A Rigorous interacting effective QFT in d=4?

[Gedankenspiel]

Sometimes one can read that constructive QFT has become somewhat superfluous with the advent of effective QFT, so there is no need anylonger to define a QFT on arbitrary small distances.

But is there a rigorous interacting effective QFT in d=4 at all? If so, how is it constructed?

 

[A. Neumaier]

is there a rigorous interacting effective QFT in d=4 at all? If so, how is it constructed?

If the theory is supposed to be local, no.

But of course one could tolerate in an effective theory a certain amount of nonlocality without affecting current experimental accuracy, and then there are plenty of constructions (using cutoffs). But this does not solve the underlying mathematical problem.

 

[Demystifier]

But of course one could tolerate in an effective theory a certain amount of nonlocality without affecting current experimental accuracy, and then there are plenty of constructions (using cutoffs). But this does not solve the underlying mathematical problem.

I wonder, if someone managed to solve quantum Yang-Mills equations with cutoffs analytically, would (s)he earned the millenium prize?

 

[A. Neumaier]

I wonder, if someone managed to solve quantum Yang-Mills equations with cutoffs analytically, would (s)he earned the millenium prize?

No. The announcement requires that the theory has to satisfy the Wightman axioms in the vacuum sector. This excludes cutoffs. It also excludes lattice approximations - otherwise Balaban would have earned the prize!

 

[Demystifier]

It also excludes lattice approximations - otherwise Balaban would have earned the prize!

What paper by Balaban do you have in mind?

 

[A. Neumaier]

What paper by Balaban do you have in mind?

He had written a series of papers on this, before (in the context of moving to a new house) he lost all his notes...
https://www.physicsforums.com/posts/5443402/
https://www.physicsforums.com/posts/5443430/

 

[Gedankenspiel]

Thanks, A. Neumaier, for your answer! Accepting some amount of nonlocality, using a cutoff, how is this QFT rigorously defined? I only know of the path integral, but is its rigorous definition (d=4) possible at all, even with a cutoff and for scalar fields? Or is an operator formalism possible?

Is non-locality (to some extent, by using cutoffs or something else) equivalent to breaking Poincaré symmetry?

 

[A. Neumaier]

I am on holiday for two weeks and can respond only afterwards. Should I forget, please remind me then.

 

[A. Neumaier]

Is non-locality equivalent to breaking Poincaré symmetry?

No, Nonlocality breaks the causal commutation rules only. Regularization via cutoff breaks both locality and Poincare invariance.

Accepting some amount of nonlocality, using a cutoff, how is this QFT rigorously defined? I only know of the path integral, but is its rigorous definition (d=4) possible at all, even with a cutoff and for scalar fields? Or is an operator formalism possible?

Yes. The cutoff makes the normally ordered Hamiltonian well-defined as a self-adjoint operator on Fock space, and then canonical quantiziation applies, for scalar, spinor, and vector fields. The vacuum sector of nonrelativistic field theories is also well-defined on Fock space, even without cutoff.

 

[Gedankenspiel]

If a cutoff breaks Poincaré invariance, how else can non-locality be implemented into an operator formalism?

Another question about Fock spaces: how can they accommodate for interactions at all? They are basically superpositions of tensor products of spatio-temporal plane waves, correct? The coefficients of these superpositions refer to the plane waves on the whole of spacetime, not to a certain spatial hyperplane. In nonrelativistic quantum mechanics I can think of the states as composed of plane waves too, but here they are plane spatial waves, not spatio-temporal waves, so the coefficients describing the composition of the state can evolve in time, reflecting interactions. I do not see how this is possible for Fock spaces in relativistic theory.

 

[A. Neumaier]

how else can non-locality be implemented into an operator formalism?

Typically by making the action non-local.

Fock spaces: how can they accommodate for interactions at all?

Interactions are accounted for by changing the Hamiltonian. In nonlocal or nonrelativistic cases this is quite easy.

But interacting relativistic field theories (without cutoff) don't have a natural Fock space structure (by Haag's theorem).

 

[Gedankenspiel]

But interacting relativistic field theories (without cutoff) don't have a natural Fock space structure (by Haag's theorem).

But what chance then do we have to construct a local interacting relativistic QFT without cutoff? Any other state spaces besides Fock space?

 

[A. Neumaier]

But what chance then do we have to construct a local interacting relativistic QFT without cutoff? Any other state spaces besides Fock space?

The right Hilbert spaces are appropriate limits of Fock spaces. Proving that the limit makes sense is the hardest part.

 

[Gedankenspiel]

Can you recommend literature on this topic?

 

[A. Neumaier]

Can you recommend literature on this topic?

Usually one constructs the correct representation by working on a finite lattice, where Fock spaces are adequate, and then takes a limit where the spacing goes to zero (ultraviolet limit) and the inner diameter to infinity (infrared limit). For the 2-dimensional case, one can find all information with full mathematical detail in the book by Glimm and Jaffe. See also the discussions with @DarMM in this forum, where something is said about the 3-dimensional case.

This works in spacetime dimension 2 and 3, but seems to fail in 4 spacetime dimensions, where it is not known how to get an interacting causal quantum field theory.

 

[Gedankenspiel]

OK, that is really a book to be afraid of... I wonder if there is something more digestible, like an overview or review with less detail, for a start.

For the 2-dimensional case: is there any kind of restriction on the interactions in this case (just talking about scalar fields)? What I mean is: is there like just a handful of possible interactions or can you construct a QFT for just about any interaction that is even remotely physically reasonable (fulfilling locality, having a Hamiltonian bounded from below etc.)?

 

[A. Neumaier]

For 2d scalar fields, any polynomial, normally ordered stable interaction works.

 

[DarMM]

Just to add, for the most part, the theories that have been constructed rigorously with no cutoff are super-renormalizable.

In the end, they are defined on a non-Fock Hilbert space, with a highly non-trivial inner-product. It turns out that our renormalization schemes are equivalent to perturbatively approximating normal orderings under this inner-product.

 

[A. Neumaier]

It turns out that our renormalization schemes are equivalent to perturbatively approximating normal orderings under this inner-product.

Can you point to literature discussing this point of view?