General Solution of the Klein-Gordon Equation

Besides plane-wave solutions of definite momentum, we can have solutions which are the Fourier transform of $\tilde{\phi}^{(\pm)}(p)$ which depend only on momentum:

$$\phi^{(\pm)}(x) = \frac{1}{\sqrt{(2\pi)^3}} \int \frac{d^3p}{\sqrt{2E}} e^{-ip\cdot x} \tilde{\phi}^{(\pm)}(p).$$ (4.23)

The relativistic invariance of solutions of this form is not immediately obvious.

The general solution to the Klein-Gordon equation has either positive or negative energy. They can be written (using $\hbar = c = 1$) as a Fourier transform of $\tilde{\phi}_a^{(\pm)}(p)$,

$$\phi_a^{(\pm)} = \frac{\sqrt{2}}{\sqrt{(2\pi)^3}} \int d^4p ... ...ot x} \delta(p^2-m^2)\theta(\pm p)\tilde{\phi}_a^{(\pm)}(p) ,$$ (4.24)

where the index $a$ is used to label different solutions of identical mass. This form is manifestly Lorentz invariant. The Lorentz invariant step function is

$$\theta(p) \equiv \theta(p_0) \equiv \left\{ \begin{array}{l}... ...m{1 if $p^0>0$}, \\ \textrm{0 if $p^0<0$}. \end{array}\right.$$ (4.25)

Lorentz invariance of $\delta(p^2-m^2)$ restricts $p$ to be a time-like vector and thus $\theta(p)$ distinguishes between past and future. Thus the expression $\int dp_0\delta(p^2-m^2)\theta(\pm p)$ ensures the condition $p^0 = \pm\sqrt{\vec{p}^{\: 2}+m^2}$.

We can show that equation 4.24 gives rise to the usual form of the solutions by rewriting the general solution, using

$$\delta(p^2-m^2) = \delta(p_0^2-\vec{p}^{\: 2}-m^2) = \delta(p_0^2-E^2)$$ (4.26)

and then applying the identity equation 2.28.

For positive energy

$$\phi_a^{(+)}(x) = \frac{2^{1/2}}{(2\pi)^{3/2}} \int d^3p dp_0 \sqrt{E} \left[\frac{... ...2E}\right] \theta(p_0) e^{-i(p_0t-\vec{p}\cdot\vec{x})} \tilde{\phi}_a^{(+)}(p)$$

$$= \frac{1}{(2\pi)^{3/2}} \int \frac{d^3p}{\sqrt{2E}} e^{-i(Et-\vec{p}\cdot\vec{x})} \tilde{\phi}_a^{(+)}(p) ,$$ (4.27)

$$\phi_a^{(+)}(x) = \frac{1}{\sqrt{(2\pi)^3}} \int \frac{d^3p}{\sqrt{2E}} e^{-ip\cdot x} \tilde{\phi}_a^{(+)}(p) .$$ (4.28)

For negative energy

$$\phi_a^{(-)}(x) = \frac{2^{1/2}}{(2\pi)^{3/2}} \int d^3p dp_0 \sqrt{E} \left[\frac{... ...E}\right] \theta(-p_0) e^{-i(p_0t-\vec{p}\cdot\vec{x})} \tilde{\phi}_a^{(-)}(p)$$

$$= \frac{1}{(2\pi)^{3/2}} \int \frac{d^3p}{\sqrt{2E}} e^{i(Et+\vec{p}\cdot\vec{x})} \tilde{\phi}_a^{(-)}(p) ,$$ (4.29)

$$\phi_a^{(-)}(x) = \frac{1}{\sqrt{(2\pi)^3}} \int \frac{d^3p}{\sqrt{2E}} e^{+ip\cdot x} \tilde{\phi}_a^{(-)}(p) .$$ (4.30)

Equations 4.28 and 4.30 are identical to the previously solutions (equation 4.23).