Hi Russell and Friends,
I just ran across the following post and
thought that you might find it interesting. Any comments?
Onward!
Stephen
On Thu, 11 Aug 2005 10:32:00 + (UTC), in
sci.physics.research [EMAIL PROTECTED] wrote:
The "Minkowski" or "inertial" vacuum state seen in an
"acceleratingframe" is a thermal state at a temperature proportional to
the"acceleration"; i.e., an heat bath containing an infinite number
ofparticles (finite density) distributed in a fashion consistent with
agas at a particular temperature.
The words in quotes are very misleading however, and require a
largeamount of clarification, because the effect has little or nothing
perse to do with acceleration or being inertial; but rather with
theoccurrrence of a causal horizon.
A quantum field theory requires you first to define a "frame",
the wordwhich -- unlike in Relativity -- does NOT a coordinate system; but
a"flow of time". Quantum theory, is you recall, treats time as
aprocess, not a dimension.
The flow of time is represented by a vector field which is
timelike.
The Minkowski or inertial frame has associated with it a
constanttime-like field which (by suitable Lorentz transformation) can
berepresented as T = d/dt -- i.e. the 4-vector T = T^{mu} d/dx^{mu}
whoseonly non-zero component is T^0 (with x^0 = t).
The Unruh frame uses a time-like field which does NOT cover
all ofspace. The flow lines are all hyperbolic, each naturally
associatedwith an observer at a given acceleration. The hyperbolas all
have, asasymptotes, the 2+1 boundary given by an equation of the form x =
c|t|;the region associated with the field being x > c|t|.
The "acceleration" a is normally defined as that associated
with one ofthe worldlines in the Unruh frame. Different worldlines
have differentaccelerations associated with them.
At this boundary, the timelike field T becomes null. A
second, mirrorregion, x < -c|t| has the boundary x = -c|t|. Both
boundaries meet att = 0. In this region, the field T "flows" in the
opposite direction.
The boundary x = c|t| is the causal horizon mentioned
before.
A field is uniquely determined by its values at t = 0, and the
space ofall states of a system is generally always associated with the
initialvalues of whatever system is in question. Here, that means,
there is anatural split of the underlying state space H into H1 + H2, with
H1being the state space associated with the region x > c|t|, and H2
beingthat associated with the region x < -c|t|.
(Solving the field equation by taking its initial values (and
theinitial values of its time derivative) comprises what's called a
Cauchyproblem. For the Klein-Gordon field, the initial values play
theanalogous role of coordinates, the initial time derivative
thecongugate momenta. The state space is then a Hilbert space in
whichthese quantities act as operators satisfying the usual
Heisenbergrelations).
H1, here, is the only one of physical relevance. But a
fulldescription of the Minkowski frame requires both H1 and H2.
Inparticular, the vacuum state |0> of a Klein Gordon field -- as seen
inthe Minkowski frame -- when expressed in terms of the H1 & H2
states--becomes:
|0> = sum |n>_1 |n>_2 exp(-pi a).
This is readily identifiable in the language of
finite-temperaturequantum field theory. The states |n>_1 can be
thought of as particlestates, those |n>_2 can be thought of as states
associated with vacuumfluctuations of the corresponding heat bath (i.e.
"holes"). So, thesuperposition |n>_1 & |n>_2 has total
energy 0, since |n>_2 reflects|n>_1. All the states |n>_2 are
negative energy since the time flow inregion 2, x < -c|t|, goes the other
way.
Since only region 1, x > c|t|, is physically relevant (you
can't seepast the boundary, the causal horizon), then the actual quantum
stateassociated with it is arrived at by phase-averaging over the states
ofregion 2. This turns the Minkowski state into the region 1
state:
|0><0| --> Trace_2(|0><0|) =
V_1with |0><0| = sum
|n>_1 _2 which, after
being traced over give you
Trace_2(|0><0|) = sum |n>_1 which is a MIXED
(and thermal) state, no longer a pure state,associated with a temperature
proportional to a.
Having a mixed state means you've lost information -- this
loss beingrepresented by the coefficients of the
mixture
exp(-2 pi a)which represent (up to proportion) probabilities ... and
probabilitiesalways mean you lost information somewhere.
In fact, this general process of tracing over a causal horizon
of somesort is GENERALLY how you get probabilities out of quantum
theory.Everything is a pure state, until you do a partial
tracephase-averaging cut-off on a horizon somewhere, and the
horizon,itself, can be thought of as nothing less than a way of quantifying
theword "observer".
The loss of information is readily identified with the loss
ofinforma
