It is noted that the Jones-matrix formalism for polarization optics is a sixparameter 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 fourby-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.
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.
An attempt is made to give a physical interpretation to the phenomenological wave function of Yukawa, which gives a correct nucleon form factor in the symmetric quark model. This wave function is first compared with the Bethe-Salpeter wave function. It is shown that they have similar Lorentz-contraction properties in the high-momentum limit. A hyperplane harmonic oscillator is then introduced. It is shown that the Yukawa wave function, which is defined over the entire four-dimensional Euclidean space, can be interpreted in terms of the three-dimensional hyperplane oscillators. It is shown further that this wave function satisfies a Lorentz-invariant differential equation from which excited harmonic-oscillator states can be constructed, and from which a gauge-invariant electromagnetic interaction can be generated.
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 Wigner's 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.
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