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arXiv:2102.04286 · Reviews in Mathematical Physics

Quantum radiation from a classical point source

Jonathan Dimock

Published Jul 1, 2022• 1 citation
arXivHTMLPDF

Abstract

We study the radiation of photons from a classical charged particle. We particularly consider a situation where the particle has a constant velocity in the distant past, then is accelerated, and then has a constant velocity in the distant future. Starting with no photons in the distant past we seek to characterize the quantum state of the photon field in the distant future. Working in the Coulomb gauge and in a [Formula: see text]-algebra formulation, we give sharp conditions on whether this state is or is not in Fock space.

DOI
10.1142/s0129055x22500325

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We study the radiation of photons from a classical charged particle. We particularly consider a situation where the particle has a constant velocity in the distant past, then is accelerated, and then has a constant velocity in the distant future. Starting with no photons in the distant past we seek to characterize the quantum state of the photon field in the distant future. Working in the Coulomb gauge and in a [Formula: see text]-algebra formulation, we give sharp conditions on whether this state is or is not in Fock space.
Classical electrodynamics including classical electromagnetic zero-point radiation leads to a ground state and resonant excited states for a charged particle in a Coulomb potential. These resonant states correspond to integer values of the action variables analogous to those appearing in the Bohr-Sommerfeld theory of the hydrogen atom. The work on classical zero-point radiation reported here is a continuation of the analysis reported in 1975, but with the addition of the ideas of relativity and resonance between the charged-particle orbit and classical zero-point radiation.
The classical and the quantum expressions of the energy of the electromagnetic radiation are compared. This comparison shows that, contrary to the commonly expected, the dependency of the energy [Formula: see text] of the photon with respect to the frequency [Formula: see text] on the Einstein–Planck relation [Formula: see text] is obtainable from classical electrodynamics, a fact that may provide a new historic perspective of the birth of quantum mechanics. It is suggested that the photon is a composite particle, constituted by quanta of electric and magnetic field fluxes maybe in an entangled state — being these two quanta either real or only as pseudo particles. It is also suggested a modification on the de Broglie’s fusion principle for the photon for the specific case in consideration.
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.
According to modern quantum field theory all elementary particles are quantum excitations of an underlying quantum field. In the case of the photon the underlying field is the electromagnetic field. In photon wave mechanics the photon concept was reached by a reinterpretation of the physical meaning of the free classical electromagnetic field. Such a reinterpretation does not give us a quantum theory of light quanta because the elementary quantum of action is foreign to classical physics. Although Planck’s constant does occur in the various Schro¨dinger-like wave equations we have presented in Part III it just plays the role of a multiplicative factor, see e.g., Eq. (10.36). The “justification” for bringing ~ into the formalism is the wish to make contact to the Einstein-de Broglie relations [Eqs. (10.40) and (10.41)]. In a sense one may call the light particle appearing from classical electrodynamics the classical photon.
We present Lorentz covariant models of the radiation field, i.e., plane wave solutions of the Maxwell equations satisfying the Coulomb gauge condition. The theory is constructed along traditional lines. We apply the interpretation of the field as a collection of photons and starts by constructing the state space of the photon. The novelty of this formulation is the use of a new action of the Lorentz group on the space of circular helicities of spin 1 which permits the construction of an action on the state space of the photon. Moreover, the generators of the action provide objects that can be used to construct the field observables both in the classical and quantum case. The result is a theory with a tight structure. It is a generalization of the standard theory, a covariant generalization, and it contains this as a special case.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.