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## Main Question or Discussion Point

Hi, I've had the following problem in elementary quantum field theory. The propagator for the Klein-Gordon scalar field takes the form

[tex]

D(x-y)=\int\frac{\textrm{d}^3\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}e^{-ip\cdot(x-y)}

[/tex]

I was interested what the propagator looks like for space-like separations. Putting [tex] x^0-y^0=0 [/tex], going to the spherical polars in momentum space, and integrating out the angular coordinates gives

[tex]

D(x-y)=\frac{1}{(2\pi)^2r}\int_0^\infty\frac{p\sin(pr)}{\sqrt{p^2+m^2}}\textrm{d}p

[/tex]

but this integral doesn't converge because the function

[tex] p/\sqrt{p^2+m^2} \rightarrow 1 [/tex] as [tex] p \rightarrow \infty[/tex]

so for large [tex] p [/tex], the integrand is just [tex] \sin(pr) [/tex].

Peskin & Schroeder shift the contour in the complex plane and transform this into a convergent integral of a decaying exponential, but their integral is just different from the ill-defined integral above.

I know that it just works if we use the above expression for the Klein-Gordon propagator, but I am lead to conclude that it is nothing but a symbolic prescription, which only works if we are told how to shift the contour in the complex plane, because the shifting process changes the nature of the integral.

Or is there something that I'm missing here?

[tex]

D(x-y)=\int\frac{\textrm{d}^3\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}e^{-ip\cdot(x-y)}

[/tex]

I was interested what the propagator looks like for space-like separations. Putting [tex] x^0-y^0=0 [/tex], going to the spherical polars in momentum space, and integrating out the angular coordinates gives

[tex]

D(x-y)=\frac{1}{(2\pi)^2r}\int_0^\infty\frac{p\sin(pr)}{\sqrt{p^2+m^2}}\textrm{d}p

[/tex]

but this integral doesn't converge because the function

[tex] p/\sqrt{p^2+m^2} \rightarrow 1 [/tex] as [tex] p \rightarrow \infty[/tex]

so for large [tex] p [/tex], the integrand is just [tex] \sin(pr) [/tex].

Peskin & Schroeder shift the contour in the complex plane and transform this into a convergent integral of a decaying exponential, but their integral is just different from the ill-defined integral above.

I know that it just works if we use the above expression for the Klein-Gordon propagator, but I am lead to conclude that it is nothing but a symbolic prescription, which only works if we are told how to shift the contour in the complex plane, because the shifting process changes the nature of the integral.

Or is there something that I'm missing here?

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