Vacuum-to-vacuum transition probability and the classic radiation theory

Highlights

• Quantum viewpoint of radiation theory based on the vacuum-to-transition probabilities.

• Mathematical method in handling radiation for extended and point sources.

• Radiated energy and power for arbitrary source distribution obtained from the above.

• Explicit power of radiation for point relativistic sources from the general theory.

Abstract

Using the fact that the vacuum-to-vacuum transition probability for the interaction of the Maxwell field $A^{\mu }\left(x\right)$ with a given current $J_{\mu }\left(x\right)$ represents the probability of no photons emitted by the current of a Poisson distribution, the average number of photons emitted of given energies for the underlying distribution is readily derived. From this the classical power of radiation of Schwinger of a relativistic charged particle follows.

Introduction

The Maxwell Lagrangian density for the interaction of the vector potential $A_{\mu }\left(x\right)$ with an external current $J^{\mu }\left(x\right)$ is given by

$$\mathcal{L}\left(x\right)=-{\displaystyle \frac{1}{4}}{F}_{\mu \nu }\left(x\right)F^{\mu \nu }\left(x\right)+J^{\mu }\left(x\right)A_{\mu }\left(x\right),F_{\mu \nu }\left(x\right)={\partial }_{\mu }{A}_{\nu }\left(x\right)-{\partial }_{\nu }{A}_{\mu }\left(x\right).$$

Prior to switching on of the current, as a source of photon production, one is dealing with a vacuum state, denoted by $|0_{-}〉$, involving no photons. After switching on of the current, the state of the system may evolve to one involving any number of photons, or it may just stay in the vacuum state, involving no photons, with the latter state now denoted by $|0_{+}〉$. Quantum theory tells us that the vacuum-to-vacuum transition probability satisfies the inequality $|〈0_{+}|0_{-}〉{|}^2<1$, due to conservation of probability, allowing the possibility that the system evolves to other states as well involving an arbitrary number of photons that may be created by the current source. A very interesting property of this system is that the probability distribution of the photon number N created by the current (Schwinger, 1970) is given by the Poisson distribution (Manoukian, 2011). That is

$$\mathrm{Prob}[N=n]=\frac{{(\lambda )}^n}{n!}{\mathrm{e}}^{-\lambda },n=0,1,\dots ,\lambda =〈N〉,$$

where $\lambda =〈N〉$ denotes the average number of photons created by the current source, and

$$\exp [-〈N〉]=|〈0_{+}|0_{-}〉{|}^2,$$

denotes the probability that no photons are created by the current source, i.e., it represents the vacuum-to-vacuum transition probability $|〈0_{+}|0_{-}〉{|}^2$ as just stated.

The purpose of this communication is by using the expression of the vacuum-to-vacuum transition amplitude, derive the exact expression of the average number of photons, of a given frequency, produced by a given general current distribution, from which the classic radiation theory of radiation emitted from a relativistic charge particle may be readily obtained for the power of radiation as well as for the power of radiation emitted along a given direction.

Quantum viewpoint analysis, as discussed above, of electromagnetic phenomena and electromagnetic radiation, e.g., Manoukian and Viriyasrisuwattana (2006), and of the related applications (Feynman, 1985, Bialynicki-Birula, 1996, Manoukian, 1997, Kennedy et al., 1980, Deitsch and Candelas, 1979, Manoukian and Viriyasrisuwattana, 2006, Manoukian, 2013, Schwinger, 1973) turns out to be quite useful in applications and certainly in simplifying, to a large extent, derivations in this field as we will witness below. In particular, we note that due to the generality of the expressions leading to the total energy of radiation emitted, derived below in (18), and being valid for arbitrary current distributions, the analysis is expected to have further applications in the domain of synchrotron radiation as well as in considering quantum corrections to general physical problems in radiation theory. We are especially interested in generalizing the present analysis to radiation in the presence of obstacles as well in different media than just the vacuum and will be attempted in a future report.