Quantum Phase Space, Quantization Hierarchy, and Eclectic Quantum Many-Body System

TL;DR

This paper introduces a novel operator-valued quantum phase space formula that links different quantization levels, proposes a hierarchy extending standard quantum theory, and applies it to a many-body system with a non-exponentially scaling Hilbert space.

Contribution

It develops a new quantization hierarchy and an eclectic quantum many-body model with local interactions and constrained dynamics.

Findings

Quantum phase space formula links first and second quantization.

A new hierarchy extends standard quantum theory.

Hilbert space dimension does not grow exponentially with particles.

Abstract

An operator-valued quantum phase space formula is constructed. The phase space formula of Quantum Mechanics provides a natural link between first and second quantization, thus contributing to the understanding of quantization problem. By the combination of quantization and hamiltonization of dynamics, a quantization hierarchy is introduced, beyond the framework of first and second quantization and generalizing the standard quantum theory. We apply our quantization method to quantum many-body system and propose an eclectic model, in which the dimension of Hilbert space does not scale exponentially with the number of particles due to the locality of interaction, and the evolution is a constrained Hamiltonian dynamics.