General Frame Covariance in Achronal Quantum General Relativity (was: No new Einstein)
- From: markwh04@xxxxxxxxx
- Date: Mon, 4 Jul 2005 03:28:15 +0000 (UTC)
John Baez wrote:
> On the hundredth anniversary of Einstein's discovery of special
> relativity, why is there no new Einstein today?
How do you know there isn't?
The question, more properly, is: IS there a new Einstein lurking around
somewhere today.
The analogue, today, nearly 100 years past, is that you have a
situation today just like 1905, where you have two seemingly
irreconcilable theories, and people (unbeknownst to themselves in some
cases) have already laid out the essential foundation of its resolution
(e.g. Smolin & Markopoulou, Penrose, etc.), while in the meantime, the
mainstream is off on an modern-day Ether Primrose Path. The analogy,
in fact, is made complete by the fact that the general resolution is
coming fully in sight now, as it did then.
Let's take a look at the issue: the incompatibility of quantum theory
and general relativity, and see how its focal issue is resolved: the
issue of general coordinate covariance.
Squark wrote:
> The essence of the Unruh effect is that a uniformly accelerated
> observer sees the QED vacuum as a thermal equilibrium (black-body
> radiation). The later, as opposed to the former, is not even a
> pure state but a mixed one. This raises the interesting question
> of what is the general transformation from the inertial observer
> to the accelerating one on the space of mixed states.
Well, first you have to understand that the term "frame" has little to
do with what relativists call a frame. It refers to a timelike field.
The one associated with the Rindler vacuum (and the ordinary inertial
vacuum) is a Killing field, but I'm not sure how essential the
assumption is that it also be Killing.
The nature of the vacuum depends entirely on which frame is used. As
Penrose pointed out, there is no covariant distinction between quantum
noise and thermal noise: what appears as quantum correlations in one
frame, gets rendered as thermal noise associated with the cut-off
boundary imposed by the causal horizon.
The reason you get a thermal state is because the frame associated with
the Rindler vacuum creates a causal horizon, at the boundary which
envelopes the flowlines corresponding to that part of the spacetime
you're in. There are 2 separate sections in the spacetime, each
bounded by a null hypersurface, which functions analogously to an event
horizon.
The thermal state is directly in associated with the horizon. The
reason states are mixed is because in the Rindler vacuum they are mixed
RELATIVE states created by phase averaging (or "coarse graining", as it
is generally called) over the horizon. This is where the quantum
correlations seen in the Minkowski vacuum get rendered as thermal
correlations.
The general transformation is the Bogoliubov Transform. I think the
reason the timelike field is required to be Killing is so that the
transform be linear. But whatever it is, the key point is that the
transform is NOT complete. In particular, the operators corresponding
to the sector that lies outside the causal horizon don't get translated
properly in the Rindler frame.
The general situation is as follows: you can partition spacetime into
compact regions, each region having its own timelike field, and each
region bounded by 2 null hypersurfaces -- Alexandroff Intervals, in
essence. Between any two such regions will be a partial transform
whose respective domains in each direction overlap. The net result is
that each region sees part of what's rendered in the other ocherently,
as a mixture.
The one construction I'm looking at developing is as follows: take the
timelike fields to be such that the flow lines meet at (r=0,t=-L) and
(r=0,t=L), and such that they have tangents proportional to d/dt at
(r=R,t=0). The different flowlines correspond to different
intersection points R, where |R| < L. The enveloping hypersurfaces are
given by |R| + |t| = L, with t > 0 giving you the positive
hypersurface, t < 0 the negative hypersurface.
If the flowlines are hyperbolic, they will generally have the form (r =
a cosh(s) + b, t = c sinh(s) + d). The different selections of the
parameters (a,b,c,d) will give you the different flowlines. This is
constrained so that it passes through (0,-L), (0,L) and for some R,
with |R| < L, (R,0).
The hypersurfaces R_l, -1 <= L <= 1 are spanned, such suitable scaling
of s, by the flowlines; and the region R is the union of R_l as l
ranges from [-1,1], with the bounding hypersurfaces R_{+/-} being the
limits respectively as l -> +/- 1. Each hypersurface R_l has boundary
H = { (Rn,0): |n|=1 } -- the same 2-sphere.
Now, a natural question is to determine what the Bogoliubov transforms
look like between different Alexandroff Intervals, taking different
points (r1,t1)-(r2,t2) in place of (0,-L)-(0,L). This implements the
quantum analogue of a coordinate transform, thus recovering some
semblance of the Relativists' notion of a local coordinate frame from
the quantum notion of a frame=time-like streamline.
What's particularly interesting about these frames is that they are all
self-contained in the sense that nothing propagates outside the region,
except through the 2 boundaries, which are both null and play the role
of local causal horizons.
That means you can do mini-quantum field theory inside the Alexandroff
Interval -- except the vacuum state must now be a thermal state or some
more elaborate type of mixed state.
Each surface R_l is, in effect, a compact Cauchy surface. So, you can
write down a symplectic structure using well-established methods --
many of which have problems when the Cauchy surfaces are not compact,
that are entirely dodged in this setting.
The difference from the Rindler vacuum is that the accelerations of the
flowlines here are radial and approach infinity at the horizon. I'm
not sure, yet, of the ramifications of this and of the construction in
general. It's still something I'm looking into.
Another key feature is that this type of self-contained mini-quantum
theory has the global element removed entirely from it. You're staying
within a compact, locally hyperbolic region, of spacetime. Therefore,
invoking a correspondence principle
What goes in Vegas stays there:
What works in the region R works there, regardless of
what the global structure of the spacetime R is embedded
in is.
then you're able to do generally covariant quantum field theory in
spacetimes, both globally hyperbolic and non-hyperbolic -- even ones
beset with all sorts of causal anomalies, such as time
non-orientability, time travel loops, etc.
>
> Read what Lee Smolin has to say:
>
> http://math.ucr.edu/home/baez/no-new-einstein.pdf
>>From hillman@xxxxxxxxxxxxxxxxxxx Sun Jul 3 21:59:33 2005
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René Meyer mentioned
> http://www.sciencemag.org/cgi/content/full/309/5731/78b
Nice, but doesn't the second to last question rather miss the point?
(Of course all the zeros are -complex numbers-. I guess the editors took
their own advice: "Don't sweat the details"!)
"T. Essel"
.
- References:
- No new Einstein
- From: John Baez
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