Quantum Psychology

Fantasy Warning

This article is mostly fantasy. Do not take it too seriously.

Quantum Structure in Cognition

This partly based on Quantum Structure in Cognition.

TO DO

1. Meaning of observable $\hat L$ in the quantum love model. (Results here.)
2. Meaning and representation of $A\land B$.
3. Meaning of imaginary part of off diagonal elements in operators.
4. Meaning of eigenvalues.
5. Two eigenvectors will present sometimes. Two eigenvectors? Once measured collapse to one of them? Needs more proof.
6. Application of Statistical Mechanical stochastic matrix or similar ones.

James A. Hampton

Hampton’s result：

• For the item Cuckoo,

Item	Concept	Weight
Cuckoo	Bird	1
	Pet	0.575
	Bird & Pet	0.842

• For the item Ashtray

Item	Concept	Weight
Ashtray	Home Furnishing	0.7
	Furniture	0.3
	Home Furnishing or Furniture	0.25

Math Representations of Guppy Effect

Two concepts A and B, their weights are $\mu(A)$ and $\mu(B)$. The conjunction and disjunction of them are A and B ($A \land B$), A or B (A$\lor$B) respectively. The weights are $\mu(A\land B)$ and $\mu(A\lor B)$.

• Conjunction minimum rule deviation

$$ \begin{equation} \Delta_c = \mu(A\land B) - min(\mu(A),\mu(B)) \end{equation} $$

• Disjunction maximum rule deviation

$$ \begin{equation} \Delta_d = max(\mu(A),\mu(B)) - \mu(A\lor B) \end{equation} $$

Quantum Representation of Concepts

Now kets stands for concepts.

But why Hermition operators? Real eigen values? Wait and see.

To explain Hampton’s experiment, we use two orthogonal concept states $\ket{A}$ and $\ket{B}$ (why orthogonal? Not nessarily). For a specific experiment $\hat M$, we have

$$ \begin{align} \mu(A) & = \bra{A}\hat M\ket{A} \\ \mu(B) & = \bra{A}\hat M\ket{A} \\ \mu(A\lor B) & = \frac{\bra{A}+\bra{B} }{\sqrt{2}}\hat M \frac{\ket{A}+\ket{B}}{\sqrt{2}} \end{align} $$

Description of Items

The key point is that operators are items while concepts are states.

Here we use the idea I proposed several month ago to interpret this method. I need a boy or girl description of a human being $\hat I_D$, which is an operator. So the bases are $\ket{B}$ and $\ket{G}$.

Suppose we have some measurement results that the expectation value of me to be a girl is $\mu(G)$ and that of me to be a boy is $\mu(B)$, and that of me to be a boy and girl is $\mu(G\lor B)$. In the Boy-Girl basis, we can write down the representations of states and operators.

$$ \begin{align} \hat I_D & \rightarrow \begin{pmatrix} a & c \\ c^* & b \end{pmatrix} \\ \ket{G} & \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ \ket{B} & \rightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{align} $$

So the expectation value of me to be a girl,

$$ \begin{align} \mu(G)& := \bra{G}\hat I_D \ket{G} = \cdots (\text{insert identity}) = a \\ \mu(B)& :=\bra{B}\hat I_D \ket{B} = \cdots (\text{insert identity}) = b \\ \mu(G\lor B) &:= \frac{1}{2}(\bra{G} + \bra{B})\hat I_D (\ket{G} + \ket{B}) = \cdots (\text{insert identity}) \\ & \phantom{} = a+b+2\mathrm {Re}(c) \end{align} $$

If we can get the measurement results of $\mu(G)$, $\mu(B)$, $\mu(G\lor B)$, the elements of the human being operator under Boy-Girl representation can be calculated except the imaginary part.

1. We can show that the imaginary part of off diagonal elements doesn’t change any of the expectation values of the Hermition operator, so for expectation values, the imaginary part is of no importance.
2. The “problem” of this formalism is that the eigenvectors of operators could have negative elements which makes it hard to interpret the meaning. But the probability is alright because the probability is the square of the coefficient.
3. Notice that Pauli matrices together with identity matrix form the orthonormal complete basis of any two energy level system. So we can use Pauli matrices to write down the operators for two level system in psychology.

Measurement Experiments

To get the data of the expectation values, repeat the Hampton experiment.

Quantum Love As A Joke

In the case of love stories, we have Boy and Girl which are two items. Then we have a “experiment”, which is the relationship between the two.

So we have two states $\ket{B}$ and $\ket{G}$ and an observable $\hat L$. The meaning of this observable is not clear.

Suppose we have a girl and a boy without a third person, the bases become the direct product of the bases the boy and the bases of the girl, $\ket{S_B}\otimes \ket{S_G}$, $\ket{S_B}\otimes \ket{L_G}$, $\ket{L_B}\otimes \ket{S_G}$, $\ket{L_B}\otimes \ket{L_G}$. It’s obvious that for such a system, the square of the element at the bottom right corner of the system operator matrix is the weight of the two being in love with each other.

Perturbation Theory

As I mentioned in the previous section, Hamiltonian of two level system can be written in terms of sigmas which makes it really easy to solve. However, it might be a problem when the system becomes large. Then we might need perturbation theory.

We will use $\hat M$ as the operator of a person or system. Suppose we can expand the operator and the state to multiple orders. The first order perturbation,

\begin{align} \hat M^{(0)} \ket{n^{(1)}} + \hat M^{(1)} \ket{n^{(0)}} = m_n^{(0)} \ket{n^{(1)}} + m_n^{(1)} \ket{n^{(0)}} \end{align}

By multiply state vectors on the left we can find out that the perturbations are

Order	Energy	State
0	$m_n^{(0)}$	$\ket{n^{(0)}}$
1	$m_n^{(1)} = \bra{n^{(0)}} \hat M^{(1)} \ket{n^{(0)}}$	$\ket{n^{(1)}} = \sum_{p \neq n}\frac{\bra{p^{(0)}} \hat M^{(1)} \ket{n^{(0)}} }{m_n^{(0)} - m_p^{(0)}}\ket{p^{(0)}} $
2	$m_n^{(2)} = \sum_{p\neq n} \frac{ \left\vert \bra{n^{(0)}}\hat M^{(1)} \ket{p^{(0)}} \right\vert ^2}{m_n^{(0)} - m_p^{(0)}}$	-

It’s required in this perturbation method that the system is non degenerate and the perturbation is small, i.e.,

\begin{equation} \left\vert \frac{\bra{p^{(0)}} \hat M^{(1)} \ket{n^{(0)}} }{m_n^{(0)} - m_p^{(0)}} \right\vert \ll 1 \end{equation}

Emotions

In quantum mechanics, the first postulate is usually written as

Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product.

(note: use shankar’s form. for later modifications.)

Of course it works so well in physics and we started thinking about apply this kind of formulation to other topics of science.

Human emotions can stack together to give us unique experiences. For example, one can be both happy and sad, be both curious and fear. Similar to quantum mechanics, the first example means superposition of emotions is also a emotion that can be experienced while the second one indicated that we can use direct product of states to describe the complete emotional state.

We now propose a hypothesis of human emotions:

Each emotion of humanbeing is associated with a topologically separable complex Hilbert space $\mathbf E$ with inner product.

Love

uantum physics is very successful in describing physical world. The generalization of quantum theory to other natural science as biology and chemistry has also produced a lot of important results. What’s more exciting in the generalization of quantum theory is that quantum theories of cognition has been proposed.¹

Meanwhile, works have shown that positive and negative affect are not necessarily polar opposites. Some feelings which was regarded as polar opposites can occur at the same time.²³

This paper will concentrate on the description of dynamics of affect in Hilbert space. Ghose argued that quantum cognition is not quantum for the fact that our brain is not a quantum system. Instead of try to build a fundamental theory, this work is only a phenomenological theory, which produces results that can be accepted. This is called quantum because the method we use is borrowed from quantum mechanics.

In quantum mechanics, the first postulate is usually written as⁴

The state of the particle is represented by a vector $\ket{\psi(t)}$ in a Hilbert space.

This postulate shows that physics happens in Hilbert space. Starting from this point, physical states remains the same when multiplied by a complex number. Superposition of states in a Hilbert space is still one of the states in the same Hilbert space. Direct product of states makes it possible to describe many-particle system.

Human emotions can stack together and show new experiences. For example, one can be both happy and sad, both curious and fear. Similar to quantum mechanics, the first example means superposition of emotions is also a emotion that can be experienced while the second one indicates that we can use direct product of states to describe multiple emotional states.

Given the similarity, this paper discusses the possibility of using Hilbert space as emotion sets and demonstrates this formulation in the example of love. Moreover, the emotional states in this discussion are quantized. Non-zero commutators which quantized quantum mechanics, will come in this discussion of emotions.

In the realm of mechanics, description of states, time evolution of states and calculation of observables are the three key steps. For emotion, this procedure is followed exactly here.

Description of Emotional States

We now propose a series of hypothesis on human emotions to make it possible to have superposition of emotional states.

The first postulate is about description of emotional states.

Each emotion of human beings is associated with a complex Hilbert space $\mathbf H$ with inner product.

By this hypothesis, all human emotions of concern are embedded into the appropriate separable Hilbert space with a well defined length. Several features we need to formulate a theory of superposition emotions are achieved.

1. Emotional states in a Hilbert space can linearly form states in the same Hilbert space.

An example of this hypothesis is the state of happy and sad. Suppose we have an normalized emotional state $\ket{\psi}$ which has $\ket{\mathrm {Happy}}$ and $\ket{\mathrm{Sad}}$ in it. Pick a complete orthonormal basis set ${ \ket{\mathrm {Happy}}, \ket{\mathrm{Sad}} }$, we can write down the state as $$ \begin{align} \ket{\psi} & = \left(\ket{\mathrm{Happy}}\bra{\mathrm{Happy}} + \ket{\mathrm{Sad}}\bra{\mathrm{Sad}} \right)\ket{\psi} \\ &= C_1\ket{\mathrm{Happy}} + C_2 \ket{\mathrm{Sad}} \end{align} $$

where $C_1 = \braket{\mathrm{Happy}}{\psi}$ and $C_2 = \braket{\mathrm{Sad}}{\psi}$。

Since it is an arbitrary emotional state in this space, this equation tells us that all kinds of states composed by $\ket{\mathrm{Happy}}$ and $\ket{\mathrm{Sad}}$ can be expressed in this way. Through adjusting the coefficients, states with more $\ket{\mathrm{Happy}}$ can be achieved. The coefficients stand for the weights of corresponding states. In this example,

$$ \begin{align} \braket{\psi}{\psi}= |C_1|^2 + |C_2|^2 =1 \end{align} $$

which is the normalization condition. Another interpretation is the conserved probability. However hypothesis of measurement is needed for this interpretation.

Quantum state, as mentioned in Partha Ghose’s paper⁵, can be geometrically represented in Bloch sphere if the state is a two-level system.⁶⁷ Though there are problems generalizing this geometrical method to higher dimensions, it is not a problem for this paper since we would be dealing with pair opposites of emotions.

Paul Ekman’s emotion universal categories (citationneeded) shows that emotions can be categoried into the following:

anger, disgust, fear, happiness, sadness, and surprise + Amusement, Contempt, Contentment, Embarrassment, Excitement, Guilt, Pride in achievement, Relief, Satisfaction, Sensory pleasure, and Shame.

Measurement of Emotional States

In quantum mechanics, the hypothesis about measurement is usually about the extraction of observables using probability of a eigenstate which is possible because of the so called state collapse in popular theories. Theories like Qbism, Relative state are alternatives to the collapse of states choice.

If the particle is in a state $\ket{\psi}$, measurement of the variable (corresponding to) $\Omega$ will yield one of the eigenvalues $\omega$ with probability $P(\omega)\propto |\braket{\omega}{\psi}|^2$. The state of the system will change from $\ket{\psi}$ to $\ket{\omega}$ as a result of the measurement.⁴

Observables in emotions have no specific meaning other than operators. For emotional states, we define one class of emotions to be a complete set of basis.

All eigenvectors $\ket{i}$ of a operator $\hat E$ can form a class of emotions ${\hat E; \ket{i},\ket{\alpha} }$ if they are complete. $$ \begin{equation} \sum_i \ket{i}\bra{i} = \hat I \end{equation} $$ where $\hat I$ is identity operator.

The bases ${\ket{\mathrm {Happy}}, \ket{\mathrm{Sad}} }$ we used in the example of first hypothesis are orthonormal and complete. However, for completeness, not only discrete bases but also continuous bases can be discussed, that is the completeness condition becomes

$$ \begin{equation} \sum_i \ket{i}\bra{i} + \int_\alpha \, d\alpha \ket{\alpha}\bra{\alpha} = \hat I . \end{equation} $$

With the help of emotion class, interpretation of states is made possible.

For emotional state $\ket{\psi}$, measurement of a specific class of emotions ${\hat E; \ket{1},\ket{2},\cdots}$ will cause the state to one of the eigenstates of the class with probability of corresponding weight of this eigenstate in state $\ket{\psi}$.

Meaning of eigenvalues is not clear. Therefore this hypothesis is only for interpretation of the states. In reality, this hypothesis is essentially the procedure of measuring a person’s emotional state with infinite time without changing the environment and finally reaching an eigenstate of the measured class.

Upon infinite times of measurements by establishing a ensemble of the system, we can find out the exact value of probability of $\ket{\mathrm{Happy}}$ and $\ket{\mathrm{Sad}}$, i.e., $|C_1|^2$ and $|C_2|^2$.

States which are orthogonal and with equal weight of Happy and Sad can also be created.

$$ \begin{align} \ket{\psi_1} &= \ket{\mathrm{Happy}} + \ket{\mathrm{Sad}}\\ \ket{\psi_2} &= \ket{\mathrm{Happy}} - \ket{\mathrm{Sad}} . \end{align} $$

The two states are not the same even they have the same probability of finding out $\ket{\mathrm{Happy}}$ and $\ket{\mathrm{Sad}}$. The difference of the two states can be significant when average values of operators are investigated.

Thus operators are the key to understanding the meaning of emotional states.

Identical Person

Identical particles is an concept that we can not distinguish two particle from each other when they are mixed together. In this paper, we are going to assume that human beings can not be distinguished from each other if their emotional states and appearances are removed.

All person are identical if they have the same emotional states and appearances.

Carry out the same argument as in quantum mechanics about particle exchange, define person exchange operator

$$ \begin{equation} \hat P_{12} \ket{\psi(x_1,x_2)} = \ket{\psi(x_2,x_1)} \end{equation} $$

Two free person with states $\ket{\psi_a(x_a)}$ and $\psi_b(x_b)$ put together will be written as direct product of the two states, i.e., $\ket{\psi_a(x_a)}\otimes\ket{\psi_b(x_b)}$.

This raise a question about symmetries under exchange operator. Fermionic and Bosonic behavior are also possible. When discussing heterosexuality, the state of gender identity has two states and two person with the same gender can not be put together. It is similar to the behavior of two electrons with the same spin can not be put together. (Under exotic conditions, the two electrons can crunched. blabla)

Operators

In quantum mechanics, z-component of spin operator for a spin $1/2$ particle is $\hat S_z = \hbar/2\sigma_z$, in which \begin{equation} \sigma_z = \begin{pmatrix}1 & 0\ 0 & -1\end{pmatrix} \end{equation} is the matrix form in bases ${ \ket{1/2},\ket{-1/2} }$

Spin up and spin down are the eigenvectors of this operator. As stated in the definition of class of emotions, the bases ${\ket{\mathrm {Happy}}, \ket{\mathrm{Sad}} }$ are eigenvectors of an operator, now denoted as $\hat E$. It’s true that in this basis, $\sigma_z$ serves well as $\hat E$.

The general requirements for operators are complicated. The method needed highly dependent on the details of the problem to be discussed. So we recommend to start over on every new topic. The method is for a better description of the reality. Thus take in whatever is useful.

(Somehow, we need to show what should operators look like here. Constraints on operators? Basically we would like to follow exactly quantum mechanics.)

Similarly, $\sigma_z$ matrix can be used as the operator whose eigenvectors are $\ket{\mathrm{Happy}}$ and $\ket{\mathrm{Sad}}$ in ${\ket{\mathrm{Happy}}, \ket{\mathrm{Sad}}}$ bases. Other Pauli matrices can also be used here .

To show the difference of the two orthogonal states $\ket{\psi_1}$ and $\ket{\psi_2}$, we can calculate the average value of this $\sigma_z$ operator in ${\ket{\mathrm{Happy}}, \ket{\mathrm{Sad}}}$ bases.

For state $\psi_1$,

$$ \begin{align} \langle{\sigma}\rangle &= \begin{pmatrix} 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \\ & = 0 \end{align} $$

The independent variables $x$ and $p$ of classical mechanics are represented by Hermitian operators $\hat X$ and $\hat P$ with the following matrix elements in the eigenbasis of $\hat X$, $$ \begin{align} \bra{x}\hat X\ket{x'} &= x\delta(x-x') \\ \bra{x}\hat P\ket{x'} &= -\mathrm i \hbar \delta'(x-x') \end{align} $$

The operators corresponding to dependent variables $\omega(x,p)$ are given Hermitian operators

$$ \begin{equation} \hat \Omega(\hat X, \hat P) = \omega(x\rightarrow \hat X, p \rightarrow \hat P) \end{equation} $$

This of course is not applicable to human emotions.

References

Experiments

• Overextension of Conjunctive Concepts: Evidence for a Unitary Model of Concept Typicality and Class Inclusion: A very important experiment in cognition that vilates a lot of earlier theories. The data might be a signature of quantum structure in cognition.

Psychology facts

1. Interference effects: Partha Ghose mentioned this.
2. Affect

Emotions:

1. Ekman, Paul (1999), “Basic Emotions”, in Dalgleish, T; Power, M, Handbook of Cognition and Emotion, Sussex, UK: John Wiley & Sons: anger, disgust, fear, happiness, sadness, and surprise + Amusement, Contempt, Contentment, Embarrassment, Excitement, Guilt, Pride in achievement, Relief, Satisfaction, Sensory pleasure, and Shame.

Cognition

• Overextension of Conjunctive Concepts: Evidence for a Unitary Model of Concept Typicality and Class Inclusion: A very important experiment in cognition that vilates a lot of earlier theories. The data might be a signature of quantum structure in cognition.