Editing Propagator (section)

==== Retarded propagator ====
[[Image:CausalRetardedPropagatorPath.svg]]

A contour going clockwise over both poles gives the '''causal retarded propagator'''. This is zero if {{mvar|x-y}} is spacelike or {{mvar|y}} is to the future of {{mvar|x}}, so it is zero if {{math|''x'' ⁰< ''y'' ⁰}}.

This choice of contour is equivalent to calculating the [[Limit (mathematics)|limit]],
<math display="block">G_\text{ret}(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(x^0 - y^0)}{2\pi} \delta(\tau_{xy}^2) + \Theta(x^0 - y^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}.</math>

Here 
<math display="block">\Theta (x) := \begin{cases}
1 & x \ge 0 \\
0 & x < 0
\end{cases}</math>
is the [[Heaviside step function]],
<math display="block">\tau_{xy}:= \sqrt{ (x^0 - y^0)^2 - (\vec{x} - \vec{y})^2}</math>
is the [[proper time]] from {{mvar|x}} to {{mvar|y}}, and <math>J_1</math> is a [[Bessel function of the first kind]]. The propagator is non-zero only if <math>y \prec x</math>, i.e., {{mvar|y}} [[causal structure|causally precedes]] {{mvar|x}}, which, for Minkowski spacetime, means
:<math>y^0 \leq x^0</math> and <math>\tau_{xy}^2 \geq 0 ~.</math>

This expression can be related to the [[vacuum expectation value]] of the [[commutator]] of the free scalar field operator,
<math display="block">G_\text{ret}(x,y) = -i \langle 0| \left[ \Phi(x), \Phi(y) \right] |0\rangle \Theta(x^0 - y^0),</math>
where 
<math display="block">\left[\Phi(x), \Phi(y) \right] := \Phi(x) \Phi(y) - \Phi(y) \Phi(x).</math>