| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
06 Mar 2006 08:27:43 PM |
| Object: |
String theory as quantum gravity |
When physicists first tried to quantize gravity, the natural approach
was to create the analog of a quantum field theory of general
relativity (ie- they merely wrote down the Lagrangian of a spin two
particle and tried to calculate the values of the resulting Feynman
diagrams). What did they get? Nonsense results where intermittent
integrals over momentum diverged to infinity in a way that could not be
remedied using the usual cancellation methods. Surprisingly, effects
in this theory from very large momentum (ie- small length scales)
contributed to the theory in a nontrivial way that would seem to
destroy its mathematical soundness.
Because of this, physicists have resorted to much more complicated
areas of research. In particular, one area called "string theory" has
stood out. String theory roughly solves the problems from the higher
momentum divergences in the original theory by treating particles as
strings. A nonzero length essentially imposes a length scale (and
therefore momentum) cutoff on the same integrals that blew up in the
original theories. Without these divergences, one could, at least in
principle, use this theory to perform calculations in this theory of
gravity.
Unfortunately, string theory comes with a price. It also was plagued
with mathematical inconsistency. It turns out that the only way string
theory itself is mathematically consistent is to place it in a 10
dimensional universe (or 26 or even 11 depending on what you are
studying). Trying to reconsile this with our own 4 dimensional
universe has been a rough spot for string theorists. Although this may
not be apparent to the lay person, this issue is in fact trivial when
compared to mathematical inconsistancy, and thus string theory remains
the leading candidate of quantum gravity.
In the last few years, something interesting has been noted. Any
semiclassical theory of gravity has, in fact, a smallest length scale.
This is brought about because any attempt to measure a resolution
smaller than the plank scale does in fact cause the formation of a
black hole. Of course if the mass of the particles used in the
experiment is small enough, the resultant black hole would appear
nothing like the black holes of hollywood. A black hole formed from a
plank scale configuration would itself be very tiny, and would
immediatly disintegrate due to Hawking radiation. To the rest of the
universe, it would look not unlike any other particle collision and
decay. Internally, however, the interaction would be a very different
beast- it would have its own schwarzshield radius within which all
information would remain unreachable. For all experimental purposes,
anything occuring within this radius would be metaphysics, unreachable
and therefore nonexistent. For all practical purposes, there would be
a smallest length scale.
My question is, doesn't this smallest length scale suggest a more
elegant solution to the quantization of gravity? A simple quantization
of a spin 2 particle would not in fact diverge because length/momentum
have a cutof imposed by general relativity. It is very possible that
the existence of a smallest length scale implies that the universe is
composed of a countable number of degrees of freedom, so out universe
might not even be described by a continuous field theory.
Of course, actually calculating anything is still difficult, as a
perturbative Feynman diagram approach would probably fail miserably in
the face of the tremendous nonlinearities of general relativity. Loop
Gravity seems to be a realization of a quantized theory of gravity that
has a smallest volume, although I don't know enough about loop gravity
to judge its consistency with the notions mentioned in this posting.
Also, Holography seems to indicate that there is an alternate
description of any theory including general relativity (in a different
number of dimensions!) that automatically incorporates black hole
formation, and would perhaps be more useful in calculating perturbative
quantities.
But if in fact the details could be worked out, and be shown
consistent, this would certainly be a more attractive area or research
than 10 dimensional string theory.
thanks
-Inquirydog
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| User: "" |
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| Title: Re: String theory as quantum gravity |
09 Mar 2006 01:24:32 PM |
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wrote:
[...]
In the last few years, something interesting has been noted. Any
semiclassical theory of gravity has, in fact, a smallest length scale.
This is brought about because any attempt to measure a resolution
smaller than the plank scale does in fact cause the formation of a
black hole.
Well, maybe. There are a bunch of "thought experiments" that claim
to demonstrate this, not all so new -- the earliest reference I know
of for this kind of argument is Pauli, Helv. Phys. Acta Suppl. 4 (1956)
69, and there's a 1957 review by Deser, Rev. Mod. Phys. 29 (1957) 417.
But there are other thought experiments that suggest that a much smaller
resolution is possible -- see, for example, Amelino-Camelia and Doplicher,
http://arxiv.org/abs/hep-th/0312313, Class.Quant.Grav. 21 (2004) 4927.
[Note: the Web reference is to the preprint arXiv; other papers of the
form hep-th/xxxxxxx or gr-qc/xxxxxxx can be found at arxiv.org as well.]
The basic problem is that if you don't have a real quantum theory of
gravity in which to test these ideas, it's not clear whether this kind
of argument means "quantum gravity really has a smallest length," or
whether it means "I'm not clever enough to think of an experiment
that would probe a smaller length."
It's not even entirely clear what such a minimum length means. Already
in special relativity, length is observer-dependent (Lorentz contraction
and all that). For what observer should the Planck scale be a minimum
length? There are possible ways around this -- one can have, for example,
a Lorentz-invariant length operator whose minimum eigenvalue is the
Planck length (see Rovelli and Speziale, gr-qc/0205108, Phys. Rev. D67
(2003) 064019), but it's not obvious that this would cut off divergences.
[...]
My question is, doesn't this smallest length scale suggest a more
elegant solution to the quantization of gravity? A simple quantization
of a spin 2 particle would not in fact diverge because length/momentum
have a cutof imposed by general relativity.
This is an old hope, going back at least to the Pauli and Deser papers
I cited above. It might even be right. But a lot of people have tried
very hard, and no one has yet managed to show that it works.
For one thing...
[...]
Of course, actually calculating anything is still difficult, as a
perturbative Feynman diagram approach would probably fail miserably in
the face of the tremendous nonlinearities of general relativity.
Right. In the 1960s and 1970s, a bunch of people worked on this, and
actually showed that certain infinite sums of Feynman diagrams converged,
with divergences of individual diagrams disappearing when you summed the
whole series. See, for example, Isham, Salam, and Strathdee, Phys. Rev.
D3 (1971) 1805. The problem is that this could only be done for a certain
particular class of Feynman diagrams, and no one knew (or knows) how to
extend this to the full theory.
There's some recent work by Reuter and collaborators revisiting these
ideas in a slightly different way by looking at the renormalization
group in quantum GR. See, for example, Reuter and Schwindt, hep-th/0511021,
JHEP 0601 (2006) 070, and Lauscher and Reuter, hep-th/0110021, Class.
Quant. Grav. 19 (2002) 483. It's still very early to evaluate this
approach.
The other thing to keep in mind is that some of the problems of quantum
gravity don't have that much to do with ultraviolet divergences. For
example, the basic symmetry of quantum gravity is (presumably) general
covariance, and it can be shown that this symmetry implies that there
are no strictly local observables. (See, for example, Torre, gr-qc/9306030,
Phys. Rev. D48 (1993) 2373.) This makeslots of things difficult... There's
a huge literature on the related "problem of time" -- see, for example,
the review by Kuchar at www.phys.lsu.edu/faculty/pullin/kvk.pdf .
Loop Gravity seems to be a realization of a quantized theory of gravity
that has a smallest volume, although I don't know enough about loop
gravity to judge its consistency with the notions mentioned in this
posting.
I'm not sure anyone knows the answer to this yet. Loop quantum gravity
does have various minimum sizes, and does give finite expressions for
various operators that might be divergent in conventional quantum gravity,
but it's not clear that this is cause and effect. There's a nice "outsiders'
review" by Nicolai et al., hep-th/0501114, Class. Quant. Grav. 22 (2005)
R193, that's worth a look.
Steve Carlip
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| User: "" |
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| Title: Re: String theory as quantum gravity |
09 Mar 2006 06:03:19 PM |
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Thanks for the thought out and well cited response. I am interested in
looking through what you have pointed me to, and it will probably take
me some time to get back with more comments. There were a couple of
things that I wanted to mention right now. however.
Well, maybe. There are a bunch of "thought experiments" that claim
to demonstrate this, not all so new -- the earliest reference I know
of for this kind of argument is Pauli, Helv. Phys. Acta Suppl. 4 (1956)
69, and there's a 1957 review by Deser, Rev. Mod. Phys. 29 (1957) 417.
But there are other thought experiments that suggest that a much smaller
resolution is possible -- see, for example, Amelino-Camelia and Doplicher,
http://arxiv.org/abs/hep-th/0312313, Class.Quant.Grav. 21 (2004) 4927.
hep-th/0505144 claims to have a proof of this from first
principles, and in particular refer to the case a LIGO like apperatus
where the measuring device is much larger than the scales being
resolved. Of course they could be mistaken, but I just wanted to point
out a particularly revelant paper.
It's not even entirely clear what such a minimum length means. Already
in special relativity, length is observer-dependent (Lorentz contraction
and all that). For what observer should the Planck scale be a minimum
length?
I would think we are talking about the proper length. Any other
definition of length is coordinate system dependent and should
therefore not show up in the equations of physics. That being said,
you have me interested in looking at the corresponding referrences now.
Right. In the 1960s and 1970s, a bunch of people worked on this, and
actually showed that certain infinite sums of Feynman diagrams converged,
with divergences of individual diagrams disappearing when you summed the
whole series.
I am not even sure that we should expect the Feynman diagrams to
converge at all, even if the theory is sound. I am sure I remember
back from my math classes examples of simple well defined functions
whose Taylor series don't converge at all at points (after all, radius
of convergence can vary). Quantized GR is a much more complicated
beast, and the fact that it wouldn't supprise me at all given how
nonlinear GR is. The theory is so nonlinear that the holography
adherents claim the nonlinearities change the actual dimension of the
space the theory is in!
Again thanks for the references, I plan on looking at many of them, in
particular I am interested in the nonlocal observables paper and the
intro to loop gravity.
thanks
Inquirydog
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| User: "" |
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| Title: Re: String theory as quantum gravity |
13 Mar 2006 02:53:29 PM |
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wrote:
[...]
Again thanks for the references, I plan on looking at many of them, in
particular I am interested in the nonlocal observables paper and the
intro to loop gravity.
If you look at the "outsider's" intro to loop quantum gravity, you
ought to also look at an "insider's" version, such as gr-qc/0410054.
For a more general overview, try my paper, gr-qc/0108040. Also, since
you cited a paper claiming the existence of a minimum length on the
order of the Planck length, you should definitely look at hep-th/0312313,
which I believe provides a counterexample.
Steve Carlip
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