In a free theory, excitations (eigenstates) are harmonic oscillators. This is because the Hamiltonian of a free theory is translationally invariant, thus diagonal in momentum space. Further, an excitation can be labeled by its momentum. After quantization, exicitations are created and destroyed by the creation and annihilation operators. Once these operators are defined, the field can be written in the basis of theses operators in the so called “second quantized form”. An expansion of the field in the basis of creation and annihilation operators is called a mode expansion.

It is important to note that, with only kinetic terms in the Lagrangian, the mode expansion for the conjugate momentum field can usually be obtained simply by a time derivative of its corresponding scalar field ( for \(L=\int \frac{1}{2}\dot{\phi}^2\),\(\hat{\pi}=\frac{\delta L}{\delta \dot{\phi}}=\dot{\phi}\)).

For example, the free Klein-Gordon field has the mode expansion $$ \hat{\phi}(t,x) = \int \frac{d^Dp}{(2\pi)^D2\omega_p} \left( \hat{a}_p e^{i(p\cdot x-\omega_p t)}+\hat{a}_p^\dagger e^{-i(p\cdot x-\omega_p t)} \right), $$ where the operators \(\hat{a}_p^\dagger\) and \(\hat{a}_p\) creates and destroys a harmonic oscillator of momentum \(p\), respectively and \(\omega_p=\sqrt{p\cdot p+m^2}\) is the frequency of the oscillator. The conjugate field has a similar expansion $$ \hat{\pi}(t,x) = \int \frac{d^Dp}{(2\pi)^D2\omega_p}(-i\omega_p) \left( \hat{a}_p e^{i(p\cdot x-\omega_p t)}-\hat{a}_p^\dagger e^{-i(p\cdot x-\omega_p t)} \right).$$

The canonical quantization procedure requires the field and its conjugate to satisfy the canonical equal-time commutation relations $$ [\hat{\phi}(t,x),\hat{\pi}(t,x’)]=i\delta(x-x’),~~[\hat{\phi}(t,x),\hat{\phi}(t,x’)]=0,~~[\hat{\pi}(t,x),\hat{\pi}(t,x’)]=0.$$ One can then deduce the commutation relations of the creation and annihilation operators without knowing the Hamiltonian $$ [a_p,a_p^\dagger]=(2\pi)^D\delta(p-p’).$$