STUECKELBERG – OGIEVETSKY – POLUBARINOV – CALB – RAMOND – MAXWELL FIELDS, THE GAUGE DEGREES OF FREEDOM

Аннотация

A comparative analysis for the problem of the gauge degrees of freedom for massless particles is described by three different systems of equations: Stückelberg’s, Ogievetsky – Polubarinov – Kalb – Ramon’s, and Maxwell’s. All three systems of equations are represented in a unified matrix form, these equations are solved in the Cartesian coordinates. Solutions of the plane wave type are constructed explicitly; correspondingly, we find 5, 4 and 4 independent solutions. In order to decide in each case, which of the solutions correspond to physically observable states and which – to gauge states, we find for all three the matrix of invariant bilinear form, which permits us to fix the structure of the energy-momentum tensor. Expressions for the energy-momentum tensor are obtained in explicit form for all independent solutions (5, 4 and 4). It is shown that for the Stueckelberg field, only one solution corresponds to a state with non-zero energy density, it describes the physically observable state; for 4 remaining solutions the energy-momentum tensor turns out to be zero, which indicates the gauge nature of these solutions. For the Ogievetsky – Polubarinov – Kalb – Ramond field, only one solution turns out to be physically observable. Finally, in the case of the Maxwell field, two solutions are physically observable, and two other solutions are gauge ones. Besides, among the two gauge solutions one has the structure of the ordinary plane wave, and the other describes an unusual plane wave: for it there is no standard relationship between energy and the three components of the linear momentum.