quantum theory for psychology

7/29/2019 Quantum theory for psychology

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arXiv:1109.335

5v1

[physics.soc-ph]14Sep2011

Quantum theory as a tool for the description of simple psychological phenomena

E. D. Vol

B. Verkin Institute for Low Temperature Physics and Engineering of the National

Academy of Sciences of Ukraine 47, Lenin Ave., Kharkov 61103, Ukraine.(Dated: September 16, 2011)

We propose the consistent statistical approach for the quantitative description of simple psycho-logical phenomena using the methods of quantum theory of open systems (QTOS).Taking as the

starting point the K. Lewins psychological field theory we show that basic concepts of this theorycan be naturally represented in the language of QTOS. In particular provided that all stimuli actingon psychological system (that is individual or group of interest) are known one can associate withthese stimuli corresponding operators and after that to write down the equation for evolution ofdensity matrix of the relevant open system which allows one to find probabilities of all possiblebehavior alternatives. Using the method proposed we consider in detail simple model describingsuch interesting psychological phenomena as cognitive dissonance and the impact of competitionamong group members on its unity.

PACS numbers: 03.65.Ta, 05.40.-a

I. INTRODUCTION

It is a common opinion that in spite of various the-ories and impressive concrete results confirmed by nu-merous experiments modern psychology is still far fromstatus of exact science such as for example theoreticalphysics. The main difference between these sciences isthat in theoretical physics we have well defined conceptsand general principles such as for example action andprinciple of least action in mechanics or entropy and sec-ond law in thermodynamics which let one to obtain allresults of the theory by successive deductive procedurefrom general principles. On the other hand in psychol-ogy throughout its history many attempts were taken tobring together its concepts and facts into integral system

that would allow one to describe and explain known psy-chological phenomena and possibly to predict some neweffects. In the present paper we start from one of suchtheories namely psychological field theory (PFT) of KurtLewin and make an attempt to represent basic conceptsof PFT in the language of quantum theory of open sys-tems (QTOS). This seemingly formal representation hashowever indisputable advantage since allow one to usewell developed mathematical methods of QTOS to ana-lyze a variety of psychological effects and situations. Inparticular we show that such widely known phenomenonin psychology as cognitive dissonance(CD) can be con-sistently considered in the framework of the method pro-

posed. In addition we demonstrate that this method canbe used to describe group dynamics processes as well asindividual behavior. The rest of the paper is organizedas follows. In the Sect2 we give a brief account of in-formation from PFT and QTOS which is necessary forunderstanding of the paper. In the Sect.3 which is ma-

jor in the paper we realize representation of such basic

Electronic address: [email protected]

concepts of the PFT as the life space,regions, locomo-tions and so on in the language of QTOS and introduceessential concept of density matrix of psychological sys-tem (PS) which let one to find probabilities of all possi-ble behavior alternatives and formulate the mathematicalmethod for description of evolution this matrix. In theSect.4 we demonstrate the effectiveness of our approachin the framework of simple but useful model for describ-ing several phychological phenomena such as cognitivedissonance and so forth. Now let us turn to a detailedaccount of the paper.

II. PREPARATORY INFORMATION

The PFT was created by K.Lewin in the middle ofXX century and stated him in various books and pa-pers (see for example [1], [2]). The comprehensive re-view of this theory a reader can find in the textbook ofHall and Lindzey [3]. Let us outline the main pointsof this theory in the form which is sufficient for under-standing of the paper. The initial concept of the PFT isthe concept of the life space by which K.Lewin meant aset of psychological facts (i.e. both personal incentivesand external impacts connected with situation) acting atthe moment on PS and determining its further evolution.For the behavior description K. Lewin proposed generalbut somewhat abstract formula: B = F(S, P), where Bmeans behavior, S-situation and P-person. Note in thisconnection that the collection of all variables describingboth S andP exactly constitute the life space of PS. Inaddition the life space of PS can be divided on separateregions each of which corresponds to a single psychologi-cal fact(for example it may be separate regions connectedwith job,family,game and so on)We assume that differentregions are mutually disjoint.Another essential concept inPFT is a concept of a stimulus.Simple stimuli can be in-troduced in PFt as certain psychological forces which acton PS and stimulate it either to occupy definite region
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or to avoid it. Depending on the nature of a stimuluswe can attribute to each region definite sign (plus if astimulus is attractive and minus in opposite case). Thevalue of a stimulus that is the tendency of PS to oc-cupy given region or to avoid it determines the valenceof corresponding region.It should be noted that in ad-dition to simple stimuli there are also complex stimuliacting on PS of interest and composed from several sim-

ple ones.Another concept in PFT is locomotion whichimplies the transition (but only psychological not phys-ical!) from certain region to another one. Note thatK.Lewin proposed also simple graphic method which al-lows one to represent completely the current state of PSwithin the framework of these basic concepts. Accordingto K. Lewin such representation is sufficient to explainactual behavior of PS in future.

However it should be noted that due to extraordinarycomplexity of the psychological phenomena modern psy-chology prefers not to talk about deterministic laws ofhuman behavior but rather only on its statistical de-scription.(see for example [4]). Therefore in the rest ofthe paper we assume the task of behavior description issolved if we can indicate the distribution function whichdetermines probabilities of all behavior alternatives or byother words the probabilities of finding PS in arbitraryregion of its life space..

Since the main goal of our paper is the representationof basic concepts of the PFT in the language of quan-tum theory let us remind the necessary information fromQTOS. First of all as long as evolution of quantum opensystem is nonunitary its state should be specified withthe help of density matrix (but not wave function). Themain result from QTOS that we need in this paper is theLindblad equation which describes evolution of densitymatrix in the case of Markov open quantum system.This

equation has the next general form:

t

= i

H,+i

Ri, R++ h.c, (1)In Eq. (1) H is Hermitian operator (hamiltonian of

open system) and Ri are non-Hermitian operators thatspecify all connections of open system of interest withits environment.If the initial state of the system (0) isknown Eq. (1) allows one to find the behavior( i.e. itsstate at any time t and thus to determine average valuesof all observables relating to open system). If we are in-terested only in stationary states of open system we mustequate r.h.s. of the Eq. (1) to zero and find stationarydensity matrix from this equation. Before moving on wewant to explain one essential point namely : why quan-tum in nature Eq. (1) can be used to describe behaviorof classical systems? The answer is that density matrixof quantum system represents its classical correlations aswell as quantum.But information about classical corre-lations is contained in diagonal elements of density ma-trix while the information about quantum correlations innondiagonal ones.

Thus in the situation when we are able to write closedequations including only a set of diagonal elements ofdensity matrix and to solve them, we actually obtainclassical distribution function and required descriptionof classical analog of corresponding quantum system. Asthe author has shown in [5] such case holds for examplein the case when all operators Ri in Eq. (1) have mono-

mial form namely Ri (a+)

ki (a)li (where operators

a and a+ are bose operators with standard commuta-tion rules:

a,a+ = 1). In this case Eq. (1) can bereduced to closed system of equations for diagonal ele-

ments of N (where N = a+a is number operator).

These arguments from [5] can be extended also to thecase when operators Ri are represented as monomial

forms of some fermi operators fj

and f+k (in the case ofseveral degrees of freedom ) and corresponding number

operators Ni = f+ifi have only two eigenvalues 0 and1. In the next part we will demonstrate as formalism ofQTOS with the help of such operators can be used forthe statistical description of PS systems in the languageof PFT.

III. MAPPING BETWEEN PFT AND QTOS

In this part we propose the representation of all basicconcepts of PFT in the language of QTOS.Note, that wewill restrict ourselves to considering only such phenom-ena when behavior of the PS of interest is determinedonly by variables describing a situation while variablesassociated with a person play minor role.In addition wewill assume that in the process of behavior there is no

restructuring of the life space(that is the number of re-gions and all their characteristics remain fixed). Such PSwe denote as simple system. The generalization of themethod proposed that takes into account also personalvariables of PS will be considered elsewhere. Obviously,or PS under consideration is inside the given region ofthe life space or outside it. In accordance with this factwe can attribute to every region occupation number niwhich takes only two values ni = 1 if PS is in the i regionand ni = 0 in opposite case. Let us introduce operatorni acting in two-dimensional linear space whose eigenval-ues are 0, 1. Let us introduce also the pair of operators

f+i =

0 1

0 0and fi = 0 0

1 0 thus that relationshipni = f+ifi = 1 00 0

holds.To avoid misunderstanding

we want to emphasize that although the operators fiand f+i satisfy to relationship :f+ifi +fif+i = 1 nev-ertheless studied PS systems are of course classical (notFermi systems!). Correspodently all regions of the lifespace should be considered as distinguishable. Now weestablish the mapping between different stimuli actingon PS of interest and corresponding operators acting on


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states of relevant quantum open system. We assume thatevery operator Ri entering in the Lindblad equation forrelevant system corresponds to certain stimulus(simpleor complex) acting on PS of interest.Besides we supposethat all Ri are some monomial functions of operators f

+j

and fk (note that index i enumerates different stimuliacting on a PS).Note also that the number of regionscan differ from number of stimuli. Let us now formu-

late two main correspondence rules between acting stim-uli and operators Ri . Rule1: If a stimulus i is simplethen we associate with it either operator Ai = ki f

+i if

given stimulus is attractive i.e. stimulates PS to occupyi region or operator Bi = li fi in opposite case. Co-efficients ki and li reflect the value of acting stimulus i.Rule2: Let a stimulus is complex ( that is composed fromseveral simple ones) then in the case when among sim-ple stimuli i1, i2...are attractive and stimuli j1, j2...arerepulsive with such complex stimulus we associate theoperator C = k fi1fi2 ...f

+j1

f+j2 ...Note that sign of the co-efficient k does not affect the final result. In addition thecurrent state of PS can be represented by density ma-

trix of behavior (t) in the linear space of dimension 2Nwhich is the tensor product H1H2 ...Hr (1 r N),where N is number of different regions in the life space).Every space H ( = 1, ....N) is two-dimensional vector

space with basis states

10

and

01

connected with

region.In the language of PFT the state

10

corre-

sponds to PS which occupies region of the life space

and the state

01

corresponds to PS which is outside

of it. Guided by these simple rules of correspondenceone can easily represent any current situation with PS

as PFT draws it using the rigorous language of QTOS.If we assume besides that considered PS has no memorythen one can try to use for the description of its evolu-tion the Lindblad equation Eq. (1) for density matrixof relevant quantum open system.(with H = 0 ). Oper-ators Ri in this equation should be chosen of course incorrespondence with the rules stated above. Thus ourmain assumption in the present paper is the assertionthat detail describing of behavior of PS can be realized in

the language of QTOS at least as well as by concepts ofPFT.But now we have in hands powerful mathematicalformalism which is sufficient for quantitative descriptionof various although at present time only simple phycho-logical phenomena. Since the previous consideration toa considerable extent was a heuristic now we want withreference to concrete psychological model to demonstrateits effectiveness. We believe that the question about ap-

plicability of the method can be solved only by carefulcomparision of obtained theoretical predictions relatingto this and similar models and results observed in testexperiments .

IV. BASIC PSYCHOLOGICAL MODEL

In this part of the paper we consider simple but use-ful model illustrating in our opinion the effectiveness ofthe method proposed. We begin by considering suchwell known phenomenon in psychology as cognitive disso-nance (CD). According to definition (see e.g. [6]) cogni-tive dissonance emerges when a person has two or severalideas (cognitions) which contradict each other. The stateof CD occurs for example in a situation of choice when itis impossible to estimate for sure pros and cons of differ-ent alternatives. Clearly the behavior in such sutuationwill necessarily be random in its nature. Using the cor-respondence between PFT and QTOS specified above itis easy to formulate the situation of CD in the languageof QTOS.Let us consider for simplicity the case whena person has only two positive alternatives with differ-ent incentives (which partially contradict each other) .We can represent such situation with the help of threeoperators Ri acting on density matrix

of relevant sys-

tem describing a behavior of such PS . These operators

are: R1 = a2f+1 ,R2 = b2f+2 andR3 = c2f1f2 .Coefficients of the operators Ri characterisize values ofcorrespondent stimuli in convinient normalization.Nowaccording to our assumption ( since PS of interest has nomemory)its behavior can be described by the Lindbladequation for diagonal elements of density matrix whichcan be interpreted as distribution function for a personto make appropriate choice .

N1,N2t = a N1N1,N2 N1N1,N2+ b N2N1,N2 N2N1,N2+ c N1N2N1,N2 N1N2N1,N2 (2)

where we introduce convinient notation: Ni 1 Ni(i = 1, 2). We are interesting further only in stationary

solutions of Eq. (2) for which the condition t

= 0 issatisfied. In this case matrix equation Eq. (2) taking intoaccount the normalization condition

N1N2

(N1, N2) = 1

can be written in the form of next four linear equations:


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a (0, 0) b0, 0) + c (1, 1 ) = 0, (3)a (0, 1) + b (0, 0) = 0 (4)

a (0, 0) b (1, 0) = 0 (5)a (0, 1) + b (1, 0) c (1, 1) = 0 (6)

We can easily to write down the solution of the systemEq. (3)- Eq. (6) in explicit form:

(0, 0) =abc

1, (0, 1) =

b2c

1, (7)

(1, 0) =a2c

1, (1, 1) =

ab (a + b)

1,

where 1 = ab (a + b + c) + c

a2 + b2

. Note that weconsidered somewhat more general case then usual cog-nitive dissonance. Usually assumed that two compet-ing cognitions incompatable. Evidently this case realizedwhen fractions a

cand b

ctend to zero. In this case the

probabilities of possible outcomes are:

(0, 0) =ab

2, (0, 1) =

b2

2, (8)

(1, 0) =a2

2, (1, 1) = 0,

where 2 = a2 + b2 + ab. If in addition we assume

that cognitions 1 and 2 have identical attractiveness (thecase of Buridan donkey) then the values of different al-ternatives are: (0, 0) = (0, 1) = (1, 0) = 13 and (1, 1) = 0. Thus our statistical approach results in that

probability for a donkey to die of hunger is only1

3 . Itis worth to note also that independently from values ofcoefficients a,b,c for considered PS we have the relation

(0, 1) (1, 0) = 2 (0, 0) . (9)This simple relation which does not depend on parame-ters of the system can serve as the useful test to proveor disprove the validity of approach proposed. Now wewant to demonstrate that the method proposed can bealso applied for desribing simple processes of group dy-namics. Let us assume we have the same mathematicalmodel that we used for the description of cognitive disso-

nance but now look at it with another point of view. Webelieve now that this model could describe the behav-ior of two members of formal group which tend to reachspecific personal goal or status but in their activity com-

pete with each other .We will consider the state

10

i

as

the state of success and the state

01

i

(i = 1, 2) as the

failure state of i member. The stationary solution of themodel as before has the form Eq. (7), but now we are

interesting in the another question namely :how unitedgroup will be in such process?. Explain what we are keep-ing in mind. Assume first that parameters of the modelare connected by the relation: c = a + b. It is easy to seein this case that stationary density matrix Eq. (7) can berepresented in the form of direct production of two ma-

trixes: : (N1, N2) =1

(a+b)

2

ab 0 0 00 b2 0 0

0 0 a2

00 0 0 ab

= 1 2,

where 1 =1

(a+b)

b 00 a

and 2 =

1a+b

a 00 b

. It is

natural to interpret such decomposition as the desinte-gration of the group. Our main task now is to intro-duce a quantity which could measure the unity of thegroup, i.e. how far is the group from state of desin-tegration. For this purpose we will use the analogy ofthis problem with a similar problem in quantum the-ory of composite systems. Remind that in theory ofquantum entanglement to measure how far is given purestate of composite system from factorized one it is con-

vinient to use the quantity which called concurrence [7].In particular if we are interesting only in two qubit purestates which can be represented in the next vector form| z1, z2, z3, z4 (with normalization condition|z1|2 + |z2|2 + |z3|2 + |z4|2 = 1) the concurrence ( in ourcase we prefer to call corresponding quantity as unity) can be defined as U = 2 |z1z4 z2z3|. It is easy toprove that for all two qubit states 0 U 1. BesidesU = 0 for factorized states and U = 1 for maximally en-tangled (for example four Bells states). In our problemhowever all diagonal elements of density matrix are realmoreover they are positive.Therefore there are no explicitanalogs of Bells states for our system of interest whichis of course classical.Nevertheless the concept of groupunity (i.e. classical analog of concurrence) is still veryuseful. We are interested here how group unity dependson parameters of the model.

According to definition, group unity U =

2 a2b2c(a+bc)[c(a2+b2+ab)+ab(a+b)]2

. We consider coefficientsa, b in this expression as fixed parameters and c ascontrol parameter and we are interested in what is theeffect of competition on group unity.We can find theoptimal value of c from the condition of maximum U:Uc

= 0 which implies cop =ab(a+b)

a2+b2+3ab . Substituting

this value cop in U we obtain that Umax =ab

2(a+b)2.

But it should be note that when c tends to infinitythe corresponding value of U asymptotically tends to

U =2a2b2

(a2+b2+ab)2. Therefore we must compare two

values : Umax and U.

It is easy to see that this problem reduced to the eval-

uation of the expression: F (a, b) =

a2 + b2 + ab2

4ab (a + b)2

. ConditionF (a, b) 0 implies thatUmax U and vice versa. Let b ta, thenF (a, b) a2f(t) = a2

1 + t + t2

2 4 (1 + t)2.


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The equation f(t) = 0 has two positive roots t1,t2(connected by relation t1t2 = 1) which can be foundfrom the solution of quadratic equation :t + 1

t= 1 + 2

2.

Rough numerical values of roots are: t1 0, 3, t2 3, 3.and thus we obtain that F (a, b) 0 in interval t1 t t2).

It is interesting to note that starting from very simplevirtually toy model we nevertheless come to the conclu-

sion which was not obvious in advance. It turns outthat when abilities or efforts of two competing membersto achieve some goal differ not very essentially strongcompetition can increase the unity of the group. In op-posite case when the difference is significant it is neces-sary to establish the optimal level of competition to getthe maximal unity of the group.

Let us sum up the results of our study. Starting fromideas of PFT and QTOS we established the connectionbetween these two theories that seemed far from eachother and proposed the consistent approach for describ-ing simple phychological phenomena relating both to in-dividuals and groups of individuals.In the framework ofconcrete simple model we have demonstrated the effec-tiveness of the method proposed to calculate the proba-

bilities of behavior alternatives and also to predict somepeculiarities of behavior which can hardly be revealedby other approaches. We express the hope that furtherdevelopment of the method let one to extend its applica-bility to more broad sphere of interesting psychologicalphenomena.

[1] K. Lewin , A Dynamical theory of Personality, New York,McGraw-Hill, 1935.

[2] K. Lewin, Principles of Topological Psychology, New York,

McGraw-Hill,1936.[3] C. S. Hall, G. Lindzey, Theories of Personality, 2ed,

J.Wiley&Sons, New York,1970.[4] L. Ross, R. E. Nisbett, The Person and the Situation,

McGraw-Hill, New York,1991..[5] E. D. Vol, quant-ph/1107.0548, (2011).[6] E. Aronson, The Social Animal, 7ed, W. H. Freeman and

Company, New York,1995.[7] W. K. Wootters, Phys. Rev. Lett. 80, 2245, (1998).

quantum theory for psychology

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