QFT Recipe: Outlining the Formal Structure

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Discussion Overview

The discussion centers on the formal structure of Quantum Field Theory (QFT), particularly the challenges in axiomatization and the various approaches or "recipes" that exist for understanding QFT. Participants explore theoretical frameworks, the role of the Wightman axioms, and the practical aspects of applying QFT to experimental data.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants note that the formal structure of QFT has resisted axiomatization, suggesting it is more of a collection of recipes than a fully established theory.
  • One participant mentions the Wightman axioms, indicating that only a few carefully-constructed theories have been shown to satisfy them.
  • Another participant describes a recipe involving non-interacting particles and the use of "in" and "out" states to calculate transition probabilities via the S-matrix, highlighting the role of Feynman diagrams and renormalization.
  • A question is raised regarding the applicability of theorems derived from the Wightman axioms to commonly used theories like QED and QCD, given that these theories may not satisfy the axioms.
  • One participant expresses frustration over the lack of a clear outline of the requested recipe, questioning the state of QFT education and the subject matter itself.
  • Another suggests fitting QFT results to experimental data as part of the recipe, implying a practical approach to understanding QFT.
  • There is a discussion about the appropriateness of referring individuals to textbooks for foundational concepts, with differing opinions on whether this is a sufficient response to inquiries.

Areas of Agreement / Disagreement

Participants express varying levels of frustration regarding the clarity and accessibility of QFT concepts. There is no consensus on a definitive outline of the formal structure or recipe for QFT, and multiple competing views on the role of axioms and practical applications remain unresolved.

Contextual Notes

Some limitations are noted, including the dependence on the definitions of axioms and the unresolved nature of certain mathematical steps in the discussion. The scope of the conversation is primarily theoretical, with practical applications mentioned but not fully explored.

newbee
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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.
 
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There are the Wightman axioms, but only a handful of carefully-constructed theories have been shown to satisfy them.
 
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. :smile: 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" :smile:).
 
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?
 
You take a QFT with unknown (fitting) parameters and fit its results to the experimental data.
 
newbee said:
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.
 
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?
 
newbee said:
If I were to ask you what the axioms of QM are would you just say read a book?
Yes.
newbee said:
If so then what are you doing on this site?
Some questions require book-length answers, some do not.
 

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