Imaginary Evolution

Optimization

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$$\begin{aligned} S & = \int_{-\infty}^{+\infty}(\frac{i}{2}\sum_s \overline{\Psi}_s\dot{\Psi}_s - \frac{i}{2}\sum_s \dot{\overline{\Psi}}_s \Psi_s-\sum_{ss'}\overline{\Psi}_s\Psi_{s'} H_{ss'})dt\\ & = \int_{-\infty}^{+\infty}( \frac{i}{2}\sum_s \overline{\Psi}(\overline{W})_s\dot{\Psi}(W)_s - \frac{i}{2}\sum_s \dot{\overline{\Psi}}(\overline{W})_s \Psi(W)_s- \sum_{ss'}\overline{\Psi}(\overline{W})_s\Psi(W)_{s'} H_{ss'} )dt\\ & = \int_{-\infty}^{+\infty}( \frac{i}{2}\sum_s \overline{\Psi}_s \sum_{j}\frac{\partial \Psi_s}{\partial W_j}\dot{W_j} - \frac{i}{2}\sum_s \Psi_s \sum_j \frac{\partial \overline{\Psi}}{\partial \overline{W}_j}\dot{\overline{W}_j} - \sum_{ss'}\overline{\Psi}_s \Psi_{s'} H_{ss'} )dt\\ \end{aligned}$$

let a variation on $W$:

$$\begin{aligned} \overline{W} & \rightarrow \overline{W} + \overline{\delta W}\\ & \text{or}\\ W & \rightarrow W + \delta W \end{aligned}$$

Then, according to Euler-Lagrange equation

$$\begin{aligned} \text{for} \quad & S(q) = \int_{a}^{b} L(t, q, \dot{q}) dt\\ \text{we have} \quad & \frac{\partial L}{\partial q_i} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} = 0 \end{aligned}$$

consider $W_i$ and $\overline{W}_i$ whose functional is $L$ :

$$\frac{\partial L}{\partial W_i} - \frac{d}{dt}\frac{\partial L}{\partial \dot{W_i}} = 0$$

we have,

$$\begin{aligned} \frac{\partial L}{\partial W_i} & = \frac{i}{2}\sum_{s}\overline{\Psi}_s \sum_j \frac{\partial^2 \Psi_s}{\partial W_i \partial W_j} \dot{W_j} - \frac{i}{2}\sum_s \frac{\partial \Psi_s}{\partial W_i} \sum_j \frac{\partial \overline{\Psi}_s}{\partial \overline{W}_j}\dot{\overline{W}_j} - \sum_{ss'}\overline{\Psi}_s\frac{\partial \Psi_{s'}}{\partial W_i} H_{ss'}\\ \frac{\partial L}{\partial \dot{W_i}} & = \frac{i}{2}\sum_s \overline{\Psi}_s\frac{\partial \Psi_s}{\partial W_i} \\ \frac{d}{dt}\frac{\partial L}{\partial \dot{W}_i} & = \frac{i}{2}\sum_s \frac{\partial \Psi_s}{\partial W_i} \sum_j \frac{\partial \overline{\Psi}_s}{\partial \overline{W}_j} \dot{\overline{W}_j} + \frac{i}{2}\sum_s \overline{\Psi}_s \sum_j \frac{\partial^2 \Psi_s}{\partial W_i \partial W_j} \dot{W_j} \end{aligned}$$

Therefore,

$$\frac{\partial L}{\partial W_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot{W_i}} = -i\sum_s \frac{\partial \Psi_s}{\partial W_i} \sum_j \frac{\partial \overline{\Psi}_s}{\partial \overline{W}_j} \dot{\overline{W}_j}- \sum_{ss'}\overline{\Psi}_s\frac{\partial \Psi_{s'}}{\partial W_i} H_{ss'} = 0$$

Then we have

$$-i\sum_s \frac{\partial \Psi_s}{\partial W_i} \sum_j \frac{\partial \overline{\Psi}_s}{\partial \overline{W}_j} \dot{\overline{W}_j} = \sum_{ss'}\overline{\Psi}_s\frac{\partial \Psi_{s'}}{\partial W_i} H_{ss'}$$

Similarly, for $\frac{\partial L}{\partial \overline{W}_i}$ and $\frac{\partial L}{\partial \dot{\overline{W}}_i}$, we have its Euler-Lagrange equation:

$$+i \sum_s \frac{\partial \overline{\Psi}_s}{\partial \overline{W}_i}\sum_j \frac{\partial \Psi_s}{\partial W_j} \dot{W_j} = \sum_{ss'}\frac{\partial \overline{\Psi}_s}{\partial \overline{W_i}}\Psi_{s'} H_{ss'}$$

define $O_s^i = \frac{1}{\Psi_s}\frac{\partial \Psi_s}{\partial W_i}$, then

$$-i\sum_s |\Psi_s|^2 O_s^i \sum_j \overline{O}_s^j \dot{\overline{W_j}} = \sum_{ss'}\overline{\Psi}_s \Psi_{s'} O_{s'}^i H_{ss'}\\ -i\sum_s |\Psi_s|^2 O_s^i \sum_j \overline{O}_s^j \dot{\overline{W_j}} = \sum_s |\Psi_s|^2 O_{s}^i \frac{\sum_{s'}\overline{\Psi}_s H_{ss'}}{\overline{\Psi}_s}\\ -i\sum_s |\Psi_s|^2 O_s^i \sum_j \overline{O}_s^j \dot{\overline{W_j}} = \sum_s |\Psi_s|^2 O_{s}^i \overline{E}_{loc}\\ -i\sum_{sj} |\Psi_s|^2 O_s^i \overline{O}_s^j \dot{\overline{W_j}} = \sum_s |\Psi_s|^2 O_{s}^i \overline{E}_{loc}$$

denote $G_{ij} = \sum_s \frac{\partial \overline{\Psi}_s}{\partial \overline{W}_i}\frac{\partial \Psi_s}{\partial W_j} = \sum_s |\Psi_s|^2 \overline{O}_s^i O_s^j$ and $\overline{G}_{ij} = \sum_s \frac{\partial \Psi_s}{\partial W_i}\frac{\partial \overline{\Psi}_s}{\partial \overline{W}_j} = \sum_s |\Psi_s|^2 O_s^i \overline{O}_s^j$

thus, we have

$$-i\sum_{j} \overline{G}_{ij} \dot{\overline{W_j}} = \sum_s |\Psi_s|^2 O_{s}^i \overline{E}_{loc}$$

$$-i\sum_{j} \overline{G}_{ij} \dot{\overline{W_j}} = \langle O_{s}^i \overline{E}_{loc} \rangle$$

and similarly,

$$+i\sum_{j} G_{ij} \dot{W_j} = \langle \overline{O}_{s}^i E_{loc} \rangle$$

The linear system can be solved by calculating the inverse of $G$, thus we have

$$-i \dot{\overline{W}} = \overline{G}^{-1} \langle O_s \overline{E}_{loc}\rangle\\ +i \dot{W} = G^{-1} \langle \overline{O}_s E_{loc}\rangle$$