Quantisation of the free electromagnetic field implies quantisation of its angular momentum

Abstract

It is shown that when the gauge-invariant Bohr-Rosenfeld commutators of the free electromagnetic field are applied to the expressions for the linear and angular momentum of the electromagnetic field interpreted as operators then, in the absence of electric and magnetic charge densities, these operators satisfy the canonical commutation relations for momentum and angular momentum. This confirms their validity as operators that can be used in quantum mechanical calculations of angular momentum. IntroductionThe free electromagnetic field was first quantized in a plane wave basis in the Coulomb gauge and later in the Lorenz gauge [1,2,3,4]. The four components of the electromagnetic potential were promoted to operators and the commutation relations between them determined by analogy with those of the simple harmonic oscillator. The commutations relations are different in each gauge [3]. The quanta of the electromagnetic field are known as photons and the states that the operators act on are Fock states of photons (not necessarily plane waves) [5]. By taking appropriate combinations of the derivatives of the potentials, Bohr and Rosenfeld [6] obtained the gauge-invariant equal time commutation relations between the field operators E(x,t) and B(y,t)

Role of the nonlocality of the vector potential in the Aharonov–Bohm effect

“…The non-local nature of the potentials was anticipated by Feynman [13]. The potentials ( 9) and ( 10) at time t are determinable (at some later time) to an arbitrary accuracy (within the limits allowed by quantum mechanics [27]) to the extent that the fields are measurable with detectors that have put in place in the past. It is only when the arbitrary gauge function χ of ( 3) is included in the potentials that they become undeterminable because of its presence.…”

Comparison of Different Forms for the “Spin” and “Orbital” Components of the Angular Momentum of Light

“…( 13), was not specified by these writers. Although the total angular momentum of the field has been shown to satisfy the canonical commutation relations for angular momentum [14] the components have not [5].…”