Looking for a mathematically rigorous definition of a Quantum Field

[LarryS]

TL;DR

Mathematically rigorous definition of a Quantum Field?

I am trying to teach myself QFT. So, that includes understanding the mathematical definition of a Quantum Field. My education is in math, not physics, so for me the theory IS the math. Wikipedia defines a quantum field simply as an operator-valued distribution. But apparently, it does not end there. There are the Wightman Axioms and the Haag-Kastler Axioms and Algebraic QFT.

Is there one true definition of a quantum field? If so, where can I read about it?

Thanks in advance.

 

[martinbn]

How about this one

"Quantum Field Theory: A Tourist Guide for Mathematicians" - Gerald Folland

 

[renormalize]

Is there one true definition of a quantum field? If so, where can I read about it?

You should look at the 4 Wightman axioms W0-W3 and their consequences, as summarized for example here: https://en.wikipedia.org/wiki/Wightman_axioms

 

[Matterwave]

Wikipedia defines a quantum field simply as an operator-valued distribution.

Is there something lacking in mathematical rigor with this definition?

Is there one true definition of a quantum field? If so, where can I read about it?

What kind of thing are you looking for in a "one true definition"?

 

[loop quantum gravity]

You should look at the 4 Wightman axioms W0-W3 and their consequences, as summarized for example here: https://en.wikipedia.org/wiki/Wightman_axioms

But those axioms don't work for 3+1 QFT, at least that's what I understand from Witten's and Jaffa's Millenium prize proposal.
We don't have a constructive QFT for 3+1 dimensions.

 

[jbergman]

I would look at Folland's book or Talagrand's "What is a quantum field theory".

 

[jefferywinkler3]

I just want to let you know to look out for pitfalls in learning quantum field theory, such as what Alexandros Gezerlis outlined here.

Six Textbook Mistakes in Quantum Field Theory

https://arxiv.org/pdf/2604.24871

 

[A. Neumaier]

Is there something lacking in mathematical rigor with this definition?

It lacks the constraints that turn an operator valued distribution into a quantum field.

There is no single, all-encompassing definition of a quantum field theory. An up to date overview of the modeling choices is in https://arxiv.org/pdf/2210.03128. Talagrand's book mentioned earlier in this thread is probably the best starting point for mathematicians.

 

[A. Neumaier]

But those axioms don't work for 3+1 QFT, at least that's what I understand from Witten's and Jaffa's Millenium prize proposal.

This is a misunderstanding. Indeed, a requirement in the Yang-Mills problem is to satisfy the Wightman axioms or something stronger.

We don't have a constructive QFT for 3+1 dimensions.

We have various well-defined linear quantum field theories in 4 domensions. While we don't have a nonlinear one, we also don't have mathematical arguments that they don't exist. See, e.g., https://physicsoverflow.org/32752/

Jaffe and Witten wouldn't have posed the Yang-Mills millennium problem if they didn't expect that it has a positive solution. Thus mathematically rigorous interacting QFTs in 4D satisfying the Wightman axioms are likely to exist, but proving existence is very hard.

 

[jeffn1]

This is a misunderstanding. Indeed, a requirement in the Yang-Mills problem is to satisfy the Wightman axioms or something stronger.

We have various well-defined linear quantum field theories in 4 domensions. While we don't have a nonlinear one, we also don't have mathematical arguments that they don't exist. See, e.g., https://physicsoverflow.org/32752/

Jaffe and Witten wouldn't have posed the Yang-Mills millennium problem if they didn't expect that it has a positive solution. Thus mathematically rigorous interacting QFTs in 4D satisfying the Wightman axioms are likely to exist, but proving existence is very hard.

Get AI on it! ;)

 

[Matterwave]

It lacks the constraints that turn an operator valued distribution into a quantum field.

Those constraints being? Do you have a pointer in the linked paper perhaps? :)