*5.1. Super-Radiance*

In this section, we show the super-radiance in time evolution equations for coherent fields with the rotating wave approximations neglecting non-resonant terms and quantum fluctuations. We have used the derivations in [70,71] for background coherent fields.

We shall consider only *k*0 = 1 2*I* in this section and we expand the electric field *E*1 and the transition rate *ψ*¯0*ψ*¯∗ 1as,

$$E\_1(\mathbf{x}^0, \mathbf{x}^2) \quad = \frac{1}{2} \epsilon(\mathbf{x}^0, \mathbf{x}^2) \epsilon^{-i(k^0 \mathbf{x}^0 - k^0 \mathbf{x}^2)} + \frac{1}{2} \epsilon^\*(\mathbf{x}^0, \mathbf{x}^2) \epsilon^{i(k^0 \mathbf{x}^0 - k^0 \mathbf{x}^2)},\tag{102}$$

$$
\Psi\_1 \Psi\_0^\* \quad = \frac{1}{2} R(\mathbf{x}^0, \mathbf{x}^2) e^{-i(\mathbf{k}^0 \mathbf{x}^0 - \mathbf{k}^0 \mathbf{x}^2)},
\tag{103}
$$

We consider the following case,

$$\begin{aligned} |\partial\_0 \varepsilon| &\ll |k^0 \varepsilon|, \quad |\partial\_0 R| \ll |k^0 R|. \\ |\partial\_2 \varepsilon| &\ll |k^0 \varepsilon|. \end{aligned} \tag{104}$$

Neglect non-resonant terms like *e*±2*ik*0*x*<sup>0</sup> and quantum fluctuations (Green's functions Δ01 and Δ10) (the rotating wave approximation). Then from Equations (99)–(101), we arrive at the Maxwell–Bloch equations,

$$\frac{\partial \epsilon}{\partial \mathbf{x}^0} + \frac{\partial \epsilon}{\partial \mathbf{x}^2} = \epsilon i \epsilon d\_\epsilon \mathbf{k}^0 \mathbf{R},\tag{105}$$

$$\begin{array}{rcl}\frac{\partial \mathcal{Z}}{\partial \mathbf{x}^{0}} &=& \operatorname{iech}\_{\mathfrak{c}}(\mathfrak{c}R^{\*} - \mathfrak{c}^{\*}R),\\\frac{\partial \mathcal{X}^{0}}{\partial \mathbf{x}^{\*}} &=& \mathfrak{c}^{\*}\mathfrak{c}^{\*}R^{\*}.\end{array} \tag{106}$$

$$\frac{\partial \mathcal{R}}{\partial \mathbf{x}^0} = -i\epsilon d\_\epsilon \epsilon \mathbf{Z}.\tag{107}$$

We assume that , *Z* and *R* are independent of the spatial coordinate of the *x*2 direction. We shall change → *i* in the above equations and assume real functions *R* = *R*∗ and = <sup>∗</sup>. Then we can write,

$$\frac{\partial \epsilon}{\partial \mathbf{x}^0} = -\varepsilon d\_\epsilon k^0 R\_\prime \tag{108}$$

$$\frac{\partial Z}{\partial \mathbf{x}^0} = -2\epsilon d\_\epsilon \epsilon \, R\_\epsilon \tag{109}$$

$$\frac{\partial R}{\partial x^0} = \varepsilon d\_\ell \varepsilon Z. \tag{110}$$

We find the conservation law with the definition *B*<sup>2</sup> ≡ 2*R*<sup>2</sup> + *Z*2,

$$
\frac{\partial}{\partial \mathbf{x}^0} B^2 = \frac{\partial}{\partial \mathbf{x}^0} \left( 2R^2 + Z^2 \right) = 0. \tag{111}
$$

The relation *∂B ∂x*<sup>0</sup> = 0 represents the probability conservation since we can rewrite *B*<sup>2</sup> = 2|*ψ*¯1| 2 + |*ψ*¯0| 2 2 by Equation (103) and *Z* ≡ 2|*ψ*¯1| 2 − | *ψ*¯0| 2. We also find the following conservation law,

$$\frac{\partial}{\partial \mathbf{x}^0} \left[ \frac{1}{2} \epsilon^2 + \frac{1}{2} k^0 Z \right] = 0,\tag{112}$$

which represents the energy conservation. By this relation, we might be able to estimate the maximum energy density of electric fields,

$$\left(\frac{1}{2}\epsilon^2\right)\_{\text{max}} = -\frac{1}{2}k^0 Z\_{\text{min}} = \frac{1}{2}k^0 B\_\prime \tag{113}$$

in case there is no external energy supply. We derive the following solutions in Equations (108)–(110),

$$R(\mathbf{x}^0) = \frac{1}{\sqrt{2}} B \sin \theta(\mathbf{x}^0), \ Z(\mathbf{x}^0) = B \cos \theta(\mathbf{x}^0), \tag{114}$$

$$
\theta(\mathbf{x}^0) = \theta\_0 + \sqrt{2}cd\_\varepsilon \int\_0^{\mathbf{x}^0} d\mathbf{x}'^0 \varepsilon(\mathbf{x}'^0),\tag{115}
$$

with *∂θ ∂x*<sup>0</sup> = √2*ede* and the constant *B* in a similar way to [71]. The *<sup>θ</sup>*(*x*<sup>0</sup>) swings around the position *θ* = *π* with the frequency Ω = *ede* √ *k*0*B* in case we start with initial conditions at around *θ*0 ∼ *π* (|*ψ*¯1| 2 = 0), since we can rewrite Equation (108) as

$$\frac{\partial^2 \theta(\mathbf{x}^0)}{\partial (\mathbf{x}^0)^2} = (cd\_\epsilon)^2 k^0 B \sin \theta(\mathbf{x}^0). \tag{116}$$

The *B* is the order of the number density of dipoles.

We introduce the damping term 1 *L* for the release of radiation and the propagation length *L* in Equation (108). We can write,

$$\frac{\partial \epsilon}{\partial \mathbf{x}^0} + \frac{1}{L} \epsilon = \frac{cd\_c k^0}{\sqrt{2}} B \sin \theta(\mathbf{x}^0). \tag{117}$$

In *κ* = 1 *L* time derivative, we can neglect the first term in the above equations, then

$$\frac{\partial \theta}{\partial \mathbf{x}^0} = \frac{(ed\_\varepsilon)^2 k^0 B}{\kappa} \sin \theta(\mathbf{x}^0). \tag{118}$$

The solution is,

$$\theta(\mathbf{x}^0) = 2 \tan^{-1} \left[ \exp \left( \frac{(cd\_\epsilon)^2 k^0 B \mathbf{x}^0}{\kappa} \right) \tan \frac{\theta\_0}{2} \right],\tag{119}$$

and,

$$\epsilon = \frac{1}{\sqrt{2}ed\_{\epsilon}\tau\_{R}} \times \left[ \cosh\left(\frac{\mathbf{x}^{0} - \tau\_{0}}{\tau\_{R}}\right) \right]^{-1} \tag{120}$$

with *τR* = *κ* (*ede*)<sup>2</sup>*k*0*B* and *τ*0 = −*τR* ln(tan *θ*0 2 ). The *τR* ∝ 1/*B* ∼ 1/*N* with the number of dipoles *N* represents the relaxation time of electric fields in the super-radiance. When *N* dipoles decay within time scales 1/*N*, the intensity of electric fields becomes the order *N*<sup>2</sup> (super-radiant decay with correlation among dipoles), not *N* (spontaneous decay without correlation among dipoles).

#### *5.2. Higgs Mechanism and Tachyonic Instability*

In this section, we rewrite time evolution equations for coherent fields with only real functions. We assume the spatially homogeneous case. We do not adopt the rotating wave approximation in this section. We show how coherent electric fields *E*1 are affected by *Z* = 2|*ψ*¯1|<sup>2</sup> − |*ψ*¯0|2.

In Equation (101), the second derivatives of coherent fields on the right-hand side are written by,

$$\frac{\varepsilon d\_\varepsilon}{2I^2} \left( \bar{\psi}\_1^\* \bar{\psi} + \bar{\psi}\_0^\* \bar{\psi}\_1 \right) + \frac{2(\varepsilon d\_\varepsilon)^2 Z}{I} E\_{1\prime}$$

where we have used Equation (100). As a result, we arrive at,

$$\begin{split} \left[ (\partial\_{0})^{2} - (\partial\_{2})^{2} - \frac{2(ed\_{\varepsilon})^{2}Z}{I} \right] E\_{1} &= \frac{\mu\_{1}}{4I^{2}} + \frac{2(ed\_{\varepsilon})^{2}E\_{1}}{I} \int\_{p} (2F\_{11}(\mathbf{X}, p) - F\_{00}(\mathbf{X}, p) - \Delta\_{\mathbf{g},11,\mathbf{F}}(\mathbf{X}, p)) \\ &+ \frac{(ed\_{\varepsilon})^{2}}{I^{2}} E\_{1} \int\_{p} \left( \mathrm{Re}\Delta\_{\mathbf{g},11,\mathbf{F}}(\mathbf{X}, p) F\_{00}(\mathbf{X}, p) + \Delta\_{\mathbf{g},11,\mathbf{F}}(\mathbf{X}, p) \mathrm{Re}\Delta\_{00,\mathbf{R}}(\mathbf{X}, p) \right) \\ &+ \frac{(ed\_{\varepsilon})^{2}}{2I^{2}} \frac{\partial E\_{1}}{\partial X^{0}} \int\_{p} \left( \frac{\partial F\_{00}}{\partial p^{0}} \frac{\Delta\_{\mathbf{g},11,\mathbf{F}}}{i} + \frac{\rho\_{00}}{i} \frac{\partial \Delta\_{\mathbf{g},11,\mathbf{F}}}{\partial p^{0}} \right) + \frac{(ed\_{\varepsilon})^{2}}{4I^{2}} E\_{1} \frac{\partial}{\partial X^{0}} \int\_{p} \left( \frac{\partial F\_{00}}{\partial p^{0}} \frac{\Delta\_{\mathbf{g},11,\mathbf{F}}}{i} + \frac{\rho\_{00}}{i} \frac{\partial \Delta\_{\mathbf{g},11,\mathbf{F}}}{\partial p^{0}} \right), \tag{121} \end{split}$$

with the *x*1 direction of the dipole moment (density) given by *μ*1 = 2*ede ψ*¯<sup>∗</sup>1*ψ*¯0 + *ψ*¯<sup>∗</sup>0*ψ*¯1, *<sup>F</sup>*11(*<sup>X</sup>*, *p*) = <sup>Δ</sup>2111(*<sup>X</sup>*,*p*)+Δ1211(*<sup>X</sup>*,*p*) 2 , *<sup>F</sup>*00(*<sup>X</sup>*, *p*) = <sup>Δ</sup>2100(*<sup>X</sup>*,*p*)+Δ1200(*<sup>X</sup>*,*p*) 2 and <sup>Δ</sup>*g*,11,*F*(*<sup>X</sup>*, *p*) = <sup>Δ</sup>21*g*,<sup>11</sup>(*<sup>X</sup>*,*p*)+Δ12*g*,<sup>11</sup>(*<sup>X</sup>*,*p*) 2 . In the Appendix A we have shown the detailed derivation for the second, third, fourth and fifth terms in the above equations. We have assumed the self-energy Σ00 = Σ11 = 0 in deriving the time derivatives of Δ10 and Δ01 in Equation (101). Even if we include contributions of self-energy in Equation (121), they are higher order *O* (*ede*)<sup>4</sup> in the coupling expansion. We have neglected higher order terms in the gradient expansion for quantum fluctuations. In Equation (121), we leave the −(*∂*2)<sup>2</sup>*E*<sup>1</sup> term on the left-hand side in the above equation to compare with the sign of −<sup>2</sup>(*ede*)<sup>2</sup>*<sup>Z</sup> I E*1 term. We find the Higgs mechanism with the mass squared −<sup>2</sup>(*ede*)<sup>2</sup>*<sup>Z</sup> I* in the case of the normal population *Z* = 2|*ψ*¯1|<sup>2</sup> − |*ψ*¯0|<sup>2</sup> < 0. On the other hand, the tachyonic instability appears in the inverted population *Z* > 0 in the above equation. Then the electric field *E*1 will increase exponentially until *Z* becomes negative. In Equation (121), the second term on the right-hand side is proportional to 2*F*11 − *F*00 − <sup>Δ</sup>*g*,11,*F*. Near equilibrium states, we might find *F*00 > 2*F*11 − <sup>Δ</sup>*g*,11,*F*, where statistical functions *F*11, *F*00 and <sup>Δ</sup>*g*,11,*<sup>F</sup>* are proportional to the Bose–Einstein distribution 1 *ep*0/*<sup>T</sup>*−<sup>1</sup> plus 12 (with the Kadanoff–Baym ansatz) with different dispersion relations *p*0 ∼ **p**2 2*m* for *F*00 and *p*0 ∼ **p**2 2*m* + 12*I* for *F*11 and <sup>Δ</sup>*g*,11,*F*, due to the energy difference 1 2*I* − 0 2*I* between the ground state and first excited states. So the 2*F*11 − *F*00 − <sup>Δ</sup>*g*,11,*<sup>F</sup>* in the second term is negative near the equilibrium states, which might mean no tachyonic unstable terms appear from quantum fluctuations near equilibrium states. The contributions of quantum fluctuations on the right-hand side written by statistical functions (second, third, fourth and fifth terms) vanish at zero temperature *T* = 0. Quantum fluctuations represent finite temperature effects at equilibrium states, although we need not restrict ourselves to only the equilibrium case. We have shown general contributions of quantum fluctuations in both equilibrium and non-equilibrium case in this paper.

Finally we shall consider remaining equations for coherent dipole fields. By using Equations (99) and (100) and the definitions of real functions *μ*1 = <sup>2</sup>*ede*(*ψ*¯<sup>∗</sup>1*ψ*¯0 + *ψ*¯<sup>∗</sup>0*ψ*¯1), *P* = *iede*(*ψ*¯<sup>∗</sup>1*ψ*¯0 − *ψ*¯<sup>∗</sup>0*ψ*¯1) and *Z* = 2|*ψ*¯1|<sup>2</sup> − |*ψ*¯0|2, we can also derive,

$$
\partial\_0 Z \quad = \ \text{4E}\_1 \text{P}\_2 \tag{122}
$$

$$
\partial\_0 \mu\_1 \quad = \quad \frac{P}{I} \tag{123}
$$

$$
\partial\_0 P \quad = \quad -\frac{\mu\_1}{4I} - 2(ed\_\ell)^2 E\_1 Z. \tag{124}
$$

We can show *∂*0(2|*ψ*¯1| 2 + |*ψ*¯0| 2) = 0 by using these three equations. In these equations with initial conditions *E*1 > 0, *Z* > 0 (inverted population), *P* = 0 and *μ*1 = 0, the *P* and the *μ*1 decrease at around the initial time and *Z* starts to decrease due to *E*1*P* < 0. In initial conditions *E*1 > 0, *Z* < 0 (normal population), *P* = 0 and *μ*1 = 0, the *P* and the *μ*1 increase at around the initial time and *Z* starts to increase due to *E*1*P* > 0. The absolute values of *Z* decrease at around the initial time. We find that there is no term of quantum fluctuations in Equations (122)–(124).

We can solve Equations (121)–(124) with real functions in this section and the Kadanoff–Baym equations with real statistical functions and pure imaginary spectral functions in Section 4, simultaneously.

## **6. Discussion**

In this paper, we have derived time evolution equations, namely the Klein–Gordon equations for coherent photon fields, the Schrödinger-like equations for coherent electric dipole fields and the Kadanoff–Baym equations for quantum fluctuations, starting with the Lagrangian in quantum electrodynamics with electric dipoles in 2 + 1 dimensions. We have adopted the two-particle-irreducible effective action technique with the leading-order self-energy of the coupling expansion. We find that electric dipoles change their angular momenta due to coherent electric fields *E*1 ± *iαE*2 with *α* = ±1. They also change momenta and angular momenta by scattering with incoherent photons. The proof of H-theorem is possible for these processes as shown in Section 3. Our analysis provides the dynamics of both the order parameters with coherent fields and quantum fluctuations for incoherent particles.

In Section 2, we adopt two-energy level approximation for the angular momenta of dipoles. Then, we find that the *i***Δ**−<sup>1</sup> 0 is written by 3 × 3 matrix with zero (−1, 1) and (1, −<sup>1</sup>) components. The form of the matrix is similar to 3 × 3 matrix in the analysis in open systems, the central region, left and right reservoirs as in [59,61–63]. Hence we can simplify the Kadanoff–Baym equations for dipole fields in an isolated system with the same procedures as those in open systems. The difference between QED with dipoles and *φ*4 theory in open systems is that the coherent electric field changes the momenta of dipoles when the phase *αζ* in *E*1 ± *iαE*2 with *α* = ±1 is dependent on space–time. The space dependence of coherent electric fields might disappear in the time evolution due to the instability by the lower entropy of the system, then electric fields will change angular momenta of dipoles but not change momenta *p* due to *∂ζ* = 0. We can also trace the dynamics with *∂ζ* = 0. By setting the initial conditions with the symmetry *α* → <sup>−</sup>*α*, namely *ψ*¯(∗) *α* = *ψ*¯(∗) −*α*, Δ*α*<sup>0</sup> = Δ−*<sup>α</sup>*<sup>0</sup> and Δ0*α* = Δ0−*α*, with initial conditions *E*2 = 0 and *∂*0*E*2 = 0 in spatially homogeneous systems in *∂νFν*<sup>2</sup> = *J*2 in Equation (20), we can show *E*2 = 0 at any time. Then we can use *∂ζ* = 0. This condition simplifies numerical simulations in the Kadanoff–Baym equations since we need not estimate the momentum shift *p* → *p* ± *α∂ζ* in the finite-size lattice for the momentum space. As a result, the simulations for Kadanoff–Baym equations for dipoles and photons will be similar to those in QED with charged bosons in [72].

In Section 3, we have introduced a kinetic entropy current and shown the H-theorem in the leading-order of the coupling expansion with *ede*. This entropy approaches the Boltzmann entropy in the limit of zero spectral width as in [58]. The mode-coupling processes between dipoles and photons produce entropy. When there are deviations between (00) and (*αα*) components of distribution functions, entropy production occurs. Entropy production stops when the Bose–Einstein distribution is realized in the dynamics of Kadanoff–Baym equations.

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

We can also derive the energy shifts in dispersion relations due to nonzero electric fields by using the retarded Green's functions in Section 3. The 0th order equations for retarded Green's functions are given by,

$$\left(p^0 - \frac{\mathbf{p}^2}{2m} + 2(ed\_\epsilon)^2 E\_1^2 \Lambda\_{\S, 11, R}\right) \Delta\_{00, R} = -1,\tag{125}$$

$$\left(p^0 - \frac{\mathbf{p}^2}{2m} - \frac{1}{2I}\right)\Delta\_{11,R} + (ed\_\varepsilon)^2 E\_1^2 \Delta\_{00,R} \Delta\_{\mathfrak{g},11,R} = -1,\tag{126}$$

with <sup>Δ</sup>*g*,11,*<sup>R</sup>* = −1 *<sup>p</sup>*0− **p**2 2*m* − 12*I* . Multiply *p*0 − **p**2 2*m* − 12*I* , take the imaginary parts in the above equations and remember the imaginary parts of retarded Green's functions are the spectral functions, then we find,

$$\begin{array}{rcl} \mathcal{W}\left[\begin{array}{c} \rho\_{00} \\ \rho\_{11} \end{array}\right] &=& 0, \text{ with,} \\\\ \mathcal{W}\left[\begin{array}{c} \left(p^{0}-\frac{\mathbf{p}^{2}}{2\overline{m}}-\frac{1}{2\overline{l}}\right)\left(p^{0}-\frac{\mathbf{p}^{2}}{2\overline{m}}\right)-2(ed\_{\varepsilon})^{2}E\_{1}^{2} & \text{0} \\\\ -(ed\_{\varepsilon})^{2}E\_{1}^{2} & \left(p^{0}-\frac{\mathbf{p}^{2}}{2\overline{m}}-\frac{1}{2\overline{l}}\right)^{2} \end{array} \right] & \end{array} \tag{127}$$

By setting determinant |*W*| to be zero, we find the following solutions for dispersion relations,

$$p^0 = \frac{\mathbf{p}^2}{2m} + \frac{1}{4I} \pm \frac{1}{2}\sqrt{\frac{1}{4I^2} + 8(cd\_\epsilon)^2 E\_1}^2. \tag{128}$$

Here we assumed the symmetry for *α* = ±1 for Green's functions and zero self-energy Σ00 = Σ11 = 0. We find how electric fields shift two energy levels 0 and 12*I* . The above energy shift is similar to the energy shift given in [27] in 3 + 1 dimensions due to nonzero electric fields.

In Section 5.1, we have derived the super-radiance from time evolution equations for coherent fields. We find that it is possible to derive the Maxwell–Bloch equations from our Lagrangian with the probability conservation law and the energy conservation law. Super-radiant decay with intensity of the order ∝ *N*<sup>2</sup> (*N*: The number of dipoles) appears in a similar way to [70,71]. It is possible to derive the maximum energy of electric fields by use of Equation (113). We know that the moment of inertia of water molecule is *I* = 2*mHR*<sup>2</sup> with *mH* = 940 MeV with *R* = 0.96 × 10−<sup>10</sup> m. Hence the *k*0 = 12*I*= 1.1 × 10−<sup>3</sup> eV. Since *B* = *NV*= 3.3 × 10<sup>28</sup> /m<sup>3</sup> for liquid water, we find

$$\frac{1}{2}\epsilon\_{\text{max}}^2 = \frac{1}{2}k^0B = 1.8 \times 10^{25} \,\text{eV/m}^3. \tag{129}$$

When we multiply the volume of all microtubules (MTs) in a brain,

*V*MT = *π* × 15nm<sup>2</sup> × 1000nm × 2000 MTs/neuron × 10<sup>11</sup> neurons/brain = 1.4 × 10−<sup>7</sup> m3, (130)

we can arrive at,

$$
\frac{1}{2} \epsilon\_{\text{max}}^2 V\_{\text{MT}} = 0.41 \text{ J} = 0.1 \text{ cal.} \tag{131}
$$

If we maintain our brain 100 s without energy supply, we need at least 0.1 × 10−<sup>2</sup> cal/s or 86 cal/day to maintain the ordered states of memory. We can compare 86 cal/day with 4000 cal/day = 2000 kcal/day × 0.2 (energy consumption rate of brain) × 0.01 (energy rate to maintain the ordered system). The 86 cal/day is within the 4000 cal/day, which is consistent with our experiences. In this derivation, we have used coefficients in 2 + 1 dimensions and the number density of water molecules in 3 + 1 dimensions.

In Section 5.2, we have derived time evolution equations for electric field *E*1. The Higgs mechanism appears in this equation in normal population *Z* < 0. As a result, the dynamical mass generation occurs with the maximum mass <sup>Ω</sup>Higgs = 2*ede* √ *k*0*B* = 30*k*<sup>0</sup> where the number density of dipoles is *B* = 2|*ψ*¯1| 2 + |*ψ*¯0| 2 = *N V* . The period is 2 *<sup>π</sup>*/ΩHiggs = 1.3 × 10−<sup>13</sup> s. In normal population *Z* < 0, the Meissner effect appears with the penetrating length 1/ΩHiggs = 6.3 μm. On the other hand, the tachyonic instability occurs in inverted population *Z* > 0. The electric field *E*1 increases exponentially with exp(<sup>Ω</sup> *X*<sup>0</sup>) (with Ω ≤ Ωmax) where the time scale is 1/Ωmax = 2.1 × 10−<sup>14</sup> s with Ωmax = <sup>Ω</sup>Higgs. Due to energy conservation, since *Z* decreases as the absolute value of the electric field increases, tachyonic instability stops in *Z* < 0.

We have prepared for numerical simulations with time evolution equations, namely the Schödinger-like equations for coherent electric dipole fields, the Klein–Gordon equations for coherent electric fields and the Kadanoff–Baym equations for quantum fluctuations. Our simulations might describe the dynamics towards equilibrium states for quantum fluctuations and the dynamics of super-radiant states for coherent fields. Our analysis is also extended to simulations in open systems by preparing the left and the right reservoirs like those in [59] or networks [73].
