Pure Math Research into Operator Fields

Question

Has any work been done on operator fields in the pure math world? They are a big piece of quantum field theory, but I can't find anything about them outside of that messy subject. Of course, I mean "fields" in the physics sense.

The best I could find was this: http://www.amazon.com/Theory-Operator-Fields-HAWAII-HONOLULU/dp/B009TIRYRQ.

Answer 1

If you want to learn more about operator fields in the sense of physics, then I recommend Wolfhart Zimmermann’s papers on composite operators and the operator product expansion (OPE). Let me provide two references:

• K. Wilson, W. Zimmermann, Operator Product Expansions and Composite Field Operators in the General Framework of Quantum Field Theory, Communications in Mathematical Physics, Volume 24, Issue 2 (1972), pp. 87-106.
• W. Zimmermann, Local Operator Products and Renormalization in Quantum Field Theory, Lectures on Elementary Particles and Quantum Field Theory, Proceedings of the 1970 Brandeis Summer Institute in Theoretical Physics, S. Deser et al. (Eds.), MIT Press, Cambridge, MA, pp. 399-589.

Zimmermann is the inventor of the Zimmermann Forest Formula in the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) Renormalization Scheme, which is a method of subtracting the pesky infinities in divergent Feynman amplitudes in a combinatorial fashion.

John Collins has written a book entitled Renormalization, published by the Cambridge University Press. It treats the theory of operator fields in a mathematically respectable manner, so you should check it out.