Rindler observers, correlated states, boundary conditions, and the meaning of the thermal spectrum

Lon N. Pringle

doi:10.1103/PhysRevD.39.2178

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Rindler observers, correlated states, boundary conditions, and the meaning of the thermal spectrum

Lon N. Pringle

Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208

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Phys. Rev. D 39, 2178 – Published 15 April, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.2178

Abstract

Two new results concerning the Minkowski vacuum state as seen by Rindler observers are derived. The first result is that the thermal excitations detected by two causally separated Rindler observers are correlated in both energy and phase. Therefore, the state as seen by these observers is similar to an Einstein-Podolsky-Rosen state. The second result suggests that a Rindler observer detects the same thermal spectrum even if there is a boundary condition beyond his horizon. A discussion of what these effects tell us about the origin of the thermal spectrum follows.

References (18)

(a) S. Fulling, Phys. Rev. D 7, 2850 (1973); (b) P. C. W. Davies, J. Phys. A 8, 365 (1975).
W. G. Unruh, Phys. Rev. D 14, 870 (1976).
N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, New York, 1982).
W. Rindler, Am. J. Phys. 34, 1174 (1966).
Higher Transcendental Functions (Bateman Manuscript Project), edited by A. Erdélyi . (McGraw-Hill, New York, 1953), Vol. 2.
R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 48, 777 (1935).
Y. Aharonov (private communication).
``Any such EPR case'' refers to a situation in which the two correlated complementary observables have a discrete spectrum, or a continuous spectrum which can be taken as the limit of a discrete spectrum. By complementary I mean that the unitary matrix generated by the eigenstates of the two operators transforms no subspace (created by the eigenstates of one operator) into a subspace (created by the eigenstates of the other) of equal or lower dimension, except for the trivial case where the subspace is the entire space itself. For example, the spin operators $S_{x}$ and $S_{y}$ satisfy this condition in any finite dimension. For these cases it is easy to prove that the spectrum of the reduced density matrix of either observer will be flat.
A good review of phase variables is by P. Carruthers and M. Nieto, Rev. Mod. Phys. 40, 411 (1968).
E. C. Lerner, H. W. Huang and G. E. Walters, J. Math. Phys. 11, 1679 (1970).
The relationship between thermal states and phase states was first discussed by Y. Aharonov, E. C. Lerner, H. W. Huang and J. M. Knight, J. Math. Phys. 14, 746 (1973).
S. J. Summers and R. Werner, J. Math. Phys. 28, 2448 (1970).
R. J. Glauber, Phys. Rev. 131, 2766 (1963).
T. D. Lee, Nucl. Phys. B264, 437 (1986).
J. D. Bekenstein, Phys. Rev. D 12, 3077 (1975).
A. Gould, Phys. Rev. D 35, 449 (1987).

[c1] (a) S. Fulling, Phys. Rev. D 7, 2850 (1973); (b) P. C. W. Davies, J. Phys. A 8, 365 (1975).

[c2] W. G. Unruh, Phys. Rev. D 14, 870 (1976).

[c3] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, New York, 1982).

[c4] W. Rindler, Am. J. Phys. 34, 1174 (1966).

[c5] Higher Transcendental Functions (Bateman Manuscript Project), edited by A. Erdélyi . (McGraw-Hill, New York, 1953), Vol. 2.

[c6] R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).

[c7] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).

[c8] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 48, 777 (1935).

[c9] Y. Aharonov (private communication).

[c10] ``Any such EPR case'' refers to a situation in which the two correlated complementary observables have a discrete spectrum, or a continuous spectrum which can be taken as the limit of a discrete spectrum. By complementary I mean that the unitary matrix generated by the eigenstates of the two operators transforms no subspace (created by the eigenstates of one operator) into a subspace (created by the eigenstates of the other) of equal or lower dimension, except for the trivial case where the subspace is the entire space itself. For example, the spin operators $S_{x}$ and $S_{y}$ satisfy this condition in any finite dimension. For these cases it is easy to prove that the spectrum of the reduced density matrix of either observer will be flat.

[c11] A good review of phase variables is by P. Carruthers and M. Nieto, Rev. Mod. Phys. 40, 411 (1968).

[c12] E. C. Lerner, H. W. Huang and G. E. Walters, J. Math. Phys. 11, 1679 (1970).

[c13] The relationship between thermal states and phase states was first discussed by Y. Aharonov, E. C. Lerner, H. W. Huang and J. M. Knight, J. Math. Phys. 14, 746 (1973).

[c14] S. J. Summers and R. Werner, J. Math. Phys. 28, 2448 (1970).

[c15] R. J. Glauber, Phys. Rev. 131, 2766 (1963).

[c16] T. D. Lee, Nucl. Phys. B264, 437 (1986).

[c17] J. D. Bekenstein, Phys. Rev. D 12, 3077 (1975).

[c18] A. Gould, Phys. Rev. D 35, 449 (1987).

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Rindler observers, correlated states, boundary conditions, and the meaning of the thermal spectrum

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