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## Homework Statement

Hey guys,

So here's the problem I'm faced with. I have to show that

[itex] (\Box + m^{2})<|T(\phi(x)\phi^{\dagger}(y))|>=-i\delta^{(4)}(x-y) [/itex],

by acting with the quabla ([itex]\Box[/itex]) operator on the following:

[itex]T(\phi(x)\phi^{\dagger}(y))=\theta(x_{0}-y_{0})\phi(x)\phi^{\dagger}(y)+\theta(y_{0}-x_{0})\phi^{\dagger}(y)\phi(x)[/itex]

## Homework Equations

[itex]\partial_{0}\theta(x_{0}-y_{0})=\delta(x_{0}-y_{0})[/itex]

## The Attempt at a Solution

So I've split the quabla into its time and spatial derivatives: [itex]\Box = \partial_{0}^{2}-\nabla^{2}[/itex] and I'm applying the time derivative first, using the product rule:

[itex]

\partial_{0}T(\phi(x)\phi^{\dagger}(y))

=\delta(x_{0}-y_{0})\phi(x)\phi^{\dagger}(y) +\theta(x_{0}-y_{0})\dot{\phi}(x)\phi^{\dagger}(y)\\

+\theta(x_{0}-y_{0})\phi(x)\dot{\phi}^{\dagger}(y) -\delta(x_{0}-y_{0})\phi^{\dagger}(y)\phi(x)

+\theta(y_{0}-x_{0})\dot{\phi}^{\dagger}(y)\phi(x) +\theta(y_{0}-x_{0})\phi^{\dagger}(y)\dot{\phi}(x)

[/itex]

However, the two delta terms vanish as the commutator of two fields is 0. so I'm left with

[itex]

\partial_{0}T(\phi(x)\phi^{\dagger}(y))

=\theta(x_{0}-y_{0})\dot{\phi}(x)\phi^{\dagger}(y) +\theta(x_{0}-y_{0})\phi(x)\dot{\phi}^{\dagger}(y)\\

+\theta(y_{0}-x_{0})\dot{\phi}^{\dagger}(y)\phi(x) +\theta(y_{0}-x_{0})\phi^{\dagger}(y)\dot{\phi}(x)

[/itex]

At this point I'm meant to be using the equal-time commutation relation: [itex] [\phi(x),\dot{\phi}^{\dagger}(y)] = i\delta^{(3)}(x-y)[/itex] but all my signs are positive...so what do I do?

Thanks guys...