- #1
Dixanadu
- 254
- 2
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...