A geometric gauge for the photon position operator

J. Bertrand

Margaret Hawton, W. E. Baylis

We extend a procedure for the construction of photon position operators with transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)] to body rotations described by three Euler angles. The axial angle can be made a function of the two polar angles, and different choices of the functional dependence are analogous to different gauges of a magnetic field. Symmetries broken by a choice of gauge are reestablished by transformations within the gauge group. The approach allows several previous proposals to be related. Because of the coupling of the photon momentum and spin, our position operator, like that proposed by Pryce, is a matrix that does not commute with the spin operator. Unlike the Pryce operator, however, our operator has commuting components, but the commutators of these components with the total angular momentum require an extra term to rotate the matrices for each vector component around the momentum direction. Several proofs of the nonexistence of a photon position operator with commuting components are based on overly restrictive premises that do not apply here.

Michał Dobrski, Maciej Przanowski, Jaromir Tosiek +1 more

Karl-Peter Marzlin, Stephen D. Bartlett, Barry C. Sanders

We introduce the entanglement gauge describing the combined effects of local operations and nonlocal unitary transformations on bipartite quantum systems. The entanglement gauge exploits the invariance of nonlocal properties for bipartite systems under local (gauge) transformations. This new formalism yields observable effects arising from the gauge geometry of the bipartite system. In particular, we propose a non-Abelian gauge theory realized via two separated spatial modes of the quantized electromagnetic field manipulated by linear optics. In this linear optical realization, a bipartite state of two separated spatial modes can acquire a non-Abelian geometric phase.

Steven B. Berger

Margaret Hawton

A first quantized theory of the photon is developed that incorporates all of the usual rules of quantum mechanics. The state vector is argued to be proportional to the four-vector potential. Scalar products and the probability amplitude are invariant under gauge transformations that satisfy the Lorentz gauge condition. The eigenvectors of a recently constructed Hermitian position operator with commuting components provide a basis for the position representation.