Re: Symmetries of general relativity

on 28/11/03 11:40 pm, Matt Reece at [EMAIL PROTECTED] wrote:

<snip>

> In GR, there is a Lorentzian metric. Hence the tangent space at each
> point has an inner product with an SO(3,1) symmetry. Accordingly, on
> any given Lorentzian manifold solving the equations of GR, when doing
> particle physics, it still makes sense to classify particles (more
> precisely, classical fields) as (Lorentz) vectors, spinors, and so on.

This is true.

<snip>

> In this way the usual actions of particle physics generalize
> straightforwardly to any background that is a solution of GR, yes? So
> perturbation theory around any such background looks more or less like
> perturbation theory around Minkowski space, in the sense that the
> particles are still classified by Lorentz group representations and
> they can be coupled in the usual way. Is this correct so far?

This is sort of true. Particles can still be classified as scalar, spinor,
vector etc.

> 
> On the other hand, a given background -- say, AdS_5 -- has an isometry
> group that is not the Lorentz group. Is there anything useful to gain
> in particle physics from considering the representation theory of this
> group?

Yes, because the space of classical solutions carries a representation of
this group, and the quantum field theory is built on the space of classical
solutions.

> Apologies if this is a dumb question;

No, it's not a dumb question at all. The relationship between QFT on
Minkowski spacetime and QFT on more general spacetimes is not obvious. The
whole concept of "particle" (and, particularly, the usefulness of the
particle number operators) gets a bit shaky.

<snip>

> I suppose I should go read the parts of Wald's book on spinors and QFT
> in curved space, which I haven't yet studied. It doesn't seem like any
> of the particle physics or QFT texts I'm aware of treat issues of
> curved spacetime at all, nor do classes typically, but a lot of
> current phenomenology deals with things like AdS. Aren't there
> nontrivial issues to worry about it in doing such things? If so, where
> do people tend to learn them? Are there standard papers to consult on
> particle physics in curved spacetime?

If you are interested, you should definitely take a look at some standard
texts:

Birrell and Davies, _Quantum fields in curved space_, Cambridge University
Press, 1982 (there may be later editions -- I don't know).

Wald, _Quantum Field Theory in Curved Spacetime and Black Hole
Thermodynamics_, University of Chicago Press, 1994 (again, there may be
later editions which I don't know about).

As usual, Wald's treatment is spare and elegant. The other book has lots of
examples, but is less unified.

Tim