**7. Conclusions**

We have derived the Schrödinger equations for coherent electric dipole fields, the Klein–Gordon equations for coherent electric fields and the Kadanoff–Baym equations for quantum fluctuations in QED with electric dipoles in 2 + 1 dimensions. It is possible to derive equilibration for quantum fluctuations and super-radiance for background coherent fields simultaneously. Total energy consumption to maintain super-radiance in microtubules is consistent with energy consumption in our experiences. We can describe dynamical information transfer with super-radiance via microtubules without violation of the second law in thermodynamics. We have also derived the Higgs mechanism in normal population and the tachyonic instability in inverted population. These dynamical properties might be significant to form and maintain coherent domains composed of dipoles and photons. We are ready to describe memory formation processes towards equilibrium states in 2 + 1 dimensions with equations in this paper. Furthermore, our approach might pave the way to understand the dynamical thinking processes with memory recalling in QBD by investigating the case in open systems with the Kadanoff–Baym equations. This work will be extended to the 3 + 1 dimensional analysis to describe memory formation processes in numerical simulations. We should derive the Schödinger-like equations, the Klein–Gordon equations and the Kadanoff–Baym equations by starting with the single Lagrangian in QED with electric dipoles in 3 + 1 dimensions in the future study. These equations in 3 + 1 dimensions will describe more realistic and practical dynamics in QBD.

**Author Contributions:** Conceptualization, A.N, S.T. and J.A.T.; methodology, A.N.; software, A.N.; validation, A.N., S.T. and J.A.T.; formal analysis, A.N.; investigation, A.N.; resources, S.T. and J.A.T.; data curation, A.N.; writing-original draft preparation, A.N.; writing-review and editing, A.N., S.T. and J.A.T.; visualization, A.N.; supervision, S.T. and J.A.T.; project administration, S.T. and J.A.T.; funding acquisition, S.T. and J.A.T.

**Funding:** This work was supported by JSPS KAKENHI Grant Number JP17H06353.

**Acknowledgments:** J.A.T. is grateful for research support received from NSERC (Canada).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Quantum Fluctuations in the Klein–Gordon Equations**

In this section, we shall derive the second, third, fourth and fifth terms involving quantum fluctuations on the right-hand side in Equation (121) in spatially homogeneous systems. They correspond to the following term,

$$-2ed\_{\mathfrak{e}}(\partial\_0)^2 \left[\Delta\_{10}(\mathfrak{x}, \mathfrak{x}) + \Delta\_{01}(\mathfrak{x}, \mathfrak{x})\right] \nu$$

*Entropy* **2019**, *21*, 1066

in Equation (101) with the symmetry Δ10 = Δ−<sup>10</sup> and Δ01 = Δ0−1. It appears in taking the time derivative of *J*1 (given by Equation (21) ) in Equation (20). Here <sup>Δ</sup>10(*<sup>x</sup>*, *x*) and <sup>Δ</sup>01(*<sup>x</sup>*, *x*) can be rewritten by,

$$
\Delta\_{10}(\mathbf{x}, \mathbf{x}) \quad = \quad - \frac{ed\_{\mathcal{E}}}{\underline{i}} \int\_{\mathbf{w}} \Delta\_{\mathbf{g}, 11}(\mathbf{x}, \mathbf{w}) E\_1(\mathbf{w}) \Delta\_{00}(\mathbf{w}, \mathbf{x}), \tag{A1}
$$

$$
\Delta\_{01}(\mathbf{x}, \mathbf{x}) \quad = \quad - \frac{ed\_c}{i} \int\_w \Delta\_{00}(\mathbf{x}, w) E\_1(w) \Delta\_{\mathbf{g}, 11}(w, \mathbf{x}), \tag{A2}
$$

where we have used Equations (31) and (34) by setting *E*2 = 0. We rewrite second time derivatives of <sup>Δ</sup>10(*<sup>x</sup>*, *x*) and <sup>Δ</sup>01(*<sup>x</sup>*, *<sup>x</sup>*).

We shall rewrite Equation (30) without self-energy Σ*αα* as,

$$\left[i\frac{\partial}{\partial \mathbf{x}^0} + \frac{\nabla\_i^2}{2m} - \frac{1}{2I}\right] \Delta\_{\mathbb{S},11}(\mathbf{x}, w) \quad = \quad i\delta\_{\mathbb{C}}(\mathbf{x} - w), \tag{A3}$$

$$\left[-i\frac{\partial}{\partial \mathbf{x}^0} + \frac{\nabla\_i^2}{2m} - \frac{1}{2I}\right] \Delta\_{\mathbf{\mathcal{G}}, 11}(w, \mathbf{x}) \quad = \quad i\delta\_{\mathbf{\mathcal{C}}}(w - \mathbf{x}), \tag{A4}$$

where we have multiplied <sup>Δ</sup>*g*,<sup>11</sup> from the right and left of Equation (30). By using the above equations and Equations (32) and (33) with Equations (A1) and (A2) and Δ−<sup>1</sup> 0,00(*<sup>x</sup>*, *y*) = *i ∂∂x*0 + ∇2*i* 2*m δ*C (*x* − *y*), we can show

$$\begin{split} \frac{\partial}{\partial x^{0}} \Lambda\_{10}(\mathbf{x}, \mathbf{x}) &= \; \text{ed}\_{\varepsilon} \left[ \left[ \left( -\frac{\nabla\_{i}^{2}}{2m} + \frac{1}{2I} \right) \Delta\_{\mathcal{S}, 11} + i\delta\_{\mathcal{C}} \right] E\_{1} \Lambda\_{00} \\ &+ \Delta\_{\mathcal{S}, 11} E\_{1} \frac{\nabla\_{i}^{2}}{2m} \Delta\_{00} + 2 \Delta\_{\mathcal{S}, 11} \epsilon d\_{\varepsilon} E\_{1} \Delta\_{01} E\_{1} - \Delta\_{\mathcal{S}, 11} E\_{1} i\delta\_{\mathcal{C}} \right] \\ &= \; \text{ed}\_{\varepsilon} \left[ \left( \frac{1}{2I} \Delta\_{\mathcal{S}, 11} + i\delta\_{\mathcal{C}} \right) E\_{1} \Lambda\_{00} + 2 \Delta\_{\mathcal{S}, 11} \epsilon d\_{\varepsilon} E\_{1} \Delta\_{01} E\_{1} - \Delta\_{\mathcal{S}, 11} E\_{1} i\delta\_{\mathcal{C}} \right], \\ &\left[ \left( \frac{1}{2I} \Delta\_{\mathcal{S}, 11} + i\delta\_{\mathcal{C}} \right) \Lambda\_{00} + 2 \Delta\_{\mathcal{S}, 11} \epsilon d\_{\varepsilon} E\_{1} \Delta\_{01} E\_{1} - \Delta\_{\mathcal{S}, 11} E\_{1} i\delta\_{\mathcal{C}} \right], \end{split} \tag{A5}$$

$$\frac{\partial}{\partial \mathbf{x}^{0}} \Delta\_{01}(\mathbf{x}, \mathbf{x}) \quad = \left. \text{erf}\_{\mathbf{f}} \left[ \left( -2 \text{ed}\_{\mathbf{f}} E\_{1} \Lambda\_{10} + i \delta\_{\mathbf{C}} \right) E\_{1} \Lambda\_{\mathbf{g}, 11} + \Lambda\_{00} \text{E}\_{1} \left( -\frac{1}{2I} \Lambda\_{\mathbf{g}, 11} i \delta\_{\mathbf{C}} \right) \right] , \tag{A6}$$

where *δ*C represents the delta function in the closed-time path. Here the terms proportional to ∇2*i* are cancelled in spatially homogeneous systems. By use of the above two equations, we can show

$$\frac{\partial}{\partial \mathbf{x}^0} \left(\Delta\_{10} + \Delta\_{01}\right) = \frac{1}{2iI} \left(\Delta\_{10} - \Delta\_{01}\right),\tag{A7}$$

and,

$$\begin{array}{rcl} \frac{\partial^2}{\partial(\mathbf{x}^0)^2} \left(\Delta\_{10} + \Delta\_{01}\right) &=& \frac{ed\_c}{2iI} \left[ \left(\frac{\Delta\_{\mathcal{S},11}}{2I} + i\delta\_{\mathcal{C}}\right) E\_1 \Delta\_{00} + 2\Delta\_{\mathcal{S},11} cd\_c E\_1 \Delta\_{01} E\_1 - \Delta\_{\mathcal{S},11} E\_1 i\delta\_{\mathcal{C}} \\\\ &- \left(-2cd\_c E\_1 \Delta\_{10} + i\delta\_{\mathcal{C}}\right) E\_1 \Delta\_{\mathcal{S},11} - \Delta\_{00} E\_1 \left(-\frac{1}{2I} \Delta\_{\mathcal{S},11} - i\delta\_{\mathcal{C}}\right) \right] \\\\ &=& \frac{ed\_c}{2iI} \left[ 2iE\_1 \left(\Delta\_{00} - \Delta\_{\mathcal{S},11}\right) + \frac{1}{2I} \left(\Delta\_{\mathcal{S},11} E\_1 \Delta\_{00} + \Delta\_{00} E\_1 \Delta\_{\mathcal{S},11} \right) \\\\ &+& 2cd\_c \left(\Delta\_{\mathcal{S},11} E\_1 \Delta\_{01} E\_1 + E\_1 \Delta\_{10} E\_1 \Delta\_{\mathcal{S},11} \right) \right]. \end{array} \tag{A8}$$

Since we can rewrite Equations (35) or (36) by multiplying *<sup>i</sup>*Δ*g*,<sup>11</sup> as,

$$\begin{aligned} i\Delta\_{\mathbb{g},11} - i\Delta\_{11} &= & \epsilon d\_{\mathbb{f}} \Delta\_{\mathbb{g},11} E\_1 \Delta\_{01} \\ &= & \epsilon d\_{\mathbb{f}} \Delta\_{10} E\_1 \Delta\_{\mathbb{g},11} \end{aligned} \tag{A9}$$

we arrive at,

$$
\frac{\partial^2}{\partial(\mathbf{x}^0)^2} \left(\Delta\_{10} + \Delta\_{01}\right) \quad = \quad - \frac{1}{4I^2} \left(\Delta\_{10} + \Delta\_{01}\right) + \frac{ed\_\epsilon E\_1}{I} \left(\Delta\_{00} - 2\Delta\_{11} + \Delta\_{\mathbf{g},11}\right), \tag{A10}
$$

where we have used Equations (A1) and (A2).

Finally by rewriting the statistical parts (subscript '*F*') of Δ10 + Δ01 with Equations (A1) and (A2), and using *<sup>E</sup>*1(*w*) = *<sup>E</sup>*1(*x*)+(*w*<sup>0</sup> − *<sup>x</sup>*<sup>0</sup>)*∂*0*E*1(*x*) in,

$$\left[\int dw \left[\Delta\_{\mathbb{S},11}(\mathbf{x},w)E\_1(w)\Delta\_{00}(w,\mathbf{x}) + \Delta\_{00}(\mathbf{x},w)E\_1(w)\Delta\_{\mathbb{S},11}(w,\mathbf{x})\right]\right]\_F$$

and the relation in the first order in the gradient expansion,

$$\begin{split} \left[ \int dw \Delta\_{\mathbb{S},11}(\mathbf{x}, w) \Delta\_{00}(w, \mathbf{x}) \right]\_{\mathrm{F}} &= \quad \int\_{p} \left( \frac{\Delta\_{\mathbb{S},11, \mathbb{R}}(\mathbf{x}, p)}{i} F\_{00}(\mathbf{x}, p) + \Delta\_{\mathbb{S},11, \mathbb{F}} \frac{\Delta\_{00, \mathcal{A}}}{i} \right. \\ &\left. + \frac{i}{2} \left\{ \frac{\Delta\_{\mathbb{S},11, \mathbb{R}}(\mathbf{x}, p)}{i}, F\_{00}(\mathbf{x}, p) \right\} + \frac{i}{2} \left\{ \Delta\_{\mathbb{S},11, \mathbb{F}} \frac{\Delta\_{00, \mathcal{A}}}{i} \right\} \right). \end{split} (\text{A11})$$

with the advanced (subscript '*A*') <sup>Δ</sup>00,*A* = *i*(Δ1100 − Δ2100) = ReΔ00,*R* − *ρ*002 and the retarded Δ00,*R* = *i*(Δ1100 − Δ1200) = ReΔ00,*R* + *ρ*002 , we can derive the third, fourth and fifth terms on the right-hand side in Equation (121).
