D. Han

Stokes parameters as a Minkowskian four-vector

It is noted that the Jones-matrix formalism for polarization optics is a six-parameter two-by-two representation of the Lorentz group. It is shown that the four independent Stokes parameters form a Minkowskian four-vector, just like the energy-momentum four-vector in special relativity. The optical filters are represented by four-by-four Lorentz-transformation matrices. This four-by-four formalism can deal with partial coherence described by the Stokes parameters. A four-by-four matrix formulation is given for decoherence effects on the Stokes parameters, and a possible experiment is proposed. It is shown also that this Lorentz-group formalism leads to optical filters with a symmetry property corresponding to that of two-dimensional Euclidean transformations.

Illustrative example of Feynman’s rest of the universe

Coupled harmonic oscillators occupy an important place in physics teaching. It is shown that they can be used for illustrating an increase in entropy caused by limitations in measurement. In the system of coupled oscillators, it is possible to make the measurement on one oscillator while averaging over the degrees of freedom of the other oscillator without measuring them. It is shown that such a calculation would yield an increased entropy in the observable oscillator. This example provides a clarification of Feynman’s rest of the universe.

Little group for photons and gauge transformations

Based on Weinberg’s work on Lorentz‐transformation properties of massless particles, we discuss the little group for photons and gauge transformations from a pedagogical standpoint. It is pointed out that the ’’translational’’ degrees of freedom associated with the photon little group can generate a transformation which guarantees the transversality of the four‐vector representation for photons. This little‐group transformation, which leaves the photon momemtum invariant, can be regarded as a gauge transformation.

Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two-component spinors

A set of rotations and Lorentz boosts is presented for studying the three‐parameter little groups of the Poincare group. This set constitutes a Lorentz generalization of the Euler angles for the description of classical rigid bodies. The concept of Lorentz‐generalized Euler rotations is then extended to the parametrization of the E(2)‐like little group and the O(2,1)‐like little group for massless and imaginary‐mass particles, respectively. It is shown that the E(2)‐like little group for massless particles is a limiting case of the O(3)‐like or O(2,1)‐like little group. A detailed analysis is carried out for the two‐component SL(2,c) spinors. It is shown that the gauge degrees of freedom associated with the translationlike transformation of the E(2)‐like little group can be traced to the SL(2,c) spins that fail to align themselves to their respective momenta in the limit of large momentum and/or vanishing mass.

Jones-matrix formalism as a representation of the Lorentz group

It is shown that the two-by-two Jones-matrix formalism for polarization optics is a six-parameter two-by-two representation of the Lorentz group. The attenuation and phase-shift filters are represented, respectively, by the three-parameter rotation subgroup and the three-parameter Lorentz group for two spatial dimensions and one time dimension. The Lorentz group has another three-parameter subgroup, which is like the two-dimensional Euclidean group. Optical filters that may have this Euclidean symmetry are discussed in detail. It is shown that the Jones-matrix formalism can be extended to some of the nonorthogonal polarization coordinate systems within the framework of the Lorentz-group representation.

Polarization optics and bilinear representation of the Lorentz group

Abstract It is shown that the bilinear representation of the Lorentz group is the natural language for the polarization of light. The combined effect of attenuation and phase-shift filters leads to a two-by-two representation of the Lorentz group, which can then be translated into the bilinear representation. The coherency matrix is a representation of the six-parameter Lorentz group.

Wigner rotations and Iwasawa decompositions in polarization optics.

Wigner rotations and Iwasawa decompositions are manifestations of the internal space-time symmetries of massive and massless particles, respectively. It is shown to be possible to produce combinations of optical filters which exhibit transformations corresponding to Wigner rotations and Iwasawa decompositions. This is possible because the combined effects of rotation, phase-shift, and attenuation filters lead to transformation matrices of the six-parameter Lorentz group applicable to Jones vectors and Stokes parameters for polarized light waves. The symmetry transformations in special relativity lead to a set of experiments which can be performed in optics laboratories.

O(3,3)-like Symmetries of Coupled Harmonic Oscillators

In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) or SL(4,r) in the four‐dimensional phase space. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r), respectively. It is shown that among the six Sp(4)‐like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics.

Interferometers and decoherence matrices

It is shown that the Lorentz group is the natural language for two-beam interferometers if there are no decoherence effects. This aspect of the interferometer can be translated into six-parameter representations of the Lorentz group, as in the case of polarization optics where there are two orthogonal components of one light beam. It is shown that there are groups of transformations which leave the coherency or density matrix invariant, and this symmetry property is formulated within the framework of Wigners little groups. An additional mathematical apparatus is needed for the transition from a pure state to an impure state. Decoherence matrices are constructed for this process, and their properties are studied in detail. Experimental tests of this symmetry property are possible.

Lorentz Squeezed Hadrons and Hadronic Temperature

Abstract It is shown possible to define the temperature of Lorentz-squeezed hadrons in terms of their speed. Within the framework of the covariant harmonic oscillator formalism which is the simplest scientific language for Lorentz-squeezed hadrons, the hadronic temperature is measured through ( v c ) 2 = exp ( - h ω kT ) . As the temperature rises, the hadron goes through a transition from the confinement phase to a plasma phase.