Configuration space propagator

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ReImPlot[HankelH2[1,s]/s,{s,-10,10}]
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Plot3D[Re[HankelH2[1,Sqrt[t^2-x^2]]/Sqrt[t^2-x^2]],{x,-10,10},{t,-10,10}]
Power
:Infinite expression
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DensityPlot[Re[HankelH2[1,Sqrt[t^2-x^2]]/Sqrt[t^2-x^2]],{x,-10,10},{t,-10,10},FrameLabel->{"x","t"}]
Power
:Infinite expression
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What is the Klein-Gordan propagator in d dimensions
A convenient way to state the Klein-Gordon (Feynman) propagator for a free real scalar field of mass m in d-dimensional Minkowski space is through its momentum-space representation:​(1) D_F(x - y) = ∫ (d^d p / (2π)ᵈ) exp(-i p·(x - y)) × [ i / (p² - m² + iε) ].​Here p² = p₀² - p⃗² in (+,−,−,…) signature (or similarly with the opposite sign convention, depending on your metric), and ε > 0 is the usual infinitesimal prescription ensuring proper convergence and causality.​────────────────────────────────────────────────────────────────────────────────Position-space representation────────────────────────────────────────────────────────────────────────────────​Although (1) is often the most useful definition in practical computations (since Feynman diagrams are usually done in momentum space), one can also write the propagator in closed form in position space. In d dimensions, one obtains an expression in terms of a modified Bessel function K:​(2) DF(x) = (m^ν / (2π)^(ν+1)) × Kν( m √(-x² + iε) ) / [ √(-x² + iε) ]^ν, where • x² = xᵘ xᵘ (the Minkowski norm, which can be positive or negative depending on the signature and whether x is timelike/spacelike), • ν = d/2 - 1, • and the small +iε in the argument indicates the usual causal prescription (bypassing the singularity at x² = 0 in a way consistent with Feynman boundary conditions).​In simpler terms, for a d-dimensional Minkowski space, set ν = d/2 - 1. Then,​ D_F(x) = m^ν / (2π)^(ν+1) × ( K_ν( m √(-x² + iε) ) / ( √(-x² + iε) )^ν ).​For example, • In d = 2 dimensions, ν = 0 and one obtains K₀-type functions. • In d = 4 dimensions, ν = 1, and the propagator can be expressed in terms of K₁. ​These forms reduce to well-known simpler expressions in special cases (e.g., d = 2 or d = 4).
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Plot[BesselK[0,r],{r,0,10}]
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Plot[BesselK[1,r],{r,0,10}]
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