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Journal of Physics Communications

Causality in quantum field theory with classical sources

Bo-Sture Skagerstam, Karl-Erik Eriksson, Per Kristian Rekdal

Published Aug 1, 2019• 9 citations
Publisher

Abstract

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.

DOI
10.1088/2399-6528/ab3c1c

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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.
In an exact quantum-mechanical framework, we show that expectation values of the second-quantized electro-magnetic fields in the Coulomb gauge, and in the presence of classical sources, automatically lead to causal and retarded electro-magnetic field strengths. The classical ℏ\hbarℏ-independent Maxwell's equations naturally emerge from this fundamental quantum-mechanical approach in terms of expectation values of quantum fields, and are therefore also consistent with the special theory of relativity. The fundamental difference between interference phenomena due to the linear nature of the classical Maxwell theory as, e.g., in classical optics, and interference effects of quantum states is clarified. The framework outlined also provides for a simple approach to, e.g., spontaneous photon emission and/or absorption processes as well as to the classical Vavilov-Cherenkov radiation. The inherent and necessary quantum fluctuations, limiting a precise space-time knowledge of expectation values of the quantum fields considered, are, finally, recalled.
Relativistic formulation is presented of the second-order classical coherence theory of the electromagnetic field in the presence of random sources. It is shown that the dynamics of the second-order correlation tensors may be reduced to a simpler form in terms of certain tensor potentials. The gauge transformations of these tensor potentials are considered. In particular, two gauges of special interest are found, analogous to the Lorentz gauge and the radiation gauge in Maxwell’s electromagnetic theory. Some interesting relations and conservation laws are presented.
Maxwell’s equations, formulated in 1865, preceded quantum mechanics by six decades. When the logical direction is reversed, however, the electromagnetic field structure can be interpreted as the form required for a quantum theory of charged matter once three standard ingredients are imposed: local gauge invariance, conserved electric current, and a dynamical electromagnetic field. Quantum mechanics supplies the charged-particle state, probability amplitudes, quantum currents, and—through the Dirac equation—a relativistically consistent description of electron charge and spin. Gauge symmetry then organizes the electromagnetic interaction, while current conservation constrains the admissible source structure. The homogeneous Maxwell identities follow naturally from expressing the field through a four-potential, whereas the sourced Maxwell equations require an additional electromagnetic field action or equivalent dynamical assumption. The resulting framework clarifies both the strength and the limit of a “Maxwell from quantum mechanics” program: quantum theory and gauge structure strongly constrain electromagnetism, but Maxwell field dynamics are not obtained from the electron wave equation alone.
This paper examines how Maxwell electrodynamics can be recovered as the field-theoretic structure required to consistently couple quantum matter to electromagnetism, while distinguishing what follows from quantum mechanics from what must be supplied independently. Local phase invariance introduces electromagnetic potentials through minimal coupling and yields the homogeneous Maxwell equations as geometric identities, while quantum probability conservation provides a conserved electric current and Ehrenfest dynamics recover the Lorentz-force form. The inhomogeneous Maxwell equations require an additional Lorentz- and gauge-invariant electromagnetic field action, whose vacuum constants satisfy μ0ε0=1/c2\mu_0\varepsilon_0=1/c^2, linking the construction to the SQC/MSC electric–magnetic prefactor relation. Hydrogen is used as an explicit microscopic source: the 1s1s state generates a quantum charge distribution, 2p2p states provide orbital currents and magnetic moments, and 1s1s–2p2p superpositions produce radiative dipole transitions. The treatment is then extended from individual atoms to a hydrogen gas, where polarization and magnetization generate the macroscopic fields DD and HH. The resulting framework shows how quantum sources and material response fit consistently within Maxwell electrodynamics without claiming that quantum mechanics alone uniquely derives the complete electromagnetic field dynamics.
This chapter reviews the classical description of radiation. It describes how classical electromagnetic radiation interacts with a quantum mechanical particle. The general rule is that, even though the wave functions and the electromagnetic potentials change with the gauge, all physically observable quantities are independent of the gauge. The chapter also describes the process of the absorption of light by a system in terms of the effect of an external classical electromagnetic field on the system. The total emission rate is the sum of the induced and the spontaneous rates. Spontaneous emission is just the quantum mechanical version of the classical phenomenon of radiation from an accelerating charge. A very important consequence of the fact that the electromagnetic field is quantized is that there are vacuum fluctuations of the field, analogous to the zero point motion of a harmonic oscillator. Einstein gave a simple statistical argument to determine the rate at which spontaneous emission takes place.
The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.