How To Do (Achronal) GR + Quantum Theory ... Without Quantum Gravity
- From: whopkins@xxxxxxxxxxx
- Date: Fri, 6 May 2005 13:47:41 +0000 (UTC)
These are a few rambling remarks outlining the big picture I'm trying
put together on building up a combined infrastructure for GR and
Quantum Theory.
In a way it follows up on a recent article where I provide a rough
proof deriving the structure of a canonically quantized theory (with a
quadratic Hamiltonian) almost ab initio, using the axioms:
(1) 2nd order law of motion
q''(t) = a(q(t),q'(t))
(2) configuration space finite dimensional
q = (q1,q2,...,qN)
(3) equal-time commutators 0
[q(t),q(t)] = 0 for all t
(4) c-number coordinate-velocity commutators
[q(t),q'(t)] c-numbers.
A more detailed writeup, following up on Hojman and Shepley's 1990
result, discussing the development of the desired structure can
currently be found in
"On the Quantum Dynamics of Moving Bodies"
currently listed under
http://federation.g3z.com/Physics/Index.htm
Here, the structure of a hybrid classical quantum theory is derived in
which the quantum part has the above structure, along with
superselection sectors parametrized by the classical part. The major
deficit of the result is that the classical subsystem is essentially
external. No automorphic evolution (which includes unitary evolution
as a special case) can house communications in the direction (quantum
-> classical); only in the opposite direction.
One approach that's been taken to handle this deficit was by Jadczyk,
discussed in some detail in
http://quantumfuture.net/quantum_future/papers/9406204/topics.htm
to post, ab initio, a hybrid continuous/stochastic framework for
combining a Schroedinger and Lueders evolution for quantum states.
In this context, regarding the result I establish, he might say (for
instance) "this is nice and interesting but does not fully incorporate
the issues I've brought up in my work about the basic need to adopt a
[piecewise] stochastic non-automorphic dynamics
as a *postulate*".
And that's where I want to step off from with the following comments.
In fact, these extra considerations may very well prove to be
unnecessary. The extra stuff being put in by hand may already come out
automatically as a result (and "extra stuff" includes even the
Einstein's equations!), by covering spacetime by compact locally
hyperbolic regions, each enclosed in a local Rindler horizon and then
treating the state within a given region as if it were the
phase-averaged relative state, cut at the horizon, of a universal pure
state in a globally hyperbolic universe containing the region in
question as a submanifold.
The requirement that different overlapping regions fit together
consistently yeilds Einstein's equations, via the Raychaudhuri
equation, if one postulates the Bekenstein bound on each horizon (a
result established in a somewhat more restrictive context by Jacobson
in 1995). This extra requirement (of treating each region as if it
were embedded in a globally hyperbolic spacetime) represents a kind of
correspondence principle where, here, the corresponding is between the
general case "no global t coordinate" and the limiting special case "a
global t coordinate".
The vacuum state of each region is thus a thermal Rindler state,
evolution is already essentially stochastic, the randomness associated
with the flow of information through the horizon, and the thermal noise
associated with heat bath corresponding to the thermal state ensuring
that the regions fit together consistently, free of time-traveller
paradox, even when the spacetime is globally achronal and globally
non-hyperbolic, lacking any Cauchy surfaces or global "t" coordinate.
Evolution within each region is done through the region's homotopy
parameter, which is taken as the local "t" parameter. The regions can
always be set up in such a way that each region R has a boundary dR
that is purely spacelike, comprising a (N-1)-chain of the form dR = R+
- R-, with R+, R- spacelike (with respect to the local time causal
structure of the region), sharing the same (N-2)-boundary -- the
"anchor" H (the local Rindler horizon, itself), and being the extreme
surfaces of the homotopy with
R = union { R(t): t in [0,1] }
R(0) = R-, R(1) = R+
dR(t) = H = the local Rindler horizon for all t in [0,1].
The state space is thought, fictitiously as being that of a family of
Cauchy surfaces
C(t) = R(t) union C0
C0 = C(t) - R
of a hypothetical globally hyperbolic spacetime containing R as a
submanifold-with-boundary. The phase-averaging is that which averages
out all the "external" modes associated with the fictitious surface C0.
This gives you a thermal Rindler state based on the horizon H which
I've termed above the "anchor" of the region R.
Jacobson showed that the requirement for self-consistency, plus the
imposition of the Bekenstein bound on each H, forces a kind of
gravitational lensing effect which recovers the Einstein field
equations in the classical limit. The novelty of what I'm describing
in relation to Jacobson is that he still worked within the framework of
a global "t" coordinate, whereas I'm not only extending his result to
globally non-hyperbolic spacetimes, but *exploiting* the result with
the essential introduction of a thermal vacuum state to ensure
consistency free of time-travel paradox in the case where the global
structure is achronal.
So, in one extreme case, one can have two overlapping regions of this
type where a surface seen in one region is spacelike, and seen in the
other region is mixed or timelike. The interference associated with
the non-commutativity of operators in the second region will translate
to vacuum fluctuations in the heat bath of the first region where the
corresponding operators are spacelike-separated and commute.
The need to handle the cases of achronal spacetimes, removes the
bugaboo of the global "t" coordinate, and more than anything, is what
makes eschewing the hypothesis of a cosmological pure quantum state and
cosmological configuration space in favor of local regions, local
configuration spaces, with quasi-Rindler mixed state vacuua.
So, we have the outline of a comprehensive theory along the following
lines:
(a) the boundary principle, as described in the article "Most
Fundamental Principle of Modern Physics" Here, the boundary principle
will state that the dynamics associated with the configuration
variables of region R are uniquely determined by the boundary data on
R+ and R-.
(b) the imposition of the Bekenstein bound on each local Rindler (N-2)
horizon encompassing a compact spacetime N-region as its "anchor".
(c) the imposition of equal-time commutation relations for the
configuration coordinates of each region, with the time being the local
homotopy parameter of the region
(d) the requirement that the [q, dq/dt] commutators be c-numbers.
As Jacobson ultimately showed, the imposition of (b) yields Einstein's
equations in the classical limit. The requirement also means that the
configuration space associated with systems in each such region will be
finite dimensional.
As I described a short while ago in the article mentioned in (a), the
boundary principle leads to the existence, locally, of a 2nd order law
of motion for the configuration coordinates of the system.
And as I showed in "The Quantum Dynamics of Moving Bodies" writeup, the
existence of a 2nd order system for a finite dimensional configuration
space, combined with the requirements (c) and (d) yields, a quantum
system canonically quantized by a Hamiltonian quadratic in the
conjugate momenta, coupled to external classical superselection
parameters which are to be associated with the horizon modes.
The result is a spacetime covered by locally hyperbolic regions, each
described by a quantum theory with a thermal vacuum state, each having
Einstein's field equations in the classical limit.
The extra consideration concerning the thermal vacuum state,
additionally, provides the tie-in to the prior discussion I posted here
a while back concerning a more appropriate treatment of negative energy
states, the spectral postulate of axiomatic quantum field theory, and
particles vs. holes.
The devil in all the foregoing, of course, is in the details. But the
light at the end of the tunnel shining from the panoramic vista looking
out from its exit is now finally visible.
.
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