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Consider a real (or complex, with + in the right places) scalar

particle (a Klein-Gordon field).

Now consider the propagator (or correlation function)

G(x-y)= <0|phi(y) phi(x) |0>

(where I assume that the two operators are correctly time-ordered already)

My questions are related to how exactly to interpret this.

A standard interpretation seems to be to say that since phi(x) is (or

contains) a creation operator for a virtual particle, this describes

the creation of a virtual particle at x and its destruction at y.

My first problem is that intuitively this does not make much sense,

because the vacuum is completely Lorentz invariant, so for each particle

created at x (and later destroyed at y), there would be another

virtual particle that was created at an earlier time and is destroyed

at x. After all, the vacuum is absolutely stationary.

So here is my Question 1: In what sense is this interpretation to

be understood?

I could try to understand it in a sense from a path integral

interpretation. I look at the vacuum-to-vacuum amplitude <0|0>. There

I have to integrate over all field configurations that have different

contributions according to their action. For each configuration, I

could calculate phi(y)phi(x), and summing over all configurations with

the appropriate weights, I could see the correlation. This would give

meaning to the term "vacuum fluctuation" if applied to each

configuration separately - if the field value is very large at x and

if y is a near-by point, the field will with higher probability be

large there as well because otherwise the action would be very

large. I think this is how correlation functions are used in lattice

gauge theory.

So Question 2 is simply: Is this interpretation within the path

integral formalism correct? If so, is there a way to transfer it to

the canonical formalism?

Furthermore, on the other hand, standard lore is that for any operator Q

<0|Q|0>

is the vacuum expectation value of Q, i.e., it is the value I would

get if I were to perform a very large number of measurements of the

variable Q.

So if I now set Q=phi(y)phi(x), this would tell me that G(x-y) is the

expectation value I would get if I measure first phi(x), then phi(y)

and then calculate the product of both.

So let's create infinitely many copies of the vacuum (so I can perform

enough measurements to get an expectation value). In each of my

copies, I now perform a measurement of the field at x. This will

create a new state of my universe. In quantum mechanics, I would say

that this new state would be an eigenstate to phi(x) (which would then

evolve), but - here is my Question 3 - does phi(x) actually *have*

eigenstates? (If so, what are they, especially if phi is a complex

field?) And is there actually any way I could perform this

measurement of phi? How do you measure a field value of a quantum

field anyway? (Again, especially in the case of a complex field?)

Since I'm now in a new state phi(x)|0>, I could interpret

<0|phi(y) phi(x) |0>

as the matrix element of getting from my new state back to the vacuum

state by applying another field measurement phi(y). This would then

mean that <0|phi(y) phi(x) |0> is something like the probability of

getting from vacuum back to a vacuum if I apply two subsequent

measurements of phi (similar to Fermi's golden rule, where a matrix

element also gives a transition probability). So finally, my question

4: Would this be a correct interpretation of <0|phi(y) phi(x) |0>?

This is my first post here - I've read a lot hereabouts, but did not really find an answer to these question. Thanks for any help.