The “Galilean” components of a position operator for the photon

Michał Dobrski, Maciej Przanowski, Jaromir Tosiek +1 more

Helmut Bez

The representation theory of the Galilean group extended by a one-dimensional group of dilations (which leave invariant the free Schrodinger equation) is studied. The existence of position and time operators is investigated from the standpoint of the imprimitivity theorem, for quantum particles whose states carry a projective representation of the extended group. Position and time operators are shown to exist for some two-component direct sums with non-trivial multipliers and for vector representations with zero helicity.

J. Bertrand

R. Huegele, Z. E. Musielak, John L. Fry

A search for fundamental (Galilean invariant) dynamical equations for two- and four-component spinor wavefunctions is conducted in Galilean spacetime. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wavefunctions. A method to derive the Levy-Leblond equation for a four-component spinor wavefunction is presented. It is formally proved that the Levy-Leblond and Schrodinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schrodinger phase factor. Physical implications of the obtained results and their relationships to the Pauli–Schrodinger equation are discussed.

M. Comi

Steven B. Berger