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I am reading a paper about quantum field theory, something that I am new to. I have some experience with quantum mechanics. In the paper, it explains how a field is a function from a spacetime manifold(M) to a vector space(V):

Φ:M→V

The paper then explains the form of the observables. It explains how an observable is the value of Φ at x∈M, but our calculations will work out better if we instead look at average values of Φ in small regions of space, as infinitely much energy is required to know the value of a field at a point. To calculate the average values of the field in a small region of space, and thus our observable, the paper defines a Schwartz space, S, as a space of Schwartz functions of the form f:M→V*. It then explains that the observables are of the form:

$O_f (Φ) = \int_M 〈f,Φ〉$

So at this point I am confused. How does this integral lead to an average value of the field within a region of space? I would really appreciate any help showing me how the integral above is indeed the form of an observable for the average value of the field within a small region in space. Specifically, how does one carry out the calculation that gives them a value for $O_f$? Thanks in advance for any help.

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The following is a Schwartz function, for any $\epsilon > 0 $:

$$ f_\epsilon (x) = N \int e^{-(\|\vec{x}-\vec{y}\|^2 + (x^0 -y^0)^2)/\epsilon^2} \theta(\|\vec{y}\|-R) \theta(|y^0|-T/2) d^4 y $$

This is a smoothened version of the function which is $1$ in a ball of radius $R$ in space and for a time duration $T$, and zero outside of that. Integrating an observable against $f_\epsilon(x)$ thus, heuristically, gives the average of the observable in the ball of radius $R$, in a time interval of length $T$.

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