Non-commutative geometry and measurements of polarized two photon coincidence counts

Abstract

Employing Maxwell's equations as the field theory of the photon, quantum mechanical operators for spin, chirality, helicity, velocity, momentum, energy and position are derived. The photon "Zitterbewegung" along helical paths is explored. The resulting non-commutative geometry of photon position and the quantum version of the Pythagorean theorem is discussed. The distance between two photons in a polarized beam of given helicity is shown to have a discrete spectrum. Such a spectrum should become manifest in measurements of two photon coincidence counts. The proposed experiment is briefly described.

Dynamics of zero-point energy and two-slit phenomena for photons

“…Much of the above formalism can be found in [4]. A symmetric treatment of forward and backward in time motions is part and parcel of the Maxwell field theory.…”

“…(62) Of course, the center is unspecified, that is why two, parallel, same helicity, photons were needed in [4] to allow for a measurement of the quantization in the difference between the (transversal) positions of the two photons. Incidentally, Maxwell was well aware of two opposite screw motions (corresponding to the two helicities) about the axis of propagation.…”

“…Of course, once non-commuting spins are introduced, say for a massless photon of spin 1, the quantum and classical theories need not be equivalent. For example, the spin would precess for a given mean momentum p at an angular frequency Ω = pc/h = 2πc/λ in a plane perpendicular to the direction of motion [4]. We consider this in the following Section.…”

“…Proceeding further, in [4], using Maxwell's equations the photon Zitterbewegung motion along helical paths was explored and the resulting non-commutative geometry of photon position and, the distance between two photons in a polarized beam of a given helicity was shown to have a discrete spectrum that should become manifest in measurements of two photon coincidence counts. An experiment was proposed and its feasibility examined in [5].…”

“…However, once noncommuting spins are introduced, the quantum and classical theories need not be equivalent. For example, the spin precession needs to be considered [4]. This is studied in Sec.…”