Editing Propagator (section)

=== Position space ===
The position space propagators are [[Green's function]]s for the [[Klein–Gordon equation]]. This means that they are functions {{math|''G''(''x'', ''y'')}} satisfying
<math display="block">\left(\square_x + m^2\right) G(x, y) = -\delta(x - y),</math>
where
* {{mvar|x, y}} are two points in [[Minkowski spacetime]],
* <math>\square_x = \tfrac{\partial^2}{\partial t^2} - \nabla^2</math> is the [[d'Alembertian]] operator acting on the {{mvar|x}} coordinates,
* {{math|''δ''(''x'' − ''y'')}} is the [[Dirac delta function]].

(As typical in [[special relativity|relativistic]] quantum field theory calculations, we use units where the [[speed of light]] {{mvar|c}} and the [[reduced Planck constant]] {{mvar|ħ}} are set to unity.)

We shall restrict attention to 4-dimensional [[Minkowski spacetime]]. We can perform a [[Fourier transform]] of the equation for the propagator, obtaining
<math display="block">\left(-p^2 + m^2\right) G(p) = -1.</math>

This equation can be inverted in the sense of [[Distribution (mathematics)|distributions]], noting that the equation {{math|1=''xf''(''x'') = 1}} has the solution (see [[Sokhotski–Plemelj theorem]])
<math display="block">f(x) = \frac{1}{x \pm i\varepsilon} = \frac{1}{x} \mp i\pi\delta(x),</math>
with {{mvar|ε}} implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.

The solution is
{{Equation box 1
|indent = :
|equation = <math>G(x, y) = \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{p^2 - m^2 \pm i\varepsilon},</math>
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where 
<math display="block">p(x - y) := p_0(x^0 - y^0) - \vec{p} \cdot (\vec{x} - \vec{y})</math> 
is the [[4-vector]] inner product.

The different choices for how to deform the [[Methods of contour integration|integration contour]] in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the <math>p_0</math> integral.

The integrand then has two poles at 
<math display="block">p_0 = \pm \sqrt{\vec{p}^2 + m^2},</math> 
so different choices of how to avoid these lead to different propagators.