Quantum triviality

Remarks

For $d \ge 5$ this was proved by Aizenman and by Fröhlich in the early 1980s. The case $d = 4$ was proved by Aizenman and Duminil-Copin, and is called marginal triviality. The proofs use random current or random walk representations of the correlation functions. They bound the four-point Ursell function, which measures the deviation from Gaussian behaviour, by a quantity that tends to $0$ in the scaling limit.

So the lattice approach cannot give an interacting quartic interaction in four dimensions. In $d = 2$ and $d = 3$, interacting $\phi^4$ theories exist; they are built by the methods of constructive field theory. In perturbation theory, triviality shows up as a running coupling that grows at high energy and has a Landau pole. So the renormalized coupling must vanish when the cutoff is removed. The same is expected, but not proved, for quantum electrodynamics.

Going deeper

For a student

Triviality is a statement about the scaling limit, studied with the renormalization group. In $d > 4$ the Gaussian fixed point is stable, and in $d = 4$ the coupling is marginally irrelevant, decreasing only logarithmically. For the Higgs sector of the Standard Model, which contains a quartic coupling, triviality is read as an upper bound on the energy scale where new physics must appear.