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I am reading the book QFT for the gifted amateur and I have a question concerning how to go from the wave function picture to the Green's function as defined by equations (16.13) and (16.18) at page 147.

## \phi(x,t_{x}) = \int dy G^{+}(x,t_{x},y,t_{y})\phi(y,t_{y}) ##

##G^{+}(x,t_{x},y,t_{y}) = \theta(t_{x}-t_{y}) \langle x(t_{x})|y(t_{y})\rangle##

I tried to write ## \phi(x,t_{x})## as a function of ##\phi(y,t_{y}) ## by introducing an integral form for the identity operator:

## \phi(x,t_{x}) = \langle x|\hat{U}(t_{x})|\phi)\rangle ##

## \phi(x,t_{x}) = \int dy \langle x|\hat{U}(t_{x})|y\rangle\langle y|\phi\rangle ##

## \phi(x,t_{x}) = \int dy \langle x|\hat{U}(t_{x})\hat{U}(-t_{y})\hat{U}(t_{y})|y\rangle\langle y|\hat{U}(-t_{y})\hat{U}(t_{y})|\phi\rangle ##

## \phi(x,t_{x}) = \int dy \langle x|\hat{U}(t_{x}-t_{y})\left[\hat{U}(t_{y})|y\rangle\langle y|\hat{U}^{\dagger}(t_{y})\right]\hat{U}(t_{y})|\phi\rangle ##

Now I assume the jacobian of the unitary matrix U (time evolution operator) is 1 so that I rewrite the integral:

## \phi(x,t_{x}) = \int dy \langle x|\hat{U}(t_{x}-t_{y})|y\rangle\langle y|\hat{U}(t_{y})|\phi\rangle ##

## \phi(x,t_{x}) = \int dy \langle x|\hat{U}(t_{x}-t_{y})|y\rangle\phi(y,t_{y}) ##

## \phi(x,t_{x}) = \int dy \langle x(t_{x})|y(t_{y})\rangle\phi(y,t_{y}) ##

So by identification, I get:

##G^{+}(x,t_{x},y,t_{y}) = \langle x(t_{x})|y(t_{y})\rangle##

and my question is: where does the Heaviside function ##\theta(t_{x}-t_{y})## come from in the definition? I do not obtain it from this derivation, unless it was necessary at some point to introduce it in the integral and I failed to see it.