Thursday, May 31, 2007

Overall view about construction of S-matrix

I have the feeling that the construction of S-matrix in TGD framework is now reasonably well-understood. I however realized that the chapter whose purpose was to give overall view about situation was hopelessly messy. The idea to represent all twists of the Odysseia leading to the final enlightment was not so good as it looked first. Therefore I decided to re-organize completely the chapter representing just the ideas and results. This process led to nice new results. In particular, thermodynamics becomes in zero energy ontology unavoidably an essential part of quantum theory. I glue below the abstract of the re-organized chapter Towards S-matrix of a book with same title.
The construction of S-matrix has been key challenge of quantum TGD from the very beginning when it had become clear that path integral approach and canonical quantization make no sense in TGD framework. My intuitive feeling that the problems are not merely technical has turned out to be correct. In this chapter the overall view about the construction of S-matrix is discussed. It is perhaps wise to summarize briefly the vision about S-matrix.

  1. S-matrix defines entanglement between positive and negative energy parts of zero energy states. This kind of S-matrix need not be unitary unlike the U-matrix associated with unitary process forming part of quantum jump. There are several good arguments suggesting that that S-matrix cannot unitary but can be regarded as thermal S-matrix so that thermodynamics would become an essential part of quantum theory. In TGD framework path integral formalism is given up although functional integral over the 3-surfaces is present.

  2. Almost topological QFT property of quantum allows to identify S-matrix as a functor from the category of generalized Feynman cobordisms to the category of operators mapping the Hilbert space of positive energy states to that for negative energy states: these Hilbert spaces are assignable to partonic 2-surfaces. Feynman cobordism is the generalized Feynman diagram having light-like 3-surfaces as lines glued together along their ends defining vertices as 2-surfaces. This picture differs dramatically from that of string models. There is a functional integral over the small deformations of Feynman cobordisms corresponding to maxima of Kähler function. It is difficult to overestimate the importance of this result bringing category theory absolutely essential part of quantum TGD.
  3. Imbedding space degrees of freedom seem to imply the presence of factor of type I beside HFF of type II1 for which unitary S-matrix can define time-like entanglement coefficients. Only thermal S-matrix defines a normalizable zero energy state so that thermodynamics becomes part of quantum theory. One can assign to S-matrix a complex parameter whose real part has interpretation as interaction time and imaginary part as the inverse temperature. S-matrices and thus also quantum states in zero energy ontology possess a semigroup like structure and in the product time and inverse temperature are additive. This suggests that the cosmological evolution of temperature as T propto 1/t could be understood at the level of fundamental quantum theory.

  4. S-matrix should be constructible as a generalization of braiding S-matrix in such a manner that the number theoretic braids assignable to light-like partonic 3-surfaces glued along their ends at 2-dimensional partonic 2-surfaces representing reaction vertices replicate in the vertex.

  5. The construction of braiding S-matrices assignable to the incoming and outgoing partonic 2-surfaces is not a problem. The problem is to express mathematically what happens in the vertex. Here the observation that the tensor product of HFFs of type II is HFF of type II is the key observation. Many-parton vertex can be identified as a unitary isomorphism between the tensor product of incoming resp. outgoing HFFs. A reduction to HFF of type II1 occurs because only a finite-dimensional projection of S-matrix in bosonic degrees of freedom defines a normalizable state. In the case of factor of type II only thermal S-matrix is possible without finite-dimensional projection and thermodynamics would thus emerge as an essential part of quantum theory.

  6. HFFs of type III could also appear at the level of field operators used to create states but at the level of quantum states everything reduces to HFFs of type II1 and their tensor products giving these factors back. If braiding automorphisms reduce to the purely intrinsic unitary automorphisms of HFFs of type III then for certain values of the time parameter of automorphism having interpretation as a scaling parameter these automorphisms are trivial. These time scales could correspond to p-adic time scales so that p-adic length scale hypothesis would emerge at the fundamental level. In this kind of situation the braiding S-matrices associated with the incoming and outgoing partons could be trivial so that everything would reduce to this unitary isomorphism: a counterpart for the elimination of external legs from Feynman diagram in QFT.

  7. One might hope that all complications related to what happens for space-like 3-surfaces could be eliminated by quantum classical correspondence stating that space-time view about particle reaction is only a space-time correlate for what happens in quantum fluctuating degrees of freedom associated with partonic 2-surfaces. This turns out to be the case only in non-perturbative phase. The reason is that the arguments of n-point function appear as continuous moduli of Kähler function. In non-perturbative phases the dependence of the maximum of Kähler function on the arguments of n-point function cannot be regarded as negligible and Kähler function becomes the key to the understanding of these effects including formation of bound states and color confinement.

  8. In this picture light-like 3-surface would take the dual role as a correlate for both state and time evolution of state and this dual role allows to understand why the restriction of time like entanglement to that described by S-matrix must be made. For fixed values of moduli each reaction would correspond to a minimal braid diagram involving exchanges of partons being in one-one correspondence with a maximum of Kähler function. By quantum criticality and the requirement of ideal quantum-classical correspondence only one such diagram would contribute for given values of moduli. Coupling constant evolution would not be however lost: it would be realized as p-adic coupling constant at the level of free states via the log(p) scaling of eigen modes of the modified Dirac operator.

  9. A completely unexpected prediction deserving a special emphasis is that number theoretic braids replicate in vertices. This is of course the braid counterpart for the introduction of annihilation and creation of particles in the transition from free QFT to an interacting one. This means classical replication of the number theoretic information carried by them. This allows to interpret one of the TGD inspired models of genetic code in terms of number theoretic braids representing at deeper level the information carried by DNA. This picture provides also further support for the proposal that DNA acts as topological quantum computer utilizing braids associated with partonic light-like 3-surfaces (which can have arbitrary size). In the reverse direction one must conclude that even elementary particles could be information processing and communicating entities in TGD Universe.

Sunday, May 27, 2007

Quantum chaos in astrophysical length scales?

Kea commented about transition to quantum chaos and gave a link to the article Quantum Chaos by Martin Gurtzwiller in Matthew Watkins's home page devoted to Riemann Zeta. Occasionally even this kind of a masterpiece of scientific writing manages to stimulate only an intention to read it more carefully later. When you indeed read it again few years later it can shatter you into a wild resonance. Just this occurred at this time.

1. Brief summary about quantum chaos

The article of Gurtzwiller discusses the complex regime between quantal and classical behavior as it was understood at the time of writing (1992). As a non-specialist I have no idea about possible new discoveries since then.

The article introduces the division of classical systems into regular (R) and chaotic (P in honor of Poincare) ones. Besides this one has quantal systems (Q). There are three transition regions between these three realms.

  1. R-P corresponds to transition to classical chaos and KAM theorem is a powerful tool allowing to organize the view about P in terms of surviving periodic orbits.

  2. Quantum-classical transition region R-Q corresponds to high quantum number limit and is governed by Bohr's correspondence principle. Highly excited hydrogen atom - Rydberg atom - defines a canonical example of the situation.

  3. Somewhat surprisingly it has turned out that also P-Q region can be understood in terms of periodic classical orbits (nothing else is available!). P-Q region can be achieved experimentally if one puts Rydberg atom in a strong magnetic field. At the weak field limit quantum states are delocalized but in chaotic regime the wave functions become strongly concentrated along a periodic classical orbits.

    At the level of dynamics the basic example about P-Q transition region discussed is the chaotic quantum scattering of electron in atomic lattice. Classical description does not work: a superposition of amplitudes for orbits, which consist of pieces which are fragments of a periodic orbit plus localization around atom is necessary.

The fractal wave function patterns associated with say hydrogen atom in strong magnetic field are extremely beautiful and far from chaotic. Even in the case of chaotic quantum scattering one has interference of quantum amplitudes for classical Bohr orbits and also now Fourier transform exhibits nice peaks corresponding to the periods of classical orbits. The term chaos seems to be an unfortunate choice referring to our limited cognitive capacities rather than the actual physical situation and the term quantum complexity would be more appropriate. For a quantum consciousness theorist the challenge is to try to formulate in a more precise manner this fact. Quantum measurement theory with a finite measurement resolution indeed provide the mathematics necessary for this purpose.

2. What does the transition to quantum chaos mean?

The transition to quantum chaos in the sense the article discusses it means that a system with a large number of virtually independent degrees of freedom (in very general sense) makes a transition to a phase in there is a strong interaction between these degrees of freedom. Perturbative phase becomes non-perturbative. This means emergence of correlations and reduction of the effective dimension of the system to a finite fractal dimension. When correlations become complete and the system becomes a genuine quantum system, the dimension of the system is genuinely reduced and again non-fractal. In this sense one has transition via complexity to new kind of order.

2.1 The level of stationary states

At the level of energy spectrum this means that the energy of system which correspond to sums of virtually independent energies and thus is essentially random number becomes non-random. As a consequence, energy levels tend to avoid each other, order and simplicity emerge but at the collective level. Spectrum of zeros of Zeta has been found to simulate the spectrum for a chaotic system with strong correlations between energy levels. Zeta functions indeed play a key role in the proposed description of quantum criticality associated with the phase transition changing the value of Planck constant.

2.2 The importance of classical periodic orbits in chaotic scattering

Poincare with his immense physical and mathematical intuition foresaw that periodic classical orbits should have a key role also in the description of chaos. The study of complex systems indeed demonstrates that this is the case although the mathematics and physics behind this was not fully understood around 1992 and is probably not so even now. The basic discovery coming from numerical simulations is that the Fourier transform of a chaotic orbits exhibits has peaks the frequencies which correspond to the periods of closed orbits. From my earlier encounters with quantum chaos I remember that there is quantization of periodic orbits so that their periods are proportional to log(p), p prime in suitable units. This suggests a connection of arithmetic quantum field theory and with p-adic length scale hypothesis. Note that in planetary Bohr orbitology any closed orbit can be Bohr orbits with a suitable mass distribution but that velocity spectrum is universal.

The chaotic scattering of electron in atomic lattice is discussed as a concrete example. In the chaotic situation the notion of electron consists of periods spend around some atom continued by a motion along along some classical periodic orbit. This does not however mean loss of quantum coherence in the transitions between these periods: a purely classical model gives non-sensible results in this kind of situation. Only if one sums scattering amplitudes over all piecewise classical orbits (not all paths as one would do in path integral quantization) one obtains a working model.

2.3. In what sense complex systems can be called chaotic?

Speaking about quantum chaos instead of quantum complexity does not seem appropriate to me unless one makes clear that it refers to the limitations of human cognition rather than to physics. If one believes in quantum approach to consciousness, these limitations should reduce to finite resolution of quantum measurement not taken into account in standard quantum measurement theory.

In the framework of hyper-finite factors of type II1 finite quantum measurement resolution is described in terms of inclusions N subset M of the factors and sub-factor N defines what might be called N-rays replacing complex rays of state space. The space M/N has a fractal dimension characterized by quantum phase and increases as quantum phase q=exp(iπ/n), n=3,4,..., approaches unity which means improving measurement resolution since the size of the factor N is reduced.

Fuzzy logic based on quantum qbits applies in the situation since the components of quantum spinor do not commute. At the limit n→∞ one obtains commutativity, ordinary logic, and maximal dimension. The smaller the n the stronger the correlations and the smaller the fractal dimension. In this case the measurement resolution makes the system apparently strongly correlated when n approaches its minimal value n=3 for which fractal dimension equals to 1 and Boolean logic degenerates to single valued totalitarian logic.

Non-commutativity is the most elegant description for the reduction of dimensions and brings in reduced fractal dimensions smaller than the actual dimension. Again the reduction has interpretation as something totally different from chaos: system becomes a single coherent whole with strong but not complete correlation between different degrees of freedom. The interpretation would be that in the transition to non-chaotic quantal behavior correlation becomes complete and the dimension of system again integer valued but smaller. This would correspond to the cases n=6, n=4, and n=3 (D=3,2,1).

3. Quantum chaos in astrophysical scales?

3.1 Quantum criticality

  1. TGD Universe is quantum critical. The most important implication of quantum criticality of TGD Universe is that it fixes the value of Kähler coupling strength, the only free parameter appearing in definition of the theory as the analog of critical temperature. The dark matter hierarchy characterized partially by the increasing values of Planck constant allows to characterize more precisely what quantum criticality might means. By quantum criticality space-time sheets are analogs of Bohr orbits. Since quantum criticality corresponds to P-Q region, the localization of wave functions around generalized Bohr orbits should occur quite generally in some scale.

  2. Elementary particles are maximally quantum critical systems analogous to H2O at tri-critical point and can be said to be in the intersection of imbedding spaces labelled by various values of Planck constants. Planck constant does not characterize the elementary particle proper. Rather, each field body of particle (em, weak, color, gravitational) is characterized by its own Planck constant and this Planck constant characterizes interactions. The generalization of the notion of the imbedding space allows to formulate this idea in precise manner and each sector of imbedding space is characterized by discrete symmetry groups Zn acting in M4 and CP2 degrees of freedom. The transition from quantum to classical corresponds to a reduction of Zn to subgroup Zm, m factor of n. Ruler-and-compass hypothesis implies very powerful predictions for the remnants of this symmetry at the level of visible matter. Note that the reduction of the symmetry in this chaos-to-order transition!

  3. Dark matter hierarchy makes TGD Universe an ideal laboratory for studying P-Q transitions with chaos identified as quantum critical phase between two values of Planck constant with larger value of Planck constant defining the "quantum" phase and smaller value the "classical" phase. Dark matter is localized near Bohr orbits and is analogous to quantum states localized near the periodic classical orbits. Planetary Bohr orbitology provides a particularly interesting astrophysical application of quantum chaos.

  4. The above described picture for chaotic quantums scattering applies quite generally in quantum TGD. Path integral is replaced with a functional integral over classical space-time evolutions and the failure of the complete classical non-determinism is analogous to the transition between classical orbits. Functional integral also reduces to perturbative functional integral around maxima of Kähler function.

3.2 Rings and spokes as the basic building blocks of dark matter structures

The Bohr orbit model for the planetary orbits based on the hierarchy of dark matter relies in an essential manner on the idea that macroscopic quantum phases of dark matter dictate to a high degree the behavior of the visible matter. Dark matter is concentrated on closed classical orbits in the simple rotationally symmetric gravitational potentials involved. Orbits become basic structures instead of points at the level of dark matter. A discrete subgroup Zn of rotational group with very large n characterizes dark matter structures quite generally. At the level of visible matter this symmetry can be broken to approximate symmetry defined by some subgroup of Zn.

Circles and radial spokes are the basic Platonic building blocks of dark matter structures. The interpretation of spokes would be as (gravi-)electric flux tubes. Radial spokes correspond to n=0 states in Bohr quantization for hydrogen atom and orbits ending into atom. Spokes have been observed in planetary rings besides decomposition to narrow rings and also in galactic scale. Also flux tubes of (gravi-)magnetic fields with Zn symmetry define rotational symmetric structures analogous to quantized dipole fields.

Gravi-magnetic flux tubes indeed correspond to circles rather than field lines of a dipole field for the simplest model of gravi-magnetic field, which means deviation from GRT predictions for gravi-magnetic torque on gyroscope outside equator: unfortunately the recent experiments are performed at equator. The flux tubes be seen only as circles orthogonal to the preferred plane and planetary Bohr rules apply automatically also now.

A word of worry is in order here. Ellipses are very natural objects in Bohr orbitology and for a given value of n would give n2-1 additional orbits. In planetary situation they would have very large eccentricities and are not realized. Comets can have closed highly eccentric orbits and correspond to large values of n. In any case, one is forced to ask whether the exactly Zn symmetric objects are too Platonic creatures to live in the harsh real world. Should one at least generalize the definition of the action of Zn as symmetry so that it could rotate the points of ellipse to each other. This might make sense. In the case of dark matter ellipses the radial spokes with Zn symmetry representing radial gravito-electric flux quanta would still connect dark matter ellipse to the central object and the rotation of the spoke structure induces a unique rotation of points at ellipse.

3.3. Dark matter structures as generalization of periodic orbits

The matter with ordinary or smaller value of Planck constant can form bound states with these dark matter structures. The dark matter circles would be the counterparts for the periodic Bohr orbits dictating the behavior of the quantum chaotic system. Visible matter (and more generally, dark matter at the lower levels of hierarchy behaving quantally in shorter length and time scales) tends to stay around these periodic orbits and in the ideal case provides a perfect classical mimicry of quantum behavior. Dark matter structures would effectively serve as selectors of the closed orbits in the gravitational dynamics of visible matter.

As one approaches classicality the binding of the visible matter to dark matter gradually weakens. Mercury's orbit is not quite closed, planetary orbits become ellipses, comets have highly eccentric orbits or even non-closed orbits. For non-closed quantum description in terms of binding to dark matter does not makes sense at all.

The classical regular limit (R) would correspond to a decoupling between dark matter and visible matter. A motion along geodesic line is obtained but without Bohr quantization in gravitational sense since Bohr quantization using ordinary value of Planck constant implies negative energies for GMm>1. The preferred extremal property of the space-time sheet could however still imply some quantization rules but these could apply in "vibrational" degrees of freedom.

3.4 Quantal chaos in gravitational scattering?

The chaotic motion of astrophysical object becomes the counterpart of quantum chaotic scattering. By Equivalence Principle the value of the mass of the object does not matter at all so that the motion of sufficiently light objects in solar system might be understandable only by assuming quantum chaos.

The orbit of a gravitationally unbound object such as comet could define the basic example. The rings of Saturn and Jupiter could represent interesting shorter length scale phenomena possible involving quantum scattering. One can imagine that the visible matter object spends some time around a given dark matter circle (binding to atom), makes a transition along radial spoke to the next circle, and so on.

The prediction is that dark matter forms rings and cart-wheel like structures of astrophysical size. These could become visible in collisions of say galaxies when stars get so large energy as to become gravitationally unbound and in this quantum chaotic regime can flow along spokes to new Bohr orbits or to gravi-magnetic flux tubes orthogonal to the galactic plane. Hoag's object represents a beautiful example ring galaxy. Remarkably, there is also direct evidence for galactic cart-wheels. There are also polar ring galaxies consisting of an ordinary galaxy plus ring approximately orthogonal to it and believed to form in galactic collisions. The ring rotating with the ordinary galaxy can be identified in terms of gravi-magnetic flux tube orthogonal to the galactic plane: in this case Zn symmetry would be completely broken.

For more details see the new chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-Time".

Friday, May 25, 2007

Two objections against planetary Bohr orbitology

There are two objections against planetary Bohr orbitology.
  1. Kea already mentioned in her comment to the previous posting the first objection. The success of this approach in the case solar system is not enough. In particular, it requires different values of v0 for inner and outer planets.

  2. The basic objection of General Relativist against the planetary Bohr orbitology model is the lack of the manifest General Goordinate and Lorentz invariances. In GRT context this objection would be fatal. In TGD framework the lack of these invariances is only apparent.

1. Also exoplanets obey Bohr rules

In the previous posting I proposed a simple model explaining why inner and outer planets must have different values of v0 by taking into account cosmic string contribution to the gravitational potential which is negligible nowadays but was not so in primordial times. Among other things this implies that planetary system has a finite size, at least about 1 ly in case of Sun (nearest star is at distance of 4 light years).

I have also applied the quantization rules to exoplanets in the case that the central mass and orbital radius are known. Errors are around 10 per cent for the most favored value of v0=2-11 (see this). The "anomalous" planets with very small orbital radius correspond to n=1 Bohr orbit (n=3 is the lowest orbit in solar system). The universal velocity spectrum v= v0/n in simple systems perhaps the most remarkable prediction and certainly testable: this alone implies that the Bohr radius GM/v02 defines the universal size scale for systems involving central mass. Obviously this is something new and highly non-trivial.

The recently observed dark ring in MLy scale is a further success and also the rings and Moons of Saturn and Jupiter obey the same universal length scale (n≥ 5 and v0→ (16/15)×v0 and v0→ 2×v0).

There is a further objection. For our own Moon orbital radius is much larger than Bohr radius for v0=2-11: one would have n≈138. n≈7 results for v0 →v0/20 giving r0≈ 1.2 RE. The small value of v0 could be understood to result from a sequence of phase transitions reducing the value of v0 to guarantee that solar system participates in the average sense to the cosmic expansion and from the fact inner planets are older than outer ones in the proposed scenario.

Remark: Bohr orbits cannot participate in the expansion which manifests itself as the observed apparent shrinking of the planetary orbits when distances are expressed in terms Robertson-Walker radial coordinate r=rM. This anomaly was discovered by Masreliez and is discussed here. Ruler-and-compass hypothesis suggests preferred values of cosmic times for the occurrence of these transitions. Without this hypothesis the phase transitions could form almost continuum.

2. How General Coordinate Invariance and Lorentz invariance are achieved?

One can use Minkowski coordinates of the M4 factor of the imbedding space H=M4×CP2 as preferred space-time coordinates. The basic aspect of dark matter hierarchy is that it realizes quantum classical correspondence at space-time level by fixing preferred M4 coordinates as a rest system. This guarantees preferred time coordinate and quantization axis of angular momentum. The physical process of fixing quantization axes thus selects preferred coordinates and affects the system itself at the level of space-time, imbedding space, and configuration space (world of classical worlds). This is definitely something totally new aspect of observer-system interaction.

One can identify in this system gravitational potential Φgr as the gtt component of metric and define gravi-electric field Egr uniquely as its gradient. Also gravi-magnetic vector potential Agr and and gravimagnetic field Bgrcan be identified uniquely.

3. Quantization condition for simple systems

Consider now the quantization condition for angular momentum with Planck constant replaced by gravitational Planck constant hbargr= GMm/v0 in the simple case of pointlike central mass. The condition is

m∫ v•dl = n × hbargr

The condition reduces to the condition on velocity circulation

∫ v•dl = n × GM/v0.

In simple systems with circular rings forced by Zn symmetry the condition reduces to a universal velocity spectrum


so that only the radii of orbits depend on mass distribution. For systems for which cosmic string dominates only n=1 is possible. This is the case in the case of stars in galactic halo if primordial cosmic string going through the center of galaxy in direction of jet dominates the gravitational potential. The velocity of distant stars is correctly predicted.

Zn symmetry seems to imply that only circular orbits need to be considered and there is no need to apply the condition for other canonical momenta (radial canonical momentum in Kepler problem). The nearly circular orbits of visible matter objects would be naturally associated with dark matter rings or more complex structures with Zn symmetry and dark matter rings could suffer partial or complete phase transition to visible matter.

4. Generalization of the quantization condition

  1. By Equivalence Principle dark ring mass disappears from the quantization conditions and the left hand side of the quantization condition equals to a generalized velocity circulation applying when central system rotates

    ∫ (v-Agr)•dl .

    Here one must notice that dark matter ring is Zn symmetric and closed so that the geodesic motion of visible matter cannot correspond strictly to the dark matter ring (perihelion shift of Mercury). Just by passing notice that the presence of dark matter ring can explain also the complex braidings associated with the planetary rings.

  2. Right hand side would be the generalization of GM by the replacement

    GM → ∫ e•r2Egr × dl .

    e is a unit vector in direction of quantization axis of angular momentum, × denotes cross product, and r is the radial M4 coordinate in the preferred system. Everything is Lorentz and General Coordinate Invariant and for Schwartschild metric this reduces to the expected form and reproduces also the contribution of cosmic string to the quantization condition correctly.

For more details see the new chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-Time".

A simple quantum model for the formation of astrophysical structures

The stimulus to the latest little calculation in astro-Bohr-orbitology came when Kea mentioned Saturn's rings in her blog. These rings look indeed very quantal and after having recovered from the week-end in the wedding of the son of an old friend, I decided to look for a simple model for them.

Quite generally, the mechanisms behind the formation of planetary systems, galaxies and larger systems are poorly understood but planar structures seem to define a common denominator and the recent discovery of dark matter ring in a galactic cluster in Mly scale (see this) suggest that dark matter rings might define a universal step in the formation of astrophysical structures.

Also the dynamics in planet scale is poorly understood. In particular, the rings of Saturn and Jupiter are very intricate structures and far from well-understood. Assuming spherical symmetry it is far from obvious why the matter ends up to form thin rings in a preferred plane. The latest surprise is that Saturn's largest, most compact ring consist of clumps of matter separated by almost empty gaps. The clumps are continually colliding with each other, highly organized, and heavier than thought previously.

The situation suggests that some very important piece might be missing from the existing models, and the vision about dark matter as a quantum phase with a gigantic Planck constant (see this and this) is an excellent candidate for this piece. The vision that the quantum dynamics for dark matter is behind the formation of the visible structures suggests that the formation of the astrophysical structures could be understood as a consequence of Bohr rules.

1. General quantum vision about formation of structures

The basic observation is that in the case of a straight cosmic string creating a gravitational potential of form v12/ρ Bohr quantization does not pose any conditions on the radii of the circular orbits so that a continuous mass distribution is possible.

This situation is obviously exceptional. If one however accepts the TGD based vision (see this) that the very early cosmology was cosmic string dominated and that elementary particles were generated in the decay of cosmic strings, this situation might have prevailed at very early times. These cosmic strings can transform to strings with smaller string tension and magnetic flux tubes can be seen as their remnants dark energy being identifiable as magnetic energy. If so, the differentiation of a continuous density of ordinary matter to form the observed astrophysical structures would correspond to an approach to a stationary situation governed by Bohr rules and in the first approximation one could neglect the intermediate stages.

Cosmic string need not be infinitely long: it could branch into n return flux tubes, n very large in accordance with the Zn symmetry for the dark matter but also in this case the situation in the nearby region remains the same. For large distances the whole structure would behave as a single mass point creating ordinary Newtonian gravitational potential. Also phase transitions in which the system emits magnetic flux tubes so that the contribution of the cosmic string to the gravitational force is reduced, are possible.

What is of utmost importance is that the cosmic string induces the breaking of the rotational symmetry down to a discrete Zn symmetry and in the presence of the central mass selects a unique preferred orbital plane in which gravitational acceleration is parallel to the plane. This is just what is observed in astrophysical systems and not easily explained in the Newtonian picture. In TGD framework this relates directly to the choice of quantization axis of angular momentum at the level of dark matter. This mechanism could be behind the formation of planar systems in all length scales including planets and their moons, planetary systems, galaxies, galaxy clusters in the scale of Mly, and even the concentration of matter at the walls of large voids in the scale of 100 Mly.

The Zn symmetry for the dark matter with very large n suggests the possibility of more precise predictions. If n is a ruler-and-compass integer it has as factors only first powers of Fermat primes and a very large power of 2. The breaking of Zn symmetry at the level of visible matter would naturally occur to subgroups Zm subset Zn. Since m is a factor of n, the average number of matter clumps could tend to be a factor of n, and hence a ruler-and-compass integer. Also the hexagonal symmetry discovered near North Pole of Saturn (see this)could have interpretation in terms of this symmetry breaking mechanism.

2. How inner and outer planets might have emerged?

The Bohr orbit model requires different values of the parameter v0 related by a scaling v0→v0/5 for inner and outer planets. It would be nice to understand why this is the case. The presence of cosmic string along rotational axis implied both by the model for the asymptotic state of the star and TGD based model for gamma ray bursts might allow to understand this result.

One can construct a simple modification of the hydrogen atom type model for solar system by including the contribution of cosmic string to the gravitational force. For circular orbits the condition identifying kinetic and gravitational radial accelerations plus quantization of angular momentum in units of gravitational Planck constant are used. The prediction is that only a finite number of Bohr orbits are possible. One might hope that this could explain the decomposition of the planetary system to inner and outer planets.

String tension implies anomalous acceleration of same form as the radial kinetic acceleration implying that for given radius kinetic energy per mass is shifted upwards by a constant amount. This acceleration anomaly is severely bounded above by the constant acceleration anomaly of space-crafts (Pioneer anomaly) and for the recent value of the cosmic string tension the number of allowed inner planets is much larger than 3.

The situation was however different in the primordial stage when cosmic string tension was much larger and gradually reduced in phase transitions involving the emission of closed magnetic flux tubes. Primordial Sun could have emitted the seeds of the two planetary systems related by scaling and that this might have happened in the phase transition reducing magnetic flux by the emission of closed magnetic flux tube structure.

3. Models for the interior of astrophysical object and for planetary rings

Using similar quantization conditions one can construct a very simple model of astrophysical object as a cylindrically symmetric pancake like structure. There are three basic predictions which do not depend on the details of the mass distribution.

  1. The velocity spectrum for the circular orbits is universal and given as v=v0/n and that only the radii of Bohr orbits of particles depend on the form of average mass distribution which can vary in wide limits.

  2. Velocity does not decrease with distance and is constant in the presence of only cosmic string.

  3. The size of the system is always finite and increasing values of n correspond to decreasing radii. This came as a complete surprise, and is a complete opposite for what hydrogen atom like model without cosmic string predicts (when cosmic string is introduced planetary system for given value of v0 has necessarily a finite size).

Four mass distributions were tested.

  1. First corresponds to a power law, second to a logarithmic velocity distribution and third to a spectrum of orbital radii coming as powers of 2 in accordance with p-adic length scale hypothesis. Fourth mass distribution corresponds to evenly spaced Bohr radii below certain radius.

  2. Only the third option works as a model for say Earth and predicts that dark matter forms an onionlike structure with the radii of shells coming as powers of 2 (of 21/2 in the most general formulation of p-adic length scale hypothesis). This prediction is universal and means that dark matter part of stellar objects would be very much analogous to atom having also shell like structure. Actually this is not surprising.

  3. Second and fourth option could define a reasonable model for ring like structures (Saturn's and Jupiter's rings). The predicted universal velocity spectrum for dark rings serves as a test for the model.

For more details see the new chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-Time".

P. S. In New Scientist there is an article Do magnetic fields drive dark energy?. Unfortunately, I do not have access to it. Suffice it to say that I have made this identification of dark energy long ago (see for instance this and this), much before the idea about dark matter as large Planck constant phase.

Thursday, May 17, 2007

Category theoretical formulation of quantum TGD in nutshell

For long time it has been clear that category theory might provide a fundamental formulation of quantum TGD. The problem has been that category theory seems to postulate quite too many objects. The reading of Quantum Quandaries by John Baez helped to see the situation in all its simplicity.

  1. Topological Quantum Field theories have extremely simple formulation as a functor from the category of cobordisms (topological evolutions between n-1-manifolds by connecting n-manifold) to the category of Hilbert spaces assignable to n-1-manifolds.

  2. Since light-like partonic 3-surfaces correspond to almost topological QFT, with the overall important "almost" coming just from the light-likeness in the induced metric, the theory is non-trivial physically and nothing of the beauty of TQFT as a functor is lost. Cobordisms are however replaced by what I have christened Feynmann cobordisms generalizing the Feynman diagrams to 3-D context: the ends of light-like 3-manifolds meet at the vertices which correspond to 2-dimensional partonic surfaces.

  3. Also the counterparts of ordinary string diagrams having interpretation as ordinary cobordisms are possible but have nothing to do with particle reactions: the particle simply decomposes into several pieces and spinor fields propagate along different routes. This is the space-time correlate for what happens in double slit experiment when photon travels along two different paths simultaneously.

  4. The intriguing results is that for n<4-dimensional cobordisms unitary S-matrix exists only for trivial cobordisms. I wonder whether string theorists have considered the possible catastrophic consequences concerning the non-perturbative dream about the unique stringy S-matrix. In the zero energy ontology of TGD S-matrix appears as time-like entanglement coefficients and need not be unitary. I have already proposed that p-adic thermodynamics and thermodynamics in general could be regarded as an exact part of quantum theory in this framework and the basic mathematics of hyper-finite factors provides strong technical support for this idea. It could be that one cannot require unitarity in the case of Feynman cobordisms and that only the condition that S-matrix for a product of Feynman cobordisms is a product of S-matrices for composites. Hence the time parameter in S-matrix can be replaced with complex time parameter with imaginary part in the role of temperature without losing the product structure. p-Adic thermodynamics and particle massication might be topological necessities in this framework.

For more details see the chapter Construction of Quantum Theory: S-Matrix of "Towards S-Matrix".

Tuesday, May 15, 2007

The dark matter ring found by NASA corresponds to the lowest Bohr orbit

I told yesterday about dark matter ring found by NASA. TGD based model for dark matter inspires the hypothesis that it corresponds to Bohr orbit for macroscopically quantum coherent dark matter with gigantic value of Planck constant predicted by the model. The article about finding is now in archive and contains the data making possible to test the model. I am grateful for Kea for providing the link. The ring corresponds with a good accuracy to the lowest Bohr orbit for v0= 3×2-11, which is 3 times the favored value but allowed by the general hypothesis for the favored values of Planck constant.

I added the first version of the little calculation to the previous posting. Unfortunately it contained besides an innocent error in the formula of Bohr radius also a numerical error giving a result which was exactly 10 times too small. The erratic calculation however happened to give the correct result for v0=2-11, which is the preferred value. In some magic manner mistakes conspire to give the desired result and ridicule the poor theoretician! To minimize confusion I deleted the original calculation and added the corrected calculation here.

The number theoretic hypothesis for the preferred values of Planck constants states that the gravitational Planck constant

hbar= GMm/v0

equals to a ruler-and-compass rational which is ratio q= n1/n2 of ruler-and-compass ni integers expressible as a product of form n=2k∏ Fs, where all Fermat primes Fs are different. Only four of them are known and they are given by 3, 5, 17, 257, 216+1. v0=2-11 applies to inner planets and v0=2-11/5 to outer planets and the conditions from the quantization of hbar are satisfied.

The obvious TGD inspired hypothesis is that the dark matter ring corresponds to Bohr orbit. Hence the distance would be

r= n2 r0,

where r0 is Bohr radius and n is integer. n=1 for lowest Bohr orbit. The Bohr radius is given


where M the total mass in the dense core region inside the ring. This would give distance of about 2000 times Schwartschild radius for the lowest orbit for the preferred value of v0=2-11.

This prediction can be confronted with the data since the article Discovery of a ringlike dark matter structure in the core of the galaxy cluster C1 0024+17 is in the archive now.

  1. From the Summary and Conclusion part of the article the radius of the ring is about .4 Mpc, which makes in a good approximation 1.2 Mly (I prefer light years). More precisely - using arc second as a unit - the ring corresponds to a bump in the interval 60''-85'' centered at 75''. Figure 10 of of the article gives a good idea about the shape of the bump.

  2. From the article the mass in the dense core within radius which is almost half of the ring radius is about M=1.5×1014× MSun. The mass estimate based on gravitational lensing gives M=1.5×1014× MSun. If the gravitational lensing involves dark mass not in the central core, the first value can be used as the estimate. The Bohr radius this system is r0=GM/v02= 1.5×1014× r0(Sun),

    where I have assumed v0=2-11 as for the inner planets in the model for the solar system.

  3. The Bohr orbit for our planetary system predicts correctly Mercury's orbital radius as n=3 Bohr orbit for v0 =2-11 so that one has


    where rM is Mercury's orbital radius. One obtains

    r0= 1.5×1014× rM/9.

  4. Mercury's orbital radius is in a good approximation rM=.4 AU, and AU (the distance of Earth from Sun) is 1.5×1011 meters. 1 ly corresponds to .95×1016 meters. This gives

    r0 =11 Mly to be compared with 1.2 Mly deduced from observations. The result is by a factor 9 too large.

  5. If one replaces v0 with 3v0 one obtains downwards scaling by a factor of 1/9, which gives r0=1.2 Mly. The general hypothesis indeed allows to scale v0 by a factor 3.

  6. If one considers instead of Bohr orbits genuine solutions of Schrödinger equation then only n> 1 structures can correspond to rings like structures. Minimal option would be n=2 with v0 replaced with 6v0.

The conclusion would be that the ring would correspond to the lowest possible Bohr orbit for v0=3× 2-11. I would have been really happy if the favored value of v0 had appeared in the formula but the consistency with the ruler-and-compass hypothesis serves as a consolation. Skeptic can of course always argue that this is a pure accident. If so, it would be an addition to long series of accidents (planetary radii in solar system and radii of exoplanets). One can of course search rings at radii corresponding to n=2,3,... If these are found, I would say that the situation is settled.

For more details see the new chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-time"

Quantum Quandaries

This morning I encountered something which I should have read for long time ago. John Baez's Quantum Quandaries discusses in a physicist friendly manner the possible application of category theory to physics. The lessons obtained from the construction of topological quantum field theories (TQFTs) suggest that category theoretical thinking might be very useful in attempts to construct theories of quantum gravitation.

The point is that the Hilbert spaces associated with the initial and final state n-1-manifold of n-cobordism indeed form in a natural manner category. Morphisms of Hilb in turn are unitary or possibly more general maps between Hilbert spaces. TQFT itself is a functor assigning to a cobordism the counterpart of S-matrix between the Hilbert spaces associated with the initial and final n-1-manifold. The surprising result is that for n<4 the S-matrix can be unitary S-matrix only if the cobordism is trivial. This should lead even string theorist to raise some worried questions.

In the hope of feeding some category theoretic thinking into my spine, I briefly summarize some of the category theoretical ideas discussed in the article and relate it to the TGD vision, and after that discuss the worried questions from TGD perspective. That space-time makes sense only relative to imbedding space would conform with category theoretic thinking.

1. The *-category of Hilbert spaces

Baez considers first the category of Hilbert spaces. Intuitively the definition of this category looks obvious: take linear spaces as objects in category Set, introduce inner product as additional structure and identify morphisms as maps preserving this inner product. In finite-D case the category with inner product is however identical to the linear category so that the inner product does not seem to be absolutely essential. Baez argues that in infinite-D case the morphisms need not be restricted to unitary transformations: one can consider also bounded linear operators as morphisms since they play key role in quantum theory (consider only observables as Hermitian operators). For hyper-finite factors of type III inclusions define very important morphisms which are not unitary transformations but very similar to them. This challenges the belief about the fundamental role of unitarity and raises the question about how to weaken the unitarity condition without losing everything.

The existence of the inner product is essential only for the metric topology of the Hilbert space. Can one do without inner product as an inherent property of state space and reduce it to a morphism? One can indeed express inner product in terms of morphisms from complex numbers to Hilbert space and their conjugates. For any state Ψ of Hilbert space there is a unique morphisms TΨ from C to Hilbert space satisfying TΨ(1)=Ψ. If one assumes that these morphisms have conjugates T*Ψ mapping Hilbert space to C, inner products can be defined as morphisms T*Φ TΨ. The Hermitian conjugates of operators can be defined with respect to this inner product so that one obtains *-category. Reader has probably realized that TΨ and its conjugate correspond to ket and bra in Dirac's formalism.

Note that in TGD framework based on hyper-finite factors of type II1 (HFFs) the inclusions of complex rays might be replaced with inclusions of HFFs with included factor representing the finite measurement resolution. Note also the analogy of inner product with the representation of space-times as 4-surfaces of the imbedding space in TGD.

2. The monoidal *-category of Hilbert spaces and its counterpart at the level of nCob

One can give the category of Hilbert spaces a structure of monoid by introducing explicitly the tensor products of Hilbert spaces. The interpretation is obvious for physicist. Baez describes the details of this identification, which are far from trivial and in the theory of quantum groups very interesting things happen. A non-commutative quantum version of the tensor product implying braiding is possible and associativity condition leads to the celebrated Yang-Baxter equations: inclusions of HFFs lead to quantum groups too.

At the level of nCob the counterpart of the tensor product is disjoint union of n-1-manifolds. This unavoidably creates the feeling of cosmic loneliness. Am I really a disjoint 3-surface in emptiness which is not vacuum even in the geometric sense? Cannot be true!

This horrifying sensation disappears if n-1-manifolds are n-1-surfaces in some higher-dimensional imbedding space so that there would be at least something between them. I can emit a little baby manifold moving somewhere perhaps being received by some-one somewhere and I can receive radiation from some-one at some distance and in some direction as small baby manifolds making gentle tosses on my face!

This consoling feeling could be seen as one of the deep justifications for identifying fundamental objects as light-like partonic 3-surfaces in TGD framework. Their ends correspond to 2-D partonic surfaces at the boundaries of future or past directed light-cones (states of positive and negative energy respectively) and are indeed disjoint but not in the desperately existential sense as 3-geometries of General Relativity.

This disjointness has also positive aspect in TGD framework. One can identify the color degrees of freedom of partons as those associated with CP2 degrees of freedom. For instance, SU(3) analogs for rotational states of rigid body become possible. 4-D space-time surfaces as preferred extremals of Kähler action connect the partonic 3-surfaces and bring in classical representation of correlations and thus of interactions. The representation as sub-manifolds makes it also possible to speak about positions of these sub-Universes and about distances between them. The habitants of TGD Universe are maximally free but not completely alone.

2. TQFT as a functor

The category theoretic formulation of TQFT relies on a very elegant and general idea. Quantum transition has as a space-time correlate an n-dimensional surface having initial final states as its n-1-dimensional ends. One assigns Hilbert spaces of states to the ends and S-matrix would be a unitary morphism between the ends. This is expressed in terms of the category theoretic language by introducing the category nCob with objects identified as n-1-manifolds and morphisms as cobordisms and *-category Hilb consisting of Hilbert spaces with inner product and morphisms which are bounded linear operators which do not however preserve the unitarity. Note that the morphisms of nCob cannot anymore be identified as maps between n-1-manifolds interpreted as sets with additional structure so that in this case category theory is more powerful than set theory.

TQFT is identified as a functor nCob → Hilb assigning to n-1-manifolds Hilbert spaces, and to cobordisms unitary S-matrices in the category Hilb. This looks nice but the surprise is that for n<4 unitary S-matrix exists only if the cobordism is trivial so that topology changing transitions are not possible unless one gives up unitarity.

This raises several worried questions.

  1. Does this result mean that in TQFT sense unitary S-matrix for topology changing transitions from a state containing ni closed strings to a state containing nf≠ ni strings does not exist? Could the situation be same also for more general non-topological stringy S-matrices? Could the non-converging perturbation series for S-matrix with finite individual terms matrix fail to no non-perturbative counterpart? Could it be that M-theory is doomed to remain a dream with no hope of being fulfilled?

  2. Should one give up the unitarity condition and require that the theory predicts only the relative probabilities of transitions rather than absolute rates? What the proper generalization of the S-matrix could be?

  3. What is the relevance of this result for quantum TGD?

3. The situation is in TGD framework

The result about the non-existence of unitary S-matrix for topology changing cobordisms allows new insights about the meaning of the departures of TGD from string models.

  1. When I started to work with TGD for more than 28 years ago, one of the first ideas was that one could identify the selection rules of quantum transitions as topological selection rules for cobordisms. Within week or two came the great disappointment: there were practically no selection rules. Could one revive this naive idea? Could the existence of unitary S-matrix force the topological selection rules after all? I am skeptic. If I have understood correctly the discussion of what happens in 4-D case (see this), only the exotic diffeo-structures of n=4 dimensional spaces modify the situation in 4-D case.

  2. In the physically interesting GRT like situation one would expect the cobordism to be mediated by a space-time surface possessing Lorentz signature. This brings in metric and temporal distance. This means complications since one must leave the pure TQFT context. Also the classical dynamics of quantum gravitation brings in strong selection rules related to the dynamics in metric degrees of freedom so that TQFT approach is not expected to be useful from the point of view of quantum gravity and certainly not the limit of a realistic theory of quantum gravitation.

    In TGD framework situation is different. 4-D space-time sheets can have Euclidian signature of the induced metric so that Lorentz signature does not pose conditions. The counterparts of cobordisms correspond at fundamental level to light-like 3-surfaces, which are arbitrarily except for the light-likeness condition (the effective 2-dimensionality implies generalized conformal invariance and analogy with 3-D black-holes since 3-D vacuum Einstein equations are satisfied). Field equations defined by the Chern-Simons action imply that CP2 projection is at most 2-D but this condition holds true only for the extremals and one has functional integral over all light-like 3-surfaces. The temporal distance between points along light-like 3-surface vanishes. The constraints from light-likeness bring in metric degrees of freedom but in a very gentle manner and just to make the theory physically interesting.

  3. In string model context the discouraging results from TQFT hold true in the category of nCob, which corresponds to trouser diagrams for closed strings or for their open string counterparts. In TGD framework these diagrams are replaced with a direct generalization of Feynman diagrams for which 3-D light-like partonic 3-surfaces meet along their 2-D ends at the vertices. In honor of Feynman one could perhaps speak of Feynman cobordisms. These surfaces are singular as 3-manifolds but vertices are nice 2-manifolds. I contrast to this, in string models diagrams are nice 2-manifolds but vertices are singular as 1-manifolds (say eye-glass type configurations for closed strings).

    This picture gains a strong support for the interpretation of fermions as light-like throats associated with connected sums of CP2 type extremals with space-time sheets with Minkowski signature and of bosons as pairs of light-like wormhole throats associated with CP2 type extremal connecting two space-time sheets with Minkowski signature of induced metric. The space-time sheets have opposite time orientations so that also zero energy ontology emerges unavoidably. There is also consistency TGD based explanation of the family replication phenomenon in terms of genus of light-like partonic 2-surfaces.

    One can wonder what the 4-D space-time sheets associated with the generalized Feynman diagrams could look like? One can try to gain some idea about this by trying to assign 2-D surfaces to ordinary Feynman diagrams having a subset of lines as boundaries. In the case of 2→ 2 reaction open string is pinched to a point at vertex. 1→ 2 vertex, and quite generally, vertices with odd number of lines, are impossible. The reason is that 1-D manifolds of finite size can have either 0 or 2 ends whereas in higher-D the number of boundary components is arbitrary. What one expects to happen in TGD context is that wormhole throats which are at distance characterized by CP2 fuse together in the vertex so that some kind of pinches appear also now.

  4. Zero energy ontology gives rise to a second profound distinction between TGD and standard QFT. Physical states are identified as states with vanishing net quantum numbers, in particular energy. Everything is creatable from vacuum - and one could add- by intentional action so that zero energy ontology is profoundly Eastern. Positive resp. negative energy parts of states can be identified as states associated with 2-D partonic surfaces at the boundaries of future resp. past directed light-cones, whose tips correspond to the arguments of n-point functions. Each incoming/outgoing particle would define a mini-cosmology corresponding to not so big bang/crunch. If the time scale of perception is much shorter than time interval between positive and zero energy states, the ontology looks like the Western positive energy ontology. Bras and kets correspond naturally to the positive and negative energy states and phase conjugation for laser photons making them indeed something which seems to travel in opposite time direction is counterpart for bra-ket duality.

  5. In TGD framework one encounters two S-matrix like operators.

    • There is U-matrix between zero energy states. This is expected to be rather trivial but very important from the point of view of description of intentional actions as transitions transforming p-adic partonic 3-surfaces to their real counterparts.

    • The S-matrix like operator describing what happens in laboratory corresponds to the time-like entanglement coefficients between positive and negative energy parts of the state. Measurement of reaction rates would be a measurement of observables reducing time like entanglement and very much analogous to an ordinary quantum measurement reducing space-like entanglement. There is a finite measurement resolution described by inclusion of HFFs and this means that situation reduces effectively to a finite-dimensional one.

  6. p-Adic thermodynamics strengthened with p-adic length scale hypothesis predicts particle masses with an amazing success. At first the thermodynamical approach seems to be in contradiction with the idea that elementary particles are quantal objects. Unitarity is however not necessary if one accepts that only relative probabilities for reductions to pairs of initial and final states interpreted as particle reactions can be measured.

    The beneficial implications of unitarity are not lost if one replaces QFT with thermal QFT. Category theoretically this would mean that the time-like entanglement matrix associated with the product of cobordisms is a product of these matrices for the factors. The time parameter in S-matrix would be replaced with a complex time parameter with the imaginary part identified as inverse temperature. Hence the interpretation in terms of time evolution is not lost. In the theory of hyper-finite factors of type III1 the partition function for thermal equilibrium states and S-matrix can be neatly fused to a thermal S-matrix for zero energy states and one could introduce p-adic thermodynamics at the level of quantum states. It seems that this picture applies to HFFs by restriction. Therefore the loss of unitarity S-matrix might after all turn to a victory by more or less forcing both zero energy ontology and p-adic thermodynamics.

Monday, May 14, 2007

NASA Hubble Space Telescope Detects Ring of Dark Matter

The following catched my attention during this morning's webwalk.
NASA Hubble Space Telescope Detects Ring of Dark Matter

NASA will hold a media teleconference at 1 p.m. EDT on May 15 to discuss the strongest evidence to date that dark matter exists. This evidence was found in a ghostly ring of dark matter in the cluster CL0024+17, discovered using NASA's Hubble Space Telescope. The ring is the first cluster to show a dark matter distribution that differs from the distribution of both the galaxies and the hot gas. The discovery will be featured in the May 15 issue of the Astrophysical Journal.

"Rings" puts bells ringing! In TGD Universe dark matter characterized by a gigantic value of Planck constant making dark matter a macroscopic quantum phase in astrophysical length and time scales. Rotationally symmetric structures - such as rings- with an exact rotational symmetry Zn, n very very large, of the "field body" of the system, is the basic prediction. In the model of planetary orbits the rings of dark matter around Bohr orbits force the visible matter at the Bohr orbit (see this).

For more details see the chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-time".

Sunday, May 13, 2007

Platonism, Structuralism, and Quantum Platonism

I have been trying to understand how Category Theory and Set Theory relate to quantum TGD inspired view about fundamentals of mathematics. I managed to clarify my thoughts about what these theories are by reading the article Structuralism, Category Theory and Philosophy of Mathematics by Richard Stefanik (Washington: MSG Press, 1994). The reactions to postings in Kea's blog and email correspondence with Sampo Vesterinen have been very stimulating and inspired the attempt to represent TGD based vision about the unification of mathematics, physics, and consciousness theory in a more systematic manner.

Before continuing I want to summarize the basic ideas behind TGD vision. One cannot understand mathematics without understanding mathematical consciousness. Mathematical consciousness and its evolution must have direct quantum physical correlates and by quantum classical correspondence these correlates must appear also at space-time level. Quantum physics must allow to realize number as a conscious experience analogous to a sensory quale. In TGD based ontology there is no need to postulate physical world behind the quantum states as mathematical entities (theory is the reality). Hence number cannot be any physical object, but can be identified as a quantum state or its label and its number theoretical anatomy is revealed by the conscious experiences induced by the number theoretic variants of particle reactions. Mathematical systems and their axiomatics are dynamical evolving systems and physics is number theoretically universal selecting rationals and their extensions in a special role as numbers, which can can be regarded elements of several number fields simultaneously.

1. Platonism and structuralism

There are basically two philosophies of mathematics.

  1. Platonism assumes that mathematical objects and structures have independent existence. Natural numbers would be the most fundamental objects of this kind. For instance, each natural number has its own number-theoretical anatomy decomposing into a product of prime numbers defining the elementary particles of Platonia. For quantum physicist this vision is attractive, and even more so if one accepts that elementary particles are labelled by primes (as I do)! The problematic aspects of this vision relate to the physical realization of the Platonia. Neither Minkowski space-time nor its curved variants understood in the sense of set theory have no room for Platonia and physical laws (as we know them) do not seem to allow the realization of all imaginable internally consistent mathematical structures.

  2. Structuralist believes that the properties of natural numbers result from their relations to other natural numbers so that it is not possible to speak about number theoretical anatomy in the Platonic sense. Numbers as such are structureless and their relationships to other numbers provide them with their apparent structure. According to the article structuralism is however not enough for the purposes of number theory: in combinatorics it is much more natural to use intensional definition for integers by providing them with inherent properties such as decomposition into primes. I am not competent to take any strong attitudes on this statement but my physicist's intuition tells that numbers have number theoretic anatomy and that this anatomy can be only revealed by the morphisms or something more general which must have physical counterparts. I would like to regard numbers are analogous to bound states of elementary particles. Just as the decays of bound states reveal their inner structure, the generalizations of morphisms would reveal to the mathematician the inherent number theoretic anatomy of integers.

2. Structuralism

Set theory and category theory represent two basic variants of structuralism and before continuing I want to clarify to myself the basic ideas of structuralism: the reader can skip this section if it looks too boring.

2.1. Set theory

Structuralism has many variants. In set theory the elements of set are treated as structureless points and sets with the same cardinality are equivalent. In number theory additional structure must be introduced. In the case of natural numbers one introduces the notion of successor and induction axiom and defines the basic arithmetic operations using these. Set theoretic realization is not unique. For instance, one can start from empty set Φ identified as 0, identify 1 as {Φ}, 2 as {0,1} and so on. One can also identify 0 as Φ, 1 as {0}, 2 as {{0}},.... For both physicist and consciousness theorist these formal definitions look rather weird.

The non-uniqueness of the identification of natural numbers as a set could be seen as a problem. The structuralist's approach is based on an extensional definition meaning that two objects are regarded as identical if one cannot find any property distinguishing them: object is a representative for the equivalence class of similar objects. This brings in mind gauge fixing to the mind of physicists.

2.2 Category theory

Category theory represents a second form of structuralism. Category theorist does not worry about the ontological problems and dreams that all properties of objects could be reduced to the arrows and formally one could identify even objects as identity morphisms (looks like a trick to me). The great idea is that functors between categories respecting the structure defined by morphisms provide information about categories. Second basic concept is natural transformation which maps functors to functors in a structure preserving manner. Also functors define a category so that one can construct endless hierarchy of categories. This approach has enormous unifying power since functors and natural maps systemize the process of generalization. There is no doubt that category theory forms a huge piece of mathematics but I find difficult to believe that arrows can catch all of it.

The notion of category can be extended to that of n-category: in the previous posting I described a geometric realization of this hierarchy in which one defines 1-morphisms by parallel translations, 2-morphisms by parallel translations of parallel translations, and so on. In infinite-dimensional space this hierarchy would be infinite. Abstractions about abstractions about.., thoughts about thoughts about, statements about statements about..., is the basic idea behind this interpretation. Also the hierarchy of logics of various orders corresponds to this hierarchy. This encourages to see category theoretic thinking as being analogous to higher level self reflection which must be distinguished from the direct sensory experience.

In the case of natural numbers category theoretician would identify successor function as the arrow binding natural numbers to an infinitely long string with 0 as its end. If this approach would work, the properties of numbers would reflect the properties of the successor function.

3. The view about mathematics inspired by TGD and TGD inspired theory of consciousness

TGD based view might be called quantum Platonism. It is inspired by the requirement that both quantum states and quantum jumps between them are able to represent number theory and that all quantum notions have also space-time correlates so that Platonia should in some sense exist also at the level of space-time. Here I provide a brief summary of this view as it is now. The articles TGD, and TGD inspired theory of consciousness provide an overview about TGD and TGD inspired theory of consciousness.

3.1 Physics is fixed from the uniqueness of infinite-D existence and number theoretic universality

  1. The basic philosophy of quantum TGD relies on the geometrization of physics in terms of infinite-dimensional Kähler geometry of the "world of classical worlds" (configuration space), whose uniqueness is forced by the mere mathematical existence. Space-time dimension and imbedding space H=M4×CP2 are fixed among other things by this condition and allow interpretation in terms of classical number fields. Physical states correspond to configuration space spinor fields with configuration space spinors having interpretation as Fock states. Rather remarkably, configuration space Clifford algebra defines standard representation of so called hyper finite factor of II1, perhaps the most fascinating von Neumann algebra.

  2. Number theoretic universality states that all number fields are in a democratic position. This vision can be realized by requiring generalization of notions of imbedding space by gluing together real and p-adic variants of imbedding space along common algebraic numbers. All algebraic extensions of p-adic numbers are allowed. Real and p-adic space-time sheets intersect along common algebraics. The identification of the p-adic space-time sheets as correlates of cognition and intentionality explains why cognitive representations at space-time level are always discrete. Only space-time points belonging to an algebraic extension of rationals associated contribute to the data defining S-matrix. These points define what I call number theoretic braids. The interpretation in of algebraic discreteness terms of a physical realization of axiom of choice is highly suggestive. The axiom of choice would be dynamical and evolving quantum jump by quantum jump as the algebraic complexity of quantum states increases.

3.2 Holy trinity of existence

In TGD framework one would have 3-levelled ontology numbers should have representations at all these levels (see this).

  1. Subjective existence as a sequence of quantum jumps giving conscious sensory representations for numbers and various geometric structures would be the first level.

  2. Quantum states would correspond to Platonia of mathematical ideas and mathematician- or if one is unwilling to use this practical illusion- conscious experiences about mathematic ideas, would be in quantum jumps. The quantum jumps between quantum states respecting the symmetries characterizing the mathematical structure would provide conscious information about the mathematical ideas not directly accessible to conscious experience. Mathematician would live in Plato's cave. There is no need to assume any independent physical reality behind quantum states as mathematical entities since quantum jumps between these states give rise to conscious experience. Theory-reality dualism disappears since the theory is reality or more poetically: painting is the landscape.

  3. The third level of ontology would be represented by classical physics at the space-time level essential for quantum measurement theory. By quantum classical correspondence space-time physics would be like a written language providing symbolic representations for both quantum states and changes of them (by the failure of complete classical determinism of the fundamental variational principle). This would involve both real and p-adic space-time sheets corresponding to sensory and cognitive representations of mathematical concepts. This representation makes possible the feedback analogous to formulas written by mathematician crucial for the ability of becoming conscious about what one was conscious of and the dynamical character of this process allows to explain the self-referentiality of consciousness without paradox.

This ontology releases a deep Platonistic sigh of relief. Since there are no physical objects, there is no need to reduce mathematical notions to objects of the physical world. There are only quantum states identified as mathematical entities labelled naturally by integer valued quantum numbers; conscious experiences, which must represent sensations giving information about the number theoretical anatomy of a given quantum number; and space-time surfaces providing space-time correlates for quantum physics and therefore also for number theory and mathematical structures in general.

3.3 Factorization of integers as a direct sensory perception?

Both physicist and consciousness theorist would argue that the set theoretic construction of natural numbers could not be farther away from how we experience integers. Personally I feel that neither structuralist's approach nor Platonism as it is understood usually are enough. Mathematics is a conscious activity and this suggests that quantum theory of consciousness must be included if one wants to build more satisfactory view about fundamentals of mathematics.

Oliver Sack's book The man who mistook his wife for a hat [Touchstone books. (First edition 1985), see also this ] contains fascinating stories about those aspects of brain and consciousness which are more or less mysterious from the view point of neuroscience. Sacks tells in his book also a story about twins who were classified as idiots but had amazing number theoretical abilities. I feel that this story reveals something very important about the real character of mathematical consciousness.

The twins had absolutely no idea about mathematical concepts such as the notion of primeness but they could factorize huge numbers and tell whether they are primes. Their eyes rolled wildly during the process and suddenly their face started to glow of happiness and they reported a discovery of a factor. One could not avoid the feeling that they quite concretely saw the factorization process. The failure to detect the factorization served for them as the definition of primeness. For them the factorization was not a process based on some rules but a direct sensory perception.

The simplest explanation for the abilities of twins would in terms of a model of integers represented as string like structures consisting of identical basic units. This string can decay to strings. If string containing n units decaying into m> 1 identical pieces is not perceived, the conclusion is that a prime is in question. It could also be that decay to units smaller than 2 was forbidden in this dynamics. The necessary connection between written representations of numbers and representative strings is easy to build as associations.

This kind theory might help to understand marvellous feats of mathematicians like Ramanujan who represents a diametrical opposite of Groethendienck as a mathematician (when Groethendienck was asked to give an example about prime, he mentioned 57 which became known as Groethendienck prime!).

The lesson would be that one very fundamental representation of integers would be, not as objects, but conscious experiences. Primeness would be like the quale of redness. This of course does not exclude also other representations.

3.4 Experience of integers in TGD inspired quantum theory of consciousness

In quantum physics integers appear very naturally as quantum numbers. In quantal axiomatization or interpretation of mathematics same should hold true.

  1. In TGD inspired theory of consciousness quantum jump is identified as a moment of consciousness. There is actually an entire fractal hierarchy of quantum jumps consisting of quantum jumps and this correlates directly with the corresponding hierarchy of physical states and dark matter hierarchy. This means that the experience of integer should be reducible to a certain kind of quantum jump. The possible changes of state in the quantum jump would characterize the sensory representation of integer.

  2. The quantum state as such does not give conscious information about the number theoretic anatomy of the integer labelling it: the change of the quantum state is required. The above geometric model translated to quantum case would suggest that integer represents a multiplicatively conserved quantum number. Decays of this this state into states labelled by integers ni such that one has n=∏ ni would provide the fundamental conscious representation for the number theoretic anatomy of the integer. At the level of sensory perception based the space-time correlates a string-like bound state of basic particles representing n=1.

  3. This picture is consistent with the Platonist view about integers represented as structured objects, now labels of quantum states. It would also conform with the view of category theorist in the sense that the arrows of category theorist replaced with quantum jumps are necessary to gain conscious information about the structure of the integer.

3.5 Infinite primes and arithmetic consciousness

Infinite primes were the first mathematical fruit of TGD inspired theory of consciousness and the inspiration for writing this posting came from the observation that the infinite primes at the lowest level of hierarchy provide a representation of algebraic numbers as Fock states of a super-symmetric arithmetic QFT so that it becomes possible to realize quantum jumps revealing the number theoretic anatomy of integers, rationals, and perhaps even that of algebraic numbers.

  1. Infinite primes have a representation as Fock states of super-symmetric arithmetic QFT and at the lowest level of hierarchy they provide representations for primes, integers, rationals and algebraic numbers in the sense that at the lowest level of hierarchy of second quantizations the simplest infinite primes are naturally mapped to rationals whereas more complex infinite primes having interpretation as bound states can be mapped to algebraic numbers. Conscious experience of number can be assigned to the quantum jumps between these quantum states revealing information about the number theoretic anatomy of the number represented. It would be wrong to say that rationals only label these states: rather, these states represent rationals and since primes label the particles of these states.

  2. More concretely, the conservation of number theoretic energy defined by the logarithm of the rational assignable with the Fock state implies that the allowed decays of the state to a product of infinite integers are such that the rational can decompose only into a product of rationals. These decays could provide for the above discussed fundamental realization of multiplicative aspects of arithmetic consciousness. Also additive aspects are represented since the exponents k in the powers pk appearing in the decomposition are conserved so that only the partitions k=∑i ki are representable. Thus both product decompositions and partitions, the basic operations of number theorist, are represented.

  3. The higher levels of the hierarchy represent a hierarchy of abstractions about abstractions bringing strongly in mind the hierarchy of n-categories and various similar constructions including n:th order logic. It also seems that the n+1:th level of hierarchy provides a quantum representation for the n:th level. Ordinary primes, integers, rationals, and algebraic numbers would be the lowest level, -the initial object- of the hierarchy representing nothing at low level. Higher levels could be reduced to them by the analog of category theoretic reductionism in the sense that there is arrow between n:th and n+1:th level representing the second quantization at this level. On can also say that these levels represent higher reflective level of mathematical consciousness and the fundamental sensory perception corresponds the lowest level.

  4. Infinite primes have also space-time correlates. The decomposition of particle into partons can be interpreted as a infinite prime and this gives geometric representations of infinite primes and also rationals. The finite primes appearing in the decomposition of infinite prime correspond to bosonic or fermionic partonic 2-surfaces. Many-sheeted space-time provides a representation for the hierarchy of second quantizations: one physical prediction is that many particle bound state associated with space-time sheet behaves exactly like a boson or fermion. Nuclear string model is one concrete application of this idea: it replaces nucleon reductionism with reductionism occurs first to strings consisting of A≤4 nuclei and which in turn are strings consisting of nucleons. A further more speculative representation of infinite rationals as space-time surfaces is based on their mapping to rational functions.

3.6 Number theoretic Brahman=Atman identity

The notion of infinite primes leads to the notion of algebraic holography in which space-time points possess infinitely rich number-theoretic anatomy. This anatomy would be due to the existence of infinite number of real units defined as ratios of infinite integers which reduce to unit in the real sense and various p-adic senses. This anatomy is not visible in real physics but can contribute directly to mathematical consciousness (see this).

The anatomies of single space-time point could represent the entire world of classical worlds and quantum states of universe: the number theoretic anatomy is of course not visible in the structure of these these states. Therefore the basic building brick of mathematics - point- would become the Platonia able to represent all of the mathematics consistent with the laws of quantum physics. Space-time points would evolve, becoming more and more complex quantum jump by quantum jump. Configuration space and quantum states would be represented by the anatomies of space-time points. Some space-time points are more "civilized" than others so that space-time decomposes into "civilizations" at different levels of mathematical evolution.

Paths between space-time points represent processes analogous to parallel translations affecting the structure of the point and one can also define n-parallel translations up to n=4 at level of space-time and n=8 at level of imbedding space. At level of world of classical worlds whose points are representable as number theoretical anatomies arbitrary high values of n can be realized (see this).

It is fair to say that the number theoretical anatomy of the space-time point makes it possible self-reference loop to close so that structured points are able to represent the physics of associated with with the structures constructed from structureless points. Hence one can speak about algebraic holography or number theoretic Brahman=Atman identity.

3.7. Finite measurement resolution, Jones inclusions, and number theoretic braids

In the history of physics and mathematics the realization of various limitations have been the royal road to a deeper understanding (Uncertainty Principle, Gödel's theorem). The precision of quantum measurement, sensory perception, and cognition are always finite. In standard quantum measurement theory this limitation is not taken into account but forms a corner stone of TGD based vision about quantum physics and of mathematics too as I want to argue in the following.

The finite resolutions has representation both at classical and quantum level.

  1. At the level of quantum states finite resolution is represented in terms of Jones inclusions N subset M of hyper-finite factors of type II1 (HFFs)(see this). N represents measurement resolution in the sense that the states related by the action of N cannot be distinguished in the measurement considered. Complex rays are replaced by N rays. This brings in noncommutativity via quantum groups. Non-commutativity in TGD Universe would be therefore due to a finite measurement resolution rather than something exotic emerging in the Planck length scale. Same applies to p-adic physics: p-adic space-time sheets have literally infinite size in real topology!

  2. At the space-time level discretization implied by the number theoretic universality could be seen as being due to the finite resolution with common algebraic points of real and p-adic variant of the partonic 3-surface chosen as representatives for regions of the surface. The solutions of modified Dirac equation are characterized by the prime in question so that the preferred prime makes itself visible at the level of quantum dynamics and characterizes the p-adic length scale fixing the values of coupling constants. Discretization could be also understood as effective non-commutativity of imbedding space points due to the finite resolution implying that second quantized spinor fields anticommute only at a discrete set of points rather than along stringy curve.

In this framework it is easy to imagine physical representations of number theoretical and other mathematical structures.
  1. Every compact group corresponds to a hierarchy of Jones inclusions corresponding to various representations for the quantum variants of the group labelled by roots of unity. I would be surprised if non-compact groups would not allow similar representation since HFF can be regarded as infinite tensor power of n-dimensional complex matrix algebra for any value of n. Somewhat paradoxically, the finite measurement resolution would make possible to represent Lie group theory physically (see this and this)

  2. There is a strong temptation to identify the Galois groups of algebraic numbers as the infinite permutation group S consisting of permutations of finite number of objects, whose projective representations give rise to an infinite braid group B. The group algebras of these groups are HFFs besides the representation provided by the spinors of the world of classical worlds having physical identification as fermionic Fock states. Therefore physical states would provide a direct representation also for the more abstract features of number theory (see this).

  3. Number theoretical braids crucial for the construction of S-matrix (see this) provide naturally representations for the Galois groups G associated with the algebraic extensions of rationals as diagonal imbeddings G×G×.... to the completion of S representable also as the action on the completion of spinors in the world of classical worlds so that the core of number theory would be represented physically (see this). At the space-time level number theoretic braid having G as symmetries would represent the G. These representations are analogous to global gauge transformations. The elements of S are analogous to local gauge transformations having a natural identification as a universal number theoretical gauge symmetry group leaving physical states invariant.

3.8. Hierarchy of Planck constants and the generalization of imbedding space

Jones inclusions inspire a further generalization of the notion of imbedding space obtained by gluing together copies of the imbedding space H regarded as coverings H→H/Ga×Gb. In the simplest scenario Ga×Gb leaves invariant the choice of quantization axis and thus this hierarchy provides imbedding space correlate for the choice of quantization axes inducing these correlates also at space-time level and at the level of world of classical worlds (see this)

Dark matter hierarchy is identified in terms of different sectors of H glued together along common points of base spaces and thus forming a book like structure. For the simplest option elementary particles proper correspond to maximally quantum critical systems in the intersection of all pages. The field bodies of elementary particles are in the interiors of the pages of this "book".

One can assign to Jones inclusions quantum phase q =exp(i2π/n) and the groups Zn acts as exact symmetries both at level of M4 and CP2. In the case of M4 this means that space-time sheets have exact Zn rotational symmetry. This suggests that the algebraic numbers qm could have geometric representation at the level of sensory perception as Zn symmetric objects. We need not be conscious of this representation in the ordinary wake-up consciousness dominated by sensory perception of ordinary matter with q=1. This would make possible the idea about transcendentals like π, which do not appear in any finite-dimensional extension of even p-adic numbers (p-adic numbers allow finite-dimensional extension by since ep is ordinary p-adic number). Quantum jumps in which state suffers an action of the generating element of Zn could also provide a sensory realization of these groups and numbers exp(i2π/n).

Planck constant is identified as the ratio na/nb of integers associated with M4 and CP2 degrees of freedom so that a representation of rationals emerge again. The so called ruler and compass rationals whose definition involves only a repeated square root operation applied on rationals are cognitively the simplest ones and should appear first in the evolution of mathematical consciousness. The successful quantum model for EEG is only one of the applications providing support for their preferred role. Other applications are to Bohr quantization of planetary orbits interpreted as being induced by the presence of macroscopically quantum coherent dark matter (see tthis).

The pdf version of this posting can be found at my homepage. I have added the text also to the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness of "TGD as a Generalized Number Theory".