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 +0000 (UTC), in
sci.physics.research [EMAIL PROTECTED] wrote:
The "Minkowski" or "inertial" vacuum state seen in an
"accelerating
frame" is a thermal state at a temperature proportional to the "acceleration"; i.e., an heat bath containing an infinite number of particles (finite density) distributed in a fashion consistent with a gas at a particular temperature. The words in quotes are very misleading however, and require a
large
amount of clarification, because the effect has little or nothing per se to do with acceleration or being inertial; but rather with the occurrrence of a causal horizon. A quantum field theory requires you first to define a "frame",
the word
which -- unlike in Relativity -- does NOT a coordinate system; but a "flow of time". Quantum theory, is you recall, treats time as a process, 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
constant
time-like field which (by suitable Lorentz transformation) can be represented as T = d/dt -- i.e. the 4-vector T = T^{mu} d/dx^{mu} whose only non-zero component is T^0 (with x^0 = t). The Unruh frame uses a time-like field which does NOT cover
all of
space. The flow lines are all hyperbolic, each naturally associated with an observer at a given acceleration. The hyperbolas all have, as asymptotes, 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 of
the worldlines in the Unruh frame. Different worldlines have different accelerations associated with them. At this boundary, the timelike field T becomes null. A
second, mirror
region, x < -c|t| has the boundary x = -c|t|. Both boundaries meet at t = 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 of
all states of a system is generally always associated with the initial values of whatever system is in question. Here, that means, there is a natural split of the underlying state space H into H1 + H2, with H1 being the state space associated with the region x > c|t|, and H2 being that associated with the region x < -c|t|. (Solving the field equation by taking its initial values (and
the
initial values of its time derivative) comprises what's called a Cauchy problem. For the Klein-Gordon field, the initial values play the analogous role of coordinates, the initial time derivative the congugate momenta. The state space is then a Hilbert space in which these quantities act as operators satisfying the usual Heisenberg relations). H1, here, is the only one of physical relevance. But a
full
description of the Minkowski frame requires both H1 and H2. In particular, the vacuum state |0> of a Klein Gordon field -- as seen in the 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-temperature
quantum field theory. The states |n>_1 can be thought of as particle states, those |n>_2 can be thought of as states associated with vacuum fluctuations of the corresponding heat bath (i.e. "holes"). So, the superposition |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 in region 2, x < -c|t|, goes the other way. Since only region 1, x > c|t|, is physically relevant (you
can't see
past the boundary, the causal horizon), then the actual quantum state associated with it is arrived at by phase-averaging over the states of region 2. This turns the Minkowski state into the region 1 state: |0><0| --> Trace_2(|0><0|) = V_1 with |0><0| = sum |n>_1 <n|_1 |n>_2 <n|_2 exp(-2 pi a) which, after being traced over give you Trace_2(|0><0|) = sum |n>_1 <n|_1 exp(-2 pi a) 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 being
represented by the coefficients of the mixture exp(-2 pi a) which represent (up to proportion) probabilities ... and probabilities always mean you lost information somewhere. In fact, this general process of tracing over a causal horizon
of some
sort is GENERALLY how you get probabilities out of quantum theory. Everything is a pure state, until you do a partial trace phase-averaging cut-off on a horizon somewhere, and the horizon, itself, can be thought of as nothing less than a way of quantifying the word "observer". The loss of information is readily identified with the loss
of
information of what's going on in the other parts of spacetime outside the region x > c|t|. The general lesson is that what appears as quantum noise in
one frame
becomes thermal noise seen from another frame; and there is no longer any covariant distinction between quantum and thermal noise. There's an unlimited number of ways to define time-like fields
in a
region of spacetime, and you can always have time-like fields defined in such a way that they have causal horizons somewhere. A more dramatic example of this is where the region in question is actually finite in size: |r| < 1 - |t| (using units where c = 1). Then defining coordinates (R, T) by r = R (1 - T^2)/(1 - (RT)^2) t = T (1 - R^2)/(1 - (RT)^2) this produces a metric ds^2 = dT^2 ((1 - R^2)/(1 - (RT)^2))^2 - ds_3^2 ((1 - T^2)/(1 - (RT)^2)^2 where ds_3 = dR^2 + R^2 ((d theta)^2 + (sin theta d psi)^2) The worldlines associated with (R, theta, psi) = constant
are
hyperbolic worldlines that meet at (t,r) = (-1,0), (t,r) = (+1,0), and cross the t = 0 hyperplane at r = R; and the timelike field is just d/dT, that associated with the coordinate "T", itself. Here, the causal horizon is |r| = 1 - |t| and the region
enclosed is
finite, which means that the modes associated with the Klein-Gordon field will form a discretely spaced set, rather than continuous (just as if you were to quantize "in a box"). So, there is a HUGE cut-off of field modes here, since the globally defined quantum field has a continuously distributed set of modes. As of yet, I still don't know what the Minkowski vacuum looks
like in
this frame. The wave equation has a particularly bizarre feature that half of the initial values become redundant, when the initial values are taken on the 1/2 causal horizon |r| = 1 - t, 0 < t < 1; or the other 1/2; |r| = 1 + t, -1 < t < 0; and the Cauchy problem for the field becomes a Dirichlet problem (in part because there are no time-derivatives to define, since "time is frozen" on the causal horizon). |

- What is the 'Unruh Effect'? Stephen Paul King
- Re: What is the 'Unruh Effect'? Bob Hearn