Return to Answer

The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spacing $a$ goes to zero. A rigorous proof for $d=4$ has been published only recently (2021), by Aizenman and Duminil-Copin in Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models. A proof for $d>4$ was obtained earlier (1981), by Aizenman.

For finite $a$ interaction terms persist, and are relevant on energy scales below $1/a$, see Lüscher and Weisz (1987).

The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spacing $a$ goes to zero. A rigorous proof for $d=4$ has been published only recently (2021), by Aizenman and Duminil-Copin in Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models. A proof for $d>4$ was obtained earlier (1981), by Aizenman.

For finite $a$ interaction terms persist, see Lüscher and Weisz (1987).

The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spacing goes to zero. A rigorous proof for $d=4$ has been published only recently (2021), by Aizenman and Duminil-Copin in Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models. A proof for $d>4$ was obtained earlier (1981), by Aizenman.

The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spacing $a$ goes to zero. A rigorous proof for $d=4$ has been published only recently (2021), by Aizenman and Duminil-Copin in Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models. A proof for $d>4$ was obtained earlier (1981), by Aizenman.

For finite $a$ interaction terms persist, and are relevant on energy scales below $1/a$, see Lüscher and Weisz (1987).

The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the continuum limit. This has been rigorously proven for $d>4$, with numerical evidence for $d=4$. For small but nonzero lattice spacing $a$ at energies well below the cutoff mass $1/a$, the theory effectively behaves like a continuum theory with particle interactions.

See Triviality of four dimensional $\phi^4$ theory on the lattice for references to the literature.

The $\phi^4$ theory on a hypercubic lattice with $d$ space-time dimensions is "trivial" for $d\geq 4$, in the sense that it reduces to a free non-interacting theory in the limit that the lattice spacing goes to zero. A rigorous proof for $d=4$ has been published only recently (2021), by Aizenman and Duminil-Copin in Marginal triviality of the scaling limits of critical 4D Ising and $\phi^4$ models. A proof for $d>4$ was obtained earlier (1981), by Aizenman.