# Time-ordered product of real scalar fields

1. Feb 17, 2015

Hi guys,

So I've got a real scalar field which is the sum of the positive frequency part and negative frequency part:

$\phi(x)=\phi^{(+)}(x)+\phi^{(-)}(y)$

and I'm looking at the time-ordered product:

$T(\phi(x)\phi(y))=\theta(x^{0}-y^{0})\phi(x)\phi(y)+\theta(y^{0}-x^{0})\phi(y)\phi(x)$

for the two cases where $x^{0}>y^{0}$ and $x^{0}<y^{0}$. So in the first case:

$T(\phi(x)\phi(y))=:\phi(x)\phi(y):+[\phi^{(+)}(x),\phi^{(-)}(y)]$, that's all good.

However for the second case, here is what my lecturer has written:

$T(\phi(x)\phi(y))=:\phi(x)\phi(y):+\theta(x^{0}-y^{0})<0|\phi(x)\phi(y)|0>+\theta(y^{0}-x^{0})<0|\phi(y)\phi(x)|0>$.

I have no idea how/why all of a sudden the expectation value is being taken using the vacuum states. Can someone please explain, why is this so different from the first case?

2. Feb 22, 2015