QFT and Representation Theory
Today I started reading, out of simple curiosity, a rather long paper by paper Peter Woit, "Quantum Field Theory and Representation Theory: A Sketch." The paper is available from arxiv.org at the URL: http://arxiv.org/abs/hep-th/0206135 . 56 pages is a lot, especially when packed with data. But the subject is interesting and also somehow related to the other material discussed in my blog. I thought I will be picky, like others are picky when jumping on Bogdanovs. But right after starting reading Peter’s masterpiece I realized that it will not be that easy to be picky! Indeed, right at the end of the Introduction, just before starting the main part of the text, the author writes:
"The reader is to be warned that the present version of this document suffers from sloppiness on several levels. If factors of $2$, $\pi$ and $i$ seem to be wrong, they probably are. Many of the mathematical statements are made with a blatant disregard for mathematical precision. Sometimes this is done out of ignorance, sometimes out of a desire to simply get to the heart of the matter at hand. The goal has been to strike a balance such that physicists may have a fighting chance of reading this while mathematicians may not find the level of imprecision and simplification too hard to tolerate."
If only Bogdanovs were that smart! They would write in their papers and in the introduction to their book: "The reader is to be warned that the present version of this document suffers from sloppiness on several levels." But they were not, and so their critics rightly complain about their "blatant disregard for mathematical precision." Jokes aside, at least it seems to me that I can understand Peter’s ideas, and so to disgree with him, or to criticise his statements, gives some kind of a satisfaction. So I will be only moderately picky, because how can you be mad at someone who is humble ("sloppiness on several levels") the way Peter Woit is…..
I do not know yet neither how far will I go, nor which part of the paper I will focus on. Time (which, as we know from Connes and Rovelli, is the one parameter group of automorphisms of some algebra, the group associated to a stationary equilibrium state of a something) will tell. I will move slowly, step by step, pausing here and there if I see something that attracts particularly my attention.
Let us start with the introductory section.
Ever since the early days of theory there has been a close link between representation theory and quantum mechanics.
This seems to be correct. However, this is only true if we will consider Mackey’s theory of "imprimitivity systems" as a part of the representation theory. In quantum mechanics we are dealing with imprimitivity systems all the time, while groups enter only occassionaly, in special cases. It is true that Mackey’s theory of imprimitivity systems is important for the theory of group representations, but it can not be replaced by the theory of group representations - unless we move into the realms of infinite dimensional groups of maps - where the ground becomes shaky. I will return to this point pretty soon below.
The Hilbert space of quantum mechanics is a (projective) unitary representation of the symmetries of the classical mechanical system being quantized.
OK. Here we have the first sloppiness. What a pleasure for a referee of a paper to find such a pearl! The Hilbert space is NOT a representation! It carries a represenation! (Refereess love finding such totally inessential "errors". Correcting them proves that they really read the paper!) But even when read correctly the statement is not quite correct. The key problem is: how do we define a "symmetry of a classical mechanical system"? For instance, when we use a Lagrangian formalism, do we assume that the Lagrangian must be invariant? Or can it be invariant up to a divergence? Etc. Etc. And what if the classical systems is such that has no symmetries at all? What sunitary representation are we talking about in that case?
The fundamental observables of quantum mechanics correspond to the infinitesimal generators of these symmetries (energy corresponds to time translations, momentum to space translations, angular momentum to rotations, charge to phase changes).
Again incorrect statement. The fundamental observable in quantum mechanics is the position. Some authors go even that far they claim that it is the only observable, and that all other observables can or should be expressed in terms of positions. Bohmian mechanics, for instance, is all based on such an assumption. Path integral formalism works well in the configuration space, but needs special care when we move to the "momentum space", not to mention the fact that the very concept of the "momentum space" in general relativistic quantum mechanics asks for a serious attention. It is true that in case of a Galilei invariant quantum system the position observable happens to be mysterously related to a boost generator. But when the system is not Galilei invariant, when there are external forces, or when we move to special or general relativity, the position becomes the fundamental observable, independent of any group considerations.
To be continued
Chronostalker
