Continuous plane wave solutions to Klein Gordon Field Equation

Question

The continuous plane wave solution to the Klein Gordon Field Equation can be written as

$\phi(x) = \int\frac{d^3\vec{k}}{\sqrt{2(2\pi)^3w_\vec{k}}} a(\vec k) e^{-ikx} + \int\frac{d^3\vec{k}}{\sqrt{2(2\pi)^3w_\vec{k}}} b(\vec k) e^{ikx}$

where $x$ and $k$ are 4-vectors.

The LHS is a function of 4-vector $x$, i.e. it is a function of the 4 components $x^0,x^1,x^2$ and $x^3$.

Similarly, the RHS is a function of 8 components $k^0,k^1,k^2,k^3$ and $x^0,x^1,x^2,x^3$, but integrated over $k^1,k^2,k^3$, meaning it reduces to a function of 5 components ($k^0$, $x^0,x^1,x^2,x^3$).

So apparently the LHS and RHS don't agree with each other. One side is a function of 4 components and the other is a function of 5 components.

What am I seeing wrong here?

Answer 1

These solutions are on-shell, meaning that they maintain ${\bf{k}}^2=m^2$ which gives $k_0$ as function of the 3-vector $k$. This is why we need to integrate only over $k_1,k_2,k_3$ (and also the reason for the measure of $\omega_{\vec{k}}^{-1/2}$).