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From path integral quantization to stochastic quantization: a pedestrian's journey

Dario Benedetti, Ilya Chevyrev, Razvan Gurau

Abstract

We give two novel proofs that the path integral and stochastic quantizations of generic scalar Euclidean quantum field theories are equivalent. Our proofs rely on Taylor interpolations indexed by forests, in the fashion of constructive field theory. The first proof works at the level of individual terms in the Feynman expansion, with the forests appearing as spanning forests in Feynman graphs. The second one works at the level of the path integral and avoids the full expansion of the Feynman perturbation series.

From path integral quantization to stochastic quantization: a pedestrian's journey

Abstract

We give two novel proofs that the path integral and stochastic quantizations of generic scalar Euclidean quantum field theories are equivalent. Our proofs rely on Taylor interpolations indexed by forests, in the fashion of constructive field theory. The first proof works at the level of individual terms in the Feynman expansion, with the forests appearing as spanning forests in Feynman graphs. The second one works at the level of the path integral and avoids the full expansion of the Feynman perturbation series.
Paper Structure (47 sections, 2 theorems, 124 equations, 18 figures)

This paper contains 47 sections, 2 theorems, 124 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Path integral and stochastic quantization
  4. Euclidean quantum field theory.
  5. Path integral quantization.
  6. Stochastic quantization.
  7. Main result
  8. Preliminaries
  9. Combinatorial structures
  10. Graphs.
  11. Combinatorial maps.
  12. Unlabeled combinatorial maps.
  13. Tree picking
  14. Trees and ordinary differential equations
  15. Recursive trees.
  16. ...and 32 more sections

Key Result

Theorem 1

The quantum field theoretical $n$-point functions coincide with the stochastic $n$-point functions in the limit of large, coincident, fictitious times: Our first result is that this equality holds at all orders in a (formal) perturbative expansion in the vertex kernels. Our second result is that this equality holds at the level of formal path integrals.

Figures (18)

  • Figure 1: A combinatorial map with half-edges ${\cal D} =\{r,v^1,v^2,v^3,v^4,w^1,w^2,w^3,u^1,u^2\}$, root $r$, and permutations, written in cycle notation, $\sigma = (r) (v^1v^2v^3v^4)(w^1w^2w^3)(u^1u^2)$ and $\alpha=(rv^1) (v^2w^1) (v^3u^1)(v^4w^3) (u^2w^2)$. The root $r$ is a fixed point $(r)$ of the permutation $\sigma$.
  • Figure 2: Unlabeled rooted map corresponding to the map in Fig. \ref{['fig:combimap']}.
  • Figure 3: Examples of unlabeled abstract graphs rooted at the external vertex on the left. For each graph we have displayed the number of distinct possible embedding in the plane, that is the number of unlabeled rooted combinatorial maps associated to it.
  • Figure 4: Unlabeled rooted maps with up to three 3-valent vertices and the distinguished "keep to the right" tree.
  • Figure 5: Recursive trees with up to four non-root vertices, where we have added a root $r$. Notice that these are non-embedded trees, meaning that the order of children at each vertex does not matter.
  • ...and 13 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1: Main theorem
  • Remark 2
  • Theorem 2
  • Remark 3