Non-Equilibrium φ 4 Theory in a Hierarchy: Towards Manipulating Holograms in Quantum Brain Dynamics

: We describe non-equilibrium φ 4 theory in a hierarchical manner to develop a method for manipulating coherent ﬁelds as a toy model of introducing control into Quantum Field Theory (QFT) of the brain, which is called Quantum Brain Dynamics (QBD). We begin with the Lagrangian density of φ 4 model, where we adopt 2-Particle-Irreducible (2PI) effective action, and derive the Klein–Gordon equation of coherent ﬁelds with a damping term as an input–output equation proposed in areas of morphological computation or reservoir computing. Our analysis is extended to QFT in a hierarchy representing multiple layers covering cortex in a brain. We ﬁnd that the desired target function is achieved via time-evolution in the Klein–Gordon Eqs. in a hierarchy of numerical simulations when a signal in both the input and output prevails over noise in the intermediate layers. Our approach will be applied to control coherent ﬁelds in the systems (in a hierarchy) described in the QFT framework, with potential applications allowing the manipulation of quantum ﬁelds, especially holograms in QBD. We could then provide realistic physical degrees of freedom of a light–matter system in the contexts of quantum cognition and the associated free-energy principle.


Introduction
Quantum Field Theory (QFT) is a powerful tool to describe various phenomena in nature. It is applied in cosmology, elementary particle physics, nuclear physics, condensed matter physics, and biology. It can describe both microscopic degrees of freedom in Quantum Mechanics and macroscopic matter in Classical mechanics. (QFT is distinguished from Quantum mechanics, which can be applied only to the microscopic world.) In particular, its application to the brain might be a promising approach in describing the mechanism of memory formation in the human brain [1,2].
Quantum Field Theory of the brain or Quantum Brain Dynamics (QBD) can be traced back to the monumental work by Ricciardi and Umezawa in 1967 [3]. The brain is a mixed system of classical neurons and quantum degrees of freedom in QFT, namely corticons and exchange bosons [4,5]. The macroscopic vacua emerging in the spontaneous breakdown of symmetry (SBS) in QFT correspond to memory storage, and the creation of Nambu-Goldstone (NG) bosons emerging in SBS corresponds to memory retrieval. Since the vacua are maintained by long-range correlations by massless NG bosons, they describe nonlocal and stable features of memory. In 1968, Fröhlich suggested that the coherence with long-range correlation might occur in biological systems, resulting in boson condensation. As a result, the system with boson condensation operates as a single entity, called the Fröhlich condensate [6,7]. In 1976, Davydov and Kislukha proposed a model for a solitary wave propagation along protein α-helical structures, called the Davydov soliton [8]. The Fröhlich condensate and the Davydov soliton emerge as mirror images of each other, spontaneous photon emissions from microtubules in the brain's neurons. Interference of reference and object waves induces patterns of aligned water dipoles and dipoles in random directions. Then, step-function-like patterns of aligned and random patterns of dipoles might represent long-term holographic memory storage.
To propose an experimental design for the verification of QBD and holography hypothesis, we introduce control theory to manipulate coherent fields, which might correspond to subjective experiences, by external input fields. In experiments, visual subjective experiences can be controlled by dynamical stimulations of the visual cortex as shown in [31]. Our approach refers to morphological computation using an input-output equation to manipulate or control soft materials in [32], where it is possible to derive input function realizing the desired target function which can be time-dependent. We substitute the output function by target function in the input-output Equation to derive input function; next, we solve the input-output Equation with a derived input function and finally check whether a desired target function is achieved. The morphological computing corresponds to the reservoir computing theory [33,34], where we set input, spatio-temporal patterns in reservoirs and in the output, since soft materials correspond to physical reservoirs.
The aim of this paper is to describe a non-equilibrium φ 4 theory in a hierarchical manner for reservoir computing as a toy model for the control theory of QBD. In QBD, we adopt dipole-photon interactions as shown in Figure 1a, where dipoles in the ground state absorb photons and dipoles then transition to the first excited states, and reverse processes are allowed to occur [35]. Using this diagram, we can depict the loop-diagram shown in Figure 1b, which represents photon-photon interaction with four external lines. This diagram can be represented by the interaction term in the φ 4 model in Figure 1c. Then, the φ 4 might represent photon-photon interactions in QBD. Beginning with the Lagrangian density in the φ 4 theory, we derive the Klein-Gordon (KG) equations for the coherent field φ(x), the expectation value of the quantum field φ(x). The equations involve a damping term due to quantum fluctuations, which originates from the diagram in Figure 1d in the loop-expansion [36][37][38]. This damping term represents a field-particle conversion where the energy of coherent fields is transferred to that of incoherent particles. We also add an input function to the KG equations. Solving the equations with the damping term and the input function, we show how a target function is achieved by an input function. Our analysis is extended to KG equations in a hierarchy representing multiple layers covering cortex in a brain. We adopt the Klein-Gordon equations in a hierarchy involving the input, layers (reservoirs), and the output. We derive an input function to achieve the desired target function, then solve the Klein-Gordon Eqs. with a derived input function and check whether the target function is achieved as an output function in the time-evolution dynamics of the system. The target function is found to be achieved in a hierarchy for transmission less than its threshold where signal transfer prevails above noise in intermediate layers or reservoirs. Our approach might be applied to manipulating subjective experiences in the brain by external electromagnetic fields as input functions representing external stimuli for the visual cortex, auditory cortex, somatosensory area, and so on. It might also be applied to writing or controlling our hologram memory by external electromagnetic fields. It represents non-invasive control theory for holograms or subjective experiences in Quantum Brain Dynamics, which will be the subject of a future study. If our brain would adopt the language of holography, we could also find realistic physical degrees of freedom in the contexts of quantum cognition [39][40][41][42][43][44] and the free-energy principle [45,46]. This paper is organized as follows. In Section 2, we begin with the Lagrangian density and derive KG equations with an input function in a hierarchical manner. In Section 3, we show how a target function is achieved in time-evolution of the KG equations by numerical simulations. In Section 4, we discuss our numerical results and applications to manipulate holograms in QBD. In Section 5, we provide concluding remarks and perspectives. In this paper, we use the metric tensor η µν = η µν = diag(1, −1, −1) in 2 + 1 dimensions with the Greek letters (µ, ν) running over 0 to d in d + 1 dimensions and the subscripts (i, j) running over 1 to d. The subscripts (I, J) run over 1 to N layers in a hierarchy. We set the light speed, the Planck constant divided by 2π and the Boltzmann constant as 1.

Lagrangian Density and Klein-Gordon Equation
In this section, we begin with the Lagrangian density in the φ 4 theory, show a 2-Particle-Irreducible (2PI) effective action, and derive an input-output equation, namely the Klein-Gordon (KG) equation involving input function and a damping term due to quantum fluctuations of fields obeying the Kadanoff-Baym equation. We also extend our approach to KG equations in a hierarchy.
We begin with the Lagrangian density of the φ 4 model given by where m represents the mass of particles and λ represents the coupling constant of the interaction. We adopt the closed-time path (CTP) formalism to describe non-equilibrium phenomena in QFT. The CTP represents the path '1' from −∞ to ∞ and the path '2' from ∞ to −∞ [47,48]. Beginning with the Lagrangian density, we derive a generating functional for the connected loops of Feynman diagrams in the path integral, adopt the Legendre transformation of the generating functional, and then we can derive the 2PI effective action [49,50]. The 2PI effective action for this Lagrangian is derived by with the expectation value of the background quantum fieldφ = φ and the Green's function G(x, y) = T C (δφ(x)δφ(y)) with the notation for the expectation value · , timeordered product T C in CTP and fluctuations δφ(x) ≡ φ(x) −φ(x). The inverse of Green's function iG −1 0 in the above equation is written as with the delta function δ C (x − y) in CTP. Due to the Legendre transformation from the generating functional to the 2PI effective action, we can derive the following relations, for the vanishing source terms on the right-hand side. The Equation (5) represents the Kadanoff-Baym equation for Green's function G(x, y) [36][37][38]51,52] written as with self-energy Σ ≡ iδΓ 2 /δG. The Green's function G(x, y) has two independent components for anti-commutation and commutation relations of δφ(x) and δφ(y), namely the statistical function F(x, y) including information of particle distributions and the spectral function ρ(x, y) involving information of dispersion relations and the spectral width, defined as These functions can be Fourier-transformed as in spatially homogeneous systems with times x 0 = t and y 0 = t . The statistical function F(t, t ; k) is symmetric F(t, t ; k) = F(t , t; k) for the interchange of t and t , while the spectral function ρ(t, t ; k) is anti-symmetric ρ(t, t ; k) = −ρ(t , t; k). The Equation (4) provides the Klein-Gordon equation for coherent fieldφ as The term λ 2 F(x, x)φ(x) in Equation (11) can be absorbed to the term m 2φ . Here the Feynman diagram of iΓ 2 /2 is depicted in Figure 1d.
as shown in [37,38]. The term − δΓ 2 δφ(x) in Equation (11) can be rewritten as where we have used the relations ReΣ 0 . The 2nd term in the above equation represents the damping term in the Klein-Gordon equation with the damping factor γ. The damping factor appears due to the field-particle conversion where coherent fields are damped and incoherent particles described by Green's functions are produced. It is dependent on the coupling constant λ and temperature of incoherent particles with thermal equilibrium distributions or Bose-Einstein distributions. The larger the λ and the temperature are, the larger the damping factor γ is.
The KG equation which we should solve is given by with the spatial label i = 1, 2, the input function u(x) as an external source and the coupling constant v. The 2nd line represents an input-output equation. When the stationary target function is given byφ s (x), the input function u(x) is written as Next, we show the Klein-Gordon equation in a hierarchy in Figure 2. The timeevolution equations in a hierarchy are, where I = 1, 2, · · ·N with fixedφ N+1 = 0 andφ 0 = u 0 representing the input u 0 . The origin of the term v(φ I+1 +φ I−1 ) is the transmission between φ I 's in Ith system represented by the transmission Lagrangian term L tra = ∑ I vφ I φ I+1 with the transmission parameter v.
The input functions are calculated by where J = 1, 2, · · ·, N, fixed u N+1 = 0 and target function u N =φ s . We use the input functionφ 0 = u 0 derived in Equation (17) as a boundary condition in solving Equation (16).

Numerical Results
In this section, we show how the desired target function is achieved using the Klein-Gordon equations in a hierarchy. Non-equilibrium processes are described in the timeevolution where coherent fieldsφ(x), expectation values in coherent states, evolve in time. Since the time-evolving coherent fields correspond to time-evolving coherent states, dynamically evolving coherent states, involving boson condensation of an infinite number of particles on the vacua [53], are traced in our non-equilibrium approach.
We set a two-dimensional spatial lattice by x i = −N s a s , −(N s − 1)a s , · · ·, n i a s , · · ·(N s − 1)a s , N s a s with discrete labels n i for x i with i = 1, 2, lattice size 2N s = 128, and lattice spacing a s m = 1.0 scaled by mass m. Periodic boundary conditions for spatial coordinates x i are adopted. We prepare time-step a t as a t /a s = 0.001. We shall investigate the number of layers N = 5. We then prepare coupling λ/m = 1.0, transmission v s /m 2 = 0.4 and damping factor γ/m = 0.2. To determine the time-evolution of the system, the fourth-order Runge-Kutta method is adopted.
The desired target functionφ s is set as where we omit the scaling factor 1/ √ m. We calculate u 0 using Equation (17). The input function u 0 for N = 5 is shown in Figure 3. We find sawtooth waveform in u 0 compared with the cosine curve multiplied by 0.1 in Equation (18). The maximum value in u 0 is approximately 20 since the input function can be approximately derived by multiplyingφ s by the factor (1/v) N , with which we find 0 We set initial conditions ofφ I for Ith layers (I = 0, 1, · · ·N) as Initial conditions for the time derivatives are for J = 0, · · ·, N + 1. We fixφ 0 (x) = u 0 (x), ∂φ 0 ∂x 0 = 0,φ N+1 = 0 and ∂φ N+1 ∂x 0 = 0 for any time point.
We solve Klein-Gordon equations in a hierarchy (16) with I = 1, 2, · · ·, N with N = 5. In Figure 4, we show the time-evolution of distributionsφ N depicted every three spatial points. At mx 0 = 0.0, we prepare a square-shaped distribution as an initial condition in Equation (19). Its maximum value at the peaks is 0.1. Theφ values at around the peaks at mx 0 = 0.0 start oscillating with frequency m in their time-evolution and tend to become negative at mx 0 = 3.0, which is nearly equal to π. The negative bottom is approximately −0.054. At mx 0 = 4.0, we find that the cosine curve gradually appears although the shape of the initial condition is still unchanged. At mx 0 = 5.0, the shape of the initial condition tends to disappear since the inverse of the damping factor γ = 0.2m is m/γ = 1/0.2 = 5.0. The sawtooth waveform appears at mx 0 = 5.0 similar to the waveform u 0 . The shape gradually approaches the cosine curve at mx 0 = 40. The fluctuations around the cosine curve are found to appear at this time. The values of peaks at mx 0 = 40 are larger than the amplitude 0.1 in target functionφ s in Equation (18). At mx 0 = 80, the shape tends to be the target function or a cosine curve with amplitude 0.1. In Figure 6, we show the distribution of the coherent field for v/m 2 = 0.4, 0.5, 0.57, 0.58, 0.59 and 0.6 for N = 5, We find thatφ N 's at mx 0 = 80 for v/m 2 = 0.4, 0.5, 0.57 and 0.58 converge to the target function. The deviations from the target function cannot be clearly seen. Contrary to that,φ N 's at mx 0 = 80 for v/m 2 = 0.59 and 0.6 do not have the waveform of the target function. Comparing it with the cosine function −0.3 cos(2πn 1 /N s ) for v/m 2 = 0.6, we find that the shape seems to be near step-function-like forms. The threshold of whether the convergence to the target function is achieved is approximately 0.58. We shall also investigate cases for N = 6 for the target function in Equation (18) and initial conditions in Equations (19) and (20). We investigate several cases of transmissions v/m 2 = 0.5, 0.55, 0.56, and 0.57 with λ/m = 1.0 and γ/m = 0.2. In Figure 7, we show the time-evolution ofφ N (x 0 , n 1 = 32, n 2 = 0) for v/m 2 = 0.5, 0.55, 0.56, and 0.57 for N = 6. For v/m 2 = 0.5, 0.55, and 0.56, theφ N (x 0 , n 1 = 32, n 2 = 0) starts from 0.1 and tends to converge to −0.1. As we decrease v/m 2 , fluctuations in the time-evolution become larger. On the contrary,φ N (x 0 , n 1 = 32, n 2 = 0) tends to converge to 0.21 for v/m 2 = 0.57. This means thatφ N does not converge to the target function for v/m 2 = 0.57. The threshold of whether the convergence to the target function is achieved in the time-evolution is approximately 0.56. As the number of layers N increases, the threshold for v seems to decrease gradually.

Discussion
In this paper, we have investigated non-equilibrium φ 4 theory in a hierarchy as a toy model of control theory to manipulate holograms in Quantum Brain Dynamics (QBD). We have introduced the Lagrangian density of φ 4 theory and derived the Klein-Gordon (KG) equation with a damping term, which originated from the field-particle conversion where damped oscillations of coherent fields occur and incoherent particles are produced from coherent fields. We have added an input function u as an external source of coherent fields, which might represent an external electromagnetic field in QBD. We have subsequently extended the equation to the KG equations in a hierarchy representing layers covering the cortex area in the human brain, where the number of layers is N. Then, we have derived the input function u 0 to achieve the convergence to the desired target function in Nth layer. Using the derived input function and solving KG equations, we have investigated whether the coherent fieldφ N in the Nth layer converges to the target function in time-evolution using numerical simulations. We have found that the convergence to the target function can be achieved for the transmission parameter v whose value is below the threshold.
We discuss the convergence and uniqueness in time-evolution of coherent fields. It is straightforward to investigate the case N = 1. Substituting the input function in Equation (15) into the input-output Equation (14), we can derive the following timeevolution equation, with ∆φ ≡φ −φ s . The above equation, except for the nonlinear term ∼φ 3 , represents the damped oscillation for ∆φ in time-evolution. Since ∆φ converges to zero in time-evolution, theφ converges to the target functionφ s . Even if a nonlinear term exists, the results for convergence do not change. Since m 2φ + λ 6φ 3 is a monotonically increasing function forφ, φ has one-to-one correspondence to m 2φ + λ 6φ 3 . When m 2φ + λ 6φ 3 is equal to m 2φ s + λ 6φ 3 s , φ is equal toφ s . (Or, since we can write φ 3 −φ 3 s = ∆φ φ 2 +φφ s +φ 2 s , a nonlinear term will be a correction term to m 2 ∆φ in damped oscillations.) Then, the uniqueness of the output function in convergence is achieved for a given target functionφ s .
For the number of layers N > 1 in a hierarchy, we found the threshold for the transmission parameter v of whether the convergence to target function is achieved. When transmission v is less than the threshold, the convergence to the target function is achieved. The threshold in a hierarchy might be estimated as follows. We first investigate the timeevolution equation, The solution of this equation can be written as with φ c = A cos(mx 0 ) + B sin(mx 0 ) with constants A and B, and with the Fourier transfor- due toũ(ω) = 2πδ(ω)u(x). The outputφ is expressed as the sum ofφ c and the external input u. We next investigate the time-evolution equation, in a hierarchy. In a similar way to the above derivation, we find, whereφ I,c represents A I cos(mx 0 ) + B I sin(mx 0 ) with constants A I and B I and we have omitted the scaling factor 1/m 2 . In N = 5, the output functionφ N can be expanded as Here, due to u 0 ∼ 1 v 5φs , we can show v 5 u 0 ∼ 1 ×φ s . In the above equation, the first and second terms on the right-hand side represent the signal information. On the other hand, the third, fourth, fifth and sixth terms in intermediate layers are regarded as noise.
We shall estimate the order of signal and noise by taking a sum of coefficients where the coefficient in v 5 u 0 is 1. The signal and noise function with f signal and f noise are written as as a function of transmission parameter v. We show the values of the signal and noise function in Table 1.
58. Yet, f noise (v) prevails over f signal (v) in v = 0.59 and 0.6. The threshold for whether the convergence to the target function is achieved is v = 0.58 in numerical simulations for N = 5 in the previous section. The threshold for numerical simulations corresponds to the upper limit where f signal (v) prevails over f noise (v) in Table 1. We also investigate the case N = 6. In a similar way to the above derivation, we can derive, We then find for N = 6. We show f signal and f noise in Table 2. In this table, we find that f signal (v) prevails over f noise (v) in v ≤ 0.56. This value corresponds to the upper limit for the convergence to the target function 0.56 in the previous section. We might be able to derive the threshold in the above analysis. As the number of layers increases, the threshold decreases gradually. To achieve the convergence to the target function, small transmission parameters where signal prevails over noise are required. In controlling holograms or subjective experiences in the brain, we might qualitatively need to select electromagnetic waves with smaller transmission and external electromagnetic fields above the threshold. If transmission of magnetic fields is large in the brain where the noise prevails over signal, we propose to select electric fields with small transmission.
The geometry is also of significance for control theory. In this paper, we investigate flat two-dimensional surfaces or multiple flat layers in a hierarchy. Manipulating holograms in a flat surface, we propose to control our subjective experiences and to overwrite memory by external electromagnetic fields non-invasively. However, if holographic memory is not on flat surfaces, the situations for control theory will change. There are several structures that might memorize information, such as spherical, toroidal and cylindrical shapes for neurons, glia cells and microtubules as a candidate structures to record memory. To control holograms with these structures, we might need a hemispherical surface headset covering our head to achieve the convergence to the target functions on three-dimensional structures for cells and their cytoskeletons. We then need three-dimensional control theory to construct target functions for target areas in the brain.
McFadden proposed a conscious electromagnetic information field in [54,55]. Electromagnetic fields might be a candidate to solve the binding problem, namely how the brain integrates parallel processing in various diffused areas. Since electromagnetic fields without media propagate at the speed of light and are not localized, we need to introduce media with water dipoles so that photons acquire a mass in the brain [1]. We can adopt the Higgs mechanism in QBD where Nambu-Goldstone modes are absorbed by photons fields and photons acquire mass. The massive photons are called evanescent photons. The maximum mass is found to be · N V = 50 meV with the dipole moment of a water molecule 2ed e (elementary charge e = 0.3 and the distance d e = 0.2 Å), the moment of inertia I with 1/I = 4 meV and the number density of coherent water N/V = 3.3 × 10 28 m −3 by estimating the Meissner effect of electric fields in the Klein-Gordon equations [35]. Then, we might be able to adopt the integrated version of the holographic brain theory proposed by Pribram and QBD with water dipoles and evanescent photons. Our subjective experiences and memories might be represented by holograms within this type of theory. To investigate whether our brain encodes information using the language of holography, we need the control theory of holograms using external electromagnetic fields.
When our memory-inducing subjective experiences are manipulated, we might find one-to-one correspondence between subjective experiences in the mind and holograms constructed in physical light-matter systems in the brain. Our approach might represent a reductionism of subjective experiences to holograms induced by quantum fields with light-matter, or contribute to mind-matter unification in panpsychism. Mind-light-matter unification is proposed in [56], where light plays a role of the bridge between mind and matter. Mind-matter unification would be impossible without the bridging role of light. As suggested in [57], "quantum intelligence" (QI), a novel quantum-computing prototype, aims to clarify the concept of causality. We could then reach logically definable causality and mind-light-matter unity. Holographic extension using bipolar qubits in QI is also possible. One could adopt a holographic approach to brain dynamics involving logical reasoning provided by logically definable causality. The mind-brain relationship might be described in quantum theory where measurement processes convert several possibilities in superposition states to actual observed events acausally, as proposed in [58]. Quantum approaches break the causal closure of deterministic Newtonian or Classical mechanics and provide new explanations of mind and matter.
When our brain adopts the language of holography, we can provide realistic physical degrees of freedom for quantum cognition and the free-energy principle. In quantum cognition, quantum-like mathematical models are adopted for our decision-making not as physico-chemical approach but as information-theoretical approach [39][40][41][42][43][44]. Quantum interference between states describes irrational decision-making with fallacy, the violation of the sure-thing principle, quantum-like information processing, and so on. The freeenergy principle as a mathematical approach suggests that our brain function adopts the minimization of free energy in biological information processing [45,46]. The minimization of free energy is related to Bayesian inference in which causes of events c with the prior probability p(c) are inferred by given data d in resultant events in Bayes' theorem. This minimization indicates the case that the posterior probability p(c|d) (conditional probability of causes c in given data d occurring) = the prior probability p(c). In adopting the excess Bayesian inference involving a quantum logic implemented in [59] instead of classical Bayesian inference, the free-energy principle is connected to quantum cognition. Although quantum-like states are adopted in a mathematical model on quantum cognition, we can assign holograms (constructed by realistic physical degrees of freedom of light-matter) to quantum-like states in quantum cognition. We consider the superposition state of a photon propagating in two pathways |0 and |1 . If the 0 and 1 pathways are exposed on holograms 0 and 1, respectively, we can consider the entanglement given by |0 + |1 → |0 |holo 0 + |1 |holo 1 , for photons in propagating through holograms 0 and 1. In holography, the optical information propagating through holograms can perform information processing by filtering. After filtering processes, the quantum state of a photon might be processed as |0 |holo 0 + |1 |holo 0 → |0 |holo 0 |decision 0 + |1 |holo 1 |decision 1 .
Due to the measurement processes, the decision is made. Increasing the number of photons, we then take the ensemble average and find the probability of our decisions. Although we have investigated the control theory in φ 4 model as a toy model of a lightmatter system in this paper, our approach will be extended to more realistic QBD. We could then provide realistic physical degrees of freedom in the contexts of quantum cognition and the free-energy principle.

Concluding Remarks and Perspectives
We have introduced control theory within Quantum Field Theory in a hierarchy, namely an input layer, intermediate layers and an output layer, using the Klein-Gordon equations with external input function. We have referred to morphological computation or reservoir computing using input-output equations. When signal in the input and output layers prevails over noise in intermediate layers, the convergence to the target function is achieved. Our analysis will be extended to control theory in holograms within Quantum Brain Dynamics in a future study to investigate whether our brain employs the language of holography for information storage and retrieval. It could provide realistic physical degrees of freedom in the contexts of quantum cognition and the free-energy principle.