Causality in Quantum Field Theory with Classical Sources Bo-Sture K. Skagerstam1;a), Karl-Erik Eriksson2;b), Per K. Rekdal3;c) a)Department of Physics, NTNU, Norwegian University of Science and Technology, N-7491 Trondheim, Norway b) Department of Space, Earth and Environment, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden c)Molde University College, P.O. Box 2110, N-6402 Molde, Norway 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 source, automatically lead to causal and properly retarded elec- tromagnetic field strengths. The classical ~-independent and gauge invariant Maxwell's equations then naturally emerge and are therefore also consistent with the classical spe- cial 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 is- sues, 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-Cherenkovˇ radiation. The inherent and necessary quan- tum uncertainty, limiting a precise space-time knowledge of expectation values of the quantum fields considered, is, finally, recalled. arXiv:1801.09947v2 [quant-ph] 30 Mar 2019 1Corresponding author. Email address:
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[email protected] 1. Introduction The roles of causality and retardation in classical and quantum-mechanical versions of electrodynamics are issues that one encounters in various contexts (for recent discussions see, e.g., Refs.[1]-[14]).