# Definition of Local Function

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?

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.