Zee, Quantum Field Theory in a Nutshell, problem 1.3.1

[Maurice7510]

Homework Statement

I'm working through Zee for some self study and I'm trying to do all the problems, which is understandably challenging. Problem 1.3.1 is where I'm currently stuck: Verify that D(x) decays exponentially for spacelike separation.

Homework Equations

The propagator in question is
$$ D(x) = -i \int \frac{d^3k}{2(2\pi)^3} \frac{e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}}{\sqrt{\boldsymbol{k}^2+m^2}} $$

The Attempt at a Solution

Presumably, I would have to solve the integral and show that it decays exponentially (the spacelike aspect has already been taken into account for the above integral) and what I did was switch to spherical coordinates and integrated over the azimuthal:

$$ D = \frac{-i}{8\pi^2}\int dr\,d\theta\frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}}r\cos\theta $$

This is where I'm stuck. The square root in the denominator suggests this is a branch cut integral but I haven't been able to find a source that explains it sufficiently. If anyone could help me figure this out I'd appreciate it. Thanks.

 

[stevendaryl]

Homework Statement

I'm working through Zee for some self study and I'm trying to do all the problems, which is understandably challenging. Problem 1.3.1 is where I'm currently stuck: Verify that D(x) decays exponentially for spacelike separation.

Homework Equations

The propagator in question is
$$ D(x) = -i \int \frac{d^3k}{2(2\pi)^3} \frac{e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}}{\sqrt{\boldsymbol{k}^2+m^2}} $$

I feel that there is something wrong with this equation. Does $x$ mean a spatial vector, or a 4-vector? If it's a 4-vector, then there is a problem, because on the right-hand side of the equals sign, there is no mention of the time component of $x$. If $x$ is a spatial vector vector, then it doesn't make any sense to talk about spacelike separations.

 

[Maurice7510]

That is curious, though I wrote exactly what he has in the book. Initially, we have
$$ D(x) = -i \int \frac{d^3k}{(2\pi)^32\omega_k}[e^{-i(\omega_kt - \boldsymbol{k}\cdot\boldsymbol{k})}\theta(x^0)+e^{i(\omega_kt - \boldsymbol{k}\cdot\boldsymbol{k})}\theta(-x^0)] $$
which he reduces to the above equation in consideration of spacelike separation where $x^0= 0$. So I suppose the notation on the left is a four vector, it just happens to be $x = (0, x^1, x^2, x^3) \equiv \boldsymbol{x}$.

 

[Maurice7510]

I now realize I made a mistake in the spherical coordinate substitution, the integral should be
$$ D(x) = \frac{-i}{8\pi^2}
int dr\,d\theta \frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}} r^2\sin\theta $$
The integral in $\theta$ is fairly straight forward at this point, and I got
$$ D(x) = \frac{-i}{4\pi x}\int dr \frac{r\sin(rx)}{\sqrt{r^2+m^2}} $$
which i now have no idea how to solve

 

[malawi_glenn]

I now realize I made a mistake in the spherical coordinate substitution, the integral should be
$$ D(x) = \frac{-i}{8\pi^2}
int dr\,d\theta \frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}} r^2\sin\theta $$
The integral in $\theta$ is fairly straight forward at this point, and I got
$$ D(x) = \frac{-i}{4\pi x}\int dr \frac{r\sin(rx)}{\sqrt{r^2+m^2}} $$
which i now have no idea how to solve

That is because that integral has no solution in terms of elementary functions.

 

[Fred Wright]

If you assume that the magnitude of the radial velocity is much less than the speed of light, then you can expand the denominator in the integral to second order in $p_r$.
$\sqrt{p^2 + m^2} = m\sqrt{1 + \frac {p^2} {m^2}} \approx m(1 + \frac {p^2} {m^2}) = \frac {(p - im)(p + im)} {m}$
The integral becomes
$D(x) \approx \frac {1} {2(2\pi)^2} \frac {m} {|x|} \int_{0}^\infty dr \frac {r[(-i) \sin(r|x|)]} {(r - im)(r + im)} = \frac {1} {2(2\pi)^2} \frac {m} {|x|} \mathcal {Im} \int_{0}^\infty dr \frac {r e^{-i|x|r}} {(r - im)(r + im)}$
We have simple poles at $z_0 = \pm im$. We use the Cauchy Integral theorem and the residue theorem, closing a semi-circular contour in the lower half-plane, and noting that the integral along the arc tends to zero as the radius tends to $\infty$ and the integral along the real axis from $-\infty$ to $+\infty$ is twice the integral from 0 to $\infty$. We find,
$D(x) \approx \frac {1} {16\pi}\frac{m} {|x|}e^{-|x|m}$