TOPOLOGICAL CHARGES

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TOPOLOGICAL CHARGES Particles and fields are the solutions to the fundamental equations of physics. Thoughts are the solutions to the fundamental equations of logic. The existence of the thinking brain 'proves' that common solutions do exist, and our underlying hypothesis is that these solutions are t o p o l o g i c a l . In accordance with this idea, the key goal of this work is to provide a topological solution to consciousness. The language of topology is the new language for brain science, as well as for physics. Since Einstein the majority of physicists believe that physical forces can be explained using pure geometry, if necessary, the geometry of higher dimensions. The main thrust of this study is to show that as much as geometry is essential for understanding the physical universe, topology is essential for understanding the phenomenon of consciousness [Ref 78]. The development of geometry preceded the development of topology, and due to historical reasons and education our concept of the world, including the brain, was and continues to be primarily geometrical. However, looking at a moving amoeba or considering the liquid flexibility of a developing embryo, one gets a strong feeling that for living matter and for biology in general the concepts of geometry are not enough. Bioscience clearly lacks insight into some unknown laws of topology which can explain a wide range of life phenomena. Geometry is concerned with the properties of figures in space and with the properties of space itself. Originally, it started as a practical subject in ancient Egypt and Babylonia, used in surveying and building. The ancient Greeks realized that the properties of geometrical figures could be deduced logically from other properties. Pythagorus proved his celebrated theorem, and around 300 BC Euclid produced one of the most famous texts in the whole of mathematics, Elements. Despite difficulties with the fifth postulate, the Euclidean geometry of Elements survived as an unquestioned canon until the non-Euclidean geometries were discovered. Prior to that, in 1637, Descartes in his La Gdomdtrie introduced into mathematics the fundamental principles and techniques of coordinate geometry, where points could be represented by numbers and lines by equations. This gave mathematicians a new analytical tool with which to attack geometric problems algebraically. The 19th century saw major advances in geometry. Cayley developed the algebraic geometry of n-dimensional spaces and Lobachevsky non-Euclidian geometry. Finally, in 1854, Riemann put forward a view of geometry as the study of any kind of space of any number of dimensions. The important development of the past two decades has been noncommutative geometry, which extends the geometry to the context of noncommutative rings.

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A notion of invariant distance is essential for geometry. A set of points is a metric space if there is a metric p which gives to any pair of points x, y a nonnegative number p(x, y), their distance or separation, and is such that (1) p ( x , y ) = 0 iff x=y,

(2) p(x, y) : p(y, x) (3) p(x, y) + p(x, z) >_p(x, z) Choosing a pseudo-Euclidian signature (+ + + -) for a metric we obtain the geometrical model for special relativity, known as Minkowski space. Relativity theory rejects the idea of space and time as separate entities, and is extended in four dimensions compared with the three dimensions of ordinary space. Space and time are joined into a single concept of spacetime in order to describe the geometry of the universe, effectively reducing the study of physics to the study of geometry. In special relativity spacetime is flat, much as space is flat in Euclidian geometry. General relativity is concerned with gravitational effects of matter, which cause spacetime to curve: massive objects produce distortions and ripples in local spacetime, and the motion of bodies are then dictated by the curvature. The curved spacetime is described by means of Riemann geometry. A description of spacetime in terms of Minkowski and Riemann geometries and the fundamental link between geometry and physical laws in general gained greater clarity after Noether in 1917 proved a theorem showing that the conservation laws of physics are in fact consequences of moie fundamental laws of symmetries. The conservation of energy and momentum follow from the symmetry (isotropy) of time and space. The conservation of electrical charge follows from a symmetry of a particle's wavefunction. In general, we say that particles such as the electron and proton carry Noether charges, the attributes that are maintained because of geometrical symmetries. But the attributes and properties of objects may also stay invariant under topological deformations. The corresponding conservation laws are topological as opposed to conservation due to geometry. Unlike the geometer, who is typically concerned with questions of congruence or similarity, the topologist is not at all concerned with distances, shapes and angles, and will for example regard (this has nothing to do with the decline of moral values) a wedding ring and a tea cup as equivalent, since either can be continuously deformed into the other. A set, together with sufficient extra structure to make sense of the notion of continuity, is called a topological space. More formally, a set X is a topological space if a collection T of subsets of X is specified, satisfying the following axioms: (1) the empty set and X itself belong to T O E T and X e T (2) the intersection of two sets in T is again in T X~T,

Y~T

~

XnY

~T

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(3) the union of any collection of sets in T is again in T. X ~ T, Y ~ T ~ XuY~ T The sets in T are called open sets and T is referred to as a topology on X. For example, the real line R~ becomes a topological space if we take as open sets those subsets U for which, given any x ~ U there exists e >0 such that { y ~ R l :l x-y I< E ) is contained in U. A similar definition is valid in a metric space, but it is not in general required of every topological space that it should be metric. A well-known correspondence exists between algebraic geomeh'y and physical objects. A space gives rise to a function algebra; a vector bundle over the space corresponds to a projective module over this algebra; cohomology can be read off as the de Rham complex; and so on. We will take steps to establish a different type of correspondence, the correspondence between the elements of logic and the elements of topology. Since our main objective is to show that the laws of topology hold the key to the laws of the thinking brain and that the information physics of the consciousness is rooted in topology, the focus of this study is not the neurological brain. What we want is to understand the topological brain and its intelligence-supporting logic. Many attempts to explain the cognizing phenomenon and to understAND consciousness neurophysically lead to a dead end. No knowledge about the neural or biophysical processes in the brain can satisfactorily answer the hard question: what is the actual mechanism of consciousness? Those who try to answer this fundamental question in the mechanistic framework of the interaction of neurons, the brain's electricity, neurochemistry or quantum mechanics are often as unproductive as those who offer purely philosophical, spiritualistic or theological explanations only. Somehow human thought, even though connected to processes in the brain matter, seems to be intractable, almost immaterial. Abstractions, on the other hand, often have great physical power. Words and thoughts alone can induce measurable changes in the brain, can alter the state of consciousness, with the amplified effects being visible to the naked eye in the state of hypnosis, for example. As we mentioned earlier the laws of conservation in physics are consequences of corresponding symmetries: the conservation of energy follows from the symmetry of time, the conservation of momentum is due to the isotropy of space. These attributes and others like mass or the charges of elementary particles are conserved due to geometric properties, and can be defined as metric charges. Mental or logical attributes are m a i n t a i n e d not as geometric but as topological objects. It may happen that the field line of a logical exciton ties a knot in cognitive space which cannot be smoothed out. As a result, it is prevented from dissipating and will behave much like a particle. A parallel example from physics is a magnetic monopole - the isolated pole of a magnet - which has not been detected in nature but shows up as twisted configurations in field theory. In the traditional view, particles such as electrons and quarks, which carry geometric charges, are seen as fundamental, whereas particles such as magnetic monopoles, are derivative

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particles, to which we can assign topological charges. What is important is that a topologically nontrivial field configuration, such as soliton, exchanges roles with ordinary quanta. To describe consciousness one does not really need spacetime, or more radically, does not really have spacetime any more, but just a tensor product of two-dimensional topologies, much as with strings where one does not have a classical spacetime but only the corresponding twodimensional theory describing the propagation of strings. Worldlines are replaced by worldsheets, the interaction vertices in the Feynman diagrams are smoothed out, and spacetime exists only to the extent that it can be extracted from that two-dimensional field which encodes information. Although we are all familiar with notion of thoughts, in reality we never observe an isolated thought in particular locations of the brain. It is everywhere and nowhere. A thought for the brain is like a neutrino for the universe. The organization of the brain is distinguished by extraordinary plasticity, with one region of the brain smoothly taking the role of the other if the need arises. Following an immediate reflex, one is tempted to connect thoughts with quantum nonlocality. But there is a more fundamental concept, the concept of the topological charge, which brings greater clarity to the question of the nonlocality of thoughts. To understand that we must understand a key difference between topological and Noether charges. A topological charge is a knot which is essentially nonlocal. It is a defect on a field line which characterises it as a whole. A geometrical Noether charge, in contrast, is local. It can be localized in a particular spacetime point, to a degree allowed by the uncertainty relation. We can in principle localize an electron in the brain, but we cannot, even in principle, localize a thought. When a thought emerges, a (topo)logical knot is tied up, and the knot by its very definition is a spatially extended object. This (topo)logical approach to the problem of consciousness offers a new understanding of the phenomenon. Nature obeys mathematical laws, but while for the physical brain these laws are primarily geometrical, both in the commutative and noncommutative spaces, for the cognitive brain the underlying mathematical theory is essentially and fundamentally topological. We pursue this viewpoint to an even greater extreme: geometry cannot be used to describe logical consciousness. Thought is essentially a topological effect, connected to the brain by means of duality, much as the magnetic monopole, a collective excitation, is related to the dual electric charge. In the actual brain there are Noether charges and these are converted into (topo)logical excitons that move freely through the neuronal medium, decaying into their constituent parts and recombining back. A (topo)logical exciton emerges as a fundamental quantum of consciousness, forming coherent waves that run through the brain matter. However, unlike electrons, (topo)logical excitons do not carry mass across the brain. They carry topological energy, and in spite of almost classical propagation regime, their spectra remain highly coherent, because a coherent superposition of true and false underscores the very existence of a (topo)logical exciton.

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Application of this model to the brain-mind duality offers a fundamental explanation of consciousness. It suggests that there exist two equivalent formulations of the logical brain in which the roles of geometric charges and (topo)logical charges are reversed, just as we exchange electric charge and magnetic charge in field theory. In such a dual picture of the brain either charge, (topo)logical or geometrical, can be taken as elementary, and then a dual charge arises as derivative. In quantum field theory a fundamental particle with charge e is equivalent to a solitonic particle with charge 1/e. This leads to a vast mathematical simplification. For instance, in the theory of quarks we can hardly make any calculations when the quarks interact strongly. But monopoles in the theory must interact weakly, and by doing calculations with a theory based on monopoles one automatically get all the answers for quarks. The duality principle, applied to the problem of the thinking brain, provides a promising theoretical framework. For a very long time we have been struggling to understand the intractable mechanism of consciousness which somehow converts physical to mental and mental to physical. The duality between (topo)logical charges and Noether charges removes the impediments to understanding how the thought process is able to induce controlled changes in the brain matter. When we think, the brain transforms (topo)logical charges, which are fundamental primitives of thoughts, interacting weakly. When such a transformation is completed we automatically gain the answers for the 'strongly' interacting neurological brain. A charge is a measure of the strength of an interaction but physical and logical charges obey opposite laws of attraction and repulsion. Identical Noether charges, like those of two electrons, repel,

while identical logical charges gravitate towards each other and merge: xANDx-x

-

the absorption law

The opposite physical charges, like those of an electron and a proton, attract,

but the opposite logical charges are mutually excluding and repel each other: x AND $ = 0

-

the contradiction law

90 No two identical Fermi particles, such as an electron or proton, can ever be in the same quantum state, but 'logical fermions' would not follow the Pauli exclusion principle. This example touches on fundamental aspects of the brain-mind duality, which connects the strong coupling of one theory with the weak coupling of another. Consciousness is a topological effect' the brain decides g e o m e t r i c a l l y ; the mind decides topologically. Topology is not a matter of choice but is fundamental. Consequently, there are two dual theories of the brain: the geometrical theory which we used until now and a topological theory whose underlying principles we intend to formulate. When the brain is described in terms of the Noether charges, the dual (topo)logical charges emerge as derivative [Ref 77]. Quite symmetrically one can choose the (topo)logical charges to be fundamental, and then to treat the biophysical electrophysiological brain as derivative,

(Topo)iogical charge

r

Noether c h a r g e

The notion of topological charges as the physical basis of consciousness naturally leads to the notion of topological waves or currents which carry the charges. The charges are nontrivial dynamical topological configurations that exchange with ordinary quanta. A (topo)logical current propagating along a closed information loop (knot) manifests itself as the thought process. The knot may have various configurations, but a particular geometry of the knot is irrelevant, as long as it retains the same (topo)logical charge. The (topo)logical currents are effectively isolated from the outside universe and cannot be subjected to ordinary physical measurement. The most we can achieve with state-of-the-art Hermitian devices is to measure the dual Noether currents, and the attempt to do so is made indirectly when we measure the electrical and neurochemical activity of the brain in the laboratory. However, a (topo)logical charge maps to a corresponding Noether charge and vice versa. Making use of this duality we can influence the (topo)logical current and with it the inner content of consciousness. This effectively was done in the classical experiments of Wilder Penfield, performing neurosurgery on conscious patients" electrical stimulation of the brain can induce virtual cognitive states and memories which the subject treats as real. In the Penfield effect the Noether charges are converted into (topo)logical charges but there must be and there is the reverse transformation whereby the (topo)logical charges are mapped onto electrical Noether charges of the brain. There is no mystery in the ability of the mind to influence the physical states of the brain, an idea which strangely enough continues to be treated as theological and nonscientific in some circles. But anyone who doubts that the mind acts on matter, exerting control over the physical brain, may recall, for example, the experiments carried out with paralyzed patients. Patients who couldn't communicate at all were taught to write sentences on a computer screen via electrodes implanted in their brain by

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changing their brain waves at will. Signals from an EEG are used to control a simple switch. The patients learn to make their cortical potential more negative or positive, in this manner moving a cursor up or down on a computer screen, writing messages. This system, being based on simple binary choices, has limited scope, but there is no doubt that neuroscience one day will make use of the ' t h o u g h t translation machines', for example to exert fine control over devices that simulate muscles, allowing paralyzed patients to regain control over their bodies.

LOGICAL V A C U U M One of the basic tenets of quantum physics is that all particles have positive energy. According to Dirac the negative levels are filled up and by definition the vacuum has zero energy. This positivity of energy imposes fundamental limitations on what can and what cannot occur in physics. If we were able to produce objects with negative energy, which is less than vacuum, then we could generate exotic field configurations in which time is bent into a circle. However, this effect, much loved by science fiction writers, has become actuality in quantum field theory with the introduction of antiparticles. A closed vacuum diagram describes the birth of particles emerging out of a vacuum and submerging back into the void. At the fundamental level particles and antiparticles moves freely backwards and forwards in time, violating no macroscopic laws of causality. In the framework of the two alternative dual theories of the brain, the geometrical and the topological, we now introduce notion of a logical vacuum 1~ which carries a nontrivial topological charge. The logical vacuum is not physically empty, but a connection-empty space, meaning that there is no interaction between the elements of the system, and as a direct consequence of that, the system has no deductive or inductive logical (reasoning) capabilities. The closest analogy from physics would be a system of noninteracting particles in quantum mechanics. The logical vacuum could have different degrees, reflected in part by different IQ. Indeed, there are, as we all know, some big but empty heads with a lot of'intellectual vacuum'. The presence of a logical connection can be interpreted differently. In the classical picture a connection is a convex link which can be thought of as a line joining two points in a topological space. The fundamental matrix-logical picture is less intuitive' the creation of topological links now signifies the creation of a matrix particle or n i b b l e . If the particle is annihilated, submerging into the vacuum, the logical links which were created vanish in concert too. The role of nibbles is similar to the role of elementary particles in physics and to genes in biology. They are fundamental elementary logical