ACT-09/95 CERN-TH.127/95 CTP-TAMU-24/95 ENSLAPP-A|-/95 hep-ph/9505401 Non-Critical String Theory Formulation of Microtubule Dynamics and Quantum Asp ects of Brain Function a; b;c N.E. Mavromatos and D.V. Nanop oulos a Lab oratoire de Physique Theorique ENSLAPP (URA 14-36 du CNRS, asso ciee a l' E.N.S de Lyon, et au LAPP (IN2P3-CNRS) d'Annecy-le-Vieux), Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France. b Texas A & M University, College Station, TX 77843-424 2, USA and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, Wo o dlands, TX 77381, USA. c CERN, Theory Division, Geneva 23 Switzerland Abstract Microtubule (MT) networks, subneural paracrystalline cytosceletal structures, seem to play a fundamental role in the neurons. We cast here the complicated MT dynamics in processed by the SLAC/DESY Libraries on 30 May 1995. 〉 the form of a 1 + 1-dimensional non-critical string theory,thus enabling us to provide a consistent quantum treatment of MTs, including enviromental friction e ects. We suggest, thus, that the MTs are the microsites, in the brain, for the emergence of stable, macroscopic PostScript quantum coherent states, identi able with the preconscious states. Quantum space-time e ects, as describ ed by non-critical string theory, trigger then an organizedcol lapse of the coherent states down to a sp eci c or conscious state. The whole pro cess we estimate to take O (1 sec), in excellent agreement with a plethora of exp erimental/observational ndings. The microscopic arrow of time, endemic in non-critical string theory, and apparent here in the self-collapse pro cess, provides a satisfactory and simple resolution to the age-old problem of how the, central to our feelings of awareness, sensation of the progression of time is generated. ACT-09/95 CERN-TH.127/95 HEP-PH-9505401 CTP-TAMU-24/95 ENSLAPP-A|-/95 May 1995 On leave from P.P.A.R.C. Advanced Fellowship, Dept. of Physics (Theoretical Physics), University of Oxford, 1 Keble Road, Oxford OX1 3NP, U.K. 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 19 magnitude G m 10 , with G Newton's gravitational constant and m the p N p N 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 12 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- 8 ments have shown that there exist N =10 tubulins p er neuron, and that there are 11 10 neurons in the brain, it follows that that this order of magnitude for N refers 7 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 1 higher spin target-space states[11 ]. They are resp onsible for string-level mixing, and hence they are broken in anylow-energy approximation.