Causality in quantum field theory with classical sources

Abstract

In an exact quantum-mechanical framework we show that space-time expectation values of the second-quantized electromagnetic fields in the Coulomb gauge in the presence of a classical conserved source automatically lead to causal and properly retarded ÿ-independent electromagnetic field strengths. The classical ÿ-independent and gauge invariant Maxwell's equations naturally emerge in terms of quantum-mechanical expectation values and are therefore also consistent with the classical special theory of relativity. The fundamental difference between interference phenomena due to the linear nature of the classical Maxwell theory as considered in, e.g., classical optics, and interference effects of quantum states is clarified. In addition to these issues, the framework outlined also provides for a simple approach to invariance under time-reversal, some spontaneous photon emission and/or absorption processes as well as an approach to Vavilov-Čherenkov radiation. The inherent and necessary quantum uncertainty, limiting a precise space-time knowledge of expectation values of the quantum fields considered, is, finally, recalled.

Quantum field theory with classical sources—linearized quantum gravity

“…In the present paper we extend the work of Ref. [1] to linear gravitational quantum deviations around a flat Minkowski space-time in a Coulomb-like gauge in the presence of a classical and conserved energy-momentum tensor. The time-evolution as dictated by quantum mechanics, for a suitably defined second-quantized gravitational field, then automatically solves the issues of causality and retardation in a similar manner as in the case of electro-dynamics.…”

“…In a previous publication [1] we have shown that the time-evolution as dictated by quantum mechanics, for second-quantized electro-magnetic fields in the presence of a classical conserved electric current, automatically solves these issues. The classical theory of Maxwell then naturally emerges in line with more general S-matrix arguments due to Weinberg [11].…”

“…The dynamical equations can, as in the case of electro-dynamics [1], be reduced to a system of decoupled harmonic oscillators with space-time dependent external forces. No pre-defined global causal order is assumed other than the deterministic time-evolution as prescribed by the Schrödinger equation.…”

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In a previous work [1] and in terms of an exact quantum-mechanical framework, -independent causal and retarded expectation values of the second-quantized electromagnetic fields in the Coulomb gauge were derived in the presence of a conserved classical electric current. The classical -independent Maxwell's equations then naturally emerged.

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On the three-slit experiment and quantum mechanics

“…Apart from an overall factor, the interference pattern will therefore be the same for a source prepared in, e.g., a quantum-mechanical Fock-state of photons or for a conventional coherent state. As is well-known, the use of coherent states naturally leads to the interference of classical electro-magnetic fields (see, e.g., [13] and references cited therein). The observation of a non-zero value of κ S is therefore not exclusively related to quantum-mechanical interference effects.…”