Mathematical foundation of quantum field theory

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Discussion Overview

The discussion revolves around the mathematical foundations of quantum field theory (QFT), exploring its consistency, the challenges it faces, and the current focus of research in theoretical physics. Participants question whether QFT is as mathematically robust as quantum mechanics and discuss specific issues such as renormalization and perturbative expansions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants express uncertainty about the mathematical foundations of QFT, suggesting that it may still be "iffy" in certain areas.
  • There is mention of a significant monetary reward for proving the consistency of QFT, indicating that this is an open question in the field.
  • One participant identifies the primary "iffy" points as related to infinities and renormalization, while another challenges this view, suggesting that the issues lie more with perturbative expansions and Haag's theorem.
  • Concerns are raised about the definitions of particles and fields, particularly in relation to interactions, with some suggesting that particles may be well-defined only in specific contexts (distant past or future).
  • A participant notes that modern perspectives on QFT view it as an approximation of a potentially unknown theory, which may not be a QFT, complicating the relevance of questions about its foundational status.
  • There is a suggestion that the behavior of a system at lower energies does not inform its behavior at higher energies, which adds to the complexity of understanding QFT.
  • One post includes a link that appears unrelated to the main discussion, prompting a request for clarification on its relevance.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the mathematical foundations of QFT, with multiple competing views on its consistency and the nature of its challenges. The discussion remains unresolved regarding the implications of these issues for current research.

Contextual Notes

Participants highlight limitations in understanding related to perturbative expansions, Haag's theorem, and the implications of energy scale behavior, but do not resolve these complexities.

Kalimaa23
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Greetings,

I have question regarding the mathematica foundations of QFT. As I understand, the "regular" QM (Schrödinger, Heisenberg...) been developped so that the math underlying it checks out. Is this the case for QFT, or is the theory still "iffy" at points? I know it works well experimentally, but are the theories in itself consistent and well-known?

I would really like to know if current research in theoretical physics focusses mainly on quantum gravity, or if there are still a lot of people completing QFT.

Cheers
 
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Originally posted by Dimitri Terryn
Greetings,

I have question regarding the mathematica foundations of QFT. As I understand, the "regular" QM (Schrödinger, Heisenberg...) been developped so that the math underlying it checks out. Is this the case for QFT, or is the theory still "iffy" at points?
still iffy.

I know it works well experimentally, but are the theories in itself consistent and well-known?
it is not known whether the theories are consistent. anyone who shows that they are (or even just makes significant progress in this area) is in for a million bucks from clay math.

I would really like to know if current research in theoretical physics focusses mainly on quantum gravity, or if there are still a lot of people completing QFT.

Cheers
certainly there are people working on the mathematical foundations of QFT. i just think those people are mathematicians, not physicists.
 
The "iffy" points are primarily hand-waving at infinities, or renormalization, as I understand.
 
Strange that you would say that only mathematicians are working on it. It seems to me that this would be an interesting topic for theoretical, or at least mathematical physicists.

Besides the renormalizations, are there other major inconsistencies?
 
Originally posted by Loren Booda
The "iffy" points are primarily hand-waving at infinities, or renormalization, as I understand.

Not at all; this is a popular misconception. Regularization and renormalization are not the problem, perturbative expansion is. And also the handling of interacting fields (Haag's theorem).

There are ways to get around Haag's theorem but the results as to the definitions of particles and fields are pretty iffy themselves: you can have fairly well-defined particles in the distant past or in the distant future, but not, or not exactly, in the interaction itself.

The perturbative expansion problem is that the series may not converge. There is some (shaky) evidence that it doesn't; this goes by the name "Landau Pole".
 
Because of the modern view of QFTs as approximations at lower energies of an as yet unknown or unproven "correct" theory which probably isn't a QFT, together with what we've learned from renormalization group ideas about the relation between a theory's behaviour at different energy scales (namely, that the behaviour of a system at lower energies doesn't depend on it's behaviour at higher energies. Unfortunately this also means that inferences can't be safely drawn about the behaviour of a system at high energies from it's behaviour at lower energies), questions about the ultimate status of QFT as a basis for physical theories don't seem as relevant as they did as late as 30 years ago, and whatever residual concern remains about such issues certainly isn't driving mainstream research in high energy theory.
 
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