SIGMA 18 (2022), 044, 15 pages arXiv:2203.09296
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action
Field Calculus: Quantum and Statistical Field Theory without the Feynman Diagrams
John E. Gough
Department of Physics, Aberystwyth University, SY23 3BZ, Wales, UK
Received March 18, 2022, in final form June 12, 2022; Published online June 14, 2022
For a given base space $M$ (spacetime), we consider the Guichardet space over the Guichardet space over $M$. Here we develop a ''field calculus'' based on the Guichardet integral. This is the natural setting in which to describe Green function relations for Boson systems. Here we can follow the suggestion of Schwinger and develop a differential (local field) approach rather than the integral one pioneered by Feynman. This is helped by a DEFG (Dyson-Einstein-Feynman-Guichardet) shorthand which greatly simplifies expressions. This gives a convenient framework for the formal approach of Schwinger and Tomonaga as opposed to Feynman diagrams. The Dyson-Schwinger is recast in this language with the help of bosonic creation/annihilation operators. We also give the combinatorial approach to tree-expansions.
Key words: quantum field theory; Guichardet space; Feynman versus Schwinger; combinatorics.
pdf (433 kb)
tex (33 kb)
- Cvitanović P., Gyldenkerne E., Field theory, Nordita, Copenhagen, 1983.
- Feynman R.P., The theory of positrons, Phys. Rev. 76 (1949), 749-759.
- Glimm J., Jaffe A., Quantum physics. A functional integral point of view, 2nd ed., Springer-Verlag, New York, 1987.
- Gough J., Kupsch J., Quantum fields and processes. A combinatorial approach, Cambridge Studies in Advanced Mathematics, Vol. 171, Cambridge University Press, Cambridge, 2018.
- Guichardet A., Symmetric Hilbert spaces and related topics. Infinitely divisible positive definite functions. Continuous products and tensor products. Gaussian and Poissonian stochastic processes, Lecture Notes in Math., Vol. 261, Springer-Verlag, Berlin -- New York, 1972.
- Maassen H., Quantum Markov processes on Fock space described by integral kernels, in Quantum Probability and Applications, II (Heidelberg, 1984), Lecture Notes in Math., Vol. 1136, Springer, Berlin, 1985, 361-374.
- Meyer P.-A., Quantum probability for probabilists, Lecture Notes in Math., Vol. 1538, Springer-Verlag, Berlin, 1993.
- Parthasarathy K.R., An introduction to quantum stochastic calculus, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1992.
- Rivers R.J., Path integral methods in quantum field theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987.
- Schweber S.S., QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1994.
- Schwinger J., A report on quantum electrodynamics, in The Physicist's Conception of Nature, Editor J. Mehra, Springer, Dordrecht, 1973, 413-429.
- The on-line encyclopedia of integer sequences, https://oeis.org/.
- Veltman M., Diagrammatica. The path to Feynman rules, Cambridge Lecture Notes in Physics, Vol. 4, Cambridge University Press, Cambridge, 1994.