Re: Looking for a concept

<thomas_larsson_01@xxxxxxxxxxx> schrieb
Ilja Schmelzer wrote:

AFAIU, the extended group is a different group, therefore
the resulting quantum theory does not have the same symmetry
group as the classical theory. Doesn't that mean that you also
have to give up "Einstein's insight" about the symmetry group?

You have to give up, or at least relax, that diffeomorphisms form a
*gauge* symmetry.

Fine. But how to make this sure?

To clarify: I believe, for (maybe) completely different reasons,
that we have to give up diffeomorphism symmetry in quantum
gravity. I would like to understand your reasons for giving it
up.

But the extended diffeomorphism group is still a
symmetry.

Of course. If there is some theory, there is also some symmetry
of the theory.

From my viewpoint, it is just the fundamental quantum form
of diffeomorphism symmetry. So I don't think of it as another
symmetry, but rather as the correct way to describe diffeomorphisms
on the quantum level.

Whatever the correct group is, we share the belief that it is different
from the diff group. We also share the belief that in the classical limit
the diff group should somehow appear as an (approximate) symmetry.

The relation between the diffeomorphism algebra and the
multi-dimensional Virasoro algebra is completely analogous to the
relation between conformal symmetry and the ordinary Virasoro algebra
(well, two copies thereof). People still use the phrase conformal
symmetry when they really mean Virasoro symmetry, and I follow the
same practise.

Is it possible to explain the difference in terms of configurations
equivalent in one theory but not equivalent in the other?

In the classical limit, the extension vanishes, and some physical
degrees of freedom decouple and become redundant gauge variables.

I'm not about the classical limit hbar->0.

More about a description in classical terms similar to the path
integral as a description in terms of trajectories.

This is the sort of argumentation I have used for myself.
If there would be complete diff invariance, the path
integral for finite times

I(t0,t1) = int_t0^t1 something D(g_mn(x,t))

would be nonsensical, because it could not depend
on t1-t0. I propose to give up diff invariance and to
use harmonic gauge to define this integral.

But weaker forms of breaking the symmetry are possible
too, for example, introducing some oberser trajectory and
then

I(t0,t1) = int_t0^t1 something D(g_mn(x,t),x(t))

with the condition that t1-t0 is the proper time along x(t).

The free string in D dimensions is an excellent toy model. It is also
very relevant, because in the Polyakov formulation it is nothing but
2D gravity coupled to D scalar fields. Classically, all three
components of the world*** metric are gauge variables, i.e.
redundancies. After quantization, the trace of the metric becomes
physical, unless D=26.

Means, that is a quite subtle effect, impossible to understand without
doing the computations which lead to the exceptional D=26?

By analogy, one
would expact that QG in 4D has more physical degrees of freedom
than the two transverse graviton polarizations.

Means, there is also not more than "by analogy"?

Ilja

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