There is a analog between harmonic oscillator $x=\frac{1}{\sqrt{2\omega}}(a+a^\dagger)$ and quantum field $\phi=\int dp^3\frac{1}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_p e^{ipx}+a^\dagger e^{-ipx})$, which is used to quantize the field operator.
However, one thing confuse me is about the coefficient $\frac{1}{\sqrt{2\omega}}$. For field operator, this comes from Lorentz invariance, just because we have integrated time t. However, for harmonic oscillator, there seems no apparent Lorentz symmetry give me this. Is there any hidden symmetry behind the harmonic oscillator?