Reducible Quantum Electrodynamics. I. The Quantum Dimension of the Electromagnetic Field

Abstract

In absence of currents and charges the quantized electromagnetic field can be described by wave functions which for each individual wave vector are normalized to one. The resulting formalism involves reducible representations of the Canonical Commutation Relations. The corresponding paradigm is a space-time filled with two-dimensional quantum harmonic oscillators. Mathematically, this is equivalent with two additional dimensions penetrated by the electromagnetic waves.

Emergent Coulomb Forces in Reducible Quantum Electrodynamics

“…The vectors u (1) , u (2) , v (3) , v (4) are the analogues of the polarization vectors of the photon. They are partly fixed by the requirement that the vector with components ψr satisfies Dirac's equation…”

“…Similarly is γ 0 v (s) (k) = −v (s) (−k). Both relations are used in v (4) (k)|γ 0 γ µ v (4) (k ′ ) = Cu (1) (−k)|γ 0 γ µ Cu (1) (−k ′ ) = u (1) (−k)|C † γ 0 γ µ Cu (1) (−k ′ ) = u (1) (−k)|(γ µ γ 0 ) T u (1) (−k ′ ) = u (1) (−k ′ )|γ µ γ 0 u (1) (−k) = u (1) (k ′ )|γ 0 γ µ u (1) (k) .…”

“…In the first of this series of papers [1] free electromagnetic fields are treated in a reducible representation. The methodology developed there is used here to present a theory of free fermionic fields.…”

“…In the present approach the anti-commutation relations are non-canonical. Typical for the approach of [1] is that the integrations over wave vectors move from operator expressions towards the evaluation of expectation values. The equivalent of (1) becomes { φk (x), φk ′ (x ′ )} + = 0 and { φk (x), φ † k ′ (x ′ )} + = e ikν x ν e −ik ′ ν x ′ν .…”

On the Emergence of the Coulomb Forces in Quantum Electrodynamics

“…See [7,8,9] and references given in these papers. This formalism can be simplified along the lines worked out by the author in a series of papers [10,11,12,13].…”