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| Arnold Neumaier... |
 Posted: Thu Nov 05, 2009 5:58 pm |
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Igor Khavkine wrote:
[quote]On Nov 4, 9:06 am, Arnold Neumaier <Arnold.Neuma... at (no spam) univie.ac.at
wrote:
Igor Khavkine wrote:
And, it seems that he is saying we can do a Fourier-analog analysis on
the curved space, so long as it possesses requisite symmetry. So, the
question then becomes, "what symmetries are required?"
The manifold has to be a "symmetric space", which is a technical
mathematical term. There are only about as many of those as you can
count on your fingers.
You seem to have infinitely many fingers. There are many symmetric
spaces. Only upon restricting to 4 dimensions the number becomes finite.
I do indeed have only finitely many fingers. :-)
But for a group representation approach, it is enough to have a
homogneous space (still a highly symmetric space but less than a
symmetric space), and there are infinitely many of these even in 4D
(one just needs 4 independent Killing fields), some of them of
high interest to cosmology.
Do you mean de Sitter or anti-de Sitter spaces,
[/quote]
These are indeed the only symmetric spaces, apart from Minkowski space,
that figure in general relativity. There are mor homogeneous spaces, though.
[quote]or something else? The
spatial slices of FRW cosmologies are usually treated as homogeneous
spaces. But their time evolution is put together in such a way
that breaks any kind of time translation invariance.
[/quote]
Yes, you are right; I was too quick.
Because of the big bang, there cannot be time invariance, and
realistic cosmological models with symmetry only have 3 independent
Killing fields.
Arnold Neumaier |
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| Jay R. Yablon... |
| Posted: Thu Nov 05, 2009 6:00 pm |
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"Arnold Neumaier" <Arnold.Neumaier at (no spam) univie.ac.at> wrote in message
news:4AF133DB.70702 at (no spam) univie.ac.at...
[quote]Jay R. Yablon wrote:
"Arnold Neumaier" <Arnold.Neumaier at (no spam) univie.ac.at> wrote in message
news:4AED7E86.9060801 at (no spam) univie.ac.at...
Igor Khavkine wrote:
On Nov 1, 10:05 am, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
Would you please check out this link:
http://en.wikipedia.org/wiki/Pontrjagin_duality#Fourier_transform
Then, please advise whether in your view this approach is
applicable to
curved spacetime, and, more generally, please advise what the
limitations may be on this approach, if any.
This approach is limited to locally compact abelian groups. R^n
happens to be one, but in general a curved space-time is not.
In general, finding the Green functions for a wave equation in a
curved space-time is hard and there is no magic wand like "a
generalized Fourier transform" that you can wave to quickly find
them.
Generalizations of the Fourier transform exist for locally compact
homogeneous spaces, which one can find under the heading of
noncommutative harmonic analysis. It involves detailed knowledge
of the representation theory of the associated symmetry groups.
Thus one can get closed form Green functions for any sufficiently
symmetric space-time, not only for Minkowski space (which has a
particularly simple - abelian - transitive symmetry group, whose
representation theory is determined by the ordinary Fourier
transform).
But, of course, this doesn't help much for studying general
general relativity.
Arnold Neumaier
I do not know if it was intentional or a typo that you referred to
"general general relativity," with an intention to suggest that for
some subset of general curved spacetime manifolds, one could do a
proper harmonic analysis
it was intentional. One can do it for highly symmetric space-times,
assuming that gravitational distortions that violate the symmetry can
be ignored. Then one gets a contracted approximate description in
terms of a simplified dynamics. Indeed, this is what happens in the
Post-Newton approximation, where the highly symmetric space-time is
taken to be Minkowski space. But nothing forbids to develop simial
approximations for other highly symmetric space-times.
I haven't followed the literature on this closely, so can't give
references. But Volume 1 of Thirring's treaatise on math physics
gives a classification of highly symmetric space-times.
the question is one of what subset manifolds from among all possible
mathematically-permitted curved spacetime manifolds which might
otherwise be permitted by general relativity, have sufficient
symmetry to allow the derivation of closed form Green functions. Are
you saying that there is *no curved manifold at all* for which this
will work,
With the standard dynamics of general relativity, any space-time
symmetry will be instantly destroyed by the dynamics, once there are
more than three sources of gravitation.
If there is *no curved manifold at all* for which this can be done,
then that would seem to be saying that path integral quantization
only works for a flat spacetime background,
This argument is not conclusive. As long as the space-time is
diffeomorphic to a homogeneous space one can use a diffeomorphism to
transform coordinates to that space, then do the Fourier analysis
there, then transform back. One can do this even locally (with
Minkowski space) and then patch things together - this is called
microlocal analysis, see H"ormander's books.
The corresponding transformations for path integrals get additional
determinants from the transformations, whcih are not well understood
rigorously (but informally handled with ghost fields).
Thus all this is just open territory, not a clsed road with valid
no-go theorems.
Arnold Neumaier
Hi Arnold,[/quote]
First, let me say that I appreciate your dialogue with both me and Igor.
It has been very helpful in trying to clarify the issues involved with
doing path integral quantization, which implicitly requires some form of
harmonic analysis, in curved spacetime. I think a separate thread might
be also suitable, titled "Does, and if so under what conditions does,
path integration quantization apply to curved spacetime?"
On the supposition that path integration does and should apply to curved
spacetime, or at least to a symmetry-restricted subset of curved
spacetime manifolds, I have posted an exercise paper at
http://jayryablon.files.wordpress.com/2009/11/path-integration-of-the-maxwell-action-2-11.pdf.
If you can be so kind as to briefly take a look at this, it would be
much appreciated.
In the the first three sections 1-3 I calculate the QED Green functions
in curved spacetime, with boundary terms included not discarded. This
can be done fully and successfully, but only up to the point of finding
an explicit expression for the propagator.
To make further progress on an explicit expression for the propagator,
it seems unavoidable that one must of necessity do some type of
"Fourier" analysis analog, which we all have been calling "harmonic
analysis," in curved spacetime. I am glad to hear that you regard this
as "open territory." I am hoping that sections 4 through 7 might help
to better define that territory.
*If nothing else, I would ask you to please look at section 4.* Section
4 mirrors the discussions we have been having here, and even quotes some
of the very helpful statements you have provide in this thread. I
expect that this is a "first draft" of whatever it eventually becomes,
but I would like to know if I am at least talking basic sense in this
section. After some extended discussion of the issues in this thread, I
arrive in section 4 at the point where I progress to calculations of the
path integral by using Pontryagin duality, and thereby accept whatever
restrictions are placed on spacetime manifolds when one uses that
particular analysis technique which -- form what I can tell -- is the
closest we can get to Fourier analysis.
In section 5, I then attempt to how that gauge symmetry itself, greatly
facilitates the ability to conduct harmonic analysis in curved
spacetime, and does allow exact calculations to be done with Pontryagin
duality. By equation (5.34), I develop forward and inverse Pontryagin
duality transformations, which differ from but can be expressed in terms
of ordinary Fourier transformations, and which allow for explicit
calculation of the propagator in any curved spacetime manifold that has
the requisite symmetries to make Pontryagin duality an admissible
technique.
Section 6 veers off into Yang-Mills theory, which seems to emerge
naturally out of the gauge symmetry analysis in Section 5.
In section 7, I finally get to an explicit Pontryagin duality
calculation of the propagator in the spacetime manifold under
consideration, which, by definition, must be a manifold to which
Pontryagin duality applies.
Once again, I very much appreciate the information you have provided and
the discussions so far, and look forward to further discussion and
exploration of this "open territory."
Thanks,
Jay |
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| Arnold Neumaier... |
| Posted: Thu Nov 05, 2009 10:42 pm |
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Jay R. Yablon wrote:
[quote]"Arnold Neumaier" <Arnold.Neumaier at (no spam) univie.ac.at> wrote in message
news:4AF133DB.70702 at (no spam) univie.ac.at...
The corresponding transformations for path integrals get additional
determinants from the transformations, whcih are not well understood
rigorously (but informally handled with ghost fields).
Thus all this is just open territory, not a closed road with valid
no-go theorems.
[/quote]
I had meant with the open territorry the problem of a rigorous
mathematical definition of the path integral. This is an exceedingly
difficult problem, already for flat spacetime. It is unlikely to be
solved in the curved case before the flat case is understood.
On the handwaving level (i.e., in the way how all path integral stuff
is currently dealt with in physics), the path integral in curved
space is already well understood, and people work with this for a
long time.
Thus further progress in understanding the path integral must
come from understanding the flat case first, in a nonperturbative,
rigorous fashion.
Arnold Neumaier |
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| Igor Khavkine... |
| Posted: Fri Nov 06, 2009 11:26 pm |
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On Nov 6, 5:00 am, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
[quote]First, let me say that I appreciate [Arnold's] dialogue with both me
and Igor. It has been very helpful in trying to clarify the issues
involved with doing path integral quantization, which implicitly
requires some form of harmonic analysis, in curved spacetime.
[/quote]
Jay, I'm sorry to say this, but you are still missing a large point in
this discussion. Path integrals in curved space-time and harmonic
analysis are not contingent upon each other. You keep asking about
generalizations of Fourier transforms, so you keep getting answers
about harmonic analysis. However, none of this discussion is moving
you closer to a better understanding of path integrals in curved space-
time.
[quote]In the the first three sections 1-3 I calculate the QED Green functions
in curved spacetime, with boundary terms included not discarded. This
can be done fully and successfully, but only up to the point of finding
an explicit expression for the propagator.
[/quote]
You've just contradicted yourself. The Green function and the
propagator are essentially one and the same object. If you are missing
an "explicit expression" for one, you could not have "fully and
successfully" "calculated" the other.
[quote]After some extended discussion of the issues in this thread, I
arrive in section 4 at the point where I progress to calculations of the
path integral by using Pontryagin duality, and thereby accept whatever
restrictions are placed on spacetime manifolds when one uses that
particular analysis technique which -- form what I can tell -- is the
closest we can get to Fourier analysis.
[/quote]
Congratulations, you've restricted yourself to Minkowski space-time.
And you don't just get close to Fourier analysis in that case, you get
Fourier analysis.
[quote]Once again, I very much appreciate the information you have provided and
the discussions so far, and look forward to further discussion and
exploration of this "open territory."
[/quote]
Neither QFT in curved space-time nor harmonic analysis are "open
territory". Both are filled with large industrial installations, with
open territory only way beyond these.
Igor |
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| Jay R. Yablon... |
| Posted: Sat Nov 07, 2009 10:37 pm |
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"Igor Khavkine" <igor.kh at (no spam) gmail.com> wrote in message
news:bbf271a9-b242-4428-9a3f-21d63fecd933 at (no spam) g27g2000yqn.googlegroups.com...
[quote]On Nov 6, 5:00 am, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
First, let me say that I appreciate [Arnold's] dialogue with both me
and Igor. It has been very helpful in trying to clarify the issues
involved with doing path integral quantization, which implicitly
requires some form of harmonic analysis, in curved spacetime.
Jay, I'm sorry to say this, but you are still missing a large point in
this discussion. Path integrals in curved space-time and harmonic
analysis are not contingent upon each other. You keep asking about
generalizations of Fourier transforms, so you keep getting answers
about harmonic analysis. However, none of this discussion is moving
you closer to a better understanding of path integrals in curved
space-
time.
In the the first three sections 1-3 I calculate the QED Green
functions
in curved spacetime, with boundary terms included not discarded.
This
can be done fully and successfully, but only up to the point of
finding
an explicit expression for the propagator.
You've just contradicted yourself. The Green function and the
propagator are essentially one and the same object. If you are missing
an "explicit expression" for one, you could not have "fully and
successfully" "calculated" the other.
[/quote]
Igor,
I think this may be just a question of my not using the right
terminology.
However one divides up the terminology between "Green function" and
"propagator," the calculation of the path integral has two parts to it
and that is the way in which I was using different terminologies for
each part. Specifically:
First, one must calculate path integral Z as a series expansion of the
form:
Z = sum G^(s) (1/n!) J^n (1)
where G^(s) includes what I refer to as the "propagator" D(x-y) written
as a function of (x-y) in spacetime and not yet as a function of the
momentum variable of integration (p) in momentum space. An example of
this part of the overall calculation is (3.39) in
http://jayryablon.files.wordpress.com/2009/11/path-integration-of-the-maxwell-action-2-11.pdf.
In this particular case of (3.39), it happens that one only has even
powers of J, and I have kept one power of D(x-y) grouped with the J^2
and the multiple powers of D(x-y) inside G^(s). This is what I refer to
as "calculating the Green functions" though perhaps it is better
referred to as calculating the series expansion for the Green functions
or some such thing.
Secondly, one must obtain an explicit expression for D(p) rather than
D(x-y), which I call the "spacetime propagator" and the "momentum space
propagator" respectively, and that requires us to have some way to go
back and forth between D(x-y) <--> D(p) and in flat spacetime we use
Fourier analysis but in curved spacetime we need something else that we
gave been talking about as "harmonic analysis." That is what I address
starting in section 4.
I do not think there is anything wring in my calculation of (3.39) which
builds on everything up to that point, but only in the words I am using
for this. Perhaps you can make clear what the right language is in
which to discuss this.
[quote]
After some extended discussion of the issues in this thread, I
arrive in section 4 at the point where I progress to calculations of
the
path integral by using Pontryagin duality, and thereby accept
whatever
restrictions are placed on spacetime manifolds when one uses that
particular analysis technique which -- form what I can tell -- is the
closest we can get to Fourier analysis.
Congratulations, you've restricted yourself to Minkowski space-time.
And you don't just get close to Fourier analysis in that case, you get
Fourier analysis.
[/quote]
Here, you may or may not have a point on the substance, which goes
beyond the words we are using. I'd like to find this out for sure:
On this question, may I refer you to (5.34) of
http://jayryablon.files.wordpress.com/2009/11/path-integration-of-the-maxwell-action-2-11.pdf,
which are the transformations I use to go between D(x-y) <--> D(p).
Clearly, there is Fourier analysis involved but not the same Fourier
analysis which occurs in Minkowski spacetime because one has the
sqrt(-g) and the Fourier(sqrt(-g(x))) and delta^(4)(p) in the
transformations. Thus, we are using compositions -- actually
convolutions which contain sqrt(-g), see (5.2 -- which do not show up
in Minkowski space. And so, I am entertaining the idea that what
happens in curved spacetime are not the same Fourier transformations
which happen in flat spacetime, but do continue to employ the same
Fourier analysis, where the sqrt(-g) gets convolved with the functions
of (x-y) and (p) we are transforming. What may be an key question is
whether (5.34) in fact preserves the mapping between x <--> p spaces in
the way that is required, and I do not know how to rigorously assess
this point. Here, I have in mind Arnold's earlier point that " As long
as the space-time is diffeomorphic to a homogeneous space one can use a
diffeomorphism to transform coordinates to that space, then do the
Fourier analysis there, then transform back. "
Now, I do in (5.34) use the same kernel exp[-ipx] as in flat spacetime,
and that may and probably will strike you on first impression as wrong.
In general non-Euclidean geometry, this would indeed be wrong. But, in
physics, we also have gauge symmetry, and I believe that gauge symmetry
permits one to do this, because one can always "gauge out" certain
arbitrariness from the kernel including the arbitrariness of a general
coordinate transformation, in gauge theory. This is a central part of
my argument, and the purpose of my section 5 is to explain why I believe
this is so.
Thanks,
Jay |
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| X-Phy... |
| Posted: Sun Nov 08, 2009 7:53 am |
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On 6 nov, 05:00, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
[quote]First, let me say that I appreciate your dialogue with both me and Igor.
It has been very helpful in trying to clarify the issues involved with
doing path integral quantization, which implicitly requires some form of
harmonic analysis, in curved spacetime. I think a separate thread might
be also suitable, titled "Does, and if so under what conditions does,
path integration quantization apply to curved spacetime?"
[/quote]
Sure path integral quantization applies to curved spacetime. That it
be tractable is quite another issue, but is it in flat space?
Basically, path integral is the integral formulation of the Huygens
principle, and is not the only route to quantization. Indeed, it is
not the easiest. It simply leads to the wave equation, also in second
quantization even if the equation never appears explicitely. Though,
curved space provide difficulties only, or first, in first
quantization.
--
X-Phy |
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| Rock Brentwood... |
| Posted: Mon Nov 09, 2009 12:24 pm |
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On Oct 23, 6:24 am, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
[quote]In flat spacetime, one may take the forward Fourier transform, for
example, of a vector J^u, according to:
J^u(k) = $ d^4x J^u(x) exp[-i p_s x^s] (1)
Does this change at all in curved spacetime?
[/quote]
Another result, which I didn't mention in the previous replies links
manifods to their function spaces. It is also in this sense that the
Fourier transform generalizes. Here, it generalizes as a kind of
holistic top-down representation of a geometry.
The relevant theorem is one which equates the space C^{infinity}(M, C)
of functions on a manifold M to the complex numbers C to a C*-algebra
and -- conversely -- identifies a commutative C*-algebra as a function
space over a manifold.
The correspondence, roughly speaking, is x <-> delta_{x} where delta_x
is the (singular) function which yields integrals
integral_M f(y) delta_x(y) dy = f(x).
More precisely, the map
delta_x: f |-> f(x)
yields a "minimal" homomorphism from the function space C^{infinity}
(M, C) to each point x in M. In the converse direction, a commutative
C*-algebra A has a family of minimal 1-dimensional ideals, A_x = C,
each represented by a map d_x: A -> A_x = C, which is linear d_x(f +
g) = d_x(f) + d_x(g) and preserves products d_x(fg) = d_x(f) d_x(g).
All this comes under the header of the GNS correspondence.
The exercise to follow may or may not be related to the above theorem.
It is as follows.
In field theory, one has an issue integrating over momentum space.
This shows up most prevalently when doing the Wick time-ordered
operator expansion for a term that represents a loop diagram.
The traditional approach is to yank out an ax and chop off the space
at some radius |p| = Lambda.
Bear in mind, as this is done, that the space is endowed with an IN-
definite metric. So, what the "scale cut off" actually is, is NOT a
cut off of "small scale" (as you'll read in almost all textbook
presentations and hear of in almost all explanations of what' s
happening here), but a cut-off away from the light cone p^2 = 0.
So, now the exercise is this: replace the cut-off with a
compactification of momentum space -- a redefinition of momentum space
onto a curved compact geometry.
Bear in mind that momenta are not things to be considered in
isolation. Fundamental systems in particle and field theories have
state spaces that are irreducible representations of the underlying
space-time symmetry group. This means, the Lie algebra (and more
generally, the Poisson-Lie manifold) for the momentum generators
{P_{mu}, P_{nu}} = 0
is part of a larger structure that also contains the generators {J_{mu
nu} = -J_{nu mu}} for rotations and boosts.
There are two ways to go with this:
(1) something along the lines, {P_{mu}, P_{nu}} = lambda J_{mu nu}
or
(2) a "Penrose" compactification of the P's, themselves, into a
teleparallel geometry (which, here, means a curved geometry that is
the group manifold of a non-Abelian Lie group), with the J's combined
with the P's via a semi-direct product.
In both cases, the subtlety is how the compactification will work with
the fact that the metric is indefinite. In case (2), you're talking
about a compactification of Minkowski space (e.g. Penrose's
compactification into a projective space).
In case (1), lambda has units of 1/Area; e.g. lambda = 1/A_P = c^3/(h-
bar G). This might be termed the "Planck cut-off" and the example 1/
A_P corresponds to a momentum cut-off of Lambda given by Lambda^2 = h-
bar c^3/G. |
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