# The complex form of Hamilton canonical equations

I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that

If, however, the conjugate variables $$q_k$$, $$p_k$$ are replaced by the complex variables $$\frac{q_k+ip_k}{\sqrt{2}}=u_k$$ and $$\frac{q_k-ip_k}{\sqrt{2}}=u_k^*,$$ then the double set of canonical equations $$\dot q_i=\frac{\partial H}{\partial p_i}, \qquad\dot p_i=-\frac{\partial H}{\partial q_i}$$ can be replace by the following single set of complex equations $$\frac{du_k}{idt}=-\frac{d H}{d u_k^*}.$$

I did not understand that why shall we introduce the complex conjugate of $$u_k$$ and the coefficient $$\sqrt 2$$. It seems to me that the equation $$\frac{du_k}{idt}=-\frac{d H}{d u_k}$$ with $$u_k=q_k+ip_k$$ can already fulfill the task. Could you please tell me why my thinking is wrong?

P.S. I found another link (Relationship between complex analysis and Hamilton's canonical equation) related to this problem but not exactly the same.

1. We need the complex conjugated variable $$\bar{u}$$ because the Hamiltonian $$H(u,\bar{u})$$ is typically not holomorphic in $$u$$.

2. The normalization of the CCR$$^1$$
$$\{u,\bar{u}\} ~=~-i \tag{1}$$ explains the $$\sqrt{2}$$-normalization in $$u=\frac{q+ip}{\sqrt{2}}.\tag{2}$$

3. Recall that Hamilton's equations can be written in terms of the Poisson bracket $$\dot{u}~=~\{ u, H\}~=~\{ u, u\}\partial H + \{ u, \bar{u}\}\bar{\partial} H. \tag{3}$$ In the last equality in eq. (3) we used that the Poisson bracket $$\{ \cdot, \cdot\}$$ is a first-order differential operator wrt. its second entry.

4. We can now derive the complex form of Hamilton's equations $$\dot{u} ~\stackrel{(1)+(3)}{=}~-i\bar{\partial}H.\tag{4}$$

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$$^1$$ Quantum mechanically, the Poisson bracket (1) should be replaced with $$(i\hbar)^{-1}$$ times the corresponding commutator
$$[\hat{u},\hat{u}^{\dagger}] ~=~\hbar \hat{\mathbb{1}}, \tag{5}$$ in accordance with the correspondence principle between classical and quantum physics. Here the operator $$\hat{u}$$ and the Hermitian adjoint operator $$\hat{u}^{\dagger}$$ are an annihilation and a creation operator, respectively.

• Is there really an interest in writing the Hamilton equations in complex form? What gives? – Cham Aug 8 '19 at 2:13
• Yes. The complex Hamiltonian formulation is used in e.g. coherent state path integrals. – Qmechanic Aug 8 '19 at 5:48