Non-Equilibrium Quantum Brain Dynamics: Water Coupled with Phonons and Photons

Abstract

We investigate Quantum Electrodynamics (QED) of water coupled with sound and light, namely Quantum Brain Dynamics (QBD) of water, phonons and photons. We provide phonon degrees of freedom as additional quanta in the framework of QBD in this paper. We begin with the Lagrangian density QED with non-relativistic charged bosons, photons and phonons, and derive time-evolution equations of coherent fields and Kadanoff–Baym (KB) equations for incoherent particles. We next show an acoustic super-radiance solution in our model. We also introduce a kinetic entropy current in KB equations in 1st order approximation in the gradient expansion and show the H-theorem for self-energy in Hartree–Fock approximation. We finally derive conserved number density of charged bosons and conserved energy density in spatially homogeneous system.

1. Introduction

The physical mechanism of memory involving our subjective experience in a brain is still an open question. According to generally accepted understanding in neuroscience [1], learning and memory are both linked to synaptic plasticity, which are represented at a neuronal level by long-term potentiation (LTP). However, there is a logical gap in this type of explanation because while synaptic membrane components are transient, memories of events and experience can last many years, even a lifetime. Therefore, a mechanism behind memory-related synaptic activity must be physically converted into a more enduring form, e.g., at a molecular level within post-synaptic dendritic spines, shafts and cell bodies. A specific memory code was proposed in a computational paper [2] to involve phosphorylation of neuronal microtubules (MTs) using the CaMKII enzyme. This is inspired by a mechanism present in communication technology, whereby a code converts information from one form of representation to another. The CaMKII-MT interaction was hypothesized to provide such a biomolecular code for memory in brain neurons and a conclusion was drawn that the information capacity of neuronal MTs is enormous and can easily explain the number of bits of information that could encode all sensory input data over a human lifetime. However, there could be an even more fundamental physical mechanism of information encoding in the human brain that could surpass that proposed by Craddock et al. and can act at the quantum level [3], which would also be of fundamental importance to our understanding of higher cognitive functions such as qualia [4]. The phrase ‘higher cognitive functions’ is meant to represent an evolutionary development not specific human activities. Qualia are universal but subjective instances of conscious experience such as the perception of redness of an apple or bitterness of the taste of an espresso. It is generally argued that qualia are perceptions of the human mind whose physics-based explanation so far has eluded scientists and hence has been termed the hard problem of consciousness by Chalrmers [5]. Currently, the most generally accepted view of memory encoding is by long-term potentiation of synaptic connections being strengthened when activated by a signal being transmitted between neurons [6]. The hypothesis formulated by Craddock et al delves deeper into this issue by proposing a specific encoding of information in terms of phosphorylation patterns on neuronal microtubule surfaces implemented by the enzyme CaMKII which has been experimentally validated as an active motor protein involved during the process of memory formation [7]. Here, as first suggested by Umezawa et al. [8], we explore the possibility of an even more fundamental approach to memory formation in terms of quantum fields generated by the brain. To the best of our knowledge, there is no other specific physics-based theory of memory encoding which would explore a fundamental mechanism at a quantum level with a possible exception of OrchOR [9] which links it to quantum wave function collapse due to gravitational self-interactions involving tubulin units of microtubules. Our model can be viewed as somewhat related to this perspective since we consider microtubules as the fundamental substrates of the quantum fields but we also include phonons and photons as well as the water environment, all of which makes it a more realistic setting for this physical model. It is important to note in this connection that several properties of memory in a brain different from computer memory are observed [10], namely sequential patterns, auto-associative recalling, storage in a hierarchy, memorized patterns in an invariant form. Memory is diffused in a whole brain, as a result memory is robust against lesions in a brain [11,12]. To explain these features of memory, Pribram proposed the holographic brain theory as a candidate of theory of memory and perception [13,14]. Holography, a technique to record 3-dimensional information on medium invented by Gabor [15], can describe various properties of memory in a brain as listed above. Jibu and Yasue, who collaborated with Pribram, studied Quantum Brain Dynamics (QBD), that is Quantum Field Theory (QFT) of the brain involving water electric dipole fields and photon fields [16], known as Jibu–Yasue approach distinguished from Penrose–Hameroff approach [17].

Quantum Field Theory provides a fundamental approach to describe the nature [18]. It is applied to a variety of phenomena in elementally particle physics, nuclear physics, cosmology, condensed matter physics and furthermore biology. QFT is distinguished from Quantum Mechanics since QFT describes both macroscopic matter in classical mechanics and microscopic degrees of freedom in Quantum Mechanics. QFT approach to memory in a brain is originated with the monumental work by Ricciardi and Umezawa in 1967 [19] by adopting the concept of spontaneous breakdown of symmetry (SBS). Memory is represented by vacua emerging in SBS, or macroscopic ordered patterns described in the framework of QFT. The theory is further developed by Stuart et al. [8,20], namely non-local memory storage, stability of vacua for long-term memory, metastable states for short-term memory, and memory recalling mechanism due to excitation of Nambu–Goldstone quanta emerging in SBS, which are described in Takahashi model. They proposed quantum degrees of freedom, corticons and exhange bosons, to describe quantum phenomena in a brain. What concrete degrees of freedom for corticons and exchange bosons are was not given in this stage. In 1968, the Bose–Einstein condensation in biological systems involving coherence with long-range correlations is proposed by Fröhlich, referred as Fröhlich condensation [21,22]. In 1976, Davydov and Kislukha proposed a soliton solution propagating along the alpha-helix structures of protein chains and DNA, called the Davydov soliton [23]. The Fröhlich condensation and the Davydov soliton solution emerge as static and dynamical properties, respectively, in a non-linear Schrödinger equation with an equivalent quantum Hamiltonian [24]. Around the same time, Pribram proposed the holographic brain theory [13,14]. Del Giudice et al. proposed to apply QFT to biological systems in 1980s [25,26,27,28,29]. In 1990s, Jibu and Yasue, collaborators of Pribram, proposed quantum concrete degrees of freedom in a brain, namely water rotational dipole fields and photon fields [16,30,31,32,33,34,35,36]. When water dipoles are aligned in the same direction, the spontaneous breakdown of rotational symmetry occurs and new vacua in SBS representing memory storage in a brain emerges. Excitation of incoherent photons on the vacua represents memory recalling in this theory. Vitiello showed that a huge memory capacity is achieved in the vacua by regarding the brain as an open system and adopting two-mode squeezed coherent states for Nambu–Goldstone bosons [37]. Vitiello also showed that squeezed coherent states are isomorphic to fractals in the nature [38,39]. For example, trajectories on logarithmic spirals are written by both damped and amplified oscillators with the closed system Lagrangian which is transformed to the Hermite Hamiltonian. In imposing the quantization on positions and momenta, and rewriting creation and annihilation operators of particles, the quantum states are described by two-mode squeezed coherent states in time-evolution. Quantum states composed of Nambu–Goldstone bosons in an open system evolve among squeezed coherent states corresponding to fractal. Holographic approach is also proposed by adopting coherent waves of super-radiance [33,40]. Combining QBD and holography, we can describe holographic memory storage in QBD involving properties of diversity, non-locality, stability, sequential patterns, auto-associative recalling, storage in a hierarchy, and memorized patterns in an invariant form [40].

Whether or not our brain can be described by the language of holography will be investigated by external stimuli to brain. Beauchamp et al. have shown a recent experiment using invasive stimulation to change our visual subjective experience [41]. We can also propose non-invasive approach. Non-invasive stimulation to our brain has been proposed for several decades [42], starting with transcranial magnetic stimulation (TMS) by Barker [43]. There are several non-invasive stimulation methods, such as transcranial electric stimulation with direct current [44] and alternating current [45], photonic approach involving near-infrared photons [46,47], and ultrasound approach [48,49,50] applied to treat neuropsychiatric diseases. The ultrasound-mediated drug delivery and biomarker release are also proposed as the ultrasound therapy [51]. The ultrasound therapy is adopted to destruct tumors and generate anti-tumor immune responses. When we adopt ultrasound approach, we need QFT of phonons [52,53,54,55,56] which should be extended for non-invasive neural stimulation of water-phonon-photon systems. We will further adopt reservoir computing or morphological computation [57,58,59] as a control theory of holograms via ultrasound by developing QFT in a hierarchy representing multiple layers, such as scalp, skull, dura, celebrospinal fluid covering our neocortex.

We aim to provide a theoretical formulation of water coupled with phonons and photons in QBD. This paper provides the extension of our previous approaches to QBD involving phonons, quanta of sound. We can trace a full dynamics of coherent sound fields and incoherent phonons in a brain, where sound fields decay to incoherent phonons in non-equilibrium QFT approach, for example. First we introduce the Lagrangian density of charged bosons as water degrees of freedom, phonons and photons. We refer to Keppler’s approach to describe glutamate by charged Bose fields [60]. We describe water degrees of freedom by charged Bose fields. We also refer to [61] to represent phonon fields. Time-evolution equations of coherent fields and quantum fluctuations are derived in this framework. Next we derive an acoustic super-radiance solution in our model by assuming coherence inside a microtubule [27]. Green’s functions for quantum fluctuations obey the Kadanoff–Baym equations [62,63,64] for charged bosons, phonons and photons. We also introduce a kinetic entropy current for incoherent charged bosons (water), phonons and photons in 1st order approximation in the gradient expansion, and provide a proof of H-theorem for Hartree–Fock approximation of interaction. Finally we provide time-evolution equations in spatially homogeneous systems and show concrete forms of conserved charge and energy. Our theory will be extended to the control theory of holograms in QBD using external ultrasound waves. Using reservoir computing or morphological computation theory, we can develop a non-invasive method to manipulate our subjective experiences and check whether or not our brain adopts the language of holography.

This paper is organized as follows. In Section 2, we provide background and motivation of application of QFT to biology, especially a brain. In Section 3, we introduce a Lagrangian density of QBD with phonons and show 2-Particle-Irreducible effective action. In Section 4, we show an acoustic super-radiance solution in this model. In Section 5, we show a summary for a kinetic entropy current in 1st order approximation in the gradient expansion and the H-theorem for self-energy in Hartree–Fock approximation. In Section 6, we write time-evolution equations in spatially homogeneous system and show conserved charge and energy density. In Section 7, we discuss our results. In Section 8, concluding remarks and perspectives are provided. The natural unit with the light speed, the Planck constant ℏ and Boltzmann constant set to be 1 is adopted. The metric tensor is ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ with space-time subscript $\mu ,\nu =0,1,2,3$ and spatial subscript $i,j,k=1,2,3$.

2. Quantum Field Theory to Brain

In this section, we provide background and motivation to apply Quantum Field Theory (QFT) to a brain.

In conventional neuroscience, the synaptic plasticity between neurons describes memory in a brain. However, the synaptic plasticity is transient, and synaptic connections change in a few days [65,66]. In addition, this type of memory might not be robust against lesions in a brain since memory will be lost when one of the synaptic connections is destructed. If we adopt the synaptic plasticity, we cannot explain memory in a single cell organism [67]. We then need cytoskeletons inside cells to explain the mechanism of memory and information processing in a single cell organism. One of the candidates in cytoskeletons corresponds to microtubules. Phosphorylation in microtubules in [2] can be described as bits, which can be enhance the memory capacity in a brain compared with that of the synaptic plasticity.

However, the phosphorylation in microtubules appears in local events. We require diffused non-local features of memory in a brain. Non-locality of memory is suggested by Lashley [12] using lesions of brains of rats. Even if parts of the brain are damaged, rats can perform tasks. We require the processes to extend microscopic events to macroscopic features of the size of the brain. We might need tryptophan mega-networks for macroscopic events [68]. Or, we can also consider water molecules coupled with photons and phonons covering the whole brain, in which local memories in phosphorylation of microtubules are converted to optical information, such as holography. We then need coherent light (and sound) to achieve interference patterns in holography. Quantum Brain Dynamics (QBD) adopts water degrees of freedom and photons, and suggests that the new vacua emerging in spontaneous symmetry breaking where water dipoles are all aligned in the same direction. As diamond crystals and magnets emerging in breakdown of symmetry are stable at a room temperature, aligned water dipoles might be stable. The Exclusion Zone (EZ) water around hydrophilic surfaces is discovered in experiments [69,70,71]. The EZ water corresponds to the coherent water as suggested by Del Giudice et al. [72]. The size of domains composed of the EZ water is the order of $50\mathsf{\mu }\mathrm{m}$. This value corresponds to the inverse of energy difference $1/I=2/2I-0/2I=4$ meV (I: the moment of inertia of a water molecule) between excited states of rotational motion with angular momentum squared $=2$ and the ground state with angular momentum zero, which suggests the significance of the molecular orientations of water. The EZ water with the size $50\mathsf{\mu }\mathrm{m}$ might emerge around hydrophilic surfaces in and around neurons.

When we adopt the new vacua for aligned dipoles, we can propose holographic approach in the framework of QBD [40]. Holography adopts recording of interference patterns of two incident coherent lights imposed with different angles on holograms. The holograms need not to be patterns composed of curves for interference of waves, such as sine curves. We can also use binary patterns with small and large transmittance, called binary holograms [73,74,75]. Holographic memory storage with two patterns of aligned dipoles in coherent domains and water dipoles in the random orientations can be used as binary holograms. Both light or sound holography are possible with changing transmittance of light or sound for water system. These binary holograms with stable vacua involving aligned dipoles and states of random dipoles will be more stable than interference patterns with sine curves. Holographic memories are robust against damages of parts, which suggests that the whole image will be reconstructed by undamaged parts in holograms. Memory retrieval is achieved by imposing photons (or phonons for sound holography) on the holograms with the same angle as coherent lights (or sounds) in recording. Finite number of excitations of photons cannot break the vacua of aligned dipoles [18]. Human memories are modified by recalling and thinking. To rewrite the holograms involving the vacua of aligned dipoles, we require to impose coherent light or sound fields involving condensation of an infinite number of photons or phonons on holograms. Or we need to impose coherent lights or sounds in the different angles on holograms where multiple memories are recorded in the same recording media composed by water molecules. Changing the angles of incident photons or coherent light, our memory retrieval will be modified.

Tegmark suggested that coherence cannot be maintained in physiological conditions [76]. However, his expected value for coherent time is extremely short $\sim {10}^{-20}\mathrm{s}$. The problem of his analysis is first to adopt the mass of a water molecule as $\sim 18\times 940\mathrm{MeV}$. To investigate the macroscopic vacua of aligned water dipoles in QBD, we need to adopt the inverse of the moment of inertia $\sim 4\mathrm{meV}$, or the mass of polaritons emerging in water fields and photon fields. Furthermore, time scales for decoherence are divided by the number of surrounding Na ions ${10}^6$, which is a strange procedure. He lacks the idea that the brain is an open system, with continuous energy supply to physiological system. In the Fröhlich model, the physical system connected with an energy supply and a heat bath is considered as significant component [24]. The Fröhlich condensates might emerge in microtubules [77,78]. In the open system, the balance of decoherence and error corrections for quantum coherence can be used to maintain coherence of the physical system with continuous flow of external energy. His estimations lack these types of analysis, as a result the his estimated value of the coherence times are unrealistically small. His criticism of the quantum coherence in microtubules has been strongly rebutted by Hagan et al. [79]. Moreover, in a recent experimental paper [80], evidence was presented for the presence of long-lived (5 ns) collective quantum excitations in microtubules.

In the analysis for quantum systems, we can adopt Schrödinger equations in quantum mechanics for the atomic and molecular system. We then need to describe absorption and emission of photons and phonons. Cooperative behaviors of many-body systems for atoms and molecules can be described by QFT involving Gross–Pitaevskii equation extended from the one-body Schrödinger equation. The QFT approach includes both microscopic quantum mechanical degrees of freedom and macroscopic cooperative behaviors of quanta. We consider effective charges for molecules involving their polarization and dipole moment [60], where we can also adopt vanishing charge. Our approach includes any effective charges where we can change several charge parameters for quantum states of water molecules, such as rotations, stretching, vibrations, and so on. These water molecules are coupled with photons and phonons, or cooperatively coherent light and sound. We especially describe aspects of sound with quanta, phonons in this paper. This paper is the extension of our QBD approach with including phonons. Our approach adopts both coherent photon and phonon fields and their quantum fluctuations which are required for a full understanding the Fröhlich condensate [81]. Furthermore non-equilibrium collective behaviors for phonon condensates described in [82] are also included in our approach.

3. Lagrangian Density and 2-Particle-Irreducible Effective Action

In this section, we provide a Lagrangian density for charged bosons, phonons and photons, show 2-Particle-Irreducible Effective Action and derive time-evolution equations. We adopt the background field method [83,84,85,86]. Using the Lagrangian, we can derive the Gross–Pitaevskii equation as extension of Schrödinger equation to describe collective properties of water molecules. The equation describes various quantum states of water molecules, including conformational states related with strongly bonded or less bonded states observed in ranges of wavelength from $1300\mathrm{nm}$ to $1600\mathrm{nm}$ as shown in [87], and rotational degrees of freedom in wavelength $310\mathsf{\mu }\mathrm{m}$, for example.

The Lagrangian density is given by,

$$\begin{array}{ccc}\hfill \mathcal{L}& =& -{\displaystyle \frac{1}{4}}{\mathcal{F}}^{\mu \nu }[A+a]{\mathcal{F}}_{\mu \nu }[A+a]-{\displaystyle \frac{1}{2\xi }}{\left({\partial }^{\mu }{a}_{\mu }\right)}^2\hfill \\ & & +{\psi }^{*}\left(i{\displaystyle \frac{\partial }{\partial x^0}}+e(A^0+a^0)+{\displaystyle \frac{{\left({\nabla }_i-ie(A_i+a_i)\right)}^2}{2m}}\right)\psi \hfill \\ & & +{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial Q_a^i}{\partial x^0}}\right)}^2-v_{2,ij}\left({\displaystyle \frac{\partial Q_a^i}{\partial x^k}}\right)\left({\displaystyle \frac{\partial Q_a^j}{\partial x^k}}\right)\right]+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial Q_o^i}{\partial x^0}}\right)}^2-{\Omega }_{2,ij}{Q}_o^i{Q}_o^j\right]\hfill \\ & & -g_{aL}{\psi }^{*}\psi {\nabla }_i{Q}_{aL}^i+i{g}_{aT}\left({\psi }^{*}\left({\partial }_i\psi \right)-\left({\partial }_i{\psi }^{*}\right)\psi -2ie{\psi }^{*}\psi (A_i+a_i)\right)Q_{aT}^i\hfill \\ & & -g_{oL}{\psi }^{*}\psi {\nabla }_i{Q}_{oL}^i+i{g}_{oT}\left({\psi }^{*}\left({\partial }_i\psi \right)-\left({\partial }_i{\psi }^{*}\right)\psi -2ie{\psi }^{*}\psi (A_i+a_i)\right)Q_{oT}^i,\hfill \end{array}$$

(1)

with electromagnetic tensor ${\mathcal{F}}_{\mu \nu }\left[A\right]={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ with background photon fields $A_{\mu }$ and quantum fluctuations $a_{\mu }$, complex charged Bose fields ${\psi }^{*}$ and $\psi$, acoustic phonon fields $Q_a^i=-Q_{a,i}$, optical phonon fields $Q_o^i=-Q_{o,i}$, gauge fixing parameter $\xi$, elementary charge e, sound velocity matrix $v_{2,ij}$ including eigenvalues of velocity squared for transverse and longitudinal modes for acoustic phonons ($v_T^2$ and $v_L^2$), frequency matrix ${\Omega }_{2,ij}$ including eigenvalues of frequency squared for transverse and longitudinal modes for optical phonons (${\Omega }_T^2$ and ${\Omega }_L^2$), coupling constants between charged bosons and acoustic longitudinal phonons $g_{aL}$ and transverse phonons $g_{aT}$, and coupling constants between charged bosons and optical longitudinal phonons $g_{oL}$ and transverse phonons $g_{oT}$. Longitudinal phonons are coupled with the gradient of density of charged bosons ${\nabla }_i\left({\psi }^{*}\psi \right)$, while transverse phonons coupled with the flow of charged bosons $\left({\psi }^{*}\left({\partial }_i\psi \right)-\left({\partial }_i{\psi }^{*}\right)\psi -2ie{\psi }^{*}\psi (A_i+a_i)\right)$. Our Lagrangian is based on Quantum Electrodynamics for non-relativistic charged bosons. We can also charge e to any effective values. We include phonon degrees of freedom in QBD theory in this paper to investigate coherent sound emitted from microtubules as super-radiance as shown in the next section and provide non-equilibrium theory for incoherent phonons. Both acoustic and optical phonons are introduced as quantum fields and coupled with density of water molecules and flow of water molecules.

The above Lagrangian is invariant under the type I gauge transformation given by,

$$\begin{array}{c}\hfill \psi \left(x\right)\to e^{i\alpha \left(x\right)}\psi \left(x\right),{\psi }^{*}\left(x\right)\to e^{-i\alpha \left(x\right)}{\psi }^{*}\left(x\right),A_{\mu }\left(x\right)\to A_{\mu }\left(x\right)+{\displaystyle \frac{1}{e}}{\partial }_{\mu }\alpha \left(x\right),a_{\mu }\left(x\right)\to a_{\mu }\left(x\right).\end{array}$$

(2)

We shall set gauge fixing $a^0=0$ and $\xi =1$. Closed-time path formalism is adopted to describe non-equilibrium quantum dynamics [88,89]. Starting with the Lagrangian density in Equation (1), we can write 2-Particle-Irreducible (2PI) Effective Action [90,91,92] as,

$$\begin{array}{ccc}\hfill {\Gamma }_{2\mathrm{PI}}[A,{\overline{a}}_i,\overline{\psi },{\overline{\psi }}^{*},{\overline{Q}}_a,{\overline{Q}}_o,\Delta ,\mathbf{D}]& =& {\int }_{\mathcal{C}}{d}^{4}x\mathcal{L}[A,{\overline{a}}_i,\overline{\psi },{\overline{\psi }}^{*},{\overline{Q}}_a,{\overline{Q}}_o]\hfill \\ & & +{\displaystyle \frac{i}{2}}\mathrm{Tr}ln{\mathbf{D}}^{-1}+{\displaystyle \frac{1}{2}}\mathrm{Tr}\left(i{\mathbf{D}}_0^{-1}\mathbf{D}\right)+i\mathrm{Tr}ln{\Delta }^{-1}+\mathrm{Tr}(i{\Delta }_0^{-1}\Delta )\hfill \\ & & +{\displaystyle \frac{1}{2}}{\Gamma }_2[A_i+{\overline{a}}_i,\overline{\psi },{\overline{\psi }}^{*},{\overline{Q}}_a,{\overline{Q}}_o,\Delta ,\mathbf{D}],\hfill \end{array}$$

(3)

where $\mathcal{C}$ represents the closed-time path contour in path ‘1’ from $-\infty$ to ∞ and path ‘2’ from ∞ to $-\infty$. The bar represents expectation values $\overline{\psi }=\langle \psi \rangle =\mathrm{Tr}\left(\mathcal{R}\psi \right)$ for an arbitrary density matrix $\mathcal{R}$. When we set $\mathcal{R}\sim e^{-\mathcal{H}/T}$ with Hamiltonian $\mathcal{H}$ and temperature $T=310\mathrm{K}$, we can include contributions of finite-temperature medium. The $\mathbf{D}(x,y)$ represents matrix of Green’s functions for photons and phonons given by,

$$\begin{array}{ccc}\hfill \mathbf{D}(x,y)& =& \left(\begin{array}{ccc}{D}_{ij}(x,y)& D_{\gamma a,ij}(x,y)& D_{\gamma o,ij}(x,y)\\ D_{a\gamma ,ij}(x,y)& D_{aa,ij}(x,y)& D_{ao,ij}(x,y)\\ D_{o\gamma ,ij}(x,y)& D_{oa,ij}(x,y)& D_{oo,ij}(x,y)\end{array}\right),\hfill \end{array}$$

(4)

with definitions $D_{ij}(x,y)=\langle {\mathrm{T}}_{\mathcal{C}}\delta a_i\left(x\right)\delta a_j\left(y\right)\rangle$ with time-ordered product ${\mathrm{T}}_{\mathcal{C}}$ and $\delta a_i=a_i-{\overline{a}}_i$, $D_{a\gamma ,ij}(x,y)=\langle {\mathrm{T}}_{\mathcal{C}}\delta Q_{a,i}\left(x\right)\delta a_j\left(y\right)\rangle$ with $\delta Q_{a,i}=Q_{a,i}-{\overline{Q}}_{a,i}$, $D_{aa,ij}(x,y)=\langle {\mathrm{T}}_{\mathcal{C}}\delta Q_{a,i}\left(x\right)\delta Q_{a,j}\left(y\right)\rangle$, $D_{oo,ij}(x,y)=\langle {\mathrm{T}}_{\mathcal{C}}\delta Q_{o,i}\left(x\right)\delta Q_{o,j}\left(y\right)\rangle$ with $\delta Q_{o,i}=Q_{o,i}-{\overline{Q}}_{o,i}$ and so on. In matrix notation of closed-time path, $D_{a\gamma ,ij}(x,y)$ represents,

$$\begin{array}{ccc}\hfill D_{a\gamma ,ij}(x,y)& =& \left(\begin{array}{cc}{D}_{a\gamma ,ij}^{11}(x,y)& D_{a\gamma ,ij}^{12}(x,y)\\ D_{a\gamma ,ij}^{21}(x,y)& D_{a\gamma ,ij}^{22}(x,y)\end{array}\right)=\left(\begin{array}{cc}\langle \mathrm{T}\delta Q_{a,i}\left(x\right)\delta a_j\left(y\right)\rangle & \langle \delta a_j\left(y\right)\delta Q_{a,i}\left(x\right)\rangle \\ \langle \delta Q_{a,i}\left(x\right)\delta a_j\left(y\right)\rangle & \langle \tilde{\mathrm{T}}\delta Q_{a,i}\left(x\right)\delta a_j\left(y\right)\rangle \end{array}\right),\hfill \end{array}$$

(5)

with time-ordered product $\mathrm{T}$ and anti-time-ordered product $\tilde{\mathrm{T}}$. The $\Delta (x,y)$ represents,

$$\begin{array}{c}\hfill \Delta (x,y)=\langle {\mathrm{T}}_{\mathcal{C}}\delta \psi \left(x\right)\delta {\psi }^{*}\left(y\right)\rangle ,\end{array}$$

(6)

with $\delta {\psi }^{(*)}={\psi }^{(*)}-{\overline{\psi }}^{(*)}$. The $i{\mathbf{D}}_0^{-1}(x,y)$ represents the matrix,

$$\begin{array}{ccc}\hfill i{\mathbf{D}}_0^{-1}(x,y)& =& \left(\begin{array}{ccc}-\left({\partial }_x^2+{\displaystyle \frac{e^2{\overline{\psi }}^{*}\left(x\right)\overline{\psi }\left(x\right)}{m}}\right){\delta }_{ij}& -2e{g}_{aT}{\overline{\psi }}^{*}\overline{\psi }\left({\delta }_{ij}-{\displaystyle \frac{{\partial }_i{\partial }_j}{{\partial }_k^2}}\right)& -2e{g}_{oT}{\overline{\psi }}^{*}\overline{\psi }\left({\delta }_{ij}-{\displaystyle \frac{{\partial }_i{\partial }_j}{{\partial }_k^2}}\right)\\ -2e{g}_{aT}{\overline{\psi }}^{*}\overline{\psi }\left({\delta }_{ij}-{\displaystyle \frac{{\partial }_i{\partial }_j}{{\partial }_k^2}}\right)& -\left({\partial }_0^2{\delta }_{ij}-v_{2,ij}{\partial }_k^2\right)& 0\\ -2e{g}_{oT}{\overline{\psi }}^{*}\overline{\psi }\left({\delta }_{ij}-{\displaystyle \frac{{\partial }_i{\partial }_j}{{\partial }_k^2}}\right)& 0& -\left({\partial }_0^2{\delta }_{ij}+{\Omega }_{2,ij}\right)\end{array}\right)\hfill \\ & & \times {\delta }_{\mathcal{C}}(x-y),\hfill \end{array}$$

(7)

$i{\Delta }_0^{-1}(x,y)={\displaystyle \frac{{\delta }^2{\int }_w\mathcal{L}\left(w\right)}{\delta {\psi }^{*}\left(x\right)\delta \psi \left(y\right)}}$ represents,

$$\begin{array}{ccc}\hfill i{\Delta }_0^{-1}(x,y)& =& (i{\displaystyle \frac{\partial }{\partial x^0}}+e{A}^0+{\displaystyle \frac{{({\nabla }_i-ie(A_i+{\overline{a}}_i))}^2}{2m}}\hfill \\ & & -g_{aL}\left({\nabla }_{x,i}{\overline{Q}}_{aL}^i\left(x\right)\right)+i{g}_{aT}\left({\overline{Q}}_{aT}^i\left(x\right)+{\overline{Q}}_{aT}^i\left(y\right)\right)\left({\partial }_{x,i}-ie(A_i\left(x\right)+{\overline{a}}_i\left(x\right))\right)\hfill \\ & & -g_{oL}\left({\nabla }_{x,i}{\overline{Q}}_{oL}^i\left(x\right)\right)+i{g}_{oT}\left({\overline{Q}}_{oT}^i\left(x\right)+{\overline{Q}}_{oT}^i\left(y\right)\right)\left({\partial }_{x,i}-ie(A_i\left(x\right)+{\overline{a}}_i\left(x\right))\right)){\delta }_{\mathcal{C}}(x-y),\hfill \end{array}$$

(8)

and ${\displaystyle \frac{i{\Gamma }_2}{2}}$ represents all the 2-Particle-Irreducible loop diagrams corresponding to $\Phi$-derivable loop expansion technique in [64].

We adopt Hartree–Fock approximation for the ${\displaystyle \frac{i{\Gamma }_2}{2}}$. The 2-loop diagrams in ${\displaystyle \frac{i{\Gamma }_2}{2}}$ is depicted in Figure 1. We find local terms in Figure 1a inducing mass shift of charged bosons and photons and terms in Figure 1b inducing the coupling for phonon-photon exchange. Non-local terms in Figure 1c–g are depicted to represent interaction among charged bosons, photons and phonons. Differentiating ${\displaystyle \frac{i{\Gamma }_2}{2}}$ by Green’s functions, we can derive self-energy.

The time-evolution equations are given by differentiating ${\Gamma }_{2\mathrm{PI}}$ in Equation (3) by fields ${\overline{a}}_i\left(x\right)$, ${\overline{\psi }}^{(*)}\left(x\right)$, Green’s functions $\mathbf{D}(x,y)$ and $\Delta (x,y)$. They are given in Appendices Appendix A, Appendix C and Appendix E.

4. Acoustic Super-Radiance

In this section, we show a solution of acoustic super-radiance, cooperative coherent spontaneous sound emission from water, shown in Figure 2. We adopt a microtubule, cytoskeleton involving a cylindrical structure, as a coherent sound source. We neglect contributions of quantum fluctuations in this section. The size of coherent domains can estimated as $15\mathrm{nm}$ [27]. This size corresponds to the inner diameter of microtubules. Hence microtubules are regarded as devices achieving coherent domains. We can derive acoustic super-radiance from coherent domains in microtubules emitted in a radial direction in Figure 2. It might be observed around microtubules with continuous external energy supply from mitochondria [93].

We use Equations (A2) and (A6) in Appendix A by assuming $e=0$, $g_{aL}=0$, $g_{oT}=0$, and $g_{oL}=0$ only for interaction between water and transverse acoustic phonons. The relation (A2) is rewritten by,

$$\begin{array}{c}\hfill \left(i{\displaystyle \frac{\partial }{\partial x^0}}+{\displaystyle \frac{{\nabla }_i^2}{2m}}-2i{g}_{aT}{\overline{Q}}_{aT,i}{\nabla }_i+\tilde{U}\left(x\right)\right)\overline{\psi }\left(x\right)=0,\end{array}$$

(9)

where we added potential energy term $\tilde{U}\left(x\right)$ representing water molecular states. The relation (A6) is rewritten by,

$$\begin{array}{ccc}\hfill \left({\partial }_0^2-v_T^2{\partial }_j^2\right)Q_{aT,i}+i{g}_{aT}\left({\overline{\psi }}^{*}{\nabla }_i\overline{\psi }-\left({\nabla }_i{\overline{\psi }}^{*}\right)\overline{\psi }\right)& =& 0.\hfill \end{array}$$

(10)

We adopt two-energy level approximation for water molecular states, namely the ground state and the 1st excited state. Water absorbs the various ranges of wavelength of photons [94], which correspond to various water molecular states including symmetric and anti-symmetric stretching modes with ranges of wavelength from $1300\mathrm{nm}$ to $1600\mathrm{nm}$, and rotational motions with the order of a few meV with the wavelength $\sim 300\mathsf{\mu }\mathrm{m}$, for example. We can set $\tilde{U}\left(x\right)$ in Equation (9) representing energy of rotational motion of water molecules or mean field energy for molecular stretching, and so on, given by approximations of interaction with photons and phonons. Since water absorbs both light and sound [87], water is also regarded as coherent light and sound sources. The Equation (9) covers those water molecular states, while the dynamics of coherent sound fields is given by Equation (10).

Using the calculations in Appendix B, we arrive at the amplitude of coherent phonon fields $\mathcal{Q}$ given by,

$$\begin{array}{c}\hfill \mathcal{Q}={\displaystyle \frac{g\overline{\Omega }N}{4\pi }}{\left[cosh\left({\displaystyle \frac{x^0-{\tau }_0}{{\tau }_R}}\right)\right]}^{-1},\end{array}$$

(11)

where the $\overline{\Omega }$ represents the energy difference between the ground state and 1st excited state, the N represents the number of water molecules, the g is given by $g\equiv m{g}_{aT}{J}_{01}$ with Equation (A28) related with transition dipole moment of water molecules, and ${\tau }_R$ represents,

$$\begin{array}{c}\hfill {\tau }_R={\displaystyle \frac{2\pi }{g^2\overline{\Omega }N}}\propto {\displaystyle \frac{1}{N}},\end{array}$$

(12)

and ${\tau }_0=-{\tau }_{R}lntan{\displaystyle \frac{{\theta }_0}{2}}$. Due to N water molecules cooperatively decaying in $1/N$ time scale, the coherent sound field has intensity of the order of $N^2$ instantly, representing acoustic super-radiance. Substituting the mode $\overline{\Omega }$ corresponding to water molecular states, N representing the number of water molecules in the states, and g for transition of water molecules in Equations (11) and (12) by their concrete values, we can derive the amplitude of coherent sound fields and the time scale of the acoustic super-radiance for each water molecular state.

5. Kinetic Entropy Current and the H-Theorem

In this section, we summarize the introduction of a kinetic entropy current and the H-theorem for 1-loop self-energy in Hartree–Fock approximation. We refer to the preceding works in [95,96,97,98,99,100,101]. The detailed calculations are given in Appendix D.

We finally arrive at,

$$\begin{array}{c}\hfill {\partial }_{\mu }{s}^{\mu }=\left(\mathrm{PP}\right)+\left(\mathrm{Dia}\left(\mathrm{c}\right)\right)+\left(\mathrm{Dia}\left(\mathrm{d}\right)\right)+\left(\mathrm{Dia}\left(\mathrm{e}\right)\right)+\left(\mathrm{Dia}\left(\mathrm{f}\right)\right)+\left(\mathrm{Dia}\left(\mathrm{g}\right)\right)+\left(\mathrm{Dia}\left(\mathrm{h}\right)\right)\ge 0,\end{array}$$

(13)

where $\left(\mathrm{PP}\right)$ represents the photon-phonon exchange via water molecules. The total entropy current $s^{\mu }$ is given by Equation (A216). The H-theorem is proved in the Hartree–Fock approximation for Kadanoff–Baym equations in 1st order in the gradient expansions.

Entropy production stops in the condition,

$$\begin{array}{ccc}\hfill f\left(p\right)& =& {\displaystyle \frac{1}{e^{\frac{p^0-{\mu }_c}{T}}-1}},\hfill \end{array}$$

(14)

with temperature T and chemical potential ${\mu }_c=e\left(A^0-{\displaystyle \frac{{\partial }^0\beta }{e}}\right)$ for distribution function of charged bosons $f\left(p\right)$, and,

$$\begin{array}{c}\hfill f_T\left(k\right)=f_L\left(k\right)=f_{aa,T}\left(k\right)=f_{aa,L}\left(k\right)=f_{oo,T}\left(k\right)=f_{oo,L}\left(k\right)={\displaystyle \frac{1}{e^{\frac{k^0}{T}}-1}},\end{array}$$

(15)

with distribution functions for transverse photons $f_T$, longitudinal photons $f_L$, transverse acoustic phonons $f_{aa,T}$, longitudinal acoustic phonons $f_{aa,L}$, transverse optical phonons $f_{oo,T}$ and longitudinal optical phonons $f_{oo,L}$. The system composed by charged bosons, photons and phonons evolve in time with entropy producing processes.

6. Time-Evolution Equations in Spatially Homogeneous Systems

In this section we show time-evolution equations in spatially homogeneous system and show concrete forms of conserved charge and energy density.

First we shall rewrite Kadanoff–Baym Equations (A65), (A68), (A97), (A102), (A105), (A109), (A112), (A116) and (A126), in Appendix C. We use statistical functions $F(X,p)={\displaystyle \frac{G^{12}(X,p)+G^{21}(X,p)}{2}}$, $F_L(X,k)={\displaystyle \frac{D_L^{12}(X,k)+D_L^{21}(X,k)}{2}}$, $F_T(X,k)={\displaystyle \frac{D_T^{12}+D_T^{21}}{2}}$, $F_{aa,L}(X,k)={\displaystyle \frac{D_{aa,L}^{12}+D_{aa,L}^{21}}{2}}$, $F_{aa,T}(X,k)={\displaystyle \frac{D_{aa,T}^{12}+D_{aa,T}^{21}}{2}}$, $F_{oo,L}(X,k)={\displaystyle \frac{D_{oo,L}^{12}+D_{oo,L}^{21}}{2}}$, $F_{oo,T}(X,k)={\displaystyle \frac{D_{oo,T}^{12}+D_{oo,T}^{21}}{2}}$, $d_{aa,F,T}={\displaystyle \frac{d_{aa,T}^{12}+d_{aa,T}^{21}}{2}}$ and $d_{oo,F,T}={\displaystyle \frac{d_{oo,T}^{12}+d_{oo,T}^{21}}{2}}$, spectral functions $\rho =i(G^{21}-G^{12})$, ${\rho }_L=i(D_L^{21}-D_L^{12})$, ${\rho }_T=i(D_T^{21}-D_T^{12})$, ${\rho }_{aa,L}=i(D_{aa,L}^{21}-D_{aa,L}^{12})$, ${\rho }_{aa,T}=i(D_{aa,T}^{21}-D_{aa,T}^{12})$, ${\rho }_{oo,L}=i(D_{oo,L}^{21}-D_{oo,L}^{12})$, ${\rho }_{oo,L}=i(D_{oo,L}^{21}-D_{oo,L}^{12})$, $d_{aa,\rho ,T}=i(d_{aa,T}^{21}-d_{aa,T}^{12})$, and $d_{oo,\rho ,T}=i(d_{oo,T}^{21}-d_{oo,T}^{12})$. Spectral functions represent which states particles are occupied, while statistical functions represent particle number density or how many particles are in the states. We also use statistical parts of self-energy ${\Sigma }_F={\displaystyle \frac{{\Sigma }_{\mathrm{nonl}}^{12}+{\Sigma }_{\mathrm{nonl}}^{21}}{2}}$, ${\Pi }_{F,L}={\displaystyle \frac{{\Pi }_{\mathrm{nonl},L}^{12}+{\Pi }_{\mathrm{nonl},L}^{21}}{2}}$, ${\Pi }_{F,T}={\displaystyle \frac{{\Pi }_{\mathrm{nonl},T}^{12}+{\Pi }_{\mathrm{nonl},T}^{21}}{2}}$, ${\Pi }_{aa,F,L}={\displaystyle \frac{{\Pi }_{\mathrm{nonl},aa,L}^{12}+{\Pi }_{\mathrm{nonl},aa,L}^{21}}{2}}$, ${\Pi }_{aa,F,T}={\displaystyle \frac{{\Pi }_{\mathrm{nonl},aa,T}^{12}+{\Pi }_{\mathrm{nonl},aa,T}^{21}}{2}}$, ${\Pi }_{oo,F,L}={\displaystyle \frac{{\Pi }_{\mathrm{nonl},oo,L}^{12}+{\Pi }_{\mathrm{nonl},oo,L}^{21}}{2}}$, ${\Pi }_{oo,F,T}={\displaystyle \frac{{\Pi }_{\mathrm{nonl},oo,T}^{12}+{\Pi }_{\mathrm{nonl},oo,T}^{21}}{2}}$, $U_{aa,F,T}={\displaystyle \frac{U_{aa,T}^{12}+U_{aa,T}^{21}}{2}}$, $U_{oo,F,T}={\displaystyle \frac{U_{oo,T}^{12}+U_{oo,T}^{21}}{2}}$, $V_{aa,F,T}={\displaystyle \frac{V_{aa,T}^{12}+V_{aa,T}^{21}}{2}}$, and $V_{oo,F,T}={\displaystyle \frac{V_{oo,T}^{12}+V_{oo,T}^{21}}{2}}$, and spectral parts ${\Sigma }_{\rho }=i({\Sigma }_{\mathrm{nonl}}^{21}-{\Sigma }_{\mathrm{nonl}}^{12})$, ${\Pi }_{\rho ,L}=i({\Pi }_{\mathrm{nonl},L}^{21}-{\Pi }_{\mathrm{nonl},L}^{12})$, ${\Pi }_{\rho ,T}=i({\Pi }_{\mathrm{nonl},T}^{21}-{\Pi }_{\mathrm{nonl},T}^{12})$, ${\Pi }_{aa,\rho ,L}=i({\Pi }_{\mathrm{nonl},aa,L}^{21}-{\Pi }_{\mathrm{nonl},aa,L}^{12})$, ${\Pi }_{aa,\rho ,T}=i({\Pi }_{\mathrm{nonl},aa,T}^{21}-{\Pi }_{\mathrm{nonl},aa,T}^{12})$, ${\Pi }_{oo,\rho ,L}=i({\Pi }_{\mathrm{nonl},oo,L}^{21}-{\Pi }_{\mathrm{nonl},oo,L}^{12})$, ${\Pi }_{oo,\rho ,T}=i({\Pi }_{\mathrm{nonl},oo,T}^{21}-{\Pi }_{\mathrm{nonl},oo,T}^{12})$, $U_{aa,\rho ,T}=i(U_{aa}^{21}-U_{aa}^{12})$, $U_{oo,\rho ,T}=i(U_{oo}^{21}-U_{oo}^{12})$, $V_{aa,\rho ,T}=i(V_{aa}^{21}-V_{aa}^{12})$, and $V_{oo,\rho ,T}=i(V_{oo}^{21}-V_{oo}^{12})$.

The Kadanoff–Baym equations for charged bosons for real F and ${\Sigma }_F$ and pure imaginary $\rho$ and ${\Sigma }_{\rho }$ are,

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left\{i{G}_0^{-1}\left(p\right)-{\Sigma }_{\mathrm{loc}}-\mathrm{Re}{\Sigma }_R,F\right\}+\left\{\mathrm{Re}{G}_R,{\Sigma }_F\right\}& =& {\displaystyle \frac{1}{i}}\left(F{\Sigma }_{\rho }-\rho {\Sigma }_F\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \left\{i{G}_0^{-1}\left(p\right)-{\Sigma }_{\mathrm{loc}}-\mathrm{Re}{\Sigma }_R,\rho \right\}+\left\{\mathrm{Re}{G}_R,{\Sigma }_{\rho }\right\}& =& 0,\hfill \end{array}$$

(17)

with,

$$\begin{array}{ccc}\hfill i{G}_0^{-1}\left(p\right)& =& p^0-{\displaystyle \frac{{\mathbf{p}}^2}{2m}}-g_{aL}\left({\partial }_{X,i}{\overline{Q}}_{aL}^i\right)-g_{oL}\left({\partial }_{X,i}{\overline{Q}}_{oL}^i\right)+2g_{aT}{p}_i{\overline{Q}}_{aT}^i+2g_{oT}{p}_i{\overline{Q}}_{oT}^i.\hfill \end{array}$$

(18)

Here the subscript ‘R’ represents retarded parts for Green’s functions and self-energy. For example, its real part is given by $\mathrm{Re}{G}_R={\displaystyle \frac{i}{2}}\left(G^{11}-G^{22}\right)$. The local self-energy for charged bosons is given in Equation (A228) in Appendix E. The statistical and spectral parts in self-energy are given in Equations (A231) and (A232) in Appendix E.

Next, Kadanoff–Baym equations for photons and phonons are derived from (A65), (A68), (A97), (A102), (A105), (A109), (A112) and (A116). For statistical and spectral parts, we can derive,

$$\begin{array}{ccc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\left\{i{D}_0^{-1}\left(k\right)-{\Pi }_{\mathrm{loc}}-\mathrm{Re}{\Pi }_{R,L},F_L\right\}}_P+{\left\{\mathrm{Re}{D}_{R,L},{\Pi }_{F,L}\right\}}_P& =& {\displaystyle \frac{1}{i}}\left(F_L{\Pi }_{\rho ,L}-{\rho }_L{\Pi }_{F,L}\right),\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {\left\{i{D}_0^{-1}\left(k\right)-{\Pi }_{\mathrm{loc}}-\mathrm{Re}{\Pi }_{R,L},{\rho }_L\right\}}_P+{\left\{\mathrm{Re}{D}_{R,L},{\Pi }_{\rho ,L}\right\}}_P& =& 0,\hfill \end{array}$$

(20)

with $i{D}_0^{-1}\left(k\right)=k^2-{\displaystyle \frac{e^2{\left|\overline{\psi }\right|}^2}{m}}$, and,

$$\begin{array}{cc}& {\left\{i{D}_0^{-1}\left(k\right)-{\Pi }_{\mathrm{loc}}-\mathrm{Re}{\Pi }_{R,T}+\mathrm{Re}{U}_{aa,R,T}+\mathrm{Re}{U}_{oo,R,T},F_T\right\}}_P\\ & +{\left\{\mathrm{Re}{D}_{R,T},{\Pi }_{F,T}-U_{aa,F,T}-U_{oo,F,T}\right\}}_P\\ \hfill =& {\displaystyle \frac{1}{i}}\left(F_T{\Pi }_{\rho ,T}-{\rho }_T{\Pi }_{F,T}\right)-{\displaystyle \frac{1}{i}}\left(F_T{U}_{aa,\rho ,T}-{\rho }_T{U}_{aa,F,T}\right)-{\displaystyle \frac{1}{i}}\left(F_T{U}_{oo,\rho ,T}-{\rho }_T{U}_{oo,F,T}\right),\end{array}$$

(21)

$$\begin{array}{cc}& {\left\{i{D}_0^{-1}\left(k\right)-{\Pi }_{\mathrm{loc}}-\mathrm{Re}{\Pi }_{R,T}+\mathrm{Re}{U}_{aa,R,T}+\mathrm{Re}{U}_{oo,R,T},{\rho }_T\right\}}_P\\ & +{\left\{\mathrm{Re}{D}_{R,T},{\Pi }_{\rho ,T}-U_{aa,\rho ,T}-U_{oo,\rho ,T}\right\}}_P=0.\end{array}$$

(22)

For acoustic phonons, we can derive,

$$\begin{array}{ccc}\hfill {\left\{i{D}_{0,aa,L}^{-1}-\mathrm{Re}{\Pi }_{aa,R,L},F_{aa,L}\right\}}_P+{\left\{\mathrm{Re}{D}_{aa,R,L},{\Pi }_{aa,F,L}\right\}}_P& =& {\displaystyle \frac{1}{i}}\left(F_{aa,L}{\Pi }_{aa,\rho ,L}-{\rho }_{aa,L}{\Pi }_{aa,F,L}\right),\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\left\{i{D}_{0,aa,L}^{-1}-\mathrm{Re}{\Pi }_{aa,R,L},{\rho }_{aa,L}\right\}}_P+{\left\{\mathrm{Re}{D}_{aa,R,L},{\Pi }_{aa,\rho ,L}\right\}}_P& =& 0,\hfill \end{array}$$

(24)

with $i{D}_{0,aa,L}^{-1}\left(k\right)={\left(k^0\right)}^2-v_L^2{\mathbf{k}}^2$,

$$\begin{array}{c}\hfill {\left\{i{D}_{0,aa,T}^{-1}-\mathrm{Re}{\Pi }_{aa,R,T},F_{aa,T}\right\}}_P+{\left\{\mathrm{Re}{D}_{aa,R,T},{\Pi }_{aa,F,T}\right\}}_P\\ \hfill +{\left\{\mathrm{Re}{V}_{aa,R,T},d_{aa,F,T}\right\}}_P-{\left\{\mathrm{Re}{d}_{aa,R,T},V_{aa,F,T}\right\}}_P\\ \hfill ={\displaystyle \frac{1}{i}}\left(F_{aa,T}{\Pi }_{aa,\rho ,T}-{\rho }_{aa,T}{\Pi }_{aa,F,T}\right)-{\displaystyle \frac{1}{i}}\left(d_{aa,F,T}{V}_{aa,\rho ,T}-d_{aa,\rho ,T}{V}_{aa,F,T}\right),\end{array}$$

(25)

$$\begin{array}{c}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\left\{i{D}_{0,aa,T}^{-1}-\mathrm{Re}{\Pi }_{aa,R,T},{\rho }_{aa,T}\right\}}_P+{\left\{\mathrm{Re}{D}_{aa,R,T},{\Pi }_{aa,\rho ,T}\right\}}_P\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\left\{\mathrm{Re}{V}_{aa,R,T},d_{aa,\rho ,T}\right\}}_P-{\left\{\mathrm{Re}{d}_{aa,R,T},V_{aa,\rho ,T}\right\}}_P=0,\end{array}$$

(26)

with $i{D}_{0,aa,T}^{-1}\left(k\right)={\left(k^0\right)}^2-v_T^2{\mathbf{k}}^2$. For optical phonons, we can write Kadanoff–Baym equations by replacing the subscript ‘$aa$’ to ‘$oo$’ in Equations (23)–(26) with $i{D}_{0,oo,L}^{-1}\left(k\right)={\left(k^0\right)}^2-{\Omega }_L^2$ and $i{D}_{0,oo,T}^{-1}\left(k\right)={\left(k^0\right)}^2-{\Omega }_T^2$. We also use the auxiliary equations,

$$\begin{array}{c}\hfill {\left\{i{D}_{0,aa,T}^{-1}-\mathrm{Re}{\Pi }_{aa,R,T},d_{aa,F,T}\right\}}_P+{\left\{\mathrm{Re}{d}_{aa,R,T},{\Pi }_{aa,F,T}\right\}}_P={\displaystyle \frac{1}{i}}\left(d_{aa,F,T}{\Pi }_{aa,\rho ,T}-d_{aa,\rho ,T}{\Pi }_{aa,F,T}\right),\end{array}$$

(27)

$$\begin{array}{c}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\left\{i{D}_{0,aa,T}^{-1}-\mathrm{Re}{\Pi }_{aa,R,T},d_{aa,\rho ,T}\right\}}_P+{\left\{\mathrm{Re}{d}_{aa,R,T},{\Pi }_{aa,\rho ,T}\right\}}_P=0,\end{array}$$

(28)

with equations where subscripts ‘$aa$’ are replaced by ‘$oo$’.

Next we write time-evolution equation for coherent charged boson fields. It is written by,

$$\begin{array}{ccc}\hfill {\partial }_0{\left|\overline{\psi }\right|}^2& =& -{\partial }_0{\int }_p\left({\displaystyle \frac{\partial \mathrm{Re}{\Sigma }_R^{\left(\mathrm{g}\right)+\left(\mathrm{h}\right)}\left(p\right)}{\partial p^0}}F\left(p\right)+\mathrm{Re}{G}_R\left(p\right){\displaystyle \frac{\partial {\Sigma }_F^{\left(\mathrm{g}\right)+\left(\mathrm{h}\right)}\left(p\right)}{\partial p^0}}\right)\hfill \\ & & -{\int }_p\left(F\left(p\right){\displaystyle \frac{{\Sigma }_{\rho }^{\left(\mathrm{g}\right)+\left(\mathrm{h}\right)}\left(p\right)}{i}}-{\displaystyle \frac{\rho \left(p\right)}{i}}{\Sigma }_F^{\left(\mathrm{g}\right)+\left(\mathrm{h}\right)}\left(p\right)\right),\hfill \end{array}$$

(29)

which provides time-evolution of the density $|\overline{\psi }{|}^2$. Here we have used Equation (A4) and Appendix F. We also encounter the constraint relation (A290) in Appendix F. We also use Equations (A295) for coherent photon fields, (A297)–(A300) for coherent phonon fields.

Finally, using the above relations, we can derive the total charge conservation,

$$\begin{array}{ccc}\hfill {\partial }_0\left[-e\left(|\overline{\psi }{\left(X\right)|}^2+{\int }_{p}F(X,p)\right)\right]& =& 0,\hfill \end{array}$$

(30)

where we can derive the right-hand side in Equation (29) and time-derivative of ${\int }_{p}F(X,p)$ with integration of Equation (16) cancel. We can also derive the total energy conservation,

$$\begin{array}{c}\hfill {\partial }_0{E}_{\mathrm{tot}}=0,\end{array}$$

(31)

with total energy density $E_{\mathrm{tot}}$ given in Appendix G.

7. Discussion

In this paper, we have investigated Quantum Electrodynamics (QED) for water coupled with phonons and photons corresponding to Quantum Brain Dynamics involving water, sound and light, and provided theoretical formulation in our model. Beginning with Lagrangian density of charged bosons representing water molecular states, phonons and photons, we have derived time-evolution equations for coherent photon, charged boson and phonon fields, and Kadanoff–Baym equations for incoherent photons, charged bosons and phonons. We have derived an acoustic super-radiance solution in our model. We have introduced a kinetic entropy current in 1st order approximation in the gradient expansion and shown the H-theorem for self-energy in Hartree–Fock approximation. Finally we have given time-evolution equations in spatially homogeneous system and shown concrete forms of conserved total charge and energy density.

We have adopted the similar Lagrangian density to that by Nguyen et al. [61] who introduced interaction Hamiltonian between transverse phonon fields and flow of charged bosons. They introduced the interaction Hamiltonian $H_I$ like,

$$\begin{array}{c}\hfill H_I\sim i\int d^{3}x\left(({\psi }^{*}{\partial }_i\psi -\left({\partial }_i{\psi }^{*}\right)\psi \right)Q_{aT,i},\end{array}$$

(32)

and derived collective behaviors of charged particles, namely plasmons. However, there is a serious problem in Nguyen’s model. Since ${\partial }_i\psi$ is not a covariant derivative like $({\partial }_i-ie{A}_i)\psi$, the Hamiltonian is not gauge-invariant. Then we find that charge conservation is no longer satisfied in their model. Hence for our model in this paper, we adopt gauge-invariant interaction Lagrangian involving covariant derivatives. Imposing gauge-invariance in Lagrangian, we encounter phonon–photon interaction via charged bosons in term ${\psi }^{*}\psi A_i{Q}_{aT,i}$ in Equation (1). Due to this term, the Kadanoff–Baym equations for photons and phonons involve off-diagonal elements of Green’s functions like $D_{a\gamma }(x,y)$ and so on. Photon–phonon energy exchange via water degrees of freedom (charged bosons) is described by off-diagonal elements in our model. Entropy production by the energy exchange is represented in Equation (A218). We encounter sound–light interaction via water in our theory.

Acoustic super-radiance solution is derived in our model. We adopt an interaction Lagrangian term between transverse acoustic phonons and charged bosons $g_{aT}({\psi }^{*}\left({\partial }_i\psi \right)-$$\left({\partial }_i{\psi }^{*}\right)\psi -2ie{\psi }^{*}\psi (A_i+a_i))Q_{aT}^i$ in Equation (1). This term resembles photon–charged boson interaction term in Quantum Electrodynamics (QED) with non-relativistic charged bosons. In QED theory, we can derive the super-radiance solution, cooperative spontaneous coherent light emission, for photon–charged boson degrees of freedom [102,103,104]. In a similar way to photon–charged boson system, we can derive acoustic super-radiance solution in phonon–charged boson system, namely we encounter phonon–water cooperative properties. Even in vanishing charge $e=0$, cooperative spontaneous decay of water molecular states from 1st excited state to the ground state occurs, and then resultant coherent sound emission emerges. Coherent sound might induce interference patterns of sound waves, resulting in pressure gradient of water systems. Interference patterns might be adopted for sound holographic information processing. Even in the tentative interference patterns in sound holography, information processing might be possible in water system in a brain. Sound holography might appear in pressure gradient in water systems, while optical holography emerges in density distribution of ionic bio-plasma or ionized water since density affects transmittance of coherent light. Hence sound holography might correlate with optical holography via water degrees of freedom. Since our model involves both photon and phonon degrees of freedom, we can investigate both sound and optical holography for information processing in a brain.

The 2nd law in thermodynamics is investigated in this paper. The kinetic entropy current for Kadanoff–Baym equations for incoherent photons, charged bosons, and phonons is introduced and the H-theorem in Hartree–Fock approximation is shown. Entropy production occurs in photon–phonon energy exchange in Equation (A218). Equation (A218) means that entropy production due to this term ceases when photon distribution $f_T$ is equal to phonon distributions $f_{aa,T}$ and $f_{oo,T}$, not necessarily Bose–Einstein distributions. Even in out of equilibrium states, phonon distribution tends to approach photon distribution. This means that both distributions might correlate with each other in out-of-equilibrium states. In the presence of only Equation (A218), the system never approaches thermal equilibrium state. To achieve Bose–Einstein distribution, we need contributions in Equations (A219)–(A224). Equation (A219) (Diagram (c) in Figure 1) represents photon–charged boson interaction, where charged bosons come in, photons are absorbed and charged bosons go out, and its inverse processes take place. Similarly, Equation (A220) represents photon–charged boson interaction induced by coherent sound waves, and Equations (A221) and (A222) represent phonon–charged boson interaction. Equation (A223) represents photon–charged boson interaction induced by coherent charged Bose fields. Coherent charged Bose fields collapse to incoherent charged bosons. Equation (A224) represents photon–phonon–charged boson interaction induced by coherent charged Bose fields. Even in the presence of Equations (A223) and (A224), total charge for coherent charged Bose fields and incoherent charged bosons is conserved. All the contributions for non-local terms in ${\displaystyle \frac{i{\Gamma }_2}{2}}$ in Figure 1 have been made to induce entropy production. Entropy production stops only when distributions are the Bose–Einstein distributions.

Charge and energy conservation law can be shown in spatially homogeneous system. In Section 6, we have given time-evolution equations for coherent charged Bose fields, photon fields and sound fields, and Kadanoff–Baym equations for incoherent charged bosons, photons, and phonons. Self-energy is derived by differentiating ${\displaystyle \frac{i{\Gamma }_2}{2}}$ by Green’s functions in Figure 1. Diagrams (c), (d), (e), and (f) never change the charge density for coherent charged Bose fields $-e{\partial }_0{\left|\overline{\psi }\right|}^2=0$ and that of incoherent charged bosons $-e{\partial }_0{\int }_{p}F=0$. Diagrams (g) and (h) change $-e|\overline{\psi }{|}^2$ or $-e{\int }_{p}F$. Although coherent charged Bose fields collapse to incoherent charged bosons, total charge can be shown to be conserved by similar analysis to that in Appendix in [101]. Similarly we can derive conserved total energy density in a similar way to derivation in [101,105,106].

For Hartree–Fock approximation, we will encounter an additional ${\displaystyle \frac{i{\Gamma }_2^{\left(\mathrm{i}\right)}}{2}}$ as shown in Figure 3. It is expressed by equation,

$$\begin{array}{c}\hfill {\displaystyle \frac{i{\Gamma }_2^{\left(\mathrm{i}\right)}}{2}}=-{\displaystyle \frac{e{g}_{aT}}{m}}{\int }_{\mathcal{C},z,w}{\overline{Q}}_{aT,j}\left(w\right)\left[2ie\left(({\partial }_{z,i}-ie{A}_i\left(z\right))\Delta (z,w)\right)\Delta (w,z)D_T^{ij}(z,w)\right]+(a\to o).\end{array}$$

(33)

In the presence of diagram (i) shown in this figure, we just modify self-energy for ${\Sigma }^{12}\left(p\right)$ and ${\Pi }^{12}\left(k\right)$ by,

$$\begin{array}{ccc}\hfill {\Sigma }^{\left(\mathrm{c}\right)+\left(\mathrm{d}\right)+\left(\mathrm{i}\right),12}\left(p\right)& =& -{\displaystyle \frac{e^2}{4m^2}}{\int }_k{G}^{12}(p-k)\hfill \\ & & \times \left[4\left({\left(\mathbf{p}+2\tilde{\mathbf{Q}}\right)}^2-{\displaystyle \frac{{\left((\mathbf{p}+2\tilde{\mathbf{Q}})·\mathbf{k}\right)}^2}{{\mathbf{k}}^2}}\right)D_T^{12}\left(k\right)+{\displaystyle \frac{{({\mathbf{k}}^2-2\mathbf{p}·\mathbf{k})}^2}{{\mathbf{k}}^2}}{D}_L^{12}\left(k\right)\right],\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill {\Pi }_T^{\left(\mathrm{c}\right)+\left(\mathrm{d}\right)+\left(\mathrm{i}\right),12}\left(k\right)& =& -{\displaystyle \frac{e^2}{2m^2}}{\int }_p\left({\left(\mathbf{p}+2\tilde{\mathbf{Q}}\right)}^2-{\displaystyle \frac{{\left((\mathbf{p}+2\tilde{\mathbf{Q}})·\mathbf{k}\right)}^2}{{\mathbf{k}}^2}}\right)G^{12}(p+k)G^{21}\left(p\right),\hfill \end{array}$$

(35)

and,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta {\Gamma }_2}{\delta {\overline{Q}}_{aT}^i}}& =& -{\displaystyle \frac{2e^2{g}_{aT}}{m}}{\int }_p\left(\mathrm{Re}{\Xi }_{R,ij}\left(p\right)F\left(p\right)+{\Xi }_{F,ij}\left(p\right)\mathrm{Re}{G}_R\left(p\right)\right)\left(p_j+2{\tilde{Q}}_j\right),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta {\Gamma }_2}{\delta {\overline{Q}}_{oT}^i}}& =& -{\displaystyle \frac{2e^2{g}_{oT}}{m}}{\int }_p\left(\mathrm{Re}{\Xi }_{R,ij}\left(p\right)F\left(p\right)+{\Xi }_{F,ij}\left(p\right)\mathrm{Re}{G}_R\left(p\right)\right)\left(p_j+2{\tilde{Q}}_j\right),\hfill \end{array}$$

(37)

with $\tilde{\mathbf{Q}}=m(g_{aT}{\overline{\mathbf{Q}}}_{aT}+g_{oT}{\overline{\mathbf{Q}}}_{oT})$. We also set,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta {\Gamma }_2}{\delta A^i}}& =& -e{\int }_p\left({\displaystyle \frac{\partial \mathrm{Re}{\Sigma }_R^{\left(\mathrm{c}\right)+\left(\mathrm{d}\right)+\left(\mathrm{e}\right)+\left(\mathrm{i}\right)}\left(p\right)}{\partial p^i}}F\left(p\right)+{\displaystyle \frac{\partial {\Sigma }_F^{\left(\mathrm{c}\right)+\left(\mathrm{d}\right)+\left(\mathrm{e}\right)+\left(\mathrm{i}\right)}\left(p\right)}{\partial p^i}}\mathrm{Re}{G}_R\left(p\right)\right),\hfill \end{array}$$

(38)

in Equation (A296). Even in the presence of diagram (i), we can prove the H-theorem since positive coefficient $\left({\left(\mathbf{p}+2\tilde{\mathbf{Q}}\right)}^2-{\left((\mathbf{p}+2\tilde{\mathbf{Q}})·\mathbf{k}\right)}^2/{\mathbf{k}}^2\right)\ge 0$ is given, and show charge and energy conservation by using the relation (A296) for the sum of diagrams (c), (d), (e) and (i).

Application to control theory can be proposed in our model. To investigate whether our brain adopts the language of holography, we manipulate holograms composed of water media in a brain and check how our subjective experiences will change. To manipulate holograms non-invasively, we can adopt external electromagnetic fields, photons, ultrasound, and so on [42]. Adopting our model involving phonons, we obtain the control theory of non-invasive neural stimulation by external sound waves. Sound waves were found to induce changes of water conformational states in experimental study [87]. The sound perturbations with frequency $432\mathrm{Hz}$ and $440\mathrm{Hz}$ affect populations of ice-like (strongly bonded) water and vapor-like (less bonded) water shown in near-infrared spectroscopy with bandwidth related with symmetric and anti-symmetric stretching of water molecules. Hence we might be able to manipulate the density distributions of water molecules, namely transmittance of light or sound required in holographic memory storage. We then adopt reservoir computing or morphological computation approach [57,58,59] to control holograms, using sound. For example, to manipulate holograms of visual cortex involving water molecular distributions induced by pressure gradient, we calculate input external sound waves in scalp and impose the sound waves on scalp. Sound waves in target neocortex change pressure gradient or water molecular distributions, namely holograms of water media. Our model provides the control theory by developing QED theory with phonons in a hierarchy representing multiple layers such as scalp, skull, dura, celebrospinal fluid, and neocortex, as shown in [107].

8. Concluding Remarks and Perspectives

We have provided a theoretical formulation of the application of QED to the description of memory formation in the human brain where we extended previous quantum field theory models to account for the active role of water molecular states coupled with both photons and phonons in order to provide a more realistic representation for the mechanism involved. It is well-documented that the organization of water molecules in biological systems is much more orderly than in a fluid state and that this organized state of water, called Exclusion Zone (EZ) water [70] interacts in a distinct way with photons and phonons. Almost all of the water molecules in a living cells are present as EZ water [71].

Beginning with Lagrangian density, we have derived time-evolution equations for coherent fields and the Kadanoff–Baym equations for incoherent particles. Next, we have derived an acoustic super-radiance solution in cytoskeletons involving cylindrical structures in a brain like microtubules. We have shown the Kadanoff–Baym equations for incoherent charged bosons, photons, and phonons. We have also introduced a kinetic entropy current for Kadanoff–Baym equations and shown the H-theorem for self-energy in Hartree–Fock approximation. Finally, we have shown time-evolution equations in spatially homogeneous system and provided concrete forms of conserved charge and energy density. Our approach will be applied to systems composed of water coupled with sound and light, and specifically applied to a control theory of memory and consciousness physically represented by holograms of water system by external sound waves in non-invasive ultrasound neural stimulation.