Non-Critical String Theory Formulation of Microtubule Dynamics And

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1 Intro duction

The interior of living cells is structurally and dynamically organized by cytoskele-

tons, i.e. networks of protein p olymers. Of these structures, MicroTubules (MT)

app ear to b e [1] the most fundamental. Their dynamics has b een studied recently by

anumb er of authors in connection with the mechanism resp onsible for dissipation-

free energy transfer. Recently, Hamero and Penrose [2] have conjectured another

fundamental ro ^le for the MT, namely b eing resp onsible for quantum computations

in the human brain, and, thus, related to the consciousness of the human mind.

The latter is argued to b e asso ciated with certain asp ects of quantum theory [3]

that are b elieved to o ccur in the cytoskeleton MT, in particular quantum sup er-

p osition and subsequent collapse of the wave function of coherentMTnetworks.

While quantum sup erp osition is a well-established and well-understo o d prop erty

of quantum physics, the collapse of the wave function has b een always enigmatic.

We prop ose here to use an explicit string-derived mechanism - in one interpreta-

tion of non-critical string theory - for the collapse of the wave function[4], involving

quantum gravity in an essential way and solidifying previous intuitively plausible

suggestions[5 , 6]. It is an amazing surprise that quantum gravity e ects, of order of

1=2

magnitude G m 10 , with G Newton's gravitational constant and m the

p N p

proton mass, can playa r^ole in suchlow energies as the eV scales of the typical

energy transfer that o ccurs in cytoskeleta. However, as we show in this article, the

ne details of the MT characteristic structure indicate that not only is this con-

ceivable, but such scenaria lead to order of magnitude estimates for the time scales

entering conscious perception that are close enough to those conjectured/\observed"

by neuroscientists, based on completely di erent grounds.

To understand how quantum space-time e ects can a ect conscious p erception,

we mention that it has long b een susp ected [7] that large scale quantum coherent

phenomena can o ccur in the interior of biological cells, as a result of the existence

of ordered water molecules. Quantum mechanical vibrations of the latter are re-

sp onsible for the app earance of `phonons' similar in nature to those asso ciated with

sup erconductivity. In fact there is a close analogy b etween sup erconductivity and

energy transfer in biological cells. In the former phenomenon electric currentis

transferred without dissipation in the surface of the sup erconductor. In biological

cells, as we shall discuss later on, energy is transferred through the cell without

loss, despite the existence of frictional forces that represent the interaction of the

cell with the surrounding water molecules [8]. Such large scale quantum coherent

states can maintain themselves for up to O (1 sec), without signi cantenvironmental

entanglement. After that time, the state undergo es self-collapse, probably due to

quantum gravity e ects. Due to quantum transitions b etween the di erent states

of the quantum system of MT in certain parts of the human brain, a sucient dis-

tortion of the surrounding space-time o ccurs, so that a microscopic (Planck size)

black hole is formed. Then collapse is induced, with a collapse time that dep ends on

the order of magnitude of the number N of coherent microtubulins. It is estimated

that, with an N = O [10 ], the collapse time of O (1 sec ), which app ears to b e a

typical time scale of conscious events, is achieved. Taking into account that exp eri-

ments have shown that there exist N =10 tubulins p er neuron, and that there are

10 neurons in the brain, it follows that that this order of magnitude for N refers

to a fraction 10 of the human brain, whichisvery close to the fraction b elieved

resp onsible for human p erception.

The self-collapse of the MT coherent state wave function is an essential step for

the op eration of the MT network as a quantum computer. In the past it has b een

suggested that MT networks pro cessed information in a way similar to classical

cellular automata (CCA) [9]. These are describ ed byinteracting Ising spin chains

on the spatial plane obtained by leting op en and attening the MT cylindrical

surface. Distortions in the con gurations of individual parts of the spin chain can

b e in uenced by the environmental spins, leading to information pro cessing. In

view of the suggestion [2] on viewing the conscious parts of the human mind as

quantum computers, one might extend the concept of the CCA to a quantum cellular

automaton (QCA), undergoing wave function self-collapses due to (quantum gravity)

enviromental entanglement.

An interesting and basic issue that arises in connection with the ab over^ole of the

brain as a quantum computer is the emergence of a direction in the ow of time

(arrow). The latter could b e the result of succesive self-collapses of the system's

wave function. In a recent series of pap ers [4] wehave suggested a rather detailed

mechanism by whichan irreversible time variable has emerged in certain mo dels of

string quantum gravity. The mo del utilized string particles propagating in singular

space-time backgrounds with event horizons. Consistency of the string approach

requires conformal invariance of the asso ciated -mo del, which in turn implies a

coupling of the backgrounds for the propagating string mo des to an in nity of global

(quasi-top ological) delo calized mo des at higher (massive) string levels. The existence

of such couplings is necessitated by sp eci c coherence-preserving target space gauge

symmetries that mix the string levels [4].

The sp eci c mo del of ref. [4] is a completely integrable string theory, in the sense

of b eing characterised by an in nity of conserved charges. This can b e intuitively

understo o d by the fact that the mo del is a (1 + 1)-dimensional Liouville string, and as

such it can b e mapp ed to a theory of essentially free fermions on a discretized world

sheet (matrix mo del approach [10 ]). A system of free fermions in (1 + 1) dimen-

sions is trivially completely integrable, the in nity of the conserved charges b eing

provided by appropriate moments of the fermion energies ab ove the Fermi surface.

Of course, formally, the precise symmetries of the mo del used in ref. [4] are much

more complicated [11], but the idea b ehind the mo del's integrability is essentially

the ab ove. It is our b elief that this quantum integrabilityisavery imp ortant feature

of theories of space-time asso ciated with the time arrow. In its presence, theories

with singular backgrounds app ear consistent as far as maintainance of quantum co-

herence is concerned. This is due to the fact that the phase-space density of the eld

theory asso ciated with the matter degrees of freedom evolves with time according

to the conventional Liouville theorem[4]

@ = f; H g (1)

t PB

as a consequence of phase-space volume-preserving symmetries. In the two-dimensional

example of ref. [4], these symmetries are known as W , and are asso ciated with

higher spin target-space states[11 ]. They are resp onsible for string-level mixing, and

hence they are broken in anylow-energy approximation. If the concept of `mea-

surementby lo cal scattering exp eriments' is intro duced [4], it b ecomes clear that

the observable background cannot contain such global mo des. The latter haveto

be integrated out in any e ectivelow-energy theory. The result of this integration

is a non-critical string theory, based on the propagating mo des only. Its conformal

invariance on the world sheet is restored by dressing the matter backgrounds by

the Liouville mo de , which plays the role of the time co ordinate. The mo de

is a dynamical lo cal world-sheet scale [4], owing irreversibly as a result of certain

theorems of the renormalization group of unitary -mo dels [12]. In this way time in

target space has a natural arrow for very sp eci c stringy reasons.

Given the suggestion of ref. [2] that space-time environmental entanglement could

b e resp onsible for conscious brain function, it is natural to examine the conditions

under which our theory [4] can b e applied. Our approach utilizes extra degrees

of freedom, the W global string mo des, which are not directly accessible to lo cal

scattering `exp eriments' that make use of propagating mo des only. Such degrees

of freedom carry information, in a similar spirit to the information loss suggested

byHawking[13] for the quantum-black-hole case. For us, such degrees of freedom

are not exotic, as suggested in ref. [14], but app ear already in the non-critical

String Universe [4, 15], and as such they are considered as `purely stringy'. In this

resp ect, we b elieve that the suggested mo del of consciousness, based on the non-

critical-string formalism of ref. [4], is physically more concrete. The idea of using

string theory instead of p oint-like quantum gravity is primarily asso ciated with

the fact that a consistent quantization of gravity is at present p ossible only within

the framework of string theory, so far. However, there are additional reasons that

make advantageous a string formalism. These include the p ossibility of construction

of a completely integrable mo del for MT s, and the Hamiltonian representation of

dynamical problems with friction involved in the physics of MTs. This leads to

the p ossibility of a consistent(mean eld) quantization of certain soliton solutions

asso ciated with the energy transfer mechanism in biologcal cells.

According to our previous discussion emphasizing the imp ortance of strings, it is

imp erative that we try to identify the completely integrable system underlying MT

networks. Thus, it app ears essential to review rst the classical mo del for energy

transfer in biological systems asso ciated with MT. This will allow the identi cation

of the analogue of the (stringy) propagating degrees of freedom, whicheventually

couple to quantum (stringy) gravity and to global environmental mo des. As we

shall argue in subsequent sections, the relevant basic building blo cks of the human

brain are one-dimensional Ising spin chains, interacting among themselves in a way

so as to create a large scale quantum coherent state, b elieved to b e resp onsible

for preconscious b ehaviour in the mo del of [2]. The system can b e describ ed in a

world-sheet conformal invariantway and is unitary. Coupling to gravity generates

deviations from conformal invariance which lead to time-dep endence, by identifying

time with the Liouville eld on the world sheet. The situation is similar to the

environemtnal entanglement of ref. [16, 17 ]. Due to this entanglement, the system

of the propagating mo des op ens up as in Markov pro cesses [18]. This leads to a

dynamical self-collapse of the wave function of the MT quantum coherent network.

In this way, the part of the human brain asso ciated with conscioussness generates,

through successive collapses, an arrow of time. The interaction among the spin

chains, then, provides a mechanism for quantum computation, resembling a planar

cellular automaton. Such op erations sustain the irreversible ow of time.

The structure of the article is as follows: in section 2 we discuss a mo del used for

the physical description of a MT, and in particular for a simulation of the energy

transfer mechanism. The mo del can b e expressed in terms of a 1 + 1-dimensional

classical eld, the pro jection of the displacement eld of the MT dimers along the

tubulin axis. There exists friction due to intreraction with the environment. How-

ever, the theory p ossesses travelling-wave solitonic states resp onsible for loss-free

transfer of energy. In section 3, we give a formal representation of the ab ove sys-

tem as a 1 + 1-dimensional c = 1 Liouville (string) theory. The advantage of the

metho d lies in that it allows for a canonical quantization of the friction problem,

thereby yielding a mo del for a large-scale coherent state, argued to simulate the

preconscious state. There is no time arrow in the ab ove system. In section 4, we

discuss our mechanism of intro ducing a dynamical time variable with an arrowinto

the system, by elevating the ab ove c = 1 Liouville theory to a c = 26 non-critical

string theory, incorp orating quantum gravity e ects. Such e ects arise from the

distortion of space-time due to abrupt conformational changes in the dimers. Such

a coupling leads to a breakdown of the quantum coherence of the preconscious state.

Estimates of the collapse times are given, with the result that in this approach con-

cious p erception of a time scale of O (1 sec), is due to a 10 part of the total brain.

In section 5 we brie y discuss growth ofaMTnetwork in our framework, which

would b e the analogue of a non-critical string driven in ation for the e ective one-

dimensional universe of the MT dimer degrees of freedom. We view MT growth as

an out-of-equilibrium one-dimensional sp ontaneous-symmetry breaking pro cess and

discuss the connection of our approach to some elementary theoretical mo dels with

driven di usion that could serve as prototyp es for the phenomenon. Conclusions

and outlo ok are presented in section 6. We discuss some technical apsects of our

approachintwo App endices.

2 Physical Description of the Microtubules

In this section we review certain features of the MT that will b e useful in subse-

quent parts of this work. MT are hollow cylinders (cf Fig 1) comprised of an exterior

surface (of cross-section diameter 25 nm) with 13 arrays (proto laments) of protein

dimers called tubulins. The interior of the cylinder (of cross-section diameter 14 nm)

contains ordered water molecules, which implies the existence of an electric dip ole

moment and an electric eld. The arrangement of the dimers is such that, if one

ignores their size, they resemble triangular lattices on the MT surface. Each dimer

consists of twohydrophobic protein p o ckets, and has an electron. There are two

p ossible p ositions of the electron, called and conformations, which are depicted

in Fig. 2. When the electron is in the -conformation there is a 29 distortion of

the electric dip ole moment as compared to the conformation.

In standard mo dels for the simulation of the MT dynamics, the `physical' degree

of freedom - relevant for the description of the energy transfer - is the pro jection

of the electric dip ole moment on the longitudinal symmetry axis (x-axis) of the

MT cylinder. The 29 distortion of the -conformation leads to a displacement u

along the x-axis, whichisthus the relevantphysical degree of freedom. This way,

the e ective system is one-dimensional (spatial), and one has a rst indication that

quantum integrability might app ear. We shall argue later on that this is indeed the

case.

Information pro cessing o ccurs via interactions among the MT proto lamentchains.

The system may b e considered as similar to a mo del of interacting Ising chains on a

trinagular lattice, the latter b eing de ned on the plane stemming from leting op en

and atening the cylindrical surface of Fig. 1. Classically, the various dimers can

o ccur in either or conformations. Each dimer is in uenced by the neighb oring

dimers resulting in the p ossibility of a transition. This is the basis for classical infor-

mation pro cessing, which constitutes the picture of a (classical) cellular automatum.

The quantum computer character of the MT network results from the assumption

that each dimer nds itself in a sup erp osition of and conformations [2]. There

is a macroscopic coherent state among the various chains, which lasts for O (1 sec )

and constitutes the `preconscious' state. The interaction of the chains with (stringy)

quantum gravity, then, induces self-collapse of the wave function of the coherentMT

network, resulting in quantum computation.

In what follows we shall assume that the collapse o ccurs mainly due to the inter-

action of eachchain with quantum gravity, the interaction from neighb oring chains

b eing taken into accountby including mean- eld interaction terms in the dynamics

of the displacement eld of eachchain. This amounts to a mo di cation of the ef-

fective p otential by anharmonic oscillator terms. Thus, the e ective system under

study is two-dimensional, p ossesing one space and one time co ordinate. The precise

meaning of `time' in our mo del will b e clari ed when we discuss the `non-critrical

string' representation of our system.

Let u b e the displacement eld of the n-th dimer in a MT chain. The continuous

approximation proves sucient for the study of phenomena asso ciated with energy

transfer in biological cells, and this implies that one can make the replacement

u ! u(x; t) (2)

with x a spatial co ordinate along the longitudinal symmetry axis of the MT. There

is a time variable t due to uctuations of the displacements u(x) as a result of

the dip ole oscillations in the dimers. At this stage, t is viewed as a reversible

variable. The e ects of the neighb oring dimers (including neighb oring chains) can

b e phenomenologically accounted for by an e ective double-well p otential [19]

1 1

2 4

U (u)= Au (x; t)+ Bu (x; t) (3)

2 4

with B>0. The parameter A is temp erature dep endent. The mo del of ferro electric

distortive spin chains of ref. [20] can b e used to give a temp erature dep endence

A = jcostj(T T ) (4)

where T is a critical temp erature of the system, and the constant is determined

phenomenologically [19 ]. In realistic cases the temp erature T is very close to T ,

which for the human brain is taken to b e the ro om temp erature T = 300K .Thus,

b elow T A>0. The imp ortant relative minus sign in the p otential (3), then,

guarantees the necessary degeneracy, which is necessary for the existence of classical

solitonic solutions. These constitute the basis for our coherent-state description of

the preconscious state.

Including a phenomenological kinetic term for the dimers, eachhaving a mass M ,

one can write down a Hamiltonian [19 ]

1 1

2 2 2 2 4

H = kR (@ u) M (@ u) Au + Bu + qE u (5)

x t

2 4

where k is a sti ness parameter, R is the equilibrium spacing b etween adjacent

dimers, E is the electric eld due to the `giant dip ole' representation of the MT

cylinder, as suggested by the exp erimental results [19], and q =182e(ethe

electron charge) is a mobile charge. The spatial-derivative term in (5) is a contin-

uous approximation of terms in the lattice Hamiltonian that express the e ects of

restoring strain forces b etween adjacent dimers in the chains [19].

The e ects of the surrounding water molecules can b e summarized by a viscuous

force term that damps out the dimer oscillations,

F = @ u (6)

with determined phenomenologically at this stage. This friction should b e viewed

as an environmental e ect, whichhowever do es not lead to energy dissipation, as

a result of the non-trivial solitonic structure of the ground-state and the non-zero

constant force due to the electric eld. This is a well known result, directly relevant

to energy transfer in biological systems [8]. The mo di ed equations of motion, then,

read

2 2

@ u @ u @u

2 3

M kR Au + Bu + qE =0 (7)

2 2

@t @x @t

According to ref. [8] the imp ortance of the force term qE lies in the fact that eq

(7) admits displaced classical soliton solutions with no energy loss. The solution

acquires the form of a travelling wave, and can b e most easily exhibited by de ning

a normalized displacement eld

u()

( )= (8)

A=B

where,

jAj

(9) (x vt)

M (v v )

with

v k=MR (10)

0 0

the sound velo city, of order 1k m=sec, and v the propagation velo city to b e deter-

mined later. In terms of the ( )variable, equation (7) acquires the form of the

equation of motion of an anharmonic oscillator in a frictional environment

00 0 3

+ + + =0

3=2

v M jAj(v v ) ; = q BjAj E (11)

which has a unique b ounded solution [19]

( )=a+ (12)

1+e

with the parameters b; a and d satisfying:

q B

( a)( b)( d)= E (13)

3=2

jAj

Thus, the kink propagates along the proto lament axis with xed velo city

2 1

(14) v = v [1 + ]

9d Mv 0

This velo city dep ends on the strength of the electric eld E through the dep endence

of d on E via (13). Notice that, due to friction, v 6= v , and this is essential for

a non-trivial second derivative term in (11), necessary for wave propagation. For

realistic biological systems v ' 2m=sec. With a velo city of this order, the travelling

waves of kink-like excitations of the displacment eld u( ) transfer energy along a

mo derately long microtubule of length L =10 m in ab out

t =510 sec (15)

This time is very close to Frohlich's time for coherent phonons in biological system.

We shall come back to this issue later on.

The total energy of the solution (12) is easily calculated to b e [19]

2 2 A A 1 1

2 2

E = dxH = + k + M v + M v (16)

3 B B 2 2

whichisconserved in time. The `e ective' mass M of the kink is given by

4 MA

M = (17)

R B

3 2

The rst term of equation (16) expresses the binding energy of the kink and the

second the resonant transfer energy. In realistic biological mo dels the sum of these

two terms dominate over the third term, b eing of order of 1eV [19]. On the other

hand, the e ective mass in (17) is[19] of order 5 10 kg, which is ab out the proton

mass (1GeV ) (!). As we shall discuss later on, these values are essential in yielding

the correct estimates for the time of collapse of the `preconscious' state due to our

quantum gravityenvironmental entangling. To make plausible a consistent study of

such e ects, wenow discuss the p ossibility of representing the equations of motions

(11) as b eing derived from string theory.

Before closing we mention that the ab ove classical kink-like excitations (12) have

b een discussed so far in connection with physical mechanisms asso ciated with the hy-

drolysis of GTP (Guanosine-ThreePhosphate) tubulin dimers to GDP (Guanosine-

DiPhosphate) ones. Because the two forms of tubulins corresp ond to di erent con-

formations and ab ove, it is conceivable to sp eculate that the quantum mechanical

oscillations b etween these two forms of tubulin dimers might b e asso ciated with a

quantum version of kink-like excitations in the MT network. This is the idea we

put forward in the presentwork. The novelty of our approach is the use of Liouville

(non-critical) string theory for the study of the dynamics involved. This is discussed in the next section.

3 Non-Critical (Liouville) String Theory Repre-

sentation of a MT

It is imp ortant to notice that the relative sign (+) b etween the second derivative

and the linear term in in equation (11) are such that this equation can b e consid-

ered as corresp onding to the tachyon -function equation of a (1 + 1)-dimensional

string theory, in a at space-time with a dilaton eld linear in the space-like

co ordinate [21],

= (18)

Indeed, the most general form of a `tachyon' deformation in such a string theory,

compatible with conformal invariance is that of a travelling wave [22] T (x ) , with

0 0

x = (x vt) ; t = (t vx)

v v

2 1=2

(1 v ) (19)

where v is the propagation velo city. As argued in ref. [22 ] these translational

invariant con gurations are the most general backgrounds, consistent with a unique

factorisation of the string -mo del theory on a Minkowski space-time G =

h i

2 (2)

S = d z @X @X G (X )+(X)R + T (x) (20)

0 0

into two conformal eld theories, for the t and x elds, corresp onding to central

charges

2 2 2

0 0

c =124v ; c =1+24 (21)

t x

v v

In our case (11), the r^ole of the space-like co ordinate x is played by , and the

velo city v is the velo city of the kink. The velo city of light in this e ective string

mo del is replaced by the sound velo city v (10), and the friction co ecient is

expressed in terms of the central charge de cit (21)

= (c( ) 1)=2 (22)

The space-like `b o osted' co ordinate ,thus, plays the r^ole of space in this e ec-

tive/Liouville mo de string theory.

The imp ortant advantage of formulating the MT system as a c = 1 string theory,

lies in the p ossibility of casting the friction problem in a Hamiltonian form. To

this end, wenow make some comments on the various non-derivative terms in the

tachyon p otential V (T ) U ( ) in the target-space e ective action [23]. Such terms

contributre higher order non-derivative terms in the equations of motion (11). The

term linear in is xed in string theory and normalized (with resp ect to the second

derivative term ) as in (11) [22 ]. The higher order terms are p olynomials in and

their co ecients can b e varied according to renormalization prescription [23 ].

The general structure of the tachyon e ective action in the target space of the

string is, therefore, of the form [23]

h i

L = e G (T c) + L(rT; r) (23)

where G is a target space metric eld, and c is a constant. The linear term

is the only universal term, showing the imp ossibility of nding a stable tachyon

background in b osonic string theory, unless it is time dep endent.

In our sp eci c mo del, wehave seeen that a term cubic in in the equation

of motion (11), with a relative minus sign as compared to the linear term, was

resp onsible for the app earance of a kink-like classical solution. Anychange in the

non-linear terms would obviously a ect the structure of the solution, and we should

understand the physical meaning of this in our biological system. To this end, we

consider a general p olynomial in T equation of motion for a static tachyon in (1+1)

string theory

00 0

T ( )+2 T ()=P(T) (24)

where is some space-like co-ordinate and P (T ) is a p olynomial of degree n,say. The

`friction' term T expresses a Liouville derivative, since the e ective string theory

of the displacement eld u is viewed as a c = 1 matter non-critical string. In our

interpretation of the Liouville eld as a lo cal scale on the world sheet it is natural to

assume that the single-derivative term expresses the non-critical string -function,

and hence is itself a p olynomial R of degree m

T ( )=R(T) (25)

Using Wilson's exact renormalization group scheme, wemay assume that R(T )=

a T+a T , where a ;a are related to the anomalous dimension and op erator pro d-

2 4 2 4

uct expansion co ecients for the tachyon couplings. The compatibility conditions

for the existence of b ounded solutions to the equation (24), then, imply the form[24]

2 4

P (T )= A +AT+AT +AT , with A = f (a ;a );i =1; :::4. Indeed such

1 2 3 4 i i 2 3

equations have b een shown [24] to lead to kink-like solutions,

1 1

T ( )= fsg n(a a )a tanh[ a (x ut)] a g (26)

2 4 2 2 2

2a 2

A 3a a

3 2 4

. This fact expresses for us a sort of universal where the velo city u =

b ehaviour for biological systems. This shows the existence of at least one class of

schemes which admit kink like solutions of the same sort as the ones of Lal [8] for

energy transfer without dissipation in cells.

The imp ortance of solutions of the form (26) lies in the fact that they can b e

derived from a Hamiltonian and, thus, can b e quantized canonically [25]. They are

connected to the solitons (12) by a Renormalization Scheme change on the world-

sheet of the e ective string theory, repro ducing (11). This amounts to the p ossibility

of casting friction problems, due to the Liouville terms, into a Hamiltonian form.

This is quite imp ortant for the quantization of the kink solution, which will provide

one with a concrete example of a large-scale quantum coherent state for the pre-

conscious state of the mind. In a pure eld-theoretic setting, a quantization scheme

has b een discussed in ref. [26 ], using a variational approachby means of squeezed

coherent states. There is a vast literature on soliton quantization using approximate

WKB metho ds. We selected this metho d for our purp oses here, b ecause it yields

more accurate results than the usual WKB metho ds of soliton quantization[27], and

it seems more appropriate for our purp oses here, due to its direct link with coherent

ground states. We shall not give details on the derivation but concentrate on the

results. We refer the interested reader to the literature[26]. A brief description of

the metho d is provided in App endix B.

The result of such a quantization was a mo di ed soliton equation for the (quantum

corrected) eld C (x; t) [26]

2 2 (1)

@ C (x:t) @ C (x; t)+ M [C (x; t)] = 0 (27)

t x

with the notation

1 @

(G(x;x;t)G (x;x))

(n) (n) (n) n n

M = e U (z)j ; U d U =dz (28)

z=C(x;t)

Ab ove, U denotes the p otential of the original soliton Hamiltonian, and G(x; y ; t)

is a bilo cal eld that describ es quantum corrections due to the mo di ed b oson

(n)

eld around the soliton. The quantities M carry information ab out the quantum

corrections, and in this sense the ab ovescheme is more accurate than the WKB ap-

proximation [26]. The whole scheme may b e thought of as a mean- eld-approachto

quantum corrections to the soliton solutions. For the kink soliton (26) the quantum

corrections (27) have b een calculated explicitly in ref. [26], thereby providing us

with a concrete example of a large-scale quantum coherent state.

The ab ove results on a consistent quantization of the soliton solutions (26), derived

from a Hamiltonian function, nd a much more general and simpler application in

our Liouville approach [28 , 29 ]. To this end, we rst note that the structure of the

equation (24), which leads to (26), is generic for Liouville strings, with arbitrary

targets. If we view the Liouville mo de as a lo cal scale of the renormalization group

on the world-sheet [4, 30 ], one can easily show that for the coupling g of any non-

marginal deformation V of the -mo del , the following Liouville renormalization

group equation holds [31, 4]

C [g ] 25

i i i ij 2

g + Qg_ = = G @ C [g ] ; Q = + ::: (29)

where the dot denotes di erentiation with resp ect to the renormalization group Li-

ouville scale t, and ::: denote terms removable by rede nitions of the couplings

For a concise review of the formalism and relevant notation see App endix A.

i ij

g (renormalization scheme changes). The tensor G is an inverse metric in eld

space[12]. Notice in (29) the r^ole of the non-criticality of the string (Q 6=0)as

providing a source of friction [4] in the space of elds g . The non-vanishing renor-

malization group -functions play the r^ole of generalized forces. The functional C [g ]

is the Zamolo dchikov c-function, which is constucted out of a particular combination

of comp onents of the world-sheet stress tensor of the deformed -mo del [12, 4].

The existence of friction terms in (29) implies a statistical description of the

temp oral evolution of the system using classical density matrices (g ;t) [4]

@ = f; H g + G (30)

t PB ij

where p are conjugate momenta to g , and G / is a metric in the space

i ij i j

of elds fg g [12]. The non-Hamiltonian term in (30) leads to a violation of the

Liouville theorem (1) in the classical phase space fg ;p g, and constitutes the basis

for a mo di ed (dissipative) quantum-mechanical description of the system[4], up on

quantization.

In string theory, summation over world sheet surfaces will imply quantum uctu-

ations of the string target-space background elds g [4, 28 ]. Canonical quantization

in the space fg g can b e achieved, given that the necessary Helmholtz conditions

[25] can b e shown to hold in the string case [28]. The imp ortant feature of the

string-lo op corrected (quantum) conformal invariance conditions is that they can

b e derived from a target-space action [32 ], whichschematically can b e represented

as[25 , 28 ]

Z Z

S = dt( d g E (t; g ; g;_ g)+total der iv ativ es ;

j i i

E (t; g ; g;_ g) G (g + Qg_ + ) (31)

i ij

where the tensor G is a (quantum) metric[12] in theory space fg g. Itischarac-

terized by a sp eci c b ehaviour [29] under the action of the renormalization group

op erator D@ +_g@,

t i

DG = QG =+::: (32)

ij ij i j

where the ::: denote di eomorphism terms in g -space, that can b e removed byan

appropriate scheme choice[29].

In this way, a friction problem (29) can b e mapp ed non-trivially onto a canonically

quantized Hamiltonian system, in similar spirit to the solitonic p oint-like eld theory

discussed in section 3. The quantum version of (30) reads [4]

i i

@ ^ = i[^; H ]+ i G [^g ; ^] (33)

t ij

where the hat notation denotes quantum op erators, and appropriate quantum or-

dering is understo o d (see b elow). We note that the equation (33) implies that @

dep endes only on (t) and not on a particular decomp osition of (t) in the pro jec-

tions corresp onding to various results of a measurement pro cess. This automatically

implies the absence of faster-then-light signals during the evolution.

The analogy with the soliton case, discussed previously,goeseven further if we

recall [4] the fact that energy is conserved in average in our approach of time as

a renormalization scale. Indeed, it can b e shown that the temp oral change of the

energy functional of the string particle, H, ob eys the equation

@ H/D <(z; z); (0) >=0 (34)

where denotes the trace of the world-sheet stress tensor; the vanishing result is

due to the renormalizability of the -mo del on the world-sheet surface, which, thus,

replaces time-translation invariance in target space.

2 2

However, the quantum energy uctuations E [<< H >> (<< H >>) ]

are time-dep endent:

2 i j j i

@(E ) = i<< [ ;H] G >>=<< G >> (35)

t ji ji

i j

Using the fact that G is a renormalization-group invariant quantity,we can

express (35) in the form

2 2 i j 2

@ (E ) = << Q G >>= << Q @ C>>0 (36)

t ij t

We, thus, observe that, for Q > 0 (sup ercritical strings), the energy uctuations

decrease with time for unitary string -mo dels [12].

Before closing this section we wish to make an imp ortant remark concerning

quantum ordering in (33). The quantum ordering is chosen in sucha way so that

energy and probability conservation, and p ositivity of the density matrix are pre-

served in the quantum case. Taking into account that in our string case G =

j j

C g :::g , with the expansion co ecients appropriate vertex op erator cor-

ij :::j

n n

relation functions [4], it is straightforward to cast the the ab ove equation into a

Lindblad form [33]

y y

_ @ = i[; H ] fB B; g +2B B (37)

t t + t

where the `environment' op erators B , B are de ned appropriately as `squared ro ots'

j i

of the various partitions of the op erator G :::g . This form mayhave imp ortant

consequences in the case one considers a wave-function representation of the density

matrix. Indeed, as discussed in ref. [34], (37) implies a stochastic di usion equation

for a state vector, which has imp ortant consequences for the lo calization of the

wave-function in a quantum theory of measurement. We stress, however, that our

approach based on density matrices (37) and renormalizability of the string -mo del

is more general than any approach assuming state vectors.

4 Quantum Gravity and Breakdown of Coherence

in the String Picture of a Microtubule

Ab ovewehave established the conditions under which a large-scale coherent state

app ears in the MT network, which can b e considered resp onsible for loss-free energy

transfer along the tubulins. As a result of conformational (quantum) transitions

of the tubulin dimers, there is an abrupt distortion of space-time. Formally this is

expressed by coupling the c = 1 string theory to two-dimensional quantum gravity.

This elevates the matter-gravity system to a critical c = 26 theory. Such a coupling,

then, causes decoherence, due to induced instabilities of the kink quantum-coherent

`preconscious state', in a way that we shall discuss b elow. As the required collapse

time of O (1 sec) of the wave function of the coherent MT network is several orders

of magnitude bigger than the energy transfer time t (15), the two mechanisms

are compatible with each other. Energy is transfered during the quantum-coherent

preconscious state, in 10 sec, and then collapse o ccurs to a certain (classical) con-

formational con guration. In this way,Frohlich's frequency asso ciated with coherent

`phonons' in biological cells is recovered, but in a rather di erent setting.

Wenow pro ceed to describ e the precise mechanism for the breakdown of coher-

ence, once the system couples to quantum gravity. First, we discuss an explicit

way of dynamical creation of black holes in two-dimensional string theory [35, 36 ]

through collapse of tachyonic matter T (x; t). This is a pro cedure that can happ en

as a result of quantum uctuations of various excitations. Consider the equations

of motion for the graviton and dilaton elds obtained by imp osing conformal invari-

, ignoring the non-universal tachyon terms in the ance in the mo del (20) to order

p otential. It can b e shown that a generic solution for the graviton deformation has

the form [35]

Z Z Z

x t t

2 0 2 2 0 2 0 2

_ _ _

0 0 0

ds = f1+ dx [(@ T ) +(T ) ] dt T@ T gdt + f1+ dt T@ T gdx (38)

x x x

1 1 1

Consider now an incoming lo calized wavepacket of the form (a const )

T = e (39)

cosh[2(x + t)]

It b ecomes clear from (38) that there will b e an horizon, obtained as a solution of

the equation derived by imp osing the vanishing of the co ecient of the dt term .

Thus, for late times t !1, the resulting metric con guration is a two-dimensional

static black hole [36 ]

4 4

2 2x 2 2 2x 1 2 2

a e )dt +(1 a e ) dx (40) ds = (1

3 3

and the whole pro cess describ es a dynamical collapse of matter. The energy of the

collapsing wavepacket gives rise to the ADM mass a of the black hole [36 ]. It is

crucial for the argument that there is no part of the wave-packet re ected. Otherwise

the resulting ADM mass of the black hole will b e the part of the energy that was

not re ected. In realistic situations, the black hole is only virtual, since low-energy

matter pulses are always re ected in two-dimensional string theories, as suggested

by matrix-mo del computations [37]. Indeed, if one discusses pulses which undergo

total re ection [35],

a a

T (x; t)= e (x; t) ; (x; t)= + (41)

cosh(2(x + t)) cosh(2(x t))

then, it can b e easily shown that there is a transitory p erio d where the space time

geometry lo oks like a black hole (40), but asymptotically in time one recovers the

linear dilaton ( at) vacuum[21]. The ab ove example (41) gives a generic wayof

a (virtual) dynamical matter collapse in a two-dimensional stringy space-time, of

the typ e that we encounter in our mo del for the brain, as a result of quantum

conformational changes of the dimers. In the case of MT, the replacement x ! ,

where is the b o osted co ordinate in (12), is understo o d.

The imp ortant p oint to notice is that the system of T ( ) coupled to a black hole

space-time (40), even if the latter is a virtual con guration, it cannot b e critical

(conformal invariant) non-perturbatively if the tachyon has a travelling wave form.

The factorisation prop erty of the world-sheet action (20) in the at space-time case

breaks down due to the non-trivial graviton structure (40). Then a travelling wave

cannot b e compatible with conformal invariance, and renormalization scale dressing

app ears necessary. The gravity-matter system is viewed as a c = 26 string [36 , 4 ],

and hence the renormalization scale is time-like [21]. This implies time dep endence

in T (; t).

A natural question arises whether there exist a deformation that turns on the

coupling T ( ) which is exactly marginal so as to maintain conformal invariance. To

answer this question, we rst note that there is an exact conformal mo del [36 ], a

Wess-Zumino SL(2;R)=U (1) coset theory, whose target space has the metric (40).

The exaclty marginal deformation of this black hole background that turns on mat-

ter, couples necessarily the propagating tachyon T ( ) zero mo des to an in nityof

higher-level string states [38 ]. The latter are classi ed according to discrete repre-

sentations of the SL(2;R) isospin, and together with the propagating mo des, form

a target-space W -algebra [4, 11 ]. This coupling of massive and massless mo des is

due to the non-vanishing Op erator Pro duct Expansion (O.P.E.) among the vertex

op erators of the SL(2;R)=U (1) theory [38]. The mo del is completely integrable, due

to an in nity of conserved charges in target space [4] corresp onding to the Cartan

subalgebra of the in nite-dimensional W [11]. This integrability p ersists quanti-

zation [39 ], and it is very imp ortant for the quantum coherence of the string black

hole space-time[4]. Due to the sp eci c nature of the W symmetries, there is no 1

information loss during a stringy black hole decay, the latter b eing asso ciated with

instabilities induced by higher-genus e ects on the world-sheet [40 , 4]. The phase-

space volume of the e ective eld theory is preserved in time, only if the in nite

set of the global string mo des is taken into account. This is due to the string-level

mixing prop erty of the W - symmetries of the target space.

However, any lo cal op eration of measurement, based on lo cal scattering of propa-

gating matter, such as the functions p erformed by the human brain, will necessarily

break this coherence, due to the truncation of the string deformation sp ectrum to

the lo calized propagating mo des T ( ). The latter will, then, constitute a subsystem

in interaction with an environment of global string mo des. The quantum integra-

bility of the full string system is crucial in providing the necessary couplings. This

breaking of coherence results in an arrow of time/Liouville scale, in the way ex-

plained brie y ab ove [4]. The black-hole -mo del is viewed as a c = 26 critical

string, while the travelling wave background is a non-conformal deformation. To

restore criticality one has to dress T ( ) with a Liouville time dep endence T (; t)

[4]. From a -mo del p oint of view, to O ( ), a non-trivial consistency check of this

approach for the black hole mo del of ref. [36 ] has b een provided in ref. [4]. We

stress once more that the Liouville renormalization scale now is time-like, in contrast

with the previous string picture of a c = 1 matter string theory, representing the

displacement eld u alone b efore coupling to gravity.

By viewing the time t as a lo cal scale on the world sheet, a natural identi cation

of t will b e with the logarithm of the area A of the world sheet, at a xed top ology.

As the non-critical string runs towards the infrared xed p oint the area expands. In

our approach to Liouville time [4], the actual ow of time is opp osite to the world

sheet renormalization group ow. This is favoured by a b ounce interpretation of the

Liouville ow due to sp eci c regularization prop erties of Liouville correlation func-

tions [4, 41 ]. This imlplies that wemay set t /lnA, with A owing always towards

the infrared A !1. In this way,a Time Arrow is implemented automatically in

our approach, without requiring the imp osition of time-asymmetric b oundary con-

ditions in the analogue of the Hartle-Hawking state. In this resp ect, our theory has

many similarities to mo dels of conventional dissipative systems, whose mathematical

formalism [42, 43] nds a natural application to our case.

With this in mind, one can examine the prop erties of the correlation functions of

V , A =, and hence the issue of coherence breakdown. In critical

i N i i

string theory such correlators corresp ond to scattering amplitudes in the target-

space theory. It is, therefore, essential to check on this interpretation in the present

situation. Since the correlators are mathematically formulated on xed area A

world-sheets, through the so-called xed area constraint formalism [44 , 45], it is

interesting to lo ok for p ossible A-dep endences in their evolution. In such a case

their interpretation as target-space scattering amplitudes would fail. Indeed, it

has b een shown [29] that there is an induced target space A-dep endence of the

regularized correlator A , which, therefore, cannot b e identi ed with a target space

S -matrix element, as was the case of critical strings [46]. Instead, one has non-

factorisable contributions to a sup erscattering amplitude S= 6= SS , as is usually the

case in open quantum mechanical systems, where the fundamental building blo cks

are density matrices and not pure quantum states[5 ]. For completeness, we describ e

in App endix A some formal asp ects of this situation, based on results of our approach

to non-critical strings [4].

It is this sort of coherence breakdown that we advo cate as happ ening inside the

part of the brain related to consciousness, whose op eration is describ ed by the dy-

namics of (the quantum version of ) the mo del (5). The e ectivetwo-dimensional

substructures, that wehave identi ed ab ove as the basic elements for the energy

transfer in MT, provide the necessary framework for coupling the (integrable) stringy

black hole space-time (40) to the displacement eld u(x; t). This allows for a qual-

itative description of the e ects of quantum gravity on the coherent sup erp osition

of the preconcious states.

One can calculate in this approach the o -diagonal elements of the density matrix

i i

in the string theory space u , with now u (t) representing the displacement eld of

the i-th dimer. In a -mo del representation (20), this is the tachyon deformation.

The computation pro ceeds analogously [4] to the Feynman-Vernon [16 ] and Caldeira

and Leggett [17 ] mo del of environmental oscillators, using the in uence functional

metho d, generalized prop erly to the string theory space u . The general theory

of time as a world-sheet scale predicts [4] the following expression for the reduced

density matrix[16, 17] of the observable states:

R R

0 2

t +

(u( )u( ))

N d d

0 2

0 ( )

(u ;u ;t)= (u ;u ;t) ' e '

I F S I F

R R

0 i j

^ ^

N d d G (S )

DN t(u u ) +:::

ij 0

I F

0 '

e ' e (42)

where the subscript \S" denotes quantities evaluated in conventional Schrodinger

quantum mechanics, and N is the numb er of the environment `atoms'[6]interacting

with the background u .For (1; 1) op erators, that we are interested in, the structure

i i ui ui 2

^ ^

of the renormalization group -functions is = u + , where = O (u ) and

! 0 is the anomalous dimension. Recalling [4] the p ole structure of the Zamolo d-

(1)

chikov metric, G = G + regular, one nds that the dominant contribution to

and the p ole term the exp onent K of the mo del (42) comes from the -term in

in G :

Z Z

t t

(1)

i j i uj

^ ^

K = N G d 3 2 u G d + O () (43)

0 0

Throughout this work, as well as our previous works on the sub ject [4], we assume that such

a space exists, and admits [12] a metric G . Indications that this is true are obtained from

p erturbative calculations in -mo del, to whichwe base our b elief.

i i

Assuming slowly varying u and over the time t, this implies that the o -

diagonal elements of the density matrix would decay exp onentially to zero, within

a collapse time of order [6]

(1)

u j 1

t = (O [ G u ]) t (44)

coll s

in fundamental string units t of time. The sup erscript (1) in the theory space

metric denotes the single residue in, say, dimensional regularization on the world

sheet [4, 30]; N is the numb er of (coherent) tubulin dimers in interaction with

the given dimer that undergo es the abrupt conformational change. Here the

2 2

-function is assumed to admit a p erturbative expansion in p owers of @ , in target

s X

space, where the fundamental string unit of length is de ned as

=( ) (45)

where v is the sound velo city (10), of order 1k m=sec [19 ]. Wework in a system

of units where the lightvelo cityis c = 1, and we use as the scale where quantum

gravity e ects b ecome imp ortant, the grand uni ed string scale, M =10 GeV ,

gus

or in length 10 cm, which is 10 times the conventional Planck scale. This is so,

b ecause our mo del is supp osed to b e an e ective description of quantum gravity

e ects in a stringy (and not p oint-like) space-time. This scale corresp onds to a time

scale of t =10 sec.Wenow observe that, to leading order in the p erturbative

gus

function expansion in (44), any dep endence on the velo city v disapp ears in favour

of the scales M and t . It is, then, straightforward to obtain a rough estimate

gus gus

for the collapse time

gus

t = O [ ] (46)

col

E N

where E is a typical energy scale in the problem. Thus, we estimate that a collapse

time of O (1 sec) is compatible with a numb er of coherent tubulins of order

N ' 10 (47)

provided that the energy stored in the kink background is of the order of eV . This

is indeed the case of the (dominant) sum of binding and resonant transfer energies

' 1eV (16) at ro om temp erature in the phenomenological mo del of ref. [19 ].

This numb er of tubulin dimers corresp onds to a fraction of 10 of the total brain,

which is pretty close to the fraction b elieved to b e resp onsible for human p erception

on the basis of completely indep endent biological metho ds.

An indep endent estimate for the collapse time t , can b e given on the basis of

col

p oint-like quantum gravity quantum gravity theory, assuming that the latter exists,

either per se, or as an approximation to some string theory. One incorp orates

quantum gravity e ects by employing wormholes [47] in the structure of space time, and then applies the calculus of ref. [6] to infer the estimates of the collapse time.

In that case, one evaluates the o -diagonal elements of the density matrix in real

con guration space x, which should b e compared to that in string theory space

(42). The result of ref. [6] for the time of collapse induced by the interaction with

a `measuring apparatus' with N units is

1 1

0 3

t ' (M =m) (48)

gus

coll

3 2

N m (x)

where m isatypical mass unit in the problem. The fundamental unit of velo cities

in (48) is provided by the velo city of light c, since the formula (48) refers to generic

four-dimensional space-time e ects. In the case of the tubulin dimers, it is reasonable

to assume that the p ertinentmoving mass is the e ective mass M (17) of the kink

background (12). This will make contact with the microscopic mo del ab ove. Tobe

sp eci c, (17) gives M ' 3m , where m is the proton mass. This makes plausible

p p

the rather daring assumption that the nucleons (protons, neutrons) themselves inside

the protein dimers are the most sensitive constituents to the e ects of quantum

gravity. This is a reasonable assumption if one takes into account that the nucleons

are much heavier than the (conformational) electrons. If true, this would really

imply, then, that elementary particle scales come into play in brain functioning.

In this picture, then, N is the numb er of tubulin dimers, and moreover in our case

x = O[4nm], since the relevant displacement length in the problem is of order of the

longitudinal dimension of each conformational p o cket in the tubulin dimer. Thus,

substituting these in (48) one derives the result N = N ' 10 , for the number of

coherent dimers that induce a collapse within O (1 sec).

It is remarkable that the nal numb ers agree b etween these two estimates. If one

takes into account the distant metho ds involved in the derivation of the collapse

times in the resp ective approaches, then one realizes that this agreement cannot b e

a coincidence. Our b elief is that it re ects the fundamental r^ole of quantum gravity

in the brain function.

At this stage, it is imp ortant to make some clarifying remarks concerning the kind

of collapse that we advo cate in the framework of non-critical string theory. In the

usual mo del of collapse due to quantum gravity [6], one obtains an estimate of the

collapse of the o -diagonal elements of the density matrix in con guration space, but

no information is given for the diagonal elements. In the string theory framework

of ref. [4] the collapse (42) also refers to o -diagonal elements of the string density

matrix, but in this case the con guration space is the string background eld space.

In this case the o -diagonal element collapse suces, b ecause it implies localization

in string background space, which means that the quantum string cho oses to settle

down in one of the classical backgrounds, which in the case at hand is the solitonic

background discussed in section 2.

Of course, the imp ortant question `which sp eci c background is selected by the

ab ove pro cess ?' cannot b e answered unless a full string eld theory dynamics is

develop ed. However, we b elieve that our approach [4] of viewing the selection of a

critical string ground state as a generalized `measurement' pro cess in string theory

space might prove advantageous over other dynamical metho ds in this resp ect.

5 Growth (Dynamical Instability) of a Microtubule

Network and Liouville Theory

The ab ove considerations are valid for MT networks whose size of individual MT

is larger than a certain critical size [19]. Kink-like excitations, that were argued to b e

crucial for the physics of the conscious functions of the brain, cannot form for small

microtubules. The question, therefore, arises whether the non-critical e ctive string

theory framework describ ed ab ove is adequate for describing the growth pro cess

asso ciated with the formation of a MT network. This phenomenon is physically and

biologically very interesting since these structures are known to b e the only ones

so far that exhibit the so-called `dynamical instability growth' [48]. This is an out-

of-equilibrium pro cess acording to which an individual MT can switch randomly

between an `assembly state' (+), in which the MT grows with a sp eed v , and

`diassembly state' (-), in which the MT shrinks with velo city v . Recently there

have b een attempts to construct simple one-dimensional theoretical mo dels with

di usion [49 ] that can describ e qualitatively the ab ove phenomenon. An interesting

feature of these mo dels, relevant to our framework, is that for a certain range of

their parameters exhibit a phase transition to an unb ounded growth state . In the

case of MT networks, it is known exp erimentally that a `sawto oth' b ehaviour in the

time-dep endence of the size of a MT (Fig. 3), o ccurs as a result of hydrolysis of

GTP nulceotides b ound to the tubulin proteins (i.e. the transformation GTP !

GDP )[48]. The hydrolysis is resp onsible for providing the necessary free energy

for the conformational changes of the tubulin dimers: the dynamical instability

phenomenon p ertains to p olymerization of the GTP tubulin, while the GDP tubulin

stays essentially unp olymerized. In view of quantum oscillations b etween the two

conformations of the tubulin, and the ab ove di erent b ehaviour of p olymerization,

it is natural to conjecture that quantum e ects may playa r^ole in the MT growth

pro cess.

This situation may also b e viewed from a sp ontaneous-symmetry breaking p oint of view along

the lines of ref. [50], which in one dimension can o ccur when the system is out of equilibrium. This

second p oint of view is more relevant to our non-critical string approach, whichaswe explained

earlier is an out-of-equilibrium pro cess. The symmetry breaking can b e exhibited easily by lo oking

at one-dimensional mo dels with driven di usion of saytwo sp ecies of particles corresp onding to

the (+) and (-) conformational states of the MT growth. In the broken phase there is a di erence

between the `currents' correpsonding to the (+) or (-) states. Under certain plaussible conditions

[50], asso ciated with formation of droplets of the `wrong sign' in any given con guration of the

ab ove states, the system can switchbetween (+) and (-) states with a switching time that dep ends

on dynamical parameters. For a certain range of these parameters the switching time is of order

e , where N is the size of the system, thereby implying sp ontaneous symmetry breaking in the

thermo dynamic limit where N !1.

Thus, our Liouville theory prepresentation of the e ective degrees of freedom u

involved in the mo del of [19], invented to explain classical asp ects of the hydrolysis of

GTP ! GDP, might, in principle, b e able of explaining qualitatively the `sawto oth'

b ehaviour of Fig. 3, even b efore the formation of kinks. To this end, we remark

that in Liouville dynamics, with the Liouville scale identi ed with the target time

[4], the inherent non-unitarity (in the world-sheet) of the Liouville mo de implies

that the central charge Q of the theory ows with the scale in sucha way that

near xed p oints it oscillates a bit b efore settling down. Indeed, for a non-critical

string with running central charge C [g; t], t is the Liouville scale/time, the following

second order equation (lo cal in target space-time) holds near a xed p ointofthe

Renormalization Group Flow [31 ]

C [g; t]+ Q[g; t]C[g; t] 0 forC 25 ; Q [g; t]= (C[g; t] 25) (49)

This is a lo cal phenomenon in target space-time. Globally, there is a preferred

direction in time along which the entropy of the system increases [12 , 4].

The small oscillations of C in (49) may b e attributed to the double direction of

Liouville time that arises as a result of imaginary parts (dynamical instabilities)

app earing due to world-sheet regularization by analytic continuation of non-critical

string correlation functions [41, 4] (Fig. 4) This p oint is discussed brie y in App endix

Along each direction of Liouville time in Fig. 4 there is an asso ciated variation

of Q[g; t] and C [g; t], and the volume of the `one-dimensional universe' of MT ei-

ther increases or decreases [28]. Thus, as a result of the oscillations of the Liouville

dynamics (49) a `sawto oth' b ehaviour of MT, which expresses the result of p oly-

merization of GTP tubulin, can b e qualitatively explained within the framework of

non-critical Liouville dynamics. The analogy of the ab ove situation to a sto chastic

expansion of the Universe (in ation) in non-critical string theory as discussed in ref.

[28] should b e p ointed out.

It should b e noted at this stage that in this e ective framework the origin of non-

criticality of the subsystem of tubulin dimers is left unsp eci ed. Quantum Gravity

uctuations app ear on an equal fo oting with the environment of the nucleating

solvant that sourrounds the MT in their physical environment. The distinction can

b e made once a detailed description of the environment is given, which of course

would sp ecify the form of the target metric coupled to the matter system of tubulins.

As wehave discussed in previous sections, in the quantum Gravity case this is

achieved by the exactly marginal op erators of the SL(2;R)=U (1) conformal eld

theory that describ es space-time singularities in one-dimensional strings [36]. The

latter involve global (non-propagating) string mo des which cannot b e detected by

lo calized scattering exp eriments [4]. On the other hand, it seems likely that an exact

conformal eld theory that describ es the nucleating solvant do es not exist. However,

the e ective Liouville string that describ es the emb edding of a MT in it, and the

asso ciated environmental entanglement, is obtained by a simple Liouvlle dressing of

the mo del discussed in section 3. The information ab out the environment is hidden

in the form of the target metric that couples to the system. The fact that both the

formation of an MT via the dynamical instability phenomenon, and the quantum

gravity e ects on MT, involve conformational changes of the tubulin supp orts the

ab ove p oint of view. Of course the strength of the dynamical instability in case

the latter is due to quantum gravity uctuations will b e much more suppressed as

compared to the hydrolysis case. In that case, there will not b e sucient time for

complete p olymerization of the GTP tubulin conformation. In such a case one would

probably exp ect `sudden' `sawto oth' p eaks of the tip of the MT whose magnitude

will b e a ected by the order of quantum gravityentanglement. The growth in this

case will b e b ounded.

It should b e mentioned that recently there have b een some exp eriments claim-

ing such length uctuations [51], in the case of carb on nanotub es. The authors of

ref. [51] claimed that they observed suchchanges in the length of the tub es, which

they attributed to sudden jumps of the resp ectivewavefunctions, according to the

approach of Ghirardi Rimini and Web er [52]. In this resp ect, one should think of

rep eating the tests but with isolated MT[53]. We should stress, however, that the

ab ove discussion is at this stage highly sp eculative and even controversial, given

that there app ear to b e conventional explanations of this phenomenon in carb on

nanotub es[54 ]. Of course such conventional explanations do not exclude the p os-

sibility of future observations of quantum-gravity induced b ounded growth in MT,

along the lines sketched ab ove.

We close this section by mentioning that in realistic situations the growth pro cess

of an MT network is not unlimited. After a critical length is exceeded, the growth

is saturated and eventually stops. The formation of kink excitations (12) mightbe

imp ortant for this auto-regulation of the MT growth [19]. For more details on the

conjectural r^ole of the kinks in this growth-control mechanism we refer the reader

to the literature [19, 49 ].

The ab ove situation should b e compared with the limitations on the (sto chastic)

Universe expansion in the non-critical-string-driven in ationary scenario of ref. [28 ].

There, it can b e shown that within our framework of identifying the Liouville eld

with the target time, the average density << >> Tr( ) of the non-critical

s s

strings that drive the in ationary scenario ob ey an equation of the form [55, 28 ]

@ << >>= aQ << >> +bQ << >> (50)

t s s s

where ,b are p ositive quantities, computable in principle in the Liouville-string

framework [28]. The rst term in (50) is due to the (exp onential) expansion of

the string-Universe volume, and the second term corresp onds to the regeneration of

strings via breaking of large strings whose size exceeds that of the Hubble horizon

[28]. This second term comes from the di usion due to the non-quantum mechanical

terms in the equation (33). At early times the di usion term balances the string

depletion e ects of the rst term and the uniform density condition for in ation

(universe exp onential expansion) is sats ed. As the time elapses, however, the

depletion term in (50) dominates, the Universe's expansion is diminished gradually,

and eventually stops. This is the case when the non-equilibrium non-critical string

approaches its (critical-string ) equilibrium state. Hence, in our case one may view

(50) as an e ective mo del for the temp oral evolution of the density of tubulin dimers.

Then, one can understand, at least qualitatively, the ab ove limitations of the MT

growth pro cess by the presence of the kinks, since the latter can b e asso ciated with

an equilibrium ground state of the e ective string theory describing a MT.

6 Conclusions

Wehave presented an e ective mo del for the simulation of the dynamics of the

tubulin dimers in the brain. Wehave used an e ective (1 + 1)-dimensional string

representation to study the dynamics of a detailed mechanism for energy transfer in

the biological cells. We argued how it can give rise to a large-scale coherent state in

the dimer lattice. Such a state is obtained from quantization of kink solitonic states

that transfer energy through the cell without dissipation. The quantization b ecame

p ossible through the freedom that string theory o ers, enabling one to cast dynam-

ical problems with friction in a Hamiltonian form. The collapse phenomenon in our

approach do es not require the existence of a wave-fucntion, and it is induced by the

formation of microscopic black holes (singularities) in the e ective one-dimensional

space-time of the tubulin chains. This is achieved by the dynamical collapse of pulses

of the displacenent eld of the MT dimers. The pulses are a result of abrupt confor-

mational changes ( $ ) that suciently distort the surrounding space-time. In

this sense, the situation is similar but not identical, to the `sudden hits' that a par-

ticle's wave function su ers o ccasionally (every 10 years) in the mo del of quantum

measurement of Girardi, Rimini and Web er [52 ]. However, contrary to these con-

ventional theories, our stringy approach to gravity-induced collapse [4] incorp orates

automatically an irreversible ow of time for sp eci cally stringy reasons, and energy

conservation. When applied to the mo del of MT, our approach implies a collapse

time of O (1 sec ), which is obtained by the interaction of a tubulin dimer with a

fraction of 10 of the total numb er of tubulin dimers in the brain. This number is

fairly close to the fraction of the brain that neuroscientists b elieve resp onsible for

human p erception. This is a very strong indication that the ab ove ideas, although

sp eculative at this stage, might b e relevant for the discovery of a physical mo del

for consciousness and its relation to the irreversible ow of time. In addition, our

mo del predicts damp ed (microscopic) quantum-gravity-induced oscillations of the

length of isolated MTs, which are due to the di erent prop erties of the two tubulin

conformations under p olymerization (phenomenon of b ounded dynamical-instability

growth).

There are certain formal asp ects of our e ective mo del, namely its two-dimensional

structure and its complete (quantum) integrability, that might turn out to b e imp or-

tant features for the construction of realistic soluble mo dels for brain function. The

quantum integrability is due to generalized in nite-dimensional symmetry struc-

tures (W -symmetries and its generalizations) which are strongly linked with issues

of quantum coherence and unitary evolution in phase-space. Such symmetries are

related to global non-propagating mo des of the e ective string theory, which do not

decouple from the propagating (observed) mo des in the presence of (microscopic)

space-time singularities. It will b e interesting to understand further the physical r^ole

of such structures in the mo dels of MT. At present they app ear as an environmentof

fundamental string mo des that are physical at Planck scales. However, such struc-

tures, may admit a less-ambitious physical meaning, asso ciated with fundamental

biological structures of the brain. We should stress that the entire picture of non-

critical string wehave describ ed ab ove, which is a mo del-indep endent picture as far

as environmental op erators are concerned, could still apply in such cases, but simply

describing purely biological environmental entanglement of the conscious part of the

brain, the latter b eing describ ed as a completely integrable mo del. We do hop e to

come back to these issues in the near future.

Acknowledgements

It is a pleasure to acknowledge discussions with J. Ellis. We also acknowledge

useful discussions and corresp ondence with S. Hamero . N.E.M. acknowledges in-

formative communications with H. Rosu on quantum jumps and exp erimental ob-

servations of dynamical instabilities in nanotub es. The work of N.E.M. is supp orted

by a EC ResearchFellowship, Prop osal Nr. ERB4001GT922259 . That of D.V. N.

is partially supp orted by D.O.E. Grant DEFG05-91-GR-40 63 3.

App endix A

Extracts from Non-Critical (Liouville) String Theory and Time as the

Liouville scale

In this subsection we shall comment on the non-factorizability of the induced

target-space S= -matrix for matter scattering in a non-conformal string background.

This re ects information leakage as a result of the non-critical character of the

string. Although for our purp oses primarily we shall b e interested in a sp eci c string

background, that of a stringy black hole, however in this section our discussion will

be kept as general as p ossible with the aim of demonstrating the generality of our

scheme.

Consider a conformal eld theory on a two-dimensional world sheet, describ ed

by an action S [g ]. The fg g are a set of space-time backgrounds. The theory is

p erturb ed by a deformation V , which is not conformal invariant

2 i

S [g ]= S[g ]+ d zg V (51)

The couplings g corrsp ond to world-sheet renormalization group -functions

i i i i j j k k

=(h 2)(g (g ) )+c (g (g ) )(g (g ) )+::: (52)

expressing the scale dep endence of the non-conformal deformations. The op erator

pro duct expansion co ecients are de ned as usual by coincident limits in the pro duct

of twovertex op erators V

lim V ( )V (0) ' c V ( )+::: (53)

!0 i j i

where the completeness of the set fV g is assumed.

Coupling the theory (51) to two-dimensional quantum gravity restores the confor-

mal invariance at a quantum level, by making the gravitationally-dressed op erators

[V ] exactly marginal, i.e. ensuring the absence of anycovariant scale dep endence

with resp ect to the world-sheet metric . Belowwe simply outline the basic results,

used in our approach here.

One rescales the world-sheet metric

= e (54)

with is kept xed, and then one integrates over the Liouville mo de . The measure

of suchanintegration[44] can b e expressed in terms of the ducial metric by means

of a determinant which is the exp onential of the Liouville action. The nal result

for the gravitationally-dressed matter theory is then

2 (2)

d z@ @ QR + ()V (55) S = S [g ]+

i Lm

where is the Regge slop e for the world-sheet theory (inverse of the string tension).

The gravitational dressing of the op erators follows from the requirement of restoring

the conformal invariance of the theory,atany given order in the coupling-constant

expansion. For instance, to order O (g ) the gravitationally-dressed coupling ()

are given by[44,56]:

i i j k

i i

()=g e + c g g e + ::: (56)

with

j25 cj

Q = ; + Q = sq n(25 c)(h 2) (57)

i i

and c is the (constant) central charge of the non-critical string. From the quadratic

equation for only the solution

Q Q

= + (h 2) (58)

i i

2 4

for c 25, is kept due to the Liouville b oundary conditions.

In ref. [4] we made an extra assumption, as compared to the ab ove standard

Liouville dynamics. We identi ed the eld with a dynamical lo cal scale on the

world sheet. This induces extra counterterms in the world-sheet renormalized action.

Consistency of the scheme required that the Liouville functions are identical with

i i

the at space renormalization co ecients up on the replacement g ! () .

The typ e of op erators that we are interested in this work, are such that h = 2 but

c 6= 0. In the language of conformal eld theory this means that these op erators

are (1; 1) but not exactly marginal.From (56), then, one obtains the simple relation

d()

(59) =

where the time t is related to the Liouville mo de as

1 Q 1

(z;z) 2

t = e^ ; = Q + 8 (60) lnA ; A d z +

Q 2 2

with A the world-sheet area. In the lo cal scale formalism of ref. [4] Q is given by

j25 C [g; ]j 1

i j

Q = + G (61)

3 2

where C [g; ] is the Zamolo dchikov C -function [12 ], which reduces to the central

charge c at a xed p oint of the ow. The extra terms in (61), as compared to (57),

are due to the lo cal character of the renormalization group scale[4]. Such terms

may always b e removed by non-standard rede nitions of C [g; ]. The quantity

G is related to divergencies of the two-p oint functions and hence to

ij i j

Zamolo dchikov metric [12, 4].

From the renormalization-group structure (56) one obtains close to a xed p oint

[31]

i i ij

() + Q = = G @ C [; ] (62)

where the dot denotes di erentiation with repsect to the Liouville lo cal scale .

For the C [g; ] (lo cal in target space-time) one obtains near a xed p oint

C [g; t]+ Q[g; t]C[g; t] 0 forC 25 ; Q [g; t]= (C[g; t] 25) (63)

The small oscillations of C [; ], b efore it settels down to a xed p oint, are due

to the `non-unitary' world-sheet contributions of the Liouville mo de ;however

globally in target space-time there is a monotonic change of the degrees of freedom

of the system, as discussed in detail in [4].

These considerations can b e understo o d more easily if one lo oks at the correlation

functions in the Liouviulle theory, viewing the liouville eld as a lo cal scale on the

world sheet . Standard computations[57] yield for an N -p oint correlation function

among world-sheet integrated vertex op erators V d zV (z; z):

i i

s 2 s

~ ~

A =(s) <( d z e^ ) V :::V > (64)

N i i i i =0

1 1

N N

where the tilde denotes removal of the Liouville eld zero mo de, which has b een

path-integrated out in (64). The world-sheet scale is asso ciated with cosmological

constant terms on the world sheet, which are characteristic of the Liouville the-

ory [44 ]. The quantity s is the sum of the Liouville anomalous dimensions of the

op erators V

Q Q 1

s = ; = + Q +8 (65)

2 2

i=1

The function can b e regularized[41, 4] (for negative-integer values of its argument)

by analytic coninuation to the complex-area plane using the the Saaschultz contour

of Fig. 4. This yields the p ossibility of an increase of the running central charge

due to the induced oscillations of the dynamical world sheet area (related to the

Liouville zero mo de). This is asso ciated with the oscillatory solution (63) for the

Liouville central charge. On the other hand, the b ounce intepretation of the infrared

xed p oints of the ow, given in refs. [41, 4], provides an alternative picture of the

overall monotonic change at a global level in target space-time.

The ab ove formalism also allows for an explicit demonstration of the non-factorizability

of the sup erscattering matrix asso ciated with target-space interactions in non-critical

string theory. This was very imp ortant for our purp oses in the context of the col-

lapse of the wave-function as a result of quantum entanglement due to quantum

gravity uctuations.

To this end, one expands the Liouville eld in (normalized) eigenfunctions f g

of the Laplacian on the world sheet

X X

(66) / A (z; z)= c =c +

n 0 n n 0 0

n6=0

with A the world-sheet area, and

= n =0;1;2;::: ( ; )= (67)

n n n n m nm

The result for the correlation functions (without the Liouville zero mo de) app earing

on the right-hand-side of eq. (64) is, then

X X

1 Q

A / dc exp( c R c +

N n6=0 n n n n

8 8

n6=0 n6=0

n n s

n6=0

e^ ) (68)

2 (2)

with R = d R ( ) .We can compute (68) if we analytically continue [57] s

n n

to a p ositiveinteger s ! n 2 Z . Denoting

(x) (y )

n m

f (x; y ) (69)

n;m 6=0

one observes that, as a result of the lack of the zero mo de,

(2)

(70) f (x; y )=4 (x; y )

Wemaycho ose the gauge condition d ^ = 0. This determines the conformal

prop erties of the function f as well as its `renormalized' lo cal limit[58]

f (x; x)= lim (f (x; y )+lnd (x; y )) (71)

R x!y

where d (x; y ) is the geo desic distance on the world sheet. Integrating over c one

obtains

exp[ f (z ;z )+

i j i j

i;j

ZZ Z

R(x)R(y)f(x; y ) R^ (x)f (x; z )] (72)

i i

Wenow consider in nitesimal Weyl shifts of the world-sheet metric, (x; y ) !

(x; y )(1 (x; y )), with x; y denoting world-sheet co ordinates. Under these, the

correlator A transforms as follows[29]

^ ~

dx ^ R (x)+ A / [ h (z )+

N i i

Z Z

q q

fQs dx ^ (x)+ (s) dx ^ (x)f (x; x)+

ZZ Z

q q

2 2 2

^ ^

Qs dxd y R^ (x)(y)G(x; y ) s ^ (x)G(x; z ) d x

i i

s f (z ;z ) d x ^ (x)

i R i i

2 2

^ ~

dxd y ^ (x)^ (y )R(x)f (x; x) (y )g]A (73)

R N

where the hat notation denotes transformed quantities, and the function G (x,y) is

de ned as

G (z; !) f (z; !) (f (z; z)+ f (!; !)) (74)

R R

and transforms simply under Weyl shifts[58]. We observe from (73) that if the sum of

the anomalous dimensions s 6= 0 (`o -shell' e ect of non-critical strings), then there

are non-covariant terms in (73), inversely prop ortional to the nite-size world-sheet

area A. In general, this is a feature of non-critical strings wherever the Liouville

mo de is viewed as a lo cal scale of the world sheet. In such a case, the central charge

of the theory ows continuously with time/scale t, as a result of the Zamolo dchikov

c-theorem [12]. In contrast, the screening op erators yield quantized values[21]. This

induced time (A-) dep endence of the correlation function A implies the breakdown

of their interpretation as factorisable S= -matrix elements.

In our framework, the e ects of the quantum-gravityentanglement induce such

A-dep endences in correlation functions of the propagating matter vertex op erators

of the string [4], corresp ondng to the displacement eld u(x; t) of the MTs. To this

end, we rst note that the physical states in such completely integrable mo dels fall

into representations of the SL(2;R) target symmetry, which are classi ed by the

non-compact isospin j and its third comp onent m[59]. There is a formal equivalence

of the physical states b etween the at-space time (1 + 1)-dimensional string and

the black-hole mo del , which con rms the p oint of view[62, 4 ] that the at-space

(1 + 1)-dimensional string theory is the spatially- (and temp orally-) asymptotic

limit of the SL(2;R)=U (1) black hole. The existence of discrete (quasi-top ological,

non-propagating) Planckian mo des in the two-dimensional string theory leads to

selection rules[62] in the number N of the scattered propagating degrees of freedom,

according to the intermediate-exchange state :

Liouv il l e ener g y (momentum)):

(76)

2 2

N 2

p = ; N 3

(77)

Such rules are obtained by imp osing the Liouville energy and momentum p con-

servation, leading to s = 0, with s the sum of Liouville anomalous dimensions as

de ned earlier. Obviously, if the exchange state is an (o -shell) propagating mo de,

Originally, there were claims [59] that there are extra states in the black hole mo dels, as

compared to the at-space time string; however, later on it has b een shown that such states can

b e either gauged away[60] or b o osted[61], and so they disapp ear from the physical sp ectrum.

b elonging to the continuous representation of SL(2;R), i.e. j 2 R, j , there are

no restrictions on N , and a convetnional S -matrix amplitude can b e de ned as the

residue of the Liouville amplitudes with resp ect to the single p oles in s [63]. How-

ever, in two space-time dimensions graviton excitations are discrete, corresp onding

to string-level-one representations of SL(2;R). Hence, once non-trivial quantum-

gravity uctuations are considered in our approach, whichintwo dimensions are

black-hole backgrounds (40), one has to takeinto account discrete on-shell exchange

mo des in the Liouville correlation functions . Such states represent excited states of

the (virtual) black-holes, created by the collapse of the propagating matter mo des

u(x; t), as describ ed in section 4 (40). In two-dimensional string theory black holes

are like particles[4], the di erence b eing their top ological nature. Such mo des consti-

tute, in our case, `the consciousness degrees of freedom', which cannot b e measured

by lo cal scattering exp eriments. Integrating them out in the `mind', implies a time

arrow as describ ed in ref. [4]. Indeed, in the correlation functions (64), as a result

of Liouville energy conservation (77), one of the mo des is necessarily discrete. If

we suppress such mo des, and consider only external propagating mo des, accessi-

ble to physical scattering pro cesses, then it is evident that s 6=0. According to

our previous analysis (73), then, this implies world-sheet-area(A) dep endence of the

correlation functions. In this picture, we also note that Quantum-Gravitational

uctuations of singular space-time form, corresp o ding to higher-genus world-sheet

e ects, have b een argued [4] to b e represented col lectively byworld-sheet instanton-

anti-instanton deformations in the stringy -mo del. It is known [64 ] that such

con gurations are resp onsible for a non-perturbative breakdown of the conformal

invariance of the -mo del. Using a dynamical (world-sheet) renormalization-group

scale (Liouville mo de) to represent all such non-conformal invariant e ects[4], and

identifying it with the target time, one, then, arrives at non-factorisable sup erscat-

tering op erators, as describ ed ab ove.

Notice that the precise microscopic nature of the environmental op erators is not

essential as long as the latter imply a conformal anomaly. There are general conse-

quences of this conformal anomaly, including dynamical collapse of the string theory

space to a certain con guration, as discused in section 4. A similar situation o ccurs

in ordinary quantum mechanics of op en systems. Once a sto chastic framework using

state vectors is adopted [34] for the description of environmental e ects, there will

always b e lo calization of the state vector in one of its channels, irresp ective of the

detailed form of the environment op erators. It should b e noted that sto chasticity

is a crucial feature of our approach to o. This follows from the sto chastic nature of

the renormalization group in two-dimensions [65, 4]. This sto chasticitywas argued

to play an imp ortantr^ole in the MT growth, discussed in section 5.

The on-shell condition imp oses algebraic relations among j and m for such mo des, involving

the string-level numb er due to the Virasoro constraints [59]. This, in turn, implies restrictions to

the number N of scattered particles in such cases.

App endix B

Variational Approach to Soliton Quantization via Squeezed Coherent States

It is the purp ose of this app endix to discuss brie y the formalism leading to the

quantization of the solitonic states discussed in section 2.

One assumes the existence of a canonical second quantized formalism for the

(1 + 1)-dimensional scalar eld u(x; t), based on creation and annihilation op erators

a , a . One then constructs a squeezed vacuum state[26]

T (t)

j (t) >= N (t)e j0 > ; T (t)= dxdy u(x) (x; y ; t)u(y ) (78)

where j0 > is the ordinary vacuum state annihilated by a , and N (t) is a normaliza-

tion factor to b e determined. (x; y ; t) is a complex function, which can b e splitted

in real and imaginary parts as

1 1

[G (x; y ) G (x; y ; t)] + 2i(x; y ; t) (x; y ; t) =

G (x; y ) = < 0ju(x)u(y )j0 > (79)

The squeezed coherent state for this system can b e then de ned as[26 ]

iS (t)

j(t) > e j (t) > ; S (t)= dx[D (x; t)u(x) C (x; t) (x)] (80)

with (x) the momentum conjugate to u(x), and D (x; t), C (x; t) real functions.

With resp ect to this state (x; t) can b e considered as a momentum canonically

conjugate to G(x; y ; t) in the following sense

< (t)ji j(t) >= G(x; y ; t) (81)

(x; y ; t)

The quantity G(x; y ; t) represents the mo di ed b oson eld around the soliton.

To determine C ,D , and one applies the Time-Dep endentVariational Approach

(TDVA) [26] according to which

dt < (t)j(i@ H )j(t) >=0 (82)

where H is the canonical Hamiltonian of the system. This leads to a canonical set

of (quantum) Hamilton equations

H H

_ _

D (x; t) = C (x; t)=

C (x; t) D(x; t)

H H

_ _

G (x; y ; t) = (x; y ; t)= (83)

(x; y ; t) G(x; y ; t)

where the quantum energy functional H is given by[26]

H<(t)jH j(t) >= dxE (x) (84)

with

1 1

2 2 (0)

E (x)= D (x; t) + (@ C (x; t)) + M [C (x; t)] +

2 2

1 1

+ 2 + lim r r (85)

x!y x y

8 2

1 1

lim r r (86)

x!y x y 0

8 2

where we use the following op erator notation in co ordinate representation A(x; y ; t) <

xjA(t)jy>, and

1 @

(G(x;x;t)G (x;x))

(n) (n) (n) n n

M = e U (z)j ; U d U =dz (87)

z=C(x;t)

Ab ove, U denotes the p otential of the original soliton Hamiltonian, H . Notice

that the quantum energy functional is conserved in time, despite the various time

dep endences of the quantum uctuations. This is a consequence of the canonical

form (83) of the Hamilton equations.

Performing the functional derivations in (83) one can get

(1)

D (x; t) = C (x; t) M [C (x; t)]

C (x; t) = D (x; t) (88)

which after elimination of D (x; t), yields the mo di ed (quantum) soliton equation

(n)

(27). We note that the quantities M carry information ab out the quantum cor-

rections, and in this sense tha ab ovescheme is more accurate than the WKB ap-

proximation [27]. The whole scheme may b e thought of as a mean- eld-approachto

quantum corrections to the soliton solutions.

In our string framework, then, these p oint-like quantum solitons can b e viewed as

alow-energy approximation to some more general ground state solutions of a non-

critical string theory, formulated in higher genera on the world sheet to account for

the quantum corrections. Wehave not worked out in this work the full string-theory

representation of the relevant quantum coherent state. This will b e an interesting

topic to b e studied in the future, which will allow for a rigorous study of the e ects

of the global string mo des on the collapse of the quantum-coherent preconscious state.

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Figure Captions

Figure 1 - Microtubular Arrangement: (a) the structure of a Microtubule

(MT), (b) cross section of a MT, (c) two neighb oring dimers along the direction of

a MT axis

Figure 2 - The two conformations and of a MT dimer. Transition (switching)

between these two conformational states can b e viewed as a quantum-mechanical

e ect. Quantum-Gravityentanglement can cause the collapse of quantum-coherent

states of such conformations, which might arise in a MT network mo delling the

preconscious state of the human brain.

Figure 3 - Illustration of the phenomenon of `dynamical instability' of a MT

network : (a) unb ounded `sawto oth' growth (b) b ounded `sawto oth' growth. Dotted

lines show the average over many MT with the same dynamical parameters.

Figure 4 - (a) Contour of integration in the analytically-continued (regularized)

version of (s) for s 2 Z . This is known in the literature as the Saalschutz

contour, and has b een used in conventional quantum eld theory to relate dimen-

sional regularization to the Bogoliub ov-Parasiuk-Hepp-Zimmermann renormaliza-

tion metho d, (b) schematic rep esentation of the evolution of the world-sheet area as

the renormalization group scale moves along the contour of g. 4(a)