Re: quantum mechanics and curvature of space-time

From: Igor Khavkine (igor.kh_at_gmail.com)
Date: 09/28/04 

Date: Tue, 28 Sep 2004 15:20:00 +0000 (UTC)

alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0409241312.3e02324@posting.google.com>...
> The amplitude of a wavefunction squared is proportional to the
> probability of finding a particle at a position in space.
> This squaring of the amplitude is often compared to the intensity
> (energy density) of an electromagnetic wave.
> The stress energy momentum tensor of general relativity contains
> the term Too - the energy density.Can Too be related to the
> square of the amplitude of the quantum mechanical wavefunction?
> Is Too (and therefore the curvature of spacetime due to energy
> density) proportional to the probability of finding a particle at a
> position in space?

Instead of wondering how T_{00} could be related to the wavefunction,
one can simply calculate it. T_{00} is a function of the dynamical variables
of the theory (a function of its momentum for a single particle) and as
such is an observable. Observables are represented as hermitian operators
that we can calculate expecration values of. What you are looking for
is <T_{00}> whose relation to the wavefunction is quite obvious.

As to how <T_{00}>, or in more generality <T_{ij}>, affects the equations
of motion for the gravitational field, I don't really know. The biggest
problem is that gravity coupled to matter must be consisten (i.e. the
induced curvature induces the right equations of motion for matter)
and that's hard to do when part of the system is treated classically and
the other quantum-mechanically. But I do remember many of the details
subtleties were hashed out between Arnold Neumaier and Arkadiusz Jadczyk
in this thread: http://groups.google.ca/groups?th=b7be680c23c06f9d

Igor

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