The WKB Method and Nonperturbative Effects

The interaction of waves on deep water with spatially varying currents may be described by a ray theory, with the wave amplitudes determined by the principle of conservation of wave action (CWA). However, all previous deep water derivations of CWA are restricted to the case of an irrotational current. In this paper, both the ray theory and CWA are derived by a WKB method without the assumption of irrotationality. Also derived is a new equation for a spatially varying phase shift which is not predicted by the usual ray theory, and which, in general, displaces the positions of the wave crests by a distance on the order of a wavelength. This phase shift, which is caused by variations of the current velocity with depth, vanishes in the irrotational case, and so is in accord with the irrotational theory.

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THE UNREASONABLE EFFECTIVENESS OF EXPONENTIALLY SUPPRESSED CORRECTIONS IN PRESERVING INFORMATION

International Journal of Modern Physics D ◽

10.1142/s0218271813420303 ◽

2013 ◽

Vol 22 (12) ◽

pp. 1342030 ◽

Cited By ~ 9

Author(s):

KYRIAKOS PAPADODIMAS ◽

SUVRAT RAJU

Keyword(s):

Hilbert Space ◽

Black Hole ◽

Quantum Mechanics ◽

Nonperturbative Effects ◽

Von Neumann Entropy ◽

Von Neumann ◽

Information Theoretic ◽

Black Hole Evaporation ◽

Strong Subadditivity ◽

Unreasonable Effectiveness

We point out that nonperturbative effects in quantum gravity are sufficient to reconcile the process of black hole evaporation with quantum mechanics. In ordinary processes, these corrections are unimportant because they are suppressed by e-S. However, they gain relevance in information-theoretic considerations because their small size is offset by the corresponding largeness of the Hilbert space. In particular, we show how such corrections can cause the von Neumann entropy of the emitted Hawking quanta to decrease after the Page time, without modifying the thermal nature of each emitted quantum. Second, we show that exponentially suppressed commutators between operators inside and outside the black hole are sufficient to resolve paradoxes associated with the strong subadditivity of entropy without any dramatic modifications of the geometry near the horizon.

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