Toward a Common Solution to the GR + QFT & "Wave function collapse"Problems

Rock Brentwood

Uncle Al wrote:
> GR (c=c, G=G, h=0) cannot be unified with quantum field theory (c=c,
> G=0, h=h).

It can. The results are surprising in several respects.

First, though it is not well-known, it's actually possible to do
Hamiltonian dynamics without splitting spacetime into space+time. This
was always one of the main obstacles to incorporating a field theory
that truly respected the spirit of relativity.

However, there are substantial differences. The covariant Hamiltonian
increases the velocities and momenta componets 4-fold, pulling a full
Legendre transform
H = sum (p^m dq/dx^m) - L.
The Poisson bracket structure has to be modified. You don't have a
nice [p,q] relation anymore, but rather, something involving
differential forms [sum p^m dS_m, q], where dS_m is the (n-1) surface
element orthogonal to coordinate axis x^m.

How quantum theory generalizes in terms of the covariant Hamiltonian
isn't yet fully known, but this is the foundation ultimately of the
resolution between how quantum theory and General Relativity handle
time. The fix, to put it simply, is on the quantum side, fixing the
way it approaches quantization through modification of its Hamiltonian
foundation.

Second, there are clear-cut issues that come about as a result of
trying to quantize gravity; these show up some novel principles. They
lead directly to the notion of hybrid classico-quantum theories (i.e.
quantum theories with superselection).

The metric is represented by the vierbein frame field. This field
determines, locally, which frames are inertial.

A fluctuation in this field leads to a fluctuation in which frames are
inertial. A superpositoin of two states with slightly different
metrics results in a superposition of two vacuua that disagree
slightly one which frames are inertial. One sees the others' frames as
accelerating.

As is well-known from the Unruh-Davies effect, an accelerating vacuum
does not reside in the same sector as the inertial vacuum. They cannot
coherently superpose! They can only participate in incoherent
superpositions.

Thus, superselection enters directly into the core of quantum theory.
The fluctuation of the vierbein field leads to an effective mean time
to superselection or (as Penrose put it, as mean time to "objective
wavefunction reduction"). Penrose's "objective reduction" is just a
fancy way of saying "gravitationally induced superselection".

Thus, the vierbein field can no more be quantized as a bona fide
quantum field, than phonons in a solid can be. They enter the picture
as quasi-particle modes.

This not only buttresses the conclusion Jacobson posed in his 1995
paper, but fall right into the came of Sardanashvily's "gauge
gravitation" formalism .. which also brings us full circle, since this
formulation is set squarely in the language of the covariant
Hamiltonian approach.

Sardanashvily adds in an extra point that had been little noted. The
vierbein is associated with the fermion field. It is the affine part
of the gauge field, while the connection coefficients are the spin or
curvature part.

However, the fermion field does not transform under the full local
GL(4) symmetry group of the manifold. It selects out an inertial frame
-- namely, the vierbein field itself. It only transforms locally under
SO(3,1). The broken symmetry GL(4) -> SO(3,1) yields 10 Higgs-like
modes that are none other than the 10 components of the metric. These
parametrize between the different vacuum sectors. Each sector being a
separate state-space that cannot coherently superpose with the states
of any other sector.

In other words, the vierbein field is essentially classical and can
only be quantized as quasi-particle modes.

In the process of uncovering this revelation one also sees a golden
path laid out to one of the major issues of quantum theory: how do
coherent superpositions undergo the effective "superselection"
associated with measurements, aka "wave function collapses"?

How is playing the role of the handmaiden of all superselection?

The answer is the vierbein. This consequently leads to what
Sardanashvily calls the "fermion complex" Each separate "setting" of
the vierbein defines a subset of possible fermion fields that reside
in the sector associated with the field. But different vierbeins give
you different sets, and the fermion fields that reside in different
sectors cannot superpose coherently.

One way this may enter quantum field theory is directly into S-matrix
theory. There may be a superselection on particle number between
Feynman different diagrams that have different external legs.

I'm not entirely sure, but I think that it's a hidden or tacit
assumption already made when carrying out actual calculations to treat
the different "tree sectors" of the Feynman diagrams as residing in
different sectors.