# Spread of the (smeared) field observable under time-evolution

Setup: Essentially, I'm interested in performing an analysis which is completely standard in QM, but I've never seen the analogue in QFT: Given I measure a system to have some value of its canonical observable at time $$t_0$$, what is the probability of measuring a different value for the canonical observable at some later time $$t_1$$?

[Technical note: In QFT, I've heard the canonical observable $$\widehat{\phi(x)}$$ is not actually well-defined as an "operator-valued function at $$x$$", but rather only as an "operator-valued distribution"---i.e., it only makes sense as an operator after being smeared with a test function. Hence, I will consider only the "smeared"/averaged field observable in what follows.]

Ok, suppose I measure a scalar quantum field to have a value $$\phi(f)$$ averaged over some spacetime region, $$U$$---more precisely, I measure the value of the field when "smeared" with a test function to be $$\int d^4x \phi(x)f(x)$$ with $$\text{supp}f\in U$$.

Now "after" my measurement---i.e., in the immediate causal future of the region $$U$$, I expect the field should have "collapsed" into the eigenstate of the operator $$\widehat{\phi(f)}=\int d^4x \widehat{\phi(x)}f(x)$$. i.e., it should be in the state $$|\phi(f)\rangle$$ defined such that

\begin{align*} \widehat{\phi(f)}|\phi(f)\rangle=\phi(f)|\phi(f)\rangle \end{align*}

For another spacetime region $$U'$$ lying entirely in the causal future of $$U$$, I then measure the average value of the field by smearing with some $$g\in U'$$.

For given $$g$$, I expect the probability of finding the average value of the field to be $$\phi(g)$$ is given by

\begin{align*} |\langle\phi(g)|\phi(f)\rangle|^2 \end{align*}

Q1: Is this reasoning correct?

Q2: I'd like to actually evaluate this for say the non-interacting Klein-Gordon field in Minkowski spacetime. How does one construct the eigenstate of the smeared operator, $$\widehat{\phi(f)}$$?