nLab holonomic quantum field

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 $n$ 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:

• Tetsuji Miwa, Michio Jimbo, Introduction to holonomic quantum fields, pp. 28–36 in: The Riemann problem, complete integrability and arithmetic applications, Lec. Notes in Math. 925, Springer (1982) $[$doi:10.1007/BFb0093497$]$

• Michio Jimbo, Tetsuji Miwa, Mikio Sato, Yasuko Môri, Holonomic Quantum Fields — The unanticipated link between deformation theory of differential equations and quantum fields —, In: K. Osterwalder (ed.), Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, 116 Springer (2005) $[$doi:10.1007/3-540-09964-6_310$]$

Original articles:

• Mikio Sato, Tetsuji Miwa, Michio Jimbo, Studies on holonomic quantum fields, I, Proc. Japan Acad. Ser. A Math. Sci. 53 1 (1977) 6-10 $[$doi:10.3792/pjaa.53.6$]$

• Mikio Sato, Tetsuji Miwa, Michio Jimbo, Holonomic quantum fields I, Publ. RIMS 14 1 (1978) 223–267 $[$pdf$]$

  • Holonomic quantum fields II — The Riemann-Hilbert Problem, Publ. RIMS 15 1 (1979) 201–278 $[$doi:10.2977/prims/1195188429, pdf$]$

  • Holonomic quantum fields III, Publ. RIMS 15 2 (1979) 577-629, $[$pdf, doi:10.3792/pjaa.53.153$]$

  • Holonomic quantum fields IV, Publ. RIMS 15 (1979) No. 3 pp.871-972, pdf;

  • Holonomic quantum fields V, Publ. RIMS 16 (1980) No. 2 pp.531-584, pdf

  • Supplement to Holonomic quantum fields IV, Publ. RIMS 17 (1981) No. 1 pp.137-151 pdf

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”