a consistent world model for consciousness and physics

BioSystems 69 (2003) 2738

Quantum monadology: a consistent world modelfor consciousness and physics

Teruaki NakagomiDepartment of Information Science, Kochi University, Kochi 780-8520, Japan

Received 6 June 2002; received in revised form 8 October 2002; accepted 5 November 2002

Abstract

The NL world model presented in the previous paper is embodied by use of relativistic quantum mechanics, which revealsthe significance of the reduction of quantum states and the relativity principle, and locates consciousness and the concept offlowing time consistently in physics. This model provides a consistent framework to solve apparent incompatibilities betweenconsciousness (as our interior experience) and matter (as described by quantum mechanics and relativity theory). Does matterhave an inside? What is the flowing time now? Does physics allow the indeterminism by volition? The problem of quantummeasurement is also resolved in this model. 2002 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Origin of consciousness; Quantum consciousness; Now in relativity; Life and matter

1. Introduction

Recently, consciousness is becoming one of themost important subjects of scientific research amonga small but increasing number of physical scien-tists in the situation in which various functions ofmind are explained in terms of material sciences andthe peculiarity of consciousness has been broughtinto relief (see e.g. Globus, 1995). Among them,quantum-mechanical approaches are particularly in-teresting. Analogy or connection between the be-havior of quantum states and that of mind has beenpointed out by several philosophers and scientists(e.g. Whitehead, 1929; Bohm, 1951; Riccardi andUmezawa, 1967; Cochran, 1971; Nakagomi, 1992,1995; Khrennikov, 2000, 2002) and moreover, explicitquantum-mechanical approaches to brain, especially,to consciousness are proposed by Jibu et al. (1994,

E-mail address: [email protected] (T. Nakagomi).

1995, 1997) and Vitiello (2001). The author also con-siders that quantum theory will play an essential rolein understanding consciousness. However, there areapparent incompatibilities appearing between funda-mental properties of consciousness and prerequisitesof physics as given below, and before proceeding tophysical study of consciousness, we must first resolvethese incompatibilities. Otherwise, physics must denyconsciousness.

1. Interiority: Consciousness is internal experience.We can experience consciousness through intro-spection, but cannot observe it externally. When wedissect the brain, we will see only material systemssuch as neurons, microtubules, proteins, molecules,and so on. What is the experience of consciousnessor the direct experiences of colors, sounds, smells,pains, and so on? If these are material phenomena,then matter must have an interior, because we areable to experience these only from the inside of

0303-2647/02/$ see front matter 2002 Elsevier Science Ireland Ltd. All rights reserved.PII: S0 3 0 3 -2647 (02 )00161 -2

28 T. Nakagomi / BioSystems 69 (2003) 2738

the brain matter, where the inside does not meanspatial concept such as the inside of the skull butsomething that cannot be explained as material sys-tems. What is the interior of matter? Physics doesnot have the concept of inside or interior of matterand cannot explain the interiorexterior mystery ofconsciousness.

2. Now: Consciousness exists now and not at anypoint of the past or the future. The interior ofmatter, if it exists, is accompanied with now andnot with the future nor the past. What is now?Now is the time point or the duration where con-sciousness exists.1 My consciousness and yourconsciousness seem to share in common the now,which is flowing or passing. The physical pictureof time is, however, the fourth component of thefour-dimensional spacetime continuum. In thispicture, our experience of time is something pass-ing along with the time axis, but what is passing?Physics cannot explain the passing now associatedwith consciousness or the interior of matter.

3. Volition: Consciousness is not only passive inreceiving senses but also active in giving indeter-ministic effects to the material world. This effectoccurs with flowing time and propagates in the fu-ture but not in the past. However, physics neitheraccept the indeterministic effects due to volitionnor asymmetry between past and future, becausethere is the deterministic and time reversible lawof change of material states at the bottom levelof physics, i.e. the Shrdinger equation or theunitary evolution law of quantum states.

In spite of these contradictions, physics remainsvalid by taking the strategy that physics concerns onlywhat are measured objectively by measuring appara-tuses and tells nothing about what are not measuredobjectively. Consciousness is experienced internallyand not observed externally, being out of the scope ofphysics. The spacetime concept is used only for de-scribing the deterministic law of change of materialstates and causes no difficulties in the proper field ofphysics.

1 When we map the now where consciousness exists into thephysical time, it can be a duration rather than an instant, assuggested by the experiments of Libet (1985). Also in quantummonadology, a duration of time is required to describe volitionalaction of monads in physical time.

This strategy of physics makes sense, so long asconsciousness is passive. However, if consciousnesshas active effects mentioned above, the strategy wouldshow some failure even in its proper field. Indeed,physics has been bothered for a long time by the fun-damental problems concerning the conflict betweendeterminism and indeterminism that appears in themeasurement problem of quantum mechanics and inthe reasoning of statistical mechanics. Many effortshave been spent in attempts to reduce the indetermin-ism to the determinism, although these efforts havenot yet been successful.

Physics consists of mathematical theories and aworld model by which the theories are interpreted.It is apparent that the incompatibilities and the prob-lems mentioned originate in the world model thatphysics adopts. Spacetime continuum and matterin it, this is the all of the world that physics sup-poses. There is no room for consciousness, interiorof matter, the flowing now, and indeterminism. So anew world model is needed that allows them to existand brings no effective change in the mathematics ofphysics except a small one that could not be detectedin the current experiments but might be observed inthe future by advanced technology.

As such a world model, the author proposed quan-tum monadology (Nakagomi, 1992) 10 yeas ago.In this paper, the model is fully revised on the basisof the NL world formulated in the previous paper(Nakagomi, 2002). In this world model, the monado-logical structure of interiorexterior reflection andthe basic theory of physics, relativistic quantum me-chanics, are incorporated consistently . It is calledthe quantum NL world in the sense of a quantumrestriction to the NL world. Its outline is as follows:

Let

W = (V, F,L, , , , , )be the quantum NL world to be constructed. The setV of monad-images can be any finite set with an ele-ment specified as the self-image vself . The main ideain defining the other items of W is that the internalworld of a monad is described by a quantum state andLorentz frames associated with the monad-images.This main idea and the purpose of the quantum NLworld almost determine the structure of W .

The set V is decomposed into two subsets {vself} andVother = V {vself}. The quantum state of a monad is

T. Nakagomi / BioSystems 69 (2003) 2738 29

also decomposed into the tensor product of the self partand the other part, and the selfother coupling of thequantum state defines the list of choices. The choicesby monads appear as quantum reduction processes ofselfother coupled states. The preferability is relatedto the reduction probability. The choice-driven part ofstate-change operator represents this reduction. Theautomatic part of is defined so as to cause inhomo-geneous change of Lorentz frames of monad-images.The interpreter is specified by frameframe relationand selfother conversion of monad-images. Finally,the appetite is given by the entropy of distributionof over in the same way as proposed in the NLworld.

The quantum NL world fulfills all the optional con-ditions in the NL world, and hence the consequencesfrom them discussed here hold also in the quantumNL world and are reinterpreted from the point of viewof relativistic quantum mechanics. Applying Theoremof the NL world to the quantum NL world yieldsthe relativity principle with inhomogeneous Lorentzgroup. The enhancement process of monads decisionby a dominant-state mechanism explains the measure-ment process in quantum mechanics. Henceforth, The-orem 1 and Conditions 14 mean those of the previouspaper.

The time parameter t introduced in the NL worldrepresents the flowing time in which monads makedecisions and cause the change of internal states.In the quantum NL world, the automatic part ofstate-change can be expressed as a motion of view-point in Minkowsky spacetime, and this motion isparameterized by t. Ordinary physics does not havesuch a flowing time parameter, and cannot describethe reduction of quantum states properly, and alsocannot deal with motion of viewpoint with whichconsciousness and free will are associated. Note thatthe time axis in Minkowsky spacetime is a mathe-matical entity needed for manipulating the Lorentzgroup and does not represent the real time. Also notethat unitary transformation of quantum state is onlythe motion of viewpoint and not the real change.Without reduction of states, the motion of viewpointgives no effect to the world.

In Section 2, we prepare relativistic quantum me-chanics in the form of continuous unitary representa-tions of the inhomogeneous SL(2,C) group. Specialnotations in this paper are also explained there. In

Section 3, each item of the quantum NL world W isspecified. In Section 4, basic rules are restated in thestyle of the quantum NL world. In Section 5, a sym-metry on S(F) induced from S(V) is introduced, anddiscussed are relativity principle, internal descriptionand the relation between the flowing time and theMinkowsky time. In Section 6, dominant-state mech-anism is dealt with, which gives an answer to theproblem of measurement in relation with enhance-ment process of monads volition.

Here it is noted that consciousness and volitionappearing in this model should not be given direct in-terpretation by our daily experience, but be consideredas an elemental and primitive origin of those of thehuman level.

2. Preliminaries for relativistic quantummechanics

In general, a relativistic quantum mechanical systemis defined by specifying a continuous unitary repre-sentation of the inhomogeneous SL(2,C) (cf. Streaterand Wightman, 1964). This section starts with a briefreview of SL(2,C). The set V of monad-images is as-sumed to be given.

2.1. SL(2,C) and Lorentz transformations

The special linear group SL(2,C) is the well-knownmatrix group defined by

SL(2,C) = { M(2,C) | det = 1},

where M(2,C) is the set of 2 2 complex matrices.Let

H(2) = {x M(2,C) | x = x}.

H(2) makes a four-dimensional real linear space be-cause any x H(2) can be written as

x=(x0 + x3 x1 ix2x1 + ix2 x0 x3

)

= x00 + x11 + x22 + x33,

where 0 = I, and 1, 2 and 3 are the Pauli spin ma-trices. For each SL(2,C) a linear transformation


() on H(2) is defined by

()x = x for x H(2).It is evident that (I) = I and ()() = (),and the mapping () defines a representationof SL(2,C) by linear transformations on H(2).2 Theinner product in H(2) is introduced by

x y = 12 (det(x+ y) det x det y).The transformation () preserves this inner product,

()x ()y = x y.The 4 4 matrix (i j)ij is equal to the Minkowskymetric matrix, i.e.

(i j)ij =

11

11

.

Hence, H(2) is regarded as the Minkowsky space and() as the Lorentz transformation on it. Here afterwe will call elements in H(2) 4-vectors. Note that1(I) = {I,I} and () is a two-to-one cor-respondence, but covers the proper Lorentz group, theconnected component of the Lorentz group containingthe identity.

2.2. Polar decomposition

Let us define subsets of H(2) and SL(2,C):

Htime(2) = {x H(2) | x x > 0, 0 x > 0},Htime,1(2) = {x H(2) | x x = 1, 0 x > 0},Hspace(2) = {x H(2) | x x < 0 or x = 0},SLpure(2,C) = { SL(2,C) | = , tr > 0},SU(2) = { SL(2,C) | = I}.Htime(2) is the set of positive time-like 4-vectors,Htime,1(2) the set of unit time-like 4-vectors, andHspace(2) the set of space-like 4-vectors. SLpure(2,C)corresponds to the set of pure Lorentz transformations,and SU(2) is known as the special unitary group. It

2 Symbol I stands for two meanings, the identity matrix inH(2) and the identity transformation on H(2), which can be easilydistinguished in context.

is evident from x x = det x and 0 x = (1/2)tr xthat SLpure(2,C) coincides with Htime,1(2).

Any Htime,1(2) has its square root

SLpure(2,C), and () is the pure Lorentz trans-

formation that brings 0 to , that is, ()0 = .

For any SL(2,C) there exist uniquely SLpure(2,C) and o SU(2) such that = o,which is called the polar decomposition of . This ex-pression is obtained by putting = and o =()1, and the uniqueness follows from the posi-

tivity of . In terms of Lorentz transformations, ()represents a pure Lorentz transformation, and (o) aspatial rotation.

For x Htime, the proper time length |x| is definedby

|x| = x x.The sets Hspace(2) and Htime(2) are characterized asfollows:

x Hspace {x H(2), Htime,1(2), x = 0.

(1)

x Htime {x H(2), Htime,1(2), x |x|.

(2)

The equality in (2) holds iff = x/|x|. Note that forx Htime the Lorentz transformation (

x/|x|)1

brings x to a 4-vector parallel to the time axis 0 forwhich the equality in (2) holds.

2.3. Linear functionals on H(2)

For a linear mapping f from H(2) to R, there existsuniquely a 4-vector f such that f(x) = f x for anyx H(2), which is explicitly written as

f =3i=0

i(i )if(i).

We will identify f with f and write as f(x) = f x.Generalizing the above fact to an arbitrary linear

space S, we will call a linear mapping q from H(2) toS a S-valued 4-vector, and ()q is defined by

(()q)(x) = q((1)x).


With the notation q(x) = q x, it becomes()q x = q (1)x.Let f be a linear mapping from S to a linear spaceS (S, R or any other). Then the x f(q(x)) definesS-valued 4-vector, which we write f(q). From thesedefinitions we have

()f(q) = f(()q).

2.4. Inhomogeneous SL(2,C)

We make a new group G = H(2)SL(2,C), inho-mogeneous extension of SL(2,C), whose group oper-ation is defined by(x, )(x, ) = (x+()x, ). (3)The identity element of G is given by (0, I) and theinverse of (x, ) by(x, )1 = ((1)x, 1). (4)Note that x H(2) can be identified with (x, I) Gand that H(2) is treated as a subgroup of G.

A V -indexed family of elements of G is written asg = {gv}vV with gv G (v V). The set of allV -indexed families of elements of G is denoted byGV . For g = {gv}vV , g = {gv}vV in GV , g, g inG and r in S(V), we define

gg = {gvgv}vV , ggg = {ggvg}vV ,gr = {grv}vV .Also we use such notation as g = (x, ), whichmeans gv = (xv, v) (v V).2.5. Unitary representations

In order to make a quantum description of the in-ternal world of a monad, we introduce a Hilbert spaceH and continuous unitary representations U and Kv,v V , of G:(x, ) G U(x, s) (whole unitary representation),(x, ) G Kv(x, s), v V(individual unitary representations).

It is assumed that H has a V -indexed tensor productexpression such thatH =

vVH0, (5)

U(x, ) = vV

U0(x, ) if x Hspace(2), (6)

Kv(x, )= vV

{ifv = v then U0(x, ) else I0},(7)

where H0 is a Hilbert space, U0(x, ) is a unitaryoperator on H0, and I0 is the identity operator on H0.This assumption implies that

U(x, )Kv(y, I) = Kv(()y, I)U(x, )if x Hspace(2), (8)

Kv(x, )Kv(x, ) = Kv(x, )Kv(x, ). (9)

The condition (6) or (8) corresponds to the causalitycondition in quantum field theory, whose significancein the NL world is shown later.

By using the above tensor product expression, a uni-tary operator (r) on H for each r S(V) is definedby

(r) vV

v = vV

r1v.

The mapping r S(V) (r) makes a unitary rep-resentation of the group S(V). Evidently,

(r)Kv(x, ) = Krv(x, )(r). (10)Commutability between (r) and U(x, ) holds in thecase of (6), but we generally assume thatU(x, )(r) = (r)U(x, ). (11)

The tensor product vVv is identified withvself (vVotherv). By this identification, we alsouse such tensor product expression of H as

H = Hself Hotherwith Hself = H0 and Hother =

vVotherH0,

and the partial trace with respect to Hother is de-noted by Trother. For any trace class operator A on H,TrotherA defines a trace class operator on Hself .

The inner product of and H is denoted by,, and [] represents the subspace spanned by. For a closed subspace S in H, the projection oper-ator to S is denoted by Q(S). In particular, Q([]) =|| if = 1.


2.6. Energymomentum operators

The whole energymomentum operator P is de-fined by the generator of U(x, I), x H(2), throughWigners theorem, i.e. an operator valued 4-vector Psatisfying

U(x, I) = eiP x

for any x H(2). The four components of P are givenby P i, i = 0, . . . , 3. The 0-th component P0 =P 0 corresponds to energy and the remaining threeP = (P 1, P 2, P 3) to momentum. The groupproperty implies

U(y, )U(x, I)U(y, )1

= U(()x, I) = eiP ()x = ei()1P x,and we obtain

U(y, )PU(y, )1 = ()1P for any (y, ) G.The assumption of (11) implies

(r)P(r) = P for any r S(V).For each v V the individual energymomentum

operator pv is defined as the generators of Kv(x, I),x H(2), in the similar way as above.Kv(x, I) = eipvx

for any x H(2). Energy p0v = pv 0 and momentumpv = (pv 1, pv 2, pv 3) are defined for eachv V .

From (8) it follows that if y Hspace(2) then

U(y, )Kv(x, I)U(y, )1

= Kv(()x, I) = eipv()x = ei()1pvx,which implies

U(y, )pvU(y, )1 = ()1pv if y Hspace(2),

and also from (10)

(r)pv(r) = pr1v for any r S(V).In general, the whole energymomentum operator isnot the total sum of the individual ones. However, the

causality assumption (6) yields

P x =vV

pv x if x Hspace(2),

which implies

P =vV

pv.

Note that this is not true for the energy component.Finally, we assume that , pv is included in

Htime(2) for any non-zero . This means positiveenergy and positive mass. The mass operator mv isdefined by

mv =pv pv =

(p0v)

2 pv pv.Now the preparations are completed.

3. Construction of the quantum NL world

The quantum NL world W is constructed by spec-ifying each item of W =(V , F , L, , , , , ), butV is already given.

3.1. Internal states (F)

F = {(, x, , ) Htime,1(2)GV H| (xvself , vself ) = (0, I), xv = 0 v V },

where (, x, , ) is abbreviation of (, (x, ), ).Henceforth, =(, g, ) =(, x, , ) is an arbi-trary element of F if not specified otherwise. rep-resents the quantum state of the internal world of amonad, xv and v are the location and Lorentz frameof a monad-image v viewed from the self-image vself ,and is a time axis which characterizes the locationsof monad-images.

3.2. Contents of consciousness (L)

L = the set of all closed subspaces of H,which is an orthomodular lattice with lattice operations

a b = a+ b and a b = a b for a, b Land with the ordinary orthocomplement operation inH. It is evident that any unitary operator on H induces


an automorphism on L. As mentioned in the previouspaper (Nakagomi, 2002), contents of consciousnessis only the name of elements of L, but the authorconsider that there is a certain relation between L andthe structure of our experience of consciousness.

3.3. List of choices ()

()= ()

={{Qi IotherH | i = 0, 1, 2, . . . } if = 0, if = 0,

where Iother is the identity operator on Hother and Qi,i 1, are projection operators on Hself defined by thespectral decomposition of TrotherQ([]), i.e.TrotherQ([]) =

i1

piQi,

with

pi > pi+1 > 0, i = 1, 2, . . . , andi1

piTrQi = 1,

and Q0 is defined by Q0 = I

i1 Qi. Here() = () means that () depends only on of = (, g, ). Such a notation will be used in thefollowing.

3.4. Preferability ()

(a |) = (a |) =

,Q(a)2 if = 0,

0 if = 0.It is evident that ( |) defines a probability measureon L if = 0 and thata()

(a |) = 1 if = 0.

3.5. Appetite ()

() = () = 0(1 e1S())withS() =

a()

(a |)log (a |),

where 0 and 1 are constants with 0 < 0 1 and0 < 1. S() means the entropy of the probability dis-tribution (a |) over a () and is non-negative.Therefore, 0 () < 0 1.

3.6. Interpreter ()

The function (r, ) depends only on r S(V) andg = (x, ) of and is defined by(r, ) = (r, g) = U(grvself )(r).

3.7. State-change operator ()

It consists of two parts, the pure change part 0(a)and the automatic change part 1(a) = 10(a).0(a) provides the reduction of quantum state as-sociated with the choice a

0(a) = (, g, )where

=

Q(a)

Q(a) if Q(a) = 0,0 if Q(a) = 0.

1 is made of two steps:

1 = 1110,which are defined by10 = ((vself ), g1vself g, U(g1vself )) (12)with

gv = (x, )

= gv

(0,

(v)1

, pv| , pv|

)if = 0,

I if = 0,(13)

and

11 = (, g1vself g, U( g1vself )) (14)

with

gv = gv(

0

(1v ) 0, I

). (15)


Evidently 0(a) F and 1 F if F , and(a) defines an operator on F . It is easily checked that0(1) = I and hence (1) = 1, which agrees withthe definition of automatic change in the NL world.Either of 10 and 11 consists of change of g followedby the change of viewpoint of the self-image vself .The transformation v v in (13) has the meaningthat it makes the average energymomentum vector, pv parallel to the time axis or equivalently let itsenergy component , p0v minimum. The minimumvalue is in general equal to or larger than the averagemass , mv. In particular, if is the eigenstate ofm then it is equal to the average mass. The second stepof (14) represents translations of frames associatedwith monad-images along respective proper time axis(v)0 of lengths 1/ (v)0 for v V . In theautomatic change 1, the quantum state undergoesonly the unitary transformation caused by the changeof viewpoint.

3.8. Check of basic conditions

Conditions C0.1, C0.2, C0.3 are evident, and C0.4is checked below.

Take = (, x, , ) and = (, x, , ) F . Any a () can be written asa =

vV{if v = vself then QH0 else H0},

where Q is a projection operator on H0. Since xv =0 is equivalent with xv = 0 or xv xv < 0, we havefrom the causality condition (6)

(r, ) =[vV

U0(xrvself , rvself )

](r).

Therefore,

(r, )a

= vVU0(xrvself , rvself )vV {if v = rvself thenQH0 else H0}

= vV {if v = rvself thenU0(xrvself , rvself )QH0else H0}.Consequently,

(r, )() (r, )() if rvself = rvself .Note that the causality condition (6) has played anessential role in deriving this commutability condition.

4. Rules

Now that the mathematical items that comprise thequantum NL world W have been specified with basicconditions satisfied, we can apply Rules 13 to W .

Rule 1 (Monads and correspondences). With W , a setMW of monads is associated, whose cardinal numberis the same as that of V . Each monad i MW has aone-to-one and onto correspondence ci: j MW cij V that satisfiescii = vself . (16)For every pair i, j MW , a correspondence rij S(V)is defined by

rij = ci c1j . (17)

Rule 2 (Current states). Each monad i MW has avariable i taking values in F , called the current stateof the monad i. The current state variable i of eachmonad i W has three main components:i = (i, i, i),and i has 2|V | subcomponents:i = {iv}vV = {Xiv, iv}vV .

Rule 3 (Choice and renewal). The current states ofmonads are renewed by iteration of the following pro-cess: Each monad decides to choose an element from(m) or to do nothing. Let A be the set of mon-ads to make decision of choosing, and am (m)be the choice by m A. This situation occurs withprobability[mA

(m)

][m/A

(1 (m))]

1|A|

iA

i,Q(a(i)A )ii2

witha(i)A =

mA

U(rimvself )(rim)am,

and yields the change of current state for each monadas follows:

(i, i, i) := 10(a(i)A )(i, i, i). (18)


where we have used the same convention for A = as in the previous paper.

The style of presenting Rule 3 is different from thatof the previous paper, but equivalent to it.

Note that in case of A = we have a(i)A = 1, andonly the automatic change occurs, in which the com-ponenti undergoes unitary transformation associatedwith the change of viewpoint of the monad i.

5. Internal description and relativity

5.1. Null monads

The null state condition () = is equivalent to = 0, and a null monad i is specified by i = 0. It isevident that the quantum NL world satisfies Condition1. Hence, a null monad remains a null monad as thecurrent state changes.

5.2. Symmetry conditions

Operation of r S(V) on = (, g, ) F isdefined byr = ((1v(r)), g1v(r)gr1, U(g1v(r))(r)) (19)withv(r) = r1vself , (20)where g = (x, ). This defines a representation ofthe group S(V) on S(F) because IV = (IV is theidentity in S(V)) andr(r)= ((1

v(rr)v(r))(1v(r)),

g1v(rr)gv(r)g

1v(r)gr1r1,

U(g1v(rr)gv(r))(r)U(g

1v(r))(r

))

= ((1v(rr)), g

1v(rr)g(rr)1,

U(g1v(rr))(rr

))

= (rr).Conditions 2 and 3 can be checked straightforwardlywith some patience.

5.3. Internal description in an active monad

Conditions 1 and 2 ensure Theorem 1. In the restof this section, we assume that the interrelation men-

tioned in Theorem 1 is achieved among all the ac-tive monads. Moreover, Condition 3 make it possibleto give the internal description by use of the quanti-ties v() and v(). By definitions these quantitiesbecome

v() = ([v, vself ], [v, vself ])([v, vself ]),v() = ([v, vself ]),where [v, vself ] S(V) is the transposition of v andvself . Explicit forms of these are as follows:

v() = v() = ([v, vself ])(([v, vself ])),v() = v() = 0(1 exp 1Sv())with

Sv()= S(([v, vself ]))=

av()

(a |)log (a |).

Let us consider the internal description in an activemonad i0 MW . Put = (, , ) = i0 . Theprobability variables J and bJ for J Vact are definedin the same way as in the previous paper. J takesvalues in 2Vact and bJ in J(i), where J() =

vJ v(). The conditional probability P( |) ofthese variables depends only on as given below:

P(J = J, bJ = a |)

=[v/J

(1 v())][

vJv()

], Q(a)

2(21)

for any J Vact and any a J(). The law of state-change is given by(, , ) := 10(bJ )(, , ). (22)Note that the case of Q(b

J) = 0 does not appear,

because the probability (21) of such a case is equalto 0.

5.4. Time parameter

In chronological description, time parameter t isintroduced by replacing the substitution formula (22)with the equality formula:

([t + 1], [t + 1], [t + 1])= 10(bJ[t][t])([t], [t], [t]). (23)


The probability law for the t-parametric variables isgiven by

P(J[t]= J, bJ [t] = a | [t])

=[v/J

(1 v( [t]))][

vJv( [t])

]

[t],Q(a) [t] [t]2 .

The above two equations with initial condition([0], [0], [0]) defines a stochastic process fort = 0, 1, 2, . . . . Note that (23) cannot be solved ininverse direction of t as t = 0,1,2, . . . , because0(a, J) is not invertible unless a = 1.

5.5. Relativity principle

In the above subsection, we have obtained the in-ternal description using the internal variables {, , , J , (bJ )JVact} of an active monad i0. Let i1 beanother active monad, and the internal variables {, , , J , (bJ )

JVact} of i1 be defined in the same

way as above. Then these two are interrelated to eachother as follows:

= (1v(r)), (24)

= 1v(r)r1, (25)

= U(1v(r))(r), (26)

J = rJ, (27)brJ = U(1v(r))(r)bJ , (28)where r = ri1i0 and v(r) is defined by (20), whichmeans the image of the monad i1 appearing in the in-ternal world of i0. The law of change given by (21) and(22) is invariant under these transformations (24)(28),which gives the relativity principle stated in a generalstyle. Let us compare this with the ordinary relativityprinciple of Einstein.

The above-mentioned interrelation has two kindsof factors. One appears in v(r), (r), and r, whoserole is reassignment of the indexes of monad-images.The other appears in (1v(r)),

1v(r) and U(v(r)),

which represent the (inhomogeneous) Lorentz trans-formation connecting the two frames associated with

the monad i0 and i1. The latter appears in the ordinaryrelativity theory, but the former does not.

The relativity principle in relativistic quantum me-chanics is expressed fully by

= U(g), (29)where and are the same quantum state viewedfrom different frames, and g G represents the inho-mogeneous Lorentz transformation that connects thetwo frames. Since the two frames can be separated bya time-like distance, the relation (29) includes rela-tivistic evolution law. On the other hand, the relation(26), though corresponding to (29), does not includean evolution law, because the distance between thetwo frames is restricted to space-like distances by thecondition Xv(r) = 0. If bJ = 1 in (22), then onlythe automatic change 1, which is unitary on , oc-curs and relation (26) can be extended to a time-likedistance, such as

[t] = U(g)(r) [t] (t = t)with a suitable g G. The ordinary relativistic quan-tum mechanics corresponds to this special case. Thisis the reason why we cannot describe the reductionof quantum states in the relativistic quantum me-chanics. To describe it, we need 0 in (22), whichbrings non-unitary change of quantum states causedby monads volition.

The reduction of quantum states does not occur inthe Minkowsky time or the time axis in space-time,but in the flowing or living time that is expressed bythe parameter t in the chronological description. Thus,we have two concepts of time. The variable hasthe role to make linkage between them. The time-liketranslation between two successive reductions by 0is given by 11, whose operation on is as follows

U

( 0 0 , I

) = exp

[ 1 0 P

0].

This means the translation of length 1/ 0 alongthe time axis in Minkowsky space-time. Let H be theenergy operator in the ordinary physical units. Thenthere is a universal constant 0 connecting the twoenergy operators as P0 = 0H . In this unit, the lengthof time translation is rewritten as0

0 , (30)


and 0 provides the maximum time length of one stepchange of the quantum NL world because 0 1with equality iff = 0.

The variables of active monads are related to eachother through the Lorentz transformations that connecttheir frames, and they provide a common time axis.This common time axis seems to have the role of theabsolute time, which necessarily implies the definitionof the simultaneity or the absolute space perpendicularto the time axis. However, it has nothing to do withthe energymomentum operators and neither appearin theory nor in experiment of the ordinary physics,and hence it does not contradict the relativity theory.

6. Enhancement processes

6.1. Dominant states

Let us consider a quantum version of dominantstates. A simple example of a dominant state =(, g, ) with respect to a non-empty D Vother isconstructed as follows: =

i1

cii i Hself Hother (31)

with

i =jSi

cijvvij Hother =

v=vselfH0, (32)

where Si are suitable index sets, and

i i and vij vijif (i, j) = (i, j) and v D. (33)

Without loss of generality, we can put

= i = i = vij = 1,and hencei1

|ci|2 =j1

|cij|2 = 1.

For simplicity we also assume that there are no acci-dental degeneracies in the coefficients {|ci|2}i and in{|cicij|2}ij. Then we have() = {[i]Hother}i,v() = {([v, vself ])[vij]Hother}ij.

For a = [i ] Hother () and b = ([v, vself ])[vij]Hother v() we have(a b |) = ,Q(a b) = | ci |2 | cij|2ii,(b |) = ,Q(b) = | ci|2 | cij | 2.Therefore, it is evident that the dominant statecondition

(a b |) = 0 (b |) = (b a |),is fulfilled.

6.2. Well-behavedness

The quantum NL world satisfies the well-behaved-ness condition. The definition of (a) = 10(a) is inaccordance with C4.1. Conditions C4.2C4.4 followsdirectly from the definitions of and , and C4.5is satisfied if the constant 0 in (30) is sufficientlysmall. Indeed, a and b given above with condition(a b |) = 0 cause the same reduction in the listof choices, that is,

(0(a)) = (0(b)) = {a} (34)though the equalities in (34) should be replaced withnearly equal if is used instead of 0.

6.3. Measurement process

The discussion of measurement process in theNL world can be paraphrased in the quantum NLworld by use of the dominant state presented before.It is evident from the form of the dominant statethat this paraphrase reproduces the quantum proba-bility of measurement. The form of (31) is similarto the quantum state representing the coupling ofmeasured object and measuring apparatus in the stan-dard measurement theory of von Neumann (1932),but the reduction process is different from it. In vonNeumanns theory, the reduction is ultimately at-tributed to observers consciousness, while in ourtheory it is to the decisions of large number of mon-ads corresponding to monad-images in D, which isconsidered as the measuring apparatus.

As discussed in the previous paper, this measure-ment process can be used to enhance the volition ofan active monad whose current state is dominant withrespect to a large set of monad-images corresponding


to active monads. The author considers that such aprocess may occur in our brain.

7. Concluding remarks

The quantum NL world allows to introduce the con-cepts of the flowing time and the interior of matterwith which monads live. The inconsistencies betweenour inner experience and physics are resolved if weadopt the quantum NL world to describe the real worldwhere we live. The hierarchy of consciousness men-tioned in the previous paper must be constructed onthe basis of the quantum NL world.

Finally, there are some remarks in relation tophysics. In the quantum NL world, after relativisticcorrespondence is achieved between all active mon-ads, fundamental physical theories hold almost asthey are in the internal world of each active monad,and the internal world can be identified with thephysical world. A modification is in the point thatthe reduction of quantum states is included in thefundamental law of state-change. The unitary evo-lution has only the role of linking two neighboringreductions. Associated with reductions, simultaneityis introduced, which is common to all active monadsor equivalently to all Lorentz frames. As mentionedbefore, this simultaneity, however, has no effect onthe Hamiltonian, and hence has no contradiction tothe ordinary relativity theory. Its effect is only on thereductions of quantum states, and might be detectedby measurement of simultaneous reductions of twoquantum systems separated by a long distance. Suchsituation of measurement appears in the EPR prob-lem (Einstein et al., 1935), in which two correlatedquantum particles separated by a long distance areexpected to cause quantum reduction simultaneously.

Another detectable effect might be in the differencebetween P0 and

v p

0v, where P0 is the whole Hamil-

tonian of the internal world and p0v the individualHamiltonians associated with monad-images v. Thelatter Hamiltonians determine how the monad-imagesappear in the internal world (or equivalently in thephysical world). Monad-images are expected to appearas coherent quantum parts in the whole state, since re-ductions of a quantum state occur so as to resolve thesuperposition that appears when written in the tensorproduct form (5) and not to destroy each component

of the tensor product. This monad-coherent structuremight throw a new light on the ensembles used in sta-tistical mechanics and provide a linkage between thehierarchy of matter and that of consciousness men-tioned in the previous paper.

Acknowledgements

Thanks are due to Prof. K. Yasue for helpful andencouraging discussions.

References

Bohm, D., 1951. Quantum Theory. Prentice-Hall, EnglewoodCliffs.

Cochran, A.A., 1971. Relationships between quantum physics andbiology. Found. Phys. 1, 235250.

Einstein, A., Podolsky, B., Rosen, N., 1935. Can quantum-mechanical description of physical reality be consideredcomplete? Phys. Rev. 47, 777780.

Globus, G.G., 1995. Advances in Consciousness Research, vol. 1.The Postmodern Brain. John Benjamins, Amsterdam.

Jibu, M., Hagan, S., Hameroff, S.R., Pribram, K.H., Yasue, K.,1994. Quantum optical coherence in cytoskeletal microtubules:Implications for brain function. BioSystems 32, 195209.

Jibu, M., Yasue, K., 1995. Advances in Consciousness Research,vol. 3. Quantum Brain Dynamics and Consciousness. JohnBenjamins, Amsterdam.

Jibu, M., Yasue, K., Hagan, S., 1997. Evanescent (tunneling)photon and cellular vision. BioSystems 42, 6573.

Khrennikov, A., 2000. Classical and quantum dynamics on p-adictrees of ideas. BioSystems 56, 95120.

Khrennikov, A., 2002. Quantum-like formalism for cognitivemeasurements. Preprint.

Libet, B., 1985. Unconscious cerebral initiative and the role ofconscious will in voluntary action. Behav. Brain Sci. 8, 529566.

Nakagomi, T., 1992. Quantum monadology: A world model tointerpret quantum mechanics and relativity. Open Syst. Inform.Dyn. 1, 355378.

Nakagomi, T., 1995. Feeling decision systems and quantummechanics. Cybernet. Syst. 26, 601619.

Nakagomi, T., 2002. Mathematical formulation of Leibnizianworld: A basic framework for consciousness and matter.BioSystems. 69, 1526.

Riccardi, L.M., Umezawa, H., 1967. Bran and physics ofmany-body problems. Kybernetik 4, 4448.

Streater, R.F., Wightman, A.S., 1964. PCT, Spin and Statistics,and All That (W.A. Benjamin, Reading).

Vitiello, G., 2001. Advances in Consciousness Research, vol. 32.My Double Unveiled, The Dissipative Quantum Model of Brain.John Benjamins, Amsterdam.

von Neumann, J., 1932. Mathematische Grundlagen der Quanten-mechanik. Springer, Beriln .

Whitehead, A.N., 1929. Process and Reality. An Essay in Cosmo-logy. Macmillan Publishing Co., New York.

Quantum monadology: a consistent world model for consciousness and physicsIntroductionPreliminaries for relativistic quantum mechanicsSL(2,C) and Lorentz transformationsPolar decompositionLinear functionals on H(2)Inhomogeneous SL(2,C)Unitary representationsEnergy-momentum operators

Construction of the quantum NL worldInternal states (F)Contents of consciousness (L)List of choices ()Preferability (rho)Appetite ()Interpreter (lambda)State-change operator (beta)Check of basic conditions

RulesInternal description and relativityNull monadsSymmetry conditionsInternal description in an active monadTime parameterRelativity principle

Enhancement processesDominant statesWell-behavednessMeasurement process

Concluding remarksAcknowledgementsReferences

a consistent world model for consciousness and physics

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