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In QFT and statistical mechanics, one is usually interested in studying integrals of the form: $$Z(\phi) =\int d\mu_{C}(\phi')e^{-V(\phi+\phi')}$$ where $\mu_{C}$ is Gaussian measure with mean zero and covariance $C$. Some authors say that, in order to implement renormalization group methods, $e^{-V}$ must have some 'locality properties'. My question concerns these locality properties. To be more specific, here are some questions:

(1) What does locality mean mathematically? What properties should one expect in order to $e^{-V}$ to be local?

(2) What does locality mean physically?

(3) Is locality a property of $e^{-V}$ itself, or is it something we demand on $V$?

Any comments are welcome!

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In the context of renormalization group analysis, locality is essentially demanding that the interaction is short-ranged (thus it is a requirement on $V$, rather than $Z$). Then, after the Fourier transform (i.e. in the plane waves representation) the interaction kernel can be expanded in wave vectors, trancating this expansion after a few terms: $$v(\mathbf{k})\approx v_0+v_1\mathbf{k}+v_2\mathbf{k}^2.$$ One is then interested in the order of the expansion that does not vanish under the RG transformations.

Remark I am largely guided here by the presentation in the Shankar's review.

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Your formulae seem to come from statistical mechanics, but I can only answer in the context of QFT, in which the formulae look a little different, and locality does not apply specifically to renormalisability, but concerns the possible form of field operators. I hope nonetheless that there is enough simularity that the answer has some applicability.

We have an interaction density, $I(x)$ (corresponding to your $V$), from the interaction Hamiltonian, and we integrate the Schrodinger equation iteratively. This leads to the perturbation expansion, or Dyson expansion, which can be written $$U(t) \approx 1+ \sum_{n=1}^\infty \frac{(-1)^n}{n!}\int{d^4x} \cdots \int{d^4x} \mathscr{T} \{I(x_1) \cdots I(x_n) \} $$

Under Lorentz transformation, the order of interactions can be changed in the time-ordered product $\mathscr{T} $ whenever $x_i - x_j$ is space-like. Conversely, Lorentz transformation cannot change the calculation of probabilities under the condition that the initial and final kets are stable states of free particles, as in scattering experiments. The locality condition, or microcausality, follows immediately as a requirement on interaction operators. It states that for any $x,y$ such that $x-y$ is space-like, the commutator of the interaction densities at $x$ and $y$ vanishes, $$[I(y), I(x)] =0 $$ We fulfil this condition by constructing the interaction densities from products of field operators for the particles, which obey the same condition for bosons, and the corresponding anti-commutation relation for fermions, which necessarily appear in pairs in the interaction density.

Physically the vanishing of (anti-)commutators for spacelike separations means that effects cannot propagate faster than the speed of light. $e^{-V}$ is an expression of the corresponding series expansion for your application. The locality condition applies to your $V$ (corresponding to my $I$).

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