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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
      lost. We shall return to this subject in the next, final

      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

      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,
      let us first return to entanglement, now in curved spacetime,
      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

      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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