QHO Field

$[\varphi (\vec{x}),\Pi (\vec{y})]=i\hslash {\delta }^3(\vec{x}-\vec{y})\mathbb{I}$

Let us try an ansatz just as for $X$ and $P$.

$\varphi (\vec{x})=\int [d\vec{k}]A(a_k{e}^{i\vec{k}.\vec{x}}+a_k^{†}{e}^{-i\vec{k}.\vec{x}})$

$\Pi (\vec{x})=-i\int [d\overrightarrow{k^{\prime }}]B(a_{k^{\prime }}{e}^{i\overrightarrow{k^{\prime }}.\vec{x}}-a_{k^{\prime }}^{†}{e}^{-i\overrightarrow{k^{\prime }}.\vec{x}})$

with the notation $[d\vec{k}]=\frac{d^{3}k}{(2\pi {)}^3{N}_k},[a_k,a_{k^{\prime }}^{†}]=(2\pi {)}^3{N}_k{\delta }^3(\vec{k}-\overrightarrow{k^{\prime }})$.

Here’s how: Solving the commutator

Now the Hamiltonian will dictate the energy relation to momentum and give us the spectrum of the theory, which should be bounded from below on energy.

• Example: Klein-Gorden Field