Quantum-like behavior without quantum physics I

Abstract

Recently there has been much interest in the possible quantum-like behavior of the human brain in such functions as cognition, the mental lexicon, memory, etc., producing a vast literature. These studies are both empirical and theoretical, the tenets of the theory in question being mainly, and apparently inevitably, those of quantum physics itself, for lack of other arenas in which quantum-like properties are presumed to obtain. However, attempts to explain this behavior on the basis of actual quantum physics going on at the atomic or molecular level within some element of brain or neuronal anatomy (other than the ordinary quantum physics that underlies everything), do not seem to survive much scrutiny. Moreover, it has been found empirically that the usual physics-like Hilbert space model seems not to apply in detail to human cognition in the large. In this paper we lay the groundwork for a theory that might explain the provenance of quantum-like behavior in complex systems whose internal structure is essentially hidden or inaccessible. The approach is via the logic obeyed by these systems which is similar to, but not identical with, the logic obeyed by actual quantum systems. The results reveal certain effects in such systems which, though quantum-like, are not identical to the kinds of quantum effects found in physics. These effects increase with the size of the system.

Appendices

Appendix A: Mathematical results

1.1 A.1 Modal identities

It will be convenient to record some modal identities. Let 〈W,≈〉 denote a proximity space, with 〈W,⊥〉 denoting the associated orthogonality space. We have defined in Section 2 for each subset E ⊆ W:

$$ \Diamond E := \{ w \in W \colon \exists \,v \in E \text{ such that} \ w \approx v \} $$

(63)

and noted that E ⊆♢E. Dually, we define $\square E$ as

$$ \square E := \left( \Diamond E^{c} \right)^{c}. $$

(64)

Then we have:

Proposition 1

For 〈W,≈ 〉as above and E, F ⊆ W :

• M1. ♢(E ∪ F) = ♢E ∪ ♢F

• M2. For a family $\mathcal {F}$ of subsets of W, $\Diamond (\bigcap _{F \in \mathcal {F}}F)\subseteq \bigcap _{F \in \mathcal {F}}(\Diamond F)$

• M3. ♢E = (E ^⊥)^c := E ^⊥c

• M4. $\square E = E^{c\perp } = \{ w \in W \colon S_{w} \subseteq E \}$

• M5. E ⊆ E ^⊥⊥

• M6. E ⊆ F implies F ^⊥ ⊆ E ^⊥

• M7. E ^⊥⊥⊥ = E ^⊥

• M8. $\square \Diamond \square E = \square E$

• M9. $\Diamond \square \Diamond E = \Diamond E$

• M10. (E ∪ F)^⊥ = E ^⊥ ∩ F ^⊥

• M11. E ^⊥ ∪ F ^⊥ ⊆ (E ∩ F)^⊥

The proofs are elementary and may be found in the references cited (or [17, 18]).

1.2 A.2 Propositions in Kripke orthomodels

For a proximity space as above Dalla Chiara et al. [16] define a proposition to be a subset X ⊆ W satisfying:

$$ \text{if \(x\in W\) is such that \(\forall \, y \approx x, \,\exists \, z\in X\) such that \(y \approx z\), then \(x \in X\).} $$

(65)

Equivalently, X is a proposition iff

$$ S_{x} \subseteq \Diamond X \text{ implies} \ x\in X. $$

(66)

Then we have:

Proposition 2

• 1. For all X ⊆ W , $\square \Diamond X = X^{\perp \perp }$ and X is a proposition iff $\square \Diamond X = X^{\perp \perp }= X$ .

• 2. X is a proposition iff x∉X implies ∃ y ≈ x with y ⊥ X .

• 3. For any Y ⊆ W , Y ^⊥ is a proposition.

• 4. If $\mathcal {C}$ is a family of propositions, then $\bigcap \mathcal {C}$ is a proposition.

• 5. If Y is a proposition, then X ⊆ Y iff ♢X ⊆♢Y .

Thus, from the first statement above, the two definitions of propositions (cf. (16)) are equivalent. Note also that it follows from M5 and M6 that for a subset E ⊆ W, E ^⊥⊥ is the smallest proposition containing E.

For proofs, see the references cited (or [17, 18]).

1.3 A.3 Proof of Theorem 5, section 5

(1) implies (2). Assume the lattice R(〈W,⊥〉) is Boolean. Then in view of theorem 4, Section 3, Part W is also Boolean, hence distributive. The join in Part W is just set union and we shall exploit its distributivity in this case. First we derive an expression for the meet in this lattice which we shall write as ⊓. Thus, for any subsets E, K of W:

$$\begin{array}{@{}rcl@{}} \Diamond E \sqcap \Diamond K & =& ((\Diamond E)^{*} \cup (\Diamond K)^{*})^{*} \end{array} $$

(67)

$$\begin{array}{@{}rcl@{}} & =& ((\Diamond (\Diamond E)^{c}) \cup (\Diamond (\Diamond K)^{c}))^{*} \end{array} $$

(68)

$$\begin{array}{@{}rcl@{}} & =& ((\Diamond E^{\perp}) \cup (\Diamond K^{\perp}))^{*} \qquad \text{from M3} \end{array} $$

(69)

$$\begin{array}{@{}rcl@{}} & =& (\Diamond (E^{\perp} \cup K^{\perp}))^{*}\quad\ \ \ \ \ {\kern1pt} \quad\text{from M1} \end{array} $$

(70)

$$\begin{array}{@{}rcl@{}} & =& \Diamond (\Diamond (E^{\perp} \cup K^{\perp}))^{c} \end{array} $$

(71)

$$\begin{array}{@{}rcl@{}} &=& \Diamond ((E^{\perp} \cup K^{\perp})^{\perp})\ \ \ \ {\kern1pt}\qquad\text{from M3} \end{array} $$

(72)

$$\begin{array}{@{}rcl@{}} &=& \Diamond (E^{\perp\perp} \cap K^{\perp\perp})\ \ \ \ \ \qquad\text{from M10}. \end{array} $$

(73)

(It is not hard to show that this meet is indeed the largest subset of ♢E ∩♢K of the form ♢(): that is to say, if ♢L ⊆♢E ∩♢K then ♢L ⊆♢(E ^⊥⊥∩ K ^⊥⊥) ⊆♢E ∩♢K.) Meet distributes over join in any lattice iff the converse also holds, so if Part W is Boolean, hence distributive, the following identity would obtain for any subsets E, F, G of W:

$$ \Diamond E \cup (\Diamond F \sqcap \Diamond G) = (\Diamond E \cup \Diamond F) \sqcap (\Diamond E \cup \Diamond G). $$

(74)

The left hand side of this last equation is

$$\begin{array}{@{}rcl@{}} \Diamond E \cup (\Diamond F \sqcap \Diamond G)& = &\Diamond E \cup \Diamond(F^{\perp\perp} \cap G^{\perp\perp})\qquad\text{from (73)} \end{array} $$

(75)

$$\begin{array}{@{}rcl@{}} & =&\Diamond (E \cup(F^{\perp\perp} \cap G^{\perp\perp}))\qquad\text{from M1} \end{array} $$

(76)

$$\begin{array}{@{}rcl@{}} & = &\Diamond ((E \cup F^{\perp\perp}) \cap (E \cup G^{\perp\perp})). \end{array} $$

(77)

The right hand side is, from (73):

$$\begin{array}{@{}rcl@{}} (\Diamond E \cup \Diamond F) \sqcap (\Diamond E \cup \Diamond G)& = &\Diamond((\Diamond E \cup \Diamond F)^{\perp\perp} \cap (\Diamond E \cup \Diamond G)^{\perp\perp}) \end{array} $$

(78)

$$\begin{array}{@{}rcl@{}} & =& \Diamond((\Diamond (E \cup F))^{\perp\perp} \cap (\Diamond (E \cup G))^{\perp\perp}). \end{array} $$

(79)

If the the left hand side equals the right hand side for any subsets, take E = ∅ and G = F. Then equality gives

$$ \Diamond(F^{\perp\perp}) = \Diamond ((\Diamond F)^{\perp\perp}) $$

(80)

or

$$ F^{\perp\perp\perp c} = F^{\perp c \perp\perp\perp c} $$

(81)

or, from M7

$$ F^{\perp c} = F^{\perp c \perp c} $$

(82)

So

$$ \Diamond F = \Diamond \Diamond F $$

(83)

or

$$ F^{\perp} = (\Diamond F)^{\perp}. $$

(84)

This proves the assertion (2).

(2) implies (3). First we note that for any subset E ⊆ W

$$ (\Diamond E)^{c} = E^{\perp} $$

(85)

from M3. So if (84) holds we have

$$ (\Diamond E)^{c} = (\Diamond E)^{\perp}. $$

(86)

So if E is a proposition

$$\begin{array}{@{}rcl@{}} E & = &\square\Diamond E \end{array} $$

(87)

$$\begin{array}{@{}rcl@{}} & = &(\Diamond E)^{c \perp}\qquad\ \text{from M3} \end{array} $$

(88)

$$\begin{array}{@{}rcl@{}} & = &(\Diamond E)^{\perp\perp}\qquad\text{from (86).} \end{array} $$

(89)

But then

$$\begin{array}{@{}rcl@{}} \Diamond E &\subseteq& (\Diamond E)^{\perp\perp}\qquad\text{from M5} \end{array} $$

(90)

$$\begin{array}{@{}rcl@{}} & = &E \qquad\ \ \ {} \qquad\text{from (89)} \end{array} $$

(91)

$$\begin{array}{@{}rcl@{}} & \subseteq& \Diamond E. \end{array} $$

(92)

Thus E = ♢E if E is a proposition. But then, since E is a proposition

$$ E = \square\Diamond E= \square E. $$

(93)

Since, for any proposition E,

$$ E = \Diamond E = E^{\perp c} $$

(94)

we have

$$ E^{c} = E^{\perp}. $$

(95)

Then the join (⊔) in R(〈W,⊥〉) is given by

$$\begin{array}{@{}rcl@{}} E \sqcup F &=& (E^{\perp} \cap F^{\perp})^{\perp} \end{array} $$

(96)

$$\begin{array}{@{}rcl@{}} & =& (E^{c} \cap F^{c})^ c\quad\text{by the above and proposition 2(4), subsection A.2} \end{array} $$

(97)

$$\begin{array}{@{}rcl@{}} & =& E \cup F \end{array} $$

(98)

so that R(〈W,⊥〉) is just a Boolean lattice of subsets of W.

This proves (3).

Assertion (1), hence the theorem, follows immediately.

1.4 A.4 Proof of Corollary 1, section 5

If R(〈W,⊥〉) is Boolean then we have shown above (95) that for any proposition E, we have E ^⊥ = E ^c. Then, since for any F ⊆ W, F ^⊥ is a proposition (proposition 2(3)), we have

$$ F^{\perp\perp} = F^{\perp c} $$

(99)

which was to be proved.

1.5 A.5 Proof of Corollary 2, section 5

First note again that for elements E, F of R(〈W,⊥〉), from M10,

$$ E\sqcup F = (E^{\perp} \cap F^{\perp})^{\perp} = (E\cup F)^{\perp\perp} \supseteq E \cup F. $$

(100)

Now suppose that for all propositions E in R(〈W,⊥〉) we have E ⊔ E ^⊥ = E ∪ E ^⊥. Since E ⊔ E ^⊥ = W this entails E ^⊥ = E ^c. Then it follows as in the proof of theorem 5(3) that ⊔ = ∪ on the whole of R(〈W,⊥〉) so that the latter lattice is a Boolean sublattice of 2 ^W, contradicting our assumption. Consequently there exists a proposition, F, say, such that

$$ W = F \sqcup F^{\perp} \ne F \cup F^{\perp} $$

(101)

so that the inclusion F ∪ F ^⊥⊂ F ⊔ F ^⊥ = W is strict and F cannot be empty (since otherwise (101) would not hold). Now consider the orthotheorem a ⊔∼ a where a is an atomic formula, and define ρ(a) to be F with any other assignments to the other atoms, extending this assignment to formulas inductively in the usual way.

Then

$$\begin{array}{@{}rcl@{}} W & = &\rho (a\,\, \sqcup \sim a) \end{array} $$

(102)

$$\begin{array}{@{}rcl@{}} & = &\rho(a) \sqcup \rho(a)^{\perp} \end{array} $$

(103)

$$\begin{array}{@{}rcl@{}} & = &F \sqcup F^{\perp} \end{array} $$

(104)

$$\begin{array}{@{}rcl@{}} & \supset& F \cup F^{\perp} \end{array} $$

(105)

$$\begin{array}{@{}rcl@{}} & = &\rho(a) \cup \rho(a)^{\perp}. \end{array} $$

(106)

So, there exists a w ∈ W such that

$w \Vdash _{\mathcal {M}}a \,\sqcup \sim a \quad$ but $\quad w\nVdash _{\mathcal {M}} a\,\,$and $\,\, w \nVdash _{\mathcal {M}}\sim a$.

Conversely, if there exists an orthotheorem of the form ${\sqcup _{i}^{n}}\,\alpha _{i}$ such that ${\bigsqcup _{i}^{n}} \rho (\alpha _{i}) \ne {\bigcup _{i}^{n}} \rho (\alpha _{i})$ then R(〈W,⊥〉) cannot be Boolean by theorem 5(3).

This proves the corollary.

1.6 A.6 Proof of Corollary 3, section 5

From corollary 2 there exists a modal theorem of the form (α ₁ ⊔ α ₂)^∘, by theorem 2, and a Kripkean model $\mathcal {M} = \langle W, \approx , \rho \rangle$. Define, for atomic formulas a _i , and w ∈ W

$$ \vartheta(a_{i}, w) := \left\{\begin{array}{ll} 1 & \text{if \(w \in \rho(a_{i})\)}\\ 0 & \text{if \(w \notin \rho(a_{i})\)} \end{array}\right. $$

(107)

and inductively extend it to all modal formulas via V1–V4 in Section 3. Then $\mathcal {B} = \langle W, \approx ,$ 𝜗〉 is a B-model and it is easily proved that for any orthoformula α

$$ \| \alpha^{\circ} \|_{\mathcal{B}} = \rho (\alpha) $$

(108)

(cf. [17], proposition 2.5.1). Consequently (dropping the $\mathcal {B}$ subscripts),

$$\begin{array}{@{}rcl@{}} \| (\alpha_{1} \sqcup \alpha_{2})^{\circ} \| & = &\rho(\alpha_{1} \sqcup \alpha_{2}) \end{array} $$

(109)

$$\begin{array}{@{}rcl@{}} & =& \rho(\alpha_{1})\sqcup \rho (\alpha_{2}) \end{array} $$

(110)

$$\begin{array}{@{}rcl@{}} & \supset& \rho(\alpha_{1})\cup \rho (\alpha_{2}) \end{array} $$

(111)

$$\begin{array}{@{}rcl@{}} & = &\| \alpha_{1}^{\circ} \| \cup \| \alpha_{2}^{\circ} \| \end{array} $$

(112)

since from corollary 2 there exists a w ∈ W such that w ∈ ρ(α ₁) ⊔ ρ(α ₂) = ∥(α ₁ ⊔ α ₂)^∘∥ but w∉ρ(α ₁) ∪ ρ(α ₂) and thus $w \models _{\mathcal {B}} (\alpha _{1} \sqcup \alpha _{2})^{\circ }$ but $w \nvDash _{\mathcal {B}}\alpha _{1}^{\circ }$ and $w \nvDash _{\mathcal {B}}\alpha _{2}^{\circ }$ which was to be proved.

Conversely, suppose $\mathcal {B} = \langle W, \approx , \vartheta \rangle$ is a B-model, that there exists a modal theorem of the type $({\sqcup _{i}^{n}}\,\alpha _{i})^{\circ }$ and a w ∈ W satisfying the stated condition. Define ρ on atoms a _i by

$$ \rho(a_{i}):= \| a_{i}^{\circ}\|_{\mathcal{B}} = \| a_{i}\|_{\mathcal{B}}^{\perp\perp} $$

(113)

and extend it inductively to all formulas. Then, for any formula α it follows from an easy induction on complexity that:

$$ \rho(\alpha)= \| \alpha^{\circ}\|_{\mathcal{B}}. $$

(114)

Then $\mathcal {M} = \langle W, \approx , \rho \rangle$ is a Kripkean model for OL, and since it is assumed that $\vdash _{\mathbf {B}} ({\sqcup _{i}^{n}}\,\alpha _{i})^{\circ }$ we have also $\vdash _{\mathbf {O}}{\sqcup _{i}^{n}}\,\alpha _{i}$ by theorem 2. Now it is clear that for all i, $w \nVdash _{\mathcal {M}} \alpha _{i}$, for otherwise there would be an i ₀ say, such that $w \in \rho (\alpha _{i_{0}}) = \| {\alpha ^{\circ }}_{\!i_{0}}\|$, i.e. $w \models _{\mathcal {B}} {\alpha ^{\circ }}_{\!i_{0}}$ contrary to our hypotheses. The non-Booleanness of R(〈W,⊥〉) now follows from the converse to corollary 2.

This proves the corollary.

1.7 A.7 Counting certain families of superpositions

A proposition in W _n is its intersection with a unique non-zero subspace of R ⁿ. Each proper non-zero subspace of R ⁿ determines an orthogonal complement which is also proper and non-zero. So each proper non-zero subspace of R ⁿ determines a proposition in W _n along with its orthogonal complement, and each such pair determines an infinite family of superpositions, namely elements of W _n not contained in either the proposition or its complement. (Note that in W ₂ there is only one such family, depicted as the meridian in Fig. 4, since each member of any pair of linearly independent vectors is expressible as a linear combination of any other independent pair, so any pair of orthogonal propositions determines the same family of superpositions.) These pairs of propositions may of course be used to represent an orthotheorem of the form a ⊔ ∼ a in any Kripkean model based on W _n . Different families of associated superpositions will now correspond two different pairs of such propositions. Thus, to count the number of such families in the case of this special orthotheorem, we need only determine the number of non-zero proper subspaces of R ⁿ and halve it, to avoid a double count of each pair. But each k-dimensional subspace of R ⁿ corresponds to a unique one-dimensional subspace of $\bigwedge ^{k} \mathbf {R}^{n}$ and there are dim$(\bigwedge ^{k} \mathbf {R}^{n}) = \binom {n}{k}$ such independent subspaces. So the number we seek is

$$\begin{array}{@{}rcl@{}} \frac{1}{2}\sum\limits_{k=1}^{n-1} \binom{n}{k}& =&\frac{1}{2} \left( \sum\limits_{k=0}^{n} \binom{n}{k} - \binom{n}{0} - \binom{n}{n} \right) \end{array} $$

(115)

$$\begin{array}{@{}rcl@{}} & =& \frac{1}{2} (2^{n} - 2) \end{array} $$

(116)

$$\begin{array}{@{}rcl@{}} & =& 2^{n-1} - 1. \end{array} $$

(117)

Note the for n = 2 we do indeed get just one family of superpositions.

Appendix B: Neuron structure and function

We give a very brief and impressionistic account of neuron structure and function. Although we will use the word “neuron,” much of what is described applies to other types of cell such as nerve cells. (For this section see [26]. For a deep mathematical study of neuron dynamics, and to get some idea of the complexity of neuronal internal states, see [27]). A neuron is a specialized cell that functions in a node-like manner, is networked with other neurons, and communicates with them via the chemical mediation of electrical impulses in a manner to be described. A neuron generally has very many input channels, but only one signal is output, though perhaps to many recipients. The inputs are branched projections, called dendrites, of the neuron body, or soma, that conduct electrochemical signals into the body of the neuron from other cells and neurons. The outgoing branch is a single pseudopod-like projection, usually on the side opposite to the dendrites, called the axon. The active site where the axon connects to the soma is called the hillock. The axon conducts a single electrochemical output pulse or train of pulses, but may have many branching outputs at its end, called the axon terminals. These then provide the input signals to the dendrites of other neurons. (The axon can be immensely long relative to the size of the cell, as in nerve cells.)

The electrochemistry underlying this activity is extremely complex. In simple terms, electrical potentials are formed across the cell membranes via the bidirectional flows through the membrane of various types of ion: in this case mainly sodium and potassium ions, though there is a multitude of other ions in this environment. These ion flows are controlled by clusters of proteins embedded in the cell membranes (as they are in all cells) called ion channels, which act upon ion flows like gates, pumps and/or valves. There is a multiplicity of varieties of ion channel.

In its “resting” state, a neuron holds a potential of about − 70 mV (millivolts) reflecting a steady state of polarization between the internal and external ion flow states. If input signals arrive from the dendrites, or groups of them, the cell depolarizes and its potential goes up. The cell may then rapidly repolarize without issue if the signal is not strong enough, in which case the signal does not penetrate very far into the soma. But if the stimulus is strong enough to enable a threshold potential to be reached (at about − 55 mV) then positive feedback kicks in and there is a rapid depolarization and concomitant rise in the potential to a peak of about 60 mV. At some point during this rise, the signal penetrates the soma and reaches the hillock, and the potential profile, or a train of copies of it, begins to be conducted along the axon via a spontaneously choreographed succession of opening and closing ion channels along the body of the axon. This is the point at which the neuron is said to fire. Meanwhile, the cell body rapidly repolarizes and the action potential now plunges down to around − 90 mV, which has the salutary effect of preventing the signal from being conducted back up the axon toward the soma. After a refractory period in this territory, the potential rises again to the resting value of − 70 mV. The time scales are roughly as follows: the duration of the “spike” is about 0.5 ms (milliseconds); the entire duration of the action potential, including the refractory period, between the two rest states, has a duration of about 5 ms. It should be mentioned that it is only the frequency of the output signal train that is determined by the strength of the input signal, not the amplitude, which is independent of the input signal strength.

(A couple of remarks are in order. Firstly, a set of equations describing the progress of the action potential, called the Hodgkin–Huxley model, was proposed in 1952 and was considered a great advance in biophysics, winning for its authors two thirds of the Nobel Prize in Physiology and Medicine in 1963. Secondly, the mode of transmission of a signal from an axon terminal of one neuron, to a dendrite of another neuron, is not by direct contact but rather through a complex intermediary mechanism involving synapses (presynaptic bodies at the ends of axon terminals, with postsynaptic receptors at the ends of dendrites) and neurotransmitters which are complex molecules of a large variety of types, including serotonin, dopamine, tryptophan, histamine, etc. The elucidation of this mechanism, by J. C. Eccles, garnered for its author the other third of the 1963 Nobel Prize in Physiology and Medicine.)