# Why "Green's function" is used more than "correlations" in QFT?

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## Main Question or Discussion Point

Very often, the term "Green's function" is used more than "correlations" in QFT. For example, the notation:

$$<\Omega|T\{...\}|\Omega> =: <...>$$

appears in Schwartz's QFT book. And it seems very natural, basically because the path integral definition of those terms "looks like" the probabilistic average of a given measure.

So, it seems it would make sense to do the following:

$$<\Omega|T\{...\}|\Omega> := <...>$$

But in general (I think Schwartz is an exception) the probabilistic/average interpretation is basically ignored.

And this is surprising, because for example in statistical physics, say with the Ising model, one has the probabilistic interpretation, but not the Hilbert space one.

As a consequence, it seems the probabilistic interpretation is wider in scope. But in general, I think it is somewhat downplayed.

Do you have this same feeling? If so, why does that happen?

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vanhees71
Gold Member
2019 Award
No, in standard textbooks rightfully the probabilistic interpretation of quantum theory is taught right away. That in vacuum QFT the time-ordered vacuum expectation values are the most important quantities comes from the solution of the time-evolution operator for states in the interaction picture, where time ordering immediately occurs.

For more general many-body states you need all kinds of correlation functions, not only time-ordered ones. Even in equilibrium when using the real-time (Schwinger-Keldysh contour) formalism you need contour-ordered Green's functions, which can be expressed in terms of all kinds of orderings (time-ordered, anti-time-ordered, fixed-ordered Wightman). For an introduction see

https://th.physik.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

atyy and jordi
atyy
In quantum mechanics, one can find that sandwiching an operator between a bra and ket gives the expectation value, so it is the same as the expectation value in probability (maybe the only difference is that it is more often called the expected value). https://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)

Some quirks are just historical like partition function instead of characteristic function, and connected correlation function instead of cumulant because of the Feynman diagrams for the correlations.

vanhees71 and jordi
I agree, I also remember thinking that QFT had a language that seemed the generalization to infinite dimensions of the characteristic function and the cumulants. With this language, it is just common sense that the correlations are everything that is needed to "reconstruct" the theory.

But I think that this "dictionary" of terms between physical theories and probability is not encouraged. Or maybe it is just me.

atyy
vanhees71 and jordi
A. Neumaier
2019 Award
partition function instead of characteristic function, and connected correlation function instead of cumulant
These are in both cases not the same object but analytic continuations of each other (corresponding to real resp. imaginary time).

atyy and jordi