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A New Look At The Path Integral Of Quantum Mechanics

Edward Witten

TL;DR

The paper develops a direct link between the Feynman path integral of quantum mechanics and the A-model by complexifying phase space and identifying integration cycles with coisotropic A-branes. By carefully employing Morse theory, boundary conditions, and sigma-model localization, Witten derives new, middle-dimensional integration cycles in the loop space whose structure mirrors A-model branes, notably the canonical coisotropic brane. This construction extends to gauge theories, establishing a path integral bridge from N=4 super Yang-Mills in four dimensions to Chern-Simons theory on the boundary, and provides a framework to understand holomorphic superpotentials and hyper-Kähler structures in quantization. The approach yields a unified view across dimensions, connecting quantization, boundary observables, and topological field theory with potential implications for knot homologies via brane dualities.

Abstract

The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for Chern-Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N=4 super Yang-Mills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N=4 path integral in four dimensions.

A New Look At The Path Integral Of Quantum Mechanics

TL;DR

The paper develops a direct link between the Feynman path integral of quantum mechanics and the A-model by complexifying phase space and identifying integration cycles with coisotropic A-branes. By carefully employing Morse theory, boundary conditions, and sigma-model localization, Witten derives new, middle-dimensional integration cycles in the loop space whose structure mirrors A-model branes, notably the canonical coisotropic brane. This construction extends to gauge theories, establishing a path integral bridge from N=4 super Yang-Mills in four dimensions to Chern-Simons theory on the boundary, and provides a framework to understand holomorphic superpotentials and hyper-Kähler structures in quantization. The approach yields a unified view across dimensions, connecting quantization, boundary observables, and topological field theory with potential implications for knot homologies via brane dualities.

Abstract

The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for Chern-Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N=4 super Yang-Mills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N=4 path integral in four dimensions.

Paper Structure

This paper contains 36 sections, 189 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Integration Cycles For Quantum Mechanics
  3. Preliminaries
  4. The Basic Feynman Integral
  5. Analytic Continuation
  6. The Simplest Integration Cycles
  7. Review Of Morse Theory
  8. A New Integration Cycle For The Feynman Integral
  9. $I$ And The $A$-Model
  10. Interpretation In Sigma-Model Language
  11. The Physical Model
  12. The Boundary Condition And Localization
  13. Recovering The Hilbert Space
  14. Hamiltonians
  15. Rerunning The Story With A Hamiltonian
  16. ...and 21 more sections

Figures (1)

  • Figure 1: Integration cycles for the integral $I_\Gamma$ of eqn. (\ref{['tmorf']}). The four cycles obey one relation $\Gamma_1+\Gamma_1+\Gamma_2+\Gamma_3=0$, and any three of them give a basis for the space of possible integration cycles.