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holonomic quantum field

Contents

What is this about

There is no single mathematical idea expressed here yet!

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.

Idea: Somebody should figure it out

References

There is a series of longer articles called Holonomic quantum fields I-V (mainly Publ. RIMS) and longer series of shorter articles called Studies in holonomic quantum fields I-XVI (the latter can be found at project euclid, see link).

  • 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

  • Mikio Sato, Tetsuji Miwa, Michio Jimbo, Holonomic quantum fields I, Publ. RIMS 14 , n.1 (1978) pp.223–267, pdf; Holonomic quantum fields II — The Riemann-Hilbert Problem, Publ. RIMS 15 (1979) No. 1, pp.201–278, pdf; Holonomic quantum fields III, Publ. RIMS 15 (1979) No. 2 pp.577-629, pdf; 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

  • Michio Jimbo, Mikio Sato, Tetsuji Miwa, Supplement to Holonomic quantum fields IV, Publ. RIMS 17 (1981) No. 1 pp.137-151 pdf

  • 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)

  • M. Jimbo, T. Miwa, M. Sato, Holonomic quantum fields — the unanticipated link between deformation theory of differential equations and quantum fields, K. Osterwalder (ed.), Mathematical problems in theoretical physics, Springer (1980) 119–142

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”

Revised on March 24, 2010 17:25:59 by Zoran Škoda (161.53.130.104)