Canonical Transformation to Normal Variables

Introduce complex normal variables $a_k$:

$$q_k = \frac{1}{\sqrt{2 \omega_k}} (a_k + a_{-k}^*), \quad p_k = i \sqrt{\frac{\omega_k}{2}} (a_k - a_{-k}^*).$$ (24)

Inverse:

$$a_k = \sqrt{\frac{\omega_k}{2}} q_k - \frac{i}{\sqrt{2 \omega_k}} p_k.$$ (25)

The quadratic Hamiltonian becomes diagonal:

$$H_2 = \sum_k \omega_k \vert a_k\vert^2.$$ (26)

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