archived as http://www.stealthskater.com/Documents/Pitkanen_05A.doc (also …Pitkanen_05A.pdf) => doc pdf URL-doc URL-pdf

more from Matti Pitkanen is on the /Pitkanen.htm page at doc pdf URL

note: because important websites are frequently "here today but gone tomorrow", the following was archived from http://tgdtheory.com/figu.html on 10/17/2007. This is NOT an attempt to divert readers from the aforementioned website. Indeed, the reader should only read this back-up copy if the updated original cannot be found at the original author's site.

Figures and Illustrations related to TGDDr. Matti Pitkänen 08/09/2000

Postal address:Köydenpunojankatu 2 D 1110940, Hanko, Finland

E-mail: matpitka@luukku.comURL-address: http://tgdtheory.com

(former address: http://www.helsinki.fi/~matpitka )"Blog" forum: http://matpitka.blogspot.com/

The understanding of the TGD-based spacetime concept relies heavily on generalizations from simple 2-dimensional concepts to higher dimensions. The mere linguistic representation leads easily to misunderstandings (as I have concretely found in case of "wormhole" concept!). Thus the following (mostly) 2-dimensional illustrations are strongly recommended.

1. 2-dimensional illustrations of TGD-based space-time concept

2. 2-dimensional illustrations related to TGD-inspired theory of brain

3. Further illustrations related to TGD-inspired theory of Consciousness

4. p-Adic fractals

5. see also partly TGD-inspired graphics of Mark Thornally

1

TGD

1. 2-dimensional illustrations of TGD-based space-time concept(http://www.helsinki.fi/~matpitka/illua.html )

A. TGD-based space-time concept

The starting point of TGD is the 'energy problem' of General Relativity. By Noether's theorem, conservation laws are in one-one correspondence with symmetries. In particular, translational invariance of the empty Minkowski space M4 implies energy and momentum conservation in Special Relativity. By the basic postulate of General Relativity, matter makes space-time curved. This means that the symmetries of the empty Minkowski space are lost as are lost also the corresponding conservation laws, in particular the conservation of energy.

The basic idea of TGD is to assume that space-time is representable as a surface of some higher dimensional space H = M ×S. And that translational symmetries and -- more generally -- Poincare invariance correspond to the symmetries of M-factor of this higher-dimensional space rather than those of space-time itself.

Hence, a fusion of Special and General Relativities in a well-defined sense is in question. In fact, mathematical and physical reasons force to replace empty Minkowski space M4 with its lightcone M4

+. Future light cone corresponds to empty Robertson-Walker cosmology and TGD-inspired cosmology has subcritical mass density as a consequence. There is small cosmological breaking of Poincare invariance since M4 is replaced by its lightcone.

Figure 1. Matter makes space-time curved and spoils translational invariance. Two-dimensional illustration.

By physical constraints (elementary particle spectrum), the space S must be CP2, the complex projective space of two complex (four real) dimensions. The size of CP2 is about 104 Planck lengths (roughly 10-30 meters). [It took long time to realize that the original assumption about size being of the order of the Planck length was not correct].

Figure 2. Future lightcone of Minkowski space

2

Figure 3. CP2 is obtained by identifying all points of C 3 , space having 3 complex dimensions, which differ by a complex scaling Lambda: z is identified with Lambda×z.

CP2 can also be regarded as a coset space SU(3)/U(2), U(2)=SU(2)×U(1). What this means is that one starts from the 8-dimensional group SU(3) of unitary 3×3 matrices of determinant one and identifies all matrices which differ by a left multiplication by an element of the 4-dimensional subgroup U(2).

One can also say that each point of SU(3) is obtained from CP2 by replacing each point of CP2 with the group U(2). U(2) in turn can be regarded as S3×S1. That is, as the space obtained by replacing each 2-dimensional disk giving a cross section of the ordinary torus (doughnut) with a 3-dimensional sphere S3. By this construction, CP2 is so called symmetric space whose all points are equivalent metrically (like those of Euclidian space) and has color group SU(3) as its group of distance preserving transformations, isometries.

Figure 4. H= M 4 +×CP2 is obtained by replacing each point of the future light cone with the 4-dimensional compact space CP2 of size R of order 10 4 Planck lengths (10 -30 meters).

The second manner to end up with TGD is to start from the old fashioned string model, which also served as a starting point of superstring models which have been in fashion during the last 10 years.

Mesons are strongly interacting particles and string model description was in terms of a string with quark and anti-quark attached to the ends of the string. A problem was encountered in an attempt to generalize this description to apply to baryons which consist of 3 quarks. One cannot put 3 quarks to the ends of the string since it has only 2 ends.

3

Figure 5. The transition from hadronic string model to TGD.

The solution of the problem is simple. Replace 1-dimensional strings with small 3-dimensional surfaces. Since the ends of the string correspond to the boundaries of a 1-dimensional manifold, they correspond in 3-dimensional case boundaries of small holes drilled in 3-dimensional space. Put quarks on these boundaries. In a 3-dimensional case, one can drill arbitrary number of these holes so that also baryons can be described in this kind of model.

The TGD-based space-time concept differs in many crucial aspects from the conventional one. In the following, this difference is visualized by replacing 3-dimensional space (now surface in H) with 2-dimensional surface whereas 8-dimensional imbedding space is replaced with 3-dimensional slab of thickness of order 104 Planck lengths. This simplification makes it possible to illustrate the most essential aspects of the generalization easily and, at least, the geometrically/topologically-oriented reader can guess the rest.

Below is a general view of what many-sheeted 3-space would look like if it were 2-dimensional

Figure 6. This is what 3-space would look if it were a 2-dimensional surface in 3-dimensional slab of thickness of order 10 4 Planck lengths.

B. Elementary particles as 3-surfaces of size of order R=104 Planck lengths: CP2 extremals.

Elementary particles have geometric representation as so called CP2-type extremals. Instead of standard imbedding of CP2 as a surface of M4

+×CP2 obtained by putting Minkowski coordinates mk

constant

mk=const.,

one considers "warped" imbedding

mk =fk(u) u is arbitrary function of CP2 coordinates with the property that the M4+ projection of the

surface is random light-like curve:

mkl dmk/du dml/du =0, mkl is flat M4 metric. (A)

4

The condition implies that induced metric is just CP2 metric. Which is Euclidian! The curve is random and therefore one has classical non-determinism. This makes sense since the solution is vacuum extremal.

Figure 7. The projection of CP2-type extremal to M 4 + is light-like curve.

Elementary particles correspond to CP2-type extremals with holes. The intersection of bound with m0=const hyperplane is sphere, torus, sphere with 2 handles, etc. shortly a surface with genus g=0,1,2, ... . Different fermion families correspond to different genera. Bosons are also predicted to have family replication phenomenon.

Figure 8. Different fermion families correspond to different genera for the boundary component of CP2-type extremal.

Feynman diagrams correspond to topological sums of CP2-type extremals: the lines of diagram being thickened to CP2-type extremals:

Figure 9. Feynman diagrams correspond to connected sums of CP2 type extremals: each line of Feynman diagram is thickened to CP-type extremal.

The quantum version of the condition (A) stating that the M4+ projection is light like curve leads to Super Virasoro conditions. It turns out that elementary particles together with their 10-4 Planck mass excitations belong to representation of p-adic Super Virasoro and Kac Moody. The p-adic mass calculations lead to excellent predictions for particle masses.

C. Induced gauge field concept implies radical generalization of space-time concept

5

The concept of connection geometrizes the concept of the parallel translation appearing already in elementary geometry. Parallel translation can be performed for vectors, tensors, spinors, etc.

In the Euclidian case, parallel translation is just what one would imagine it to be. And the parallel translation around a closed curve brings the vector back without any change in its direction. The formal definition of the parallel translation along a geodesic line (the counterpart of a straight line) in a more general context requires that the angle between the vector and geodesic line is preserved. The sphere is a simple example of a situation in which parallel translation around a closed curve changes the direction of the vector. One says that the sphere is curved and curvature is locally measured by the amount of change in the direction of a vector for very small geodesic triangle.

Figure 10. Parallel translation on sphere and on plane.

In General Relativity, the so-called "Riemann connection" defining the parallel translation in space-time leads to a beautiful geometrization of the gravitational interaction. The presence of matter makes space-time curved. And geodesics are not straight lines anymore. The advent of the gauge theories led to a partial geometrization of the boson fields. The components of the gauge potentials can be regarded as components of a connection defining parallel translation formally.

The problem is, however, that there is not direct geometric interpretation for this parallel translation. And here, TGD provides the final geometrization of classical gauge field concept. The components of electro-weak gauge potentials are obtained as projections of the spinor connection of CP2 to space-time surface:

Aμ = Ak ∂μ hk

(μ is the coordinate index for space-time coordinates, k for imbedding space coordinates)

or geometrically; parallel translation on space-time surface is performed using spinor connection of the imbedding space.

6

Figure 11. Classical electro-weak gauge potentials at space-time surface are obtained as projections of the components of CP2 spinor connection.

Classical color gauge potentials are identified as projections of Killing vector fields of SU(3) to space-time surface (very much like in Kaluza-Klein theories). The requirement is that the Standard Model electro-weak gauge group allows only M4

+×CP2 as imbedding space. Also Standard Model quantum numbers are geometrized in terms of CP2 geometry and topology of the boundary component of CP2-type extremal. The special features of CP2 -- in particular, the fact that it does not allow standard spinor structure -- are crucial for obtaining the coupling structure of Standard Model.

The induced gauge field concept differs from the ordinary one. The PRIMARY dynamical variables are the four CP2 coordinates. This implies strong constraints among classical gauge fields. For instance, classical electric field is often accompanied by classical Z0 field even in Macroscopic length scales.

There is rather precise metaphor making possible to understand the concept of induced gauge field intuitively. The shadow (projection!) of a non-dynamical solid object (↔ metric and spinor connection of H) with time-independent size and shape to a surface (↔ 3-surface) changing its size and shape is dynamical.

Figure 12. Classical electro-weak gauge potentials at space-time surface are obtained as projections of the components of CP2 spinor connection.

Even more importantly: gauge potentials are determined by the image of map

X4→ CP2

whereas ordinary gauge potentials are determined by the map

X4 → TM4,

7

where TM4 is the space of field values at given point of space-time and isomorphic to tangent space of M4. CP2 is COMPACT whereas TM4 is NON-COMPACT. A crucial difference!

This implies that general electromagnetic gauge potentials are imbeddable only in some open region surrounding given point of space-time. And that the imbeddability fails at the boundary of this region.

Figure 13. The Maxwell field associated with a given charge distribution is representable as induced gauge field only in a finite region of space-time. This implies the presence of boundaries. 2-dimensional illustration.

The failure of imbeddability leads to generation of space-time BOUNDARIES in all lengths scales. At the boundary, space-time simply ends.

There is the following problem on the boundaries. Kähler electric gauge flux must be conserved on the boundary. Since the 3-space ends at boundary, there is no other manner to cope with situation than to introduce a second -- and larger -- space-time sheet parallel to the first one and allow the gauge flux to run on this space-time sheet via tiny wormholes connecting the 2 sheets. A wormhole is constructed by drilling tiny spherical holes inside the 2 parallel space-time sheets and connecting the boundaries S2 of the holes with a cylinder S2xI having 2 ends with S2 topology. The figure below illustrates the situation if 3-space were 2-dimensional.

Figure 14. Charged wormholes feed the electromagnetic gauge flux to the 'lower' space-time sheet.

By adding these wormholes on the boundary of 3-surfaces, the gauge flux can flow to the lower space-time sheet. An interesting possibility is that wormholes are itself slightly deformed pieces of CP2-type extremals.

8

The throats of wormhole behave as classical charges -Q and Q where Q is the electric gauge flux flowing to the wormhole at upper space-time sheet and out of it at lower space-time sheet. Thus they serve as currents and sources (of opposite sign) of classical gauge fields at the two space-time sheets.

Figure 15. The two throats of wormhole behave as classical charges of opposite sign.

Wormholes couple to the DIFFERENCE OF CLASSICAL GAUGE POTENTIALS associated with the two space-time sheets since the classical charges are opposite.

It seems safe to assume that photons (the extremely small CP2-type extremals!) see wormholes from a wider perspective that is extremely small dipoles formed by the throats. The distance between charges is of order 104 Planck lengths. The direction of dipole is transversal to space-time surface so that polarization vector has very small projection in M4 where polarization vector of photon is. Thus the coupling to photons should be negligible (dipole moment satisfies p > Q×R (R the size of CP2) and thus also dissipation effects.

Figure 16. As for as coupling to photons is considered, wormoholes are expected to behave as extremely tiny dipoles.

This suggests very strongly that wormholes behave much like conduction electrons and are thus localized to the boundaries of space-time surface. If wormholes are light (as they turn out to be), they obey the d'Alembert type equation and there is large energy gap between ground state and excited states. Thus wormholes become suffer B-E condensation to ground state. Charged wormholes behave thus much like super-conductors.

Wormholes can have several topologies. In general, one can drill holes of genus g (sphere, torus, etc.) on two space-time sheets and connect them using cylinder I+genus g surface. In this, they resemble ordinary elementary particles which have also several genera (family replication phenomenon).

9

The gauge flux conservation problem is encountered also in the lower space-time sheet. And one must introduce third, fourth, etc. space-time sheet. In general, one has a hierarchy of space-time sheets with increasing sizes.

Figure 17. Many sheeted space-time structure results from the requirement of gauge flux conservation.

The conclusion is that induced gauge field concept leads unavoidably to the concept of many sheeted space-time. This has radical consequences for the structure of physical theory. One must replace thermodynamics, hydrodynamics, etc. with a hierarchy of dynamics of various types -- one for each space-time sheet in the hierarchy. This replacement must be performed in ALL length scales.

D. Matter as topology

Since many-sheetedness is encountered in all length scales, a very attractive manner to reinterpret our visual experience about world suggests itself.

Material objects having Macroscopic boundaries correspond actually to sheets of 3-space and 3-space literally ends at the boundary of object. The 3-space outside the object corresponds to the "lower" space-time sheet.

Actually, we can see this wild 3-topology every moment!! The following 2-dimensional illustration should make clear what the generalization really means.

10

Figure 18. Matter as topology

E. Join along boundaries contacts and join along boundaries condensate

The recipe for constructing many-sheeted 3-space is simple. Take 3-surfaces with boundaries … glue them by topological sum to larger 3-surfaces … glue these 3-surfaces in turn on even larger 3-surfaces, etc..

The smallest 3-surfaces correspond to CP2 type extremals that is elementary particles and they are at the top of hierarchy. In this manner You get quarks, hadrons, nuclei, atoms, molecules,... cells, organs, ..., stars, ..,galaxies, etc...

Besides this one can also glue different 3-surfaces together by tubes connecting their BOUNDARIES. This is just connected sum operation for boundaries. Take disks D2 on the boundaries of two objects and connect these disks by cylinder D2×D1 having D2:s as its ends. Or more concretely, let the two 3-surfaces just touch each other.

11

Figure 19. Join along boundaries bond in (a) 2-dimensions and (b) 3-dimensions for solid balls.

Depending on the scale join along boundaries, bonds are identified as color flux tubes connecting quarks; bonds giving rise to strong binding between nucleons inside nuclei; bonds connecting neutrons inside neutron star; chemical bonds between atoms and molecules; gap junctions connecting cells; the bond which is formed when You touch table with Your finger; etc.

One can construct from a group of nearby disjoint 3-surfaces so-called "join along boundaries condensate" by allowing them to touch each other here-and-there.

Figure 20. Join along boundaries condensate in 2 dimensions.

The formation of join along boundaries condensates creates clearly strong correlation between 2 quantum systems. And it is assumed that the formation of join along boundaries condensate is necessary prequisite for the formation of MACROSCOPIC QUANTUM SYSTEMS.

F. p-Adic numbers and vacuum degeneracy

p-Adic length scale hypothesis derives from the analogy between SPIN GLASS and TGD. Kähler action allows enormous VACUUM DEGENERACY. ANY space-time surface -- which belongs to M4

+×Y2 where Y2 is so-called Lagrangian sub-manifold of CP2 -- is vacuum due to the vanishing of induced Kähler form (recall that Kähler action is just Maxwell action for induced Kähler form which can be regarded as U(1) gauge field).

Lagrangian sub-manifolds can be written in the canonical coordinates Pi,Qi, i=1,2 for CP2 as

Pi = fi(Q1,Q2)

12

fi = ∂i f(Q1,Q2)

where ∂i means partial derivative with respect to Qi. f is arbitrary function of Qi! Lagrangian sub-manifolds are 2-dimensional. The topology of vacuum space time is restricted only by the imbeddability requirement. Vacuum space-times can have also finite extend in time direction(!!). Charge conservation does not force infinite duration.

Figure 21. Vacuum extremals can have finite time duration

This enormous vacuum degeneracy resembles the infinite ground state degeneracy of spin glasses. In the case of spin glasses, the space of free energy minima obeys ultra-metric topology. This raises the question whether the effective topology of the real space-time sheets could be also ultra-metric in some length scale range so that the distance function would satisfy

d(x,y) <= Max(d(x),d(y)) rather than d(x)+d(y).

p-Adic topologies are ultra-metric and there is p-adic topology for each prime p=2,3,5,7, etc. The classical non-determinism of the vacuum extremals implies also classical non-determinism of field equations (but not complete randomness, of course).

p-Adic differential equations are also inherently non-deterministic. This suggests that the non-determinism of Kähler action is effectively like p-adic non-determinism in some length scale range. So that that the topology of the real space-time sheet is effectively p-adic for some value p. The lower cutoff length scale could be CP2 length scale. Of course, cutoff length scales could be dynamical.

Standard representation of p-adic number is defined as generalization of decimal expansion

x= ∑n≥n0 xnpn

p-Adic norm reads as

N(x)p = p-n0

and clearly depends on the lowest pinary digit only and is thus very rough: for reals norm is same only for x and -x. Note that integers which are infinite as real numbers are finite as p-adic numbers. The p-adic norm of any integer is -- at most -- one.

Essential element is the so-called CANONICAL CORRESPONDENCE between p-adics and reals

p-Adic number

x = ∑n≥n0 xnpn

13

is mapped to real number

y = ∑n≥n0 xnp-n

Note that only the signs of powers of p are changed.

The second natural correspondence between p-adics and reals is based on the fact that both reals and p-adics are completions of rational numbers. Hence rational numbers can be regarded as common to both p-adic and real numbers.

This defines a correspondence in the set of rationals. Allowing algebraic extensions of p-adic numbers, one can regard also algebraic numbers as common to reals and algebraic extensions of p-adics. p-Adic and real transcendentals do not have anything in common. Note that rationals have pinary expansion in powers of p which becomes periodic for high pinary digits (predictability) whereas transcendentals have non-periodic pinary expansions (non-predictability).

One could say that the numbers common to reals and p-adics are like islands of order in the middle of real and p-adic seas of chaos. Both correspondences are important in the recent formulation of p-adic physics.

G. p-Adic length scale hypothesis

p-Adic mass calculations force to conclude that the length scale below which p-adic effective topology is satisfied is given

Lp ≈ p1/2R, R= 104 × G1/2 (CP2 length scale).

One has also good reasons to guess that p-Adic effective topology makes sense only above CP2 length scale. One can also define n-ary p-adic length scales

Lp(n) =p(n-1)/2Lp

It is very natural to assume that the space-time sheets of increasing size have typical sizes not too much larger than Lp(n). The following figure illustrates the situation.

Figure 22. p-Adic length scale hierarchy

14

The obvious question is "Are there some physically favored p-adic primes?" p-Adic mass calculations encourage the following hypothesis.

The most interesting p-adic primes p correspond to primes near prime powers of two.

p ≈ 2k, k prime

Especially important are physically Mersenne primes Mk for which this condition is optimally satisfied p= 2k-1 Examples: M127= 2127-1, M107 = 2107-1, M89= 289-1. Electron, hadrons, intermediate gauge bosons.

A real mathematical justification for this hypothesis is still lacking. Probably the p-adic dynamics depends sensitively of p. And this selects certain p-adic primes via some kind of "natural selection".

H. Generalization of space-time concept

One can wonder whether p-adic topology is only an effective topology. O whether one could speak about a decomposition of the space-time surface to real and genuinely p-adic regions and what might be the interpretation of the p-adic regions (note that also real space-time regions would still be characterized by some prime characterizing their effective topology).

The development of the TGD-inspired theory of Consciousness led finally to what seems to be a definite answer to this question. p-Adic physics is the physics of Cognition and Intention. p-Adic non-determinism is the classical space-time correlate for the non-determinism of imagination and cognition. p-Adic space-time sheets represent intentions and quantum jump in which p-adic space-time sheet is transformed to real one can be seen as a transformation of Intention to Action.

This forces us to generalize the notion of the imbedding space. The basic idea is that rational numbers are in a well-defined sense common to both real number field R and all p-adic number fields Rp. The generalized imbedding space results when the real H and all p-adic versions Hp of the imbedding space are glued together along rational points. One can visualize real and p-adic imbedding spaces as planes, which intersect along a common axis representing rational points of H. Real and p-adic space-time region are glued together along the boundaries of the real space-time sheet at rational points.

15

The construction of p-adic quantum physics and the fusion of real physics and p-adic physics for various primes to a larger scheme is quite a fascinating challenge. For instance, a new number theoretic view about information emerges. p-Adic entropy can be negative which means that system carries genuine information rather than entropy.

16

2. 2-dimensional illustrations related to TGD-inspired theory of brain (html)(http://www.helsinki.fi/~matpitka/illub.html )

The TGD-inspired theory of the conscious brain relies on assumption that moments of consciousness correspond to quantum jumps realized as jumps between determistic quantum HISTORIES.

Free will and non-determinism are thus outside the realm of geometrical space-time and one avoids the well-known difficulties resulting from the attempt to understand the non-determinism of quantum jump and the determinism of Schrodinger equation in standard conceptual framework of physical theories.

The hardware for conscious brain relies of various B-E condensates made possible by the special structure of TGD-ish space-time. Crucial role is played by the many-sheeted nature of TGD-ish space-time.

A. B-E Condensates of photons -- Quantum Antenna hypothesis

One of the basic differences between induced gauge field concept and ordinary is that it is possible to get classical EM and Z0 fields propagating with the velocity-of-light and with NONVANISHING vacuum gauge current. These solutions are called "MASSLESS EXTREMALS". In Maxwell electrodynamics, the solutions of free Maxwell equations have vanishing sources (currents).

The vacuum currents in turn serve as sources of B-E condensates. To each Fourier mode, there corresponds a coherent state of photons (eigenstate of photon annihilation operator). Mathematically, the situation is identical to that with harmonic oscillation in force F= A×x×cos(ωt) for each Fourier mode.

The "quantum antenna hypothesis" states that linear structures (such as microtubules, DNA, etc.) are associated with this kind of massless extremals creating B-E condensates of photons when vacuum current is non-vanishing. Vacuum gauge current can be purely electromagnetic or Z0 type or a combination of these two. Or all vacuum gauge currents vanish. But there is an electromagnetic wave propagating with velocity-of-light. In this case, no B-E condensate is created.

The massless extremal property is destroyed by the presence of charges creating Coulomb fields leading to Maxwell phase in which ordinary Maxwell equations are excellent approximate description of situation in sufficiently short length scales. Thus one should find a mechanism keeping the space-time sheet of massless extremal clean from charges.

The fact that microtubules and proteins are surrounded by ordered water suggests the mechanism. There are two space-time sheets essentially involved. "Upper" (say) space-time sheet contains ordered water, which is identified as join along boundaries condensate of water molecules. The electric gauge flux flows totally at this sheet and enters to the "lower" space-time sheet only at the boundaries of the upper space-time sheet through wormholes.

17

Figure 23. Quantum Antenna hypothesis

At the lower sheet, massless extremals create B-E condensate of photons. The frequencies are multiples of pi/L where L is the length of the microtubule or any other linear structure. For microtubules, the frequencies vary from IR to UV.

B. B-E condensate of charged wormholes

The charged wormholes feeding the electromagnetic gauge flux from a "higher" to "lower" space-time sheet behave as classical charges as far as classical EM fields are considered and couple to photons extremely weakly. Their mass can be estimated from p-adic length scale hypothesis to be of order 1/Lp

where p is the p-adic prime associated with the "lower" space-time sheet. Mass is very small in biologically interesting length scales. For Lp about 10-8 meters, the mass is about 102 eV (a fraction 10-4

of the electron mass).

Wormholes should behave like conduction electrons and thus concentrate on the boundaries of 3-surfaces. Charged wormholes are described by complex order parameter satisfying wave equation with very small mass. Thus one has energy gap. For a system with size L, the energy of the first excited state is of order 1/L. Therefore, B-E condensation to ground state occurs.

One can obtain Macroscopic B-E condensates by gluing smaller 3-surfaces by join along boundaries bonds together. Good candidates are cells: now join along boundaries bonds are gap junctions.

Figure 24. Gap junctions connect certain cells and can make them Macroscopic quantum system.

18

Gap junction connecting 2 cells can be regarded as join along boundaries bond. Cells could be glial cells in brain, cells in skin, glands, gut, heart muscle fiber. For nerve cells, gap junctions are rare. Nerve cells are connected with glial cells in Ranvier nodes.

The proteins connecting microtubules (MAPs) can also be regarded as gap junctions as well as proteins connecting different lipid layers of cell membrane.

A wormhole condensate behaves very much like a superconductor. In particular, Josephson junctions connecting two B-E condensates are possible and Josephson current of wormholes flows between these condensates. The model of nerve pulse and EEG is based on assumption that the lipid layers of cell membrane are wormhole B-E condensates and that Josephson current flows along proteins connecting the lipid layers. The situation is illustrated in the figure below.

Figure 25. The lipid layers of cell membrane as wormholes "superconductors" connected by "Josephson junctions". 2-dimensional visualization.

The model explains the difference between nerve cells and ordinary cells; explains nerve pulse as soliton and frequency coding; and explains EEG and predicts new "EG" type oscillation with frequency of order 1010 Hz possibly coordinating protein conformations as well as new ner pulse kind phenomenon with duration of order 10-10 seconds.

The model is testable. For instance, the attribute exhibitory/ inhibitory should be associated with nerve pulse (solitonic/antisolitonic) rather with axon or synaptic connection.

C. Association sequences

In TGD-inspired theory, moments of Consciousness correspond to quantum jumps. The contents of Consciousness is determined somehow by the comparison of initial and final quantum histories.

The essential point is that entire determistic quantum histories are involved. And without further assumptions, it seems that timelessness characterizes conscious experience. Thus the problem is how to understand the concept of subjective-Time and finite duration of subjective experience. Somehow the conscious experience is not localized to infinitely short time interval but has finite duration -- multi-locality in time.

19

The second problem is to understand cognitive aspects of conscious experience. A free will aspect is associated with any quantum jump. Even that performed by electron whereas cognitive aspects and thinking probably only to very specialized quantum jumps. The concept of association sequence might provide a solution of these problems.

The idea of association sequence is simply the following. In TGD, there is no reason to exclude 3-surfaces consisting of several disjoint 3-surfaces with time-like separations.

Figure 26. "Association sequence" -- a geometric model for thought as a sequence of 3-surfaces with time-like separations.

It is tempting to identify this kind of 3-surfaces -- with each 3-surface belonging to an orbit of classical space-time surface -- as a simulation of the classical history by snapshot pictures and thus as a geometric model for thought. In introduced the name "association sequence" for these objects for long time ago. My recent belief is that associations are more probably related to quantum entanglement rather than "association sequences" so that the choice for name is not the best possible.

There is, however, a problem. General coordinate invariance implies that all 3-surfaces for which the absolute minimum space-time surface is thesame are physically equivalent. In particular, association sequences in general are expected to be physically equivalent with single space-time surface in t = constant plane of M4

+.

There is a manner to overcome this difficulty based on CLASSICAL NONDETERMINISM. There are good reasons to assume that Kähler action is not completely deterministic. This means that in certain situations, the absolute minimum space-time surface X4(X3) associated with 3-surface X3 is not unique but there are several of them.

In this case, one can fix a given branch only by selecting finite number of 3-surfaces on the "orbit" of X3 besides X3 itself. In order to get rid of non-uniqueness of absolute minimum, one can generalize the concept of 3-surface by allowing association sequences! Thus thoughts emerge naturally if classical theory is non-deterministic!

D. Association sequences as binary sequences

20

One can describe the situation also in following manner. Consider non-determistic dynamical evolution (also the space-time surface can be regarded as dynamical evolution of 3-surface X3 and thus orbit in the space of 3-surfaces).

Assume that bifurcations occur for time values t1,t2,...,tn which means that at these time values, the orbits branches to two alternative branches. There are 2n possible orbits and one can specify these orbits uniquely by selecting besides x(t), n points x(τ1), x(τ2),...x(τn). where τk is suitably chosen moment of time between two successive bifurcations.

Figure 27. Association sequence as sequence of bifurcations in presence of classical non-determinism.

This set of points defines association sequence. There is extremely tight correlation between points x(τk) to the extend that the association sequence can be equivalently described by specifying x(t) and n binary digits telling which alternative is realized in each bifurcation.

Thus association sequences can be regarded as BINARY SEQUENCES! An ideal representation if one has quantum computer type operations in mind!

E. Criticality of Kähler action and non-determinism

The vacuum degeneracy of Kähler action suggests strongly non-determinism. Consider a "vacuum" 3-surface which has a vanishing induced Kähler field. There is an infinite number of vacuum extremals "going through" this 3-surface. It is plausible that the absolute minimum of Kähler action is obtained by slightly deforming some of these vacuum extremals. Suppose that this is the case.

There is an enormous number of candidates for vacuum extremals. Even their topology can wildly vary. It can well happen for two 3-surfaces not differing too much, the vacuum extremals giving rise to absolute minimum have different topology or differ in some other qualitative aspect. Then there must exist some 3-surface between them for which the vacuum extremals with both these topologies give absolute minimum with same value of action. Classical space-time surface is not unique in this case.

There is a nice analogy with thermodynamical free energy and phase transitions. Kähler action is just like free energy. Space-time surface corresponds to its absolute minimum. In phase transition, one has the situation described by cusp catastrophe (see the figure below).

21

Figure 28. Cusp catastrophe

At the Maxwell line where the phase transition occurs, both upper and lower sheet give same value of free energy and state is mixture of two phases. Elsewhere, the phase is unique.

Biosystems (or systems with cognitive abilities) would be thus critical systems for which absolute minimum of effective actions (whose bosonic part is just Kähler action) have degenerate absolute minima.

F. Thoughts and Vacuum Extremals

Here is a recipe for constructing geometric model for "thought". Take a vacuum extremal with a definite duration of time. The vacuum space-time surface begins at t1 and ends at t2. This surface behaves completely non-deterministically apart from the restriction coming from vacuum property. Thus it can be regarded as a model for completely free imagination. It need not be an absolute minimum of effective action.

Figure 29. Non-determistic vacuum extremals as model for free imagination

To construct a model of thought, let this kind of vacuum space-time surface interact with matter. That is, join this surface to space-time surface containing matter with #throats (wormholes). Some energy flows to vacuum surface and deforms it. It ceases to be vacuum. For certain deformation(s), absolute minimum of Kähler action is achieved. Typically, Kähler electric fields are generated to minimize Kähler action.

What is essential that INTERACTION WITH EXTERNAL WORLD CODES SOME PROPERTIES OF THE EXTERNAL WORLD TO THE PROPERTIES OF THE DEFORMED VACUUM. Some features of the external world is represented as properties of deformed vacuum. his is what cognitive systems do all the time. They construct representations.

22

Figure 30. Interaction with matter forms a representation of external world in the properties of vacuum extremal. There are several deformed vacuum extremals with same absolute minimum of

effective action.

This procedure can be repeated for any vacuum surface. It can, however, happen that absolute minimum value of effective action is same for several choices of vacuum extremal. Thus effective space-time surface is not unique but degenerate and one has classical non-determinism.

This in turn implies that one must form quantum superpositions of effective space-times and quantum jump. Selecting one space-time from this superposition generates conscious thought. By putting vacuum extremal of finite duration above space-time sheet containing matter, one obtains a representation for a small sample of time development as conscious experience.

More concretely, put vacuum extremal of finite duration "above" a group of neurons. The synchronous periodic firing of nerve pulses for this group of neurons could form a representation of sensory data in the properties of vacuum extremal. There is indeed some evidence that this kind of mechanism is at work. Both the spatial configuration of nerve pulses and the organization of pulse sequences in time direction seem to be code sensory experience.

Roughly, space-time is divided into 4-volumes. Each 4-volume contains nerve cell and has nerve pulse duration. To each small volume, one associates binary digit according to whether the neuron fired or not. Each nerve pulse could cause bifurcation. The time distance between surfaces of association sequence would correspond to the period between nerve pulses. EEG oscillations of glial neurons forming Macroscopic quantum system would in turn coordinate the firing of neurons to occur synchronously.

A second argument for the degeneracy of absolute minimum. Suppose an absolute minimum of effective action has been found. Call it X4. Replace this surface with a new surface, which is disjoint union of X4 and of some vacuum space-time surface having vanishing Kähler action. Also this surface is absolute minimum!

Does this mean that the each minimum of the effective action is infinitely degenerate? Is it possible to pose some conditions on the physical state eliminating this kind of degeneracy? Or should one just accept this degeneracy as a prediction of TGD? Is this free imagination aspect always present in quantum state?

23

Note: the vacuum surfaces are very much like vacuum bubbles of Quantum Field Theory appearing in any Feynman diagrams and not actually contributing to scattering amplitude.

24

3. Further illustrations related to TGD-inspired theory of Consciousness(http://www.helsinki.fi/~matpitka/illuc.html )

A. Visualizations of many-sheeted space-time concept

Figure 1: The presence of matter makes space-time curved and spoils translational and rotational symmetries. This means the loss of basic conservation laws.

Figure 2: (a) Future lightcone M 4 + of Minkowski space. (b) CP2 is obtained from C 3 by identifying the points related by complex scaling: z = lz .

Figure 3: An illustration of wormhole contacts between parallel space-time sheets having elementary particle size and of join along boundaries bonds which can have Macroscopic length.

25

Figure 4: 2-dimensional illustration of the many-sheeted space-time. Space-time sheets are connected to each other by wormhole contacts.

Figure 5: Schematic representation of the decomposition of the space-time surface to p-adic and real regions.

Figure 6: Generalized imbedding space is union of all p-adic imbedding spaces Hp and real imbedding space H intersecting along rational points common to all (Q denotes rationals, R for reals, and Rp for p-

adic numbers). In the figure, some of the imbedding spaces Hp are illustrated as half-planes.

26

Figure 7: Atomic space-time sheets are at high temperature and non-superconducting as Standard Physics predicts. But larger space-time sheets can have very low temperature and superconduct.

Figure 8: (a) 2-dimensional space-time illustration of topological light ray (massless extremal ME). Field pattern propagates with light velocity preserving its shape. (b) MEs serve as field bridges

between space-time sheets and make possible both classical and quantum communications.

Figure 9: The classical non-determinism of Kaehler action forces to generalize the notion of 3-surface by allowing sequences of space-like 3-surfaces with time-like separations. These sequences

have interpretation as linguistic expressions providing a representation for quantum jump sequence defining the contents of consciousness of self.

27

B. Visualizations related to the TGD-inspired theory of Consciousness

Figure 1: Schrödinger cat has no self identity because it entangles with a quantum bottle of poison and is simultaneously both dead and alive.

Figure 2: The simplest manner to understand psychological time is as the center of mass temporal coordinate for a mind-like space-time sheet. The arrow of psychological time results from

the drift of the mind-like space-time sheet to the direction of Future induced by the geometry of the Future lightcone.

Figure 3: Psychological now corresponds to the phase transition front at which p-adic space-time regions (blue) representing intentions are transformed to real space-time regions

representing actions and memories (green).

28

Figure 4: Fusion and sharing of mental images occurs if subselves (mental images) of two selves entangle to form a more complex 'stereo' mental image. This is not possible without length

scale dependent notion of subsystem.

C. Visualizations related to the basic quantum mechanisms of Consciousness and Bio-control

Figure 1: The ends of the magnetic flux tube can act as mirrors at which topological light rays (MEs) are reflected. Oscillations of magnetic flux tube can also amplify the signals carried by

MEs.

29

Figure 2: Mirror mechanism of long -term memory. To "remember" is to look at magnetic mirror at distance of light-years.

Figure 3: The transformation of intention to action corresponds to a quantum jump in which p-adic space-time region is transformed to a real one.

30

Figure 4: Magnetic mirrors (topological light rays associated with magnetic flux tubes) act as sensory projectors from brain (and body) to the magnetic body acting as sensory canvas.

Figure 5: MEs can serve as bridges between various space-time sheets. This induces a leakage of the supra currents from magnetic flux tubes (k=169) to the atomic space-time sheets (k=137)

and vice versa. Also, the time reversal of this process can occur (k=151 denotes cell membrane space-time sheet).

Figure 6: Many-sheeted space-time can be regarded as an extremely complex Feynman diagram with lines thickened to space-time sheets. The figure illustrates the idea for very simple Feynman

diagram describing boson exchange.

31

4. p-Adic fractals(http://www.helsinki.fi/~matpitka/padfract.pdf )

Figure 1. The real norm induced by canonical identification from 2-adic norm

Figure 2. p-Adic function x 2 for some values of p

32

Figure 3. p-Adic function 1/x for some value of p

Figure 4. The graph of the real part of 2-adically analytic z 2 = function

33

Figure 5. The graph of 2-adically analytic Im(z2) = 2xy function

34

5. see also partly TGD-inspired graphics of Mark Thornally(http://www.helsinki.fi/~matpitka/mthorn.html )

if on the Internet, Press <BACK> on your browser to return to the previous page (or go to www.stealthskater.com)

else if accessing these files from the CD in a MS-Word session, simply <CLOSE> this file's window-session; the previous window-session should still remain 'active'

35