Useful Definitions

The following notation will be handy when dealing with spin-0 fields. For scaler functions $a$ and $b$,

$$a\stackrel{\leftrightarrow}{\partial}_\mu b \equiv a \left( ... ...\right) - \left( \frac{\partial a}{\partial x^\mu} \right) b.$$ (2.25)

The Dirac delta-function can be defined using

$$\int_{-\infty}^{+\infty} dx\ e^{i(p_f-p_i)x} = 2\pi\delta(p_f-p_i) .$$ (2.26)

A useful property of the Dirac delta-function is

$$\delta[f(x)] = \frac{1}{\vert f^\prime(x)\vert _{x=x_0}} \delta(x-x_0),$$ (2.27)

where $f(x_0)=0$. One particularly useful example of the above general relationship is

$$\delta(x^2-a^2) = \frac{1}{2\vert a\vert}[\delta(x-a) + \delta(x+a)],$$ (2.28)

where $a$ is a constant.

The delta-function in three dimensions is often written as

$$\delta^3 (\vec{x} - \vec{x}^{\:\prime}) = \delta (x_1 - x_1^\prime) \delta (x_2 - x_2^\prime) \delta (x_3 - x_3^\prime) .$$ (2.29)

Cauchy's integral formula is

$$\oint_C \frac{f(z)}{z-z_0} dz = 2\pi if(z_0) ,$$ (2.30)

where the direction of the contour of integration is clockwise. A counter-clockwise direction of integration results in an over all minus sign.

Integration by parts gives

$$\int^{+\infty}_{-\infty} dx u\frac{dv}{dx} = [uv]^{+\infty}_{-\infty} - \int^{+\infty}_{-\infty} dx \frac{du}{dx} v .$$ (2.31)

The ``surface term'' $[uv]^{+\infty}_{-\infty}$ can usually be neglected.