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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?

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1 Answer

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. See also this and this Phys.SE post.

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.

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Let me make sure I'm not going crazy here. Although Srednicki does seem to mean any function that depends only on the value of the field (as opposed to say higher derivatives) at a given point, and although I agree that this definition of locality is sufficient for Lorentz-invariance, it is certainly not necessary (take for example $\partial_\mu\phi\partial^\mu\phi$. I'm used to the term locality allowing for a finite number of higher derivatives (as other SE links also seem to suggest). Qmechanic could you confirm that Srednicki's is a restrictive def of locality and perhaps nonstandard? –  joshphysics Feb 22 '13 at 20:27
    
I updated the answer. –  Qmechanic Feb 22 '13 at 20:43
    
Ok cool much appreciated. –  joshphysics Feb 22 '13 at 21:04
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