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| Juan R." Gonzlez-lvarez... |
 Posted: Tue Sep 29, 2009 2:05 am |
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Bob_for_short wrote on Thu, 24 Sep 2009 14:52:20 +0200:
[quote:85db1d3453]David wrote:
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.
Juan R. González-Álvarez wrote:
The reason is not experimental or computational but *technical*. Due to
several non-interaction theorems [1], only the free field theory is
well-defined.
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.
[/quote:85db1d3453]
The reason given above, the no-interaction theorems, has little to see
with renormalization and self-action troubles. If we introduce a minimal
scale in some /ad hoc/ way, still the resulting finite field theory
continues being unable to describe completely bound states because the
theorems hold.
Moreover, your opinions on renormalization are not well-founded.
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| Bob_for_short... |
| Posted: Tue Sep 29, 2009 11:58 am |
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On 29 sep, 14:05, "Juan R." Gonzlez-lvarez
<juanREM... at (no spam) canonicalscience.com> wrote:
[quote:af22989fc1]
The reason given above, the no-interaction theorems, has little to see
with renormalization and self-action troubles. If we introduce a minimal
scale in some /ad hoc/ way, still the resulting finite field theory
continues being unable to describe completely bound states because the
theorems hold.
Moreover, your opinions on renormalization are not well-founded.
[/quote:af22989fc1]
Briefly:
Necessity of renormalizations is a sequence of a bad choice of
interaction
term. Better choice removes renormalizations and no-interaction
theorems.
Ask yourself if no-interaction theorem appears in description of
_potential_ interactions of compound systems, like in atom-atomic
collisions or in forming molecules. |
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| Juan R. Gonzlez-lvarez... |
| Posted: Wed Sep 30, 2009 1:34 pm |
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[[Mod. note -- I am approving this posting with some reluctance,
because the discusion seems to be drifting away from math & physics
(including that in Arnold Neumaier's FAQ) into debates about who did
or didn't read one or another section. Let's stick to the math &
physics (which are both more interesting, and perhaps also easier
to reach consensus about).
-- jt]]
David Madore wrote on Thu, 24 Sep 2009 20:05:51 -0400:
[quote:7189f803b7]"Juan R." Gonzalez-eM-^Alvarez in litteris
pan.2009.09.24.08.59.03 at (no spam) canonicalscience.com> scripsit:
[/quote:7189f803b7]
(...)
[quote:7189f803b7]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).
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 >?
[/quote:7189f803b7]
Someone recommended you to read the FAQ *before* continuing posting.
If one reads the index, one would find entries for sections as
S6d. Is there a rigorous interacting QFT in 4 dimensions?
S7c. Bound states in relativistic quantum field theory
Etc.
--
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| Bob_for_short... |
| Posted: Fri Oct 02, 2009 7:05 am |
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Exctracted form FAQ S8a:
"...in relativistic QFT...local fields imply singular interactions.
But ... the
limiting process is not well understood mathematically."
As you see, the problem is in interactions. But the interaction term
is guessed, it is not given from above.
We _can_ (must) advance another interaction term and avoid
singularities and thier discarding (renormalizations).
Presently, the interaction term contains the so called "self-action".
The self-action caused physical and mathematical problems from the
very beginning. It introduced a positive (or negative, whatver)
feedback and violates the energy-momentum conservation laws. The right
approach is to use an interaction without self-action.
Many negative statements belong to only "self-action" interactions.
One should not extrapolate them to *other* interacting theories.
Also, there are many false notions, like bare pointlike particles,
infinite vacuum polarizations, "problems" at short distances, etc.,
originated from "justification" of renormalizations and having no
physical meaning. See, for example, two paragraphs below Fig. 1 in
"Atom as a "Dressed" Nucleus".
http://vladimirkalitvianski.wordpress.com |
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| Arnold Neumaier... |
| Posted: Sat Oct 03, 2009 5:33 pm |
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Bob_for_short wrote:
[quote:f96b3589d4]Exctracted form FAQ S8a:
"...in relativistic QFT...local fields imply singular interactions.
But ... the
limiting process is not well understood mathematically."
As you see, the problem is in interactions.
[/quote:f96b3589d4]
You distort my statement in the FAQ by omitting just the most important
qualification: ``in 4 dimensions''.
The limiting process _is_ well-understood in 1+0, 1+1, and 1+2
dimensions, in spite of the singularities, and makes use of
renormalization.
The case of 1+0 dimensions is ordinary quantum mechanics with singular
potentials.
In 1+1 dimensions, everything is then well-defined mathematically
in terms of rigorous renormalization theory, for arbitrary polynomial
interactions. (See the book by Glimm and Jaffe).
The 1+2-dimensional case is significantly more difficult and needs
a restriction on the polynomial degree. There is a nontrivial
renormalization theory for Phi^4 theory, which is mathematically
well-understood.
Only the 1+3 dimensional case is at present completely open.
Arnold Neumaier |
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| Bob_for_short... |
| Posted: Sat Oct 03, 2009 10:39 pm |
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On 4 oct, 05:33, Arnold Neumaier <Arnold.Neuma... at (no spam) univie.ac.at> wrote:
[quote:fedfa876d9]
You distort my statement in the FAQ by omitting just the most important
qualification: ``in 4 dimensions''.
[/quote:fedfa876d9]
Hello Arnold,
I replaced _explicitly_ the omitted parts by "..." and gave a
reference to your FAQ.
Everyone can read the entire FAQ entry with "4 dimensions", if
necessary, so no distortion of your FAQ has been done actually.
I did so in order to underline two things:
1) It is the actual interaction term which is responsible for
divergences,
2) As a remedy one tries to define "the limiting process" instead of
changing the interaction term.
I think I managed to reveal the physical and mathematical flaws of
self-action ansatz and outlined a way of better physical construction
in my publications. It is a real answer to the original post, unlike
your misleading stress on dimensions.
http://vladimirkalitvianski.wordpress.com |
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| Arnold Neumaier... |
| Posted: Sun Oct 04, 2009 6:41 am |
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Bob_for_short wrote:
[quote:0471312459]On 4 oct, 05:33, Arnold Neumaier <Arnold.Neuma... at (no spam) univie.ac.at> wrote:
You distort my statement in the FAQ by omitting just the most important
qualification: ``in 4 dimensions''.
Hello Arnold,
I replaced _explicitly_ the omitted parts by "..." and gave a
reference to your FAQ.
Everyone can read the entire FAQ entry with "4 dimensions", if
necessary, so no distortion of your FAQ has been done actually.
[/quote:0471312459]
The meaning was distorted by replacing this information by ``...''.
David Madore had explicitly said he limits his questions to the
(1+1)-dimensional case, where mathematically rigorous theories exist.
So my comment in the FAQ didn't apply to his question.
Your careful deletion of the 4-dimensions (without any significant
saving) made your quote sound as if my statements in the FAQ would
apply to general relativistic field theories, in particular to those
in 1+1 dimensions.
The fact that renormalization works correctly and rigorously in 1+1
and 1+2 dimensions is proof enough that the solution to the problems
of relativistic quantum field theory does not lie in discarding the
renormalization approach in favor of the alleged ``better physical
construction'' in your publications, which have nothing to do with
relativistic quantum field theory.
Your claims at http://vladimirkalitvianski.wordpress.com/
are at present only wishful thinking, as far as applications to
relativistic quantum field theory are concerned.
You claim to have the key for the solution of the 4D case but can't
even reproduce the well-understood 1+1-dimensional results!!!
In particular, your claim that
``This gives correct physical and mathematical description of quantum
electrodynamics: emission, absorption, scattering, bound states, and all
that -- without infinities since the electronium takes into account
exactly the charge-field coupling -- by construction.''
is completely unfounded by your work. You get low order approximations
that prove nothing. QED is checked for correctness at quite high order,
where your electronium calculations haven't even be tried, let alone
found to give a ``correct physical description of quantum
electrodynamics''. There have been other effective theories (e.g. by
Barut) that were correct at low order but failed to reproduce the
higher order terms. |
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| Gerard Westendorp... |
| Posted: Sun Oct 04, 2009 8:47 pm |
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Before looking at QFT, it might be instructive to look at something
simpler: a 2 particle wave function with 2 time coordinates.
So
Psi(x1,t1,x2,t2)
is a wave function that gives you the amplitude for finding particle 1
at x1 on time t1, and finding particle 2 at x2 on time t2.
Interpreted as a measurement outcome descriptor, this wave function
would seem to make sense. But the 2 time variables are a bit confusing
when you think of the wave function as a *state* that evolves in time.
Anyway, lets see if we can write down some wave functions.
The KG equation in 2-time would be:
(dt1^2 + dt2^2)Psi = (dx1^2 + dx2^2 - m^2)Psi
Plane wave solutions exp(i(w1*t1+w2*t2+k1*x1+k2*x2) need to satisfy
w1^2+w2^2 = k1^2+k2^2+m^2
Since we are looking at Bosons, we would have to symmetrize:
Psi(x1,t1,x2,t2)=Psi(x2,t2,x1,t1)
This is perfectly OK mathematically. I am trying to figure out if it
means anything physically.
Since |
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| Juan R." Gonzlez-lvarez... |
| Posted: Mon Oct 05, 2009 8:10 pm |
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David Madore wrote on Thu, 01 Oct 2009 19:07:51 +0000:
[quote:8bc953a751]"Juan R. González-Álvarez" in litteris
pan.2009.09.29.11.20.30 at (no spam) canonicalscience.com> scripsit:
David Madore wrote on Thu, 24 Sep 2009 20:05:51 -0400:
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 >?
Someone recommended you to read the FAQ *before* continuing posting.
I did. That's the reason why, for example, I suggested to limit my
questions to the (1+1)-dimensional case, where mathematically rigorous
theories exist.
[/quote:8bc953a751]
I am not really interested in toy universes.
[quote:8bc953a751]If one reads the index, one would find entries for sections as
S6d. Is there a rigorous interacting QFT in 4 dimensions?
S7c. Bound states in relativistic quantum field theory
The answers don't always enlighten me. Or rather, I understand
something of what they say, but I don't understand how various problems
relate. In your answer to Bob_for_short (in message
pan.2009.09.29.11.21.04 at (no spam) canonicalscience.com>) your write that "the
no-interaction theorems, has little to see with renormalization and
self-action troubles", so now I'm very confused. Could you explain what
the relations are between:
(a) the difficulties involved in renormalization (which seem to get
worse as dimension increases?),
[/quote:8bc953a751]
Integrals go as r^{2-D}. Wait point-like divergences for D>2. For our
universe D=3.
[quote:8bc953a751](b) the difficulties involved in constructing theories satisfying the
Wightman axioms (which also seem to get worse as dimension increases?),
discussed in section S6d of the FAQ,
[/quote:8bc953a751]
Field theory is a free theory. There is no proof that his axioms work for
QED and similar quantum field theories.
[quote:8bc953a751](c) the difficulty (impossibility?) to write partial differential
equations for the interacting fields, which you mentioned,
[/quote:8bc953a751]
Field theory is a free theory. Only free fields are well-defined. There is
not "interacting fields".
[quote:8bc953a751](d) the difficulties with interpolating field, discussed in section S6g
of the FAQ
[/quote:8bc953a751]
There is not interpolating field in quantum field theories as QED, because
field theory is a free theory. The parameter t may take minus plus
infinity values due to ligh cone restrictions proved by Landau and Peirls.
[quote:8bc953a751]? How are these problems connected? And, in particular, which
"problems" (of any kind) still exist or remain for a simple
(1+1)-dimensional scalar field with a \phi^4 term in the Lagrangian?
[/quote:8bc953a751]
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| Juan R." Gonzlez-lvarez... |
| Posted: Mon Oct 05, 2009 8:10 pm |
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Bob_for_short wrote on Tue, 29 Sep 2009 23:58:09 +0200:
[quote:86fe5e1362]On 29 sep, 14:05, "Juan R." González-Álvarez
juanREM... at (no spam) canonicalscience.com> wrote:
The reason given above, the no-interaction theorems, has little to see
with renormalization and self-action troubles. If we introduce a
minimal scale in some /ad hoc/ way, still the resulting finite field
theory continues being unable to describe completely bound states
because the theorems hold.
Moreover, your opinions on renormalization are not well-founded.
Briefly:
Necessity of renormalizations is a sequence of a bad choice of
interaction
term.
[/quote:86fe5e1362]
Right.
[quote:86fe5e1362]Better choice removes renormalizations and no-interaction theorems.
[/quote:86fe5e1362]
Right.
[quote:86fe5e1362]Ask yourself if no-interaction theorem appears in description of
_potential_ interactions of compound systems, like in atom-atomic
collisions or in forming molecules.
[/quote:86fe5e1362]
Precisely the theorem is what has impeded the development of the
relativistic theory of molecular systems.
The theory of Horwitz and Piron cited may be the most famous attempts to
built a relativistic theory of atoms and molecules that avoids the
theorems.
The no-interaction theorems do not apply to non-relativistic systems, for
the which the theory is well-understood and applied.
--
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| Arnold Neumaier... |
| Posted: Tue Oct 06, 2009 6:31 am |
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Juan R. Gonzlez-lvarez wrote:
[quote:fbb965014e]David Madore wrote on Thu, 01 Oct 2009 19:07:51 +0000:
"Juan R. González-Álvarez" in litteris
pan.2009.09.29.11.20.30 at (no spam) canonicalscience.com> scripsit:
David Madore wrote on Thu, 24 Sep 2009 20:05:51 -0400:
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 >?
Someone recommended you to read the FAQ *before* continuing posting.
I did. That's the reason why, for example, I suggested to limit my
questions to the (1+1)-dimensional case, where mathematically rigorous
theories exist.
I am not really interested in toy universes.
[/quote:fbb965014e]
1+1D quantum field theories do not model toy universes but quantum
wires. And 1+2D quantum field theories model quantum surfaces.
Both have nontrivial and very useful real-life applications.
[quote:fbb965014e]Could you explain what
the relations are between:
(a) the difficulties involved in renormalization (which seem to get
worse as dimension increases?),
Integrals go as r^{2-D}. Wait point-like divergences for D>2. For our
universe D=3.
(b) the difficulties involved in constructing theories satisfying the
Wightman axioms (which also seem to get worse as dimension increases?),
discussed in section S6d of the FAQ,
Field theory is a free theory. There is no proof that his axioms work for
QED and similar quantum field theories.
[/quote:fbb965014e]
There are no essential difficulties in constructing interacting quantum
field theories in 1+1D and 1+2D that satisfy the Wightman axioms.
There is no proof that interacting QFTs in 4D cannot satisfy the
Wightman axioms. The problem is completely open.
[quote:fbb965014e](c) the difficulty (impossibility?) to write partial differential
equations for the interacting fields, which you mentioned,
Field theory is a free theory. Only free fields are well-defined. There is
not "interacting fields".
[/quote:fbb965014e]
Interacting QFTs in 1+1D and 1=2D are well-defined and not free.
[quote:fbb965014e](d) the difficulties with interpolating field, discussed in section S6g
of the FAQ
There is not interpolating field in quantum field theories as QED, because
field theory is a free theory.
[/quote:fbb965014e]
QED is not the only QFT of interest, and most mathematically existing
QFTs are not free.
Arnold Neumaier |
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| Juan R." Gonzlez-lvarez... |
| Posted: Tue Oct 06, 2009 8:07 pm |
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Arnold Neumaier wrote on Tue, 06 Oct 2009 18:31:06 +0200:
[quote:7f284ab5f6]Juan R. González-Álvarez wrote:
David Madore wrote on Thu, 01 Oct 2009 19:07:51 +0000:
"Juan R. González-Ãlvarez" in litteris
pan.2009.09.29.11.20.30 at (no spam) canonicalscience.com> scripsit:
David Madore wrote on Thu, 24 Sep 2009 20:05:51 -0400:
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 >?
Someone recommended you to read the FAQ *before* continuing
posting.
I did. That's the reason why, for example, I suggested to limit my
questions to the (1+1)-dimensional case, where mathematically
rigorous theories exist.
I am not really interested in toy universes.
1+1D quantum field theories do not model toy universes but quantum
wires. And 1+2D quantum field theories model quantum surfaces.
Both have nontrivial and very useful real-life applications.
Could you explain what
the relations are between:
(a) the difficulties involved in renormalization (which seem to get
worse as dimension increases?),
Integrals go as r^{2-D}. Wait point-like divergences for D>2. For
our universe D=3.
(b) the difficulties involved in constructing theories satisfying
the Wightman axioms (which also seem to get worse as dimension
increases?), discussed in section S6d of the FAQ,
Field theory is a free theory. There is no proof that his axioms
work for QED and similar quantum field theories.
There are no essential difficulties in constructing interacting
quantum field theories in 1+1D and 1+2D that satisfy the Wightman
axioms.
There is no proof that interacting QFTs in 4D cannot satisfy the
Wightman axioms. The problem is completely open.
[/quote:7f284ab5f6]
My apologies, by field theory I mean 4D theories as QED.
[quote:7f284ab5f6]
(c) the difficulty (impossibility?) to write partial differential
equations for the interacting fields, which you mentioned,
Field theory is a free theory. Only free fields are well-defined.
There is not "interacting fields".
Interacting QFTs in 1+1D and 1=2D are well-defined and not free.
[/quote:7f284ab5f6]
Idem as above.
[quote:7f284ab5f6](d) the difficulties with interpolating field, discussed in
section S6g of the FAQ
There is not interpolating field in quantum field theories as QED,
because field theory is a free theory.
QED is not the only QFT of interest, and most mathematically
existing QFTs are not free.
[/quote:7f284ab5f6]
Here I did explicit the subclass. Also I was not worried about
"mathematical existence", but about physical existence.
It is possible mathematically build a field theory in the mathematical
sense for scalar phi(x,t) and the theory is mathematically defined and
all that, but when one tries to identify x and t with readings of
relativistic rods and clocks, then one confronts with the famous
problems of relativistic localization.
No problem for mathematicians who can do x=5 and t=-333 and play with
numbers.
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| Thomas Larsson... |
| Posted: Wed Oct 07, 2009 6:07 am |
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[quote:3a17f43959]All presentations I find of the Fock space either use momentum space
(e.g., canonical quantization) or break Lorentz invariance by handling
the time coordinate differently from the space coordinates. M-BM- Or they
mostly sidestep the Fock space and speak of the S-matrix or whatever.
Is this inevitable?
[/quote:3a17f43959]
Some years ago I spent more time than I should on something I called
MCCQ (Manifestly Covariant Canonical Quantization), which treats time
on the same footing as space. The idea was to do quantization in the
Fock space of arbitrary histories (which may or may not solve the
equations of motion), and impose dynamics afterwards as a constraint
in the history Fock space. The usual Fock space was then recovered in
cohomology a la BRST.
My reason for trying this is that I wanted to apply the projective
representation theory of the spacetime diffeomorphism algebra (i.e.
the multi-dimensional Virasoro algebra) to general relativity.
Manifest covariance is really needed here, because the constraint
algebra of GR mutates into the Dirac algebra when you fix a foliation.
Also, the Hilbert space of an interacting QFT cannot be a Fock space
(I think this is a theorem), but it might be possible to describe it
as a resolution of Fock spaces.
Alas, this programme ran into various problems, and eventually I
realized that manifest covariance was lost anyway. Not only did I have
to implement the Euler-Lagrange equations in cohomology, but also the
identification of momenta and velocities, and the latter ruined
manifest covariance. FWIW, you can find my musings at http://arxiv.org/abs/0709.2540
and references therein. |
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| Gerard Westendorp... |
| Posted: Wed Oct 07, 2009 6:08 am |
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Gerard Westendorp wrote:
[..]
[quote:d502724a3d]The 2 particle KG equation in 2-time would be:
(dt1^2 + dt2^2)Psi = (dx1^2 + dx2^2 - m^2)Psi
[/quote:d502724a3d]
We could make a change in time coordinates:
t = (t1+t2)/sqrt2
T = (t1-t2)/sqrt2
This doesn't change the equation much:
(dt^2 + dT^2)Psi = (dx1^2 + dx2^2 - m^2)Psi
But the symmetrization condition is now:
Psi(x1,x2,t,T)=Psi(x2,x1,t,-T)
This means that for T=0, we get the "ordinary" KG equation.
t is interpreted as "ordinary" time, while T is the time delay between
the measurement of particle 1 and particle 2.
In other words, we get a generalization of the KGE, with KGE as the
limiting case when T=0.
This procedure could be generalized to QFT: the creation and
annihilation operators all get an additional "delay-time parameter",
which, when zero, yields the equal time formulation.
Gerard |
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