Rigorous interacting effective QFT in d=4?

In summary: This works in spacetime dimension 2 and 3, but...In the 3-dimensional case, there is no finite limit, and the theory becomes infinitely large and complex.
  • #1
Gedankenspiel
30
0
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?
 
Physics news on Phys.org
  • #2
Gedankenspiel said:
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.
 
  • Like
Likes Demystifier and vanhees71
  • #3
A. Neumaier said:
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?
 
  • #4
Demystifier said:
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!
 
Last edited:
  • Like
Likes Keith_McClary, bhobba and vanhees71
  • #5
A. Neumaier said:
It also excludes lattice approximations - otherwise Balaban would have earned the prize!
What paper by Balaban do you have in mind?
 
  • #7
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?
 
  • #8
I am on holiday for two weeks and can respond only afterwards. Should I forget, please remind me then.
 
  • #9
Gedankenspiel said:
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.
Gedankenspiel said:
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.
 
  • Like
Likes Gedankenspiel
  • #10
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.
 
  • #11
Gedankenspiel said:
how else can non-locality be implemented into an operator formalism?
Typically by making the action non-local.

Gedankenspiel said:
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).
 
  • Like
Likes Gedankenspiel
  • #12
A. Neumaier said:
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?
 
  • #13
Gedankenspiel said:
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.
 
  • #14
Can you recommend literature on this topic?
 
  • #15
Gedankenspiel said:
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.
 
  • #16
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.)?
 
  • #17
For 2d scalar fields, any polynomial, normally ordered stable interaction works.
 
  • Like
Likes Gedankenspiel
  • #18
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.
 
  • #19
DarMM said:
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?
 

1. What is Rigorous Interacting Effective QFT in d=4?

Rigorous interacting effective quantum field theory (QFT) in d=4 is a mathematical framework for studying the behavior of interacting quantum particles in four-dimensional space-time. It provides a rigorous and systematic approach for describing and predicting the dynamics and properties of physical systems at the macroscopic scale.

2. How is Rigorous Interacting Effective QFT different from other QFT approaches?

Rigorous interacting effective QFT is based on the principles of quantum field theory, but it differs from other approaches in that it places a strong emphasis on mathematical rigor and precision. This means that its predictions and conclusions are based on solid mathematical foundations, making it a more reliable and accurate framework for studying physical systems.

3. What are the applications of Rigorous Interacting Effective QFT?

Rigorous interacting effective QFT has a wide range of applications, including in high-energy physics, condensed matter physics, and cosmology. It can be used to study the behavior of particles in different types of physical systems, such as in particle accelerators, superconductors, and the early universe.

4. How does Rigorous Interacting Effective QFT account for quantum effects?

Rigorous interacting effective QFT takes into account the effects of quantum mechanics, such as particle-wave duality and uncertainty principles, in its mathematical framework. These effects are crucial for understanding the behavior of particles at the microscopic scale, and they are incorporated into the theory through mathematical tools such as Feynman diagrams.

5. What are the challenges in studying Rigorous Interacting Effective QFT?

One of the main challenges in studying rigorous interacting effective QFT is the complexity of its mathematical formalism. It requires a deep understanding of advanced mathematical concepts such as functional analysis and renormalization theory. Additionally, due to the highly nonlinear nature of interacting systems, it can be difficult to make exact predictions, leading to the need for approximations and numerical methods.

Similar threads

  • Quantum Physics
Replies
1
Views
706
Replies
1
Views
690
Replies
36
Views
3K
  • Quantum Physics
Replies
34
Views
3K
  • Quantum Physics
3
Replies
70
Views
5K
Replies
473
Views
22K
  • Quantum Interpretations and Foundations
2
Replies
57
Views
2K
  • Quantum Physics
3
Replies
87
Views
4K
  • Quantum Physics
Replies
2
Views
1K
Replies
14
Views
1K
Back
Top