## Re: Asher Peres's important remarks on accelerated detectors - the gravitational field #1

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• D. Accelerated detectors In quantum field theory, the vacuum is defined as the lowest energy state of a field. A free field with linear equations of motion can
Message 1 of 4 , Nov 25, 2012
D. Accelerated detectors
In quantum field theory, the vacuum is defined as the
lowest energy state of a field. A free field with linear
equations of motion can be resolved into normal modes,
such as standing waves. Each mode has a fixed frequency
and behaves as a harmonic oscillator. The zero point motion
of all these harmonic oscillators is called “vacuum
fluctuations”and the latter, under suitable conditions,
may excite a localized detector that follows a trajectory
xν (τ) parametrized by its proper time τ. The internal
structure of the detector is described by non-relativistic
quantum mechanics, so that we can indeed assume that it
is approximately localized, and it has discrete energy levels
En. Furthermore, we assume the existence of a linear
coupling of an internal degree of freedom, μ, of the detector,
with the scalar field φ(x(τ)) at the position of the
detector. First-order perturbation theory gives the following
expression for the transition probability per unit
proper time

For inertial detectors (that is, x^ν = v^ντ with a constant
four-velocity vν) the transition probability is zero,
as one should expect. However, the response rate does
not vanish for more complicated trajectories. Consider
in particular one with constant proper acceleration a.
With an appropriate choice of initial conditions, it corresponds
to the hyperbola t^2 +x^2 = 1/a^2, shown in Fig. 3.

Then the transition rate between levels appears to be
the same as for an inertial detector in equilibrium with
thermal radiation at temperature T ~ ha/2πckB. This
phenomenon is called the Unruh (1976) effect. It was
also discussed by Davies (1975) and it is related to the
fluctuation-dissipation theorem (Candelas and Sciama,
1977) and to the Hawking effect that will be discussed in
the next section.17 A rigorous proof of the Unruh effect
in Minkowski spacetime was given by Bisognano and
Wichmann (1976) in the context of axiomatic quantum
field theory, thus establishing that the Unruh effect is not
limited to free field theory.

For any reasonable acceleration, the Unruh temperature
is incomparably smaller that the black-body temperature
of the cosmic background, or any temperature ever
attained in a laboratory, and is not observable. Levin,
Peleg, and Peres (1992) considered the effect of shielding
a hypothetical experiment from any parasitic sources.
This, however, creates a radically new situation, because
the presence of a boundary affects the dynamical properties
of the quantum field by altering the frequencies of
its normal modes. Finite-size effects on fields have been
known for a long time, both theoretically (Casimir, 1948)
and experimentally (Spaarnay, 1958). Levin, Peleg, and
Peres showed that if the detector is accelerated together
with the cavity that shields it, it will not be excited by
the vacuum fluctuations of the field. On the other hand,
an inertial detector freely falling within such an acceler-
ated cavity will be excited. The relevant property in all
these cases is the relative acceleration of the detector and
the field normal modes.

17 Properties of detectors undergoing circular acceleration, as in
high energy accelerators, were investigated by Bell and Leinaas
(1983), Levin, Peleg, and Peres (1993), and by Davies, Dray, and
Manogue (1996).

We now consider the evolution of an arbitrary quantum
system. An observer at rest (Alice) can describe the
quantum evolution on consecutive parallel slices of spacetime,
t = const. What can Bob, the accelerated observer,
do? From Fig. 3, one sees that there is no communication
whatsoever between him and the region of spacetime
that lies beyond both horizons. Where Alice sees a pure
state, Bob has only a mixed state. Some information is
section.

VI. BEYOND SPECIAL RELATIVITY
It took Einstein more than ten years of intensive work
to progress from special relativity to general relativity.
Despite its name, the latter is not a generalization of
the special theory, but a radically different construct:
spacetime is not only a passive arena where dynamical
processes take place, but has itself a dynamical nature.
At this time, there is no satisfactory quantum theory of
gravitation (after seventy years of efforts by leading theoretical
physicists).

In the present review on quantum information theory,
we shall not attempt to use the full machinery of general
relativity, with Einstein’s equations.18 We still consider
spacetime as a passive arena, endowed with a Riemannian
metric, instead of the Minkowski metric of special
relativity. The difference between them is essential: it
is necessary to introduce notions of topology, because it
may be impossible to find a single coordinate system that
covers all of spacetime. To achieve that result, it may be
necessary to use several coordinate patches, sewed to each
other at their boundaries. Then in each patch, the metric
is not geodesically complete: a geodesic line stops after a
finite length, although there is no singularity there. The
presence of singularities (points of infinite curvature) is
another consequence of Einstein’s equations. It is likely
that these equations, which were derived and tested for
the case of moderate curvature, are no longer valid under
such extreme conditions. We shall not speculate on this
issue, and we shall restrict our attention to the behavior
of quantum systems in the presence of horizons, in
particular of black holes. Before we examine the latter,
and to the Unruh effect, still in flat spacetime, but
described now in an accelerated coordinate system.

18 Concepts of quantum information were recently invoked in several
problems of quantum gravity and quantum cosmology, but
we restrict ourselves to conventional black hole physics.

A. Entanglement revisited
Calculations on EPRB correlations require a common
reference frame. Only then can statements such as
“if m1z = 1/2 , then m2z = – 1/2”
have an operational meaning. In a curved space
we can choose an arbitrary frame at one
spacetime point and then translate it parallel to itself
along a geodesic path. For example, spin-1/2 particles may
be sent to Alice and Bob, far away. After a reference
frame is chosen at the emission point, local frames are
established for them by parallel transport along the particles’
trajectories. However, particles only approximately
follow classical geodesic trajectories, and this inevitably
introduces uncertainties in the definition of directions.
Using path integral methods, von Borzeszkowski and
Mensky (2000) have shown that if certain conditions are
met, approximate EPR correlations still exist, but “the
longer the propagation and the stronger the gravitational
field, the poorer is the correlation”.

One of the difficulties of quantum field theory in curved
spacetimes is the absence of a unique (or preferred)
Hilbert space, the reason being that different representations
of canonical commutation or anticommutation
relations lead to unitarily inequivalent representations
(Emch, 1972; Bogolubov et al., 1990; Haag 1996). For
the Minkowski spacetime, the existence of a preferred
vacuum state enables us to define a unique Hilbert space
representation. A similar construction is also possible in
stationary curved spacetimes (Fulling, 1989;Wald, 1994).
However, in a general globally hyperbolic spacetime this
is impossible, and one is faced with multiple inequivalent
representations.

Genuinely different Hilbert spaces with different density
operators and POVMs apparently lead to predictions
that depend on the specific choice of the method of calculation.
The algebraic approach to field theory can resolve
this difficulty for PVMs. The essential ingredient
is the notion of physical equivalence (Emch, 1972; Araki,
1999; Wald, 1994), which allows to extend the formalism
of POVMs and CP maps to general globally hyperbolic
spacetimes (Terno, 2002).

As a consequence of the Reeh-Schlieder theorem, it
follows that a Minkowski vacuum  corresponds to a
mixed state in the Rindler spacetime. To relate the
Minkowski and Rindler Hilbert spaces, fields in both
wedges are required. The relation between the standard
Minkowski Fock space and a tensor product of Rindler
Fock spaces is given by a formally unitary operator U,
whose action on the Minkowski vacuum is

the previous discussion, since the energy of such a system
is also infinite.

The relationship between Minkowski and Rindler wave
packets was analyzed by Audretsch and M¨uller (1994a).
These authors also discussed local detection by Rindler
observers and EPR-like correlations (Audretsch and
M¨uller, 1994a, b).Alsing and Milburn (2002, 2003) examined the fidelity
of teleportation from Alice in an inertial frame to Bob
who is uniformly accelerated
............................................
Jack: in the special case that Alice and Bob are momentarily spatially coincident,
that is a tetrad mapping. Not so when they are separated.
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