On an Axiomatization of Path Integral Quantization and its Equivalence to Berezin's Quantization 

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Briefing Doc: Axiomatization of Path Integral Quantization and Equivalence to Berezin's Quantization
Author: Joshua Lackman, Peking University (josh@pku.edu.cn)
Source: Excerpts from "On an Axiomatization of Path Integral Quantization and its Equivalence to Berezin's Quantization" (arXiv:2410.02739v1)
Date: October 2024
Glossary of Key Terms
Abstract Coherent State Quantization: A method of quantizing a classical phase space by embedding it into a non-commutative algebra, typically a W*-algebra.
Berezin's Quantization: A specific approach to quantization using coherent states and Toeplitz operators, closely related to geometric quantization.
Coherent State: A state in quantum mechanics that minimizes the uncertainty principle for a specific pair of conjugate variables.
Deformation Quantization: A quantization scheme that deforms the commutative product of classical observables into a non-commutative product on a space of quantum observables.
Groupoid: A mathematical structure generalizing the notion of a group, where the product of two elements is not always defined.
Hamiltonian: A function on a phase space that describes the energy of a classical system, used to define the equations of motion.
Hermitian: A property of linear operators that generalizes the concept of symmetry from real numbers to complex Hilbert spaces.
Hilbert Space: A complete inner product space that provides the mathematical framework for quantum mechanics.
Kahler Manifold: A manifold with a compatible Riemannian, symplectic, and complex structure, often arising in geometric quantization.
Lagrangian Polarization: A choice of subspace of the tangent bundle of a symplectic manifold that is maximally isotropic and involutive.
Lie Algebroid: A generalization of both a Lie algebra and a tangent bundle, providing a framework for studying differentiable groupoids.
Multiplicative Line Bundle: A line bundle over a groupoid equipped with a compatible group structure on its total space.
Overcompleteness: A property of a set of vectors in a Hilbert space where removing any single vector does not change the span of the set.
Parallel Transport: A way of moving geometric objects along a path in a manifold while preserving their relationship to the underlying geometry.
Path Integral Quantization: A quantization method based on summing over all possible paths of a system, weighted by their classical action.
Prequantum Line Bundle: A Hermitian line bundle over a symplectic manifold with a connection whose curvature is proportional to the symplectic form.
Propagator: A function that describes the amplitude for a quantum particle to move from one point to another in a given time.
Reproducing Kernel Hilbert Space: A Hilbert space of functions where pointwise evaluation is a continuous linear functional.
Symplectic Manifold: A smooth manifold equipped with a closed, non-degenerate 2-form called the symplectic form.
W-Algebra:* A type of *-algebra that is also a Banach space and satisfies a weak continuity condition, important in the study of operator algebras.
Main Themes
Formalizing the Path Integral: The traditional path integral formulation lacks rigorous mathematical definition. This paper addresses this by introducing the concept of a "propagator" (Definition 3.1.1), a section of a line bundle satisfying specific properties. It formally shows the equivalence between this propagator and the path integral (Theorem 5.0.2).
Abstract Coherent State Quantization: The paper introduces "abstract coherent state quantization" (Definition 1.0.1). This framework uses a continuous injection of a manifold into a W*-algebra, satisfying axioms like projection, overcompleteness, and separation, to quantize the system.
Equivalence of Quantization Schemes: The core result is the demonstration of the equivalence between path integral quantization, defined through the propagator, and Berezin's quantization. This equivalence is established by showing that both approaches lead to the same W*-algebra representation and share essential properties.
Applications and Examples: The paper analyzes specific examples, including:
Overcomplete projective submanifolds like the 2-sphere (Section 6.1)
Quantization arising from unitary representations (Section 6.2)
Quantization using reproducing kernels (Section 6.3)
Path integral quantization of symplectic manifolds, including Berezin-Toeplitz quantization (Section 7)
The path integral in the context of Lie algebroids and Poisson manifolds (Sections 8 and 9)