Feynman quantization: an operator approach to path integration over commuting and anticommuting variables

An operator quantization scheme on a continuous direct product of Hilbert spaces over a time interval is presented as an extension of the quantization using Feynman path integrals. The continuous direct product is defined as a Hilbert space with two principal bases: the Fock and the Feynman ones. The Fock basis, defined by a complete set of commuting operators at different times, serves for a definition of the operator calculus. The Feynman basis, simultaneously diagonalizing the complete set of commuting operators, leads to path integrals constructed without time slicing as a spectral representation of certain operator functions. The construction of quantum theory and the corresponding path integrals for the harmonic oscillator is demonstrated both in the configuration and phase spaces. Extension of the theory to coherent states and anticommuting variables is performed. (author). 10 refs