Editing Propagator (section)

==== Advanced propagator ====
[[Image:CausalAdvancedPropagatorPath.svg]]

A contour going anti-clockwise under both poles gives the '''causal advanced propagator'''. This is zero if {{mvar|x-y}} is spacelike or if {{mvar|y}} is to the past of {{mvar|x}}, so it is zero if {{math|''x'' ⁰> ''y'' ⁰}}.

This choice of contour is equivalent to calculating the limit<ref>{{cite book |last1=Scharf |first1=Günter |title=Finite Quantum Electrodynamics, The Causal Approach |date=13 November 2012 |publisher=Springer |isbn=978-3-642-63345-4 |pages=89}}</ref>
<math display="block">
G_\text{adv}(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2\pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0 - i\varepsilon)^2 - \vec{p}^2 - m^2} = -\frac{\Theta(y^0-x^0)}{2\pi}\delta(\tau_{xy}^2) + \Theta(y^0-x^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}.
</math>

This expression can also be expressed in terms of the [[vacuum expectation value]] of the [[commutator]] of the free scalar field.
In this case,
<math display="block">G_\text{adv}(x,y) = i \langle 0|\left[ \Phi(x), \Phi(y) \right]|0\rangle \Theta(y^0 - x^0)~.</math>