A How many known Constructive QFTs are there?

[mad mathematician]

TL;DR Summary

I listed as advanced, though my questions aren't really that much advanced, but I need to choose some option, and I am way beyond basic level and intermediate.

I think 1+1,1+2 are known to exist.
1+3 is basically the Millenium prize, are there more options?

Is string theory basically QFT in disguise? just with more than 3+1 dimensions.
I guess the next Millenium (3000) prize would be to fill the gaps of Witten's M theory conjecture in his seminal paper from the nineties...

 

[ohwilleke]

What counts as a QFT?

There should be infinitely many of them than can be imagined.

I could imagine that you are talking about QFTs that are consistent with experimental data (which would still be infinite despite being more constrained), but that can't be what is meant, because some of the combinations of dimensions given are contrary to observation and are merely toy models.

Certainly, the descriptions given in the OP do not uniquely describe Quantum Field Theories.

 

[mad mathematician]

What counts as a QFT?

There should be infinitely many of them than can be imagined.

I could imagine that you are talking about QFTs that are consistent with experimental data (which would still be infinite despite being more constrained), but that can't be what is meant, because some of the combinations of dimensions given are contrary to observation and are merely toy models.

Certainly, the descriptions given in the OP do not uniquely describe Quantum Field Theories.

As I said, a constructive QFT, i.e axiomatic one; obviously a 3+1 if one were to prove that such a theory exists would be terrific. And for the math die-hards also D+1 axiomatic theories would be interesting (even if not necessarily physical, at least by experiments).
I am quite sure that if no one as of yet have found a 3+1 CQFT, that more than this number of dimensions would make the task even more formidable.

As for string theory(ies), I am quite a novice about it but I heard that there's string field theory (SFT) and regular string theory. In SFT do we use QFT methods and regualr string theory only relativistic QM (ala what can be found in Messiah's book or Bjorken and Drell first book).
There was a poster (I am not if he still posts here on PF) that said that relativistic QM should no longer be used since it's inconsistent.

There's a lot to read, and exercises to grind...

 

[Demystifier]

a constructive QFT, i.e axiomatic one

Did you mean non-axiomatic? Aren't constructive and axiomatic mutually exclusive?

 

[mad mathematician]

Did you mean non-axiomatic? Aren't constructive and axiomatic mutually exclusive?

https://en.wikipedia.org/wiki/Constructive_quantum_field_theory
I guess something along the book of John Baez's:
https://math.ucr.edu/home/baez/bsz.html

I do warn you, you need to know quite a lot of pure maths, though not any Metamaths...
I stopped reading this book somewhere in the first pages since I lacked the needed knowledge of Lie Algebras and groups to understand.

I will try to reread it; maybe I'll try to ask my questions here from this book.

 

[ohwilleke]

As I said, a constructive QFT, i.e axiomatic one; obviously a 3+1 if one were to prove that such a theory exists would be terrific. And for the math die-hards also D+1 axiomatic theories would be interesting (even if not necessarily physical, at least by experiments).
I am quite sure that if no one as of yet have found a 3+1 CQFT, that more than this number of dimensions would make the task even more formidable.

This still doesn't change the issue of how you can count CQFTs. There are myriad possible axioms from which you could construct a CQFT in any particular set of dimensions. There is not a unique CQFT for any given set of dimensions.

My intuition, is that you are thinking about some subset of QFTs in a given number of dimensions that has axioms corresponding to the real world in some way, perhaps continuity, locality, renormalizability, actions that correspond to the SM forces, etc. But it is still just too vague and the number is still infinite.

One can absolutely make a a 3+1 CQFT that meets the conditions of being axiomatic and being a quantum field theory. It may not look like the real world, but it can be done. Stand alone quantum electrodynamics, for example, is such a theory in Minkowski space.

But you are imposing additional, unspecified conditions that you assert make this an unsolved problem. Without knowing what those conditions are, however, it is impossible to be sure what you are really getting at.

Maybe, for example, you think it needs to be relativistic and non-trivial, although you haven't said so. Some possible conditions that would make the question tractable are suggested here.

The Millennium problem to which you are implicitly referring is much more specific than your OP.

Yang–Mills existence and mass gap, one of the Millennium Prize Problems, concerns the well-defined existence of Yang–Mills theories. The full problem statement is as follows:

Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on

and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).

In particular, Yang-Mills theories are a highly specific subset of quantum field theories with all sorts of properties in addition to being being a compact and simple gauge group, being non-trivial, and existing on

which make it much more similar to the real world described by the Standard Model than just any generic and unspecified QFT or CQFT.

In particular:

A Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)).