Graduate Why the probabilistic approach is not more prominent in QFT?

[jordi]

Of course, there is probability theory in QFT. The partition function can be understood as a characteristic functional.

What surprises me is there is little discussion about characteristic functionals, except for some 40 year old papers.

It seems surprising to me that physicists, being used to analogies from simple systems to complex ones, are not used to emphasize the existence of characteristic functions in probability, and how its obvious generalization is the partition function. Maybe this is explained somewhere, but it is not "typical".

I read that the Nelson approach, trying to define QM as "stochastic paths", à la Markov processes, was a dead end.

A bit similar with Jaynes' Max Ent principle: I see there is a guy who recently has written about recovering the Schroedinger equation from Max Ent. By looking at Jaynes arguments, it seems the Gibbs measure could be justified quite rationally under this framework. But in general (and apart from the recent trend on entropy with black holes) this idea is not used much, and the Gibbs measure is used without much justification.

Is there any referece where the concept of characteristic functionals, and measure theory for infinite dimensional systems in QFT, is discussed? Or in general, other probabilistic issues?

 

[vanhees71]

I don't know, what you mean.

All you evaluate in (vacuum) QFT is probabilistic (S-matrix elements of particle reactions or equivalently cross sections, decay widths). Also among the most convenient formulations is using generating functionals to evaluate these transition probabilities, often using path-integral and other functional methods (like the heat-kernel, Schwinger proper time, or worldline formalism).

Also the information-theoretical foundation of statistical mechanics (and in fact statistical mechanics is most coherently formulated in terms of quantum statistics and that's again most elegantly evaluated in terms of relativistic or non-relativistic QFT) becomes more and more standard.

Another approach to derive the equilibrium distributions is as old as statistical mechanics itself, i.e., the dynamical approach a la Boltzmann, leading to the H theorem, which is also closely related to the fundamental properties of the quantum-field theoretically defined S-matrix, namely its unitarity leading to the principle of detailed balance (it's not necessary to relate it to time-reversal and parity invariance as suggested by some textbooks; just unitarity of the S-matrix is sufficient; see, e.g., Landau-Lifhitz vol. 10).

 

[A. Neumaier]

Is there any referece where the concept of characteristic functionals, and measure theory for infinite dimensional systems in QFT, is discussed?

The quanum physics book by Glimm and Jaffe should satisfy your interest.