Phase-space Quantization of Field Theory

In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic tra-jectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple—indeed, classical—for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase-space. This is a pedagogical selection from work published in 1), 2) , reported at the Yukawa Institute Workshop " Gauge Theory and Integrable Models " , 26-29 January, 1999. The third complete autonomous formulation of Quantum Mechanics, distinct from conventional Hilbert space or path-integral quantization, is based on Wigner's phase-space distribution function (WF), which is a special representation of the density matrix 3), 4). In this formulation, known as deformation quantization 5) , expectation values are computed by integrating mere c-number functions in phase-space, the WF serving as a distribution measure. Such phase-space functions multiply each other through the pivotal ⋆-product 6) , which encodes the noncommutative essence of quantization. The key principle underlying this quantization is the ⋆-product's operational isomorphism 5) to the conventional Heisenberg operator algebra of quantum mechanics. Below, we address gauge invariance in phase-space through canonical transformation to and from free systems. Further, we employ the ⋆-unitary evolution operator, a " ⋆-exponential " , to specify the time propagation of Wigner phase-space distribution functions. The answer is known to be remarkably simple for the harmonic oscillator WF, and consists of classical rigid rotation in phase-space for the full quantum system. It serves as the underpinning of the generalization to field theory we consider, in which the dynamics is specified through the evolution of c-number distributions on field phase-space. We start by illustrating the basic concepts in 2d phase-space, without loss of generality; then, in generalizing to field theory, we make the transition to infinite-dimensional phase-space.