Deformation Quantization: From Quantum Mechanics to Quantum Field Theory

TL;DR

This paper provides an overview of Deformation Quantization (DQ), highlighting its development over thirty years in physics, its conceptual benefits, and its applications from quantum mechanics to quantum field theory, including specific results on dS/AdS spaces.

Contribution

It summarizes key developments in DQ, discusses its conceptual advantages, and presents previous results on Fedosov star-products on dS/AdS spaces, verifying their consistency with earlier analyses.

Findings

Fedosov star-product constructed on dS/AdS spaces

Verification of DQ results matching previous analyses

Convergence of formal series in DQ treatments

Abstract

The aim of this paper is to give a basic overview of Deformation Quantization (DQ) to physicists. A summary is given here of some of the key developments over the past thirty years in the context of physics, from quantum mechanics to quantum field theory. Also, we discuss some of the conceptual advantages of DQ and how DQ may be related to algebraic quantum field theory. Additionally, our previous results are summarized which includes the construction of the Fedosov star-product on dS/AdS. One of the goals of these results was to verify that DQ gave the same results as previous analyses of these spaces. Another was to verify that the formal series used in the conventional treatment converged by obtaining exact and nonperturbative results for these spaces.