# A Quantum-Classical Model of Brain Dynamics

Alessandro Sergi  Dipartimento di Scienze Matematiche e Informatiche, Scienze
Fisiche e Scienze della Terra, Università degli Studi di Messina,s viale F.
Stagno d’Alcontres 31, 98166 Messina, Italy; E-mail<EMAIL_ADDRESS>Institute of Systems Science, Durban University of Technology, P.O. Box 1334,
Durban 4000, South Africa  Antonino Messina  Dipartimento di Matematica ed
Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123
Palermo, Italy; E-mail<EMAIL_ADDRESS>Carmelo M. Vicario
Dipartimento di Scienze Cognitive, Psicologiche, Pedagogiche e degli Studi
Culturali, Università degli Studi di Messina, Via Concezione 6, 98121 Messina,
Italy; E-mail<EMAIL_ADDRESS>Gabriella Martino  Dipartimento
di Medicina e Clinica Sperimentale, Università degli Studi di Messina, A.O.U.
“G. Martino”, Via Consolare Valeria, 98125 Messina, Italy; E-mail:
<EMAIL_ADDRESS>
###### Abstract

The study of the human psyche has elucidated a bipartite structure of logic
reflecting the quantum–classical nature of the world. Accordingly, we posited
an approach toward studying the brain by means of the quantum–classical
dynamics of a mixed Weyl symbol. The mixed Weyl symbol can be used to describe
brain processes at the microscopic level and, when averaged over an
appropriate ensemble, can provide a link to the results of measurements made
at the meso and macro scale. Within this approach, quantum variables (such as,
for example, nuclear and electron spins, dipole momenta of particles or
molecules, tunneling degrees of freedom, and so on) can be represented by
spinors, whereas the electromagnetic fields and phonon modes can be treated
either classically or semi-classically in phase space by also considering
quantum zero-point fluctuations. Quantum zero-point effects can be
incorporated into numerical simulations by controlling the temperature of each
field mode via coupling to a dedicated Nosé-Hoover chain thermostat. The
temperature of each thermostat was chosen in order to reproduce quantum
statistics in the canonical ensemble. In this first paper, we introduce a
general quantum–classical Hamiltonian model that can be tailored to study
physical processes at the interface between the quantum and the classical
world in the brain. While the approach is discussed in detail, numerical
calculations are not reported in the present paper, but they are planned for
future work. Our theory of brain dynamics subsumes some compatible aspects of
three well-known quantum approaches to brain dynamics, namely the
electromagnetic field theory approach, the orchestrated objective reduction
theory, and the dissipative quantum model of the brain. All three models are
reviewed.

##  I Introduction

The human brain is perhaps the most complicated known condensed matter system.
It contains approximately $10^{2}$ billions of neurons and at least as many
glia cells vonbartheld . The brain is composed of 77 to 78% water, 10 to 12%
lipids, 8% proteins, 2% soluble organic substances, and 1% carbohydrates and
inorganic salts McIlwain . It is also extremely fascinating that higher brain
functions precisely define what it means to be human. Brain states and their
dynamics have so far eluded physical understanding based on molecular models.
This means that one cannot describe brain dynamics by brute force, i.e.,
starting from the behavior of all atoms and deriving macroscopic time
evolution. The problem does not only reside in the sheer number of microscopic
constituents of the brain. Some of the most complex brain functions are
delocalized over long distances and require synchronization processes that do
not seem easy to explain only by means of the classical mechanics of atoms and
molecules. In particular, the wholeness of perception requires integrating the
activity of an enormous number of brain cells. Ultimately, we would like to
develop a theory of brain processes where mesoscopic models can be constructed
from atomistic dynamics by means of controlled approximations. With respect to
this, quantum models mcfadden ; mcfadden2 ; mcfadden3 ; pockett ; pockett2 ;
pockett3 ; liboff ; liboff2 ; frohlich ; ph ; ph-2 ; ph-3 ; penrose ; penrose2
; penrose3 ; penrose4 ; penrose5 ; microtubulines_channels ;
microtubulines_channels-2 ; fisher ; fisher2 ; fisher4 ; kerskens ; qc-bneuro-
comp ; vitiello1995 ; pessa-vitiello ; alfinito-vitiello ; freeman-vitiello ;
freeman-vitiello2 ; freeman-vitiello3 ; vitiello-fractals ; vitiello-cortex ;
sabbadini-vitiello  may hold the key to a possible microscopic understanding
of some brain functions.

We have found that some psychological theories are based on a bi-partite logic
science_and_sanity ; drive_yourself_sane ; collected ;
stealing_general_semantics ; pauli ; fuchs ; lindorff ; synchronicity ; jung-
pauli ; blanco ; blanco2 ; rayner ; lombardi  that is very similar to the
logic of quantum–classical mechanics. We do wish to make clear at the very
beginning that, in this paper, the words “psyche”, “psychological”, and the
like are not used to address any metaphysical level of ‘reality’. While we
acknowledge that such concepts lack, at the moment, both quantitative
definitions and complete explanations in terms of biomolecular processes, it
must be emphasized that neuroscience neuroscience  is continuously advancing
toward the inclusion of psychological phenomena within the boundaries of
quantitative science. Thus, once our use of these words is understood, it
becomes easier to accept the idea that psychological theories and clinical
psychology could feed the synergistic growing of translational neuroscience
trans1 ; trans2 ; trans3 ; trans4 ; trans5 ; trans6 ; trans7 , quantum models
of decision-making tanaka ; khrennikov ; khrennikov2 ; busemeyer-book ;
khrennikov3 ; bond ; basieva ; busemeyer ; vandernoort , and quantum
information biology qib .

In recent years, three main quantum models of the brain have been introduced
in the literature. These are the electromagnetic field (EMF) approach mcfadden
; mcfadden2 ; mcfadden3 ; pockett ; pockett2 ; pockett3 ; liboff ; liboff2 ;
frohlich , the orchestrated objective reduction (Orch OR) theory ph ; ph-2 ;
ph-3 ; penrose ; penrose2 ; penrose3 ; penrose4 ; penrose5 ;
microtubulines_channels ; microtubulines_channels-2 ; fisher ; fisher2 ;
fisher4 ; kerskens ; qc-bneuro-comp , and the dissipative quantum model of
brain (DQMB) vitiello1995 ; pessa-vitiello ; alfinito-vitiello ; freeman-
vitiello ; freeman-vitiello2 ; freeman-vitiello3 ; vitiello-fractals ;
vitiello-cortex ; sabbadini-vitiello . Even if there are several key
differences between the EMF, the Orch OR approaches, and DQMB, these three
theories study the brain from the perspective of condensed matter physics and
matter–EMF interactions. While DQMB vitiello1995 ; pessa-vitiello ; alfinito-
vitiello ; freeman-vitiello ; freeman-vitiello2 ; freeman-vitiello3 ;
vitiello-fractals ; vitiello-cortex ; sabbadini-vitiello  is mainly concerned
with the explanation of memory storage and retrieval, long-range correlations
between brain clusters of cells and brain correlates of perception, both the
EMF mcfadden ; mcfadden2 ; mcfadden3 ; pockett ; pockett2 ; pockett3 ; liboff
; liboff2 ; frohlich  and Orch OR ph ; ph-2 ; ph-3 ; penrose ; penrose2 ;
penrose3 ; penrose4 ; penrose5 ; microtubulines_channels ;
microtubulines_channels-2 ; fisher ; fisher2 ; fisher4 ; kerskens ; qc-bneuro-
comp  models were originally introduced for explaining consciousness. With
regard to this, we want to state very clearly that our theory does not aim in
any way to explain consciousness. Instead, we stress that our target is only
to study brain dynamics in terms of physical processes. In practice, we only
consider EMF mcfadden ; mcfadden2 ; mcfadden3 ; pockett ; pockett2 ; pockett3
; liboff ; liboff2 ; frohlich  and Orch OR ph ; ph-2 ; ph-3 ; penrose ;
penrose2 ; penrose3 ; penrose4 ; penrose5 ; microtubulines_channels ;
microtubulines_channels-2 ; fisher ; fisher2 ; fisher4 ; kerskens ; qc-bneuro-
comp  insofar that they can be used as microscopic theories of physical
processes in the brain.

Motivated by the germinal considerations in Refs. mcfadden ; mcfadden2 ;
mcfadden3 ; pockett ; pockett2 ; pockett3 ; liboff ; liboff2 ; frohlich ; ph ;
ph-2 ; ph-3 ; penrose ; penrose2 ; penrose3 ; penrose4 ; penrose5 ;
microtubulines_channels ; microtubulines_channels-2 ; fisher ; fisher2 ;
fisher4 ; kerskens ; qc-bneuro-comp ; vitiello1995 ; pessa-vitiello ;
alfinito-vitiello ; freeman-vitiello ; freeman-vitiello2 ; freeman-vitiello3 ;
vitiello-fractals ; vitiello-cortex ; sabbadini-vitiello  and by the idea of a
bipartite structure of logic science_and_sanity ; drive_yourself_sane ;
collected ; stealing_general_semantics ; pauli ; fuchs ; lindorff ;
synchronicity ; jung-pauli ; blanco ; blanco2 ; rayner ; lombardi , in this
paper, we introduce an explicit quantum–classical model of brain dynamics.
Such a model is based on the hybrid quantum–classical (QC) formalism of Refs.
silin ; rukhazade ; balescu ; zhang-balescu ; balescu-zhang ; aleksandrov ;
gerasimenko ; boucher ; petrina ; prezhdo ; kapracicco ; nielsen ; as4 ; as6 ;
as14 ; osborn ; martens ; donoso ; as3 ; as13 ; as7 ; ak . In many QC
theories, the nature of the interaction between the classical and quantum
subsystems is somewhat unclear and the quantum variables are not treated on
the same footing as the classical DOF. The formulation of Refs. silin ;
rukhazade ; balescu ; zhang-balescu ; balescu-zhang ; aleksandrov ;
gerasimenko ; boucher ; petrina ; prezhdo ; kapracicco ; nielsen ; as4 ; as6 ;
as14 ; osborn ; martens ; donoso ; as3 ; as13 ; as7 ; ak  is based on mixed
Weyl symbols and is conceptually free from these drawbacks. In fact, such an
approach is founded upon a statistical operator depending parametrically on
phase space points. This implies that the dynamics must be considered at each
phase space point without the possibility of separating quantum dynamics from
the classical-like dynamics of the phase space. QC spin–boson models leggett ;
bakermeier ; plenio ; finney , and their non-linear extension ak , are
appropriate for describing a finite number of quantum variables coupled to a
classical DOF. Non-Hamiltonian deterministic thermostats as4 ; as6 ; as14 ;
nhc ; b1 ; b2  can be used to formulate the dissipative dynamics of mixed Weyl
symbols under constant temperature conditions. The QC formalism simplifies
numerical calculations of averages and response functions. In turn, response
functions can be compared to electromagnetic signals that could be provided by
macroscopic experiments on the brain riddle ; riddle2 ; abubaker ; croce ;
caruana ; nitsche-paulus ; stagg-nitsche ; papazova ; yavari ; frohlich ; koch
; martinez-banaclocha ; pinotsis ; vicario2020a ; ney2021 ; markovic2021 ;
vicario2020b ; nunez .

We are interested in studying those brain processes that can be described in
terms of a few quantum variables embedded in a classical environment. Small
numbers of quantum particles are naturally found in small biological
structures microtubulines_channels ; microtubulines_channels-2 , and from such
a scale until that of atomic nuclei fisher ; fisher2 ; fisher4 . Even a small
number of quantum variables can have a significant effect on the dynamics of
large classical systems by means of four mechanisms. One is given by non-
adiabatic transitions between energy states osborn ; martens ; donoso ; as3 ;
as13 ; as7 ; ak . The second one is caused by the stochastic collapse of the
wave function stapp ; vonneumann . The third one is generated by the motion of
quantum sources of the electromagnetic fields in the brain. The fourth one is
the famous ‘order from order’ mechanism elaborated on by Schrödinger
what_is_life , which led to the discovery of DNA dna ; dna2 . All of these
mechanisms are in agreement with Pascual Jordan’s idea beyler ; beyler2 ;
mcfadden-alkhalili  about the necessary role of the amplification of quantum
processes in order to steer classical dynamics in biological environments.
Hence, one can consider that single quantum particles, such as electrons and
protons, retain quantum properties osborn ; martens ; donoso ; as3 ; as13 ;
as7 ; ak  at every temperature mcfadden-book . For such a reason, our QC
approach can cope ‘almost by design’ with the controversial issue of
decoherence joos ; zurek-2003 ; tegmark ; microtubulines_channels  in warm and
wet environments, such as those found in biological systems. It has also been
proposed that some kind of quantum computation nielsen-chuang ; jaeger  might
take place in the brain microtubulines_channels ; fisher ; fisher2 ; kerskens
; qc-bneuro-comp . There are a few proposals regarding quantum computational
schemes performed by means of mixed states no-ent-comp ; no-ent-comp2 ; no-
ent-comp3 ; no-ent-comp4 ; however, the mainstream concept of quantum
computation requires entangled states nielsen-chuang ; jaeger . Entanglement
is fragile and, for quantum computation to be realized at biological
conditions, decoherence joos ; zurek-2003  should not destroy the phase
coherence of quantum states at a high temperature tegmark . Such a possibility
is still a matter of debate microtubulines_channels . For such a reason, in
this paper, we do not take a quantum informational perspective nielsen-chuang
; jaeger .

The paper is structured as follows. We present the historic evolution of
logic’s bipartite structure by discussing general semantics (GS) in Section
II.1, synchronicity in Section II.2, and Blanco’s bi-logic in Section II.3. In
Section III, we discuss a set of ideas in favor of quantum mechanical effects
in the brain. We review the EMF approach in Section IV, Orch OR in Section V,
and DQMB in Section VI. Our QC approach is presented in Section VII. Finally,
our conclusions are given in Section VIII.

##  II The Bipartite Structure of Psychology as the Root for Quantum–Classical
Models of the Brain

The knowledge that some theories of the human psyche suggest that logic has a
bi-partite structure science_and_sanity ; drive_yourself_sane ; collected ;
stealing_general_semantics ; pauli ; fuchs ; lindorff ; synchronicity ; jung-
pauli ; blanco ; blanco2 ; rayner ; lombardi , paralleling that of the
quantum–classical world, is, for us, one of the inspiring motivations to take
the first steps toward the elaboration of a quantum–classical model of brain
dynamics. As is clarified in the following, bi-logic comprises Aristotelian
and non-Aristotelian logics. Discussions on the bi-partite structure of logic
can also be of interest to quantum models of decision making tanaka ;
khrennikov ; khrennikov2 ; busemeyer-book ; khrennikov3 ; bond ; basieva ;
busemeyer ; vandernoort , quantum information biology qib , and translational
neuroscience trans1 ; trans2 ; trans3 ; trans4 ; trans5 ; trans6 ; trans7 .

In the field of clinical psychology, Korzybski was one of the pioneers making
use of non-Aristotelian logic science_and_sanity ; drive_yourself_sane ;
collected ; stealing_general_semantics  for therapeutical applications. A more
abstract and somewhat implicit approach to such a bi-partite structure of
logic can be found in the work of Jung and Pauli pauli ; fuchs ; lindorff ;
synchronicity ; jung-pauli . Instead, the most complete formulation of bi-
logic and its application to clinical psychology (until now) is found in the
work of Blanco blanco ; blanco2 ; rayner ; lombardi , where it is called bi-
logic. Since, in QM, the law of the excluded middle is not valid, so that, as
in the famous Schrödinger’s example cat , a cat can be both alive and dead
nielsen-chuang ; jaeger , at a fundamental level, quantum logic is non-
Aristotelian, and we propose to identify it with Blanco’s bi-logic. If the
quantum world does not follow Aristotelian logic, given that QM cannot indeed
be separated from the classical world, because of the processes of
‘measurement’ and the stochastic collapse of the wave function stapp ;
vonneumann , then physical processes must have a hybrid structure, where both
Aristotelian and non-Aristotelian logic must be employed. Thus, we sustain
that quantum physical processes can be described by hybrid quantum–classical
models. Since bi-logic blanco ; blanco2 ; rayner ; lombardi  can be likened to
a quantum–classical worldview and there are already models supporting the
quantum–classical nature of the brain mcfadden ; mcfadden2 ; mcfadden3 ;
pockett ; pockett2 ; pockett3 ; liboff ; liboff2 ; frohlich ; ph ; ph-2 ; ph-3
; penrose ; penrose2 ; penrose3 ; penrose4 ; penrose5 ;
microtubulines_channels ; microtubulines_channels-2 ; fisher ; fisher2 ;
fisher4 ; kerskens ; qc-bneuro-comp ; vitiello1995 ; pessa-vitiello ;
alfinito-vitiello ; freeman-vitiello ; freeman-vitiello2 ; freeman-vitiello3 ;
vitiello-fractals ; vitiello-cortex ; sabbadini-vitiello , the idea of
developing models by means of an explicit quantum–classical theory naturally
arises. Our parallelism between bi-logic blanco ; blanco2 ; rayner ; lombardi
and quantum–classical phenomena in the brain can also be considered as the
motivation for extending the current quantum-like models of cognition tanaka ;
khrennikov ; khrennikov2 ; busemeyer-book ; khrennikov3 ; bond ; basieva ;
busemeyer ; vandernoort  to take into account quantum–classical processes. In
the remaining part of this section, we discuss the historical development of
non-Aristotelian logic in psychology and clinical psychology.

###  II.1 General Semantics

Roughly speaking, GS is a specific instance of clinical psychology with the
specific goal of improving mental health and adaptation to the world
science_and_sanity ; drive_yourself_sane . One key aspect of this approach is
the claim that non-Aristotelian logic conforms more to reality. Once non-
Aristotelian logic is accepted as the correct way of thinking, our language
must be adjusted accordingly. GS was created with the logical structure of QM
as a template science_and_sanity . An interesting connection of GS to quantum
models of decision making, which is yet to be fully explored, may be founded
on the free energy principle friston ; sanchez-canizares . However, a first
application of this principle to quantum decision making can be found in Ref.
tanaka .

Although it constitutes the historical roots of many systems of clinical
psychology, GS is rarely acknowledged stealing_general_semantics . The
premises of GS are “A map is not the territory”, “A map does not represent all
of a territory”, and “A map is self-reflexive”, meaning that an ’ideal’ map
would include a map of the map, etc., indefinitely” collected . These
assumptions can be translated to daily life in order to improve the mental
sanity of human beings drive_yourself_sane . In this case, GS premises become
“A word is not what it represents”, “A word does not represent all of the
facts”, and “Language is self-reflexive” in the sense that, in language, we
can speak about language. Alas, human being reactions to verbal communication
are largely based on unconscious beliefs, violating the first two assumptions
and disregarding the third. Mathematics and GS are the only languages that
rigorously take into account the above non-Aristotelian premises at all times.
For such a reason, Korbizski strongly suggested to psychologists to study
mathematical structures. On page 280 of his Science and Sanity
science_and_sanity , we find a discussion of the importance of the theory of
aggregates and the theory of groups in psychology, something that will be
further examined in Blanco’s bi-logic blanco ; blanco2 ; rayner ; lombardi .

At variance with the general case stealing_general_semantics , there are some
instances in which the influence of Korzybski’s GS on various approaches is
properly acknowledged. For example, Ellis acknowledges Korzybski’s influence
on his rational emotive behavior therapy ellis . Almost similarly, Wysong pays
the dues of Gestalt therapy to GS by writing a commentary in The Gestalt
Journal wysong . How much Gestalt therapy owes to GS is also discussed in the
thesis of Allen Richard Barlow barlow , which is downloadable from The
University of Wollongong Thesis Collection online. One of many counter-
examples stealing_general_semantics  is given by family therapy minuchin ;
bowen , where it is stressed that one must be aware of abstractions leading to
disregarding the wholeness of processes minuchin  (non-elementalism
science_and_sanity ; drive_yourself_sane ), and the difference between the
verbal and the non-verbal bowen  is also underlined, but without citing GS.
Hence, GS may be considered (either directly or indirectly) as the hidden root
of various therapeutic practices.

Given the above discussion, it is not difficult to see the logical connections
between GS and QM. If we consider that scientific theories are “maps” of
reality, with classical theories providing a first level of abstraction, QM is
clearly characterized by a second level of abstraction. QM does not provide
laws for the dynamics of models of phenomena. QM gives laws for the
probability amplitudes that models of phenomena have a certain dynamics
ballentine ; weinberg , i.e., QM provides laws for models of models. GS
classifies this as the self-reflexiveness of the language. From this
perspective, we can consider GS as an application of certain QM concepts to
clinical psychology.

###  II.2 Pauli and Jung’s Synchronicity

The goal of the collaboration between Jung and Pauli was to find a unified
view of reality in terms of both the psychological and physical point of view.
Jung’s approach to the psyche was based on certain in-forming (in the sense of
having the power of giving “form”) structures that he called archetypes
archetype . As universal regulators of the psyche, archetypes transcended the
individual and belonged to a collective unconscious, common to all humankind.
From the point of view of physics, this can be deemed much less mysterious
than how it sounds. Human ideas are formulated by brains that share a common
physical structure. Although it has a great flexibility, such a structure may
be expected to constrain the type of ideas that can be formulated. In other
words, any idea that can be potentially formulated (which will be called
“archetype”) must belong to the set of all ideas permitted by the common brain
structure of humans. If we now call such a set “collective unconscious”, we
might give a biological justification to Jung’s theory percival .

Pauli was one of the founders of QM. He interpreted QM in terms of the concept
of statistical causality. This facilitated the collaboration with Jung. He
explained to Jung that QM is about ‘forms’, e.g., wave amplitudes, and it is
also intrinsically probabilistic. While the causality of the classical world
requires the exchange of physical quantities (such as energy, momentum,
angular momentum, and so on), statistical causality describes a new type of
non-local correlation between systems. Such a new type of correlation is
typically quantum in nature and is based on synchronistic events, i.e., random
non-local coincidences pauli ; fuchs ; lindorff ; synchronicity ; jung-pauli .
One example is given by the absorption of a quantum of energy from light. The
energy of the quantum is proportional to the frequency $\nu$ of the radiation
and is spread out along the whole wavefront. However, because of quantum
mechanical fluctuations, the energy $h\nu$, where $h$ is Planck’s constant,
can instantaneously disappear from the wavefront and be instantaneously
transferred to a particle with resonant de Broglie frequency $\nu=p/2mh$,
where $p$ is the momentum of the particle and $m$ is its mass ballentine ;
weinberg . The time at which the quantum of energy $h\nu$ disappears from the
wavefront and instantaneously reappears as belonging to the particle’s kinetic
energy (synchronistic events) is completely random. It seems a pure
‘coincidence’. In practice, the formalism of creation and destruction
operators of quantum field theory zee ; mandl  describes quantum processes in
terms of random disappearances and reappearances of energy quanta to and from
different systems. However, everything is probabilistic. There is no reason
for something to happen at a given time or for the quanta to reappear in the
energy of one particle instead of another one. The word ‘coincidence’ can
indeed be considered appropriate.

Analogously, Jung considered random coincidences in the classical world as the
analogue of the statistical causality in the quantum world. Moreover, in
Jung’s theory, random coincidences were also the origin of subjective meaning.
This also means that the organizing principle of reality, which Jung called
synchronicity, is found in meaningful coincidences. Afterwards, the concept of
synchronicity was further generalized to include acausal correlations without
any psychological component. We can conclude that Jung’s synchronicity
reflects the quantum–classical nature of the world.

###  II.3 The Bi-Logical Structure of Psychology

Korzybiski’s GS science_and_sanity  proposes a new psychology founded on
mathematical structures and, to this end, briefly dealt with both set and
group theory. However, it is only in the work of Blanco blanco ; blanco2 ;
rayner ; lombardi  that these ideas are fully exploited in order to generalize
Freud’s formulation of the unconscious. Whereas Freud defined the unconscious
in a qualitative way, i.e., what is hidden and repressed in the psyche, Blanco
describes it as a bipartite structure. Such a bipartite structure has one side
that is asymmetric (which we may call Aristotelian by following GS language),
pertaining man’s common-day experience, and another side that is symmetric
(which we may call non-Aristotelian), where space and time do not exist and
the logical principle of non-contradiction is no longer valid. Blanco stated
that both logics are at work in the human psyche blanco ; blanco2 ; rayner ;
lombardi  and that clinical practice must accurately take into account this
point.

Blanco’s and GS’s conceptual structures share concepts taken from QM. However,
whereas GS is fully non-Aristotelian (without any form of classical-like logic
attached to it), Blanco’s bi-logic has an Aristotelian component (congruent
with a classical worldview) and another non-Aristotelian component (in
agreement with the logic of QM). Taking both aspects into account, we conclude
that Blanco’s bi-logic blanco ; blanco2 ; rayner ; lombardi  formulates a QC
conceptual perspective of the psyche, reflecting the QC nature of the
phenomenological world. A full acknowledgement of this parallelism and its
possible consequences on clinical practice are a matter of novel research.

##  III Schrödinger’s ‘Order from Order’ and Jordan’s Quantum Amplification

In this section, we want to put into evidence two quantum mechanical effects
that we consider fundamental for biological matter mcfadden-book ; mcfadden-
alkhalili  in general and for the brain in particular. Our quantum–classical
model can be designed so that it manifestly incorporates them. One such effect
is Schrödinger’s ‘order from order’ what_is_life , and the other is Pascual
Jordan’s quantum amplification beyler ; beyler2 ; mcfadden-alkhalili .

Classical mechanics applied to biological matter, including the brain,
exploits statistical fluctuations, i.e., the mechanism that Schrödinger called
’order from disorder’ what_is_life . What Schrödinger actually wanted to
express with the expression ’order from disorder’ is that there are some
ordered macroscopic structures that can arise from the statistical disorder at
the microscopic level. In truth, ‘microscopic statistical disorder’ is a
misnomer that stands for the great number of microscopic states that
correspond to the same macroscopic state callen ; blundell . Von Neumann
entropy (and its quantum–classical generalization defined in terms of the
mixed Weyl of the statistical operator) is a property of the macrostates given
in terms of the probability of microstates ohya ; heusler . The belief that
the passage to macroscopic ‘order’ is associated with an entropy decrease is
mistaken. The first reason is that macroscopic ‘order’ is somewhat an
anthropomorphic concept that can only be defined once some macroscopic
variables are chosen. On the contrary, microscopic order is physical since it
is defined in terms of the number of microstates that are compatible with the
macroscopic constraints. A system must be considered microscopically ordered
if there is a small number of states associated to the macroscopic
constraints. In agreement with the third law of thermodynamics, for example,
this takes place at $T=0$, where there is only one accessible microstate and
the system is maximally ordered on the microscopic level. Another example is
given by the phenomenon of re-entrant phase transitions inverse_melting ;
inverse_melting2 ; inverse_melting3 ; inverse_melting4 , where the macroscopic
‘ordered’ phase has a higher entropy than the microscopic ‘disordered’ one
because of the unfreezing of certain DOF. Irreversible microscopic dynamics,
such as diffusive motion, does not conserve the number of accessible
microstates of the system conditioned by the macroscopic constraints and thus
leads to an increase in entropy ohya ; heusler . This is the essence of
Schrödinger’s ’order from disorder’ mechanism what_is_life : in our macrocosm,
we are surrounded by structures that we classify as ordered but that are based
on microscopic disorder in agreement with the second law of thermodynamics.

As discussed by Schrödinger, an ’order from disorder’ mechanism can explain
neither the stability of biological information nor the synchronization of
molecular processes. To explain living matter, Schrödinger proposed a second
mechanism that he named ’order from order’. The mechanism of ’order from
order’ is basically founded on a quantum-mechanical zero-temperature clockwork
in agreement with the third law of thermodynamics what_is_life . Only solid
forms of matter allow for quantum clockworks to exist in a high-temperature
disordered biological environment. This is caused by the existence of energy
gaps protecting, e.g., long-wavelength electronic wavefunctions in solids.
This idea led Schrödinger to predict that an aperiodic solid (ultimately
identified with DNA dna ; dna2 ) would contain, in a stable manner, the
information needed by the living organism to survive entropic decay.
Currently, the idea has become more general and is not limited to solid
structures as shields from molecular disorder. One example is found in the
Orch OR theory, according to which quantum effects are protected inside
hydrophobic regions of biological microstructures microtubulines_channels ;
microtubulines_channels-2 . Another mechanism used to protect the quantum
clockwork is provided by rigid boundaries enclosing quantum variables mukamel
; mukamel2 ; pyp .

Pascual Jordan’s idea beyler ; beyler2 ; mcfadden-alkhalili  about how quantum
mechanics can steer the dynamics of a classical biological environment also
works for systems with no genetic code and, as such, is more general than the
‘order from order’ mechanism. Jordan introduced the concept of quantum
amplification. Basically, this consists of interpreting the stochastic
collapses stapp ; vonneumann  of a quantum state as a way to funnel
information from a smaller-scale level to a classical higher-scale level,
i.e., an amplification. Even a small number of quantum variables can produce a
quantum amplification through the collapse of their state, and can have a
significant effect on the dynamics of large classical systems. Quantum
amplification is also at work in non-adiabatic transitions between energy
states of a quantum subsystem osborn ; martens ; donoso ; as3 ; as13 ; as7 ;
ak  through the back-reaction onto a classical environment, which the quantum
subsystem interacts with.

Both the mechanisms of ‘order from order’ and quantum amplification are in
some way present in the quantum approaches to brain dynamics mcfadden ;
mcfadden2 ; mcfadden3 ; pockett ; pockett2 ; pockett3 ; liboff ; liboff2 ;
frohlich ; ph ; ph-2 ; ph-3 ; penrose ; penrose2 ; penrose3 ; penrose4 ;
penrose5 ; microtubulines_channels ; microtubulines_channels-2 ; fisher ;
fisher2 ; fisher4 ; kerskens ; qc-bneuro-comp ; vitiello1995 ; pessa-vitiello
; alfinito-vitiello ; freeman-vitiello ; freeman-vitiello2 ; freeman-vitiello3
; vitiello-fractals ; vitiello-cortex ; sabbadini-vitiello  that we are going
to discuss in the following. However, since we are explicitly identifying
them, we will be able to design our quantum–classical model in order to take
them into account in a general way.

In addition to the mechanisms of ‘order from order’ what_is_life  and quantum
amplification beyler ; beyler2 ; mcfadden-alkhalili , currently, quantum
informational approaches nielsen-chuang ; jaeger  are routinely invoked to
understand, e.g., condensed matter wen  systems. However, we believe that the
idea of considering a biological systems akin to a quantum computing machine
is somewhat controversial. As we try to explain in this paper, it is even more
controversial than quantum biological theory based on the properties of a few
quantum particles, on the ’order from order’ mechanism what_is_life , and on
the quantum amplification process beyler ; beyler2 ; mcfadden-alkhalili . The
reason for this is that, despite a few suggestions no-ent-comp ; no-ent-comp2
; no-ent-comp3 ; no-ent-comp4 , entanglement is considered to be a necessary
resource for quantum computers nielsen-chuang ; jaeger . However, the main
resource of entanglement is the persistent quantum coherence that, because of
thermal disorder and decoherence, cannot commonly last for time intervals long
enough to perform robust quantum computations inside the brain. Nonetheless,
it has been proposed to interpret brain processes as a form of quantum
computing microtubulines_channels ; fisher ; fisher2 ; kerskens ; qc-bneuro-
comp . It is still a matter of debate that such a type of computation can take
place in the brain microtubulines_channels  nothwithstanding decoherence joos
; zurek-2003 ; tegmark .

Hence, we think that quantum processes in the brain can be studied from the
perspective of quantum biology mcfadden ; mcfadden-alkhalili  with a somewhat
less controversial approach. Henceforth, we will not pursue the quantum
informational perspective nielsen-chuang ; jaeger .

##  IV Electromagnetic Fields in the Brain

In this section, we discuss the EMF field approach to brain dynamics mcfadden
; mcfadden2 ; mcfadden3 ; pockett ; pockett2 ; pockett3 ; liboff  and propose
its generalization to include those quantum effects that can be influenced by
the various forms of electromagnetic brain stimulation riddle ; riddle2 ;
abubaker ; croce ; caruana ; nitsche-paulus ; stagg-nitsche ; papazova ;
yavari  and that can be observed by EEG pockett3 ; croce ; nunez .

The role of EMFs in bridging space and time scales is very important mcfadden
; mcfadden2 ; mcfadden3 ; pockett ; pockett2 ; pockett3 ; liboff ; liboff2 ;
frohlich . Brain states are routinely studied via computer simulation deco
and various noninvasive stimulation techniques, such as alternating current
stimulation (ACS) riddle ; riddle2 ; abubaker ; croce ; caruana  and
transcranial direct-current stimulation (tDCS) nitsche-paulus ; stagg-nitsche
. In particular, tDCS is one of the most investigated methods in the field of
non-invasive brain stimulation. It modulates the excitability of the cerebral
cortex with direct electrical currents (1 $\approx$ 2 mA papazova ) delivered
via two or more electrodes of opposite polarities (i.e., anode and cathode)
placed on the scalp. tDCS modulates resting neuronal membrane potentials at
sub-threshold levels nitsche-paulus , with anodal and cathodal stimulation
increasing and decreasing cortical excitability, respectively stagg-nitsche .
Although their tDCS-induced physiological mechanisms are not yet fully
understood, it is assumed that effects are based on long-term potentiation
(LTP) and long-term depression-like (LTD) mechanisms stagg-nitsche ; yavari .

In the history of brain research, it was assumed that higher brain functions,
such as learning and memory, arise from electrical impulses passing through
neurons. The physical explanation of permanent information storing was
assigned to multiple reflections of impulses through neuronal circuits
mcculloch ; caianiello . This idea is basically exemplified by the
Hodgkin–Huxley model hodgkin-huxley ; hodgkin-huxley2 . Despite its undoubted
success, some limitations of this model have been discussed hh_useful ; all-
or-none ; bodeng ; schmitt ; arbib  and possible generalizations have been
suggested agnati ; bower ; strassberg . From our perspective, the discussion
regarding the role of ion channels bower ; strassberg  is particularly
important. The relevance of quantum effects for charge transport in ion
channels has been strongly supported in Refs. ganim ; plenio_ionc . Typically,
this implies that ion channels’ selectivity may be founded on quantum dynamics
summhammer ; salari_2011 ; bernroider . Ultimately, this line of research
impinges on possible extensions of the Hodgkin–Huxley model, not only to take
into account the description of ion channels’ conductance but also to
incorporate quantum effects moradi .

The idea that other physical agents, rather that the sole dynamics of neural
networks, must be invoked to describe highly coordinated brain activity is not
new frohlich ; koch ; martinez-banaclocha ; pinotsis . Electric charges (e.g.,
electrons, protons, ions), together with their associated currents, are the
sources of EMFs mcfadden ; mcfadden2 ; mcfadden3 ; pockett ; pockett2 ;
pockett3 ; liboff ; liboff2 . In turn, these EMFs interact with water dipoles
and also influence van der Waals and Casimir interactions among brain
macromolecules. ACS has shown the importance of EMFs in the brain riddle ;
riddle2 ; abubaker ; croce ; caruana , and tDCS of human subjects nitsche-
paulus ; stagg-nitsche  has shown the importance of both cognition processes
and psychological state changes, which can be modulated. For instance, anodal
(excitatory) tDCS of the prefrontal cortex boosts affective memory, such as
fear extinction learning vicario2020a ; ney2021 ; markovic2021 . Moreover, the
cathodal (i.e., inhibitory) stimulation of the tongue motor neurons of the
primary motor cortex reduces appetite vicario2020b .

The working of tDCS might be understood through a mechanical analogy. The
complex dynamics of brain EMFs can be reduced to the time evolution of their
sources. Such dynamics can be mapped onto that of a harmonic spring mattress.
Within this pictorial description, tDCS can be equated to the nonlinear effect
generated by the application of a constant pressure to specific extended
regions of the spring mattress. The applied pressure changes the harmonic
dynamics of the mattress so that oscillations with principal frequencies
(phonons) scatter with each other. This mechanical model might be useful for
performing computer simulations of certain processes that are observed in
tDCS. We note that the same model has been used to give a pictorial
representation of quantum fields zee . Both ACS and tDCS provide evidence that
brain EMFs are not ephemeral; they are correlated to the dynamics of their
sources, but also react back and influence both cognitive functions and
emotions.

When studying brain dynamics on the mesoscopic scale of EMFs, it may seem that
there is no necessity to invoke any quantum effect. The original EMF approach
was formulated only in terms of classical physics mcfadden ; mcfadden2 ;
mcfadden3 ; pockett ; pockett2 ; pockett3 ; liboff ; liboff2 ; frohlich .
Nevertheless, our analysis below can elucidate the fundamental quantum
coherent properties of the microscopic EM fields invoked by such an approach.
mukamel ; mukamel2 ; pyp ; zee ; mandl ; romijn ; rozyk-myrta . Observable
coherent EMFs have, by definition, a well-defined phase. Quantum mechanically,
phase $\Phi$ and photon number $N$ are conjugate variables. This implies that
they obey the indeterminacy relation

$\Delta\Phi\Delta N\geq 1\;.$ (1)

According to Equation (1), when the number of quanta of the photon field $N$
is not fixed and $\Delta N$ can be large, it follows that the phase $\Phi$ is
well determined and the quantum photon field is coherent.

The only way for the number of photons $N$ to fluctuate is that photons are
continuously absorbed and re-emitted. In other words, coherent EMFs are
‘composed’ of virtual photons romijn , e.g., packets of energy in momentum
space whose existence is ephemeral. Interestingly, experimental evidence shows
that dendrimers can act as a trap for photons mukamel ; mukamel2 . According
to quantum electrodynamics mandl , a trapped photon can be represented in
terms of virtual photons continuously emitted and re-absorbed between
fermions. This picture can be developed considering that, in terms of Feynman
diagrams, a photon line connecting two fermion lines is a virtual photon
describing Møller scattering mandl . Thus, an exchange of virtual photons
along the time direction between the two fermion lines, generating a so-called
“ladder” diagram mahan ; mattuck , may very well be considered the microscopic
picture of a trapped photon.

##  V Penrose and Hameroff’s Orch OR

Orch OR theory ph ; ph-2 ; ph-3 ; penrose ; penrose2 ; penrose3 ; penrose4 ;
penrose5 ; microtubulines_channels ; microtubulines_channels-2 ; fisher ;
fisher2 ; fisher4 ; kerskens ; qc-bneuro-comp  provides a detailed molecular
mechanism for the time evolution of brain states. According to Orch OR theory
ph ; ph-2 ; ph-3 ; penrose ; penrose2 ; penrose3 ; penrose4 ; penrose5 ;
microtubulines_channels ; microtubulines_channels-2 ; fisher ; fisher2 ;
fisher4 ; kerskens ; qc-bneuro-comp , quantum effects in tubulin proteins
(which are organized in arrays of microtubules inside the cytoplasm of brain
cells) play an important role in brain function. Quantum dynamics of the
electronic orbitals of carbon rings inside tubulins, time evolution of the
nuclear spins, quantum energy transport among microtubules, and the
spontaneous collapse of microtubules’ wave function are the main ingredients
of this theory. Upon collapse of the wave function, classical brain dynamics
ensues. For example, we look at charges’ and masses’ tunneling as an event
arising from the collapse of these variables’ wavefunctions. In turn, such a
collapse induces the collapse of the environment’s and EMF’s quantum states,
triggering chemical reactions, diffusion processes, macroscopic currents, and
so on.

One peculiar characteristic of Orch OR is that neurons are not considered to
be the fundamental units of information processing frohlich . Instead, in Orch
OR, it is proposed that information processing takes place in ordered arrays
of microtubules inside the cell. This idea slowly took form during the 1980s
and the first part of the 1990s when Hameroff noticed the effects of
anesthetics on networks of microtubules inside the cell. In a series of
papers, Hameroff et al. net ; net2 ; net3 ; net4 ; net5 ; net6  proposed that
some kind of digital computation was taking place in arrays of microtubules.
Such a computation was based on nonlinear electrodynamic effects net ; net2 ;
net3 ; net4 ; net5 ; net6 . However, the question of how the results of local
digital calculations could be efficiently transferred between distant brain
regions by classical diffusive mechanisms remained. Hence, Hameroff started
his search for different mechanisms. On a different path, looking for a
fundamental explanation of wave function collapse in QM, Penrose elaborated
the theory of objective reduction (OR) penrose ; penrose2 ; penrose3 ;
penrose4 ; penrose5 .

In the standard interpretation of QM, the collapse of the wave function, i.e.,
the transition from the world of possibilities to that of classical events
kastner ; kastner2 , is explained only through the stochastic interaction of
quantum systems with classical ones. The collapse of the wave function is
called the ‘measurement’ process because of the interaction with a classical
system infamous_boundary . It is not explained within the theory but it is
assumed as a postulate. OR proposes that the superposition of different
stationary mass distributions becomes unstable because of quantum
gravitational effects, and, beyond a certain time interval threshold, it
naturally collapses according to the standard probabilistic rules of QM, but
without any external intervention of a “measuring instrument”. A simple way to
discuss this process is to consider

$\omega_{\rm Bohr}=\Delta E/\hbar\;,$ (2)

as the Bohr frequency of the energy eigenvalues of two eigenstates involved in
a certain superposition. Penrose gives a number of reasons for why the
superposition must become unstable in the presence of quantum gravitational
effects. The lifetime of the superposition is given by

$\tau\approx\frac{h}{\Delta E}\;.$ (3)

Looking at Equations (2) and (3), one might say that, in a certain sense, the
deterministic time evolution of the gravitational field acts as the instrument
measuring the superposition. However, according to Penrose penrose ; penrose2
; penrose3 ; penrose4 ; penrose5 , there is an important difference between
the measurement of the superposition by a classical instrument and by a
quantum gravitational field. A measurement performed by a quantum
gravitational field is still a fully quantum mechanical process and, as such,
is intrinsically random and absolutely non-computable. Penrose considered that
brain dynamics is interspersed with discrete events (see Ref. pockett2  for
experimental support of this idea). On a phenomenological basis, such events
parallel the discontinuity of wakefulness and awareness pockett2  and other
rhythmic phenomena in the brain. Penrose identified discrete events in the
brain with a series of wave function collapses. Between one collapse and the
other, the brain can evolve coherently so that new superpositions are formed.
We note that such a coherent evolution of the wave function, interrupted by
quantum gravitational collapses, is reminiscent of both piecewise
deterministic processes in open quantum systems ops  and nonadiabatic dynamics
of the QC system on an adiabatic basis as3 ; as13 ; as7 ; ak .

While Penrose put forth the idea that OR could have an important role in brain
dynamics, Hameroff fleshed out the detailed biomolecular mechanisms. Inside
each tubulin protein making up a given microtubule, Hameroff hypothesized the
existence of quantum matter systems able to support stable quantum dynamics in
between OR events. One example is given by carbon rings and their delocalized
molecular orbitals, which can evolve coherently in a superposition of states.
The carbon rings are pushed by hydrophobic forces into the tubulin’s interior,
shielding them from the decoherence joos ; zurek-2003  caused by the polar
environment outside the protein. The carbon rings form helical structures
inside each microtubule. They also create oriented arrangements that can act
as quantum channels microtubulines_channels ; microtubulines_channels-2
through which quantum signals travel among the lattice of microtubules inside
the cell’s cytoskeleton.

Various types of quantum oscillators are therefore found in microtubules’
ordered structures, e.g., time-dependent electric fields arising from the
dynamic polarization of molecular charges (which produce van der Waal and
Casimir–Polder forces), magnetic fields originating from electron spin
dynamics, etc. Notably, it has also been suggested fisher ; fisher2 ; fisher4
that nuclear spins can play an important role in Orch OR theory since they are
shielded from decoherence for longer time intervals than other quantum systems
in the brain. Recently, this theory fisher ; fisher2 ; fisher4  has gained
experimental support kerskens . The frequencies of all such quantum
oscillators range from kilohertz to terahertz. Orch OR theory requires the
feedback qc-bneuro-comp  between the quantum coherent evolution of
microtubules and, for example, the classical dynamics of microtubule-
associated proteins (MAPs) maps ; maps-2 . Such classical dynamics concerns
the classical evolution of MAPs maps ; maps-2  and CAMKII camkII ; camkII-2 ;
camkII-3 ; camkII-4 ; camkII-5 ; camkII-6 , viz.the direction of motion, the
place where MAPs and CAMKII halt their motion, the case in which they interact
or do not interact with the tubulins, and the precise time when they interact.
According to the Orch OR theory ph ; ph-2 ; ph-3 ; penrose ; penrose2 ;
penrose3 ; penrose4 ; penrose5 ; microtubulines_channels ;
microtubulines_channels-2 ; fisher ; fisher2 ; fisher4 ; kerskens ; qc-bneuro-
comp , the coherent evolution of the microtubule’s wave function and its OR
determine all detailed molecular events. However, we must note that the
possibility that extended brain regions may be free from decoherence joos ;
zurek-2003 ; tegmark ; microtubulines_channels  is rather controversial.

Lately, there has been a convergence of ideas between the approach to brain
dynamics via quantum EMFs mukamel2 ; mukamel ; romijn  and Orch OR
neurophotonics . The physical process underlying quantum signaling in Orch OR
has been assumed to be photon emission. Due to the work of Alexander
Gurwitsch, it has been known since the beginning of the 20th century that
tissues inside the body emit biophotons popp ; popp-2 ; cifra . Such
biophotons may be supported by the hydrophobic interior region of tubulins,
where tryptophanes, with their indole rings of $\pi$ electron orbitals forming
optically active molecular orbitals, are found. The packing of indole rings
may give rise to resonant energy transfer between molecular orbitals
neurophotonics  much in the same way that Förster resonant energy transfer
takes place between close chromophores. Kurian et al. kurian  represented the
microtubule as a chain of two-level systems and calculated the coupling
constants in the Hamiltonian by means of molecular dynamics simulations and
quantum chemical calculations. Exciton propagation was performed by means of
the Haken and Strobl method haken . Their main result is that energy transfer
occurs on a length scale of at least microns. What is even more interesting
from the quantum optical perspective is that Kurian et al.’s simulation kurian
does not consider the geometric structure of the left-handed helixes of
microtubule in mammals. There are reasons to believe that super-radiance can
be important in such complicated geometric arrangements superrad ; superrad2 ;
superrad3 . Very recently, the experimental study of Kalra et al. kalra  found
that photonic energy transfer in microtubules occurs over 6.6 nm, cannot be
explained in terms of Föster theory, and is damped by anesthetics. The idea
that electromagnetic resonance is the fundamental mechanism of communications
among molecules was first proposed by Veljkovic et al., who also suggested
that such a mechanism could provide long-range effective communication
veljkovic . At this stage, we believe that a unification of the EMF and Orch
OR theories of brain dynamics is conceptually very probable rozyk-myrta .

Nevertheless, the Orch OR model remains very controversial. It is based on
quantum gravitational effects used to objectively induce the wave function
collapse by using only a provisional theory of quantum gravity ph ; ph-2 ;
ph-3 .

##  VI The Dissipative Quantum Model of Brain

The precursor of DQMB vitiello1995  was the seminal paper umezawa  of
Ricciardi and Umezawa, where the quantum field theory model of brain (QFTMB)
was introduced thermofd ; thermofd2 . An interacting QFT can naturally
describe the creations of dynamical correlations. Whenever a quantum field has
an average value different from zero in the vacuum, the vacuum state will no
longer be unique. Instead, there will be different vacua and each of them will
spontaneously break the symmetry of the Hamiltonian density nambu ; goldstone
; goldsalw . In order to compensate for the SSB, the proliferation of bosonic
modes, establishing long-range correlations with the local configurations of
the field, sets in. The symmetry-breaking mechanism in the QFTMB umezawa  can
qualitatively describe both long-term memory storage in the ground states with
broken symmetry and long-range correlations between distant clusters of
neurons by means of the Nambu–Golstone bosons. Nambu–Goldstone bosons also act
as the agents for memory retrieval umezawa  while excited energy states of the
field describe short-term memory.

In DQMB, the dissipation is ascribed to excited thermal states, which are
represented through doubling the number of fields according to thermo field
dynamics thermofd ; thermofd2 . DQMB also predicts that long-range
correlations between distant excited areas of the brains do not occur via
chemical transport but by means of Nambu–Goldstone bosons nambu ; goldstone ;
goldsalw . One example of such long-distance correlations is observed when the
brain is locally stimulated. In this case, there is experimental evidence
riddle ; riddle2 ; abubaker ; croce ; caruana ; nitsche-paulus ; stagg-nitsche
that the response is given by simultaneous excitations in several regions
clustering ; clustering2 , which are far from one another. In DQMB, quantum
coherent fields interact with classical neurons and glia cells. DQMB presents
us with a hybrid description where memory storage finds a quantum explanation
and biochemical reactions find a classical one. Such a hybrid description
requires to coarse-grain the classical degrees of freedom (DOFs) and to
describe them in terms of some kind of waves. Only at this level of
description is it possible to formulate the interaction between the
Nambu–Goldstone bosons nambu ; goldstone ; goldsalw , the condensed quantum
field predicted by the model, and the classical waves, much in the same way
that phonons in an ordered solid interact with acoustic waves vitiello1995 ;
pessa-vitiello ; alfinito-vitiello ; freeman-vitiello ; freeman-vitiello2 ;
freeman-vitiello3 ; vitiello-fractals ; vitiello-cortex ; sabbadini-vitiello .

Due to its mesoscopic nature, DQMB does not aim at describing the behavior of
the molecular constituents of the brain with atomistic detail, e.g., neurons,
glia cells, membranes, neurotransmitters, or other macromolecules. Today, we
know that all of these structures form brain clusters clustering ; clustering2
that, once stimulated riddle ; riddle2 ; abubaker ; croce ; caruana ; nitsche-
paulus ; stagg-nitsche , can influence human behavior croce ; caruana . Since
normal mesoscopic brain dynamics is not chaotic, the brain’s response to
stimuli cannot be expected to depend on the number $N$ of the fundamental
constituents of the clusters. If $N$ is not fixed, Equation (1) is valid, the
phase $\Phi$ of the matter field is well defined, and the matter field will be
coherent. Moreover, DQMB does not specify the physical nature of the bosonic
fields of the brain. The bosonic fields in Fourier space may be identified
with the modes of the quantum oscillators considered in Orch OR theory
microtubulines_channels ; microtubulines_channels-2  and discussed in Section
V. However, another proposal suggested to interpret the bosonic fields in
terms of the dipoles of water molecules delgiudice ; delgiudice+1 ; preparata
; jibu ; jibu2 . According to the theory in Refs. delgiudice ; delgiudice+1 ;
preparata , when water molecules have a high density, the approximation of
weak coupling to the electromagnetic vacuum field qed-coherence  may not hold.
It has been suggested that water in the cytoplasm is found in a structured
state ling , so the considerations of Refs. delgiudice ; delgiudice+1 ;
preparata  are definitely relevant for brain dynamics. Since a water molecule
is dipolar, a coherent superposition of the dipoles of many water molecules
can be described by a coherent quantum dipolar field. Hence, in this model, it
is the condensation of the quantum dipolar field that produces a ground state
with broken symmetry, i.e., many unitarily inequivalent subspaces blasone .
Consequently, Nambu–Goldstone modes arise for restoring symmetry at long
range.

In the following, we use the Hamiltonian of the noninteracting dipolar wave
quanta of Ref. sabbadini-vitiello  in order to elucidate the theoretical
description of dissipation by means of doubling the DOFs as described by
Umezawa’s thermo field dynamics thermofd ; thermofd2 . The dynamical variables
of DQMB are doubled upon introducing creation and annihilation operators of
physical dipolar wave quanta, $\hat{a}^{\dagger},\hat{a}$, respectively, and
dual creation and annihilation operators of fictitious dipolar wave quanta,
$\hat{v}^{\dagger},\hat{v}$, respectively. For example, the Hamiltonian of the
noninteracting dipolar wave quanta might be defined as sabbadini-vitiello :

$\hat{H}_{0}=\sum_{k}\hbar\omega_{k}\left(\hat{a}_{k}^{\dagger}\hat{a}_{k}-\hat{v}^{\dagger}\hat{v}_{k}\right)\;,$
(4)

where $\omega_{k}$ is the oscillation frequency of each mode. The interaction
between the physical modes and their doubles can be taken as

$\hat{H}_{\rm
I}=i\sum_{k}\hbar\gamma_{k}\left(\hat{a}_{k}^{\dagger}\hat{v}^{\dagger}-\hat{a}_{k}\hat{v}_{k}\right)\;,$
(5)

where $\gamma_{k}$ is the damping constant of each mode. Finally, the total
many-body Hamiltonian of the thermal system is

$\hat{H}=\hat{H}_{0}+\hat{H}_{\rm I}.$ (6)

A thorough study of the Hamiltonian in Equation (6) and its associated
equations of motion has led to finding a number of interesting results over
the years freeman-vitiello .

DQMB has been applied by Vitiello and collaborators to study various brain
processes alfinito-vitiello . Some applications include nonlinear dynamics
freeman-vitiello2 , cortical patterns in perception freeman-vitiello3 , the
relation between fractal properties and coherent states in the brain vitiello-
fractals , rhythmic generators in the cortex vitiello-cortex , and
correlations of brain regions that are realized through entanglement
sabbadini-vitiello . DQMB dynamics has also been adopted by Nishiyama et al.
in a number of works nishiyama ; nishiyama2 ; nishiyama3 ; nishiyama4 . As
reported in Ref. nishiyama , one notes that the phenomenon of super-radiance,
which is expected to occur in complicated geometric arrangements of
microtubules, also occurs in DQMB.

##  VII The Quantum–Classical Model of Brain

Our aim is to model multi-scale brain dynamics, explicitly treating classical
DOFs and quantum variables on the same footing. To this end, our approach
considered mixed Weyl symbols of dynamical variables (represented by operators
in the standard formulation of quantum mechanics) and a mixed Weyl symbol of
the statistical operator (corresponding to the density matrix of the systems
in the standard representation of QM) silin ; rukhazade ; balescu ; zhang-
balescu ; balescu-zhang ; aleksandrov ; gerasimenko ; boucher ; petrina ;
prezhdo ; kapracicco ; nielsen ; as4 ; as6 ; as14 ; osborn ; martens ; donoso
; as3 ; as13 ; as7 ; ak . In this regard, it is worth remarking that we
introduced a quantum–classical model of brain dynamics because, while thermal
disorder, decoherence, and a short de Broglie wavelength require treating the
DOFs of massive atoms and molecules by means of classical mechanics, there are
also intrinsically quantum variables at all temperatures influencing the
dynamics of the classical DOF. Such quantum variables are, e.g., tunneling
electrons and protons, nuclear and electron spins, electronic-state-associated
phenyl rings, photons, and so on. Our working hypothesis is that certain brain
processes can most naturally be understood in terms of the above picture. In
the following, we sketch the quantum–classical theory that we used and
introduce a general Hamiltonian model that can be adapted to perform
calculations of specific phenomena at the interface between the quantum and
the classical world. For example, we plan to study if quantum dynamics can
explain ion channels’ selectivity inside the classical membrane of a brain
cells. Quantum particles that are recognized can be characterized in terms of
their vibrational spectra. Response functions can be calculated depending of
the oscillators’ frequencies and the wave vectors, and these can be compared
to the results of electromagnetic brain stimulation and EEG. Another example
is given by the trapping of photons by neurons’ dendrites. Once trapped,
photons should acquire an effective mass and the dynamics of the dendrites
should be slowed down. One could expect that some observable effect of or
alteration in neuronal dynamics should follow. The question is if this
investigation could explain the effects of electromagnetic brain stimulation.
These are just a few examples. In general, numerical simulations can be used
as thought experiments for ‘discovering’ the qualitative agreement between a
specific model and real experiments or the causal connection between the
numerical values of the parameters of the model and the behavior of the model
itself. Comparisons between different Hamiltonian models can also lead to a
better understanding of the essential features of a given phenomenon. These
are the goals of our quantum–classical model of brain dynamics. Section II has
logically sustained the choice of adopting a quantum–classical approach and
has pointed us toward the very ambitious goal of developing models that cannot
only go from the microscopic level to the results of electromagnetic brain
stimulation but that can also reach the psychological states of the brain. We
intend this as specified by a few macroscopic variables that can be related to
human behavior. If human behavior can be affected by electromagnetic brain
stimulation and if we are successful in building models going from the
quantum–classical microscopic level to brain stimulation, then our goal will
be reached, even if it is as an overall result of a research activity whose
first results are just those proposed in this article.

We imagine that the brain is described by quantum operators
$(\hat{r},\hat{p},\hat{R},\hat{P})$, where $(\hat{r},\hat{R})$ are position
operators and $(\hat{p},\hat{P})$ are the respective conjugated momenta
operators. Now, $(\hat{r},\hat{p})=\hat{x}$ corresponds to the brain variables
with a long de Broglie wavelength that, for this reason, must be treated
quantum mechanically, whereas $(\hat{R},\hat{P})=\hat{X}$ can be treated semi-
classically because of their much shorter de Broglie wavelength. A partial
Wigner transform over the $(\hat{X})$ operators kapracicco  introduces the
mixed Weyl symbols $\tilde{\cal O}(X)$ and $\tilde{\cal W}(X)$ arising from
$\hat{\cal O}(\hat{x},\hat{X})$ and $\rho(\hat{x},\hat{X})$, respectively.
Please note that the following notation is adopted: when a quantum operator
depends both on quantum variables and classical DOFs, a $\tilde{\phantom{a}}$
is written on it, whereas, if the quantum operator does not depend on $X$, a
$\hat{\phantom{a}}$ is used. No hat is used in the case of a dynamical
variable depending only on $X$. A practical example of a possible application
of this mixed QC representation can be given when considering molecular
orbitals, electron and nuclear spins, light ions, neurons, glia cells, and
electromagnetic interactions. Conformational dynamics of cells may be
represented through phonons, i.e., harmonic DOFs. Other harmonic DOFs can be
used to describe coherent EMFs. The inclusion of non-Harmonic perturbation
terms provides a description of non-trivial interactions among all of the DOFs
of the model. Zero-point effects on the motion of classical-like DOFs can be
described by means of advanced algorithms that will be explained in the
following. As in the case of DQMB, the goal is to set up a mesoscale approach
to brain dynamics, noting, however, that, in our case, the QC dynamical
variables are explicitly represented.

If we now introduce the coordinates of the EMF modes $(Q,\Pi)=\Upsilon$, a
possible model mixed Weyl symbol of the Hamiltonian $\tilde{\cal
H}(X,\Upsilon)$ can be written as

$\displaystyle\tilde{\cal H}(X,\Upsilon)$ $\displaystyle=$
$\displaystyle\hat{\cal H}_{\rm S}+{\cal H}_{\rm B}(X)+{\cal H}_{\rm
F}(\Upsilon)+\tilde{\cal V}_{\rm SB}(R)+\tilde{\cal V}_{\rm SEM}(Q)$ (7)

In Equation (7), $\hat{\cal H}_{\rm S}(t)$ is the Hamiltonian operator of the
quantum subsystem with quantum variables $\hat{x}$. The phononic Hamiltonian
is

${\cal H}_{\rm B}(X)=\sum_{J=1}^{N_{\rm
PH}}\left(\frac{P_{J}^{2}}{2}+\frac{(\omega_{J}^{\rm
PH})^{2}}{2}R_{J}^{2}\right)\;,$ (8)

where $\omega_{J}^{\rm PH}$, $J=1,...,N_{\rm PH}$ is the frequency of each
phonon. Similarly, the EMF Hamiltonian is

${\cal H}_{\rm F}(\Upsilon)=\sum_{K=1}^{N_{\rm
EM}}\left(\frac{\Pi_{K}^{2}}{2}+\frac{(\omega_{K}^{\rm
EM})^{2}}{2}Q_{K}^{2}\right)\;,$ (9)

where $\omega_{K}^{\rm EM}$, $K=1,...,N_{\rm EM}$ is the frequency of the EMF
mode. The interaction operators $\tilde{\cal V}_{\rm SB}(R)$ and $\tilde{\cal
V}_{\rm SEM}(Q)$ describe the coupling of the phonons and of the EMF to the
quantum subsystem, respectively. Assuming, for simplicity, a bilinear
approximation, these can be written as

$\displaystyle\tilde{\cal V}_{\rm SB}(R)$ $\displaystyle=$
$\displaystyle-\sum_{J=1}^{N_{\rm PH}}C_{J}R_{J}\hat{\chi}$ (10)
$\displaystyle\tilde{\cal V}_{\rm SEM}(Q)$ $\displaystyle=$
$\displaystyle-\sum_{K=1}^{N_{\rm M}}F_{K}Q_{K}\hat{\zeta}\;,$ (11)

where the $C_{J}$ and $F_{K}$ are the coupling constants of the quantum
operators $\hat{\chi}$ and $\hat{\zeta}$, respectively. The operators
$\hat{\chi}$ and $\hat{\zeta}$ act on the same space of $\hat{x}$.

The dynamics of the mixed Weyl symbol $\tilde{\cal O}(X,\Upsilon,t)$ of an
arbitrary operator $\hat{\cal O}$ is given by a QC bracket silin ; rukhazade ;
balescu ; zhang-balescu ; balescu-zhang ; aleksandrov ; gerasimenko ; boucher
; petrina ; prezhdo ; kapracicco ; nielsen ; as4 ; as6 ; as14 ; osborn ;
martens ; donoso ; as3 ; as13 ; as7 ; ak . The QC bracket is a quasi-Lie
bracket nielsen ; as4 ; as6 ; as14  that breaks the time-translation
invariance of Lie algebras because it does not satisfy the Jacobi relation. In
the case of a system with both phononic and EMF modes, it can be written by
introducing two antisymmetric matrices,
$\mbox{\boldmath$\Omega$}=-\mbox{\boldmath$\Omega$}^{-1}$ and
$\mbox{\boldmath$\Lambda$}=-\mbox{\boldmath$\Lambda$}^{-1}$:

$\mbox{\boldmath$\Omega$}=\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right]$ (12)

and

$\mbox{\boldmath$\Lambda$}=\left[\begin{array}[]{cccc}{\bf 0}&{\bf 0}&{\bf
1}&{\bf 0}\\\ {\bf 0}&{\bf 0}&{\bf 0}&{\bf 1}\\\ -{\bf 1}&{\bf 0}&{\bf 0}&{\bf
0}\\\ {\bf 0}&-{\bf 1}&{\bf 0}&{\bf 0}\end{array}\right]\;.$ (13)

The QC equation of motion in the Heisenberg picture reads

$\displaystyle\partial_{t}\tilde{\cal O}(t)$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}\left[\begin{array}[]{cc}\tilde{\cal
H}&\tilde{\cal
O}(t)\end{array}\right]\mbox{\boldmath$\Omega$}\left[\begin{array}[]{c}\tilde{\cal
H}\\\ \tilde{\cal O}(t)\end{array}\right]-\frac{1}{2}\tilde{\cal
H}\overleftarrow{\mbox{\boldmath$\nabla$}^{X,\Upsilon}}\mbox{\boldmath$\Lambda$}\overrightarrow{\mbox{\boldmath$\nabla$}^{X,\Upsilon}}\tilde{\cal
O}(t)$ (17) $\displaystyle+$ $\displaystyle\frac{1}{2}\tilde{\cal
O}(t)\overleftarrow{\mbox{\boldmath$\nabla$}^{X,\Upsilon}}\mbox{\boldmath$\Lambda$}\overrightarrow{\mbox{\boldmath$\nabla$}^{X,\Upsilon}}\tilde{\cal
H}\;,$ (18)

where $\mbox{\boldmath$\nabla$}^{X,\Upsilon}=((\partial/\partial
R),(\partial/\partial Q),(\partial/\partial P),(\partial/\partial\Pi))$ is the
phase space gradient operator.

The lhs of Equation (18) defines the quantum–classical bracket of $\tilde{\cal
O}(t)$ with $\tilde{\cal H}$. The first term in the lhs of Equation (18) is
the quantum commutator, whereas the other two terms are Poisson brackets. All
terms are written in matrix form as4 ; as6 ; as14 . The super propagator
associated to the QC bracket is

$\displaystyle\tilde{\tilde{{\cal U}}}(t)$ $\displaystyle=$
$\displaystyle\exp\left\\{(it/\hbar)\left[\begin{array}[]{cc}\tilde{\cal
H}&\ldots\end{array}\right]\mbox{\boldmath$\Omega$}\left[\begin{array}[]{c}\tilde{\cal
H}\\\ \ldots\end{array}\right]-(t/2)\left(\tilde{\cal
H}\overleftarrow{\mbox{\boldmath$\nabla$}}^{X,\Upsilon}\mbox{\boldmath$\Lambda$}\overrightarrow{\mbox{\boldmath$\nabla$}}^{X,\Upsilon}\ldots\right)\right.$
(22) $\displaystyle+$
$\displaystyle\left.(t/2)\left(\ldots\overleftarrow{\mbox{\boldmath$\nabla$}}^{X,\Upsilon}\mbox{\boldmath$\Lambda$}\overrightarrow{\mbox{\boldmath$\nabla$}}^{X,\Upsilon}\tilde{\cal
H}\right)\right\\}$ (23)

The super-operator $\tilde{\tilde{{\cal U}}}(t)$ defines the dynamics of mixed
Weyl symbols of standard operators as

$\tilde{\cal O}(t)=\tilde{\tilde{{\cal U}}}(t)\tilde{\cal O}\;,$ (24)

where $\tilde{\cal O}=\tilde{\cal O}(t=0)$. QC averages are calculated using
the formula

$\langle\tilde{\cal O}(t)\rangle={\rm Tr}^{\prime}\int
dXd\Upsilon\;\tilde{\cal W}(X,\Upsilon;t)\tilde{\cal O}(X,\Upsilon,t)\;.$ (25)

In Equation (25), the parametric time dependence of the mixed Weyl symbol of
the statistical operator of the system, $\tilde{\cal W}(X,\Upsilon;t)$,
describes possible non-equilibrium initial conditions. The formalism presented
here can be easily adapted to more general non-equilibrium situations arising
from an explicit time dependence of the mixed Weyl symbol of the Hamiltonian
in Equation (7). In such a case, it would be more convenient to adopt the
Schrödinger scheme of motion and propagate the mixed Weyl symbol of the
statistical operator. One would also have to take into account the time
ordering of the propagator, something that can be implemented by the algorithm
as13 . Non-equilibrium dynamics is important if one considers the free energy
principle proposed by Karl Friston friston ; sanchez-canizares . Recently,
such a direction of research has witnessed interesting developments tanaka .
As for QC correlation functions, they are defined in the following way:

$\langle\tilde{\cal O}_{1}(t)\tilde{\cal O}_{2}\rangle={\rm Tr}^{\prime}\int
dXd\Upsilon\;\tilde{\cal W}(X,\Upsilon;t)\tilde{\cal
O}_{1}(X,\Upsilon,t)\tilde{\cal O}_{2}(X,\Upsilon)\;.$ (26)

The operator ${\rm Tr}^{\prime}$ found in Equations (25) and (26) takes the
trace over the quantum operators $\hat{x}$, while $\tilde{\cal O}_{1}$ and
$\tilde{\cal O}_{2}$ are two arbitrary mixed Weyl symbols.

### Constant Temperature Quantum–Classical Dynamics

In order to illustrate the advanced techniques for controlling the temperature
of the harmonic modes, we consider a simple system with just two phononic
modes, with coordinates $(X_{1},X_{2})$, and two NHC chains of length one
(which is usually enough to generate ergodic dynamics for stiff harmonic
degrees of freedom) nhc ; b1 ; b2 . Thus, the extended phase space point can
be written as $X^{\rm e}$= $(R_{1}$,$\eta_{1}^{(1)},$ $\eta_{2}^{(1)},$
$R_{2},$ $\eta_{2}^{(1)},$ $\eta_{2}^{(2)},$ $P_{1},$ $P_{\eta_{1}}^{(1)},$
$P_{\eta_{1}}^{(2)},$ $P_{2},$ $P_{\eta_{2}}^{(1)},$ $P_{\eta_{2}}^{(2)});$
consequently, the extended phase space gradient is
$\mbox{\boldmath$\nabla$}^{\rm e}$= $((\partial/\partial R_{1}),$
$(\partial/\partial\eta_{1}^{(1)}),$ $(\partial/\partial\eta_{2}^{(1)}),$
$(\partial/\partial R_{2}),$ $(\partial/\partial\eta_{1}^{(2)}),$
$(\partial/\partial\eta_{2}^{(2)})$, $(\partial/\partial P_{1}),$
$(\partial/\partial P_{\eta_{1}}^{(1)}),$ $(\partial/\partial
P_{\eta_{2}}^{(1)})$, $(\partial/\partial P_{2}),$ $(\partial/\partial
P_{\eta_{1}}^{(2)})$, $(\partial/\partial P_{\eta_{2}}^{(2)}).$

If we now define the antisymmetric matrix $\mbox{\boldmath${\cal
R}$}=-\mbox{\boldmath${\cal R}$}^{-1}$ as

$\displaystyle\mbox{\boldmath${\cal
R}$}=\left[\begin{array}[]{cccccccccccc}0&0&0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&0&1\\\ -1&0&0&0&0&0&0&-P_{1}&0&0&0&0\\\
0&-1&0&0&0&0&P_{1}&0&-P_{M_{\eta_{2}}}^{(1)}&0&0&0\\\
0&0&-1&0&0&0&0&P_{M_{\eta_{2}}}^{(1)}&0&0&0&0\\\
0&0&0&-1&0&0&0&0&0&0&-P_{2}&0\\\
0&0&0&0&-1&0&0&0&0&P_{2}&0&-P_{\eta_{1}}^{(2)}\\\
0&0&0&0&0&-1&0&0&0&0&P_{\eta_{1}}^{(2)}&0\;.\end{array}\right]$ (39)

together with the mixed Weyl symbol of the extended Hamiltonian

$\displaystyle\tilde{\cal H}^{\rm e}(X^{\rm e})$ $\displaystyle=$
$\displaystyle\hat{\cal H}_{\rm S}+{\cal H}_{\rm B}(X)+\tilde{\cal V}_{\rm
SB}(R)+\sum_{I=1}^{2}\sum_{L=1}^{2}\frac{P_{\eta_{L}}^{(I)}}{2M_{\eta_{L}}}+\sum_{I=1}^{2}\sum_{L=1}^{2}k_{\rm
B}T^{(I)}\eta_{L}^{(I)}\;.$ (40)

the QC equation of motion at constant temperature can be written in compact
form as4 ; as6 ; as14  as

$\displaystyle\partial_{t}\tilde{\cal O}^{\rm e}(t)$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}\left[\begin{array}[]{cc}\tilde{\cal H}^{\rm
e}&\tilde{\cal O}^{\rm
e}(t)\end{array}\right]\mbox{\boldmath$\Omega$}\left[\begin{array}[]{c}\tilde{\cal
H}^{\rm e}\\\ \tilde{\cal O}^{\rm
e}(t)\end{array}\right]-\frac{1}{2}\tilde{\cal H}^{\rm
e}\overleftarrow{\mbox{\boldmath$\nabla$}^{\rm e}}\mbox{\boldmath$\cal
R$}\overrightarrow{\mbox{\boldmath$\nabla$}^{\rm e}}\tilde{\cal O}^{\rm e}(t)$
(44) $\displaystyle+$ $\displaystyle\frac{1}{2}\tilde{\cal O}^{\rm
e}(t)\overleftarrow{\mbox{\boldmath$\nabla$}^{\rm e}}\mbox{\boldmath$\cal
R$}\overrightarrow{\mbox{\boldmath$\nabla$}^{\rm e}}\tilde{\cal H}^{\rm e}\;,$
(45)

where $\tilde{\cal O}^{\rm e}(t)=\tilde{\cal O}^{\rm e}(X^{\rm e},t)$; the NHC
variables are $(\eta_{L}^{(I)},P_{\eta_{L}}^{(I)})$, with $I$ and $L$ running
over the phonons and the coordinates of the chain, respectively; $k_{\rm B}$
is the Boltzmann constant; $T^{(I)}$ is the temperature of each mode; and
$M_{\eta_{L}^{(I)}}$ are the inertial parameters of the NHC variables.

Constant temperature averages and correlation functions can be calculated by
choosing the mixed Weyl symbol $\tilde{\cal W}^{\rm e}(X^{\rm e})$ of the
statistical operator in extended space as

$\displaystyle\tilde{\cal W}^{\rm e}(X^{\rm e})$ $\displaystyle=$
$\displaystyle\hat{w}_{\rm S}{\cal
W}^{\beta}(X)\prod_{I=1}^{2}\prod_{L=1}^{2}\delta\left(\eta_{L}^{(I)}\right)\delta\left(P_{\eta_{L}}^{(I)}\right)\;$
(46)

where $\hat{w}_{\rm S}$ is the mixed Weyl symbol of the statistical operator
of the quantum subsystem while the thermal mixed Weyl symbol of the
statistical operators of the phonons is

$\displaystyle{\cal W}^{\beta}(X)$ $\displaystyle=$
$\displaystyle\prod_{I=1}^{2}\frac{\tanh(\beta\omega_{I}/2)}{2}\exp\left[-\frac{2\tanh(\beta\omega_{I}/2)}{\omega_{I}}\left(\frac{P_{I}^{2}}{2}+\frac{\omega_{I}^{2}}{2}R_{I}^{2}\right)\right]\;,$
(47)

where $\beta=1/k_{\rm B}T$ and $\omega_{I}$ is the frequency of phonon $I$.
If, in the mixed Weyl symbol of the Hamiltonian in Equation (40), one defines
$T^{(I)}=T~{}\forall I$, then the dynamics defined by Equation (45) defines
constant-temperature evolution. Instead, the choice of $T^{(I)}=1/k_{\rm
B}\beta^{(I)}$ with

$\beta^{(I)}=\frac{2\tanh(\beta\omega_{I}/2)}{\omega_{I}}\;,\quad\forall I\;.$
(48)

describes a time evolution of the phonons, where zero-point effects are taken
into account. The structure of the extended QC super-propagator
$\tilde{\tilde{{\cal U}}}^{\rm e}$ is similar to that displayed in Equation
(23):

$\displaystyle\tilde{\tilde{{\cal U}}}^{\rm e}(t)$ $\displaystyle=$
$\displaystyle\exp\left\\{(it/\hbar)\left[\begin{array}[]{cc}\tilde{\cal
H}^{\rm
e}&\ldots\end{array}\right]\mbox{\boldmath$\Omega$}\left[\begin{array}[]{c}\tilde{\cal
H}^{\rm e}\\\ \ldots\end{array}\right]-(t/2)\left(\tilde{\cal H}^{\rm
e}\overleftarrow{\mbox{\boldmath$\nabla$}}^{\rm e}\mbox{\boldmath$\cal
R$}\overrightarrow{\mbox{\boldmath$\nabla$}}^{\rm e}\ldots\right)\right.$ (52)
$\displaystyle+$
$\displaystyle\left.(t/2)\left(\ldots\overleftarrow{\mbox{\boldmath$\nabla$}}^{\rm
e}\mbox{\boldmath$\cal R$}\overrightarrow{\mbox{\boldmath$\nabla$}}^{\rm
e}\tilde{\cal H}^{\rm e}\right)\right\\}\;.$ (53)

Since we are interested in thermal and zero-point QC averages and correlation
functions of non-fictitious dynamical variables, we must consider mixed Weyl
symbols $\tilde{\cal O}(X)$ that, at $t=0$, do not depend on the extended
phase space point $X^{\rm e}$ but depend on the non-fictitious phase space
point $X$. However, the key to temperature control is that the phase space
variable dependence found at $t=0$ is not preserved at $t\neq 0$. We have
$\tilde{\tilde{{\cal U}}}^{\rm e}(t)\tilde{\cal O}(X)=\tilde{\cal O}(X^{\rm
e},t)$. Finally, we can write the expression for thermal (or zero-point) QC
averages as

$\displaystyle\langle\tilde{\cal O}(X,t)\rangle_{\rm e}$ $\displaystyle=$
$\displaystyle{\rm Tr}^{\prime}\int dX^{\rm e}\;{\cal W}^{\rm e}(X^{\rm
e})\tilde{\cal O}(X,t)\;,$ (54) $\displaystyle\langle\tilde{\cal
O}_{1}(X,t)\tilde{\cal O}_{2}(X)\rangle_{\rm e}$ $\displaystyle=$
$\displaystyle{\rm Tr}^{\prime}\int dX^{\rm e}\;{\cal W}^{\rm e}(X^{\rm
e})\tilde{\cal O}_{1}(X,t)\tilde{\cal O}_{2}(X)\;.$ (55)

##  VIII Conclusions

In this work, we brought to light a parallel between the bipartite structure
of human logic and the quantum–classical view of physical phenomena. We
discussed that one finds both Aristotelian logic and non-Aristotelian logic in
the human psyche. Aristotelian logic explains the behavior of the classical
world, whereas non-Aristotelian logic applies to the quantum world. We would
like to remind the reader that, in the manuscript, the word ‘psyche’ means the
set of collective brain phenomena emerging from microscopic cells’ dynamics,
and is not anything metaphysical.

We have been motivated by the analogy with bi-partite logic to propose a
quantum–classical model for studying brain processes. One idea behind this
proposal, i.e., the need to mix a quantum and classical level of description
together, had already been supported in a somewhat less explicit form by three
theoretical approaches, which we reviewed in the first part of this work. Two
approaches that we reviewed were originally designed by their authors as
theories of consciousness. However, this is not the perspective from which we
looked at them. In this paper, we were not interested in describing
consciousness. Instead, we interpreted these theories in terms of purely
physical processes, and it was only in such a respect that we considered them.

The very formulation of our model is given in terms of quantum and classical
variables that are treated on the same level. It does not need to invoke,
e.g., quantum gravitational effects in the brain. Instead, the crux is that
the quantum variables play the fundamental role of providing a quantum guiding
mechanism for the classical variables that they are coupled with. Such an idea
was originally formulated by Pascual Jordan. With respect to this, in order to
have quantum effects in brain dynamics, there is no need to invoke a highly
improbable coherent quantum state of the whole brain at a high temperature.
There are important quantum properties of few-body systems that are not lost
at a high temperature. These were discussed in the text. What is needed for
the quantum biology of the brain was again suggested long ago by Pascual
Jordan: the collapse of the wave function works as an amplification mechanism
acting as a bridge between the quantum and the classical world.

We did not perform actual numerical calculation but we introduced a general
quantum–classical Hamiltonian model, which can be specialized to describe
quantum particles and spins interacting with various types of environments,
such as those found in neurons and astrocytes, or at the sub-neuronal level
in, e.g., tubulins. Moreover, electromagnetic DOFs can also be described by
our model Hamiltonian. We showed that the quantum–classical theory provides a
statistical mechanic formulation of averages and correlation functions. In
turn, as is well-known, correlation functions lead to the definition of
response functions. Non-invasive brain stimulation techniques can provide the
numerical data to which our theory can be compared.

We took the risk to discuss many complex ideas using only logic and our
scientific knowledge. We presented a synthesis of subtle concepts, introduced
our quantum–classical model, with which we plan to take on big scientific
challenges, and declared the direction that our future work will take. We
carried this out with the belief that science is not only made by numbers, but
also made by understanding and sharing concepts with the community.
Subsequently, such concepts can be discussed and refined, possibly leading to
new advancements. Our future work will be devoted to interpreting biochemical
processes in the brain in terms of quantum–classical dynamics. This will also
require performing quantum–classical calculations of neural response
functions. The implications of the interplay between the bipartite structures
of both the world and the psyche will be investigated through the formulation
of quantum–classical models of decision making.

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