# Is the term “quantum triviality” defined by the UV or the IR behavior of the RG flow?

The Wikipedia page on quantum triviality seems to give two different definitions for the term that are not obviously equivalent.

Some parts of the page seem to define a renormalizable theory as "quantum trivial" if it has a Landau pole. In other words, if you "run the renormalization group backwards" into the UV, you find that the strength of the coupling constant diverges at a finite energy scale. (The physical significance of this process is dubious, because (a) we can't actually calculate the RG flow in practice when the coupling is strong, and (b) the RG flow is not actually reversible, as higher-mass fields that the original theory cannot predict might become important at higher energy scales.) This definition defines "quantum triviality" in terms of the theory's UV behavior.

Other parts of the page seem to define a renormalizable theory as "quantum trivial" if its IR fixed point is free. In condensed-matter terminology (which tends to only consider IR behavior, due to the atomic lattice's modifying the field theory at modest energy scales), this means that all of the interactions are marginally irrelevant. This definition defines "quantum triviality" in terms of the theory's IR behavior.

1. At the one-loop level, these two definitions are equivalent: they both amount to the coefficient of the leading-order term in the beta function being positive. But are they equivalent in general? It seems logically possible to me that the exact function $$g(\mu/\mu_0)$$ could either (a) diverge at some $$\mu/\mu_0 > 1$$ but satisfy $$g(\mu = 0) > 0$$, which would satisfy the first definition of "quantum triviality" but not the second, or else (b) have $$g(\mu = 0) = 0$$ but remain finite at all $$\mu$$, which would satisfy the second definition but not the first.

2. If these definitions are not equivalent, then is either one more standard in the physics community?

• It seems the equivalence seems to depend on where you start the RG flow. For example, let us consider the beta function $\partial_\Lambda g = g^2 - g^3$, where $\Lambda$ is a UV cutoff. If I start at a scale where $g \ll 1$, then the cubic term doesn't do anything, and one would conclude that $g$ flows to $0$ in the IR. But it has a non-trival fixed point in the UV. So it seems the terminology is more useful when one knows for sure that $\partial_\Lambda g$ is not an oscillating series. – vik May 19 '19 at 19:49

The situation is complicated, and the terminology is not uniform. Thus the wikipedia article runs into difficulties by attempting to present a mix from different sources.

Triviality is originally the name for the conjecture that a quantum field theory defined on a lattice has a trivial continuum limit, namely a free theory. This is a pure UV problem: A lattice discretization removes very fast spatial oscillations, hence is an UV regularization. The continuum limit restores the high frequencies, hence is an UV completion.

The triviality of the lattice approximation of $$\phi^4$$ is visible numerically. The standard renormalization process (fixing the physical electron mass) appears to push the interaction strength to zero. In this sense, $$\phi^4$$ theory and QED are believed to be trivial. That they have a Landau pole at very high energies, obstructing a standard continuum limit, was originally believed to be the deeper reason for this.

But other definitions of these theories, not based on a lattice approximation, may still produce a nontrivial $$\phi^4$$ theory and/or QED. For example, in causal perturbation theory, which is from the start covariant, a Landau pole is no obstruction to existence. See also Section 8, ''IS DESTRUCTIVE FIELD THEORY POSSIBLE?'' in a paper by Gallavotti and Rivasseau from 1984. This paper has an extensive discussion with many details.

Later, this catchy term was applied to all sort of loosely related things, for example, triviality may be equated to having a Landau pole. However, nobody seriously thinks of QCD as being trivial in any sense, though it has in certain approximation schemes a Landau pole at experimentally accessible energies; see https://arxiv.org/pdf/1703.04041.pdf.

A thorough discussion with many more references is here.

• @tparker: A lattice discretization removes very fast spatial oscillations, hence is an UV regularization. The continuum limit restores the high frequencies, hence is an UV completion. The triviality of the lattice approximation of $\phi^4$ is visible numerically. The standard renormalization process (fixing the physical electron mass) appears to push the interaction strength to zero. Read the references in the final link of my answer to get the detailed arguments. – Arnold Neumaier May 21 '19 at 5:58