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Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

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Abstract

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the “Minkowski vacuum” for a box at rest in an inertial frame and a “Rindler vacuum” for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. The classical analysis gives no grounds for the “heating effects of acceleration through the vacuum” which appear in the literature of quantum field theory. Thus this distinction provides (in principle) an experimental test to distinguish the two theories.

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Notes

  1. For a recent brief review, see [1]. For an extensive review of work before 1996, see [2]. See also, [3].

  2. An anonymous referee for Ref. [8] wrote: “It is also important to note that, in quantum field theory, if a box with totally reflecting walls starts off at rest with no acceleration, and with its interior in the vacuum state, then has its acceleration slowly increased to the final acceleration, its interior will be in the ‘Rindler vacuum’, not the thermal bath. Those boundaries to the box make a huge difference.” A different referee wrote: “So suppose that we now take the mirror to be inertial for t<0, and then to accelerate uniformly for t>0. Then surely the mirror again introduces extra correlations and breaks some of the supposed symmetries of the zero-point radiation. If two such mirrors form a box of size L, then after a proper time L/c has elapsed along either mirror, one would expect even the radiation in the deep interior of the box to be affected by the mirrors motion.”

  3. A typical comment is that of an anonymous referee for Ref. [8], “… (again assuming that the body is small enough so that the gravitational field does not affect it), …”

  4. See, for example, [9]. We are using unrationalized units.

  5. See for example, [14, 15].

  6. See, for example, [16] or [17] or [18].

  7. See, for example, [19] or [20].

  8. See for example, the canonical quantization discussion of [22].

  9. See the recent review by [27].

  10. An anonymous referee for Ref. [8] wrote: “It is important … to realize that the claim that an accelerated observer see a thermal bath is based on the fact that such an observer, carrying a thermometer (which is insensitive in its operation to the presence of a strong gravitational field) will find it reading a temperature. Steaks will cook, eggs will fry.” Later this same referee wrote: “I cannot recommend a paper that denies a quantum field theory direct consequence, namely, that under uniform acceleration a small thermometer will register a temperature. Accelerated enough steaks in Minkowski vacuum will cook, and eggs will fry.”

  11. See Ref. [11], p. 105024-9.

  12. This point of view was not shared by a referee for Ref. [8] who declared, “If there is any disagreement with the standard quantum treatment then this approach is surely wrong.”

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  1. Department of Physics, City College of the City University of New York, New York, NY, 10031, USA

    Timothy H. Boyer

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Boyer, T.H. Contrasting Classical and Quantum Vacuum States in Non-inertial Frames. Found Phys 43, 923–947 (2013). https://doi.org/10.1007/s10701-013-9726-4

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