On the Number of Photons in a Classical Electromagnetic Field
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On the Number of Photons in a Classical Electromagnetic Field R. M. Feshchenkoa,* and A. V. Vinogradova,** a
Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 199991 Russia *e-mail: [email protected] **e-mail: [email protected] Received March 6, 2018
Abstract—Relations determining the number of photons in an electromagnetic field are considered from the point of view of a classical electromagnetic field. A relativistically invariant expression is obtained for the number of emitted photons in terms of charges and currents producing the electromagnetic field. Examples are considered for calculating the numbers of photons in the electromagnetic field for the case of the electric dipole radiation field, as well as the field of a finite and spatially restricted electromagnetic pulse. DOI: 10.1134/S1063776118080150
1. INTRODUCTION It was shown in [1] that in a free classical electromagnetic field, the quantity
1 16π3c (1) E(t, r)E(t, r ') + H(t, r)H(t, r ') 3 3 × d rd r ', (r − r ')2 where E and H are the electric and magnetic fields, r and r' are the spatial coordinates, t is the time, and c is the velocity of light, is conserved. The quantity (1) in quantum theory has the meaning of the number of photons [1] and is referred to as the Zeldovich invariant [2, 3] in foreign literature. However, a number of related questions have remained unanswered. Among other things, expression (1) is not a relativistic invariant in explicit form, and, hence, its independence from the choice of the reference frame is not obvious. The relation between the number of photons in a free electromagnetic field and the properties of the electric charges and currents producing it is also not quite clear. The relativistic invariance of the number of photons in a free classical electromagnetic field was studied in [2], where an attempt to calculate it inside a Fabry–Perot resonator initially led to a noninvariant quantity depending on the choice of the reference frame. This drawback was partially removed by a correct summation of the number of photons over the field modes propagating in different directions in a manner similar to that described in [4]. However, the problem remains incompletely solved so far. This research is aimed at analysis of the emission of photons by the charges and currents in the sense of definition (1). It is shown that the result, i.e., the numN =
∫∫
ber of photons (1), is a relativistically invariant for a free field and can be expressed in terms of the charges and currents producing it. By way of an example, we study the dipole radiation field and the field of a finite and restricted electromagnetic pulse. 2. PHOTON EMISSION BY A SYSTEM OF CHARGES AND CURRENTS The concept of the number of photons can be introduced in classical electrodynamics by using the familiar expression for the energy–momentum emitted by a moving charged particle ([5, p. 225]). This expression can be generalized to the case of an arbitrary finite and bounded system of currents and charges as follows:
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