Deeper course for Theoretical Particle Physicists
The BPHZ theorem: a decisive turn in the history of quantum field theory
Sergey Volkov (KIT)
Abstract
At present, quantum field theory is very successful. This success is absolutely impossible without a procedure called "renormalization". This procedure absorbs all the infinities (of the ultraviolet type) contained in the bare theory by an infinite redefinition of the theory constants.
The divergent (infinite) counterterms produced by this procedure are subject to very fine tuning. For example, there are so-called subdivergences and overlapping divergences. The Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) theorem was proved in 1950-1960-x and guarantees that renormalization eliminates all ultraviolet divergences at any order of the perturbation series. The theorem raised hopes that quantum field theory made sense and could be studied mathematically rigorously. The formulation and proof of the BPHZ theorem will be explained, as well as related constructions.
Despite its success, the foundations of quantum field theory appear unsatisfactory to many scientists: it gives precise results for many observables, but the reasoning consists of a large sequence of incorrect logical transitions. In this sense, the BPHZ theorem provides an oasis of mathematical rigour. The possibility or impossibility of making something logically consistent can be considered as another very important "experimental fact". We should also consider the possibility that quantum field theory must be completely reconstructed on absolutely different principles in order to become logically correct. This reconstruction must preserve the present very precise agreement with experiments; to make this possible, some "clues" should be taken from the existing theory. We don't know, what these principles and clues are; however, the mechanism underlying divergence cancellation in the BPHZ approach may serve as a good clue.
Not only the BPHZ theorem itself, but also understanding the reasoning in its proof is very useful. The ideas and combinatorial constructions underlying it can be used to develop effective computational methods.
For computations in perturbative quantum field theory, each additional order increases the demand on computer resources by several orders of magnitude. Since high-order results are needed in many areas, calculational methods based on nontrivial mathematical ideas are very welcome. Examples will be demonstrated.
Related topics will also be discussed, such as correct definitions of integrals, regularizations, approaches to rigorous justification of dimensional regularization, Feynman and Schwinger parameters, Symanzik polynomials, power counting theorems, physical renormalization conditions, infrared divergences, mixing of infrared and ultraviolet divergences, convergence of the whole series, and also limited applicability of renormalization, relations between physics and mathematics, social processes in the society of theoretical physicists, and so on.
The materials are password-protected.
Please use your KIT-username ("xy1234", not your e-mail address) and password.
