Definition of Local Function

Question

Now a days I am studying Srednicki's QFT book. In its third chapter it is written that

Any local function of φ(x) is a Lorentz scalar, [...] .

Now my question is: What is a local function?

Answer 1

1. Well, the notion of locality depends on context. Usually in the context of QFT, a local function means a function of the form $$ f(\varphi(x), \partial\varphi(x), \partial^2\varphi(x), \ldots,\partial^N\varphi(x) ;x), $$ where $N\in\mathbb{N}_0$ is some finite order. This is sometimes called perturbative local. (If $N=0$ then the function $f$ is called ultra-local.) See also this and this Phys.SE posts.

2. Concretely, in the mentioned place almost at the end of chapter 3 in Srednicki's book, the phrase

   any local function of $\varphi(x)$

   is used (in a non-standard way) to denote

   any function of the form $f(\varphi(x))$,

   as opposed to, e.g.,

   1. functions of the form $f(\varphi(x),x)$ with explicit $x$-dependence, which may not be a Lorentz scalar,

   2. functions of the form $f(\partial\varphi(x))$, which may not be a Lorentz scalar,

   3. bi-local functions $f(\varphi(x), \varphi(y))$,

   4. functionals $F[\varphi]$,

   5. etc.