Vision about quantization of Planck constant
The quantization of Planck constant has been the basic them of TGD since 2005 and the perspective in the earlier version of this chapter reflected the situation for about year and one half after the basic idea stimulated by the finding of Nottale that planetary orbits could be seen as Bohr orbits with enormous value of Planck constant given by hbargr = GM1M2/v0, v0 ≈ 2-11 for the inner planets. The general form of hbargr is dictated by Equivalence Principle. This inspired the ideas that quantization is due to a condensation of ordinary matter around dark matter concentrated near Bohr orbits and that dark matter is in macroscopic quantum phase in astrophysical scales.
The second crucial empirical input were the anomalies associated with living matter. Mention only the effects of ELF radiation at EEG frequencies on vertebrate brain and anomalous behavior of the ionic currents through cell membrane. If the value of Planck constant is large, the energy of EEG photons is above thermal energy and one can understand the effects on both physiology and behavior. If ionic currents through cell membrane have large Planck constant the scale of quantum coherence is large and one can understand the observed low dissipation in terms of quantum coherence.
1. The evolution of mathematical ideas
From the beginning the basic challenge -besides the need to deduce a general formula for the quantized Planck constant- was to understand how the quantization of Planck constant is mathematically possible. From the beginning it was clear that since particles with different values of Planck constant cannot appear in the same vertex, a generalization of space-time concept is needed to achieve this.
During last five years or so many deep ideas -both physical and mathematical- related to the construction of quantum TGD have emerged and this has led to a profound change of perspective in this and also other chapters. The overall view about TGD is described briefly here.
2. The evolution of physical ideas
- For more than five years ago I realized that von Neumann algebras known as hyperfinite factors of type II1 (HFFs) are highly relevant for quantum TGD since the Clifford algebra of configuration space ("world of classical worlds", WCW) is direct sum over HFFs. Jones inclusions are particular class of inclusions of HFFs and quantum groups are closely related to them. This led to a conviction that Jones inclusions can provide a detailed understanding of what is involved and predict very simple spectrum for Planck constants associated with M4 and CP2 degrees of freedom (later I replaced M4 by its light cone M4� and finally with the causal diamond CD defined as intersection of future and past light-cones of M4).
- The notion of zero energy ontology replaces physical states with zero energy states consisting of pairs of positive and negative energy states at the light-like boundaries δM4�×CP2 of CDs forming a fractal hierarchy containing CDs within CDs. In standard ontology zero energy state corresponds to a physical event, say particle reaction. This led to the generalization of S-matrix to M-matrix identified as Connes tensor product characterizing time like entanglement between positive and negative energy states. M-matrix is product of square root of density matrix and unitary S-matrix just like Schrödinger amplitude is product of modulus and phase, which means that thermodynamics becomes part of quantum theory and thermodynamical ensembles are realized as single particle quantum states. This led also to a solution of long standing problem of understanding how geometric time of the physicist is related to the experienced time identified as a sequence of quantum jumps interpreted as moments of consciousness in TGD inspired theory of consciousness which can be also seen as a generalization of quantum measurement theory (see this) .
- Another closely related idea was the emergence of measurement resolution as the basic element of quantum theory. Measurement resolution is characterized by inclusion M subset N of HFFs with M characterizing the measurement resolution in the sense that the action of M creates states which cannot be distinguished from each other within measurement resolution used. Hence complex rays of state space are replaced with M rays. One of the basic challenges is to define the nebulous factor space N/M having finite fractional dimension N:M given by the index of inclusion. It was clear that this space should correspond to quantum counterpart of Clifford algebra of world of classical worlds reduced to a finite-quantum dimensional algebra by the finite measurement resolution (see this).
- The realization that light-like 3-surfaces at which the signature of induced metric of space-time surface changes from Minkowskian to Euclidian are ideal candidates for basic dynamical objects besides light-like boundaries of space-time surface was a further decisive step or progress. This led to vision that quantum TGD is almost topological quantum field theory ("almost" because light-likeness brings in induced metric) characterized by Chern-Simons action for induced Kähler gauge potential of CP2. Together with zero energy ontology this led to the generalization of the notion of Feynman diagram to a light-like 3-surface for which lines correspond to light-like 3-surfaces and vertices to 2-D partonic surface at which these 3-D surface meet. This means a strong departure from string model picture. The interaction vertices should be given by N-point functions of a conformal field theory with second quantized induced spinor fields defining the basic fields in terms of which also the gamma matrices of world of classical worlds could be constructed as super generators of super conformal symmetries (see this).
- By quantum classical correspondence finite measurement resolution should have a space-time correlate. The obvious guess was that this correlate is discretization at the level of construction of M-matrix. In almost-TQFT context the effective replacement of light-like 3-surface with braids defining basic objects of TQFTs is the obvious guess. Also number theoretic universality necessary for the p-adicization of quantum TGD by a process analogous to the completion of rationals to reals and various p-adic number fields requires discretization since only rational and possibly some algebraic points of the imbedding space (in suitable preferred coordinates) allow interpretation both as real and p-adic points. It was clear that the construction of M-matrix boils to the precise understanding of number theoretic braids (see this).
- The interaction with M-theory dualities (see this) led to a handful of speculations about dualities possible in TGD framework, and one of these dualities- M8-M4×CP2 duality - eventually led to a unique identification of number theoretic braids. The dimensions of partonic 2-surface, space-time, and imbedding space strongly suggest that classical number fields, or more precisely their complexifications might help to understand quantum TGD. If the choice of imbedding space is unique because of uniqueness of infinite-dimensional Kähler geometric existence of world of classical worlds then standard model symmetries coded by M4×CP2 should have some deeper meaning and the most obvious guess is that M4×CP2 can be understood geometrically. SU(3) belongs to the automorphism group of octonions as well as hyper-octonions M8 identified by subspace of complexified octonions with Minkowskian signature of induced metric. This led to the discovery that hyper-quaternionic 4-surfaces in M8 can be mapped to M4×CP2 provided their tangent space contains preferred M2 subset M4 subset M4×E4. Years later I realized that the map generalizes so that M2 can depend on the point of X4. The interpretation of M2(x) is both as a preferred hyper-complex (commutative) sub-space of M8 and as a local plane of non-physical polarizations so that a purely number theoretic interpretation of gauge conditions emerges in TGD framework. This led to a rapid progress in the construction of the quantum TGD. In particular, the challenge of identifying the preferred extremal of Kähler action associated with a given light-like 3-surface X3l could be solved and the precise relation between M8 and M4×CP2 descriptions was understood (see this).
- Also the challenge of reducing quantum TGD to the physics of second quantized induced spinor fields found a resolution recently (see this). For years ago it became clear that the vacuum functional of the theory must be the Dirac determinant associated with the induced spinor fields so that the theory would predict all coupling parameters from quantum criticality. Even more, the vacuum functional should correspond to the exponent of Kähler action for a preferred extremal. The problem was that the generalized eigenmodes of Chern-Simons Dirac operator allow a generalized eigenvalues to be arbitrary functions of two coordinates in the directions transversal to the light-like direction of X3l. The progress in the understanding of number theoretic compactification however allowed to understand how the information about the preferred extremal of Kähler action is coded to the spectrum of eigen modes.
The basic idea is simple and I actually discovered it for more than half decade ago but forgot! The generalized eigen modes of 3-D Chern-Simons Dirac operator DC-S correspond to the zero modes of a 4-D modified Dirac operator defined by Kähler action localized to X3l so that induced spinor fields can be seen as 4-D spinorial shock waves. The led to a concrete interpretation of the eigenvalues as analogous to cyclotron energies of fermion in classical electro-weak magnetic fields defined by the induced spinor connection and a connection with anyon physics emerges by 2-dimensionality of the evolving system. Also it was possible to identify the boundary conditions for the preferred extremal of Kähler action -analog of Bohr orbit- at X3l and also to the vision about how general coordinate invariance allows to use any light-like 3-surface X3 subset X4(X3l) instead of using only wormhole throat to second quantize induced spinor field.
- It became as a total surprise that due to the huge vacuum degeneracy of induced spinor fields the number of generalized eigenmodes identified in this manner was finite. The good news was that the theory is manifestly finite and zeta function regularization is not needed to define the Dirac determinant. The manifest finiteness had been actually must-be-true from the beginning. The apparently bad news was that the Clifford algebra of WCW world constructed from the oscillator operators is bound to be finite-dimensional. The resolution of the paradox comes from the realization that this algebra represents the somewhat mysterious coset space N/M so that finite measurement resolution and the notion inclusion are coded by the vacuum degeneracy of Kähler action and the maximally economical description in terms of inclusions emerges automatically.
- A unique identification of number theoretic braids became also possible and relates to the construction of the generalized imbedding space by gluing together singular coverings and factor spaces of CD\M2 and CP2\S2I to form a book like structure. Here M2 is preferred plane of M4 defining quantization axis of energy and angular momentum and S2I is one of the two geodesic sphere of CP2. The interpretation of the selection of these sub-manifolds is as a geometric correlate for the selection of quantization axes and CD defining basic sector of world of classical worlds is replaced by a union corresponding to these choices. Number theoretic braids come in too variants dual to each other, and correspond to the intersection of M2 and M4 projection of X3l on one hand and S2I and CP2 projection of X3l on the other hand. This is simplest option and would mean that the points of number theoretic braid belong to M2 (S2I) and are thus quantum critical although entire X2 at the boundaries of CD belongs to a fixed page of the Big Book. This means solution of a long standing problem of understanding in what sense TGD Universe is quantum critical. The phase transitions changing Planck constant correspond to tunneling represented geometrically by a leakage of partonic 2-surface from a page of Big Book to another one.
- Many other steps of progress have occurred during the last years. Much earlier it had become clear that the basic difference between TGD and string models is that in TGD framework the super algebra generators are non-hermitian and carry quark or lepton number (see this). Super-space concept is un-necessary because super generators anticommute to Hamiltonians of bosonic symmetries rather than corresponding vector fields. This allows to avoid the Majorana condition of super string models fixing space-time dimension to 10 or 11. During last years a much more precise understanding of super-symplectic and super Kac-Moody symmetries has emerged. The generalized coset representation for these two Super Virasoro algebras generalizes Equivalence Principle and predicts as a special case the equivalence of gravitational and inertial masses. Coset construction also provides justification for p-adic thermodynamics in apparent conflict with super-conformal invariance. The construction of the fusion rules of symplectic QFT as analog of conformal QFT led to the notion of number theoretic braid and to an explicit construction of a hierarchy of algebras realizing symplectic fusion rules and the notion of finite measurement resolution (see this). This approach led to the formulation of generalized Feynman diagrams and coupling constant evolution in terms of operads Taylor made for a mathematical realization of the notion of coupling constant evolution. One of the future challenges is to combine symplectic fusion algebras with the realization for the hierarchy of Planck constants.
The evolution of physical ideas related to the hierarchy of Planck constants and dark matter as a hierarchy of phases of matter with non-standard value of Planck constants was much faster than the evolution of mathematical ideas and quite a number of applications have been developed during last five years.
- The basic idea was that ordinary matter condenses around dark matter which is a phase of matter characterized by non-standard value of Planck constant.
- The realization that non-standard values of Planck constant give rise to charge and spin fractionization and anyonization led to the precise identification of the prerequisites of anyonic phase (see this). If the partonic 2-surface, which can have even astrophysical size, surrounds the tip of CD, the matter at the surface is anyonic and particles are confined at this surface. Dark matter could be confined inside this kind of light-like 3-surfaces around which ordinary matter condenses. If the radii of the basic pieces of these nearly spherical anyonic surfaces - glued to a connected structure by flux tubes mediating gravitational interaction - are given by Bohr rules, the findings of Nottale can be understood. Dark matter would resemble to a high degree matter in black holes replaced in TGD framework by light-like partonic 2-surfaces with minimum size of order Schwarstchild radius rS of order scaled up Planck length: rS ~ (hbar G)1/2. Black hole entropy being inversely proportional to hbar is predicted to be of order unity so that dramatic modification of the picture about black holes is implied.
- Darkness is a relative concept and due to the fact that particles at different pages of book cannot appear in the same vertex of the generalized Feynman diagram. The phase transitions in which partonic 2-surface X2 during its travel along X3l leaks to different page of book are however possible and change Planck constant so that particle exchanges of this kind allow particles at different pages to interact. The interactions are strongly constrained by charge fractionization and are essentially phase transitions involving many particles. Classical interactions are also possible. This allows to conclude that we are actually observing dark matter via classical fields all the time and perhaps have even photographed it (see this).
- Perhaps the most fascinating applications are in biology. The anomalous behavior ionic currents through cell membrane (low dissipation, quantal character, no change when the membrane is replaced with artificial one) has a natural explanation in terms of dark supra currents. This leads to a vision about how dark matter and phase transitions changing the value of Planck constant could relate to the basic functions of cell, functioning of DNA and aminoacids, and to the mysteries of bio-catalysis. This leads also a model for EEG interpreted as a communication and control tool of magnetic body containing dark matter and using biological body as motor instrument and sensory receptor. One especially shocking outcome is the emergence of genetic code of vertebrates from the model of dark nuclei as nuclear strings (see this).
3. Brief summary about the generalization of the imbedding space concept
A brief summary of the basic vision in order might help reader to assimilate the more detailed representation about the generalization of imbedding space.
- The hierarchy of Planck constants cannot be realized without generalizing the notions of imbedding space and space-time since particles with different values of Planck constant cannot appear in the same interaction vertex. This suggests some kind of book like structure for both M4 and CP2 factors of the generalized imbedding space is suggestive.
- Schrödinger equation suggests that Planck constant corresponds to a scaling factor of M4 metric whose value labels different pages of the book. The scaling of M4 coordinate so that original metric results in M4 factor is possible so that the scaling of hbar corresponds to the scaling of the size of causal diamond CD defined as the intersection of future and past directed light-cones. The light-like 3-surfaces having their 2-D and light-boundaries of CD are in a key role in the realization of zero energy states. The infinite-D spaces formed by these 3-surfaces define the fundamental sectors of the configuration space (world of classical worlds). Since the scaling of CD does not simply scale space-time surfaces, the coding of radiative corrections to the geometry of space-time sheets becomes possible and Kähler action can be seen as expansion in powers of hbar/hbar0.
- Quantum criticality of TGD Universe is one of the key postulates of quantum TGD. The most important implication is that Kähler coupling strength is analogous to critical temperature. The exact realization of quantum criticality would be in terms of critical sub-manifolds of M4 and CP2 common to all sectors of the generalized imbedding space. Quantum criticality would mean that the two kinds of number theoretic braids assignable to M4 and CP2 projections of the partonic 2-surface belong by the definition of number theoretic braids to these critical sub-manifolds. At the boundaries of CD associated with positive and negative energy parts of zero energy state in given time scale partonic two-surfaces belong to a fixed page of the Big Book whereas light-like 3-surface decomposes into regions corresponding to different values of Planck constant much like matter decomposes to several phases at thermodynamical criticality.
- The connection with Jones inclusions was originally a purely heuristic guess based on the observation that the finite groups characterizing Jones inclusion characterize also pages of the Big Book. The key observation is that Jones inclusions are characterized by a finite subgroup G subset SU(2) and that this group also characterizes the singular covering or factor spaces associated with CD or CP2 so that the pages of generalized imbedding space could indeed serve as correlates for Jones inclusions. The elements of the included algebra M are invariant under the action of G and M takes the role of complex numbers in the resulting non-commutative quantum theory.
- The understanding of quantum TGD at parton level led to the realization that the dynamics of Kähler action realizes finite measurement resolution in terms of finite number of modes of the induced spinor field. This automatically implies cutoffs to the representations of various super-conformal algebras typical for the representations of quantum groups closely associated with Jones inclusions. The Clifford algebra spanned by the fermionic oscillator operators would provide a realization for the factor space N/M of hyper-finite factors of type II1 identified as the infinite-dimensional Clifford algebra N of the configuration space and included algebra M determining the finite measurement resolution. The resulting quantum Clifford algebra has anti-commutation relations dictated by the fractionization of fermion number so that its unit becomes r=hbar/hbar0. SU(2) Lie algebra transforms to its quantum variant corresponding to the quantum phase q=exp(i2p/r).
- Jones inclusions appear as two variants corresponding to N:M < 4 and N:M=4. The tentative interpretation is in terms of singular G-factor spaces and G-coverings of M4 or CP2 in some sense. The alternative interpretation in terms of two geodesic spheres of CP2 would mean asymmetry between M4 and CP2 degrees of freedom.
- Number theoretic Universality suggests an answer why the hierarchy of Planck constants is necessary. One must be able to define the notion of angle -or at least the notion of phase and of trigonometric functions- also in p-adic context. All that one can achieve naturally is the notion of phase defined as root of unity and introduced by allowing algebraic extension of p-adic number field by introducing the phase if needed. In the framework of TGD inspired theory of consciousness this inspires a vision about cognitive evolution as the gradual emergence of increasingly complex algebraic extensions of p-adic numbers and involving also the emergence of improved angle resolution expressible in terms of phases exp(i2p/n) up to some maximum value of n. The coverings and factor spaces would realize these phases geometrically and quantum phases q naturally assignable to Jones inclusions would realize them algebraically. Besides p-adic coupling constant evolution based on hierarchy of p-adic length scales there would be coupling constant evolution with respect to hbar and associated with angular resolution.
For the updated version of the chapter see Does TGD Predict a Spectrum of Planck Constants?.