QFT Recipe: Outlining the Formal Structure

[newbee]

I understand that the formal structure of QFT has resisted axiomatization (so far) and that what formal structure presently exists is really a set of recipes. Can somebody outline the recipe please.

 

[Ben Niehoff]

There are the Wightman axioms, but only a handful of carefully-constructed theories have been shown to satisfy them.

 

[Fredrik]

I can't really tell you since it's been too long since I studied QFT. I intend to learn it again though, so ask me again in a year. What I can tell you now is that the recipe exploits the fact that you can define the concept of non-interacting particles rigorously. You consider only those situations where it's OK (approximately) to assume that the interaction occurs over a finite time. This enables you to specify the states before and after the interaction ("in" and "out" states) by specifying the number of particles of each species and the momentum of each particle. You then calculate the probability amplitudes of a transition from a given "in" state to a given "out" state. The matrix that has these amplitudes as its elements is called the S-matrix. The calculation involves a bunch of Feynman diagrams, renormalization, and all that stuff. Then you use the results (the S-matrix elements) to calculate the probability of each interesting result of some experiment.

If I understand the Wightman axioms correctly, their purpose is to define what a "relativistic quantum field theory with interactions" is. I'm a bit puzzled by this though. Those axioms are used in the proofs of a bunch of theorems (CPT, spin-statistics,...), right? So if the theories that we actually use (QED, QCD, the standard model) don't satisfy the axioms, can we really assume that the theorems can be applied to those theories? (This question is obviously not for "newbee" ).

 

[newbee]

I find it distressing, to say the least, that nobody can satisfy my request. What is going on in QFT classes or the subject matter itself that such a simple request goes unfilled?

 

[Bob_for_short]

You take a QFT with unknown (fitting) parameters and fit its results to the experimental data.

 

[Avodyne]

I find it distressing, to say the least, that nobody can satisfy my request. What is going on in QFT classes or the subject matter itself that such a simple request goes unfilled?

Try reading one of the many books on the subject. The definitive one so far is the three-volume set by Weinberg. A more user-friendly treatment is by Srednicki.

 

[newbee]

If I were to ask you what the axioms of QM are would you just say read a book? If so then what are you doing on this site?

 

[Avodyne]

If I were to ask you what the axioms of QM are would you just say read a book?

Yes.

If so then what are you doing on this site?

Some questions require book-length answers, some do not.