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arXiv:1104.3818 · Physica Scripta

Operator ordering and causality

L I Plimak, Stig Stenholm, W P Schleich

Published Feb 1, 2012• 9 citations
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Abstract

A formal implementation of the concepts of mesoscopic electromagnetic interaction and of the propagating wave in quantum electrodynamics beyond the rotating wave approximation is discussed. Used as a guide, these concepts lead to a natural resolution of a long-standing controversy: causality violations in the Glauber–Kelley–Kleiner photodetection theory. The Glauber–Kelley–Kleiner definition of the time-normal operator ordering must be amended without the rotating wave approximation, which eliminates all causality problems.

DOI
10.1088/0031-8949/2012/t147/014026

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A formal implementation of the concepts of mesoscopic electromagnetic interaction and of the propagating wave in quantum electrodynamics beyond the rotating wave approximation is discussed. Used as a guide, these concepts lead to a natural resolution of a long-standing controversy: causality violations in the Glauber–Kelley–Kleiner photodetection theory. The Glauber–Kelley–Kleiner definition of the time-normal operator ordering must be amended without the rotating wave approximation, which eliminates all causality problems.
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The theory of photodetection and quantum-optical coherence is formulated in the Heisenberg picture and in such a way that the causal propagation of fields at the velocity of light is manifest. Objections to the standard theory, based on a putative violation of causality, are shown to be unfounded, as is the notion that normal ordering is essential for the elimination of an infinite contribution from the vacuum field. In a similar vein we revisit the Fermi two-atom problem and explicitly demonstrate the causal nature of the interaction without invoking simplifying approximations that have recently been called into question.
Generalised phase-space techniques for electromagnetic interactions beyond the rotating wave approximation [L.P. and S.S., arXiv:1104.3825 (2011)] is applied to interactions of distinguishable devices. The paper is built around the concept of "doing quantum electrodynamics while thinking classically," which is a generalisation of Sudarshan's renowned optical equivalence theorem [E.C.G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963)] to the interacting electromagnetic field. This concept allows one to reduce inherently quantum problems to semiclassical considerations.
We present new results on the causality violations introduced by the rotating wave approximation commonly used in quantum optics and high-energy physics. We find that the causality violations and faster-than-light signaling induced by the approximation have ``fat tails,'' i.e., they are polynomially decaying with the distance from the light-cone of the emitter. Furthermore, we also show that the fundamental problems with the incompatibility between the approximation and relativity are not cured even in the long interaction time regime (where the approximation is often taken). This renders the approximation unsuitable for any regime where we are concerned about relativistic causality and information transmission via the electromagnetic field.
The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition. Causal order is a concept that is engrained in the standard understanding of time, both in classical and quantum mechanics. Oreshkovet al.generalize the standard formalism of quantum theory to a framework with no pre-existing causal order, and find a new class of correlations that have no analogue in the classical world.