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