What is the 'Unruh Effect'?

From: Stephen Paul King <stephenk1.domain.name.hidden>
Date: Fri, 7 Oct 2005 21:04:31 -0400

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 markwh04.domain.name.hidden 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).
Received on Fri Oct 07 2005 - 21:06:15 PDT