Second derivative of Heaviside step function

[abhinavabhatt]

TL;DR Summary

Identifying Klein Gordon propagator as Green's Function

In QFT by peskin scroeder page 30 the action of Klein Gordon Operator on propagator
(∂²+m²)D_R(x-y)=∂²θ(x⁰-y⁰)...

how to compute this
∂²θ(x⁰-y⁰)?

 

[anuttarasammyak]

First derivative is delta function. The second derivative $\delta'(x)$ has property
$$\int \delta'(x) f(x) dx = -f'(0)$$

 

[abhinavabhatt]

Thanks for the answer.

 

[vanhees71]

Note that Peskin and Schroeder write in fact the correct equation, i.e.,
$$(\Box+m^2) D_{R}(x-y)=-\mathrm{i} \delta^{(4)}(x-y).$$
Since by definition $D_R(x-y) \propto \Theta(x^0-y^0)$ this function is the retarded Green's function of the Klein-Gordon operator (modulo the usual conventional factor $-\mathrm{i}$ on the right-hand side).