Abstract
Photons, i.e. the basic energy quanta of monochromatic waves, are highly non-localized and occupy all available space in one dimension. This non-local property can complicate the modelling of the quantized electromagnetic field in the presence of optical elements that are local objects. Therefore, in this paper, we take an alternative approach and quantize the electromagnetic field in position space. Taking into account the negative- and the positive-frequency solutions of Maxwell's equations, we construct annihilation operators for highly localized field excitations with bosonic commutator relations. These provide natural building blocks of wave packets of light and enable us to construct locally acting interaction Hamiltonians for two-sided semi-transparent mirrors.
Highlights
In classical electrodynamics, we often characterise light by its local properties such as local amplitudes, direction of propagation and polarisation
As we shall see below, considering both positive- and negative-frequency photons allows us to construct a local description of the quantised EM field which assumes that its basic building blocks are highly-localised field excitations
We assume that the natural basic building blocks of light are highlylocalised field excitations, since these can be combined into spread-out wave packets
Summary
We often characterise light by its local properties such as local amplitudes, direction of propagation and polarisation. As we shall see below, considering both positive- and negative-frequency photons allows us to construct a local description of the quantised EM field which assumes that its basic building blocks are highly-localised field excitations. (1) Without the presence of the negative-frequency photons, we could not define annihilation operators for truly-localised field excitations with bosonic commutator relations which allow us to construct locally-acting Hermitian mirror Hamiltonians. This is done without specifying their commutator relations and without identifying their dynamical Hamiltonian.
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