Covariant Green's function for wave equation

[jason12345]

This comes from Jackson's Classical Electrodynamics 3rd edition, page 613. He finds the Green's function for the covariant form of the wave equation as:

$$D(z) = -1/(2\pi)^{4}\int d^{4}k\: \frac{e^{-ik\cdot z}}{k\cdot k}$$

Where z = x - x' the 4 vector difference, $$k\cdot z = k_0z_0 - \mathbf{k \cdot z}$$

He then performs the integral over k0 first by considering it as a complex variable to give:

$$D(z) = -1/(2\pi)^{4}\int d^{3}k e^{i\mathbf{k\cdot z}}\int_{-\infty }^{\infty } dk_0\: \frac{e^{-ik_0z_0}}{k^2_0 - \kappa^2}$$

where $$\kappa = |\mathbf{k}|$$

But then he says for $$z_0>0, e^{-ik_0z_0}$$ increases without limit in the upper half plane.

Is this correct? There's a minus in the exponential so i would have thought it's the exact opposite.

He considers a contour r in the upper half of the k0 plane from +oo to -oo, closed by a semicircle also in the upper half of k0 and says that for z0 < 0, the resulting integral vanishes, whereas for z0 > 0, the integral over k0 is:

$$\oint_{r} dk_0\: \frac{e^{-ik_0z_0}}{k^2_0 - \kappa^2} = -2\pi i\: Res \left (\frac{e^{-ik_0z_0}}{k^2_0 - \kappa^2} \right) = \frac{-2\pi}{\kappa} sin(\kappa z_0)$$

The Green function is then:

$$D_r(z) = \frac{\theta(z_0)}{(2\pi)^3} \int d^3k \: e^{i\mathbf{k \cdot z}}\ \frac{sin(\kappa z_0)}{\kappa}$$

Where does the $$\theta(z_0)$$ come from?

Thanks in advance for your interest.

 

[maverick280857]

The Green function is then:

$$D_r(z) = \frac{\theta(z_0)}{(2\pi)^3} \int d^3k \: e^{i\mathbf{k \cdot z}}\ \frac{sin(\kappa z_0)}{\kappa}$$

Where does the $$\theta(z_0)$$ come from?

Thats the Heaviside step function, which is zero for $z_0 <0$ and unity for $z_0 > 0$. The integral is zero for $z_0 <0$ as you pointed out yourself, so this is just a convenient way of writing it.

 

[George Jones]

But then he says for $$z_0>0, e^{-ik_0z_0}$$ increases without limit in the upper half plane.

Is this correct? There's a minus in the exponential so i would have thought it's the exact opposite.

Take $z_0 = a + ib$ with $a$ and $b$ real, and $b > 0$ (upper half-plane). Then

$$e^{-i k_0 z_0} =e^{-i a k_0} e^{b k_0},$$

which blows up as $b$ becomes large.

 

[George Jones]

Take $z_0 = a + ib$ with $a$ and $b$ real, and $b > 0$ (upper half-plane). Then

$$e^{-i k_0 z_0} =e^{-i a k_0} e^{b k_0},$$

which blows up as $b$ becomes large.

Oops, I wrote this without reading. Often, $z$ denotes a complex intregration variable, and I made the mistake of taking this to be so here. What I wrote above should, however, make clear what happens for complex $k$ and $z_0$ a positive real.