Re: Canonical quantization and covariance

markwh04@xxxxxxxxx wrote in message news:<1119908430.922891.116590@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>...
> Thomas Larsson wrote:
> > What I am after is a new formulation of old quantum mechanics which
> > is manifestly compatible with relativity.
>
> How compatible? In the general case, manifolds can be anything,
> including globally non-hyperbolic. In the presence of global
> non-hyperbolicity, there can be no "t" coordinate. In the presence of
> a operator metric, global [non-]hyperbolicity becomes state-dependent
> so you don't even have the ability to define "global hyperbolicity",
> itself!
>

I'm not saying anything about global issues, which are tricky with
operator-valued fields anyway. Locally, the conventional Hamiltonian
formulation singles out a privileged time direction, which path integrals
do not. My point is that it is possible to construct phase space locally
in a way which does not single a time coordinate.

Let me say something about the context. If we forget about the embedding
in spacetime, the world*** theory of the bosonic string can be viewed as
a toy model of 2D gravity. As such it is very important, because it may be
a blueprint for how real 4D gravity should be quantized. This is the
viewpoint e.g. in section 5.1 of http://www.arxiv.org/abs/hep-th/0501114 .
An obstacle is that the constraint algebra in Hamiltonian gravity is a
mess, and its representations are not understood. However, if we instead
work in the covariant phase space, the constraint algebra consists of
arbitrary 4-diffeomorphisms. Which is good for me, since I understand its
representation theory.

Thus, understanding how to build quantum reps of the 4-diffeo algebra may
be viewed as first step in the programme implicitly suggested by
Nicolai-Peeters-Zamaklar in the reference above. Another step is to
explicitly construct the covariant phase space. To that end, I start from
the antifield formalism, well described e.g. in
http://www.arxiv.org/abs/hep-th/9412228 . My formalism differs from the
antifield formalism in that I do not only include fields, ghosts and
antifields, but also their conjugate momenta. This means that there is a
Poisson bracket, so we can quantize canonically. One can readily check
that the Poisson bracket on the extended phase spaces defines a good
bracket in cohomology.

.

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