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| Jay R. Yablon... |
 Posted: Fri Oct 23, 2009 2:24 am |
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Guest
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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? In particular, is it
necessary to generalize the Fourier Kernel exp[-i p_s x^s], to something
else, for example:
exp[-i p_s x^s] --> exp[-i p_s x^s + F(x)] (2)
where F(x) is some function of the coordinates x^u?
Or, does (1) still suffice?
Thanks,
Jay.
____________________________
Jay R. Yablon
Email: jyablon at (no spam) nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm |
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| Raymond Manzoni... |
| Posted: Sat Oct 24, 2009 11:44 pm |
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Jay R. Yablon a écrit :
[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? In particular, is it
necessary to generalize the Fourier Kernel exp[-i p_s x^s], to something
else, for example:
exp[-i p_s x^s] --> exp[-i p_s x^s + F(x)] (2)
where F(x) is some function of the coordinates x^u?
Or, does (1) still suffice?
Thanks,
Jay.
[/quote]
According to Robert Wald (author of "Quantum field theory in curved
spacetime and black hole thermodynamics"
<http://books.google.com/books?id=Iud7eyDxT1AC>) in
<http://www.phys.lsu.edu/mog/mog20/node16.html> the answer could be
"Microlocal analysis"
Richard Melrose's lecture notes are available here :
<http://www-math.mit.edu/~rbm/iml90.pdf>
Hoping it helped even... if I know nearly nothing about this! 
Raymond |
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| Igor Khavkine... |
| Posted: Sun Oct 25, 2009 7:00 am |
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On Oct 23, 2:24 pm, "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]
That is not the right question to ask, because it has a false premise
embedded in it. The Fourier transform on R^n has many desirable
properties, some of which may be specific to this setup. The first
question to ask is Does an analogous transform with as many nice
properties *actually exist* in other contexts?
Depending on the nice properties of the transform you want to preserve
and depending on the context (e.g., replacing R^n by a Riemannian
manifold), in many cases an analog of the Fourier transform does not
exist at all.
That is not to say that there are no useful generalization of the
Fourier transform; there are many. The kinds of generalizations that
people find useful are often grouped under the moniker of "harmonic
analysis". Here's a old post by John Baez that gives examples of some
of these generalizations:
news:7t8eac$abk at (no spam) charity.ucr.edu
http://groups.google.com/group/sci.physics.research/msg/3b85eb9ddc0ad4e3
Igor |
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| X-Phy... |
| Posted: Tue Oct 27, 2009 3:03 am |
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[quote]On Oct 23, 2:24 pm, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
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]
On 25 oct, 18:00, Igor Khavkine <igor... at (no spam) gmail.com> wrote:
[quote]That is not the right question to ask, because it has a false premise
embedded in it. The Fourier transform on R^n has many desirable
properties, some of which may be specific to this setup. The first
question to ask is Does an analogous transform with as many nice
properties *actually exist* in other contexts?
[/quote]
The false premise is, at Fourier's time there were no curved space.
But the gist of Fourier transform still exists, namely a basis change
in the linear space of functions.
[quote]Depending on the nice properties of the transform you want to preserve
and depending on the context (e.g., replacing R^n by a Riemannian
manifold), in many cases an analog of the Fourier transform does not
exist at all.
That is not to say that there are no useful generalization of the
Fourier transform; there are many. The kinds of generalizations that
people find useful are often grouped under the moniker of "harmonic
analysis".
[/quote]
Well, call it harmonic analysis, but that's the gist of Fourier
transform: another basis made of harmonic functions. An example is
the expansion in spherical harmonics, for curved coordinates but flat
space. As for generalizations, they don't necessarily have the same
application.
[quote]Here's a old post by John Baez that gives examples of some
of these generalizations:
news:7t8eac$abk at (no spam) charity.ucr.eduhttp://groups.google.com/group/sci.physics.research/msg/3b85eb9ddc0ad4e3[/quote] |
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| Jay R. Yablon... |
| Posted: Sat Oct 31, 2009 11:05 pm |
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Guest
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"Igor Khavkine" <igor.kh at (no spam) gmail.com> wrote in message
news:a53c2aa1-8563-48ee-b872-4f75234b4d4e at (no spam) a6g2000vbp.googlegroups.com...
[quote]On Oct 23, 2:24 pm, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
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?
That is not the right question to ask, because it has a false premise
embedded in it. The Fourier transform on R^n has many desirable
properties, some of which may be specific to this setup. The first
question to ask is Does an analogous transform with as many nice
properties *actually exist* in other contexts?
Depending on the nice properties of the transform you want to preserve
and depending on the context (e.g., replacing R^n by a Riemannian
manifold), in many cases an analog of the Fourier transform does not
exist at all.
That is not to say that there are no useful generalization of the
Fourier transform; there are many. The kinds of generalizations that
people find useful are often grouped under the moniker of "harmonic
analysis". Here's a old post by John Baez that gives examples of some
of these generalizations:
news:7t8eac$abk at (no spam) charity.ucr.edu
http://groups.google.com/group/sci.physics.research/msg/3b85eb9ddc0ad4e3
Igor
[/quote]
Thanks, Igor, I enoyed the Baez article.
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.
Also, am I correct to read this so as to say that xi and xi-bar are be
inverses, such that:
xi xi-bar = 1 ?
Thanks,
Jay |
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| Igor Khavkine... |
| Posted: Sat Oct 31, 2009 11:28 pm |
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Guest
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On Nov 1, 10:05 am, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
[quote]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.
[/quote]
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.
If you look at any paper on QFT on a specific space-time, a large part
of the work is always devoted to finding the appropriate Green
function, using every available trick from the book, exactly because
there is no one method that always works.
Igor |
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| Jay R. Yablon... |
| Posted: Sun Nov 01, 2009 7:36 am |
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Guest
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"Igor Khavkine" <igor.kh at (no spam) gmail.com> wrote in message
news:99f283b1-2498-4519-b7a2-a1359976c77f at (no spam) 15g2000yqy.googlegroups.com...
[quote]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.
If you look at any paper on QFT on a specific space-time, a large part
of the work is always devoted to finding the appropriate Green
function, using every available trick from the book, exactly because
there is no one method that always works.
Igor
I have been trying to gain some further familiarity with compact spaces,[/quote]
topological spaces, and the like. R^4 is a Euclidean space. Without
all of the jargon, and speaking in terms of properties of the metric
tensor g_uv, what sorts of non-Euclidean, 4-dimensional Abelian
(commutative element) spaces would fit the requirement of local
compactness, and so make Pontrjagin duality applicable ? And, are these
reasonable spaces within which to consider electrodynamics? What does
one lose or omit by restricting oneself to locally-compact space?
It seems that the requirement that curved spacetime with metric g_uv
must have Minkowski space as a tangent space, i.e. with vierbein V:
g^\mu\nu = V^mu_a V^nu_a eta_ab (1)
may qualify many of the curved, smooth, simply-connected spacetime
manifolds of general relativity as locally compact, and they are
certainly Abelian. Is it right to think that "locally compact" =
"locally Euclidean," and if not, what does such a view either include
that it should not or exclude that it should not?
Also, was I correct to read
http://en.wikipedia.org/wiki/Pontrjagin_duality#Fourier_transform so as
to say that xi and xi-bar are be inverses, such that:
xi xi-bar = 1 ? (2)
Thanks,
Jay |
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| Igor Khavkine... |
| Posted: Sun Nov 01, 2009 7:52 am |
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Guest
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On Nov 1, 6:36 pm, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
[quote]"Igor Khavkine" <igor... at (no spam) gmail.com> wrote in message
news:99f283b1-2498-4519-b7a2-a1359976c77f at (no spam) 15g2000yqy.googlegroups.com...
On Nov 1, 10:05 am, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
http://en.wikipedia.org/wiki/Pontrjagin_duality#Fourier_transform
This approach is limited to locally compact abelian groups. R^n
happens to be one, but in general a curved space-time is not.
I have been trying to gain some further familiarity with compact spaces,
topological spaces, and the like. R^4 is a Euclidean space. Without
all of the jargon, and speaking in terms of properties of the metric
tensor g_uv, what sorts of non-Euclidean, 4-dimensional Abelian
(commutative element) spaces would fit the requirement of local
compactness, and so make Pontrjagin duality applicable ?
[/quote]
You are missing the trees for the forest. In particular, from the
above, I can only conclude that your understanding of the term Abelian
is flawed. More importantly, in the generalization from the *vector
space* R^4 to curved space-times, the property that fails first the
property of being a group. One does not add or multiply points of a
manifold, simply because these operations are not part of its
definition.
Igor |
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| Jay R. Yablon... |
| Posted: Sun Nov 01, 2009 9:42 am |
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Guest
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"Igor Khavkine" <igor.kh at (no spam) gmail.com> wrote in message
news:7ea45173-0a6b-486a-a9b0-3872bf84816b at (no spam) l13g2000yqb.googlegroups.com...
[quote]On Nov 1, 6:36 pm, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
"Igor Khavkine" <igor... at (no spam) gmail.com> wrote in message
news:99f283b1-2498-4519-b7a2-a1359976c77f at (no spam) 15g2000yqy.googlegroups.com...
On Nov 1, 10:05 am, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
http://en.wikipedia.org/wiki/Pontrjagin_duality#Fourier_transform
This approach is limited to locally compact abelian groups. R^n
happens to be one, but in general a curved space-time is not.
I have been trying to gain some further familiarity with compact
spaces,
topological spaces, and the like. R^4 is a Euclidean space. Without
all of the jargon, and speaking in terms of properties of the metric
tensor g_uv, what sorts of non-Euclidean, 4-dimensional Abelian
(commutative element) spaces would fit the requirement of local
compactness, and so make Pontrjagin duality applicable ?
You are missing the trees for the forest. In particular, from the
above, I can only conclude that your understanding of the term Abelian
is flawed. More importantly, in the generalization from the *vector
space* R^4 to curved space-times, the property that fails first the
property of being a group. One does not add or multiply points of a
manifold, simply because these operations are not part of its
definition.
Igor
I am gathering that Abelian here is use differently than it is in[/quote]
contrasting, say, U(1) with the SU(N) Yang Mills groups. I would like
to understand this, though as you say, that is not the most important
thing.
More importantly, let me go back to to what Baez said in the link you
gave earlier:
"Some of the nice stuff works just because R^n is a Lie group
- this stuff is called harmonic analysis on Lie groups. A
good example is how you can decompose any function on a
compact Lie group like SU(n) or SO(n) into a linear
combination of matrix elements of irreducible representations.
This is called the Peter-Weyl theorem. . . .
The idea here is that even if your manifold doesn't have much
symmetry at all, it still looks *locally* like R^n, so you can
do a kind of local analogue of Fourier analysis on it.
Basically, the more symmetry your space has, the easier
it is to do something like Fourier analysis on it. Above
I listed 4 of the main branches of harmonic analysis, in
order of decreasing symmetry."
It seems to me like Baez is suggesting that a curved spacetime manifold
which is locally Minkowski "still looks *locally* like R^n," (here, R^4)
and so we can do "a kind of local analogue of Fourier analysis on it."
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?"
It also looks like if we restrict ourselves to Lie Groups on a curved
manifold with is locally Minkowskian and use a suitable Haar measure,
that we can get some generalization of Fourier analysis to work there.
And, finally, of course two points displaced on a manifold cannot be
dealt with as if they were in a Euclidean vector space. But, does not
the whole parallel transport analysis which underlies the curvature
tensor R^u_abv supplant and generalize the ability to deal with vectors
in a consistent way, on a curved manifold?
Thanks,
Jay |
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| Arnold Neumaier... |
| Posted: Sun Nov 01, 2009 10:31 am |
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Guest
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Igor Khavkine wrote:
[quote]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.
[/quote]
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 |
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| Jay R. Yablon... |
| Posted: Tue Nov 03, 2009 10:59 am |
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"Arnold Neumaier" <Arnold.Neumaier at (no spam) univie.ac.at> wrote in message
news:4AED7E86.9060801 at (no spam) univie.ac.at...
[quote]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[/quote]
"general general relativity," with an intention to suggest that for some
subset of general curved spacetime manifolds, one could do a proper
harmonic analysis and arrive at closed form Green functions (which I am
in the midst of attempting at another post here started on 10/30, see
http://jayryablon.files.wordpress.com/2009/10/path-integration-of-the-maxwell-action-2-4.pdf).
But, as I read the Baez post at
http://groups.google.com/group/sci.physics.research/msg/3b85eb9ddc0ad4e3
which Igor recommended, I am of the impression that 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, or
that any curved space for which this will work must have certain
symmetries, and, if the latter, then the question becomes, "what are the
requisite symmetries?"
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, and that we need to find some other
foundation for quantum field theory if we wish to reconcile quantum
theory with gravitation. If on the other hand, there is some subset of
manifolds for which this works, then perhaps what this means is that the
path integral formulation remains valid in curved spacetime, but in the
process forces the elimination of certain curved manifolds from
consideration which do not have the requisite symmetry. Given that
physics is a process of elimination of many mathematical possibilities
which are not physically permitted down to those select few which are
physically-permissible, the restriction to manifolds with certain
symmetries that do permit closed form Green function derivation may not
be a bad thing at all, and may in fact be driving us toward what can be
physically real while winnowing out that which cannot be.
Your thoughts?
Thanks.
Jay |
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| Igor Khavkine... |
| Posted: Tue Nov 03, 2009 6:04 pm |
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Guest
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On Nov 1, 8:42 pm, "Jay R. Yablon" <jyab... at (no spam) nycap.rr.com> wrote:
[quote]"Igor Khavkine" <igor... at (no spam) gmail.com> wrote in message
news:7ea45173-0a6b-486a-a9b0-3872bf84816b at (no spam) l13g2000yqb.googlegroups.com...
You are missing the trees for the forest. In particular, from the
above, I can only conclude that your understanding of the term Abelian
is flawed. More importantly, in the generalization from the *vector
space* R^4 to curved space-times, the property that fails first the
property of being a group. One does not add or multiply points of a
manifold, simply because these operations are not part of its
definition.
I am gathering that Abelian here is use differently than it is in
contrasting, say, U(1) with the SU(N) Yang Mills groups.
[/quote]
The usage is the same in both contexts. Yet you still used it
incorrectly.
[quote]It seems to me like Baez is suggesting that a curved spacetime manifold
which is locally Minkowski "still looks *locally* like R^n," (here, R^4)
and so we can do "a kind of local analogue of Fourier analysis on it."
[/quote]
Yes, the most general version of this idea that I know of is called
"microlocal analysis". It's hard.
[quote]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?"
[/quote]
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. If you want to restrict yourself to only these
manifolds, you lose, for example, most cosmological space-times along
with an infinitude of others.
[quote]And, finally, of course two points displaced on a manifold cannot be
dealt with as if they were in a Euclidean vector space. But, does not
the whole parallel transport analysis which underlies the curvature
tensor R^u_abv supplant and generalize the ability to deal with vectors
in a consistent way, on a curved manifold?
[/quote]
I presume that your rhetorical question aimed to identify a loophole
in my statement that curved manifolds are usually not groups. Well,
the point you've made has nothing to do with the reason that I've
given to back up my statement. Since that reason remains valid, you've
gained no loopholes.
I should emphasize again that QFT in curved space-times is a mature
field described in books and review papers. If you want to learn about
it, you should look them up and read them. Fixating on non-existent or
unuseful generalizations of Fourier transforms is much less
productive.
Igor |
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| Arnold Neumaier... |
| Posted: Tue Nov 03, 2009 10:06 pm |
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Guest
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Igor Khavkine wrote:
[quote]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.
[/quote]
You seem to have infinitely many fingers. There are many symmetric
spaces. Only upon restricting to 4 dimensions the number becomes finite.
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.
On the other hand, the less symmetries there are the more difficult
is the analysis, and only the symmetric space case is fully developped.
Arnold Neumaier |
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| Arnold Neumaier... |
| Posted: Tue Nov 03, 2009 10:38 pm |
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Guest
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Jay R. Yablon wrote:
[quote]"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
[/quote]
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.
[quote]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,
[/quote]
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.
[quote]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,
[/quote]
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 |
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| Igor Khavkine... |
| Posted: Tue Nov 03, 2009 11:30 pm |
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Guest
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On Nov 4, 9:06 am, Arnold Neumaier <Arnold.Neuma... at (no spam) univie.ac.at>
wrote:
[quote]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.
[/quote]
I do indeed have only finitely many fingers. :-)
[quote]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.
[/quote]
Do you mean de Sitter or anti-de Sitter spaces, 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.
Igor |
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