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, $\widehat{\phi(f)}$?



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