nLab holonomic quantum field

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

The Kyōto school of mathematical physics (Jimbo, Miwa, Sato etc.) in mid 1970-s discovered a nontrivial connection between the theory of isomonodromic deformations of differential equations (and closely related integrable systems) and the theory of a special class of quantum field theories (whose construction heavily relies upon Clifford algebras and Clifford groups?). This subject lives under the title holonomic quantum fields. The work is also relevant to the study of Painlevé transcendents.

From JMSM 2005:

Through recent study of the problems in mathematical physics, a deep, unexpected link has emerged: a link between the monodromy preserving deformation theory for linear (ordinary and partial) differential equations, and a class of quantum field operators ([1][2] [3]). The aim of this article is to give an overview to the present stage of development in the theory (see also [4]).

The fruit of the above link is multifold. On the one hand it enables one to compute exactly the nn point correlation functions of the field in question in a closed form, using solutions to certain non-linear differential equations of specific type (such as the Painlevé equations). On the other hand it provides an effective new tool of describing the deformation theory by means of quantum field operators. Thus it stands as a good example of the fact that not only the pure mathematics is applied to physical problems but also the converse is true.

References

Introduction and review:

Original articles:

See also:

  • Сато М., Дзимбо М., Мива Т. Голономные квантовые поля (a collection of the reprints in Russian of the articles of Kyōto school) vol. 30 (1983) in the series Matematika – novoe v zarubežnoj nauke (description)

One of the primary ideas stems from an observation of

  • L. Onsager, Phys. Rev. 65 (1944), 117-149, “who discovered in effect that field operators on 2-dimensional Ising lattice are elements of a Clifford group”

Last revised on June 13, 2022 at 02:08:56. See the history of this page for a list of all contributions to it.