# $\phi^{n}$ quantum scalar field theories where $n$ is not an integer

Consider a quantum scalar field theory with interaction terms of the form $\phi^{n}$, where $n$ is not an integer.

Where are some examples of widely-studied quantum field theories which involve such interaction terms?

• I suspect you won't find any theories which are `physical'. Typically $n$ is integer because you consider a truncation of a true potential $V(\phi)$ up to some order $n$ using a Taylor expansion; any $\phi^n$ theory is not really a physical theory either but rather an approximation. You will also have some serious branch-cut problems. You should get some cool dispersion relations though! Sep 19 '16 at 4:21
• Ok, I suppose it does not make sense to consider theories where $n$ is not an integer, because the motivation for the integer-$n$ interaction terms stems from expansion of an arbitrary potential $V(\phi)$. So, let me modify my question to ask if any such studies of non-integer-$n$ interaction terms have been done at all. :) Sep 19 '16 at 4:42
• As far as I understand it is generally very problematic to quantize theories with Lagrangians/equations of motion which are not polynomials. Sep 19 '16 at 9:42
• @Blazej Indeed; this is due to the fact that is in general very complicated to understand the quantization procedure in infinite-dimensional systems (essentially due to the fact that the Lebesgue measure does not exist). In addition, it is already sufficiently difficult to understand QFTs with standard polynomial interactions ;-P Sep 19 '16 at 11:18

A perturbation theory in which is the difference of the $\phi^n$ exponent from 2 is taken as the perturbation parameter has been proposed by Bender, Milton, Moshe, Pinsky, and Simmons in their article: Novel perturbative scheme in quantum field theory.

This method is called the $\delta$- expansion.

Please see the following article where the basics of the method is explained in section 2.

The basic construction of the method is based on the following Taylor decomposition of the interaction term:

$$(\phi^2)^{1+\delta} = \phi^2 + \sum_{n=1}^{\infty} \frac{\delta^n}{n!}\phi^2 (\mathrm{log}(\phi^2))^n$$

One can observe that when $\delta = 0$, the interaction term is quadratic and the theory is free, while when $\delta = 1$, the theory becomes the usual $\phi^4$ theory. In the delta expansion analysis, the effective quantum potential is computed order by order around the free theory corresponding to $\delta = 0$. The obtained theory is perturbative in $\delta$ but nonperturbative in the coupling constant and the mass.

Each order computation involves a logarithmic interaction term. This term is treated as a limit of a derivative of of a power

$$\mathrm{log}(x) = \lim_{k \to 0} \frac{d}{dk}x^k$$

Thus the computations are performed for a polynomial theory $(\phi^2)^k$ but only the leading terms in k for $k \rightarrow 0$ need to be computed. For such a theory, the vertices of the Feynman diagrams will be of $2k$ lines. Also, it is possible to perform variational calculations order by order in $\delta$.

This expansion gave excellent results for problems in quantum mechanics and solvable models in 1+1 dimensions.

The $\delta$ expansion was used to argue the triviality of $\phi^4$ theory, but did not provide a final proof. Some work was performed at its adaptation to theories with fermions (linear $\delta$ expansion), gauge theory and models in statistical physics.