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arXiv:1709.04884 · Journal of Mathematical Physics

Photon position eigenvectors, Wigner’s little group, and Berry’s phase

Margaret Hawton, Vincent Debierre

Published May 1, 2019• 10 citations
arXivHTMLPDF

Abstract

We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x^, reflects the E(2) symmetry of the photon little group. The eigenvectors of x^ form a basis of localized states that have definite angular momentum, J^, parallel to their common axis of symmetry. This basis is well suited to the description of “twisted light” that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that x^1,x^2,J^3 is a realization of the two dimensional Euclidean e2 algebra that effects genuine infinitesimal displacements in configuration space.

DOI
10.1063/1.5009073

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We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x^, reflects the E(2) symmetry of the photon little group. The eigenvectors of x^ form a basis of localized states that have definite angular momentum, J^, parallel to their common axis of symmetry. This basis is well suited to the description of “twisted light” that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that x^1,x^2,J^3 is a realization of the two dimensional Euclidean e2 algebra that effects genuine infinitesimal displacements in configuration space.
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It is noted that the symmetry of two-mode squeezed states of light is governed by the group Sp(4) that is locally isomorphic to O(3,2). This group has subgroups that are locally isomorphic to the two-dimensional Euclidean group. Two-mode states having the E(2) symmetry are constructed. The translation-like transformations of this symmetry group shear the Wigner distribution function defined over the four-dimensional phase space consisting of two pairs of canonical variables. Sheared states are constructed in the Schrödinger picture of quantum mechanics and in the Fock space for photon numbers. It is shown that the Wigner phase-space picture is a convenient representation of quantum mechanics for calculating measurable quantities of the sheared states.
A photon position operator with commuting components is constructed, and it is proved that it equals the Pryce operator plus a term that compensates for the adiabatic phase. Its eigenkets are transverse and longitudinal vectors, and thus states can be selected that have definite polarization or helicity. For angular momentum and boost operators defined in the usual way, all of the commutation relations of the Poincar\'e group are satisfied. This new position operator is unitarily equivalent to the Newton-Wigner-Pryce position operator for massive particles.
We propose a position operator for the photon. The components commute and commute with helicity, and the two operators form a complete set of commuting observables. However, the simultaneous eigenvectors of this position operator and helicity do not rotate locally. For a normalized wavepacket state vector, the position-helicity amplitudes give a complete characterization of the physical properties of the state vector, yet their moduli-squared have rotation properties that do not have a simple interpretation. Thus we call these pseudo-amplitudes. We show that there is a regime of wavepacket state vectors, those with a small fractional spread around an average momentum, where the position-helicity probability pseudo-densities rotate very nearly locally as scalar functions. This is just the regime that is needed to describe a scattering experiment involving photons. We consider other measures of localization for the photon, including those proposed by other authors. It is important to note that exactly the same problems arise if we try to characterize the state vector of a massive particle with spin by position and helicity. The problems are not the result of the masslessness of the photon but arise because of its limited helicity spectrum.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.
We have recently constructed a photon position operator with commuting components. This was long thought to be impossible, but our position eigenvectors have a vortex structure like twisted light. Thus they are not spherically symmetric and the position operator does not transform as a vector, so that previous non-existence arguments do not apply. We find two classes of position eigenvectors and obtain photon wave functions by projection onto the bases of position eigenkets that they define, following the usual rules of quantum mechanics. The hermitian position operator, r⁁(0), leads to a Landau-Peierls wave function, while field-like eigenvectors of the nonhermitian position operator and its adjoint lead to a biorthonormal basis. These two bases are equivalent in the sense that they are related by a similarity transformation. The eigenvectors of the nonhermitian operators r⁁(±½) lead to a field-potential wave function pair. These field-like positive frequency wave functions satisfy Maxwell's equations, and thus justify the supposition that MEs describe single photon wave mechanics. The expectation value of the number operator is photon density with undetected photons integrated over, consistent with Feynman's conclusion that the density of non-interacting particles can be interpreted as probability density.