Dependence of Klein-Gordon solution only on spatial coordinates

Question

I am studying QFT with the Peskin and Schroeder textbook, and I am new to this area of physics. I'm struggling with the solution of the Klein-Gordon equation using Fourier integral as a continuum set of oscillators:

\begin{equation} \phi(\mathbf{x}) = \int \frac{d^3 p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_\mathbf{p}}} \left(a_\mathbf{p} e^{i \mathbf{p} \cdot \mathbf{x}} + a^\dagger_\mathbf{p} e^{-i \mathbf{p} \cdot \mathbf{x}}\right). \end{equation}

The question is: why doesn't the field $\phi$ depend on time? My answer is that this solution is applicable only in the Schrödinger picture, where the field is a function of spatial coordinates, not time.

The postscriptum question: Can we obtain this solution without knowledge of the harmonic oscillator? Specifically, I mean that perhaps it can be obtained without the procedure of canonical quantization, can't it?

I would appreciate any assistance: from simple reasoning to a purely formal derivation of formulas.

Answer 1

It does. The bold x notation represents the spacetime 4-vector components, so depends on time and space. The solution is not at all due to second quantization: you’re putting the cart before the horse. That solution is just the solution of the classical Klein Gordon equation, in which we then do second quantization to make it an operator.