I can have a lot of classical observables. For instant "number
> 2". And also number Pi. And also the sentence that tells me what the
> theory formulation is.
Although it is a minor linguistic point, you're not using the terminology
properly. "2" and "pi" are not really observables. An observable is a
measurable property of a physical system, and it is always represented by
an operator or "q-number", if I use Dirac's notation. See e.g.
http://en.wikipedia.org/wiki/Quantum_mechanics
I did not understand your last sentence "...tells me what the theory
formulation is" and why is it related to observables.
> > Also, physics in general
> >simply does not allow a complete decoupling of two sets of degrees of
> >freedom; if some "degrees of freedom" are completely decoupled from the
> >physical ones, then they must either be unphysical, or they must describe
> >a different Universe.
>
> It depends on the definition of the universe. My definition may be
> broader than yours.
You can define your Universe to contain unphysical observables, for
example the angels on the tip of the needle, but then your definition is
unphysical, too, and you are not thinking in terms of science (physics).
> The literature shows how using the coupling to classical degrees of
> freedom using the algebraic formalism you can simulate, in real time,
> forming a particle track in a cloud chamber.
You can simulate many things, so that the things look superficially
similar to reality, but one can also prove that Nature does not work in
certain way.
> I did not not show any other theoretical model that is able to do this
> (including the random timings of events).
All attempts trying to reveal "hidden variables" or "non-probabilistic
explanations underlying quantum mechanics" can be showed to be
incompatible either with the actual experiments (showing high
correlations), or with experimentally verified physical principles such as
the Lorentz invariance. This is what Bell's inequalities guarantee for us.
> If you know one, and can rund a simulation on your computer - please
> let me know.
I don't know what you exactly want to simulate, but be sure that if you
think that you have a model that gives the actual outcome of the
experiment instead of the probabilities, then it is a wrong model.
> >As you can see, I did not succeed in my attempt to find a true sentence in
> >your first paragraph.
>
> Perhaps because of your prejudices, ...
You can call them prejudices, but it might be more reasonable to call them
"knowledge" and "flexible up-to-date objective thinking". ;-)
> and also because you do not know the
> relevant stuff that is not in your domain in expertise?
Which stuff do you exactly have in mind?
Doug Sweetser
Hello Lubos:
I wonder if people would like to expand on this comment:
> A formalism without gauge-invariant observables is not a physical
> theory.
I know the oldest, most successful field theory - the Maxwell equations
for EM - is gauge-invariant. One can add a scalar potential field in
such a way as it does not alter the field equations, or have any
physical effects. I have read that one should also think about general
relativity as a gauge theory. There is an arbitrary vector field that
can be added into the field equations in an analogous way that also has
no physical effect. Is the gauge field a higher rank tensor in GR
compared to EM because the field equations are also higher?
The strong and weak field theories, modeled after EM, I suspect are
also invariant under gauge transformation. We can thus conclude that
at our current level of understanding which is deep, all known forces
have fields that are invariant under gauge transformation.
In my work, it is clear the equations are altered by a gauge
transformation. I would think that as long as I got back to the Maxwell
equations and some sort of dynamic metric equation consistent with
tests of GR, that would constitute a physical theory. Tests of a
theory are about the equations that result, not the guiding math
principles. This kind of blanket rejection appears to assert the
lessons of today will be the lessons of tomorrow.
doug
quaternions.com
Arnold Neumaier
Doug Sweetser wrote:
>
> I wonder if people would like to expand on this comment:
>
>>A formalism without gauge-invariant observables is not a physical
>>theory.
>
> transformation. I would think that as long as I got back to the Maxwell
> equations and some sort of dynamic metric equation consistent with
> tests of GR, that would constitute a physical theory. Tests of a
> theory are about the equations that result, not the guiding math
> principles. This kind of blanket rejection appears to assert the
> lessons of today will be the lessons of tomorrow.
Of course, comparison with experiment decides what is a physical theory.
A physical theory of gravity +E/M should have a consistent quantum version
(to be an improvement over GR) and should reproduce in the
classical limit + standard approximation the post-Newton approximation of
general relativity and a few results about binary stars and early cosmology
(to be consistent with observations). If the first fails, people will
prefer the tradition, which is already in agreement with all observations.
If the second fails, there is no reason at all to consider the theory as
relevant.
Gauge invariance is no necessity for a physical theory.
Effective theories are physical (indeed, closer to experiment
than the fundamental gauge theories other than QED).
they do not have gauge symmetries but they are (at the current
level of understanding) not predictive at high energy (infinitely
many undetermined counterterms). Hence they are thought to be deficient
from a fundamental perspective.
From the perspective of currently practiced QFT, renormalizability
(i.e. predictivity at arbitrary energies) depends for spin 1 fields
on gauge invariance. Gauge invariance is the geometric equivalent of
masslessness of the corresponding particle (in the absence of
spontaneously broken symmetry).
However, massless spin 2 (gravity) is nonrenormalizable
even though gauge invariant. Hence it is unclear whether gauge
invariance has to be taken as a necessary ingredient of a fundamental
theory.
Arnold Neumaier
Arnold Neumaier
> car...@no-physics-spam.ucdavis.edu wrote in message news:<c0oo19$fs8$1...@woodrow.ucdavis.edu>...
>
>>Arnold Neumaier <Arnold....@univie.ac.at> wrote:
>>
>>
>>>This argument is far too simplistic. One could consider a unified theory
>>>based on the equation
>>> G = psi^* T psi,
>>>where psi satisfies a Schroedinger equation in the so-called
>>>functional Schroedinger picture of QFT. Then both sides have the
>>>same status.
>
>
> I beg your pardon, which status?
They are both real numbers, and they have the same units.
> You get an expectation value on the left side. How is the "classical"
> Einstein tensor to be interpreted as an expectation value?
The equation only says that it is equal to an expectation; which
is nothing objectionable. In thermodynamics we also have observables
which equl certain expectations.
> As long as this question is not answered the above equates apples and
> pears, conceptually if not formally.
> QM is a theory of measurement , so coupling GR to QM requires to
> implement space and time as observables, i.e. it requires "quantising"
> them. That's obviously hard to do , but i do not see how it can be
> avoided.
QM is essentially 1D field theory. Time is not an observable.
QFT is the generalization where time has become 4-dimensional,
since the future can be in different directions.
Therefore, now neither time nor space are observables (in the sense
of being Hermitian operators in the formalism).
Time and space are neither observables (in the pre-quantum sense)
in classical general relativity.
There is no reason to expect this to change in a unification.
Arnold Neumaier
Thomas Larsson
> > A nontrivial coupling of a classical and a quantum system needs to
> > satisfy certain consistency requirements. These have been studied
> > and described in the literature.
>
The author of http://www.arxiv.org/abs/quant-ph/0402092 claims that
such a coupling is inconsistent. I am not competent to judge how
watertight his proof is.
Haelfix
are not fundamentally the 'space' one is dealing with.
For instance, please write down the shroedinger eqn in Hilbert space..
Oh wait, its hard to get derivatives in the space of square
integrable functions.
We know that some operators and all observables must live in Hilbert
space of course. Still savy mathematicians will point out that proper
treatment of the Schroedinger eqn involves taking sequences of
functions and showing that their limiting behaviour converges to a
vector in hilbert space.
In quantum field theory (and hence much of perturbative string
theory), the space we are dealing with is near impossible to define
rigorously. Where is the hilbert space you say in all these ill
defined integrals!
As far as the complaints against AQFT, well I won't get into an
argument there. Lets just say that it should in principle be testable
by experiment ultimately.
If CPT is violated (via say neutrino experiments), 'all hell breaks
loose'.
Arrogance can kill the cat.
Arkadiusz Jadczyk
wrote:
>On 18 Feb 2004, Arkadiusz Jadczyk wrote:
>
>> Algebraic formalism does not require local gauge-invariant observables.
>> It requires an algebra of operators.
>
>Let me just recall that only gauge-invariant observables have a physical
>meaning (that is independent on your calculational tricks) and can be
>measured.
It all depends on what is your particular version of gravity theory and
what is your particular gauge group - if any. When I measure something
using a coordinate system and
miles rather than centimeters - the number measured clearly depends
on the the choosen coordinate system. Now what is exactly gauge
invariance for gravity, and whether we want to look at gravity
this way or another - is up to us. There are different theories of
gravity.
A formalism without gauge-invariant observables is not a
>physical theory.
It depends on what is your interpretation of gauge invariance.
It depends on what is your definition and a realization of an
"observable". You can have quantum theory without an observer and
without observables!
> Gravity is associated with the general covariance, and
>this symmetry implies that the gauge-invariant operators can't be local.
One theory of gravity. There are other theories. Each have its good
features and its bad features.
>> Algberaic formalism does not require ""vanishing commutators at
>> spacelike separation". It requires algebra of operators and states on
>> this algbebra.
>
>I can explicitly quote the causality axiom E, see pages 59-60 of Haag's
>"Local quantum physics". This axiom says exactly what you think that it
>does not say; spacelike-separated operators must commute.
Now: I said algebraic formalism. This is not the same as "theory of
local algebras". Think of Alain Connes. Or think of something even more
general. You have an algebra, C* or von Neumann. It is from this
structure that space-times are being formed by quantum fluctuations
- quantum jumps.
>If your "very tolerant" description of algebraic quantum field theory were
>correct, AQFT would be a harmless concept without any physical meaning
>because "everything goes".
We have serious problem with quantum theory, with understanding of the
real meaning of the Planck constant, its meaning, its relation to the
electric charge and speed of light (fine structure constant), with
understanding why Hilbert spaces must be complex rather than real, with
understanding the very concept of events. I am not an advocate
of AQFT, but I am an advocate of AQT and its variations. Fields are
something else. They seem to be better than particles, but this is not a
concept that is general enough. Field presupposes "space" and "time" -
unless we have a more general idea, when we have an algebra and a
net of its sub algebras that are not necessarily related to space-time
localizations, but to something more fundamental, pre-geometrical.
> However, it is not harmless, and quantum
>gravity cannot fit its axioms.
That is clear. I agree.
>If you like the words "algebraic quantum field theory", try to learn how
>to spell the word "algebra" correctly because you have misspelled it at
>least 3 times.
Quantum fluctuations.
>
>> >More generally, holography seems to be a general feature of quantum
>> >gravity, and it simply means that the bulk off-shell observables can't be
>> >fundamental.
>>
>> Algbebraic formalism is simply more general than Hilbert space
>> formalism. If you tell me that you want to have quantum theory
>> without Hilbert space - then please tell me what YOU mean by a "quantum
>> theory" ?
>
>I did not say that a quantum theory does not have a Hilbert space.
>Conventional quantum theories always have a Hilbert space. I said that
>quantum gravity can't be fully formulated in terms of operators associated
>with particular points in the classical geometry.
You didn't say THAT. But if you MEAN that, then I agree.
>Let me answer your question: A quantum theory is a set of rules and
>equations that allow one to calculate the probabilities of various
>alternative outcomes of the experiments (or histories, more generally) in
>terms of the expectation values of some linear operators and their
>products.
Roughly speaking I agree. But I want to calculate not only
probabilities, but to understand how Nature is "choosing" the outcomes.
What is the mechanism of the "potentials" that becomes "real."
>In the textbook examples, a quantum theory is determined by a Hilbert
>space - usually generated as a representation of a suitable set of
>observables - together with a Hamiltonian that defines the time evolution,
>or equivalently in terms of some classical action whose quantized path
>integral can be used to calculate various correlators and amplitudes.
As you know Hamiltonian does not always exist. Sometimes we have to use
a representation of the algebra in which the time translations are NOT
unitarily implemented. Moreover, the term "observable" is, as you
probably know, rather confusing. But otherwise I agree that what you
wrote is a conventional approach. Which have its problems.
Let me quote from Rudolph Haag - whom you quoted above -
"It is often said that Quantum Theory is eminently successful, that its
predictions are verified in countless cases and that no phenomenon has
been found which contradicts it. Yet to this day, there remains some
uneasiness about its status, some disagreement concerning its
interpretation. This is not restricted to crackpots. Different camps of
eminent scientists advance widely different opinions."
and also
"The use of concepts in the theory which are not directly amenable to
observation is neither forbidden nor unusual. It seems unavoidable."
and also:
"...we need (basic or effective) limitations of the superposition
principle. To fully evaluate the scope of such limitations we need the
development of a self-consistent theory of measurement. By this I mean
we cannot claim that a self-adjoint operator in Hilbert space
corresponds to an 'observable'"
>Quantum gravity requires one to fix the gauge (e.g. by light-cone gauge)
>if we want to talk about the finite time evolution. Covariantly, no such
>full gauge-fixing is possible, and the S-matrix - the evolution operator
>between the asymptotic past and the asymptotic future - is the only known
>exactly definable operator that replaces the Hamiltonian.
You are certainly right, but you are looking only within a narrow
formalism, without realizing that more fundamental changes
are needed first! See e.g. quote from Haag above.
>> >Moreover, if we deal with a quantum theory, there can't be any true
>> >"classical observables".
>>
>> And who says so? And why?
>
>Everyone who knows the principles of quantum mechanics also knows that
>once we adopt quantum mechanical principles, they can't be reconciled with
>non-quantum principles in a single coherent theory. Quantum mechanics
>means that various possibilities are not uniquely predictable; only the
>probabilities are predicted, and the total probability must be conserved.
>Quantum evolution implies that essentially every possibility gets a
>nonzero probability and every attempt to suddenly introduce classical
>physics - where the evolution is deterministic, at least in principle -
>would destroy the picture. The probabilities would not sum to one, for
>example.
Rudolph Haag, I suppose knows the principles of quantum mechanism. And
yet he writes:
"Are there serious reasons to believe in a limitation of the general
validity of the superposition principle? We know some limitations
called "superselection rules."
Now, superselection rules do exactly that: they introduce central
quantities, commuting with ALL other quantities. They form an abelian
sub algebra. They are "classical". Conservation of probability is
possible with dynamical quantities of this type as well.
>It is not possible to give a rigorous proof of the no-go theorem about
>this fictitious theory that mixes quantum mechanics with something else
>because what this theory should exactly mean has not been defined by
>anyone. Let me therefore summarize that with the same accuracy as you
>defined the problem, it is obvious that no such mixture is possible. You
>would have to say more concretely what sort of modification of QM you have
>in mind, and then we could see whether one can give you a more rigorous
>proof why it is impossible.
OK. I will not try again to give a "short proof". I better give you a
reference:
"The Piecewise Deterministic Process Associated to EEQT"
http://xxx.lanl.gov/pdf/quant-ph/9805011
and references quoted there.
>> In fact I can have a lot of classical observables. For instant "number
>> 2". And also number Pi. And also the sentence that tells me what the
>> theory formulation is.
>
>Although it is a minor linguistic point, you're not using the terminology
>properly. "2" and "pi" are not really observables. An observable is a
>measurable property of a physical system, and it is always represented by
>an operator or "q-number", if I use Dirac's notation. See e.g.
>
> http://en.wikipedia.org/wiki/Quantum_mechanics
A number c is a particular case of a q-number. Multiplication by a
scalar c is a linear operator. It commutes with any other linear
operator. It has
a spectrum (consisting of one point, namely c), and it has spectral
decomposition - a trivial one c = c I, when I is the identity operator
(projection on the whole space)
>I did not understand your last sentence "...tells me what the theory
>formulation is" and why is it related to observables.
You see, if quantum theory is universal, it should apply to its own
description, right? If you think that your description is "good" and
only your description is good, then it is like this trivial operator
above with only one point in the spectrum - it is classical.
>> > Also, physics in general
>> >simply does not allow a complete decoupling of two sets of degrees of
>> >freedom; if some "degrees of freedom" are completely decoupled from the
>> >physical ones, then they must either be unphysical, or they must describe
>> >a different Universe.
>>
>> It depends on the definition of the universe. My definition may be
>> broader than yours.
>
>You can define your Universe to contain unphysical observables, for
>example the angels on the tip of the needle, but then your definition is
>unphysical, too, and you are not thinking in terms of science (physics).
At least I am not alone! See Haag above: "The use of concepts in the
theory which are not directly amenable to observation is neither
forbidden nor unusual. It seems unavoidable."
>> The literature shows how using the coupling to classical degrees of
>> freedom using the algebraic formalism you can simulate, in real time,
>> forming a particle track in a cloud chamber.
>
>You can simulate many things, so that the things look superficially
>similar to reality, but one can also prove that Nature does not work in
>certain way.
Each proof relies on assumptions. And our assumptions about the Nature
are often questionable.
>> I did not not show any other theoretical model that is able to do this
>> (including the random timings of events).
>
>All attempts trying to reveal "hidden variables" or "non-probabilistic
>explanations underlying quantum mechanics" can be showed to be
>incompatible either with the actual experiments (showing high
>correlations), or with experimentally verified physical principles such as
>the Lorentz invariance. This is what Bell's inequalities guarantee for us.
Lorentz invariance may happen be questionable one day (it already is).
If you read the paper above:
http://xxx.lanl.gov/pdf/quant-ph/9805011
you will see that there can be non-hidden variables and "probabilistic"
explanations, and yet alternative to the standard quantum theory. And
that is what Haag is also looking for.
>> If you know one, and can rund a simulation on your computer - please
>> let me know.
>
>I don't know what you exactly want to simulate, but be sure that if you
>think that you have a model that gives the actual outcome of the
>experiment instead of the probabilities, then it is a wrong model.
It gives actual outcome, and this outcome is based on probabilities. And
it produces a track, in real time. Standard quantum theory can not do
it.
>> >As you can see, I did not succeed in my attempt to find a true sentence in
>> >your first paragraph.
>>
>> Perhaps because of your prejudices, ...
>
>You can call them prejudices, but it might be more reasonable to call them
>"knowledge" and "flexible up-to-date objective thinking". ;-)
Let us see how flexible it will remain after reading the paper ....
>> and also because you do not know the
>> relevant stuff that is not in your domain in expertise?
>
>Which stuff do you exactly have in mind?
Stuff - like one in
http://xxx.lanl.gov/pdf/quant-ph/9805011
or
http://xxx.lanl.gov/abs/quant-ph/9812081
ark
P.S. R. Haag, "Objects, Events and Localization", in "Quantum Future.
From Volta and Como to the Present and Beyond", Ph. Blanchard and A.
Jadczyk Eds., Springer LNP 517, 1999
Arkadiusz Jadczyk
wrote:
> Tests of a
>theory are about the equations that result, not the guiding math
>principles. This kind of blanket rejection appears to assert the
>lessons of today will be the lessons of tomorrow.
Concerning the "tests of a theory" - I think we need more than just
equations. We need equations and their interpretation. This
interpretation can, to some extent, depend on the guiding mathematical
principles. Some theories and some equations may contain its own
interpretation more than some other theories. Think of the geometrical
theory of general relativity, with gauge invariance (with respect to
local diffeomorphisms). Gauge invariance leads to conservation laws and
to equations for singular matter distribution - which happen to be just
geodesics. Thus in Einstein's GR equations of motion follow from gauge
invariance. You do not have even specialize your field equations!
The theory seems to be quite robust - partly because of gauge
invariance. Whether it is good or bad - that is a different question.
And I agree that the lessons of today need not be the lessons for
tomorrow.
Paul M. Koloc
> Doug Sweeter wrote:
>>Isn't this [existence of a non-quantised field] a matter for experiment?
> It is very difficult to detect the individual gravitons - although it is
> not completely impossible. But certainly, the assumption is that the
> derivation of the existence of gravitons is correct, and unless you
> propose a working alternative theory (which is very unlikely), there is
> nothing to talk about. At this moment, the alternative - the attempt to
> claim that gravity does not have to be quantum at all - is ruled out by
> pure thought, and we don't even have to make an experiment.
Is this logical conclusion somehow entangled with the fact that our
thoughts are quantized?
Arkadiusz Jadczyk
wrote:
(snipped)
>> It may well happen that 1 can be quantized while 9 can be kept
>> classical.
>>
>>
>> ark
>
>Dear Ark,
>
>Could you please elaborate more on the above statement?
>especially the very last one.
The splitting is explained with details (even more details than really
necessary) in
http://www.cassiopaea.org/quantum_future/jadpub.htm#jad78b
and
http://www.cassiopaea.org/quantum_future/jadpub.htm#jad79e
The situation is similar to the case of electromagnetic field: you need
to know only the conformal structures (i.e. the light cones) to write
down the Lagrangian and field equations for the Maxwell field coupled to
space-time geometry. Similarly the volume form needs only to know about
the light cones - we can then write Lagrangian and field equations for
the volume. We can also try to quantize it - modulo standard problems
with zero-mass fields. For instance we take the causal structure of the
compactified Minkowski space (with the topology of (S^3 x S^1)/Z_2 )
and try to quantize the volume form alone (scalar density field) , with
this light cone structure.
Lubos Motl
> You don't really give me any hard evidence.
Mathematical inconsistency is as hard evidence that something is wrong as
you can get. What harder evidence can you imagine?
> When you say "there is no logically consistent way" do you really mean that
> we have a proof that says that classical and quantum cannot exist and it
> extends to the case of gravity coupling to other fields?
It is not a proof that would satisfy rigorous mathematicians, because the
statement itself is not sharply formulated. But using physics standards,
let me answer your question: yes, we can prove that it is impossible to
combine quantum physics with exactly classical physics in a single theory.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Only two things are infinite, the Universe and human stupidity,
and I'm not sure about the former. - Albert Einstein
Lubos Motl
> I know the oldest, most successful field theory - the Maxwell equations
> for EM - is gauge-invariant.
I did not say that a theory must have a nontrivial gauge invariance. I
said that it is only the gauge-invariant operators that have a physical
meaning. If you have a theory with no gauge invariance (no redundancy of
your degrees of freedom), then all operators are automatically
gauge-invariant because this condition becomes vacuous. If you work with a
gauge theory, e.g. Maxwell's theory, then only the gauge-invariant
observables, e.g. the electric and magnetic field, can be measured and
have invariant meaning (as opposed to the vector potential, for example,
which is not gauge-invariant and we have certain freedom to choose it).
> I have read that one should also think about general
> relativity as a gauge theory.
That's certainly right.
> There is an arbitrary vector field that can be added into the field
> equations in an analogous way that also has no physical effect. Is
> the gauge field a higher rank tensor in GR compared to EM because the
> field equations are also higher?
If you consider infinitesimal values of this vector field, it has a simple
interpretation: it gives you the infinitesimal change of the coordinates
delta x^m in the formula
x^m -> x'^m = x^m + delta x^m
The parameter of the gauge transformation has 1 vector index, and the
corresponding gauge field is the metric with 2 indices. In
electromagnetism, the parameters of the gauge transformations have 0
vector indices (scalars), and the corresponding gauge field (potential)
has 1 index (vector gauge field). There is 1 more index in gravity
because the spin of gravitons is 2 as opposed to spin 1 of photons.
In perturbative string theory, gravitons come from closed strings that
have 2 possible directions where the wave can move (left and right), while
open strings have 1 set of standing waves and heterotic strings only have
1 set of left-moving vector excitations of the right energy. This is the
explanation why open and heterotic strings can carry photons and gluons
with spin 1, while more general closed strings can carry spin 2 gravitons.
> The strong and weak field theories, modeled after EM, I suspect are
> also invariant under gauge transformation. We can thus conclude that
> at our current level of understanding which is deep, all known forces
> have fields that are invariant under gauge transformation.
Yes. The strong force has an SU(3) gauge invariance, and the
electromagnetic and weak force share an SU(2) x U(1) gauge invariance,
while gravity has the general diffeomorphism algebra. String theory
unifies all these symmetries, although all these symmetries become
*derived* objects in string theory.
> In my work, it is clear the equations are altered by a gauge
> transformation.
If your theory is not invariant under some transformations XY, then you
should not call them gauge transformations because they are not symmetries
of your theory.
> I would think that as long as I got back to the Maxwell
> equations and some sort of dynamic metric equation consistent with
> tests of GR, that would constitute a physical theory.
Gauge invariance is essential for all theories whose fundamental fields
have at least one vector index. It is because in quantum theory, this
vector index can be both spacelike as well as spacelike, and these two
possibilities lead to creation operators that creates the states with
positive and negative norm, respectively. Negative norm states imply
negative probabilities and they are unacceptable. The only way to make the
theory meaningful (how to kill the ghosts) is to show that the
negative-norm states are partially forbidden and partially decoupled (they
don't interact with the physical matter), and it can only be guaranteed by
an appropriate gauge symmetry.
> Tests of a theory are about the equations that result, not the guiding
> math principles.
If you can show that your theory is mathematically inconsistent - e.g.
because it has ghosts (negative norm states), then you don't even have to
do any experiments to rule it out.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Doug Sweetser
Hello Arnold:
Thanks for that summary on gauge invariance. I have printed it out, it
is a keeper :-)
Here was my odd thought for the week, and I would appreciate some
comments. Let's say one represents a spin-2 field with a symmetric
rank 2 tensor (that is what is normally done, no?). A symmetric tensor
will have an invariant trace. That trace could be viewed as a scalar
field. What physical thing would a scalar field represent? The scalar
field, arising from a symmetric tensor, would have nothing to do with
EM which needs an antisymmetric rank 2 tensor. Could this scalar field
be doing the same thing as the Higgs scalar field? If the trace was
zero, that would deal with a massless particle like the graviton. If
the trace was not zero, then it could provide a mechanism for mass
without destroying the symmetry of EM, a key feature of the Higgs
mechanism. It would be neat if the spin 2 field for gravity was this
directly related to giving mass to particles without breaking the
symmetry of the standard model, no Mexican hats needed :-)
doug
quaternions.com
Doug Sweetser
Hello Lubos:
I understood most of what you wrote, which is unusual for me :-)
I want to look at this point more closely:
> Gauge invariance is essential for all theories whose fundamental
> fields have at least one vector index. It is because in quantum
> theory, this vector index can be both spacelike as well as [time]like,
> and these two possibilities lead to creation operators that creates
> the states with positive and negative norm, respectively. Negative
> norm states imply negative probabilities and they are unacceptable.
> The only way to make the theory meaningful (how to kill the ghosts) is
> to show that the negative-norm states are partially forbidden and
> partially decoupled (they don't interact with the physical matter),
> and it can only be guaranteed by an appropriate gauge symmetry.
I was under the impression that this was a problem for a spin 1
particle. In the Gupta/Bleuler method for quantizing the EM
field, there are four modes: two transverse modes, a scalar mode and a
longitudinal mode. It is the scalar mode that has the negative norm
issue, requiring a supplementary condition which then neatly cancels
out the scalar and longitudinal modes, making the two modes forever
virtual.
One thing that bothers me is that a "scalar photon" is utter nonsense,
so I don't care that it has a problem with the norm. Photons flip
signs, the scalar mode won't, under a spatial inversion. Because basic
symmetry is wrong, the scalar mode cannot be spin 1. Now, if the
scalar mode was spin 2, am I correct to say that it would no longer
have the negative norm issue?
doug
quaternions.com
mike.james
So can we all (well nearly all) agree that:
1) It is desirable that gravity is quantised and there are many indications
that it should be
but
2) so far there is no PROOF that is HAS to be and no proof that a working
system with non-quantum gravity cannot work.
?
mikej
John Baez
In article <c0j90m$k88$1...@lfa222122.richmond.edu>,
<eb...@lfa221051.richmond.edu> wrote:
>In article <c0gdpg$ltu$1$8302...@news.demon.co.uk>,
>mike.james <mike....@infomax.demon.co.uk> wrote:
>>What I want to know is - is there anything that forces gravity to be a
>>quantum theory?
>The problem is that gravity couples to matter.
>Wald's book "General Relativity" goes into some detail about the
>problems that arise if you try to treat gravity classically while
>coupling it to quantum matter.
For what it's worth, there's a recent article on the arXiv
about precisely this issue. I haven't read it carefully,
but at the very least it has some good references:
http://www.arxiv.org/abs/quant-ph/0402092
Inconsistency of quantum--classical dynamics, and what it implies
Daniel R. Terno
Abstract: A new proof of the impossibility of a universal quantum-
classical dynamics is given. It has at least two consequences.
The standard paradigm "quantum system is measured by a classical
apparatus" is untenable, while a quantum matter can be consistently
coupled only with a quantum gravity.
Danny Ross Lunsford
John Baez wrote:
>>Wald's book "General Relativity" goes into some detail about the
>>problems that arise if you try to treat gravity classically while
>>coupling it to quantum matter.
>
> For what it's worth, there's a recent article on the arXiv
> about precisely this issue. I haven't read it carefully,
> but at the very least it has some good references:
>
> http://www.arxiv.org/abs/quant-ph/0402092
Assumes a Hamiltonian formulation for the "classical" theory. Since this
doesn't really exist for gravity (because it gives a special role to t),
this paper can't have much of a direct bearing on GR as such. He seems
to acknowledge this is his last sentence. Also, since the worst that can
happen is "energy non-conservation", and since energy conservation is
somewhat problematical for ordinary GR in any case, this adds more doubt
about the relevance of this paper to the issue.
Finally, I don't really understand the statement "couple gravity to
quantum matter". Is this supposed to be interpreted as allowing an
ensemble of curvatures corresponding to the quantum state?
-drl
Arnold Neumaier
> Arnold Neumaier <Arnold....@univie.ac.at> wrote:
>
>>This argument is far too simplistic. One could consider a unified theory
>>based on the equation
>> G = psi^* T psi,
>>where psi satisfies a Schroedinger equation in the so-called
>>functional Schroedinger picture of QFT.
>
>>why this leads to contradictions with logic or experiment.
> This idea was suggested by Moller and Rosenfeld about 40
> years ago (and possibly by Potier a decade earlier, but I
> haven't yet checked the reference). The first thing to
> note is that it makes the Schrodinger equation nonlinear
> -- the Schrodinger equation in a curved spacetime depends
> on the metric, which, in this formulation of ``semiclassical
> gravity,'' depends in turn on the state. It's not clear
> whether this nonlinearity is large enough to be ruled out
> by experiment; one of my students is working on this now.
> But it would at least require a drastic revision of quantum
> mechanics, since the principle of superposition would fail
> (and with it much of the rationale for describing observables
> as linear operators).
>
> A second problem was analyzed by Eppley and Hannah, Found.
> Phys. 7 (1977) 51. If gravity is ``classical,'' it seems
> likely that one can use gravitational measurements to
> simultaneously determine an object's position and momentum,
> violating the uncertainty principle. Eppley and Hannah
> show that if a gravitational measurement causes wave
> function collapse, one must either give up the uncertainty
> principle or momentum conservation, while if a gravitational
> measurement does not wave function collapse, gravitational
> interactions can be used to transmit information faster
> than light.
>
> None of this is conclusive, of course (unless my student
> or someone else can rule out the required nonlinearity
> experimentally). But it does mean that treating gravity
> classically is not simple, and would require a serious
> revision of quantum mechanics itself.
Indeed, these are also the objections given in section 2
of Carlip's review, http://www.arxiv.org/abs/gr-qc/0108040
quoted by Thomas Larsson.
But I consider both to be very weak objections.
The first, the need for nonlinear Schroedinger equations,
is not really an objection since such equations are
already in use in the QM of semiconductors - in agreement
with experiment. Essentially, eliminating the E/M field
from QED (and ignoring photons) and making a few-particle
approximation of the Fock space gives a nonlinear
Dirac equation with a retarded Coulomb
interaction, and in further approximations nonlinear
Dirac-Poisson and Schroedinger-Poisson equations,
the latter being used for numerical simulations of
semiconductor properties.
I had given some other references in
http://www.lns.cornell.edu/spr/2003-10/msg0054701.html
http://www.lns.cornell.edu/spr/2003-09/msg0054428.html
Based on this, the worst one could say against a nonlinear
Schroedinger equation is that it might be a sign that
the theory is not fundamental but an approximation of
something deeper. Nonlinearity has nothing to do with
representing observables by linear operators.
The second seems to me less a problem of classical gravity
+ quantum field theory than a problems of the foundations
of quantum measurement theory, since it argues with the collapse,
which causes already interpretation difficulties in
ordinary QM. Collapse in relativistic theories
(even without gravitation) is _very_ poorly understood.
Arnold Neumaier
Italo Vecchi
> >>>This argument is far too simplistic. One could consider a unified theory
> >>>based on the equation
> >>> G = psi^* T psi,
> >>>where psi satisfies a Schroedinger equation in the so-called
> >>>functional Schroedinger picture of QFT. Then both sides have the
> >>>same status.
> >
> > Italo Vecchi wrote:
> > You get an expectation value on the left side. How is the "classical"
> > Einstein tensor to be interpreted as an expectation value?
>
> The equation only says that it is equal to an expectation; which
> is nothing objectionable. In thermodynamics we also have observables
> which equl certain expectations.
>
An expectation value (both in classical and in quantum physics) is
defined with respect to a measurement outcome. The above equation
equates expectations relative to two different measurements. The
right-hand side is well defined. The left-hand side is not, since it
involves quantities for which, as you point out, no quantum
measurement theory is defined.
The problem is not just conceptual. We simply do not know what value
we should give to the left-hand side. How is time to be measured?
Reading from a clock? Which clock?
Best regards,
IV
Lubos Motl
> It all depends on what is your particular version of gravity theory ...
No. The statement that only gauge-invariant observables have physical
meaning and are measurable is independent of "your" particular version of
gravity theory, and in fact it holds for all theories.
> and what is your particular gauge group - if any.
For example, gravity in AdS5 x S5 can be described in various ways, e.g.
as SU(N) N=4 gauge theory in d=4 with large N - it has a different gauge
group (than the general covariance), but it is still true that only
gauge-invariant quantities have physical meaning.
> When I measure something using a coordinate system and miles rather
> than centimeters - the number measured clearly depends on the the
> choosen coordinate system. Now what is exactly gauge invariance for
> gravity, and whether we want to look at gravity this way or another -
> is up to us.
There might be different, dual descriptions of gravity that arise from
string theory, for example, but their existence is certainly not "up to
us", it is determined (or at least strongly constrained) by the objective
world of mathematics.
> There are different theories of gravity.
But it seems increasingly clear that there is only one consistent quantum
theory of gravity in d>3, and it is usually called string theory.
> A formalism without gauge-invariant observables is not a
> >physical theory.
>
> It depends on what is your interpretation of gauge invariance.
No, it does not. There is only one interpretation of gauge invariance, and
the only thing that the answer can depend on is whether one knows what
gauge invariance means or not.
> It depends on what is your definition and a realization of an
> "observable". You can have quantum theory without an observer and
> without observables!
You can have a quantum theory without explicit active observers, but you
certainly can't have a nontrivial quantum theory without observables. ;-)
Maybe you're confused by the similarity between the words "observer" and
"observable"? It's not the same thing and the latter can exist even
without the latter.
> > Gravity is associated with the general covariance, and
> >this symmetry implies that the gauge-invariant operators can't be local.
>
> One theory of gravity. There are other theories. Each have its good
> features and its bad features.
No, all low-energy descriptions of any relativistic theory of gravity
on any background share the general covariance gauge symmetry, if we want
to write them in a way that makes the equivalence principle manifest.
> Now: I said algebraic formalism.
Not really. You said "algbebraic formalism".
> This is not the same as "theory of local algebras". Think of Alain
> Connes.
OK, you must first decide what you really want to talk about, so far it is
totally fuzzy and you are jumping from one book to another and changing
your definitions along the way.
> We have serious problem with quantum theory, with understanding of the
> real meaning of the Planck constant, its meaning, ...
There does not exist any problem of this sort in real physics, it might be
someone's personal serious problem with quantum theory, but not an
objective one.
> Quantum fluctuations.
With a perfect EPR entanglement.
> You didn't say THAT. But if you MEAN that, then I agree.
Great.
> Roughly speaking I agree. But I want to calculate not only
> probabilities, but to understand how Nature is "choosing" the outcomes.
> What is the mechanism of the "potentials" that becomes "real."
There cannot be any such a mechanism as long as Lorentz invariance is
preserved at least locally. This is guaranteed by Bell's inequalities -
that are satisfied by any theory with such a "mechanism" but not by
Nature.
> As you know Hamiltonian does not always exist. Sometimes we have to use
> a representation of the algebra in which the time translations are NOT
> unitarily implemented.
Violation of unitarity of time translations means that the probability is
not conserved (or it can be negative) and the theory is inconsistent.
> Moreover, the term "observable" is, as you
> probably know, rather confusing.
No, it is not confusing.
> "It is often said that Quantum Theory is eminently successful, that its
> predictions are verified in countless cases and that no phenomenon has
> been found which contradicts it. Yet to this day, there remains some
> uneasiness about its status, some disagreement concerning its
> interpretation. This is not restricted to crackpots. Different camps of
> eminent scientists advance widely different opinions."
I disagree with this statement.
> "The use of concepts in the theory which are not directly amenable to
> observation is neither forbidden nor unusual. It seems unavoidable."
No problem. Many things can only be measured indirectly.
> You are certainly right, but you are looking only within a narrow
> formalism, without realizing that more fundamental changes
> are needed first! See e.g. quote from Haag above.
The changes that you want can't work, and we have theorems about it.
> Rudolph Haag, I suppose knows the principles of quantum mechanism.
I avoid such assumptions and judge any ideas by the available facts and by
logical reasoning, and this method leads to a very different conclusion
from yours.
> Now, superselection rules do exactly that: they introduce central
> quantities, commuting with ALL other quantities. They form an abelian
> sub algebra. They are "classical". Conservation of probability is
> possible with dynamical quantities of this type as well.
Whether or not you allow one to consider superpositions of the states from
different superselection sectors depends on your personal choice, and it
is often useful and realistic to forbid such superpositions. But in
principle you can allow even such superpositions, and it certainly does
not diminish the validity of any principle of quantum mechanics.
> "The Piecewise Deterministic Process Associated to EEQT"
> http://xxx.lanl.gov/pdf/quant-ph/9805011
It's not a paper that can be treated seriously, but I don't have time to
clarify 26 pages of misconceptions.
> A number c is a particular case of a q-number.
Dirac explicitly eliminated the c-numbers from the set of q-numbers, but
that's an unimportant terminological issue. If we say that "something is a
q-number", we always want to say that it is not just a "c-number".
> >I did not understand your last sentence "...tells me what the theory
> >formulation is" and why is it related to observables.
>
> You see, if quantum theory is universal, it should apply to its own
> description, right?
Not sure what you mean. Quantum theory is a physical theory, a way to
describe the real world, not a confusing philosophical circular argument
meant to describe itself. ;-)
> If you think that your description is "good" and
> only your description is good, then it is like this trivial operator
> above with only one point in the spectrum - it is classical.
Nice. ;-) That's right - the operator that answers the question whether
quantum mechanics is OK is more or less a c-number, and its value is YES. :-)
Well, yes, quantum mechanics also works with c-numbers sometimes. Did you
also mean this comment as a joke, or do you have something serious in mind?
> >You can define your Universe to contain unphysical observables, for
> >example the angels on the tip of the needle, but then your definition is
> >unphysical, too, and you are not thinking in terms of science (physics).
>
> At least I am not alone! See Haag above: "The use of concepts in the
> theory which are not directly amenable to observation is neither
> forbidden nor unusual. It seems unavoidable."
But if you use the angels on the tip of a needle, it's not quite the same
like calculating the properties of D-branes because the latter have
measurable consequences, at least indirect ones.
> >You can simulate many things, so that the things look superficially
> >similar to reality, but one can also prove that Nature does not work in
> >certain way.
>
> Each proof relies on assumptions. And our assumptions about the Nature
> are often questionable.
Well, I would not recommend anyone to rely on such assumptions, because
the conclusions are then almost guaranteed to be questionable themselves. ;-)
> Lorentz invariance may happen be questionable one day (it already is).
Feel free to question it, but don't expect others to consider your
conclusions seriously unless you have a point.
> you will see that there can be non-hidden variables and "probabilistic"
> explanations, and yet alternative to the standard quantum theory. And
> that is what Haag is also looking for.
Sad.
> It gives actual outcome, and this outcome is based on probabilities. And
> it produces a track, in real time. Standard quantum theory can not do
> it.
I have already commented on it.
All the best
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Lubos Motl
> I was under the impression that this was a problem for a spin 1
> particle.
You may call it [removing the unphysical polarizations of a photon] a
"problem", but in that case it is certainly an easy problem to solve, and
it has been solved 60 years ago. Since that time, people have found many
other ways to solve this "problem" that lead to the same physical results.
> In the Gupta/Bleuler method for quantizing the EM field, there are
> four modes: two transverse modes, a scalar mode and a longitudinal
> mode. It is the scalar mode that has the negative norm issue,
> requiring a supplementary condition which then neatly cancels out the
> scalar and longitudinal modes, making the two modes forever virtual.
This is a rough description, but the correct description requires a
different choice of the basis. A photon with 4-momentum "k" and a
polarization 4-vector "p" must satisfy "p.k=0" - the scalar product must
vanish as a consequence of the Gauss's law (div D=rho, and the related
vector equation with "j" on the right hand side), and the physical modes
where "p" is proportional to "k" are pure gauge, and they decouple (their
norm is zero anyway because "k" is null). These two things that we killed
could both have been chosen in a null direction, associated with the
photon's momentum and the conjugate direction, and the separation to the
"scalar (timelike)" and "longitudinal spacelike" mode is less natural.
> One thing that bothers me is that a "scalar photon" is utter nonsense,
> so I don't care that it has a problem with the norm.
There are only three reasons why it is nonsense. First, it has not been
seen experimentally. Second, it is not associated with a wave-like
solution of Maxwell's equations for E,B - equations that are confirmed by
experiments. The only new reason why it is nonsense in quantum mechanics
is that it would have a negative norm - which means that one could compute
negative probabilities. Such a problem would be the most serious one of
all - it is the only true theoretical inconsistency that we could derive
from such a theory; the only reason why we can abandon such a theory even
without doing an experiment. If you think that the "scalar photon" is
utter nonsense because of another reason, you are probably doing some
silly error in your reasoning.
> Photons flip signs, the scalar mode won't, under a spatial inversion.
> Because basic symmetry is wrong, the scalar mode cannot be spin 1.
Right. A scalar photon would not have spin 1 - scalars are *defined* to
have spin 0 - but it does not mean that spin 0 particles are "utter
nonsense"!
> Now, if the scalar mode was spin 2, am I correct to say that it would
> no longer have the negative norm issue?
Once again, the adjective "scalar" is a synonymum for "spin 0", and
therefore the assumption of your question can never be satisfied, and
consequently you can substitute anything for XY to the sentence "if the
scalar has spin 2, then XY". This is called "logic". ;-)
OK, I know what you wanted to ask.
The polarizations (components) of a tensor that have a negative norm are
those that contain an odd number of indices equal to 0 (the timelike
direction). The 00 component would have a positive norm, but again, this
is not the right basis to study spin 2 fields. General covariance and
Einstein's constraints can be used to kill all 10 polarizations of the
metric except for two - only the transverse traceless tensor survives
(with components g_{xy} and g_{xx}=-g_{yy} where the last identity follows
from the tracelessness, and where I considered the direction of the
graviton's motion to be "z").
Cheers,
Lubos
Arnold Neumaier
> On 18 Feb 2004, Arkadiusz Jadczyk wrote:
>>>Moreover, if we deal with a quantum theory, there can't be any true
>>>"classical observables".
>>
>>And who says so? And why?
>
>
> Everyone who knows the principles of quantum mechanics also knows that
> once we adopt quantum mechanical principles, they can't be reconciled with
> non-quantum principles in a single coherent theory. Quantum mechanics
> means that various possibilities are not uniquely predictable; only the
> probabilities are predicted, and the total probability must be conserved.
> Quantum evolution implies that essentially every possibility gets a
> nonzero probability and every attempt to suddenly introduce classical
> physics - where the evolution is deterministic, at least in principle -
> would destroy the picture. The probabilities would not sum to one, for
> example.
There is a large class of consistent quantum-classical theories, which have
classical phase space variables p,q and in addition a quantum wave function
psi and a Hamiltonian H(p,q) which is an operator-valued function of p and q.
The dynamics is given by
i hbar psidot = H(p,q) psi,
qdot = psi^* dH(p,q)/dp psi
pdot = - psi^* dH(p,q)/dq psi
psi^*psi is easily seen to be conserved and can be taken as =1.
psi^* H(p,q) psi is also conserved.
That this is consistent follows from the fact that it is obtained by
varying the real-valued action
S = integral dt (p qdot + i hbar psi^* psidot - psi^* H(p,q) psi).
This is used for real applications in quantum chemistry where one wants
to treat most degrees of freedom classically but selected degrees of
freedom (e.g., delocalized electrons) quantum mechanically. See, e.g.,
http://citeseer.nj.nec.com/229870.html
> You
> would have to say more concretely what sort of modification of QM you have
> in mind, and then we could see whether one can give you a more rigorous
> proof why it is impossible.
Try with the above!
>>In fact I can have a lot of classical observables. For instant "number
>>2". And also number Pi. And also the sentence that tells me what the
>>theory formulation is.
>
>
> Although it is a minor linguistic point, you're not using the terminology
> properly. "2" and "pi" are not really observables. An observable is a
> measurable property of a physical system,
pi is measurable as the quotient of areas of real plane figures.
> and it is always represented by
> an operator or "q-number",
pi is a well-defined operator that sends psi to pi*psi.
There are less trivial classical observables in respectable theories such as QED,
like the electron mass, the fine structure constant, or the Lamb shift.
They have indeed be measured to quite high precision, and the latter is widely
regarded as the master-piece of accuracy in comparing quantum field theory with
experiment.
> I don't know what you exactly want to simulate, but be sure that if you
> think that you have a model that gives the actual outcome of the
> experiment instead of the probabilities, then it is a wrong model.
There are experiments to determine the values of the electron mass,
the fine structure constant, and the Lamb shift.
Neither of these is a probability.
Traditional textbook measurement theory gives only a caricature of real
experiments. Scattering probabilities are by far not the only observable
items in our universe.
Arnold Neumaier
Urs Schreiber
> Quantum gravity requires one to fix the gauge (e.g. by light-cone gauge)
> if we want to talk about the finite time evolution. Covariantly, no such
> full gauge-fixing is possible, and the S-matrix - the evolution operator
> between the asymptotic past and the asymptotic future - is the only known
> exactly definable operator that replaces the Hamiltonian.
I am not sure if the following really pertains to what you are saying
here, but anyway:
I have once tried to cook up, for Type II strings, a covariant analog
of the light-cone gauge Hamiltonian.
The basic idea was to note that the anticommutator of the 0-mode of
the worldsheet supercurrent with a suitably chosen worldsheet fermion
yields essentially a Lie derivative on loop space (the configuration
space of the string) along a given timelike Killing vector.
From the Dirac-Ramond equation it then follows that the commutator
(instead of anti-commutator) of the 0-mode current with that fermion
constitutes the generator of superstring evolution along that Killing
vector field, i.e. the associated Hamiltonian.
This is morally just the generalization to strings of the construction
principle of the Hamiltonian for the ordinary Dirac particle.
The Hamiltonian thus obtained is available for the worldsheet
supersymmetric NSR string without any need to fix a worldsheet gauge,
nor even the need to have a lightlike Killing vector on target space.
It should hence be more generally applicable than light-cone gauge
formalism.
My original motivation for this construction was to attempt a
covariant perturbative calculation of curvature corrections to
superstring spectra around the pp-wave limit of AdS5 x S5. But in
order for this to be possible I'd have to figure out how to
incorporate Ramond-Ramond backgrounds in this formalism, which is well
known to be very difficult in the NSR framework (hep-th/9910145).
I am still hoping that this can be handled by combining the technique
of superconformal deformations that I use to incoprorate massless
NS-NS backgrounds in the above mentioned perturbation method
(hep-th/0401175) with I. Giannakis' idea to generate gravitino and RR
backgrounds by similar deformations of the BRST operator
(hep-th/0205219). I have talked about this with Arvind over at
http://golem.ph.utexas.edu/string/archives/000266.html, who raised
doubts due to the arguments in hep-th/9910145. But I. Giannakis
himself now told me that he thinks his approach to incorporate
gravitino and RR backgrounds in the covariant formulation is not
affected by the arguments in this paper. We'll see.
In any case, since AdS5 is still out of reach for me (greetings to
Niklas Beisert ;-) I have at least tested the covariant Hamiltonian
construction that I described above, as well as its perturbation
theory, in the toy example of the pp-wave limit of AdS3 x S3 x T4.
Since this background can be quantized exactly it can be used to test
methods that calculate curvature corrections to string spectra
(similar in spirit to the bosonic calculation in hep-th/0205015).
Following a referee's suggestion, this example calculation using a
covariant superstring Hamiltonian is now included in the second part
of the new version of hep-th/0311064. Unfortunately it turns out that,
while some aspects of the covariant Hamiltonian formalism are rather
elegant, others become quite technical and tedious. But of course the
same is true for light-cone formalism, as shown in hep-th/0307032.
Arkadiusz Jadczyk
On Mon, 23 Feb 2004 23:46:29 +0000 (UTC), Lubos Motl
<mo...@feynman.harvard.edu> wrote:
>> As you know Hamiltonian does not always exist. Sometimes we have to use
>> a representation of the algebra in which the time translations are NOT
>> unitarily implemented.
>
>Violation of unitarity of time translations means that the probability is
>not conserved (or it can be negative) and the theory is inconsistent.
Incorrect. Let me explain it to you why, because here is one particular
example where you have an evident hole in your physics and mathematics
education.
There is a whole branch of physics known as "Open systems dynamics"
or "Dissipative dynamics".
States of a given quantum system are described by density matrices -
trace class operators of trace=1. It is the condition trace=1 that
guarantees that probabilities add to 1. Gorini, Sudarshan and
Kossakowski, and also Lindblad, asked the following question: what is
the most general linear evolution of a quantum system that
preserves positivity and preserve the condition tr(rho(t))=1. The answer
was (with assuming a "complete positivity", which is somewhat stronger
and more stable than just "positivity"): "Lindblad's type generator".
The time evolution is then, in general, irreversible. Pure states
transform into mixed states. Yet probabilities are conserved. The
theories with dissipative dynamics are consistent.
ark
Gerard Westendorp
>>Wald's book "General Relativity" goes into some detail about the
>>problems that arise if you try to treat gravity classically while
>>coupling it to quantum matter.
The original reason for introducing quantum theory was to avoid
the uv-catastrophe in electromagnetic blackbody radiation.
I wonder if anything like gravitational blackbody radiation makes
sense. Like a 4K background radiation you could have a background
radiation of gravitons that are left over from the big bang.
If anything like thermal equilibrium were to apply to gravitons,
perhaps at a time near the big bang, classical theory would have
a problem. But maybe quantum theory also would...
Gerard
Lubos Motl
On Mon, 23 Feb 2004, Arnold Neumaier wrote:
> The dynamics is given by
> i hbar psidot = H(p,q) psi,
> qdot = psi^* dH(p,q)/dp psi
> pdot = - psi^* dH(p,q)/dq psi
Understood - you just want to make your quantum mechanical Hamiltonian
depend on some classical background given by the variables p,q. This
itself would be OK as long as this classical background would be
independent of the quantum events that happen on it.
However: A totally confusing aspect is that your classical observables q,p
are also supposed to depend on the "averaged" wavefunction, aren't they?
Does it mean that you think that the classical motion of p,q can
immediately "see" the collapse of the wavefunction when it's measured?
That would certainly violate locality and relativity. The "collapse" of
the wavefunction must have no physical consequences, otherwise the theory
is unphysical. For example, we can always imagine that the cat inside the
box remains in a superposition of the macroscopic states, and only
perform the collapse when we want to perceive what we see.
If equations like that could operate in Nature, you could materially -
without probabilities - measure, without repeating anything, some aspects
of the wavefunction itself, which would make it material. In other words,
you would have to substitute these equations by a materialist mechanism of
the wavefunction collapse, otherwise the equations are incomplete (because
the details of the collapse directly affect the classical variables p,q).
The classical observables p,q in these equations certainly do depend on
whether the wavefunction is thought of as collapsed or not (unlike the
case of orthodox quantum mechanics where the probabilities are the only
measurable quantities, and they don't depend on whether or not you have
already projected out the unrealized possibilities).
Therefore you must say exactly when the collapse happens, and according to
what rules it happens. Be sure that you won't get any consistent picture.
These equations can't be taken seriously because whoever wrote them
pretends that the wavefunction is a real "wave" in the space. It's not. It
only gives the probabilites. If the wavefunction for a particle has a a
small lump 10^{-7} at Mars (and the rest is in California), it does not
mean that such a wavefunction behaves as a particle that is 100 kilometers
from the Earth even though the average position might be there.
> varying the real-valued action
> S = integral dt (p qdot + i hbar psi^* psidot - psi^* H(p,q) psi).
This is not real, you've forgotten various complex conjugate terms. But
even if you complete them, what sort of action is it? Does it explicitly
depend on p,q,psi? Why don't you, at least, integrate the psi-dependent
terms over x?
> > Although it is a minor linguistic point, you're not using the terminology
> > properly. "2" and "pi" are not really observables. An observable is a
> > measurable property of a physical system,
>
> pi is measurable as the quotient of areas of real plane figures.
This is not the physical meaning of the word "measurement". A measurement
is something that can give you, a priori, at least two different outcomes.
It is much like "elections".
> > and it is always represented by
> > an operator or "q-number",
>
> pi is a well-defined operator that sends psi to pi*psi.
Yes, it is an *operator*, but we just don't count it as an *observable*.
> There are less trivial classical observables in respectable theories
> such as QED, like the electron mass, the fine structure constant, or
> the Lamb shift.
According to standard terminology, these constants are *not* observables
in the field theories; they are constant parameters. By observables we
mean the operators that can have various eigenvalues, i.e. the q-numbers.
> > I don't know what you exactly want to simulate, but be sure that if you
> > think that you have a model that gives the actual outcome of the
> > experiment instead of the probabilities, then it is a wrong model.
>
> There are experiments to determine the values of the electron mass,
> the fine structure constant, and the Lamb shift.
> Neither of these is a probability.
No way. ;-) In the quantum world, the only thing you can measure are the
probabilities, by repeating the same experiment many times, and the
measurement of the masses, fine structure constant and other parameters
are done using the same experiments whose outcome is uncertain and
probabilistic - because this is the case of *all* events in this Universe.
It is only because the experiments are smartly designed so that they can
"zoom in" the various graphs, and repeat the same experiment many times so
that one gets a good idea about the probability distribution of a certain
observable; in other words, the classical approximation becomes very
useful in various limits. The mass is determined from this probability
distribution (i.e. as the position of the peak of a cross section, or from
a path of the electron that can be approximated by a classical path), and
once again, this probability distribution can only be "measured" by
repeating the same experiment many times.
If you think that a single experiment can give you a fundamentally
non-probabilistic and exact answer about a question (in the realm affected
by quantum physics), you're missing the whole point of quantum mechanics.
> Traditional textbook measurement theory gives only a caricature of real
> experiments. Scattering probabilities are by far not the only observable
> items in our universe.
If you perform and interpret your experiments properly, they can be
reduced to a calculation of some quantum amplitudes. Everything else -
like "directly seeing the classical outcome of an experiment" - is just an
approximation. The more one focuses on the microworld and the elementary
particles, the more complete and usable description one gets by studying
the S-matrix.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Lubos Motl
> I am not sure if the following really pertains to what you are saying
> here, but anyway:
I don't think so: I was writing about fixing the general covariance in
spacetime.
> I have once tried to cook up, for Type II strings, a covariant analog
> of the light-cone gauge Hamiltonian.
Which Hamiltonian do you exactly mean, P^0 in spacetime, P^- in spacetime,
or some worldsheet Hamiltonian? In your construction, at least so far, I
misunderstood not only your answer, but even the question that you were
trying to solve, sorry. Can you formulate the question that you were
attempting to answer? ;-)
Urs Schreiber
news:Pine.LNX.4.31.040224...@feynman.harvard.edu...
> Which Hamiltonian do you exactly mean, P^0 in spacetime, P^- in spacetime,
> or some worldsheet Hamiltonian? In your construction, at least so far, I
> misunderstood not only your answer, but even the question that you were
> trying to solve, sorry. Can you formulate the question that you were
> attempting to answer? ;-)
Consider Type II strings in an arbitrary stationary background. Fix a
timelike Killing vector k on that spacetime. Let |psi> be a state of the NSR
string. From the super Virasoro constraints derive an equation of the form
i d_k |psi> = H |psi>
where d_k is the derivative along k and H is an operator on the OCQ Hilbert
space which is independent of the parameter along k ('time') as well as of
d_k .
Just like the Hamiltonian in the light cone gauge formalism measures the
string's energy with respect to a lightlike Killing vector on target space,
the above H measures the string's energy along a timelike Killing vector on
target space. But no worldsheet gauge needs to be fixed.
Lubos Motl
On 24 Feb 2004, Arkadiusz Jadczyk wrote:
> Incorrect. Let me explain it to you why, because here is one particular
> example where you have an evident hole in your physics and mathematics
> education.
Feel free to assume that I don't understand open system dynamics - unless
you realize that such an assumption is very unreasonable, which you
certainly should. At any rate, you're not adding anything relevant to the
discussion. The fundamental equations describing the whole system (such as
the Universe), not just an open system in it, must be unitary to preserve
the probabilities, and tracing over the "external" degrees of freedom is
just a pragmatic subjective approximation that amounts to forgetting a
part of the information about the system, and making less precise
predictions. The evidence is now very strong that the strong form of
unitarity must hold even in the black hole evolution.
Even in the case of the black holes, where the evolution of pure states
into mixed states had the biggest possible support, no one has been able
to present a working microscopic theory that could modify quantum
mechanics in this way. On the other hand, a big progress has been done
towards formulating black hole dynamics in the absolutely standard and
orthodox quantum mechanical framework.
Arnold Neumaier
Lubos Motl wrote:
> On Mon, 23 Feb 2004, Arnold Neumaier wrote:
>
>>The dynamics is given by
>> i hbar psidot = H(p,q) psi,
>> qdot = psi^* dH(p,q)/dp psi
>> pdot = - psi^* dH(p,q)/dq psi
>
>
> Understood - you just want to make your quantum mechanical Hamiltonian
> depend on some classical background given by the variables p,q. This
> itself would be OK as long as this classical background would be
> independent of the quantum events that happen on it.
>
> However: A totally confusing aspect is that your classical observables q,p
> are also supposed to depend on the "averaged" wavefunction, aren't they?
In a sense, yes, though my interpretation is a little different.
This dynamics treats p, q, psi as classical observables, though because of
a phase gauge symmetry, only expressions involving phi and phi^* in pairs are
physical. That psi^* A psi can be considered as an average is irrelevant
for the mathematical consistency.
> Does it mean that you think that the classical motion of p,q can
> immediately "see" the collapse of the wavefunction when it's measured?
How do you mean? The model assumes no collapse at all. The dynamics
is infinitely differentiable, and no measurement is involved.
> That would certainly violate locality and relativity.
I didn't claim the model above to be local or relativistic;
only as an example that classical and quantum can coexist consistently.
It is a model of nonrelativistic quantum-classical motion as compatible
with (certain approximations of) quantum chemistry, and in accord with
experiment.
Covariant versions of this (which are less easy to understand, but
there are some) are Poincare-invariant, hence cannot violate relativity.
Locality is indeed violated, but we know since the Aspect experiments that
full locality is inconsistent with quantum mechanics anyway.
> If equations like that could operate in Nature, you could materially -
> without probabilities - measure, without repeating anything, some aspects
> of the wavefunction itself, which would make it material.
The equations I mentioned do operate in Nature; they can be derived as
a certain scaling limit from the Schroedinger equation, as proved
rigorously in the paper quoted (I believe; maybe it was a different paper).
> In other words,
> you would have to substitute these equations by a materialist mechanism of
> the wavefunction collapse, otherwise the equations are incomplete (because
> the details of the collapse directly affect the classical variables p,q).
I don't see why the equations should be incomplete.
The equations as they are completely define a conservative
classical dynamical system in which psi^*H(p,q)psi plays the role of a
Hamiltonian. That the Schroedinger dynamics can be viewed as a classical
symplectic system is well-known, see e.g. Marsden and Ratio's book on
mechanics and symmetry.
> The classical observables p,q in these equations certainly do depend on
> whether the wavefunction is thought of as collapsed or not (unlike the
> case of orthodox quantum mechanics where the probabilities are the only
> measurable quantities, and they don't depend on whether or not you have
> already projected out the unrealized possibilities).
Yes.
> Therefore you must say exactly when the collapse happens, and according to
> what rules it happens. Be sure that you won't get any consistent picture.
Never, just as with the Schroedinger equation. Then everything is consistent.
> These equations can't be taken seriously because whoever wrote them
> pretends that the wavefunction is a real "wave" in the space.
Not at all. In some applications, psi is a vector in a finite-dimensional
vector space, which has no spatial interpretation. In others, it is a
function of k real arguments, where k is the number of classical degrees
of freedom which are treated quantummechanically, and as k can take any value
there is again no relation to space.
Chemists use them to compute charge distributions and equilibrium positions
of molecules such as small polypeptides.
> It's not. It only gives the probabilites.
And it gives ensemble expectations. These are measurable classical as
thermodynamic quantities. That's of interest to chemists.
>>varying the real-valued action
>> S = integral dt (p qdot + i hbar psi^* psidot - psi^* H(p,q) psi).
>
>
> This is not real, you've forgotten various complex conjugate terms.
Right. The corrected formula is below.
> But
> even if you complete them, what sort of action is it? Does it explicitly
> depend on p,q,psi? Why don't you, at least, integrate the psi-dependent
> terms over x?
In full,
S(p,q,psi) = Re integral dt L(t),
where
L(t) = p(t) qdot(t) + i hbar psi(t)^* psidot(t) - psi(t)^* H(p(t),q(t)) psi(t).
> Why don't you, at least, integrate the psi-dependent terms over x?
Because there is nowhere an x. In the simplest case, psi is a vector
with just two (time-dependent) components; the Hilbert space is that of
a single qubit, and H is a Hermitian 2x2-matrix. In a more complicated
case, psi might be a Schwartz function on some manifold, H a linear operator
on the vector space of Schwartz functions, with an integral defined by a
volumes form on the manifold.
>>>Although it is a minor linguistic point, you're not using the terminology
>>>properly. "2" and "pi" are not really observables. An observable is a
>>>measurable property of a physical system,
>>
>>pi is measurable as the quotient of areas of real plane figures.
>
> This is not the physical meaning of the word "measurement". A measurement
> is something that can give you, a priori, at least two different outcomes.
Well, if you try to measure it, you get each time slightly different answers,
depending on the accuracy with which you prepare the system. That theory
predicts you always get the same value is just like in the attempts to measure
the fine structure constant.
>>pi is a well-defined operator that sends psi to pi*psi.
>
> Yes, it is an *operator*, but we just don't count it as an *observable*.
I don't know what you call an observable, but according to the standard
textbook explanations, self-adjoint Hermitian operators are called observables.
This clearly includes pi.
>>There are less trivial classical observables in respectable theories
>>such as QED, like the electron mass, the fine structure constant, or
>>the Lamb shift.
>
> According to standard terminology, these constants are *not* observables
> in the field theories; they are constant parameters.
I have never seen the Lamb shift referred to as a constant parameter of
QED; it is a nontrivial theoretical prediction, and it certainly is
observable in the sense that it can been measured.
> By observables we
> mean the operators that can have various eigenvalues, i.e. the q-numbers.
Well, some observables simply have a single eigenvalue only.
>>There are experiments to determine the values of the electron mass,
>>the fine structure constant, and the Lamb shift.
>>Neither of these is a probability.
>
>
> No way. ;-) In the quantum world, the only thing you can measure are the
> probabilities,
No, in the quantum world, nothing at all can be measured. No one ever has
been able to produce a consistent theory of quantum measurement from within
a quantum universe.
But in the real world, the only world that exists, physicist measured and
measure a lot of things, and probabilities are just a small part of what
people are interested in and are able to measure.
For example, the collapse of the world trade towers a few years ago was a
real physical event, of which many records are available now to certify
it.
Probabilities are only measured for systems that are dynamically so
fragile that uncontrollable details influence the outcome significantly.
> If you think that a single experiment can give you a fundamentally
> non-probabilistic and exact answer about a question (in the realm affected
> by quantum physics),
No answer about nature is exact, of course. But answers can be highly
accurate. All of nature is determined quantum physics, and we can
measure many things to quite high precision, given good enough equipment.
What is a 'single' experiment? Experiments do not have an atomic structure
so that you could identify the smallest units.
One may define the whole CERN set-up as a single experiment to determine
the mass of the top quark (and other things). One can also split it up into
smaller experiments. But a 'single' experiment?
Observing the track of a particle in a wire chamber is already an art in
itself - it takes a lot of physics and numerical analysis to 'measure'
the 4-momentum of the particles produced in a collision experiment;
thus it can hardly be viewed as a 'single' experiment. Is perhaps the
detection of the particle passing a particular wire a 'single' experiment?
Hmmm, it involves a nontrivial detection mechanism that again is not
simple. Almost any of our modern high precision measurements is the result
of a complicated array of interlocking aspects...
> you're missing the whole point of quantum mechanics.
Well - I probably published more than you on quantum mechanics close to
experiment, and you want to teach me the whole point of quantum mechanics?
I know as much about quantum mechanics as you, and probably more.
And I can see behind the scenes of the official doctrine. The textbook
dogma is not the only way to view quantum mechanics, and it is not
the best way, in my opinion.
>>Traditional textbook measurement theory gives only a caricature of real
>>experiments. Scattering probabilities are by far not the only observable
>>items in our universe.
>
>
> If you perform and interpret your experiments properly, they can be
> reduced to a calculation of some quantum amplitudes. Everything else -
> like "directly seeing the classical outcome of an experiment" - is just an
> approximation. The more one focuses on the microworld and the elementary
> particles, the more complete and usable description one gets by studying
> the S-matrix.
The S-matrix is usable only for studying scattering experiments and the
observables that are directly connected to it. If QED were only about the
S-matrix, it would be of little value. But quantum optics studies lots of
very interesting things which make very little contact with the S-matrix.
It pays to keep contact to the real world and not be lost in
abstractions...
Arnold Neumaier
Lubos Motl
> From the super Virasoro constraints derive an equation of the form
> i d_k |psi> = H |psi> (###)
That's OK, the super-Virasoro constrain does look as a Dirac equation. But
what you wrote above - (###) - is simply an equation of motion. First of
all, you don't gain anything if you pick a priviliged component of the
derivative d_k and move it to the opposite side of the equation that the
rest (H). The Virasoro and super-Virasoro constraints are *covariant*
equations, and the only "virtue" of your way of writing is to obscure
that it is covariant.
Second of all, and more importantly, your "gauge" - even though it is not
gauge, it is just writing the equations in a special way - does not solve
anything that the light-cone gauge solves. In the light-cone gauge, one
identifies the worldsheet time "tau" with a spacetime light-like
coordinate (X^+), and removes the longitudinal and timelike components of
the gauge fields, graviton (metric) and everything else. You have no such
identification. You said explicitly:
> But no worldsheet gauge needs to be fixed.
Well, if you don't fix the worldsheet gauge symmetry, then it's not fixed!
;-) Consequently, you can still reparameterize the worldsheet coordinates
in any (conformal) way. If you consider wave packets - particles with the
exp(i.p.x) profile - the worldsheet conformal symmetry generators, acting
on a physical state, can give you an unphysical state - for example the
photon with a polarization vector proportional to the momentum, or
corresponding components of the graviton.
I think that you must see totally clearly that you have not done anything,
and therefore you could not have solved anything either. In the covariant
treatment - and you are still doing the standard covariant quantization -
there are D^2 components of the metric; B-field; and the dilaton, and just
like the worldsheet conformal symmetry is unfixed, also the spacetime
gauge symmetries are unfixed.
Let me say it for the case of the graviton: the "Dirac" equation that you
wrote - namely the Virasoro constraints - constrain the possible vectors
in the first-quantized stringy Hilbert space. But they don't constrain it
completely: any vector in the Hilbert space, describing e.g. a graviton
with a momentum and some polarization tensor, can be transformed via the
spacetime gauge invariance, e.g. the general covariance, to get another
solution. It's just like in usual classical GR - the equations of motion
can't tell you any priviliged definition of the time coordinate "t", until
you gauge-fix the gauge invariance - which you have not.
You start with a definition of "t", but the pure gauge fluctuations of the
metric, which are still allowed solutions, have the effect of changing
this definition of "t" arbitrarily. You have not done anything, you have
not solved anything, and the action of the spacetime Hamiltonian on the
physical spectrum is simply not well defined; you can always add zillions
of BRST-exact (pure gauge) states to your state which are equivalent to
redefining "t" arbitrarily.
Lubos Motl
On 24 Feb 2004, Arnold Neumaier wrote:
>> LM: Therefore you must say exactly when the collapse happens, and
>> according to what rules it happens. Be sure that you won't get any
>> consistent picture.
>
> Never, just as with the Schroedinger equation. Then everything is consistent.
OK, if you never make any "collapse", then your picture has absolutely
nothing to do with quantum mechanics. Your wavefunction is obviously a
real, classical wave in space - because it directly affects the classical
observables p,q - nevertheless you don't allow this wavefunction to
collapse. So the wavefunction of an electron will spread indefinitely: an
electron will be a real, classical and deterministic wave that will be
increasingly diluted in space. This is, of course, not what experimentally
happens, and such a picture has nothing to do with quantum mechanics. You
just created another purely classical theory, described by some classical
differential equations with purely classical interpretation, and it has no
capacity to agree with the real phenomena. It's only the Greek letter
"psi" that you used for one of your classical fields, and perhaps this
letter confused you and made you think that your theory has something to
do with quantum physics. It does not. What you offer us is a standard 18th
century-style classical theory. (Now we live in the 21st century, by the
way.)
In quantum mechanics, the wavefunction is not a real wave, and it can't be
directly measured. It only encodes the probabilities that the outcome of a
measurement will be X or Y. Once we know which outcome became real, we can
forget about the part of the wavefunction that was not realized, and
continue with the part of the wavefunction that was measured. This event
is sometimes referred to as the "collapse", but in reality, it is just a
psychological step because the measurement only increased our level of
knowledge about the system. We don't need to imagine that the collapse is
real, and we don't need to say when and how it precisely happens. The only
meaningful prediction of quantum mechanics is the probability of one
consistent history out of some ensemble of alternatives, and everything
else can be considered unphysical. Nevertheless, once you use in terms of
the wavefunction, at least the "psychological collapse" is totally
necessary, otherwise all wavefunctions in the Universe would be just
spreading and diluting indefinitely.
> Chemists use them to compute charge distributions and equilibrium positions
> of molecules such as small polypeptides.
They can compute anything, but what you're doing is a completely incorrect
interpretation of the wavefunction. Well, it is clear that the chemists
also often misinterpret the physical objects and they use many simplified,
approximate ways to think about a physical system. But this is a physics
newsgroup, and therefore the low-brow and inaccurate ways to teach
chemists how to imagine what a wavefunction means should not be enough.
I don't want to continue the arguments whether "pi" should be counted as
an observable, and so on.
Urs Schreiber
> On 24 Feb 2004, Urs Schreiber wrote:
>
> > From the super Virasoro constraints derive an equation of the form
> > i d_k |psi> = H |psi> (###)
>
> That's OK, the super-Virasoro constrain does look as a Dirac equation. But
> what you wrote above - (###) - is simply an equation of motion. First of
> all, you don't gain anything if you pick a priviliged component of the
> derivative d_k and move it to the opposite side of the equation that the
> rest (H). The Virasoro and super-Virasoro constraints are *covariant*
> equations, and the only "virtue" of your way of writing is to obscure
> that it is covariant.
>
> Second of all, and more importantly, your "gauge" - even though it is not
> gauge, it is just writing the equations in a special way - does not solve
> anything that the light-cone gauge solves.
Yes, so far all this is a way of rewriting parts of the full set of
constraints in a special way. This rewriting by itself does not 'solve'
anything yet, true.
The point of the rewriting is that for some questions the rewritten formula
(###) is the prescription for how to extract certain informations about the
string.
One application of (###) that I mentioned is the perturbative calculation of
string spectra:
Say you want to compute the energy of string states on some non-exactly
solvable background B = B^(infty) to which corresponds an operator H =
H^(infty) in (###). Since solving the constraints for B is hard, consider a
limit B^(0) of B on which the string is exactly solvable (e.g. a Penrose
limit). The question then is: What is the first order shift in the energy of
the exact string states on B^(0) as we take higher order corrections
B^(0) -> B^(1) into account? The answer is given by formula (###), which
says that the first order shift of the energy is <H^(1)>, where the
expectation value is taken with respect to the exact states on B^(0). So the
operator H (or its first order perturbation in this case) contains the
answer to a certain question. Finding H in (###) for general backgrounds is
nontrivial.
Compare the basic idea to the toy model of the Dirac particle:
The constraint is, say,
gamma^mu (partial_mu - iA_mu)|psi> = 0 .
and can equivalently be rewritten as
i partial_0 |psi> = [ gamma^0 gamma^j (i partial_j + A_j) - A_0]|psi> =:
H|psi> .
Even though this is just a rewriting of the original equation and solves
nothing by itself, the latter form is the right starting point to answer
some interesting questions. For instance if the background field A_mu is
perturbed, we would like to know how the energy of the particle shifts. The
answer is of course given by the expectation value <H^(1)>. For questions
like these it is useful to know the form of the operator H.
Kefka G
>On 18 Feb 2004, Arkadiusz Jadczyk wrote:
[snip]
>> I did not not show any other theoretical model that is able to do this
>> (including the random timings of events).
>
>All attempts trying to reveal "hidden variables" or "non-probabilistic
>explanations underlying quantum mechanics" can be showed to be
>incompatible either with the actual experiments (showing high
>correlations), or with experimentally verified physical principles such as
>the Lorentz invariance. This is what Bell's inequalities guarantee for us.
>
[snip]
This is a very minor point, probably irrelevant to this particular discussion,
but for completeness it should be noted that Bell's inequalities actually
guarantee us a great deal less than they are usually assumed to. In fact, they
do not rule out all reasonable hidden variable theories, since they assume a
very strict sort of locality that goes much further than requiring no signal to
be transmitted faster than the speed of light. In fact, Bell himself spent a
great deal of time driving the point home that his inequalities are less
powerful than most physicists interpret them to be, mainly since the
assumptions are too restrictive. I don't know much about this stuff myself, so
I will defer to those who are more expert than I - for more information on this
see "Speakable and Unspeakable in Quantum Mechanics" by Bell (there's some
other interesting stuff in there, too, so it's a worthwhile read), or for an
example of a theory that slips through the assumptions required to obtain
Bell's inequality, look at Bohm's causal interpretation of quantum mechanics,
which reproduces all the predictions of the usual theory while allowing us to
define particle trajectories and whatnot. It's admittedly somewhat out there,
and to me it goes against my aesthetic views on quantum theory, but as Bohm
points out, I'm a child of the Copenhagen interpretation and it's been bred
into me to hate hidden variables, so I could be wrong. I think it's at least
something people should be familiar in passing with - I'm pretty sure Bohm
explains the theory in "The Undivided Universe," although it's been a few years
and I don't have the book, so perhaps someone who does will correct/verify
this.
>> If you know one, and can rund a simulation on your computer - please
>> let me know.
>
>I don't know what you exactly want to simulate, but be sure that if you
>think that you have a model that gives the actual outcome of the
>experiment instead of the probabilities, then it is a wrong model.
Again, the Bohm model gives the actual outcome of any given experiment, and
these outcomes are shown to be statistically identical to the usual predictions
of QM. If I remember, this is actually fairly simple to show (a few lines,
even), but again, I haven't read this in several years, so I don't remember the
equations well enough to do them here. In any case, the statement that any
causal model necessarily gives wrong answers is just incorrect. The reason
Bell's inequality doesn't apply here is that the Bohm model is necessarily
non-local, even though no signal can be sent faster than light. I don't know,
however, if anyone has ever generalized this to a fully relativistic theory. I
know that the fundamental description (the quantum potential? I don't remember
so well...) cannot be relativistically invariant, but if I remember correctly,
the physics itself can be, even if some of the variables underlying it have
nonlocal dependence on particles. Anybody know more about this?
More on topic, does anyone know if there has ever been an effort to put quantum
gravity into a Bohm-type setting? I don't even know if we can extend Bohm's
ideas to the usual quantum field theories, so perhaps it would be too Herculean
a task to expect gravity to succumb when even E/M is too difficult, but I don't
know if that's the case. Any enlightenment would be appreciated.
Still too naieve to keep my mouth shut,
Eric
eric.jordan@the school where G. Bush Jr. got worse grades than I do.edu
(Spammers already have my AOL name, they won't get this one, too!)
Arnold Neumaier
>>Arkadiusz Jadczyk wrote:
>
>
>>>A nontrivial coupling of a classical and a quantum system needs to
>>>satisfy certain consistency requirements. These have been studied
>>>and described in the literature.
>>
>
> The author of http://www.arxiv.org/abs/quant-ph/0402092 claims that
> such a coupling is inconsistent. I am not competent to judge how
> watertight his proof is.
>
Any no-go theorem rests on assumptions. The assumptions in that paper
(and a number of others on the arxiv) are very restrictive.
They do not cover the class of quantum-classical theories used
in real applications, e.g., http://citeseer.nj.nec.com/229870.html
I described this class in more detail in another mail in this thread,
which did not yet appear on s.p.r.
Thus I think quant-ph/0402092 is irrelevant.
Arnold Neumaier
Arnold Neumaier
> Here was my odd thought for the week, and I would appreciate some
> comments. Let's say one represents a spin-2 field with a symmetric
> rank 2 tensor (that is what is normally done, no?). A symmetric tensor
> will have an invariant trace.
No. In general, only tensor of type (s,s) (s covariant and s contravariant
indices) have a trace. To regard a symmetric rank 2 tensor as a tensor of
type (1,1) you need to have a metric so that you can change the character
of the indices. If you apply this to the metric itself, you get the
identity (1,1)-tensor delta^mu_nu, which has constant trace 4, hence gives
no information.
> It would be neat if the spin 2 field for gravity was this
> directly related to giving mass to particles without breaking the
> symmetry of the standard model, no Mexican hats needed :-)
Well, all the simplistic ideas have been tried for long, and found not
working. You cannot pull out relevant nontrivial structure out of nowhere,
using simple magic (sorcerer's hat in place of Mexican hat).
So if a simple idea seems to work without any effort, it is extremely
likely that the reason for this is a basic mistake in the interpretation.
To a large extent, successful science is the result of a self-critical
attitude. I think no one ever earned a Nobel prize for nearly trivial
observations.
Success usually requires a very thorough understanding of everything
already existing in the area of one's activity.
Arnold Neumaier
Italo Vecchi
> You can have quantum theory without an observer and
> without observables!
>
Maybe, but then it's not physics. A physical theory is about
measurement outcomes. You need observables and observers to get
measurement outcomes.
iv
Paul M Koloc
Lubos Motl wrote:
> On Tue, 17 Feb 2004, mike.james wrote:
>>.. .
>>When you say "there is no logically consistent way" do you really mean that
>>we have a proof that says that classical and quantum cannot exist and it
>>extends to the case of gravity coupling to other fields?
>
> It is not a proof that would satisfy rigorous mathematicians, because the
> statement itself is not sharply formulated. But using physics standards,
> let me answer your question: yes, we can prove that it is impossible to
> combine quantum physics with exactly classical physics in a single theory.
Reality is the master of physics, but not mathematics. Classical
physics does not apply itself to truly exact solutions, and it does not
bother to deal with reality on the quantum mechanical scale. So, it
seems to me you are over reaching when you use the term "exactly
classical physics". Perhaps when viewed through a Gaussian filter, your
mathematical insistence will not be so exactly certainly provable.
> ______________________________________________________________________________
> E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
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> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> Only two things are infinite, the Universe and human stupidity,
> and I'm not sure about the former. - Albert Einstein
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Arnold Neumaier
>>Wald's book "General Relativity" goes into some detail about the
>>problems that arise if you try to treat gravity classically while
>>coupling it to quantum matter.
>
>
> For what it's worth, there's a recent article on the arXiv
> about precisely this issue. I haven't read it carefully,
> but at the very least it has some good references:
>
> http://www.arxiv.org/abs/quant-ph/0402092
>
> Inconsistency of quantum--classical dynamics, and what it implies
>
> Daniel R. Terno
>
> Abstract: A new proof of the impossibility of a universal quantum-
> classical dynamics is given.
The assumptions are far too strong - the proof excludes well-known,
consistent quantum-classical dynamical systems used in molecular
modeling, and thus is not convincing. The same holds for a bunch of
other papers at arxiv which work along similar lines.
There is a large class of consistent quantum-classical theories, which have
classical phase space variables p,q and in addition a quantum wave function
psi and a Hamiltonian H(p,q) which is an operator-valued function of p and q.
The dynamics is given by
i hbar psidot = H(p,q) psi,
qdot = psi^* dH(p,q)/dp psi
pdot = - psi^* dH(p,q)/dq psi
psi^*psi is easily seen to be conserved and can be taken as =1.
psi^* H(p,q) psi is also conserved.
That this is consistent follows from the fact that it is obtained by
varying the real-valued action
S = integral dt (p qdot + i hbar psi^* psidot - psi^* H(p,q) psi).
This is used successfully for real applications in quantum chemistry
where one wants to treat most degrees of freedom classically
but selected degrees of freedom (e.g., delocalized electrons)
quantum mechanically.
See, e.g., http://citeseer.nj.nec.com/229870.html
This reference is just one of many that use the above dynamical model.
Arnold Neumaier
Alfred Einstead
> 2) so far there is no PROOF that is HAS to be and no proof that a working
> system with non-quantum gravity cannot work.
However, that's not the correct question. The correct question is,
what's is all that stuff on the right hand side?
R_{mn} - 1/2 g_{mn} R
= -k (g^{pq} F_{mp} F_{nq} - 1/4 g_{mn} g^{pq} g^{rs} F_{pr} F_{qs}).
In QED, the F's are q-numbers, not c-numbers; as is the combination
on the right.
But that's not even the problem I'm referring to.
The q-number algebra for the F's is defined by commutator relations.
The prescription that yields these relations requires you to have
a Cauchy surface to formulate them on.
But the defiition of a Cauchy surface is framed (ultimately) in
terms of statements such as:
"ds^2 = g_{mn} dx^m dx^n < 0"
or
"ds^2 = g_{mn} dx^m dx^n > 0".
But the g's are q-numbers. So, neither of the statements above
are even meaningful.
So you can't even define "Cauchy surface".
In turn, you can't even define the commutator relations for the
F's, since you the question of when a surface is a Cauchy surface
or not isn't meaningful.
So, there's no way at the outset to define the q-number algebra
for the F's -- so that you don't even know what the F's are...
... which is the question posed: what are those things on the
right hand side.
Doug Sweetser
I am not questioning the solutions people have proposed for 60 years.
I think the statement of the problem itself is a problem. As you
pointed out, in an if/then statement, if the if clause is false,
anything can follow in the then clause. The many solutions to "If the
scalar mode of polarization appears for these field equations as a
spin 1 field" is false, so the published solutions are of no value
even if frequently cited. The if clause should be: "If the scalar mode
of polarization appears for these field equations as a spin 2 field."
The scalar polarization mode of emission from a 4D wave equation field
(Nabla^2 A^u = J^u) cannot be a photon. You list your three reasons.
Great, we both find a scalar polarization mode of photon emission to
be bunk.
So let me ask the question correctly by tossing in a few more words in
the right places (It should have been clear I was not talking about a
spin 0 field, that would be silly falling so close to the spin 2
field):
If the scalar polarization mode of transmission of a 4D wave equation
was spin 2, am I correct to say that it would no longer have the
negative norm issue?
You reply was:
> "The 00 component would have a positive norm" [followed by homage to
general covariance].
Good to see we agree.
doug
quaternions.com
Arnold Neumaier
> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<4033BFA0...@univie.ac.at>...
>
>
>>>>>This argument is far too simplistic. One could consider a unified theory
>>>>>based on the equation
>>>>> G = psi^* T psi,
>>>>>where psi satisfies a Schroedinger equation in the so-called
>>>>>functional Schroedinger picture of QFT. Then both sides have the
>>>>>same status.
>>>
>
>>>Italo Vecchi wrote:
>>>You get an expectation value on the left side. How is the "classical"
>>> Einstein tensor to be interpreted as an expectation value?
>>
>>The equation only says that it is equal to an expectation; which
>>is nothing objectionable. In thermodynamics we also have observables
>>which equl certain expectations.
>>
>
> An expectation value (both in classical and in quantum physics) is
> defined with respect to a measurement outcome.
No. Expectation values are defined formally without referencce to
the measurement process. Most of quantum theory is applied successfully
without caring at all about the foundational problem of measurement.
> The above equation
> equates expectations relative to two different measurements. The
> right-hand side is well defined. The left-hand side is not, since it
> involves quantities for which, as you point out, no quantum
> measurement theory is defined.
One does not need a measurement theory to give these expressions a
consistent mathematical meaning.
> The problem is not just conceptual. We simply do not know what value
> we should give to the left-hand side. How is time to be measured?
> Reading from a clock? Which clock?
As in all applications of theory to experiment, experimenters
calibrate whatever they use to measure a quantity so that it
corresponds reasonably closely to the ideal situation described by
the theory. With your requirements, physics would be impossible,
since your objections hold for any theory.
Arnold Neumaier
Arnold Neumaier
Lubos Motl wrote:
> On 24 Feb 2004, Arnold Neumaier wrote:
>>>LM: Therefore you must say exactly when the collapse happens, and
>>>according to what rules it happens. Be sure that you won't get any
>>>consistent picture.
>>
>>Never, just as with the Schroedinger equation. Then everything is consistent.
>
> OK, if you never make any "collapse", then your picture has absolutely
> nothing to do with quantum mechanics.
This is a very strange view of quantum mechanics. In orthodox quantum mechanics,
the dynamics is completely free of any reference to the collapse, which is only
invoked to interpret results.
Please tell me exactly when the collapse happens in the Schroedinger dynamics,
and I'll tell you exactly when it happens in my model.
> Your wavefunction is obviously a real, classical wave in space -
No. In case of a qubit, it has nothing wavelike to it.
Your use of 'obviously' only shows that you don't understand.
> because it directly affects the classical observables p,q -
> nevertheless you don't allow this wavefunction to collapse.
If one would model the environment, there would be additional stochastic and
dissipative terms that produce the collapse. But in the conservative case,
there is no collapse.
> it has no capacity to agree with the real phenomena.
This is not true. It is used in quantum chemistry, and these people surely
know better than you what is real.
> In quantum mechanics, the wavefunction is not a real wave, and it can't be
> directly measured. It only encodes the probabilities that the outcome of a
> measurement will be X or Y.
These probabilities can be measured, and that's how we obtain information
about the wave function of an ensemble. It is not very different from any
other indirect measurement, that of the mass distribution in the sun, say.
> We don't need to imagine that the collapse is real,
How does this square with your above remark?
''OK, if you never make any "collapse", then your picture has absolutely
nothing to do with quantum mechanics.''
If the collapse is not real it can be ignored, which is what I did
> and we don't need to say when and how it precisely happens.
But you demanded me to say precisely this, while you cop out!
It is very strange that you argue both for that the collapse is essential
(and must be described exactly) and that it is irrelevant (and can be treated
by hand-waving).
> The only
> meaningful prediction of quantum mechanics is the probability of one
> consistent history out of some ensemble of alternatives, and everything
> else can be considered unphysical.
Mainstream physics considers the Lamb shift to be a meaningful physical
prediction of quantum mechanics, even one of the most important ones,
in direct opposition to your claim.
>>Chemists use them to compute charge distributions and equilibrium positions
>>of molecules such as small polypeptides.
>
> They can compute anything, but what you're doing is a completely incorrect
> interpretation of the wavefunction. Well, it is clear that the chemists
> also often misinterpret the physical objects and they use many simplified,
> approximate ways to think about a physical system. But this is a physics
> newsgroup, and therefore the low-brow and inaccurate ways to teach
> chemists how to imagine what a wavefunction means should not be enough.
This stuff is not in the curriculum for chemistry students, but is
high level research, and is conceptually and numerically as accurate
as any realistic approximation of quantum mechanics for real applications.
In this context, low-brow is a mark of quality rather than a reason for put-down.
Low-brow quantum chemists are surely closer to reality than high-brow
string theorists, and their physics is more real than anything string theory
has ever produced. Certainly they know much more than you about extracting
experimentally relevant information from quantum theory.
You may wish to study the work of Raymond Kapral (who has a Princeton Ph.D.
degree, and hence is perhaps sufficiently high-brow for you),
for example his paper
Mixed Quantum-Classical Dynamics
R. Kapral and G. Ciccotti, J. Chem. Phys., 110, 8919 (1999).
It is published by the American Institute of Physics, hence even according to
your arbitrary standards appropriate for sci.phys.research.
The paper is available online at
http://www.chem.utoronto.ca/~rkapral/Research.html
where further quantum-classical papers by his group can be found.
Arnold Neumaier
Arkadiusz Jadczyk
On 26 Feb 2004 05:38:30 -0500, Arnold Neumaier
<Arnold....@univie.ac.at> wrote:
>Low-brow quantum chemists are surely closer to reality than high-brow
>string theorists, and their physics is more real than anything string theory
>has ever produced. Certainly they know much more than you about extracting
>experimentally relevant information from quantum theory.
Yet they may be using this particular formalism not because it is really
"good", but because they do not know anything better.
In fact, this is my believe based on my discussions with "Low-brow
quantum chemists"....
Alfred Einstead
>From Lubos Motl:
> Therefore you must say exactly when the collapse happens, and
> according to what rules it happens. Be sure that you won't get any
> consistent picture.
It's not necessary.
A "collapse" rule (meaning, actually: a Lueder's Rule for measurement
sequences) can easily be stated for Relativistic QM when you're
in the Heisenberg Picture:
(Generalized) Lueder's Rule:
For a set of local observables A1,...,An
for a given (Heisenberg) state W
P(A1 in D1, ..., An in Dn | W)
= W[T'[P1 ... Pn] T[P1 ... Pn]]
where
W[] the positive, normalized, linear functional
corresponding to the state W
T[] forward time-ordered product operator
(e.g., T[A(x) B(y)] = A(x) B(y) if x-y is future-directed)
T'[] the reverse time-ordered product operator
Pi the projector P_{Ai,Di} corresponding to the observable
Ai and interval Di; i = 1,...,n.
This works fine, as long as you're dealing with discrete spectra.
The reason it's consistent (and also, the reason you don't need
to know "when" a collapse occurs (which is not meaningful anyway
in a Relativistic context)) is because "ties" are already broken
by Microcausality:
T[A(x)B(y)] = T'[A(x)B(y)] = A(x)B(y) = B(y)A(x),
when x-y spacelike.
Observables with continuous spectra, of course, present a
problem here, as they already do anyhow in non-relativistic QM
and everywhere else.
Arkadiusz Jadczyk
On 26 Feb 2004 12:54:58 -0500, whop...@csd.uwm.edu (Alfred Einstead)
wrote:
>The reason it's consistent (and also, the reason you don't need
>to know "when" a collapse occurs (which is not meaningful anyway
>in a Relativistic context)) is because "ties" are already broken
>by Microcausality:
Whether we are in relativistic context or non-relativistic, detectors
do not care, they register "clicks", and these clicks are space-time
events. So, a collapse (idealized) should be also a space-time event.
A space-time event has a "when" associated to it, though this "when",
numerically, will depend on the chosen set of clocks or, on the
arbitrarily chosen zero of the proper time of the detector. Yet, apart
of that, each event has an invariant "when attached" - it is this
event rather than some other event.
The problem is that there are not too many models in which a "collapse"
produces an "event". Like a particle in a cloud chamber - producing a
sequence of events. Models that does not allow us to understand and to
compute such events are defective. They are not able to reproduce
elementary fact about our Reality.
Joe Rongen
"Arkadiusz Jadczyk" <arkREM...@ANDTHIScassiopaea.org>
wrote in message news:cu4s30pbequ8h639b...@4ax.com...
>
>
> On 26 Feb 2004 05:38:30 -0500, Arnold Neumaier
> <Arnold....@univie.ac.at> wrote:
>
> >Low-brow quantum chemists are surely closer to reality than high-brow
> >string theorists, and their physics is more real than anything string
> >theory has ever produced. Certainly they know much more than you
> >about extracting experimentally relevant information from quantum theory.
>
> Yet they may be using this particular formalism not because it is really
> "good", but because they do not know anything better.
Very interesting!
Would you be kind enough to have a look at this free program called
"Gamess"? (General Atomic and Molecular Electronic Structure System)
At: http://classic.chem.msu.su/gran/gamess/index.html
or http://quantum-2.chem.msu.ru/gran/gamess/index.html
or http://www.msg.ameslab.gov/GAMESS/GAMESS.html
Please, point out just where one or more
of your improvement(s) should be made?
Gamess offers:
"A wide range of quantum chemical computations are
possible using GAMESS, which
1. Calculates RHF, UHF, ROHF, GVB, or MCSCF
selfconsistent field molecular wavefunctions.
2. Calculates CI or MP2 corrections to the energy
of these SCF functions.
3. Calculates semi-empirical MNDO, AM1, or PM3
RHF, UHF, or ROHF wavefunctions.
4. Calculates analytic energy gradients for all SCF
wavefunctions, plus closed shell MP2 or CI.
5. Optimizes molecular geometries using the energy
gradient, in terms of Cartesian or internal coords.
6. Searches for potential energy surface saddle points.
7. Computes the energy hessian, and thus normal modes,
vibrational frequencies, and IR intensities. The
Raman intensities are an optional follow-on job.
8. Obtains anharmonic vibrational frequencies and
intensities (fundamentals or overtones).
9. Traces the intrinsic reaction path from a saddle
point to reactants or products.
10. Traces gradient extremal curves, which may lead from
one stationary point such as a minimum to another,
which might be a saddle point.
11. Follows the dynamic reaction coordinate, a classical
mechanics trajectory on the potential energy surface.
12. Computes radiative transition probabilities.
13. Evaluates spin-orbit coupled wavefunctions.
14. Applies finite electric fields, extracting the
molecule's linear polarizability, and first and
second order hyperpolarizabilities.
15. Evaluates analytic frequency dependent non-linear
optical polarizability properties, for RHF functions.
16. Obtains localized orbitals by the Foster-Boys,
Edmiston-Ruedenberg, or Pipek-Mezey methods, with
optional SCF or MP2 energy analysis of the LMOs.
17. Calculates the following molecular properties:
a. dipole, quadrupole, and octupole moments
b. electrostatic potential
c. electric field and electric field gradients
d. electron density and spin density
e. Mulliken and Lowdin population analysis
f. virial theorem and energy components
g. Stone's distributed multipole analysis
18. Models solvent effects by
a. effective fragment potentials (EFP)
b. polarizable continuum model (PCM)
c. conductor-like screening model (COSMO)
d. self-consistent reaction field (SCRF)
19. When combined with the add-on TINKER molecular
mechanics program, performs Surface IMOMM or
IMOMM QM/MM type simulations. (Anonymous FTP to
www.msg.ameslab.gov, directory tinker, file
tinker.tar.Z, see simomm.doc contained therein)."
> In fact, this is my believe based on my discussions with "Low-brow
> quantum chemists"....
>
> ark
> --
Many thanks, Joe ( an occasional user of the Gamess for PC's )
---
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Danny Ross Lunsford
> You may call it [removing the unphysical polarizations of a photon] a
> "problem", but in that case it is certainly an easy problem to solve, and
> it has been solved 60 years ago. Since that time, people have found many
> other ways to solve this "problem" that lead to the same physical results.
Feynman himself considered it unsovled, and he said so in his Nobel
lecture. Moreover it was an exceedingly hard problem to even partially
solve, as you will see if you read Tomonaga's lecture given at the same
time.
Furthermore, if by "many other ways" you mean in differing gauges, then
this is a trivial point. If you mean the negative norm of Gupta-Bleuler,
then you have replaced one mystery by another. So as Feynman said,
although there are nice rules for calculating the results of
experiments, QED is still and unsolved problem and will remain so as
long as it is based on unphysical ideas.
Nobel lectures of Feynman, Schwinger, and Tomonaga can be found here:
http://www.nobel.se/physics/laureates/1965/
-drl
Danny Ross Lunsford
> The original reason for introducing quantum theory was to avoid
> the uv-catastrophe in electromagnetic blackbody radiation.
>
> I wonder if anything like gravitational blackbody radiation makes
> sense. Like a 4K background radiation you could have a background
> radiation of gravitons that are left over from the big bang.
Interesting question, but I'd guess that the non-linearity means that
the spectrum is dependent on more than just the temperature. The
blackbody argument depends on all the cavity modes being independent.
-drl
Arkadiusz Jadczyk
<mo...@feynman.harvard.edu> wrote:
>> "The Piecewise Deterministic Process Associated to EEQT"
>> http://xxx.lanl.gov/pdf/quant-ph/9805011
>
>It's not a paper that can be treated seriously, but I don't have time to
>clarify 26 pages of misconceptions.
Not only that, but to read this paper you would have to know open
systems dynamics etc. If the paper above is too difficult for you to
follow, I would suggest you read another paper on the same subject
R. Olkiewicz, ``Some mathematical problems related to classical-quantum
interactions'', Rev. Math. Phys. 9 (1997), 719
It is a more of a review type, and it can serve you as a nice
introduction to completely positive semigroups and their application in
"classical-quantum interactions."
Even if you will consider it as another 30 pages of misconceptions -
there are things there that perhaps you can learn ...
Arkadiusz Jadczyk
wrote:
>n 24 Feb 2004, Arkadiusz Jadczyk wrote:
>
>> Incorrect. Let me explain it to you why, because here is one particular
>> example where you have an evident hole in your physics and mathematics
>> education.
>
>Feel free to assume that I don't understand open system dynamics - unless
>you realize that such an assumption is very unreasonable, which you
>certainly should.
The fact that you do not understand opens system dynamics is not an
assumption. It is a proven fact.
> At any rate, you're not adding anything relevant to the
>discussion.
Incorrect again. Concerning open system dynamics and whether it is
relevant or not - this is an interesting and open problem. As john Baez
write some time ago:
" It's an open and hotly debated question, and one should not let ones
distaste for mixed states unduly prejudice the matter. "
and also
">Interestingly, Hawking showed that particles can go from pure to mixed
>states, even on a fundamental level! So the idea that a mixture is a
>macroscopic ensemble may give way--perhaps on a fundamental level, particles
>should be described by density operators which do not have to represent
>pure states.
Let's note that Hawking's work while brilliant is highly controversial
and certainly without experimental verification: the idea is that
virtual black holes are constantly popping into existence, munching
(possibly virtual) particles, and disappearing in a burst of thermal
Hawking radiation, thus increasing the entropy of the world. This
builds upon Hawking's earlier and already controversial approach on the
black hole information loss problem. (See the latest "This Week's
Finds".) So we can't really say he "showed" this, only that he
suggested it might be the case."
> The fundamental equations describing the whole system (such as
>the Universe), not just an open system in it, must be unitary to preserve
>the probabilities,...
Incorrect. The evolution does not have to be unitary to conserve
probabilities. The evolution should be trace preserving positive.
Next: Understand that there is no proof whatsoever that the universe
must be governed by unitary dynamics. It can be governed by dissipative
dynamics. There is no reason to consider the universe as a closed
system. There is an alternative - to consider it as an open system.
Perhaps, as John Baez says, it is "highly controversial", perhaps. But
it often happened in the past that the ideas that were highly
controversial at the beginning produced something new and valuable at
the end. What's the point of writing papers that are not even wrong?
So, if you call open system dynamics "controversial" - that's fine. But
if you insist that it is "unreasonable" - I will debate. But please,
first learn that: " The evolution does not have to be unitary to
conserve probabilities."
Arnold Neumaier
> On 26 Feb 2004 05:38:30 -0500, Arnold Neumaier
> <Arnold....@univie.ac.at> wrote:
>
>>Low-brow quantum chemists are surely closer to reality than high-brow
>>string theorists, and their physics is more real than anything string theory
>>has ever produced. Certainly they know much more than you about extracting
>>experimentally relevant information from quantum theory.
>
> Yet they may be using this particular formalism not because it is really
> "good", but because they do not know anything better.
They use it because they need it.
The quantum-classical dynamics I talked of has been discovered
several times independently, as far as I can judge
(though I didn't do a systematic search). This is always a sign
that a concept is important.
Moreover, it works, and allows the quantitative treatment of
systems which would be untractable in a full quantum setting.
Finally, there are rigorous mathematical studies (by excellent
mathematicians) that have theorems that this quantum-classical dynamics
is a well-defined limit of a full quantum system. These theorems require
nontrivial functional analysis since a singular limit must be taken.
If one does not take this singular limit but retains a little more
information, one gets (nonrigorously, but at the customary level of
plausibility theoretical physicists are used to) dissipative
stochastic versions of quantum-classical dynamics.
(These are not far off from your event-enhanced quantum mechanics,
so it might in fact be quite interesting for you to study the papers
I had referred to and the surrounding literature.)
In any case, this quantum-classical dynamics is (like classical
mechanics and quantum mechanics) mathematically consistent,
has [in the conservative case that I discussed here]
conservation of energy, interpolates 'smoothly' between the
classical and quantum regime (in a quite strong sense that I could
explain if desired), and has many tractable special cases
that can be related to experiment.
Thus it is a respectable generalization of quantum mechanics.
As conventional quantum theory (as far as the computational aspects are
concerned), it is not concerned with the measurement process (which
is a dissipative phenomenon that is - in both theories - a level more
involved). Measuring a system given by a quantum-classical dynamics
requires as in a full quantum system the introduction of an observing
subsystem whose dynamics is then projected away, resulting after the
usual approximations in decoherence phenomena. However, the system as
a whole preserves energy just like the Schroedinger equation, and
if interpreted probabilistically, probability (i.e. psi^*psi) is fully
conserved.
Whether it is relevant at a fundamental level is a completely
different question, but one that, in view of the above, should not
be completely dismissed.
> In fact, this is my believe based on my discussions with "Low-brow
> quantum chemists"....
One has to judge the quality by the contents of the papers,
not by the label 'quantum chemist'.
There are people of different quality, like in any field.
I know excellent physicists who do quantum chemistry. A significant part
of the work in J. Chem. Physics could as well have been published
in Phys. Rev., except that there the audience would be somewhat
different (and probably less interested).
This journal treats physics with chemical origin, just as a journal on
astrophysics treats physics with stellar origin. No one apart from
Lubos Motl would conclude that the physics done there is less relevant
or that they ''also often misinterpret the physical objects and
they use many simplified, approximate ways to think about a physical system.''
Of course, they need to approximate if they want to solve their problems;
only string theorists can afford to work on theories that do not
approximate anything. In our modern world, this is the difference between
speculations and real applications.
Arnold Neumaier
Italo Vecchi
> Italo Vecchi wrote:
> > Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<4033BFA0...@univie.ac.at>...
> >
> >
> >>>>>This argument is far too simplistic. One could consider a unified theory
> >>>>>based on the equation
> >>>>> G = psi^* T psi,
> >>>>>where psi satisfies a Schroedinger equation in the so-called
> >>>>>functional Schroedinger picture of QFT. Then both sides have the
> >>>>>same status.
> >>>
>
> >>>Italo Vecchi wrote:
> >>>You get an expectation value on the left side. How is the "classical"
> >>> Einstein tensor to be interpreted as an expectation value?
> >>
> >>The equation only says that it is equal to an expectation; which
> >>is nothing objectionable. In thermodynamics we also have observables
> >>which equl certain expectations.
> >>
> >
> > An expectation value (both in classical and in quantum physics) is
> > defined with respect to a measurement outcome.
>
> No. Expectation values are defined formally without referencce to
> the measurement process. Most of quantum theory is applied successfully
> without caring at all about the foundational problem of measurement.
Without caring about the uncertainty principle and wave-particle
duality there's no QM. The Bohr-Heisenberg model provides an adequate
measurement theory on a fixed space-time metric.
> > The above equation
> > equates expectations relative to two different measurements. The
> > right-hand side is well defined. The left-hand side is not, since it
> > involves quantities for which, as you point out, no quantum
> > measurement theory is defined.
>
> One does not need a measurement theory to give these expressions a
> consistent mathematical meaning.
One needs a measurement theory to give them a physical meaning.
Physics is about experimental outcomes. The point here is not
mathematical ginmickry, but its physical meaning. Mathematics may
yield a convenient language, but you must know what physics you are
talking about.
> > The problem is not just conceptual. We simply do not know what value
> > we should give to the left-hand side. How is time to be measured?
> > Reading from a clock? Which clock?
>
> As in all applications of theory to experiment, experimenters
> calibrate whatever they use to measure a quantity so that it
> corresponds reasonably closely to the ideal situation described by
> the theory.
Physical quantities must be consistently bound to experimental
outcomes, otherwise you are literally talking about nothing. The
history of physics is littered with conceptualisations which didn't
fit experiments or which ceased to be adequate. BTW the problem of
time has been an issue in QG since its inception.
> With your requirements, physics would be impossible,
> since your objections hold for any theory.
>
> Arnold Neumaier
My objection is that fiddling with symbols without a clear idea of
the physical meaning of what we are doing is unlikely to bring us far.
GR is built on a great and strikingly simple physical idea; the
equivalence principle. Now, that's a source for inspiration.
Thanks for the good discussion.
IV
---------------------------------
"Consider a spherical cow."
John Harte, UCB
Arkadiusz Jadczyk
>Very interesting!
>
>Would you be kind enough to have a look at this free program called
>"Gamess"? (General Atomic and Molecular Electronic Structure System)
>At: http://classic.chem.msu.su/gran/gamess/index.html
>or http://quantum-2.chem.msu.ru/gran/gamess/index.html
>or http://www.msg.ameslab.gov/GAMESS/GAMESS.html
>
>Please, point out just where one or more
>of your improvement(s) should be made?
>
>Gamess offers:
One possible improvement is in urging the users of Gamess to always
check the links before they give these links to other people. ;-)
(the first two links are outdated)
As for improvements - it all depends on what is you goal.
If your goal is to compute what CAN be computed using a
limited number of available algorithms, then Gamess does
not need any improvements.
But if your goal is to simulate timing of a sequence of events - as in
real experiments, then you will not be able to do with Gamess. Gamess
does not offer a simulation of a real time chemical experiment. It
offers some "classical trajectory", but it is a fake one. You can't
obtain this way a cloud chamber track, when you have discrete events
at discrete times, and where the timing of these events depends
on the relation of the particle wave function to the detectors.
It follows that you will not be able to obtain an electron emission
event simulated in real time. You will not be able to obtain water
becoming ice in real time - as it happens in reality.
You may always say: I don't care. But there are those who do care.
ksh95
> <Arnold....@univie.ac.at> wrote:
>
> >Low-brow quantum chemists are surely closer to reality than high-brow
> >string theorists, and their physics is more real than anything string theory
> >has ever produced. Certainly they know much more than you about extracting
> >experimentally relevant information from quantum theory.
>
> Yet they may be using this particular formalism not because it is really
> "good", but because they do not know anything better.
>
> In fact, this is my believe based on my discussions with "Low-brow
> quantum chemists"....
>
> ark
No, No, No.....I'm not a quantum chemist, but I know what they do.
In atomic/molecular/condensed matter calculations one must add the
constraint of "agreement with experiment". This constraint coupled
with the lack of infinite computing power necessitates a pragmatic
approach.
For example, everybody knows a density functional really isn't a
single particle state, but the fact of the matter is that using single
particle states in place of many body states allows one to calculate
feynman diagrams and get accurate excited states.
It's interesting how freedom from experimental verification leads to a
culture where some ambigious notions of elegance or difficulty
supercede the concrete notion of agreement with experiment.
PS
Regarding the below mentioned program Gamess. We could complain all
day about using gaussian orbitals (I wouldn't want to began to count
all the problems with this), but the fact of the matter is that one
can make spectacularly accurate calculations on large numbers of
particles with minimal computing time.....In complete agreement with
experiment!!! ......after all this is the whole point? or am I
mistaken?
Lubos Motl
> From the super Virasoro constraints derive an equation of the form
> i d_k |psi> = H |psi> (###)
That's OK, the super-Virasoro constrain does look as a Dirac equation. But
what you wrote above - (###) - is simply an equation of motion. First of
all, you don't gain anything if you pick a priviliged component of the
derivative d_k and move it to the opposite side of the equation that the
rest (H). The Virasoro and super-Virasoro constraints are *covariant*
equations, and the only "virtue" of your way of writing is to obscure
that it is covariant.
Second of all, and more importantly, your "gauge" - even though it is not
gauge, it is just writing the equations in a special way - does not solve
Lubos Motl
Lubos Motl
Lubos Motl
Arnold Neumaier
Italo Vecchi wrote:
>>>An expectation value (both in classical and in quantum physics) is
>>>defined with respect to a measurement outcome.
>>
>>No. Expectation values are defined formally without referencce to
>>the measurement process. Most of quantum theory is applied successfully
>>without caring at all about the foundational problem of measurement.
>
>
> Without caring about the uncertainty principle and wave-particle
> duality there's no QM.
These are both consequences of the formalism without any reference to
measurement.
> The Bohr-Heisenberg model provides an adequate
> measurement theory on a fixed space-time metric.
A kind of caricature. For example, it assumes instantaneous meaurements
of exact numbers...
Real measurements are much more complex things, and are not easily handled
in terms of the simplified textbook quantum measurement theory.
> One needs a measurement theory to give them a physical meaning.
> Physics is about experimental outcomes. The point here is not
> mathematical ginmickry, but its physical meaning. Mathematics may
> yield a convenient language, but you must know what physics you are
> talking about.
I do know it. Still, the formalism is independent of it.
I don't think measurement theory helps you the slightest thing if
you want to understand why certain experiemnts at CERN count as
experimental verification of the mass of the top quark, say.
But knowing the formalism of quantum theory and the standard model
helps you a lot.
> Physical quantities must be consistently bound to experimental
> outcomes, otherwise you are literally talking about nothing. The
> history of physics is littered with conceptualisations which didn't
> fit experiments or which ceased to be adequate.
But the ones that are successful are tidied up to the point where they
can be understood and applied without the burden of the complex problems
involved in measurement.
Arnold Neumaier
Lubos Motl
On Sun, 29 Feb 2004, Danny Ross Lunsford wrote:
> Feynman himself considered it unsovled, and he said so in his Nobel
> lecture.
I happen to have all the Nobel lectures, and I don't see it [Feynman's
comment that removing the unphysical states in gauge theories was an open
problem]. Could you please be more specific where do you see such a
comment? Have not you confused this "problem" with another "open" problem
mentioned by Feynman, namely how to remove the infinities?
What I see about Feynman's comments on QED is the punchline of his talk:
So what happened to the old theory that I fell in love with as a youth?
Well, I would say it=92s become an old lady, that has very little attract=
ive
left in her and the young today will not have their hearts pound when the=
y
look at her anymore. But, we can say the best we can for any old woman,
that she has been a very good mother and she has given birth to some very
good children. And, I thank the Swedish Academy of Sciences for
complimenting one of them. Thank you.
_________________________________________________________________________=
_____
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Danny Ross Lunsford
Lubos Motl wrote:
>>Feynman himself considered it unsovled, and he said so in his Nobel
>>lecture.
>
> I happen to have all the Nobel lectures, and I don't see it [Feynman's
> comment that removing the unphysical states in gauge theories was an open
> problem]. Could you please be more specific where do you see such a
> comment? Have not you confused this "problem" with another "open" problem
> mentioned by Feynman, namely how to remove the infinities?
No, I haven't confused it with that (what he calls a "dippy process",
that is renormalization, in his little book). Quoting from the *opening
paragraph* of the Nobel lecture:
"So, what I would like to tell you about today are the sequence of
events, really the sequence of ideas, which occurred, and by which I
finally came out the other end with an unsolved problem for which I
ultimately received a prize."
Feynman of course had no alternative to renormalization. In fact he went
to great length in his "Lectures" to show all the crazy ideas people
tried to get things to work out, without success. In his great way as a
teacher, he considered it a rather considerable achievement to see
exactly why the theory was fundamentally broken - this being the next
best thing to having a theory that wasn't. So the student is not left
frustrated, rather challenged to try his own hand at it.
This latter is a fundamental point. Neither Feynman nor Dirac *ever
accepted* QED as it finally stood, and both thought about the problem
until the end of their lives. Dirac was *still* having new ideas as late
as the mid-70s about fixing field theory. In a way, the only honorable
course is to take up the challenge of Feynman and Dirac in earnest
instead of simply announcing that "the matter is closed, it's all easy
and done with 60 years ago". It's just as undone today as it was in 1948.
-drl
Arkadiusz Jadczyk
<Arnold....@univie.ac.at> wrote:
(snip)
>
>The quantum-classical dynamics I talked of has been discovered
>several times independently, as far as I can judge
>(though I didn't do a systematic search). This is always a sign
>that a concept is important.
I have no doubts for it.
>Moreover, it works, and allows the quantitative treatment of
>systems which would be untractable in a full quantum setting.
>
>Finally, there are rigorous mathematical studies (by excellent
>mathematicians) that have theorems that this quantum-classical dynamics
>is a well-defined limit of a full quantum system. These theorems require
>nontrivial functional analysis since a singular limit must be taken.
I understand. But the fact that some formalism is obtained by some
rigorous means from some other formalism does not mean that the
formalism is "good" for all important purposes. If I understand it
correctly the formalism neglects back action quantum->classical and as
such is completely useless when we want to know how observing the
evolution of the classical part we can learn something about the state
of the quantum part. Therefore it is useless in explaining the timing
of "events".
Now, you say it comes as a rigorous limit from quantum theory. Well, it
comes also as a rigorous limit of my own theory! Namely, when I put the
coupling constant to zero (usually I call it kappa in my papers) or,
more generally, when I put all back-action operators g_\alpha\beta = 0,
I will get, in this limit, the formalism that you are talking about. So
the formalism currently used in quantum chemistry can also be considered
as a crippled version of EEQT. That is why I wrote before that quantum
chemists are using their particular formalism "because they do not know
anything better". By better I mean taking into account the back action -
which, as we all know, do exist.
You have mentioned that there is another possibility, another limit,
namely of taking into account some of the back action and that it leads
to open system dynamics that is similar to EEQT. That sounds interesting
to me, though I do not think that it can be derived rigorously and
without some "cheating".
There is also another issue. For me EEQT is closer to reality than the
standard quantum theory. Namely because I can answer within EEQT
questions that standard quantum theory can't answer. And it can't answer
NOT because the answers would be computationally too difficult, but
because it does not have the means to even state these questions using
the accepted formalism. This is not the same with the approximations
made in quantum chemistry. There chemists are asking questions that
could have been answered using pure quantum formalism, but to
ease the computations they are making approximations and consider
parts of the system as classical.
Otherwise I agree with you that the approximation and classical-quanto
formalism used in quantum chemistry is nice and consistent.
But, as I wrote above, I also consider it being a crippled version of
EEQT, where the most essential and interesting part, which I call
a "binamics" (exchange of information, or "bits") has been neglected.
Lubos Motl
> The fact that you do not understand opens system dynamics is not an
> assumption. It is a proven fact.
I just want to mention that I have read your answer, and only found
personal insults and out-of-context quotes of a John Baez. Let me repeat
that your texts add nothing new to the discussion, and the "open systems"
are not examples of the fundamental theory in action and cannot be used
to answer the question whether the evolution of the Universe is (and must
be) unitary. It is just a matter of approximation that you imagine that
the dynamics of an open subsystem is non-unitary - it is an approximation
to the full truth that the dynamics of the whole system *is* unitary.
> Let's note that Hawking's work while brilliant is highly controversial...
Hawking's work is a firmly established calculation of the behavior near
the horizons, it is a sort of theorem and as long as its assumptions
(about the approximate locality etc.) are preserved - which is likely to
be the case in the real world - the results are guaranteed to be true. You
might find it controversial, but it does not change anything about the
fact that Hawking's work is *not* controversial.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
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Lubos Motl
> > OK, if you never make any "collapse", then your picture has absolutely
> > nothing to do with quantum mechanics.
>
> This is a very strange view of quantum mechanics.
No, it was not a view of quantum mechanics. It was a view of your theory
that has nothing to do with quantum mechanics. In your theory, x,p were
defined to be classical observables, and because these classical
observables are affected by "psi", "psi" cannot have a probabilistic
interpretation, and the only way how to make it "look" like a wavefunction
is to define a classical mechanism of collapse.
I thought that your question was "why any theory that mixes quantum
mechanics and classical physics must be wrong", and I think that I've
answered it more than clearly.
> Please tell me exactly when the collapse happens in the Schroedinger dynamics,
> and I'll tell you exactly when it happens in my model.
That's a wrong rule of the game. In regular quantum mechanics, I don't
need to ask and answer this question because "psi" only has a
probabilistic interpretation. In your theory "psi" obviously does not have
a probabilistic interpretation - it is a classical object - because the
values of "psi" directly and deterministically affect the classical
observables x,p.
The only way how can you make "psi" in your theory behave like a
wavefunction in quantum mechanics is to postulate a mechanism for its
collapse - but this mechanism is in no way necessary in correct quantum
mechanics.
Sorry that I don't have time to read all of your confusing statements.
Arnold Neumaier
> In any case, this quantum-classical dynamics is (like classical
> mechanics and quantum mechanics) mathematically consistent,
> has [in the conservative case that I discussed here]
> conservation of energy, interpolates 'smoothly' between the
> classical and quantum regime (in a quite strong sense that I could
> explain if desired), and has many tractable special cases
> that can be related to experiment.
>
> Thus it is a respectable generalization of quantum mechanics.
>
> As conventional quantum theory (as far as the computational aspects are
> concerned), it is not concerned with the measurement process (which
> is a dissipative phenomenon that is - in both theories - a level more
> involved). Measuring a system given by a quantum-classical dynamics
> requires as in a full quantum system the introduction of an observing
> subsystem whose dynamics is then projected away, resulting after the
> usual approximations in decoherence phenomena. However, the system as
> a whole preserves energy just like the Schroedinger equation, and
> if interpreted probabilistically, probability (i.e. psi^*psi) is fully
> conserved.
> There are people of different quality, like in any field.
> I know excellent physicists who do quantum chemistry. A significant part
> of the work in J. Chem. Physics could as well have been published
> in Phys. Rev., except that there the audience would be somewhat
> different (and probably less interested).
For example, there is a paper
Phys. Rev. E 56 (1997), 894.
about quantum-classical dynamics for describing what happens
to the energy released in the picosecond to nanosecond time
scale in proteins, which is related to a similar problem for
the interaction of electrons with a polarizable lattice.
(Actually the paper never addresses _real_ proteins; the
application only serves to sell the paper, it seems.)
It starts off quite high-brow with a second-quantized Hamiltonian.
At some point, the quantum-classical versdion is introduced.
Equations (26)-(27) are a modified version of the equations I gave,
and (28)-(29) is a stochastic version.
The system admits soliton solutions which are discussed, e.g., in
Physica D, 142 101-112, (2000).
www.ma.hw.ac.uk/~chris/paper-1.pdf
(from the site of Chris Eilbeck, a mathematical physicist
who wrote a book on solitons at a time where this was still
a fairly new topic).
Arnold Neumaier