Re: Symmetries of general relativity

[EMAIL PROTECTED] (Matt Reece) wrote in message news:<[EMAIL PROTECTED]>...
> Marc Nardmann <[EMAIL PROTECTED]> wrote in message news:<[EMAIL PROTECTED]>...
> I had misspoken in my own response to one of Lubos's
> posts, by saying that the Lorentz group is a local symmetry of GR. Let
> me try to say instead what I really meant by this statement:
> 
> 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.
> More precisely, these are sections of the tangent bundle of the
> manifold, or of the spinor bundle (providing there is no obstruction
> to its existence -- I'm ignorant of when this is true -- is there some
> characteristic class one must compute?),

Yes, it's called the second Stiefel-Whitney class.

> or of some other relevant
> bundle.
> 
> 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?
> 

So far, this sounds correct. This is indeed how there is
a local action of the Lorentz group.


> 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?
> 

Of course. The particle spectrum must decompose properly into
representations of the isometry group. e.g. in Minkowski space
we should have particles of all possible momenta. In AdS_5, the
particle spectrum must fall into representations of SO(4,2).
This must be true for each of the fields.



> Apologies if this is a dumb question; I haven't thought about this
> issue much. But I have recently become interested in the Higgsless
> orbifold theories of Csaki et. al., which live in a braneworld model
> in AdS_5 (like Randall-Sundrum) and this made me wonder how particle
> physics should work in curved spaces, and in particular whether the
> isometry group of the space plays any role.

If you're referring to the papers where the EW symmetry is
broken by noncommuting boundary conditions, the problem is that
the boundary conditions on the fields are very complicated and
non-geometrical. So the spectrum is not simply related to the
geometry any more. For example, suppose the boundary condition
sets the field to zero. Then there is no particle spectrum
associated to this field, no matter what the isometry.


> 
> 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?
> 

The standard book is Birrell and Davies, "Quantum Fields
in Curved Spacetime".

The issues are (mostly) not particularly difficult. The main problem
is solving the wave equation for whatever geometry you're
concerned with. If you can solve the wave equation with a delta
function source, you've got the propagator, and the vertices
are straightforward. 


> Thanks,
> 
> Matt Reece

Arvind.