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 ℏ-independent electromagnetic field strengths. The classical ℏ-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.