# Dependence of Klein-Gordon solution only on spatial coordinates

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:

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

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.

• I've seen in «Notations and conventions» that bold letters are for spatial coordinates and ordinary letters are for $D = 3+1$ spacetime. May 27 at 21:39