## Quantum field theory and triviality

In the comments on the post where Peter Woit kindly pointed to my blog (see here) there is some point about when a quantum field theory can be considered “trivial”.  Indeed, this question is fairly well acquired for the expert of quantum field theory as there exists a theorem due to Michael Aizenman (see here) for the scalar field theory but this result only applies to theory with dimensions greater or equal 5. If the scalar theory is trivial for D=4 is an open and hotly debated question still far from being settled.

So, what should one mean by a “trivial” field theory? We can say that a quantum field theory becomes trivial when in some limit is a free theory. To become free a theory should have the coupling going to zero in the considered limit and only this that is meaningful. As it is largely known, for a gauge theory a propagator can depend on the choice of the gauge. But it is also true that for some theories the appearence of a free propagator and the corresponding behavior of higher n-point funtions is already a proof of triviality.

What does it happen to the running coupling of the Yang-Mills theory at lower momenta?  A long time expectation has been that the running coupling of Yang-Mills theory should reach a non-trivial infrared fixed point, that is, it should have had a fixed value as the energy goes to zero. Lattice results said the opposite. But there is a glitch here. The problem is that we have not a clear understanding of what should be a proper definition of the running coupling in the infrared limit. People working on the lattice have chosen the definition taken from functional methods arosen with Alkofer and von Smekal. The computation on the lattice of this quantity shows that it goes to zero and no fixed point emerges. Different definitions have been proposed (see here for a review) but there are also experimental evidences, due to Giovanni Prosperi and his group and presented here, that the coupling at low momenta goes to zero.

All this gives a strong clue that Yang-Mills theory is a trivial theory in the infrared and shares a similar fate with the scalar theory.

### 6 Responses to Quantum field theory and triviality

1. […] This is exactly what one presently sees on the lattice and several authors agree upon (see this post). He derives this assuming that the beta function remains negative for the all values of the […]

2. strangerep says:

> We can say that a quantum field theory becomes trivial
> when in some limit is a free theory.

I’m wondering whether this is really an adequate definition.
What about discontinuous perturbations? E.g., it is known
in some cases that the Hilbert spaces of a free theory
and an interacting theory remain disjoint no matter how
small we make the coupling constant. There is a discontinuous
change between g=0 and g=\epsilon.

I would have thought that a definition of triviality should involve
some statement about the indistinguishability of the free and
interacting theories, i.e., same S-matrix, same spectrum, or
something along those lines.(?)

3. mfrasca says:

Just for my knowledge. What are these cases? Are just mathematical pathologies or well acquired field theories? For my definition to work I have to assume some smoothness on the way the coupling depends on momentum and is generally so for the theories we commonly use.

Marco

4. strangerep says:

Hi Marco,

John Klauder, in his book “Beyond Conventional Quantization”, (ch8), gives a reasonably
pedagogical introduction to the distinction between continuous and discontinuous
perturbations, with examples. I’m not sure what you mean by “well-acquired” field
theories, though. (BTW, this whole area is related to Haag’s Theorem, which applies to a very
large class of interacting QFTs.)

But after further reflection, perhaps such things actually support your thesis…
If it is indeed possible to find a *continuous* limit in which the theory becomes truly
free, then maybe it is indeed trivial — because the interesting cases tend to be plagued
by Haag’s Theorem and discontinuous perturbations.

Cheers.

5. Peter Morgan says:

Hey Marco: If a theory is supposed to be trivial, I suppose it would still be interesting to see what limit has to be taken to obtain a free theory. If a rather subtle limit has to be taken to obtain the free theory from a given apparently interacting theory, distinct from the more straightforward limits that can be taken to obtain a free theory in other cases, I suppose we have to classify the different types of trivial theories? However, I’m somewhat out of my depth here, and the post is a week old.

6. mfrasca says:

Peter,

a theory becomes free when any coupling goes to zero in some limit of the momentum. Presently, this is what is seen for Yang-Mills on the lattice at lower momenta and can be intepreted as if the color charge is completely screened in the infrared.

Marco