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| Hans Aberg... |
 Posted: Sun Sep 06, 2009 4:00 am |
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David Madore wrote:
[quote:c9daf51380]Hi. This is from a mathematician struggling to achieve some kind of
mental grasp of quantum field theory. :-)
I'd like to know if there exists a presentation of the Fock space of
states of a quantum field theory that uses a "multi-time"[#]
(Schroedinger picture like, but for multiple particles) presentation
so as to be manifestly Lorentz invariant, and so that the field
equations become an infinite number of coupled linear PDE's (probably
of a very singular kind, of course). Let me explain what I mean.
[/quote:c9daf51380]
I'm not sure if that is what you are looking for, but there is a rather
mathematical description in the book by Streater & Wightman, "PCT, Spin
& Statistics, and all that", ch. 3.
It is not known how to make relativistic Schroeder equation, only some
sporadic ones have been found, like the Dirac equation, which are
essentially one particle equations. (Or possibly two particle in view of
certain Bremsstrahlung effects of this equation - see the book by
Itzykson & Zuber, "Quantum field theory".)
So instead one passes directly to a Hilbert space formulation (like in
the book above), but still no universal differential equation. Feynman
found his path integral method, which from the mathematical point of
view is essentially finding the kernel of the differential equation
without having the equation itself. So in the absence of finding such a
relativistic universal differential equation, there was a movement
attempting to take the Feynman path integral method as the basis of QM.
So what you are looking for might be attempting to find the basis for QM
that no-one else has  .
Hans |
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| David Madore... |
| Posted: Wed Sep 09, 2009 7:01 am |
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Moderator, in litteris <h813hs$2p53$1 at (no spam) nef.ens.fr> scripsit:
[quote:93a4626287]========Moderator's note==================================
Under Lorentz (or better Poincare) invariance one usually understands the
invariance of natural laws under proper orthochronous Poincare
transformations. As we well know, nature does for sure not obey invariance
under C, P, CP and (since the CPT theorem of microcausal local QFT seems to hold)
thus also not under T reflection symmetries. Thus the choice of an
order of time does not break Lorentz invariance, but is to the very heart
of any sense of causality as is necessary for all physics, as we know it today,
to make sense at all.
[/quote:93a4626287]
Sorry for causing this confusion: when I wrote "a preferred time
direction", I meant to say, a preferred time *coordinate*, i.e., a
preferred axis in Minkowski space (aka, inertial frame). I wasn't
referring to the orientation of that axis, nor to time reversal.
So again: usual descriptions of Fock space are not manifestly
Lorentz-invariant in that they give the t coordinate a different role
from x,y,z, not just the partial ordering of events by light cones
(which is something the actual physics can depend on), but the actual
coordinate t.
--
David A. Madore
( http://www.madore.org/~david/ )
(Do _NOT_ remove the "+news" extension to email me.) |
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| Arnold Neumaier... |
| Posted: Wed Sep 09, 2009 7:04 am |
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David Madore schrieb:
[quote:e8100d90be]Arnold Neumaier in litteris <4AA23CF2.2030804 at (no spam) univie.ac.at> scripsit:
Canonical quantization is as Lorentz-invariant aas one can wish.
The measure of the 3D 1-particle space is already covariant, and
this property is inherited by the Fock space constructions.
OK, I meant to say, it is not manifestly Lorentz-invariant, as one has
to choose a preferred time direction, even though the final results do
not depend on it.
What happens if I try the following trick to attempt to force manifest
Lorentz-invariance?
Add a fictitious new time coordinate, perhaps calling it s, and let
that one play the role of the preferred time coordinate for the
evolution. So now instead of having (1+3)-dimensional space-time, we
have (2+3)-dimensional space time, and we are considering states with
(in QFT, multiple) (1+3)-dimensional coordinates evolving as a
function of the new time coordinate s.
[/quote:e8100d90be]
Well, the 'proper time quantum mechanics' or 'manifestly covariant
quantum mechanics' of
L.P. Horwitz and C. Piron, Helv. Phys. Acta 48 (1973) 316,
tried to do this and failed. Google.scholar.com lists 199 citations
of this paper, which shows that the idea is attractive at first
sight. But you are welcome to study the outcome and its (absent) impact.
Your suggestion looks like a variation of theirs, and I do not give
it any more chances of success.
[quote:e8100d90be]Attempts with a multi-time picture never got far since they lead to
more degrees of freedom than are physically observed.
One would need to kill the extra degrees of freedom with just as many
equations, of course. I don't understand why that should be
impossible.
[/quote:e8100d90be]
One understands by trying to do it - then one meets the obstacles.
[quote:e8100d90be]Is there a deep reason why
quantum field theory can't be made multi-time (not necessarily using
the trick I sketched above), or is that just a stupid question?
[/quote:e8100d90be]
Deep or not, the consensus is that any composite physical system
has just a single time.
Multiple times are approximations of idealized multiparticle scenarios
- like having multiple observers who behave like massive point
particles. But real observing systems are intrinsically entangled
with the system they observe....
[quote:e8100d90be]Anyhow, thanks for that useful FAQ, which contains the answer to many
questions with which I will now refrain from embarrassing myself!
[/quote:e8100d90be]
Thanks for the compliment; that's what the FAQ is for!
Arnold Neumaier |
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| Bob_for_short... |
| Posted: Thu Sep 10, 2009 8:23 pm |
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What kind of problem are you going to solve?
One time corresponds to one reference frame. The particle or field
space distribution may be described with as many variables as
necessary and all this in one reference frame with one time.
Bob. |
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| Arnold Neumaier... |
| Posted: Mon Sep 14, 2009 2:29 am |
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David Madore schrieb:
[quote:f2ab69c15e]So again: usual descriptions of Fock space are not manifestly
Lorentz-invariant in that they give the t coordinate a different role
from x,y,z, not just the partial ordering of events by light cones
(which is something the actual physics can depend on), but the actual
coordinate t.
[/quote:f2ab69c15e]
Physics need not be manifestly Lorentz invariant.
Simple Lorentz invariance is sufficient, and is experimentally
indistiguishable from manifest Lorentz invariance.
And irreducible unitary representations of the Poincare group are
_naturally_
not manifestly Lorentz invariant, as the proof of Wigner's
classification theorem shows.
Since one needs to build the Fock space from 1-particle representations
(i.e., irreducible unitary representations of the Poincare group),
this natural lack of mansifestness generalizes.
But any manifestly Lorentz invariant definition of a single particle
Hilbert space extends naturally to a manifestly Lorentz invariant
Fock space.
Thus if you like you may consider the set of solutions of the
free Dirac equations, turn it into a Hilbert space by defining
inner products using the invariant meassure, and you get the Fock space
you desire. But it is canonically isomorphic to the standard
construction that views it from an arbitrary but fixed timelike unit
vector. Since the manifestly invariant version is clumsier to work
with (since a bit too abstract), there is no real advantage in
proceeding this way.
Arnold Neumaier |
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| David Madore... |
| Posted: Mon Sep 14, 2009 12:21 pm |
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Arnold Neumaier in litteris <4AA7EAAE.4060004 at (no spam) univie.ac.at> scripsit:
[quote:94c9c7ceb5]Physics need not be manifestly Lorentz invariant.
[/quote:94c9c7ceb5]
....assuming that even means anything. Only the presentation of
physics can make Lorentz invariance manifest or not.
[quote:94c9c7ceb5]But any manifestly Lorentz invariant definition of a single particle
Hilbert space extends naturally to a manifestly Lorentz invariant
Fock space.
Thus if you like you may consider the set of solutions of the
free Dirac equations, turn it into a Hilbert space by defining
inner products using the invariant meassure, and you get the Fock space
you desire. But it is canonically isomorphic to the standard
construction that views it from an arbitrary but fixed timelike unit
vector. Since the manifestly invariant version is clumsier to work
with (since a bit too abstract), there is no real advantage in
proceeding this way.
[/quote:94c9c7ceb5]
This is the _sort_ of thing I was hoping for: I realize it would be
clumsy to work with, but I'm not so much trying to "work" with it as
to visualize it, or to get some sort of mental grasp of what can and
what cannot be done. (I find it hard to picture a function of several
space coordinates but only one time coordinate as something natural.)
So, what you say does formally answer my question, but I guess I was
implicitly hoping for a bit more:
* for the (interacting!) field equations to be more or less implied by
the description of such a "multi-time Fock space" as a subspace of
some larger space (rather than putting the free field equations in
an ad hoc way, as you suggest), and/or
* for evolution of the standard (non-multi-time) Fock space, subject
to the field equations, to be somehow readable as the "diagonal"
(set all time coordinates equal) of the "muti-time Fock space".
However, you have fairly convinced me that the question isn't very
interesting (and, pragmatically, if the "multi-time Fock space" object
is at least very difficult to construct, it won't be of much use to
help me visualize QFT)...
But no worry, I have many other equally stupid questions to ask. :-)
--
David A. Madore
( http://www.madore.org/~david/ )
(Do _NOT_ remove the "+news" extension to email me.) |
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| Arnold Neumaier... |
| Posted: Mon Sep 14, 2009 10:18 pm |
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Guest
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David Madore wrote:
[quote:475bc9a2d1]Arnold Neumaier in litteris <4AA7EAAE.4060004 at (no spam) univie.ac.at> scripsit:
Thus if you like you may consider the set of solutions of the
free Dirac equations, turn it into a Hilbert space by defining
inner products using the invariant meassure, and you get the Fock space
you desire. But it is canonically isomorphic to the standard
construction that views it from an arbitrary but fixed timelike unit
vector. Since the manifestly invariant version is clumsier to work
with (since a bit too abstract), there is no real advantage in
proceeding this way.
This is the _sort_ of thing I was hoping for: I realize it would be
clumsy to work with, but I'm not so much trying to "work" with it as
to visualize it, or to get some sort of mental grasp of what can and
what cannot be done. (I find it hard to picture a function of several
space coordinates but only one time coordinate as something natural.)
[/quote:475bc9a2d1]
The natural covariant picture is that of a vector bundle on Minkowski
space-time, with a standard Fock space attached to each point.
One moves in space-time via the Poincare group, and this action extends
to the bundle by means of the representation defining the Fock space.
This view shows that the single time in Minkowski space is very natural.
There is no natural room for multiple times.
[quote:475bc9a2d1]But no worry, I have many other equally stupid questions to ask.
[/quote:475bc9a2d1]
But after having tried to understand the theoretical physics FAQ,
please!
Arnold Neumaier |
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| Bob_for_short... |
| Posted: Tue Sep 15, 2009 6:53 pm |
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On 14 sep, 14:29, Arnold Neumaier <Arnold.Neuma... at (no spam) univie.ac.at>
wrote:
[quote:30af6ab107]
Physics need not be manifestly Lorentz invariant.
..[/quote:30af6ab107]
..
..
[quote:30af6ab107]Thus if you like you may consider the set of solutions of the
free Dirac equations, turn it into a Hilbert space by defining
inner products using the invariant meassure, and you get the Fock space
you desire. But it is canonically isomorphic to the standard
construction that views it from an arbitrary but fixed timelike unit
vector. Since the manifestly invariant version is clumsier to work
with (since a bit too abstract), there is no real advantage in
proceeding this way.
Arnold Neumaier
[/quote:30af6ab107]
This all is true but let us not discurage David.
David, what particular problem do you intend to resolve with help of
multi-time formulation? I hope it is divergences.
Vladimir. |
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| ... |
| Posted: Tue Sep 15, 2009 8:30 pm |
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On Sep 4, 10:18 am, david+n... at (no spam) madore.org (David Madore) wrote:
[quote:bd321238be]I'd like to know if there exists a presentation of the Fock space of
states of a quantum field theory that uses a "multi-time"[#]
(Schroedinger picture like, but for multiple particles) presentation
so as to be manifestly Lorentz invariant, and so that the field
equations become an infinite number of coupled linear PDE's (probably
of a very singular kind, of course). ...
[/quote:bd321238be]
Your question brings to mind the "Tomonaga-Schwinger formalism"
for QFT. I (along with M. Varadarajan) have played with this a
little.
Relatively recent articles on this (with references to the
original papers of Tomonaga and Schwinger) can be found at
Classical and Quantum Gravity, Vol. 16, 2651–2668, (1999).
Physical Review. D Vol. 76, 125012 (2007).
Physical Review D Vol. 58, 064007 (1998).
charlie torre |
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| Juan R." González-Álvarez... |
| Posted: Tue Sep 15, 2009 8:30 pm |
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David Madore wrote on Mon, 14 Sep 2009 22:21:25 +0000:
[quote:650624f2f2]Arnold Neumaier in litteris <4AA7EAAE.4060004 at (no spam) univie.ac.at> scripsit:
Physics need not be manifestly Lorentz invariant.
...assuming that even means anything. Only the presentation of physics
can make Lorentz invariance manifest or not.
But any manifestly Lorentz invariant definition of a single particle
Hilbert space extends naturally to a manifestly Lorentz invariant Fock
space.
Thus if you like you may consider the set of solutions of the free
Dirac equations, turn it into a Hilbert space by defining inner
products using the invariant meassure, and you get the Fock space you
desire. But it is canonically isomorphic to the standard construction
that views it from an arbitrary but fixed timelike unit vector. Since
the manifestly invariant version is clumsier to work with (since a bit
too abstract), there is no real advantage in proceeding this way.
This is the _sort_ of thing I was hoping for: I realize it would be
clumsy to work with, but I'm not so much trying to "work" with it as to
visualize it, or to get some sort of mental grasp of what can and what
cannot be done. (I find it hard to picture a function of several space
coordinates but only one time coordinate as something natural.)
[/quote:650624f2f2]
You may find unnatural quantum mechanics or statistical physics :-D
[quote:650624f2f2]So, what you say does formally answer my question, but I guess I was
implicitly hoping for a bit more:
* for the (interacting!) field equations to be more or less implied by
[/quote:650624f2f2]
There is not interacting field equations, only free fields are
well-defined in RQFT.
[quote:650624f2f2]the description of such a "multi-time Fock space" as a subspace of
some larger space (rather than putting the free field equations in an
ad hoc way, as you suggest), and/or
* for evolution of the standard (non-multi-time) Fock space, subject
to the field equations, to be somehow readable as the "diagonal" (set
all time coordinates equal) of the "muti-time Fock space".
[/quote:650624f2f2]
It makes little sense to build a formalism where one dummy parameter of
field theory is substituted with many more dummy parameters without
physical relation to measurements.
[quote:650624f2f2]However, you have fairly convinced me that the question isn't very
interesting (and, pragmatically, if the "multi-time Fock space" object
is at least very difficult to construct, it won't be of much use to help
me visualize QFT)...
[/quote:650624f2f2]
Multi-time formalisms are more related to distorting relativity.
In special relativity the pair (x,t) means position of the particle and
*laboratory* time. Adding a second particle mean adding a second position
(x_1, x_2, t) but time is the *same* for both. This is why wavefunctions
in QM are of type Psi = Psi(x_1, x_2, , x_N, t) for a N-particle system.
The multi-time formalisms do not use laboratory "t" but some 'time'
property associated to the particle (a kind of proper time) and then add a
new "t" for each particle. For two particles
(x_1, t_1, x_2, t_2)
but neither t_1 nor t_2 have the same meaning than t in relativistic
theories.
Even for a particle (x_1, t_1) is not the spacetime associated to SR or to
RQFT.
Moreover, since none of those t_N in multi-time theories is laboratory
time t, it is needed to introduce a new overall time for studying
*evolution*. This is why the Stuckelberg theory cited by Neumaier uses N+1
times for N particles (as was said to you before).
[quote:650624f2f2]But no worry, I have many other equally stupid questions to ask.
[/quote:650624f2f2]
--
http://www.canonicalscience.org/
BLOG:
http://www.canonicalscience.org/en/publicationzone
/canonicalsciencetoday/canonicalsciencetoday.html |
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| David Madore... |
| Posted: Sat Sep 19, 2009 4:14 am |
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"Juan R." Gonz=E1lez-=C1lvarez in litteris
<pan.2009.09.15.09.18.34 at (no spam) canonicalscience.com> scripsit:
[quote:19fd9f03df]David Madore wrote on Mon, 14 Sep 2009 22:21:25 +0000:
(I find it hard to picture a function of several space
coordinates but only one time coordinate as something natural.)
=20
You may find unnatural quantum mechanics or statistical physics 
[/quote:19fd9f03df]
I meant the above for a relativistic theory, of course. I know
nothing of relativistic statistical physics, and relativistic quantum
mechanics has its own problems. But I guess you're right.
[quote:19fd9f03df]There is not interacting field equations, only free fields are
well-defined in RQFT.
[/quote:19fd9f03df]
That's something else I've always had problems with: what does it mean
exactly, that there "are no" interacting field equations? That the
equations in question cannot be written for some theoretical reason?
That it is not known how to write them? That it wouldn't make any
sense to write them? That they could be written in theory, but they
couldn't be used to perform any computation? That they can't be local
equations like PDE's? Something else?
I always have a hard time sorting out, when one says something cannot
be done in theoretical physics, whether it means it cannot be done in
practice, whether it cannot be done in theory, whether it doesn't make
sense, or whether it's just pointless to do it.
[quote:19fd9f03df]Multi-time formalisms are more related to distorting relativity.
=20
In special relativity the pair (x,t) means position of the particle and
*laboratory* time. Adding a second particle mean adding a second positi=
on
(x_1, x_2, t) but time is the *same* for both. This is why wavefunction=
s
in QM are of type Psi =3D Psi(x_1, x_2, , x_N, t) for a N-particle syst=
em.
=20
The multi-time formalisms do not use laboratory "t" but some 'time'
property associated to the particle (a kind of proper time) and then ad=
d a
new "t" for each particle. For two particles
=20
(x_1, t_1, x_2, t_2)
=20
but neither t_1 nor t_2 have the same meaning than t in relativistic
theories.
=20
Even for a particle (x_1, t_1) is not the spacetime associated to SR or=
to
RQFT.
=20
Moreover, since none of those t_N in multi-time theories is laboratory
time t, it is needed to introduce a new overall time for studying
*evolution*. This is why the Stuckelberg theory cited by Neumaier uses =
N+1
times for N particles (as was said to you before).
[/quote:19fd9f03df]
Ah. Then that's certainly not the sort of "multi-time" I was hoping
for, you're right.
Let me try another angle of approach, then: if |S> is a state in some
quantum field theory, say that of a single (possibly self-interacting)
spinless boson \phi, one can define for each n a function F_n (or,
more probably, distribution) of n space-time coordinates x_1,...,x_n
by
F_n(x_1,...,x_n) =3D <0| \phi(x_1) ... \phi(x_n) |S>
where |0> is the vacuum.
Unless I am mistaken, giving all the F_n determines the state |S>
(since the vectors \phi(x_n) ... \phi(x_1) |0> ought to be dense in
the Fock space), i.e., the space of all |S> can be viewed as a
subspace of the space of all possible sequences (F_n) of functions of
n space-time variables. Of course, not all sequences (F_n) are
possible, so there must be some relations among the (F_n), which exist
in some abstract sense even if it turns out to be impossible to write
them down. Furthermore, the F_n appear to transform under the
Poincar=E9 group in the way I'd expect them to.
Now, probably these F_n aren't what I had in mind (in the case of free
fields, the N-particle states don't have every F_n zero except F_N,
and the F_n aren't symmetric in the n variables). But maybe they can
be made into something better (by taking the symmetric part?
time-ordering the \phi(x_i)? normal-ordering? subtracting the
corresponding functions for the vacuum? some combination of all
this?). I don't know. But at this point I don't care much that it is
impossible to write down specific equations satisfied by the
functions, so long as they exist in theory.
(I understand that some construction similar to the above, but with
|S>=3D|0>, and with a time-ordering to make the functions symmetric, is
used under the name of "n-point correlation functions". But why are
so many functions constructed, in QFT, without reference to a specific
system state?)
Again, I should emphasize that I'm not so much trying to solve a
specific problem, even less perform a computation, than simply to
achieve a mental representation of what it means to describe a
physical system, and the evolution of a system, in quantum field
theory. (And trying to figure out why, after trying to read several
vastly different approaches to QFT, such as Bjorken&Drell, Weinberg,
Zee, and more recently Zeidler's books on the subject, it [QFT] still
seems to me so utterly impenetrable that I don't even know where to
start my questions.)
--=20
David A. Madore
( http://www.madore.org/~david/ )
(Do _NOT_ remove the "+news" extension to email me.) |
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| Hans Aberg... |
| Posted: Sat Sep 19, 2009 9:10 am |
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David Madore wrote:
[quote:6d86d3387b]That's something else I've always had problems with: what does it mean
exactly, that there "are no" interacting field equations? That the
equations in question cannot be written for some theoretical reason?
That it is not known how to write them? That it wouldn't make any
sense to write them? That they could be written in theory, but they
couldn't be used to perform any computation? That they can't be local
equations like PDE's? Something else?
[/quote:6d86d3387b]
I indicated that in a reply a long time ago: none such have relativistic
Schroedinger equation has been found, leading to the Hilbert space
formulation, but no differential equation there either, so the Feynman
path integral method is used instead. It is unknown from the
mathematical point of view as to why it works.
Hans |
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| Juan R." González-Álvarez... |
| Posted: Thu Sep 24, 2009 12:17 am |
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David Madore wrote on Sat, 19 Sep 2009 16:14:09 +0200:
[quote:85d7433eb7]"Juan R." Gonz=E1lez-=C1lvarez in litteris
pan.2009.09.15.09.18.34 at (no spam) canonicalscience.com> scripsit:
David Madore wrote on Mon, 14 Sep 2009 22:21:25 +0000:
(I find it hard to picture a function of several space
coordinates but only one time coordinate as something natural.)
=20
You may find unnatural quantum mechanics or statistical physics :-D
I meant the above for a relativistic theory, of course.
[/quote:85d7433eb7]
What I said is also valid for relativistic theory. Consider two
non-interacting Dirac electrons, the two-body relativistic wavefunction is
built as a direct product of one-electron Dirac wavefuntions and the
functional form is
Psi = Psi(x_1, x_2, t)
there is two x and one t as in non-relativistic quantum mechanics. Why you
find this unnatural is something I do not understand!
[quote:85d7433eb7]I know nothing of relativistic statistical physics, and relativistic
quantum mechanics has its own problems. But I guess you're right.
[/quote:85d7433eb7]
The problems of relativistic quantum mechanics are not solved by adding
more times and relativistic statistical physics has even more serious
problems.
[quote:85d7433eb7]There is not interacting field equations, only free fields are
well-defined in RQFT.
That's something else I've always had problems with: what does it mean
exactly, that there "are no" interacting field equations? That the
equations in question cannot be written for some theoretical reason?
That it is not known how to write them? That it wouldn't make any sense
to write them? That they could be written in theory, but they couldn't
be used to perform any computation? That they can't be local equations
like PDE's? Something else?
I always have a hard time sorting out, when one says something cannot be
done in theoretical physics, whether it means it cannot be done in
practice, whether it cannot be done in theory, whether it doesn't make
sense, or whether it's just pointless to do it.
[/quote:85d7433eb7]
The reason is not experimental or computational but *technical*.
Due to several non-interaction theorems [1], only the free field theory is
well-defined.
Most particle physicists and string theorists do not like to say this in
public. Weinberg only *weakly* comments this in his modern book, when he
states that the relativistic field theory of bound states is not in
satisfactory shape [2]...
(...)
[quote:85d7433eb7]Let me try another angle of approach, then: if |S> is a state in some
quantum field theory, say that of a single (possibly self-interacting)
spinless boson \phi, one can define for each n a function F_n (or, more
probably, distribution) of n space-time coordinates x_1,...,x_n by
F_n(x_1,...,x_n) =3D <0| \phi(x_1) ... \phi(x_n) |S
where |0> is the vacuum.
[/quote:85d7433eb7]
Notice that |S> = |S(t)>
[quote:85d7433eb7]Unless I am mistaken, giving all the F_n determines the state |S
[/quote:85d7433eb7]
How does (t_1,...,t_n) specify the value of t for which |S(t)> is
evaluated?
[quote:85d7433eb7](since
the vectors \phi(x_n) ... \phi(x_1) |0> ought to be dense in the Fock
space), i.e., the space of all |S> can be viewed as a subspace of the
space of all possible sequences (F_n) of functions of n space-time
variables. Of course, not all sequences (F_n) are possible, so there
must be some relations among the (F_n), which exist in some abstract
sense even if it turns out to be impossible to write them down.
Furthermore, the F_n appear to transform under the Poincar=E9 group in
the way I'd expect them to.
Now, probably these F_n aren't what I had in mind (in the case of free
fields, the N-particle states don't have every F_n zero except F_N, and
the F_n aren't symmetric in the n variables). But maybe they can be
made into something better (by taking the symmetric part? time-ordering
the \phi(x_i)? normal-ordering? subtracting the corresponding
functions for the vacuum? some combination of all this?). I don't
know.
[/quote:85d7433eb7]
I fail to understand your motivation and goal.
[quote:85d7433eb7]But at this point I don't care much
that it is impossible to write down specific equations satisfied by the
functions, so long as they exist in theory.
(I understand that some construction similar to the above, but with
|S>=3D|0>, and with a time-ordering to make the functions symmetric, is
used under the name of "n-point correlation functions". But why are so
many functions constructed, in QFT, without reference to a specific
system state?)
[/quote:85d7433eb7]
What do you think that |0> is?
[quote:85d7433eb7]Again, I should emphasize that I'm not so much trying to solve a
specific problem, even less perform a computation, than simply to
achieve a mental representation of what it means to describe a physical
system, and the evolution of a system, in quantum field theory.
[/quote:85d7433eb7]
A physical system in RQFT is described by its Hamiltonian (or Lagrangian)
plus the state |Psi> for a N-collection of particles. The dynamical
variables are momenta, spin(helicity), and mass of the particles. x and t
are not observable but parameters in RQFT.
The system's state *formally* evolves according to a Schrodinger-like
equation
d/dt |Psi> = H |Psi>
where, for instance, H = H_QED.
But the interacting region is not defined so from here one extracts the
S-matrix for the *asymptotic* states (free fields).
[1] Here I mean the traditional non-interacting theorems but also
other theorems related.
[2] Superstring theory does not solve this. Indeed, only S-matrix for
free superstrings is again well-defined.
--
http://www.canonicalscience.org/
BLOG:
http://www.canonicalscience.org/en/publicationzone/canonicalsciencetoday/
canonicalsciencetoday.html |
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| Bob_for_short... |
| Posted: Thu Sep 24, 2009 2:52 am |
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Guest
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David wrote:
[quote:cd249b9486]That's something else I've always had problems with: what does it mean
exactly, that there "are no" interacting field equations? That the
equations in question cannot be written for some theoretical reason?
That it is not known how to write them? That it wouldn't make any
sense to write them? That they could be written in theory, but they
couldn't be used to perform any computation? That they can't be local
equations like PDE's? Something else?
I always have a hard time sorting out, when one says something cannot
be done in theoretical physics, whether it means it cannot be done in
practice, whether it cannot be done in theory, whether it doesn't make
sense, or whether it's just pointless to do it.
[/quote:cd249b9486]
Juan R. González-Álvarez wrote:
[quote:cd249b9486]The reason is not experimental or computational but *technical*.
Due to several non-interaction theorems [1], only the free field theory is
well-defined.
[/quote:cd249b9486]
I have to note here that we _can_ obtain solutions of equations with
interactions in two limiting cases:
1) Current is a known function of space-time. Then the radiated field
is calculated (coherent states, for example),
2) Field space-time distribution is a known function of time. Then we
can find “particle” equation solutions (bound states, for example).
The problem is in building a self-consistent theory. H. Lorentz,
motivated with preserving the energy-momentum conservation laws,
proposed a self-action mechanism of field-particle coupling. This
approach led immediately to mathematical and physical problems. In
other words, this approach failed. Nevertheless the self-action ansatz
was implemented in QED too. Its implementing led also to impossibility
to perform self-consistent calculations for about 20 years. A
temporary resort proposed was renormalizations. Renormalization of
fundamental constants m and e is just discarding perturbative
corrections to them because first, these corrections (if finite)
worsen agreement with experiment, next, they are so big that are good
for nothing. Discarding corrections is not legitimate mathematical
action but it removes the self-action contribution. As a result, in
several theories one can (luckily) obtain finite results comparable
with experiment but one stays alone with the huge conceptual and
mathematical problems - bare particles, infinite vacuum polarization,
Landau pole, etc. There are several “renormalization” ideologies but
none is satisfactory from practical point of view. Besides, applied
blindly to other theories (quantum gravity), the renormalization
prescription fails. This is an answer why QFT is “difficult” to
understand.
In fact, the energy-momentum conservation laws can be preserved in a
different, much more physical way. For example, formula (60) in
“Reformulation instead of Renormalizations” represents a total
Hamiltonian of interacting fields and there is no problem in
calculations – the perturbation theory if finite and the physical
phenomena are described in a routine way. So not everything is so
hopeless as many think. You can find details in my weblog:
http://vladimirkalitvianski.wordpress.com.
Cheers,
Vladimir Kalitvianski. |
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| David Madore... |
| Posted: Thu Sep 24, 2009 2:05 pm |
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Guest
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"Juan R." González-Ãlvarez in litteris
<pan.2009.09.24.08.59.03 at (no spam) canonicalscience.com> scripsit:
[quote:48d4f2e29b]What I said is also valid for relativistic theory. Consider two
non-interacting Dirac electrons, the two-body relativistic wavefunction is
built as a direct product of one-electron Dirac wavefuntions and the
functional form is
Psi = Psi(x_1, x_2, t)
there is two x and one t as in non-relativistic quantum mechanics. Why you
find this unnatural is something I do not understand!
[/quote:48d4f2e29b]
I can try to explain why I find this unnatural, but I don't think it
will be very convincing. A point in space-time is something I can
visualize, n points in space-time I can visualize, a function of n
points in space-time again, that seems to make sense. But a function
of n space coordinates and 1 time coordinate I can only visualize as a
function of [n points in space-time which happen to be simultaneous in
some reference frame]: that's very odd, why should it matter that they
are simultaneous? And if I try to imagine what goes on when
space-time is curved (not that I *want* to do QFT in curved
space-time, but I'd like it to be imaginable), I'm completely
confused.
[quote:48d4f2e29b]That's something else I've always had problems with: what does it mean
exactly, that there "are no" interacting field equations? That the
equations in question cannot be written for some theoretical reason?
...
The reason is not experimental or computational but *technical*.
Due to several non-interaction theorems [1], only the free field theory is
well-defined.
Most particle physicists and string theorists do not like to say this in
public. Weinberg only *weakly* comments this in his modern book, when he
states that the relativistic field theory of bound states is not in
satisfactory shape [2]...
[/quote:48d4f2e29b]
I had seem to understand that at least (a) certain theories which are
simple enough and/or in a reduced number of dimensions (such as,
perhaps, a single scalar field with a with a \phi^4 self-interaction
term in the Lagrangian, in 1+1 dimensions, perhaps even 1+2
dimensions, and certainly at least in 1+0 dimensions) have a
(mathematically) well-defined interacting theory, a fact which is
proven rigorously, and (b) certain asymptotically free theories (such
as, perhaps, QCD) presumably have a well-defined interacting theory,
something which appears to be confirmed by numerical evidence, even
though a mathematical proof of this is not known (and perhaps worth
10^6 dollars). Is this correct? I am quite happy to work with such a
"well behaved" theory if it helps.
Or am I confusing two different problems here, and the problem with
divergences (or the continuum limit) is wholly a different one from
the problem of describing interacting states?
In the worst case, I'm willing to discretize space-time, if it helps
at all. For the sort of questions I'm asking, determining whether the
state can be given a function of n discrete space-time coordinates
instead of n discrete space coordinates and 1 discrete time
coordinate, is just as satisfactory.
[quote:48d4f2e29b](...)
Let me try another angle of approach, then: if |S> is a state in some
quantum field theory, say that of a single (possibly self-interacting)
spinless boson \phi, one can define for each n a function F_n (or, more
probably, distribution) of n space-time coordinates x_1,...,x_n by
F_n(x_1,...,x_n) = <0| \phi(x_1) ... \phi(x_n) |S
where |0> is the vacuum.
Notice that |S> = |S(t)
[/quote:48d4f2e29b]
Well, yes, but |S(0)> determines |S(t)> for all t and vice versa. So
the vector |S> can also stand for the collection of all |S(t)>. The
important thing is that the scalar product <S'|S> is <S'(t)|S(t)> for
an arbitrary t.
[quote:48d4f2e29b]Unless I am mistaken, giving all the F_n determines the state |S
How does (t_1,...,t_n) specify the value of t for which |S(t)> is
evaluated?
[/quote:48d4f2e29b]
It does not: (unless I a mistaken,) the F_n determine |S(t)> for all
t, without singling out a particular t.
[quote:48d4f2e29b]I fail to understand your motivation and goal.
[/quote:48d4f2e29b]
My goal would be something like the following question:
"Consider a simple enough and sufficiently well-defined QFT of, say, a
single scalar field (perhaps the \phi^4 theory in 1+1 dimensions). Is
there a natural linear embedding of its space of states (where a
"state" |S> means the whole collection of |S(t)> for all t, but this
is really equivalent to just giving |S(0)>) inside the space of
families of functions of n space-time coordinates (presumably one
function for each n), in such a way that the action of the Poincaré
group on states is just given its natural action on the corresponding
functions?"
I'm not sure I haven't left out some conditions to avoid trivial and
uninteresting answers (probably I have), but this question should at
least make sense.
On the other hand, maybe I should start by asking more fundamental
questions, because there is evidently a lot about RQFT which I don't
understand (part of the problem being that I don't know exactly what).
[quote:48d4f2e29b]What do you think that |0> is?
[/quote:48d4f2e29b]
That was something I planned to ask in a future post. But for the
sort of things I was writing, I believe it's irrelevant what |0> is,
any reference state would do.
[quote:48d4f2e29b]Again, I should emphasize that I'm not so much trying to solve a
specific problem, even less perform a computation, than simply to
achieve a mental representation of what it means to describe a physical
system, and the evolution of a system, in quantum field theory.
A physical system in RQFT is described by its Hamiltonian (or Lagrangian)
plus the state |Psi> for a N-collection of particles. The dynamical
variables are momenta, spin(helicity), and mass of the particles. x and t
are not observable but parameters in RQFT.
The system's state *formally* evolves according to a Schrodinger-like
equation
d/dt |Psi> = H |Psi
where, for instance, H = H_QED.
But the interacting region is not defined so from here one extracts the
S-matrix for the *asymptotic* states (free fields).
[/quote:48d4f2e29b]
Does this last remark remark to sections S6g and S9c of Arnold
Neumaier's FAQ at <URL:
http://www.mat.univie.ac.at/~neum/physics-faq.txt >?
--
David A. Madore
( http://www.madore.org/~david/ )
(Do _NOT_ remove the "+news" extension to email me.) |
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