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Topological Geometrodynamics: an Overview?
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I have been reformulating basic quantum TGD using partonic formulation based on light-like 3-surfaces identifiable as parton orbits. This provides a precise and rigorous identification of various conformal symmetries which have been previously identified as mathematical necessities. Also concrete geometric picture emerges by using quantum classical correspondence. This kind of reformulation of course means that some stuff appears to be obsolete or simply wrong.
1. About the construction of physical states
The previous construction of physical states was still far from complete and involved erraneous elements. The partonic picture confirms however the basic vision. Super-canonical Virasoro algebra involves only generators Ln, n<0, and creates tachyonic ground states required by p-adic mass calculations. These states correspond to null states with conformal weight h<0 and annihilated by Ln, n<0. The null state property saves from an infinite degeneracy of ground states and thus also of exotic massless states. Super-canonical generators and Kac-Moody generators applied to this state give massless ground state and p-adic thermodynamics for SKM algebra gives mass squared ientified as the thermal expectation of conformal weight. The non-determinism of almost topological parton dynamics partially justifies the use of p-adic thermodynamics.
The hypothesis that the commutator of super-canonical and SKM algebras annihilates physical states seems attractive and would define the analog of Dirac equation in the world of classical worlds and eliminate large number of exotic states.
2. Consistency with p-adic thermodynamics
The consistency with p-adic thermodynamics provides a strong reality test and has been already used as a constraint in attempts to understand the super-conformal symmetries at the partonic level. In the proposed geometric interpretation inspired by quantum classical correspondence p-adic thermal excitations could be assigned with the curves ζ(n+1/2+iy) at S2subset CP2 for CP2 degrees of freedom and S2 subset δ M4+/- for M4 degrees of freedom so that a rather concrete picture in terms of orbits of harmonic oscillator would result.
There are some questions which pop up in mind immediately.
It is relatively straightforward to deduce the detailed form of the TGD countetpart of Kac-Moody algebra identified as X3-local infinitesimal transformations of H_+/-=M4+/-× CP2 respecting the lightlikeness of partonic 3-surface X3. This involves the identification of Kac-Moody transformations and corresponding super-generators carrying now fermion numbers and anticommuting to a multiple of unit matrix. Also the generalization the notions of Ramond and N-S algebra is needed. Especially interesting is the relationship with the super-canonical algebra consisting of canonical transformations of δ M4+/-× CP2.
1. Bosonic part of the algebra
The bosonic part of Kac-Moody algebra can be identified as symmetries respecting the light-likeness of the partonic 3-surface X3 in H=M4+/-× CP2. The educated guess is that a subset of X3-local diffeomorphisms of is in question. The allowed infinitesimeal transformation of this kind must reduce to a conformal transformation of the induced metric plus diffeomorphism of X3. The explicit study of the conditions allows to conclude that conformal transformations of M4+/- and isometries of CP2 made local with respect to X3 satisfy the defining conditions. Choosing special coordinates for X3 one finds that the vector fields defining the transformations must be orthogonal of the light-like direction of X3. The resulting partial differential equations fix the infinitesimal diffeomorphism of X3 once the functions appearing in Kac-Moody generator are fixed. The functions appearing in generators can be chosen to proportional to powers of the radial coordinate multiplied by functions of transversal coordinates whose dynamics is dictated by consistency conditions
The resulting algebra is essentially 3-dimensional and therefore much larger than ordinary Kac-Moody algebra. One can identify the counterpart of ordinary Kac-Moody algebra as a sub-algebra for which generators are in one-one correspondence with the powers of the light-like coordinate assignable to X3. This algebra corresponds to the stringy sub-algebra E2× SO(2)×SU(3) if one selects the preferred coordinate of M4 as a lightlike coordinate assignable to the lightlike ray of δ M4+/- defining orbifold structure in M4+/- ("massless" case) and E3× SO(3)×SU(3) if the preferred coordinate is M4 time coordinate (massive case).
The local transformation in the preferred direction is not free but fixed by the condition that Kac-Moody transformation does not affect the value of the light-like coordinate of X3. This is completely analogous to the non-dynamical character of longitudinal degrees of freedom of Kac-Moody algebra in string models.
The algebra decomposes into a direct sum of sub-spaces left invariant by Kac-Moody algebra and one has a structure analogous to that defining coset space structure (say SU(3)/U(2)). This feature means that the space of physical states is much larger than in string models and Kac Moody algebra of string models takes the role of the little algebra in the representations of Poincare group. Mackey's construction of induced representations should generalized to this situation.
Just as in the case of super-canonical algebra, the Noether charges assignable to the Kac-Moody transformations define Hamiltonians in the world of classical worlds as integrals over the partonic two surface and reducible to one-dimensional integrals if the SO(2)× SU(3) quantum numbers of the generator vanish. The intepretation is that this algebra leaves invariant various quantization axes and acts as symmetries of quantum measurement situation.
2. Fermionic sector
The zero modes and generalized eigen spinors of the modified Dirac equation define the counterparts of Ramond and N-S type super generators.
The hypothesis inspired by number theoretical conjectures related to Riemann Zeta is that the eigenvalues of the generalized eigen modes associated with ground states correspond to non-trivial zeros of zeta. Also non-trivial eigenvalues must be considered.
3. Super Virasoro algebras
The identification of SKM Virasoro algebra as that associated with radial diffeomorphisms is obvious and this algebra replaces the usual Virasoro algebra associated with the complex coordinate of partonic 2-surface X2 in the construction of states and mass calculations. This algebra does not not annihilate physical states and this gives justification for p-adic thermodynamics. The commutators of super canonical and super Kac-Moody (and corresponding super Virasoro) algebras would however annihilate naturally the physical states. Four-momentum does not appear in the expressions for the Virasoro generators and mass squared is identified as p-adic thermal expectation value of conformal weight. There are no problems with Lorentz invariance.
One can wonder about the role of ordinary conformal transformations assignable to the partonic 2-surface X2. The stringy quantization implies the reduction of this part of algebra to algebraically 1-D form and the corresponding conformal weight labels different radial SKM representations. Conformal weights are not constants of motion along X3 unlike radial conformal weights. TGD analog of super-conformal symmetries of condensed matter physics rather than stringy super-conformal symmetry would be in question.
The last section of the chapter The Evolution of Quantum TGD gives a more detailed summary of the recent picture.
The modified Dirac action for the light-like partonic 3-surfaces is determined uniquely by the Chern-Simons action for the induced Kähler form (or equivalently classical induced color gauge field possessing Abelian holonomy) and by the requirement of super-conformal symmetry. This action determines quantum physics of TGD Universe at the fundamental level. The classical dynamics for the interior of space-time surface is determined by the corresponding Dirac determinant. This classical dynamics is responsible for propagators whereas stringy conformal field theory provides the vertices. The theory is almost topological string theory with N=4 super-conformal symmetry.
The requirement that the super-Hamiltonians associated with the modified Dirac action define the gamma matrices of the configuraion space in principle fixes the anticommutation relations for the second quantized induced spinor field when one notices that the matrix elements of the metric in the complexified basis for super-canonical Killing vector fields of the configuration space ("world of classical worlds") are simply Poisson brackets for complexified Hamiltonians and thus themselves bosonic Hamiltonians. The challenge is to deduce the explicit form of these anticommutation relations and also the explicit form of the super-charges/gamma matrices. This challenge is not easy since canonical quantization cannot be used now. The progress in the understanding of the general structure of the theory however allows to achieve this goal.
1. Two options for fermionic anticommutators
The first question is following. Are anticommutators proportional
It turns out that stringy option is possible if the Poisson bracket of Hamiltonian with the Kähler form of δ M4×CP2 vanishes. The vanishing states that the super-canonical algebra must commute with the Hamiltonians corresponding to rotations around spin quantization axis and quantization axes of color isospin and hypercharge. Therefore hese quantum numbers must vanish for allowed Hamiltonians and super-Hamiltonians acting as symmetries. This brings strongly in mind weak form of color confinement suggested also by the classical theory (the holonomy group of classical color gauge field is Abelian).
The result has also interpretation in terms of quantum measurement theory: the isometries of a given sector of configuration space corresponding to a fixed selection of quantization axis commute with the basic measured observables (commuting isometry charges) and configuration space is union over sub-configuration spaces corresponding to these choices.
It is possible to find the explicit form of super-charges and their anticommutation relations which must be also consistent with the huge vacuum degeneracy of the bosonic Chern-Simons action and Kähler action.
2. Why stringy option is so nice?
An especially nice outcome is that string has purely number theoretic interpretation. It corresponds to the one-dimensional set of points of partonic 2-surface for which CP2 projection belongs to the image of the critical line s=1/2+iy containing the non-trivial zeros of ζ at the geodesic sphere S2 of CP2 under the map s→ ζ(s).
The stimulus that led to the idea that braids must be essential for TGD was the observation that a wide class of Yang-Baxter matrices can be parametrized by CP2, that geodesic sphere of S2 of CP2 gives rise to mutually commuting Y-B matrices, and that geodesic circle of S2 gives rise to unitary Y-B matrices. Together with braid picture also unitarity supports the stringy option, as does also the unitarity of the inner product for the radial modes rΔ, Δ=1/2+iy, with respect to inner product defined by scaling invariant integration measure dr/r. Furthermore, the reduction of Hamiltonians to duals of closed 2-forms conforms with the almost topological QFT character.
3. Number theoretic hierarchy of discretized theories
Also the hierarchy of discretized versions of the theory which does not mean any approximation but a hierarchy of physics characterizing increasing resolution of cognition can be formulated precisely. Both
The mode expansion of the second quantized spinor field has a natural cutoff for angular momentum l and isospin I corresponding to the integers na and nb characterizing the orders of maximal cyclic subgroups of groups Ga and Gb defining the Jones inclusion in M4 and CP2 degrees of freedom and characterizing the Planck constants. More precisely: one has l≤ na and I≤ nb. This means that the the number modes in the oscillator operator expansion of the spinor field is finite and the delta function singularity for the anticommutations for spinor field becomes smoothed out so that theory makes sense also in the p-adic context where definite integral and therefore also delta function is ill-defined notion.
The almost topological QFT character of theory allows to choose the eigenvalues of the modified Dirac operator to be of form s= 1/2+i∑knkyk, where sk=1/2+iykare zeros of ζ. This means also a cutoff in the Dirac determinant which becomes thus a finite algebraic number if the number of zeros belonging to a given algebraic extension is finite. This makes sense if the theory is integrable in the sense that everything reduces to a sum over maxima of Kähler function defined by the Dirac determinant as quantum criticality suggests (Duistermaat-Heckman theorem in infinite-dimensional context).
What is especially nice that the hierarchy of these cutoffs replaces also the infinite-dimensional space determined by the configuration space Hamiltonians with a finite-dimensional space so that the world of classical worlds is approximated with a finite-dimensional space.
The allowed intersection points of real and p-adic partonic 2-surface define number theoretical braids and these braids could be identified as counterparts of the braid hierarchy assignable to the hyperfinite factors of type II1 and their Jones inclusions and representing them as inclusions of finite-dimensional Temperley-Lieb algebras. Thus it would seem that the hierarchy of extensions of p-adic numbers corresponds to the hierarchy of Temperley-Lieb algebras.
For more details see the chapter Construction of Configuration Space Spinor Structure.
In the previous posting the possibility that zeros of Riemann Zeta could define quantum critical conformal weights associated with phase transitions between different values of Planck constants was discussed. The obvious question is whether also some zeros of Riemann polyzetas might have similar interpretation.
According to earlier considerations Riemann poly-zetas ζn (Δ1,...,Δn) could allow to generalize the notion of binding energy to that of binding conformal weight. In this case zeros form a continuum so that the set of points (Δ1,...,Δn)= ζn-1(z=ξ1/ξ2) forms a n-1 complex dimensional surface in Cn.
Completely symmetrized polyzetas are expressible using products of Riemann Zetas for arguments which are sums of arguments for polyzeta. If Δi are linear combinations of zeros of Zeta, polyzeta involves Riemann Zeta only for arguments which are sums of zeros of ζ. Symmetrized polyzeta is non-vanishing when Δi are non-trivial zeros of Zeta but vanishes for trivial zeros at Δi=-2ni. Also the zeros of symmetrized polyzeta would have interpretation in terms of quantum criticality.
An interesting question is whether ζn has a discrete subset of zeros for which pΔi is algebraic number for all primes p and Δi. This could be the case. For instance, suitable linear combinations of zeros of ζ define zeros of polyzeta. For instance, (a,b)=(s1,s1-s2) for any pair of zeros of zeta is zero of P2(a,b)= ζ (a)ζ (b)- ζ (a+b) whereas (a1,a2,a3)= (s1,s2-s1,s2,c-s1) defines a zero of
P3(a1,a2,a3)= 2ζ(a1+a2+a3) +ζ(a1)P2(a2,a3)+ζ(a2)P2(a3,a1)+ζ(a3)P2(a1,a2) -2ζ(a1)ζ(a2)ζ (a3)
for any pair (s1,s2) of zeros of ζ (subscript c refers to complex conjugation).
The conditions state that all Pm:s, m<n in the decomposition of Pn vanish separately. Besides this one ak, say a1=s1 must correspond to a zero of ζ. Same is true for the sum ∑ ak and sub-sums involving a1. The number of conditions increases rapidly as n increases. In the case of P4 the three triplets (a1,ai,aj) must be of same form as n=3 case and this allows only the trivial solution with say a4=0. Thus it would seem that only n=2 and n=3 allow non-trivial solutions for which bound state conformal weights are expressible in terms of differences of zeros of Riemann ζ. What is nice that the linear combinations of these conformal multi-weights give total conformal weights which are linear combinations of zeros of zeta.
The special role of 2- and 3-parton states brings unavoidably in mind mesons and baryons and the fact that hadrons containing larger number of valence quarks have not yet been identified experimentally.
If conformal confinement holds true then physical particles have vanishing conformal weights. This would require that ordinary baryons and mesons have real conformal weights and cannot therefore correspond to this kind of states. One must however take this idea very critically. The point is that the one-dimensional waves x1/2+iy have unitary inner product with respect to the scaling invariant inner product defined by the integration measure dx/x. For this inner product, the real part of the conformal weight should be 1/2 as it indeed is for the solutions of the conditions. If this interpretation is correct, then hadrons would represent states with non-vanishing conformal weight.
If one accepts complex conformal weights one must have some physical interpretation for them. The identification of conjugation of zeros of zeta as charge conjugation does not look promising since it would not leave neutral pion invariant. Of course, critical configurations with real conformal weight are possible at least formally and would correspond to trivial zeros s2= -2n of ζ but s1 arbitrary zero. These configuration would not however define logarithmic plane waves.
Laser physics might come in rescue here. So called phase conjugate photons are known to behave differently from photons. I have already proposed that all particles possess phase conjugates in TGD Universe. Phase conjugation is identified as reversal of time arrow mapping positive energy particles to negative energy particles. At space-time level this would mean an assignment of time orientation to space-time sheet. This is consistent with the fact that energy momentum complex consists of vector currents rather than forming a tensor. The implication is that in S-matrix positive energy particles travelling towards geometric future are not equivalent with negative energy particle travelling towards geometric past. This is essential for the notions like remote metabolism and time mirror mechanism.
The precise definition of phase conjugation at quantum level has remained obscure. The identification of phase conjugation as conjugation for the zeros of Zeta looks however very natural.
For more details see the chapter Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD.
Absolute extremum property for Kähler action implies dynamical Kac-Moody and super conformal symmetries
The absolute extremization of Kähler action in the sense that the value of the action is maximal or minimal for a space-time region where the sign of the action density is definite, is a very attractive idea. Both maxima and minima seem to be possible and could correspond to quaternionic (associative) and co-quaternionic (co-associative) space-time sheets emerging naturally in the number theoretic approach to TGD.
It seems now clear that the fundamental formulation of TGD is as an almost-topological conformal field theory for lightlike partonic 3-surfaces. The action principle is uniquely Chern-Simons action for the Kähler gauge potential of CP2 induced to the space-time surface. This approach predicts basic super Kac Moody and superconformal symmetries to be present in TGD and extends them. The quantum fluctuations around classical solutions of these field equations break these super-symmetries partially.
The Dirac determinant for the modified Dirac operator associated with Chern-Simons action defines vacuum functional and the guess is that it equals to the exponent of Kähler action for absolute extremal. The plausibility of this conjecture would increase considerably if one could show that also the absolute extrema of Kähler action possess appropriately broken super-conformal symmetries. This has been a long-lived conjecture but only quite recently I was able to demonstrate it by a simple argument.
The extremal property for Kähler action with respect to variations of time derivatives of initial values keeping hk fixed at X3 implies the existence of an infinite number of conserved charges assignable to the small deformations of the extremum and to H isometries. Also infinite number of local conserved super currents assignable to second variations and to covariantly constant right handed neutrino are implied. The corresponding conserved charges vanish so that the interpretation as dynamical gauge symmetries is appropriate. This result provides strong support that the local extremal property is indeed consistent with the almost-topological QFT property at parton level.
The starting point are field equations for the second variations. If the action contain only derivatives of field variables one obtains for the small deformations δhk of a given extremal
∂α Jαk = 0 ,
Jαk = (∂2 L/∂ hkα∂ hlβ) δ hlβ ,
where hkα denotes the partial derivative ∂α hk. A simple example is the action for massless scalar field in which case conservation law reduces to the conservation of the current defined by the gradient of the scalar field. The addition of mass term spoils this conservation law.
If the action is general coordinate invariant, the field equations read as
DαJα,k = 0
where Dα is now covariant derivative and index raising is achieved using the metric of the imbedding space.
The field equations for the second variation state the vanishing of a covariant divergence and one obtains conserved currents by the contraction this equation with covariantly constant Killing vector fields jAk of M4 translations which means that second variations define the analog of a local gauge algebra in M4 degrees of freedom.
∂αJA,αn = 0 ,
JA,αn = Jα,kn jAk .
Conservation for Killing vector fields reduces to the contraction of a symmetric tensor with Dkjl which vanishes. The reason is that action depends on induced metric and Kähler form only.
Also covariantly constant right handed neutrino spinors ΨR define a collection of conserved super currents associated with small deformations at extremum
Jαn = Jα,knγkΨR .
Second variation gives also a total divergence term which gives contributions at two 3-dimensional ends of the space-time sheet as the difference
Qn(X3f)-Qn(X3) = 0 ,
Qn(Y3) = ∫Y3 d3x Jn ,
Jn = Jtk hklδhln .
The contribution of the fixed end X3 vanishes. For the extremum with respect to the variations of the time derivatives ∂thk at X3 the total variation must vanish. This implies that the charges Qn defined by second variations are identically vanishing
Qn(X3f) = ∫X3fJn = 0 .
Since the second end can be chosen arbitrarily, one obtains an infinite number of conditions analogous to the Virasoro conditions. The analogs of unbroken loop group symmetry for H isometries and unbroken local super symmetry generated by right handed neutrino result. Thus extremal property is a necessary condition for the realization of the gauge symmetries present at partonic level also at the level of the space-time surface. The breaking of super-symmetries could perhaps be understood in terms of the breaking of these symmetries for light-like partonic 3-surfaces which are not extremals of Chern-Simons action.
For more details see the chapter TGD and Astrophysics .
Zeros of Riemann Zeta as conformal weights, braids, Jones inclusions, and number theoretical universality of quantum TGD
Quantum TGD relies on a heuristic number theoretical vision lacking a rigorous justification and I have made considerable efforts to reduce this picture to as few basic unproven assumptions as possible. In the following I want briefly summarize some recent progress made in this respect.
1. Geometry of the world of classical worlds as the basic context
The number theoretic conjectures has been inspired by the construction of the geometry of the configuration space consisting of 3-surfaces of M4× CP2, the "world of classical worlds". Hamiltonians defined at δM4+/-× CP2 are basic elements of super-canonical algebra acting as isometries of the geometry of the "world of classical worlds". These Hamiltonians factorize naturally into products of functions of M4 radial coordinate rM which corresponds to a lightlike direction of lightcone boundary δM4+/- and functions of coordinates of rM constant sphere and CP2 coordinates. The assumption has been that the functions in question are powers of form (rM/r0)Δ where Δ has a natural interpretation as a radial conformal conformal weight.
2. List of conjectures
Quite a thick cloud of conjectures surrounds the construction of configuration space geometry and of quantum TGD.
3. The unifying hypothesis
The most recent progress in TGD is based on the finding that these separate hypothesis can be unified to single assumption. The radial conformal weights Δ are not constants but functions of CP2 coordinate expressible as
where ξ1 and ξ2 are the complex coordinates of CP2 transforming linearly under subgroup U(2) of SU(3). The choice of this coordinate system is not completely unique and relates to the choice of directions of color isospin and hyper charge. This choice has a correlates at space-time and configuration space level in accordance with the idea that also quantum measurement theory has geometric correlates in TGD framework. This hypothesis obviously generalizes the earlier assumption which states that Δ is constant and a linear combination of zeros of Zeta.
A couple of comments are in order.
The quantization of hbar in multiples of integer n characterizing the quantum phase q=exp(iπ/n) in M4 and CP2 degreees of freedom separately means also separate scalings of covariant metrics by n2 in these degrees of freedom. The question is how these copies of imbedding spaces are glued together. The gluing of different p-adic variants of imbedding spaces along rationals and general physical picture suggest how the gluing operation must be carried out.
Two imbedding spaces with different scalings factors of metrics are glued directly together only if either M4 or CP2 scaling factor is same and only along M4 or CP2. This gives a kind of evolutionary tree (actually in rather precise sense as the quantum model for evolutionary leaps as phase transitions increasing hbar(M4) demonstrates!). In this tree vertices represent given M4 (CP2) and lines represent CP2:s (M4:s) with different values of hbar(CP2) (hbar(M4)) emanating from it much like lines from from a vertex of Feynman diagram.
Concerning the mathematical description of this process, the selection of origin of M4 or CP2 as a preferred point is somewhat disturbing. In the case of M4 the problem disappears since configuration space is union over the configuration spaces associated with future and past light cones of M4: CH= CH+U CH-, CH+/-= Um in M4 CH+/-m. In the case of CP2 the same interpretation is necessary in order to not lose SU(3) invariance so that one would have CH+/-= Uh in H CH+/-h. A somewhat analogous but simpler book like structure results in the fusion of different p-adic variants of H along common rationals (and perhaps also common algebraics in the extensions).
For details see the chapter Does TGD Predict the Spectrum of Planck Constants.
Connes tensor product for free fields provides an extremely promising manner to define S-matrix and I have worked out the master formula in a considerable detail. The subfactor N subset of M in Jones represents the degrees of freedom which are not measured. Hence the infinite number of degrees of freedom for M reduces to a finite number of degrees of freedom associated with the quantum Clifford algebra N/M and corresponding quantum spinor space.
The previous physical picture helps to characterize the notion of unitarity precisely for the S-matrix defined by Connes tensor product. For simplicity restrict the consideration to configuration space spin degrees of freedom.
For details see the chapter Was von Neumann Right After All.
Does the quantization of Planck constant transform integer quantum Hall effect to fractional quantum Hall effect?
The TGD based model for topological quantum computation inspired the idea that Planck constant might be dynamical and quantized. The work of Nottale (astro-ph/0310036) gave a strong boost to concrete development of the idea and it took year and half to end up with a proposal about how basic quantum TGD could allow quantization Planck constant associated with M4 and CP2 degrees of freedom such that the scaling factor of the metric in M4 degrees of freedom corresponds to the scaling of hbar in CP2 degrees of freedom and vice versa (see the new chapter Does TGD Predict the Spectrum of Planck constants?). The dynamical character of the scaling factors of M4 and CP2 metrics makes sense if space-time and imbedding space, and in fact the entire quantum TGD, emerge from a local version of an infinite-dimensional Clifford algebra existing only in dimension D=8.
The predicted scaling factors of Planck constant correspond to the integers n defining the quantum phases q=exp(iπ/n) characterizing Jones inclusions. A more precise characterization of Jones inclusion is in terms of group
Gb subset of SU(2) subset of SU(3)
in CP2 degrees of freedom and
in M4 degrees of freedom. In quantum group phase space-time surfaces have exact symmetry such that to a given point of M4 corresponds an entire Gb orbit of CP2 points and vice versa. Thus space-time sheet becomes N(Ga) fold covering of CP2 and N(Gb)-fold covering of M4. This allows an elegant topological interpretation for the fractionization of quantum numbers. The integer n corresponds to the order of maximal cyclic subgroup of G.
In the scaling hbar0→ n× hbar0 of M4 Planck constant fine structure constant would scale as
α= (e2/(4πhbar c)→ α/n ,
and the formula for Hall conductance would transform to
σH =να → (ν/n)× α .
Fractional quantum Hall effect would be integer quantum Hall effect but with scaled down α. The apparent fractional filling fraction ν= m/n would directly code the quantum phase q=exp(iπ/n) in the case that m obtains all possible values. A complete classification for possible phase transitions yielding fractional quantum Hall effect in terms of finite subgroups G subset of SU(2) subset of SU(3) given by ADE diagrams would emerge (An, D2n, E6 and E8 are possible). What would be also nice that CP2 would make itself directly manifest at the level of condensed matter physics.
For more details see the chapter Topological Quantum Computation in TGD Universe, and the chapters Was von Neumann Right After All? and Does TGD predict the Spectrum of Planck Constants?.
There has been intensive evolution of ideas induced by the understanding of large values of Planck constants. This motivated a separate chapter which I christened as "Does TGD Predict the Spectrum of Planck Constants?". I have commented earlier about various ideas related to this topic and comment here only the newest outcomes.
1. hbargr as CP2 Planck constant
What gravitational Planck constant means has been somewhat unclear. It turned out that hbargr can be interpreted as Planck constant associated with CP2 degrees of freedom and its huge value implies that also the von Neumann inclusions associated with M4 degrees of freedom meaning that dark matter cosmology has quantal lattice like structure with lattice cell given by Ha/G, Ha the a=constant hyperboloid of M4+ and G subgroup of SL(2,C). The quantization of cosmic redshifts provides support for this prediction.
2. Is Kähler coupling strength invariant under p-adic coupling constant evolution
Kähler coupling constant is the only coupling parameter in TGD. The original great vision is that Kähler coupling constant is analogous to critical temperature and thus uniquely determined. Later I concluded that Kähler coupling strength could depend on the p-adic length scale. The reason was that the prediction for the gravitational coupling strength was otherwise non-sensible. This motivated the assumption that gravitational coupling is RG invariant in the p-adic sense.
The expression of the basic parameter v0=2-11 appearing in the formula of hbargr=GMm/v0 in terms of basic parameters of TGD leads to the unexpected conclusion that αK in electron length scale can be identified as electro-weak U(1) coupling strength αU(1). This identification, or actually something slightly complex (see below), is what group theory suggests but I had given it up since the resulting evolution for gravitational coupling predicted G to be proportional to Lp2 and thus completely un-physical. However, if gravitational interactions are mediated by space-time sheets characterized by Mersenne prime, the situation changes completely since M127 is the largest non-super-astrophysical p-adic length scale.
The second key observation is that all classical gauge fields and gravitational field are expressible using only CP2 coordinates and classical color action and U(1) action both reduce to Kähler action. Furthermore, electroweak group U(2) can be regarded as a subgroup of color SU(3) in a well-defined sense and color holonomy is abelian. Hence one expects a simple formula relating various coupling constants. Let us take αK as a p-adic renormalization group invariant in strong sense that it does not depend on the p-adic length scale at all.
The relationship for the couplings must involve αU(1), αs and αK. The formula 1/αU(1)+1/αs = 1/αK states that the sum of U(1) and color actions equals to Kähler action and is consistent with the decrease of the color coupling and the increase of the U(1) coupling with energy and implies a common asymptotic value 2αK for both. The hypothesis is consistent with the known facts about color and electroweak evolution and predicts correctly the confinement length scale as p-adic length scale assignable to gluons. The hypothesis reduces the evolution of αs to the calculable evolution of electro-weak couplings: the importance of this result is difficult to over-estimate.
For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.
If the v0 deduced to have value v0=2-11 appearing in the expression for gravitational Planck constant hbargr=GMm/v0 is identified as the rotation velocity of distant stars in galactic plane, it is possible to express it in terms of Kähler coupling strength and string tension as v0-2= 2×αKK,
αK(p)= a/log(pK) , K= R2/G .
The value of K is fixed to a high degree by the requirement that electron mass scale comes out correctly in p-adic mass calculations. The uncertainties related to second order contributions in p-adic mass calculations however leave the precise value open. Number theoretic arguments suggest that K is expressible as a product of primes p ≤ 23: K= 2×3×5×...×23 .
If one assumes that αK is of order fine structure constant in electron length scale, the value of the parameter a cannot be far from unity. A more precise condition would result by identifying αK with weak U(1) coupling strength αK = αU(1)=αem/cos2(θW)≈ 1/105.3531 ,
αem= 0.00729735253327 .
Here the values refer to electron length scale. If the formula v0= 2-11 is exact, it poses both quantitative and number theoretic conditions on Kähler coupling strength. One must of course remember, that exact expression for v0 corresponds to only one particular solution and even smallest deformation of solution can change the number theoretical anatomy completely. In any case one can make following questions.
The basic condition stating that gravitational coupling constant is renormalization group invariant dictates the dependence of the Kähler coupling strength of p-adic prime exponent of Kähler action for CP2 type extremal is rational if K is integer as assumed: this is essential for the algebraic continuation of the rational physics to p-adic number fields. This gives a general formula αK= a π/log(pK), a of order unity. Since K is integer, this means that for rational value of a one would have
v02= qlog(pK)/π, q rational.
The condition for v0=2-m, m=11, allows to deduce the value of a as
a= (log(pK)/π) × (22m/K).
The condition that αK is of order fine structure constant for p=M127= 2127-1 defining the p-adic length scale of electron indeed implies that m=11 is the only possible value since the value of a is scaled by a factor 4 in m→ m+1.
The value of αK in the length scale Lp0 in which condition of the first equation holds true is given by
1/αK= 221/K≈ 106.379 .
2. What is the value of the preferred prime p0?
The condition for v0 can hold only for a single p-adic length scale Lp0. This correspondence would presumably mean that gravitational interaction is mediated along the space-time sheets characterized by p0, or even that gravitons are characterized by p0.
For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.
The understanding of how the quantum octonionic local version of infinite-dimensional Clifford algebra of 8-dimensional space (the only possible local variant of this algebra) implies entire quantum and classical TGD led also to the understanding of the quantization of Planck constant. In the model for planetary orbits based on gigantic gravitational Planck constant this means powerful constraints on the number theoretic anatomy of gravitational Planck constants and therefore of planetary mass ratios. These very stringent predictions are immediately testable.
1. Preferred values of Planck constants and ruler and compass polygons
The starting point is that the scaling factor of M4 Planck constant is given by the integer n characterizing the quantum phase q= exp(iπ/n). The evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible p-adically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant.
One expects that quantum phases q=exp(iπ/n) which are expressible using only square roots of rationals are number theoretically very special since they correspond to algebraic extensions of p-adic numbers involving only square roots which should emerge first and therefore systems involving these values of q should be especially abundant in Nature.
These polygons are obtained by ruler and compass construction and Gauss showed that these polygons, which could be called Fermat polygons, have
nF= 2k ∏s Fns
sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes Fn=22n+1 correspond to n=0,1,2,3,4 with F0=3, F1=5, F2=17, F3=257, F4=65537. It is not known whether there are higher Fermat primes. n=3,5,15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter.
2. Application to planetary Bohr orbitology
The understanding of the quantization of Planck constants in M4 and CP2 degrees of freedom led to a considerable progress in the understanding of the Bohr orbit model of planetary orbits proposed by Nottale, whose TGD version initiated "the dark matter as macroscopic quantum phase with large Planck constant" program.
Gravitational Planck constant is given by
where an estimate for the value of v0 can be deduced from known masses of Sun and planets. This gives v0≈ 4.6× 10-4.
Combining this expression with the above derived expression one obtains
GMm/v0= nF= 2k ∏ns Fns
In practice only the Fermat primes 3,5,17 appearing in this formula can be distinguished from a power of 2 so that the resulting formula is extremely predictive. Consider now tests for this prediction.
To sum up, it seems that everything is now ready for the great revolution. I would be happy to share this flood of discoveries with colleagues but all depends on what establishment decides. To my humble opinion twenty one years in a theoretical desert should be enough for even the most arrogant theorist. There is now a book of 800 A4 pages about TGD at Amazon: Topological Geometrodynamics so that it is much easier to learn what TGD is about.
The reader interested in details is recommended to look at the chapter Does TGD Predict the Spectrum of Planck Constants? of this book and the chapter TGD and Astrophysics of "TGD and Astro-Physics".
Connes tensor product as universal interaction, quantization of Planck constant, McKay correspondence, etc...
It seems that discussion both in Peter Woit's blog, John Baez's This Week's Findings, and in h Lubos Motl's blog happen to tangent very closely what I have worked with during last weeks: ADE and Jones inclusions.
1. Some background.
2. How to localize infinite-dimensional Clifford algebra?
The basic new idea is to make this algebra local: local Clifford algebra as a generalization of gamma field of string models.
3. Connes tensor product for free fields as a universal definition of interaction quantum field theory
This picture has profound implications. Consider first the construction of S-matrix.
4. The quantization of Planck constant and ADE hierarchies
The quantization of Planck constant has been the basic them of TGD for more than one and half years and leads also the understanding of ADE correspondences (index ≤ 4 and index=4) from the point of view of Jones inclusions.
For details see the chapter Was von Neumann Right After All?.
Configuration space spinor fields live in "the world of classical worlds", whose points correspond to 3-surfaces in H=M4×CP2. These fields represent the quantum states of the universe. Configuration space spinors (to be distinguished from spinor fields) have a natural interpretation in terms of a quantum version of Boolean algebra obtained by applying fermionic operators to the vacuum state. Both fermion number and various spinlike quantum numbers can be interpreted as representations of bits. Once you have true and false you have also beliefs and the question is whether it is possible to construct a quantum model for beliefs.
1. Some background about number theoretic Clifford algebras
Configuration space spinors are associated with an infinite-dimensional Clifford algebra spanned by configuration space gamma matrices: spinors are created from vacuum state by complexified gamma matrices acting like fermionic oscillator operators carrying quark and lepton numbers. In a rough sense this algebra could be regarded as an infinite tensor power of M2(F), where F would correspond to complex numbers. In fact, also F=H (quaternions) and even F=O (octonions) can and must(!) be considered although the definitions involve some delicacies in this case. In particular, the non-associativy of octonions poses an interpretational problem whose solution actually dictates the physics of TGD Universe.
These Clifford algebras can be extended local algebras representable as powers series of hyper-F coordinate (hyper-F is obtained by multiplying imaginary part of F number with a commuting additional imaginary unit) so that a generalization of conformal field concept results with powers of complex coordinate replaced with those of hyper-complex numerg, hyper-quaternion or octonion. TGD could be seen as a generalization of superstring models by adding H and O layers besides C so that space-time and imbedding space emerge without ad hoc tricks of spontaneous compactification and adding of branes non-perturbatively.
The inclusion sequence C in H in O induces generalization of Jones inclusion sequence for the local versions of the number theoretic Clifford algebras allowing to reduce quantum TGD to a generalized number theory. That is, classical and quantum TGD emerge from the natural number theoretic Jones inclusion sequence. Even more, an explicit master formula for S-matrix emerges consistent with the earlier general ideas. It seems safe to say that one chapter in the evolution of TGD is now closed and everything is ready for the technical staff to start their work.
2. Brahman=Atman property of hyper-finite type II1 factors makes them ideal for realizing symbolic and cognitive representations
Infinite-dimensional Clifford algebras provide a canonical example of von Neumann algebras known as hyper-finite factors of type II1 having rather marvellous properties. In particular, they possess Brahman= Atman property making it possible to imbed this kind of algebra within itself unitarily as a genuine sub-algebra. One obtains what infinite Jones inclusion sequences yielding as a by-product structures like quantum groups.
Jones inclusions are ideal for cognitive and symbolic representations since they map the fermionic state space of one system to a sub-space of the fermionic statespace of another system. Hence there are good reasons to believe that TGD universe is busily mimicking itself using Jones inclusions and one can identify the space-time correlates (braids connecting two subsystems consisting of magnetic flux tubes). p-Adic and real spinors do not differ in any manner and real-to-p-adic inclusions would give cognitive representations, real-to-real inclusions symbolic representations.
3. Jones inclusions and cognitive and symbolic representations
As already noticed, configuration space spinors provide a natural quantum model for the Boolean logic. When you have logic you have the notions of truth and false, and you have soon also the notion of belief. Beliefs of various kinds (knowledge, misbelief, delusion,...) are the basic element of cognition and obviously involve a representation of the external world or part of it as states of the system defining the believer. Jones inclusions for the mediating unitary mappings between the spaces of configuration spaces spinors of two systems are excellent candidates for these maps, and it is interesting to find what one kind of model for beliefs this picture leads to.
The resulting quantum model for beliefs provides a cognitive interpretation for quantum groups and predicts a universal spectrum for the probabilities that a given belief is true following solely from the commutation relations for the coordinates of complex quantum plane interpreted now as complex spinor components. This spectrum of probabilities depends only on the integer n characterizing the quantum phase q=exp(iπ/n) characterizing the Jones inclusion. For n < ∞ the logic is inherently fuzzy so that absolute knowledge is impossible. q=1 gives ordinary quantum logic with qbits having precise truth values after state function reduction.
One can make two conclusions.
The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?
The idea that the notion of space-time somehow from quantum theory is rather attractive. In TGD framework this would basically mean that the identification of space-time as a surface of 8-D imbedding space H=M4× CP2 emerges from some deeper mathematical structure. It seems that the series of inclusions for infinite-dimensional Clifford algebras associated with classical number fields F=R,C,H,O defining von Neumann algebras known as hyper-finite factors of type II1, could be this deeper mathematical structure.
1. Quaternions, octonions, and TGDThe dimensions of quaternions and octonions are 4 and 8 and same as the dimensions of space-time surface and imbedding space in TGD. It is difficult to avoid the feeling that TGD physics could somehow reduce to the structures assignable to the classical number fields. This vision is already now rather detailed. For instance, a proposal for a general solution of classical field equations is one outcome of this vision.
TGD suggests also what I call HO-H duality. Space-time can be regarded either as surface in H or as hyper-quaternionic sub-manifold of the space HO of hyper-octonions obtained by multiplying imaginary parts of octonions with a commuting additional imaginary unit.
The 2-dimensional partonic surfaces X2 are of central importance in TGD and it seems that the inclusion sequence C in H in O (complex numbers, quaternions, octonions) somehow corresponds to the inclusion sequence X2 in X4 in H. This inspires the that that whole TGD emerges from a generalized number theory and I have already proposed arguments for how this might happen.
2. Number theoretic Clifford algebrasHyper-finite factors of type II1 defined by infinite-dimensional Clifford algebras is one thread in the multiple strand of number-theoretic ideas involving p-adic numbers fields and their fusion with reals along common rationals to form a generalized number system, classical number fields, hierarchy of infinite primes and integers, and von Neumann algebras and quantum groups. The new ideas allow to fuse von Neumans strand with the classical number field strand.
Physics as a generalized number theory vision suggests that TGD physics is contained by the Jones inclusion sequence Cl(C) in Cl(H) in Cl(O) induced by C in H in O. This sequence could alone explain partonic, space-time, and imbedding space dimensions as dimensions of classical number fields. The dream is that also imbedding space H=M4× CP2 would emerge as a unique choice allowed by mathematical existence.
4. Number-theoretic localization of infinite-dimensional number theoretic Clifford algebras as a lacking piece of puzzle The lacking piece of the big argument is below.
5. Explicit general formula for S-matrix emerges also
This picture leads also to an explicit master formula for S-matrix.
The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?
Genus-generation correspondence is one of the basic ideas of TGD approach. In order to answer various questions concerning the plausibility of the idea, one should know something about the dependence of the elementary particle vacuum functionals on the vibrational degrees of freedom for the boundary component. The construction of the elementary particle vacuum functionals based on Diff invariance, 2-dimensional conformal symmetry, modular invariance plus natural stability requirements indeed leads to an essentially unique form of the vacuum functionals and one can understand why g >2 bosonic families are experimentally absent and why lepton numbers are conserved separately.
An argument suggesting that the number of the light fermion families is three, is developed. The argument goes as follows. Elementary particle vacuum functionals represent bound states of g handles and vanish identically for hyper-elliptic surfaces having g > 2. Since all g≤ 2 surfaces are hyper-elliptic, g≤ 2 and g > 2 elementary particles cannot appear in same non-vanishing vertex and therefore decouple. The g>2 vacuum functionals not vanishing for hyper-elliptic surfaces represent many particle states of g≤ 2 elementary particle states being thus unstable against the decay to g≤ 2 states. The failure of Z2 conformal symmetry for g>2 elementary particle vacuum functionals could in turn explain why they are heavy: this however not absolutely necessary since these particles would behave like dark matter in any case.