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

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?