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European Journal of Applied Sciences – Vol. 12, No. 3
Publication Date: June 25, 2024
DOI:10.14738/aivp.123.16966
Englman, R., & Yahalom, A. (2024). A Dual Hilbert-space Formalism for Consciousness; Memory Experiments. European Journal of
Applied Sciences, Vol - 12(3). 29-46.
Services for Science and Education – United Kingdom
A Dual Hilbert-space Formalism for Consciousness
Memory Experiments
R. Englman
Ariel University, Ariel 40700, Israel
A. Yahalom
Ariel University, Ariel 40700, Israel
ABSTRACT
Inspired by works of W. H. Zurek and others, a mathematical, physical theory,
entirely within a quantum mechanical formalism, is proposed for cognitive
processes in terms of an abstract Hilbert-space for the conscious state that is an
exact replica of the Hilbert-space for the neuronic physical state. Thus, any actual
state of consciousness arises by its formal alignment (identification) with some-one
in the set of the neuronic states, with the latter undergoing perpetual changes in the
wake of life-long experiences. It is posited that these changes become expressed by
an increase in the number of ordered, coherent neuronic states at the expense of
the preordinal random neuronic states. Changes, transitions between states are
induced by a Gorini-Kossakowski-Sudarshan-Lindblad formalism, which is also
instrumental in the effect of the conscious state on bodily actions. The paradigmatic
findings of R.N. Shepard (1958 - 2011) and of S. Sternberg (1966 - 2016) for long- and short-term recalls are interpreted within the model.
Keywords: Quantum Mechanics, mental states, awareness, mind-body interaction, sense- impressions, memory.
INTRODUCTION
[Definition: Briefly, Hilbert space is an ordered collective in quantum mechanics (q.m.) of all
possible states of a given system.]
Consciousness was defined in NEB (1980), quoting John Locke, as ”the perception of what
passes in one’s mind”, and similarly as ”the intuitive perception of experience and the flow of
inner time” by Atkinson (2011), or in a more nuanced form as ”sentience, awareness,
subjectivity, qualia, the ability to experience or to feel, wakefulness, having a sense of self-hood,
and the executive control system of the mind” (Farthing 1992). Regarding consciousness in
animals other than humans, the jury is still out, notwithstanding a weighty declaration that the
neural conditions in animals are also favorable for it (Low 2012).
Historically, Tononi and Edelman (1998) with their concept of a re-entrant functioning of the
mind (reminiscent of ”mean field” theories of phase transitions in physics), may be regarded as
the prime movers of a scientific, mathematical study of consciousness, an approach that has
further been continued more recently by e.g., Lamme (2006). With a growth of research
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activities in the field, both experimental and theoretical, more kinds of different theories of
consciousness have come to be recognized: e.g., nominally three in one version (Link 2022) and
over twenty in another survey (Seth and Bayne 2022). The theorizing concern with
consciousness spans several disciplines: philosophy, psychology, neurology, biology,
chemistry, information sciences and, more recently, physics. Distinct approaches include
higher order theories (HOT), by which consciousness emerges as the result of some higher
order mental activity (Rosenthal 2005). Then, one has global workspace theories (Baars 1988,
Dehaene and Nacache 2001), which associate with the conscious state the propensity to give it
an outward expression, e.g., by talking, deliberate acting, etc. Additionally, other views relate
consciousness to a unique combination of electro- and chemical processes, these taking is
placed exclusively in the brain (Levine 1983, Block 2009). A more exhaustive account of the
further, numerous theoretical approaches to interpret mental activities erstwhile in terms of
the neural framework is beyond the scope of this paper and so is the assignment of the different
types of mental events to their location in the brain or the specification of any brain component
[e.g., synapses, microtubules (Hameroff and Penrose 2014)] primarily involved in cognition.
Attributions of quantum aspects to mental activities, as in the present study, are not new and
are on the increase as of late, but it is here that a formal and inclusive formalism is offered anew.
In the past, looking backwards over several decades, one might recall a proposed interaction in
reverse between cognition and q.m. due to E. P. Wigner, who sought the solution for the infinite
regress problem of quantum mechanical observation in the mind of the observer (Wigner
1961). An association of intelligent decisions with quantum mechanics was proposed early by
Asano et al (2011), continued by Khrennikov (2019) and Khrennikov and Asano (2020).
Previous work on quantum mechanical aspects of cognitive processes was categorized by Hu
and Wu (2010). Notably, a formal identification between quantum mechanics and psychology
was made by Khrennikov (2019), where, in equation (20), the wave-mechanical function Ψ
made his appearance for the mental state. Quantum entanglement in bipartite systems and
violations of Bell-inequalities were pinpointed in psychological events by Aerts and
collaborators (2019). Noting some phenomenological aspects of decision making in human
affairs both by human individuals and collectives, the authors of a series of papers, in e.g.,
Pothos and Busemeyer (2013), Busemeyer and Bruza (2012), Wang et al (2013) showed that
these obeyed quantum mechanical rules of probabilities, rather than classical, Boolean or
Bayesian ones. Specifically, this conclusion was reached on the basis of empirical evidences
relating to non-commutativity (i.e., temporal order effects) in decision making,
complementarity (interference between different decision path-ways, such as in binocular
rivalry), super-positions (of alternative choices) and the bistable perception of ambiguous
events, all of which feature in processes of choice and decision. Other instances mentioned were
the so-called conjunctions fallacy, for whose resolution the authors in Pothos and Busemeyer
(2013) resort to an expression in terms of a quantum state function. The” Linda paradox” of
Kahanman and Tversky (1983), illustrating a falsely conceived dichotomy between her being a
feminist and a bank-teller, is further quoted there as pertaining to the quantum domain. A
related outcome of the identification with quantum mechanics is the derivation of a generalized
Jarzynski equality and a generalized Crooks’ theorem for decision-making (Braun-Moya,
Kru ̈ger and Braun 2018), in which the energy is replaced by the” utility function”.
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Englman, R., & Yahalom, A. (2024). A Dual Hilbert-space Formalism for Consciousness; Memory Experiments. European Journal of Applied Sciences,
Vol - 12(3). 29-46.
URL: http://dx.doi.org/10.14738/aivp.123.16966
All these motivate the present authors to propose a formulation of mental activities in a fully
quantum setting, touching also, but not solving (or intending to solve), the mind-body problem.
What follows, then, is a quantum mechanical description of cognitive processes, with prospects
of future extensions and empirical predictions, not as yet provided.
FORMALISM
We formalize these findings, by postulating three disjoint Hilbert spaces, together with their
representatives (state functions) at any instant of time (t), consisting of a) the physical state of
the neurons [φf(n,t)], b) the actual mental state involved in any cognitive process [ψp(m,t)], c)
the physical state of the rest of the bodily organs, excluding the neurons [χk(b,t)]. Thus, the
state-function is written in a basis of a product:
Ψf,p,k(n,m,b,t) = φf(n,t)ψp(m,t)χk(b,t) ≡ Ψi (1)
with the vectorial sub-indexes f,p,k mapped onto the single vectorial index i and omission of
time, for brevity.
There being a multitude of neurons and of bodily organs, each with its own multitude of DOF,
the functions in a) and c) are representations of long product functions (assuming at this stage
independence between the DOF), with the possibility of entangled super-positions of product
functions taking, as bases, the following form:
φf(n,t) = φf1(q1)φf2(q2)...φfNneu(qNneu) (2)
for Nneu neural DOF (the q’s). Analogous products form the bases for the function χ in c),
involving the bodily organs numbering, say, Norg and it will turn out after our introduction of
the” alignment” later in this section that for the mental state functions represented in b) the
bases are in correspondence with (in fact, duals of) those in the equation 2 above.
The employment of a quantum mechanical formalism for the first and third factors in the above
need hardly to be commented on, insomuch as they relate to physical objects (essentially,
molecules although of an immensely large number and of a complicated kind). The following
quote by the noted cognitive psychologist, R.N. Shepard is in order:”Human beings and all other
animals are themselves physical systems and, as such, operate in compliance with physical
laws” (Shepard 2004). A more explicit quote stressing the physics background of neural
activities is due to Zurek (2014): ” ... the higher mental processes all correspond to well-defined
but, at present, poorly understood information processing functions that are carried out by
physical systems, our brains. Described in this manner, awareness becomes susceptible to
physical analysis. In particular, the process of decoherence is bound to affect the states of the
brain: Relevant observables of individual neurons, including chemical concentrations and
electrical potentials, are macroscopic. They obey classical, dissipative equations of motion.
Thus, any quantum superposition of the states of neurons will be destroyed far too quickly for
us to become conscious of quantum goings-on: Decoherence applies to our own” state of
mind”.”. This is followed there by an evolutionary hypothesis:
” One might still ask why the preferred basis of neurons becomes correlated with the classical
observables in the familiar universe. The selection of available interaction Hamiltonians is
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limited and must constrain the choices of the detectable observables. There is, however,
another process that must have played a decisive role: Our senses did not evolve for the
purpose of verifying quantum mechanics. Rather, they developed through a process in which
survival of the fittest played a central role. And when nothing can be gained from prediction,
there is no evolutionary reason for perception. Moreover, only classical states are robust in
spite of decoherence and therefore have predictable consequences. Hence one might argue that
we had to evolve to perceive classical reality.”.
[Though, not everyone shares this view. Thus T. Nagel (2012) writes:” The existence of
consciousness seems to imply that the physical description of the universe, in spite of its
richness and explanatory power, is only part of the truth, and that the natural order is far less
austere than it would be if physics and chemistry accounted for everything. If we take this
problem seriously, and follow out its implications, it threatens to unravel the entire naturalistic
world picture”.]
What needs to be stated clearly is that the first factor-function φ, representing the neuronic
state is the potential and actual repository of all human experiences. The Hilbert space for this
factor is of course of a prodigiously large dimensionality: thus, by supposing just two possible
states for every neuron (e.g., off and on, (Amit1989), section 6.5.1), the Hilbert space of the first
factor is 2 to-the-power of the number of neurons in the brain (estimated at 86 billion for
humans). Furthermore, it seems possible to posit as a guiding idea, that a ”virgin”, un- experienced brain is composed of collective neuronic states in which diverse neurons are
connected by the connectional architecture in the brain, possibly in an entangled form, but in
essentially random, disorganized arrangements, i.e., with no regular correlations between
individual neuron states in a collective state. An experience or external stimulus causes a
collective state to become organized, regular, orderly in a way that resembles a paramagnetic- to-ferromagnetic phase-transition. Further experiences do the same for additional collective
states, so that there is a one-to-one correspondence between the experiences and the number
of orderly collective states, all this happening without regard to the energy of the states or to
the prevailing temperature.
This model of cooperative transition is supported by researchers of an extensive observation
on 40 retinal ganglion cells of salamander (Schneidmanet et al. 2006), (further confirmed for
monkey eyes (Shlens et al. 2014)), who found that a model of weakly correlated neuronic units
is equivalent to the Ising model with pairwise coupling in Statistical Physics. Their conclusion
is reached after noting that the observed frequency of collective spiking is many orders of
magnitudes larger than that arising from a model of independent neurons. Clearly, such
coordinated spiking is a hallmark of a coherent collective neuronic state. Subsequent work in
Tkacik et al. (2014) on up to 200 retinal cells reinforced the model and led to the authors’
writing:” It is widely agreed that .... percepts, thoughts, and actions require the coordinated
activity of many neurons in a network, not the independent activity of many individual
neurons”. Neuronal ferromagnetic versus paramagnetic phases had been found (Rotondo et al.
2018), with the former favored by weak quantum effects and low temperatures.
The quoted dictum of R.N. Shepard (2004) holds equally well for the third factor-function χ,
that represents the bodily state after the mental process, such as uttering a sound (in speaking),
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Englman, R., & Yahalom, A. (2024). A Dual Hilbert-space Formalism for Consciousness; Memory Experiments. European Journal of Applied Sciences,
Vol - 12(3). 29-46.
URL: http://dx.doi.org/10.14738/aivp.123.16966
moving a finger or taking a step. Admittedly these activities are more parsimoniously described
classically, e.g., by Mechanics of muscular action, nevertheless, for consistency we choose to
stay within a quantum formalism to formulate also bodily activities. In this formalism, the
dynamics is represented as transitions between quantum states, of which later. Alternatively, a
mental process may be followed not by a bodily change, but by a subsequent mental activity
(like thinking, contemplating) or an altered feeling (e.g., being afraid or elated); in this case
third factor stays constant and it is the middle state-factor that undergoes change(s). We now
turn to this item, which, in its form, is essentially the newcomer in the theory (although the
several cited works in the” Introduction” have paved the way to its inclusion into a fully
quantum formulation.)
The middle factor-function ψ is a vector in the Hilbert-space of all mental states. In the theory
this space is posited to be in a one-to-one correspondence with the actual, momentarily
prevailing neuronic Hilbert space (and therefore possesses the same dimensionality as that).
At any moment the mental state occupied (meaning, sensed by the person) is one that is the
reflection of one, single neuronic state. If this neuronic state is the creation of a past experience,
etc., that is to say ordered, the mental state will qualify as a conscious state; if, however the
mental state is aligned with a random, un-organized neuronic state, then the possessor of this
mental state will be lacking consciousness at that moment.
Putting in numbers, if one associates the conscious states with thoughts in the mind, one may
find an estimate for the maximal number of conscious states achievable in a human lifetime, as
108 - 109, based on 10k-100k thoughts per day. Now, this number is infinitesimally small
compared to the astronomically large number of neuronic states. In an alternative estimation,
if one may associate the collective coherent states with the meta-stable correlated states, the
number of the former grows exponentially with the number of neurons Nneu, roughly as 0.1 x
100.03Nneu, which again is much smaller than the number of neuronic states. [This is based on
Figure 10 in Tkacik et al (2014). An early estimate, due to von Neumann, based on the observed
spike frequency is about 1020 (Amit 1989), section 6.1.1.] How is it, then, that human beings are
thoughtful most of the day? The answer seems to lie in the evolutionary hypothesis of Zurek
(2014) quoted above.
The postulated neuron-consciousness correspondence has its analogue in the mapping,
erstwhile put forward by John Maynard-Smith, between genotypes (the stored biological
information) and phenotypes (functional or observed properties), allowing in that case for a
redundancy (many-to-one correspondence) quantified in Greenbury et al (2016) and recently
discussed critically in Sappington and Mohanty (2023). Our model of a single-mode alignment
is a null hypothesis, which may be modified without essentially altering the main concept.
The above process of alignment between the state of consciousness and a neuronic state, or
their duality, that is central to the theory, is just one of the transitions taking place and will be
treated in the next section under” Dynamics of Cognition”. Alignment between states in
different Hilbert spaces is not without parallel, in as much as it forms the critical step in von
Neumann’s quantum measurement theory, through the process of aligning the states of the
microscopic observable with the states of the pointer (E.g., equation (2.5) in Leggett (1980). As
is well known (e.g., Moreira-Almeida et al. 2018), there are two main approaches to the mind-
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body problem: Physicalism, according to which mind is a material or physical process, a
product of the functioning of the brain. In contrast, dualism claims that mind is something other
than, and could exist apart from, the brain. The here proposed alignment mechanism bridges
between the two approaches.
DYNAMICS OF COGNITION
Life means changes (activity) (Marx 1867). Irreversible changes in physical systems are
currently described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) formalism, which
is an extension of the time-dependent Schrödinger equation for the density matrix of systems
in interaction with their surroundings, such that the densities are non-negative and their trace
sums to unity at all times (Gorini et al 1976, Lindblad 1976). The density matrix arises as the
components of the density operator being a function of time.
ρ ≡ |Ψ >< Ψ| (3)
The range and conditions of validity of the GKSL-equation have been the subject of numerous
investigations, in which the equation is derived for a microscopic system coupled to a
macroscopic environment. In most of the treatments this coupling is postulated as weak, in the
sense of its lying below the energy spread of the system’s Hamiltonian. In the derivation of the
GKSL-equation in Pearle (2012), this requirement is not noted. But in Kosloff (2013) the weak
system-bath coupling limit is explicitly postulated, additionally to a thermalization time that is
slower than the inverse energy spread in the system. In this limit, the system-environment
interaction energy may be neglected. The uniqueness of the Lindblad formalism was studied in
Barra and Liedo (2018). In an extensive review by Rivas and Huelga (2018), one finds a sally in
their section 7.3 to the strong coupling, non-Markovian regime with a generator of the
Lindbladian form that preserves positivity.
We recall Lindblad’s equation for the time varying density of states operator ρ as being of the
following form:
(4)
in which the gammas are cofactors and Ln are the Lindblad operators for a set of n’s. We now
postulate having a single summand in the second term, conventionally named” Dissipator” D(ρ),
and rewrite this in the bra-ket form as:
(5)
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1. Perception, change in the neuronic state: neuronic state index in Eq 1changing from f to
f
0, with other indices invariant.
2. Consciousness, recognition of neuronic change: p (aligned to f) → p0 (aligned to f
0)
3. Bodily activity: k → k
0 → k” ....
4. Instinctive action: Process I → process III without intervention of process II.
5. Other unconscious activity, including dreaming and one satisfying a Turing test: Process
I → process III without intervention of process II.
AN ILLUSTRATION
We assume that the density of states ρ for the neuronic states Ψi in equation 1 may be written
at some initial time t = −∞ in a diagonal representation (and, similarly, for the analogous
functions of the consciousness). To mimic the effect of an experience starting at t = 0 on ρ, we
suppose that one of the factors, say the first, in Ψi represents a partially disordered neuronic
state, with the rest of the factors spanning any arbitrary number of ordered states, bearing the
imprint of previous sense impressions. Suppose now, further, that the new experience changes
this first factor into an ordered neuronic state. If the two, initial and final, neuronic states are
labelled respectively ρin and ρfin, this simply means that the occupation numbers (probabilities)
1 and 0 for ρin and for ρfin are interchanged due to the new experience. In the present theory this
transition is driven by a Lindblad operator.
In Figure 1 we illustrate the working of the Lindblad operator for a toy example involving only
a paltry of five neuronic states (instead of the myriad in a realistic situation). The occupation
probabilities of the neuronic states are shown by a yellos line for the initially excited state, being
replaced upon some experience by the ultimately occupied component ρfin, whose probabilities
are drawn in red. This process is then followed, after a delay of about one-tenth of a second, by
the acquisition of an awareness (consciousness) of the new neuronic state, with the respective
occupation probabilities drawn using a blue line which is horizontal initially and rises to one
finally, for before and after the acquisition of awareness. The cognitive delay 0.1s is in line with
the empirical observations,” the Libet delay” in Libet et al. (1983). As an outcome, the actual
state of consciousness is aligned with the actual neuronic state (i.e., identical in their respective
Hilbert-spaces).
THE COUPLING
At the backbone of the theory lies the role of the Lindbladian term, which primes each of the
stages I -IV, bringing about the transitions. The source of this term as arising from an interaction
between the neuronic framework and the surrounding is not identified at this stage, but seems
to reside in the unity of the neuronic system with its broad surrounding. Such unity has been
called into play for the interpretation of the quantum mechanical measurement process already
by N. Bohr (1958): ”The answer that we get is built up from the combined interaction of [the
observer’s] state and the object of interrogation.” and more recently in several works: Thus,
one may find from John Bell that ”The answer that we get is built up from the combined
interaction of [the observer’s] state and the object of interrogation.” (Bell (1990); from Asher
Peres ”A measurement both creates and records a property of the system” (Peres 1998), and
further by Anthony Leggett: ”...under these conditions (without the environment interaction)
the macroscopic apparatus, and more generally any part of the macro-world which has suffered
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Griffiths 2001). A further generalization in terms of an algorithmic information distance is due
to Chater and Vit ́anyi (2003).
Another remarkable development, also due to Shepard and summarized in Shepard (2004), is
that the time required to determine that two things are identical, in spite of their apparent
difference, linearly increases with their transformational separation (” distance”) in the space
of possible positions. On the other hand, the time required to determine that two things are
different, despite their apparent similarity, decreases non-linearly, as function of their
separation in the space of possible objects. As the guiding principle for clustering together in
human perception of widely different stimuli (sources of experience), Shepard (2001)
proposed that, ”[these] generally form a connected local region in the space of possible objects,
despite appreciable differences among individual objects of that kind in size, shape, position,
motion, or color”. We shall reformulate this broad” juxtaposition” idea of Shepard, originally
pertaining to some undefined space (his” appropriate representational space”) to apply to the
Hilbert spaces that constitute the basic elements of the present theory. Passing from a
continuous representation of different stimuli (as in rotationally different stimuli) to those that
differ in a discontinuous, discrete way, Shepard notes that one would still obtain an exponential
type of fall-off of generalization with distance, where distance is now defined in terms of the
sum of the weights of the features that differ between the two objects or, if the features are all
equal in weight, simply in terms of the number of differing features.
Noting the above two apparently discordant assertions by Shepard, [that the reaction time (RT)
for affirmative response increases linearly with distance, whereas on the contrary, the
discriminatory RT falls off according to a decreasing, non-linear function of distance between
stimuli in representational space], we shall now demonstrate how in the present formulation
of transitions the two findings may be reconciled.
As a preliminary, one bears in mind that quantitatively the RT differs by a so-called” response
time” from (it is less than) the actual experimental delay time (the time that has elapsed
between the presentation of the task to the subject and the instant of the subject giving his
decision). Then, for the explanation of affirmative responses, we simply utilize a protocol due
to Sternberg (Sternberg 1966), of which more later, which predicates that for a decision
involving several items, the system declares its decision after scanning all items, and not at any
interim stage, thereby resulting in a RT that is proportional to the number of items. If we now
conjecture that the distance in Shepard’s experiments is represented by the number of states
in the Hilbert spaces, the proportionality of RT to the distance follows for affirmative decisions.
When an affirmative answer is not forthcoming, as when the subject discriminates between the
presented and the task items, the decision time is composed of two times; the first, during which
the subject scans all the relevant items, as in the case of an affirmative decision, but now
reaching a negative result followed by a further search. Such a search had been investigated
some long time ago, e.g., for discriminating sounds as functions of both frequencies and
intensities. It was found, e.g., by Chocholle (1940), that the discrimination time decreases as
some inverse power of the sound intensity, with the exponent varying between −0.50 and −0.25
(with a dependence on the sound frequencies). Other experiments (both pre-dating and post- dating Chocholle’s), including also those with light, gave similar results. A detailed theoretical
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Englman, R., & Yahalom, A. (2024). A Dual Hilbert-space Formalism for Consciousness; Memory Experiments. European Journal of Applied Sciences,
Vol - 12(3). 29-46.
URL: http://dx.doi.org/10.14738/aivp.123.16966
study (Luce and Green 1972) interpreted the stage of decision making by the neural system as
one at which the incoming pulse-signals in the brain reach some saturation. A stochastically
varying mode was analyzed which then in some limiting approximation led to the result for the
mean reaction time:
RT = 2 arrival time of the signals/Sound intensity (6)
The intuitive meaning of the result, derived by the authors of (Luce and Green 1972) is that
when the intensity is large, proportionately less time is needed to saturate the system with
information. The inverse power relation in the above expression is to be noted, agreeing with
the empirical findings.
In a further series of studies of the latency time of short-term memory, pioneered by S.
Sternberg (1966), subjects were first given a set of terms or symbols and then tested by a
further term or symbol, and asked whether or not this was one of the test terms? Remarkably,
their mean reaction time (RT) increased linearly with the length of the sequence. The linearity
and slope of the reaction time was hypothesized by the author as implying the existence of an
internal serial-comparison process with an average rate of 25-30 symbols per second. Similar
rates were found for positive and negative answers by the subjects, which finding led the author
to his theory that in short-term memory experiments the subject scans all preliminarily
presented items, before making a response. Subsequently, some discrepancies from linearity
were noted, e.g., when the experiments were repeated with differing ordering of the test terms,
as in Derosa and Morin (1970), with Sternberg (2016) replying with a vigorous defence of his
position.
THEORETICAL
Short-Term Recall Experiments
Saul Sternberg’s serial exhaustive scanning paradigm, proposed for the explanation of short -
term memory experiments reconciling positive and negative responses, implies an information
transfer between that part of the collectives of neuronal states which was created by the pre- presented items and the consciousness state in the brain that elicits the response. Formally, in
the dual Hilbert space model, this may be represented in the most transparent way in terms of
a coupling between the neural and consciousness spaces through interaction terms in the
Hamiltonian, written in a second quantization description, in the form, nin†
jcic
†
j for which the
letters and their daggered version in n,n†,c,c† represent respectively destruction and creation
operators in the neuronic and consciousness Hilbert spaces. In physical terms, the information
transfer is affected by some further entities, whose nature is not known as of now, these being
the cognitive counterparts of mRNA featuring in the central dogma of biology for the formation
of peptides from DNA. In this the translational elongation is achieved through the scanning of
the full length of mRNA by the ribosome, this being done at a rate of about 1.5 codon per second
(Heinrich and Rapoport 1980). It stands to reason that through some entities Nature employs
the same stratagem for the analogous scanning in the brain for the information transfer, this
being done at a rate of items per second higher by a factor of about 20. (The factor is likely due
to the correspondingly higher complexity along a linear dimension in the brain, in comparison
to other bodily organs.)
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Long-Term Memory Results
By the time that the recall experimentation takes place at some instant after the short-term
memory times (this being of the order of 10ms), as in the Shepard type of investigations of
identification or discrimination between tests and probe items, the neuronic wave functions
have changed their character, due to the decohering interaction with the environment, from the
coherent entangled form of the foregoing section, to mixed quantum states. These are now
subject to a diffusive process, whose time dependent occupation probability was given in Eq.
[13] of Shepard (1958) as function of the distance Dik = |xi − xk| between test and probe items
and of time t by the Gaussian
(7)
in which V is a diffusion constant. A convolution integration is then shown by Shepard (1987)
to lead to a negative exponential dependence of the time integrated probability for the
affirmative identification between the test and probe items as function of their distance. This
establishes the theoretical basis of Shepard’s ”Universal Law of Generalization”.
The foregoing Gaussian (modified, as described below) is now proposed as the criterion for the
time (RT)-distance relation found in identification discrimination experiments (resulting in
positive-negative response, respectively), distance being a measure of dissimilarity between
sample and probe. For a qualitative understanding one notes that in a diffusion process along
an infinite line (and similarly for higher dimensions) starting at the origin, the occupation
probability at some distance away at first increases, as the diffuser makes its way randomly
away from the origin to this point, but then decreases as the diffuser proceeds to farther away.
It is now postulated that as the growing occupation probability exceeds a certain significantly
large critical value (to be denoted p0, meaning that Pr−>p0+0) a recognition is attained and
likewise when it later again decreases to beyond a critical small value p1 (which is different
from the former critical value) of the occupation probability, Pr−>p1−0 a discrimination
(negative response) has come into force. In the dual Hilbert space model of this article, the
diffusive motion takes place among the locations of the states in the neural function space, with
the consciousness’ Hilbert space” contacted”, aligned whenever the critical occupation values
are in place. To account for the empirical finding that the reaction time (RT) for affirmative
response increases linearly with distance, whereas on the contrary, the discriminatory RT falls
off according to a decreasing, concave-upward function of distance between stimuli in
representational space, we hypothesize a time dependent diffusion parameter V (t) in Shepard’s
Gaussian probability function, replacing his constant V. This assumed variable diffusion
parameter is such that the width of the Gaussian, instead of steadily increasing as in standard
diffusion theories, first increases and then drops off. The physical meaning of this is that the
basic diffusion step is not constant, but rather diminishes as the scanning proceeds. Variable
diffusion steps of a different kind are known as Levy-walks.
Examples of the postulated widths are depicted in Figure 2. Algebraically they take the form:
(8)
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Englman, R., & Yahalom, A. (2024). A Dual Hilbert-space Formalism for Consciousness; Memory Experiments. European Journal of Applied Sciences,
Vol - 12(3). 29-46.
URL: http://dx.doi.org/10.14738/aivp.123.16966
Figure 2. Postulated root - mean -square - width in diffusion as function of time in milliseconds.
Parameters in Eq. 8 are as follows. Blue: t0 =0.05; Red: t0 = 0.1.
Figure 3. Early reaction times in milliseconds for confirmation. Blue: p0 (critical probability
density) = 0.5, t0 = 2; Red: p0 =0.1, t0 = 8. A linear relation is shown with broken lines for
comparison.
Affirmative answers for identifying probes slightly dissimilar from samples take place when
the diffusional probability reaches a critical value p0. The two curves in figure 3, with the
choices p0 = 0.5 in blue and p0 = 1 in red show reaction times (RT) in milliseconds, as function
of distance. A linear relation is shown with a broken line. At a contrary extreme of large
dissimilarities, the RT for discrimination takes large values. These are shown in figure 4, again
with two choices of the p1 criterion, p1 = 0.5, and 0.1 meaning that when the diffusive occupation
probability decreases down to this value, the subject concludes that there is no similarity. Due
to the time dependent Width, this discriminatory RT falls off according to a decreasing,
concave-upward function of distance between stimuli and probe in representational space, as
found empirically by Shepard (1987). Qualitatively, the curves are robust in both parameters,
p1 and t0.
0 10 20 30 40
time
0.5
1.0
1.5
2.0
2.5
3.0
Width
0.00 0.05 0.10 0.15 0.20
distance
0.2
0.4
0.6
0.8
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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 3, June-2024
Figure 4. Reaction times for discrimination. Parameters: p1(critical probability density) = 0.05, t0
= 2 (blue); p1 = 0.1, t0 = 8 (red). An inverse power of distance curve is shown by broken lines for
comparison.
CONCLUSION
Mental states are postulated to be represented by quantum states, thereby constituting a
Hilbert space of their own, distinct from the physical Hilbert-space of the neural network but
being a mirror image (dual) of the latter. Thus, any actual mental state is at any time in
correspondence with (aligned to) a neural state, this being a particular vector in the neural
Hilbert space. The neural space is under perennial development under external experiences,
such that out of the multitude of initially disordered neuronic set of states ordered, coherent
neuronic sets are formed, one at a time, for each external stimulus. Then the states of mental
awareness (consciousness) are associated with alignment to the ordered neuronic quantum
states. Changes in both the neuronic and mental states are affected by a Gorini-Kossakowski- Sudarshan-Lindblad formalism, originally formulated to describe changes (dissipations) of
quantum systems induced by a macroscopic environment. Several components in the
postulated theory (as e.g., the quantum formulation of mental states, the transition to ordered
neuronic sets) are based on experimental evidence, as documented by the publications (and
quotes) cited in this article, but their sewing together as attempted here to represent generic
mental processes as fully quantum-like is novel, and requires factual confirmation.
Two kinds of memory experiments, for short- and long-term memory recalls, had led S.
Sternberg and R. N. Shepard, respectively, to semi-quantitative descriptions regarding
cognitive processes in the brain. These are interpreted within the dual Hilbert space model of
this paper, coupled with a proposal of a copying process analogous to the DNA-to-peptide
translation in biology and, further, through a diffusive process with time variant decrease of the
diffusion parameter. The latter leads to an experimentally verifiable prediction of a time
asymptotic narrowing of the probability-distribution of recalls.
0.5 1.0 1.5 2.0
distance
100
200
300
400
500
RT
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Englman, R., & Yahalom, A. (2024). A Dual Hilbert-space Formalism for Consciousness; Memory Experiments. European Journal of Applied Sciences,
Vol - 12(3). 29-46.
URL: http://dx.doi.org/10.14738/aivp.123.16966
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