Understanding ground state expectation values of time ordered products of operators

Question

Working through chapters 6 through 8 in Srednicki's "quantum field theory", I am having some trouble conceptually understanding what is happening when we take time ordered products of operators in expectation values.

For example, when considering the path integral of a free field theory what I picture the functional integral doing is providing an amplitude for the ground state to ground state transition. That is:

$$ Z_0 (J) = <0|0>_J = \int \mathcal{D} \phi \; exp[i \int d^4 x [L_0 + J\phi]].$$

Without the sources, we would have unity: $<0|0>_{J=0} = 1$, since without the presence of an external force a system in its ground state will remain in its ground state. However, once we add sources, the system has the potential to be driven from the ground state and so this functional integral is no longer necessarily unity. As we develop the mathematics, we see that the path integral can be expressed in terms of a propagator that describes this new ground state to ground state amplitude. Everything good so far?

What I'm struggling with then is what is going on when after all of this we take the following:

$$<0|T\phi(x_1) \phi(x_2)|0>.$$

After consulting Zee's book I believe I am to interpret this as the following. We are now asking for the probability amplitude for a ground state to ground state transition, where in which within the ground state a disturbance in the field is brought into play at $x_1$ and then propagates to $x_2$ - or perhaps instead at some time $t_1$ in the ground state there is a disturbance in the field at $x_1$, and then at some later time $t_2 > t_1$ there is a disturbance at $x_2$. We then calculate this using functional derivatives of the generating functional with respect to now the sources, since - as I understand - the sources are responsible for creating the disturbances.

Is this the correct interpretation? I think some of my difficulty lies in the propagator being present already in $ Z_0 (J) = <0|0>_J$, although in my mind nothing is propagating yet until we consider something like $<0|T\phi(x_1) \phi(x_2)|0>$, but this could just be a misunderstanding in the language choice.

Answer 1

Knowing the numeric value of $$ \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \langle 0|0\rangle_J \tag{1} $$ for every function $J$ gives us a lot of information about the theory. For any single function $J$, we can interpret (1) as the amplitude for a ground-state to ground-state transition, like you said. If we know that amplitude for every function $J$, then we can ask questions like this: How is the dependence on $J$ at a spacetime point $x_1$ correlated with its dependence on $J$ at a spacetime point $x_2$? The quantity $\langle 0|T\phi(x_1)\phi(x_2)|0\rangle_J$ is the answer to one version of that type of question, and the same quantity without the subscript $J$ is a special case.

In other words, if somebody gives you an oracle that returns the number $\la 0|0\ra_J$ for any given function $J$, but nobody tells you how that oracle is implemented (in other words, you don't get to see the lagrangian or the path-integral formulation or even what the fields are), you can still learn a lot about the theory just by querying the oracle for all different functions $J$. You can define the operator $\phi(x)$ using $\delta/\delta J(x)$, without knowing that the lagrangian was expressed in terms of such a field. You can do this even if the oracle's maker did not express the path integral in terms of such a field.

Maybe the conceptual difficulty in thinking of the propagator as being already present in $\la 0|0\ra_J$ comes from thinking too highly of the field $\phi$. The role of the fields in quantum field theory is to establish an association between operators on the Hilbert space and points in spacetime, and to implement the action principle. Sometimes (always?) the same quantum field theory can be written using different lagrangians with different fields, so as far as observables are concerned, the fields really don't have any special privileges other than implementing locality and the action principle. With that perspective, maybe the question can be answered just by striking the word "field" from what you already wrote:

We are now asking for the probability amplitude for a ground state to ground state transition, where in which within the ground state a disturbance in the field is brought into play at $x_1$ and then propagates to $x_2$ — or perhaps instead at some time $t_1$ in the ground state there is a disturbance in the field at $x_1$, and then at some later time $t_2>t_1$ there is a disturbance at $x_2$.

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Vocabulary

A comment asked for insight about the definitions of "generating functional", "partition function," and "correlation function." This appendix tries to provide some definitions that are probably consistent with most usages of those terms in quantum field theory.

• The name "generating functional for [something]" is used for any functional from which you can get all of the [somethings] by taking derivatives. In our case, $\langle 0|0\rangle_J$ is the generating functional for time-ordered vacuum expectation values of products of the field $\phi$.

• The name "correlation function" is used for any expectation value of a product of operators. In our case, we're using this name for time-ordered vacuum expectation values of products of the field operators.

• The name "partition function" is used for a functional integral that has the form $\sum_\sigma e^{f(\sigma)}$, where the sum over $\sigma$ is a sum (or functional integral) over configurations of something (in our case, configurations of the field $\phi$) and the function $f$ is something that depends on the configuration and maybe also on some other "external" items (like $J$ in our case). The name "partition function" comes from statistical mechanics, and QFT inherited the name because QFT also deals with quantities of that form (although often with a factor of $i$ multiplying the function $f$).

In our case, the quantity $\langle 0|0\rangle_J$ is both a generating functional and a partition function. The former name describes what we intend to do with it, and the latter name describes how we constructed it.