A Note on the Scalar Product in Relativistic Quantum Mechanics

G. P. Pronko

I. Gottlieb

Hartland S. Snyder

T. S. Chang

The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.

K. V. Roberts

Eduard Prugovečki

It is shown that the new interpretation of the scalar product in Hilbert space recently proposed by Aharanov, Albert, and Au is in fact the one underlying the stochastic phase-space formulation of nonrelativistic quantum mechanics. The extrapolation of this interpretation to the relativistic domain and its relationship to the program of geometrizing quantum theory and quantizing space-time are discussed.