Волновая механика для полей

Abstract

We present a detailed discussion of the Schrodinger approach to wave mechanics applied to free fields instead of particles. The state vector is a functional of the fields and a function of time, and the operators are constructed from the fields as the co-ordinates and functional derivatives with respect to these fields as the momenta, which thus obey the canonical commutation relations. We first study the complex scalar field along the lines of the harmonic oscillator in ordinary quantum mechanics, and we indicate briefly what changes are introduced when the field is real. Next we discuss the massive Lorentz vector field, examining the usual difficulties related to the scalar part, such as negative energies and the indefinite metric. We proceed to discuss the electromagnetic field, with special attention given to the constraints of the theory; we do this both in the usual manner where the potentials are the generalized co-ordinates and in a variation where the fields themselves are the co-ordinates. In both ways we obtain a gauge-independent theory with no redundant variables and no indefinite metric in the space of state vectors. We also discuss the spinor field briefly, but we soon realize that this method is not applicable in its present form. We do not obtain this way any new results, but we explore a new method that brings fresh insights to the problems that beset the quantum theory of fields.