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BRAINS AS QUANTUM MECHANICAL SYSTEMS $--$ A NEW MODEL

Chris Allen Broka\\
2002 Granemore St. Las Vegas, NV 89135\\
(chris.broka@gmail.com)

\pmb{ Abstract}

We consider the possibility that the brain functions in the manner of a conscious quantum computer. The processes that instantiate its consciousness
$--$ the physical correlates of consciousness $--$ are suggested to be quantum mechanical in nature rather than entirely classical. This idea is
by no means new. But specific physical models are hard to come by; it is not obvious what kind of \textit{ physical} process might give us something
like qubits. We begin by approximating a synapse as a small, parallel-plate capacitor. We find that the classical electromagnetic energy stored in
such a synapse corresponds closely to the spacing of energy levels we would obtain were the capacitor to be quantized. Intrigued by this surprising
observation, we propose a new model of the brain as a partially quantum mechanical system. Its potential evolutionary benefits are discussed briefly.\\
\\
\pmb{ Keywords:} Brains as Quantum Computers, Quantum Measurements, von Neumann-Wigner Interpretation, { }Synapses.

Introduction.

The author ([1], [2], [3]) has recently advanced a variant of the von Neumann-Wigner Interpretation of Quantum Mechanics [4]. We work in the language
of field theory where the state of the entire universe, at time \textit{ t}, $|\Psi $\textit{ (t)}$>$, is a vector in the Fock space of the Standard
Model. Consciousness is, here, assigned the role of $\texttt{"}$classifier.$\texttt{"}$ Some $|\Psi $\textit{ (t)}$>$s are classified as admissible.
The others are inadmissible. We gave a simple example where an electron is passed through a Stern-Gerlach apparatus. If it comes in spin-up a green
light is triggered. If it is down we get a red light. A conscious observer watches all of this. Were the electron to come in in a superposed spin-state
{ }unitary evolution would have our observer seeing a 'green-red' qualia. Wigner called this situation $\texttt{"}$absurd$\texttt{"}$ and we call
it inadmissible. $|\Psi $\textit{ (t)}$>$ can never enter into such a state. To ensure that it does not we introduce a new operator $--$ $\mathfrak{S}$
$--$ and demand that { }\(\mathfrak{S} |\Psi (t)> = |\)$\Psi $\textit{ (t)}$>$ always. If $|\Psi $\textit{ (t)}$>$ is admissible $\mathfrak{S}$ does
nothing. If it is not, $\mathfrak{S}$ looks at all the amplitudes $<$\(\Psi _a\)$|\Psi $\textit{ (t)}$>$ for every admissible $<$\(\Psi _a\)$|$.
It will square these amplitudes and, using these values as \textit{ relative} probabilities, convert { }$|\Psi $\textit{ (t)}$>$ into one of the
$|$\(\Psi _a\)$>$ at random. \\
\hspace*{2.5ex} This takes care of the $\texttt{"}$absurd$\texttt{"}$ 'green-red' qualia. But we wondered whether $\mathfrak{S}$ might not perform
another $--$ and more important $--$ function. Let $|\Psi $\textit{ (t)}$>$ describe a simple universe consisting of a single conscious brain that
is experiencing qualia. Suppose, further, that the brain instantiates its consciousness through a physical mechanism that it, in one way or another,
quantum mechanical in nature $--$ it does not behave as a totally classical system. $|\Psi $\textit{ (t)}$>$ starts out in an admissible state. But
the brain is a warm, wet, noisy environment. It may be that, as $|\Psi $\textit{ (t)}$>$ evolves unitarily, interaction with the environment begins
to carry it into an inadmissible state no longer compatible with consciousness or its functioning as a workable $\texttt{"}$quantum computer.$\texttt{"}$
$\mathfrak{S}$ then projects it back into an admissible state. It then evolves unitarily with no trouble until, after a while, it tries to, once
again, become inadmissible. We (rather loosely) described this process as quantum decoherence. There are not a great many \textit{ physical }theories
of brains as quantum computers and we used, as an example, the Hameroff-Penrose Microtubule Hypothesis [5]. Tegmark [6] has calculated that their
model would decohere in about \(10^{-13}\) - \(10^{-20}\) sec. This is quite rapid relative to the time-scales normally associated with conscious
processes. We did not, ourselves, propose any physical model of the brain as a quantum computer. But we will try to do so here.

The Independent Oscillator Model.

We begin by approximating a synapse as a capacitor consisting of two circular, parallel, plates of radius \textit{ R} separated by a gap of width
\textit{ d}. (In real synapses these are about 0.5 $\mu $m and 30 nm, respectively.) This system is studied in every introductory course on electromagnetism.
Between the plates is a potential difference $\varphi $. There will be a uniform electric field (\(- \varphi \)\(/d\)) between the plates. Assume
this varies. A circulating magnetic field (\(\frac{1}{2 c^2}\)\textit{  r} \(\partial _t E\)) will be induced between the plates. As this changes,
a correction will be introduced into the electric field. This will, in turn, induce a new correction to the magnetic field. We could carry this process
on indefinitely but we will not do this. We will drop all but the original (homogeneous) electric field, its first-order magnetic correction, and
the first-order magnetic field. As long as the potential is not varying quickly, we are justified in doing this. As a practical matter, potentials
in synapses do not change much over time-scales less than about \(10^{-5}\) sec. Given the dimensions of a typical synapse, the higher-order corrections
drop to zero very quickly. We know that the Lagrangian density for this system is given by \(\frac{\epsilon _0}{2}\)\(\left(E ^2- c^2 B^2\right)\).
\\
$\unicode{0009}$\\
1) { } \(\int \frac{\epsilon _0}{2}\left(E ^2- c^2 B^2\right)dV\) = - \(\frac{\mu }{2}\) \(\varphi _{,t}{}^2\) + \(\frac{\mu  \alpha ^2}{2}\)\(\varphi
^2\) + \(\mu  \varphi  \varphi _{,t t}\) where $\mu $ = \(\frac{\pi  R^4 \epsilon _0}{8 c^2 d}\) and $\alpha $ = { }\(\frac{\sqrt{8} c}{R}\). The
Euler-Lagrange equation, and hence the physics, is unchanged if we subtract the (useless) total derivative \(\mu \partial _t\) ($\varphi $ \(\varphi
_{,t}\)) from the above. We could just as well have written our result as: - \(\frac{3\mu }{2}\) \(\varphi _{,t}{}^2\) + \(\frac{\mu  \alpha ^2}{2}\)\(\varphi
^2\).\\
\\
\hspace*{2.5ex} Either way, variation of 1)\textit{  }by\textit{  }$\varphi $\textit{  }leads to a harmonic oscillator solution with a resonant frequency
equal to $\alpha $/\(\sqrt{3 }. \text{This} \text{is} \text{the} \text{same} \text{result} \text{we} \text{would} \text{obtain} \text{were} \text{we}
\text{to} \text{write}:\)\\
\\
2) { } $\mathcal{H}$ = { }\(\frac{1}{2 m}\)\(p^2\) + \(\frac{m \omega ^2}{2}\)\(\varphi ^2\) where { }\textit{ m} = \(\frac{3 \pi  R^4 \epsilon _0}{8
c^2 d}\) and $\omega $ = { }\(\frac{\sqrt{8/3} c}{R}\). \\
\\
We have a classical harmonic oscillator with resonant frequency $\omega $ and the inconvenient \(\varphi _{,t t}\) term has disappeared.\textit{
 }Of course, we do not know over what range $\varphi $ oscillates. It would oscillate quickly, however; $\omega $ $\approx $ \(10^{15}\) \(\sec ^{-1}\).\\
\hspace*{2.5ex} We will proceed to quantize our system. \textit{ p} and $\varphi $ are, now, to be interpreted as operators where [$\varphi $, \textit{
p}] = i $\hbar $. This leads to a wave function for $\varphi $:\\
\\
3) { } $\psi $\textit{ ($\varphi $, t)} = \(\sum _n c_n \psi _n\text{\textit{$($}}\text{\textit{$\varphi , t$}}\text{\textit{$)$}}\) where the \(\psi
_n\text{\textit{$($}}\text{\textit{$\varphi , t$}}\text{\textit{$)$}}\) are the familiar energy eigenstates for a quantized harmonic oscillator.\\
\\
\hspace*{2.5ex} We now make a very interesting observation: We look at $\hbar \omega $ (the energy of a quantized excitation of our oscillator) and
compare it to the total classical energy in our synapse which is equal to \(\frac{\epsilon _0}{2}\)\(\frac{\pi  R^2 }{d}\)\(\varphi _{\max }{}^2\)
where \(\varphi _{\max }\) is the greatest classical potential difference achieved by our oscillator. If the latter is vastly greater than the former,
we know we are \textit{ not} dealing with a quantum mechanical problem. Equating the two gives us \(\left|\varphi _{\max }\right|\) $\approx $ 30
mV. This is exactly similar to the potential differences typically seen in real synapses and neurons. Resting potentials, for instance, are generally
about -70 mV in magnitude and this can change to about +40 mV with the appearance of an action potential. We could, I suppose, write this off as
a strange coincidence. But we could, also, suspect that Nature has engineered this result carefully and for a purpose.\\
\hspace*{2.5ex} We are no longer interested in the classical physics and are, certainly, not trying to suggest that there is a classical $\varphi
$ in the synapse that oscillates\(\sim 10^{15}\) times a second. That would be ridiculous. $\psi $\textit{ ($\varphi $, t) }simply tells us the likelihood
of finding a particular value of $\varphi $ were it to be measured. And this will affect how the brain functions. \\
\hspace*{2.5ex} Let us imagine a scenario in which there is an action potential propagating down an axon towards a synapse that is in \(\psi \text{\textit{$($}}\text{\textit{$\varphi
$}}\text{\textit{$)$}}\). We envision action potentials as purely classical things. When it arrives at the synapse it must decide what to do. Since
it is a classical object it will have to perform a quantum measurement to determine the value of $\varphi $. If it finds a very small value, things
go on as usual. But suppose it measures a $\varphi $ that is large and opposite in sign. The net potential difference would then be reduced. The
action potential may find itself unable to fire the synapse and will stop there. If the synapse does fire, the resulting synaptic potential would,
likewise, be augmented or reduced. The value found for $\varphi $ is, of course, determined randomly. The overall effect is to introduce an element
of randomness into the process of neurotransmission. Action potentials will, sometimes, fail to cross a synapse although we would expect them to.
Other times they would get across even though we might not expect them to.\\
\hspace*{2.5ex} When a measurement is performed and $\varphi $ found to be \(\varphi _0\), the wave function of the synapse will want to collapse
into \(a \text{state} \)with no uncertainty in $\varphi $. The uncertainty in \textit{ p} would then be infinite. The state of the synapse would
instantly evolve into one where there would be an equal probability of a new measurement giving \textit{ any} result for $\varphi $ (even physically
impossible ones). This situation is almost as $\texttt{"}$absurd$\texttt{"}$ as the 'green-red' qualia. Let $\{$\(\psi _a\)\textit{ ($\varphi $)}$\}$
denote the set of all admissible $\psi $\textit{ ($\varphi $)}s. $\mathfrak{S}$ would project the (inadmissible) state into one of the \(\psi _a\)\textit{
($\varphi $)} with a relative probability given by \(\left|\psi _a\text{\textit{$($}}\text{\textit{$\text{\textit{$\varphi $}}_0$}}\text{\textit{$)$}}|^2\right.\).
Since each synapse constitutes its own, independent, Hilbert space, we do not think that $\mathfrak{S}$ would affect any other synapses (provided
they were already in admissible states). Here we encounter a real example where $\mathfrak{S}$ does, in fact, function to allow the brain to continue
its quantum mechanical operations (although quantum decoherence is not involved). There are thought to be up to \(10^{15}\) synapses in a human brain.
If each synapse encounters an action potential 10 times a second, $\mathfrak{S}$ would have to project the system every \(10^{-16}\) sec. We also
mention that several other models have been proposed that try to impute quantum mechanical properties to synapses ([7], [8]).

The Evolution of Consciousness.\\
\\
Whatever the process underlying consciousness, Evolution seems to have put a great deal of effort into optimizing it. It must confer a powerful benefit
upon the animal possessing it. What could this be? At first glance it might be hard to see how what we have proposed would be useful at all. As mentioned,
it leads to a certain randomization of neurotransmission and this might appear to be detrimental. But suppose there were a primitive animal in which
a particular stimulus resulted in the firing of a synapse between neurons A and B. This, in turn, leads to a behavior and an outcome. If the brain
were a completely classical system this behavior would always result from the stimulus. But, if we are right, on some random occasions the synapse
will fail to fire. The result would be a different behavior and outcome. If the different outcome was favorable the brain would $\texttt{"}$notice$\texttt{"}$
this and be inclined to weaken or delete the synapse so as to ensure that the favorable outcome always occurred. Or suppose the different outcome
was unfavorable. The brain would try to strengthen the synapse or, perhaps, create more synapses between A and B to make sure that the unfavorable
outcome never occurred again. The animal will, therefore, have\textit{  learned} something. We think the advantage of our proposed model lies in
its ability to allow the animal to (randomly) experiment with a wider range of behaviors than would otherwise be possible. Learning would take place
and the animal's survival chances would be improved.

Conclusion.

In [3] we speculated that the brain functions as a sort-of quantum computer and that $\mathfrak{S}$ may be necessary for this to be possible. We
mentioned the Microtubule Hypothesis but were unable to offer any model of our own. We have now provided one.\\
\hspace*{2.5ex} By far the strongest evidence suggesting that our model might be on the right track comes from the $\texttt{"}$coincidental$\texttt{"}$
agreement between the classical energy stored in our capacitor-like synapse and the quantum mechanical spacing of the energy levels obtained by quantizing
it. This is a truly remarkable result and hints at some very clever evolutionary design-work. Neurons, given their small size and metabolic limitations,
probably, have more-or-less to function with potential differences of about 30 mV. It is hard to see how they could generate hugely greater potentials.
This being the case, were the radius of the synapse to be ten times larger than it is, the synaptic cleft would have to be 30 $\mu $m wide for our
$\texttt{"}$coincidental$\texttt{"}$ equality to hold. Neurotransmitters would be unable to diffuse across it quickly or in significant concentrations.
And the synapse would no longer much resemble a parallel-plate capacitor. If the radius were ten times smaller, the synaptic cleft would have to
be narrower than an atom.

References. 

[1] { } { }Broka, C. A. \textit{ Consciousness and the Problem of Quantum Measurement}. arXiv:1911.01823 (2019).\textit{ \\
\\
}[2] { } { }Broka, C. A. \textit{ Gravitation and the Problem of Quantum Measurement. }arXiv:2010.14965 (2020). \\
\\
[3] { } { }Broka, C. A. \textit{ Quantum Measurements and Their Place in Nature. }Philsci-Archive.pitt.edu/21677 (2023). \\
\\
[4] { } Schr{\" o}dinger, E. \textit{ The Present Situation in Quantum Mechanics. Naturwissenschaften} \pmb{ 23} (49); 807 (1935); Wigner, E. P.
\textit{ Remarks on the Mind-Body Question.} In: I. J. Good, $\texttt{"}$The Scientist Speculates,$\texttt{"}$ (London, Heinemann, 1961); von Neumann,
J. \textit{ The} \textit{ Mathematical} \textit{ Foundations of Quantum Mechanics }(1932). \\
\\
[5] { } Hameroff, S., Penrose, R. \textit{ Consciousness in the Universe: a Review of the 'Orch OR' theory. Phys Life Rev. }\pmb{ 11} (1), 39 (2014).
doi: 10.1016/j.plrev.2013.08.002.\\
\\
[6] { }Tegmark, M. \textit{ The importance of quantum decoherence in brain processes}. \textit{ Physical Review E}. \pmb{ 61} (4), 4194, { }(2000).
doi: https://doi.org/10.1103/PhysRevE.61.4194.\\
\\
[7] { } Beck, F., Eccles, J. C. \textit{ Quantum aspects of brain activity and the role of consciousness.} \textit{ Proc. Natl. Acad. Sci. USA}.
\pmb{ 89 }(23), 11357 (1992). doi: 10.1073/pnas.89.23.11357.\\
\\
[8] { }Georgiev, D. D., Glazebrook, J. F. \textit{ The quantum physics of synaptic communication via the SNARE protein complex. Prog. Biophys. Mol.
Biol. }\pmb{ 135}, 16 (2018). doi: 10.1016/jpbiomol/bio.2018.01.006.

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