Question

Wightman Function for complex scalar field - timelike separations?

Answer 1

. How to compute this function for time-likeseparations? If (x−y)2<0(x−y)2<0 is it simply true that ⟨0|Φ∗(x)Φ(y)|0⟩=m4π2−(x−y)2√K1(m−(x−y)2−−−−−−−−√)⟨0|Φ∗(x)Φ(y)|0⟩=m4π2−(x−y)2K1(m−(x−y)2)?

2. Weinberg says that for spacelike separations (x−y)2>0(x−y)2>0, this function is symmetric under (x−y)↦(y−x)(x−y)↦(y−x). Meaning ⟨0|Φ∗(x)Φ(y)|0⟩=⟨0|Φ∗(y)Φ(x)|0⟩⟨0|Φ∗(x)Φ(y)|0⟩=⟨0|Φ∗(y)Φ(x)|0⟩. I see that this is obvious from the formula involving the Bessel K1K1, but how does one see this from the integral representation ⟨0|Φ∗(x)Φ(y)|0⟩=∫d3p(2π)312Epe−iEp(x0−y0)+ip⋅(x−y)⟨0|Φ∗(x)Φ(y)|0⟩=∫d3p(2π)312Epe−iEp(x0−y0)+ip⋅(x−y)? To me this is not obvious from the integral representation for non-zero x0−y0x0−y0...Is it still true that the Wightman function is symmetric under (x−y)↦(y−x)(x−y)↦(y−x)for timelike separations?