Monday, January 19, 2009

The recent view about the construction of configuration space spinor structure

During the last five years both the mathematical and physical understanding of quantum TGD has developed dramatically. Some ideas have died and large number of conjectures have turned to be un-necessary strong, un-necessary, or simply wrong. The outcome is that the books about basic TGD do not correspond the actual situation in the theory. Therefore I decided to perform a major cleaning operation throwing away the obsolete stuff and making good arguments more precise. Good household is not my only motivation: this kind of process, although it challenges the ego, is always extremely fruitful. The basic goal has been to replace the perspective as it was for five years ago with the one which is outcome of the development of visions and concepts like fundamental description of quantum TGD as almost topological QFT in terms of modified Dirac action for fermions at light-like 3-surfaces identified as the basic objects of the theory, zero energy ontology, finite measurement resolution as a fundamental physical principle realized in terms of Jones inclusions and having number theoretic braids as space-time correlate, generalization of S-matrix to M-matrix, number theoretical universality and number theoretical compactification reducing standard model symmetries to number theory and allowing to solve some basic problems of quantum TGD, realization of the hierarchy of Planck constants in terms of the generalization of imbedding space concept, discovery of a hierarchy of symplectic fusion algebras provided concrete understanding of the super-symplectic conformal invariance, and so on.

I started the cleaning up process from the chapter Configuration Space Spinor Structure and I glue below the abstract.

Quantum TGD should be reducible to the classical spinor geometry of the configuration space. In particular, physical states should correspond to the modes of the configuration space spinor fields. The immediate consequence is that configuration space spinor fields cannot, as one might naively expect, be carriers of a definite spin and unit fermion number. Concerning the construction of the configuration space spinor structure there are some important clues.

1. Geometrization of fermionic statistics in terms of configuration space spinor structure

The great vision has been that the second quantization of the induced spinor fields can be understood geometrically in terms of the configuration space spinor structure in the sense that the anti-commutation relations for configuration space gamma matrices require anti-commutation relations for the oscillator operators for free second quantized induced spinor fields.

  1. One must identify the counterparts of second quantized fermion fields as objects closely related to the configuration space spinor structure. Ramond model has as its basic field the anti-commuting field Gk(x), whose Fourier components are analogous to the gamma matrices of the configuration space and which behaves like a spin 3/2 fermionic field rather than a vector field. This suggests that the complexified gamma matrices of the configuration space are analogous to spin 3/2 fields and therefore expressible in terms of the fermionic oscillator operators so that their anti-commutativity naturally derives from the anti-commutativity of the fermionic oscillator operators.

    As a consequence, configuration space spinor fields can have arbitrary fermion number and there would be hopes of describing the whole physics in terms of configuration space spinor field. Clearly, fermionic oscillator operators would act in degrees of freedom analogous to the spin degrees of freedom of the ordinary spinor and bosonic oscillator operators would act in degrees of freedom analogous to the 'orbital' degrees of freedom of the ordinary spinor field.

  2. The classical theory for the bosonic fields is an essential part of the configuration space geometry. It would be very nice if the classical theory for the spinor fields would be contained in the definition of the configuration space spinor structure somehow. The properties of the modified massless Dirac operator associated with the induced spinor structure are indeed very physical. The modified massless Dirac equation for the induced spinors predicts a separate conservation of baryon and lepton numbers. The differences between quarks and leptons result from the different couplings to the CP2 Kähler potential. In fact, these properties are shared by the solutions of massless Dirac equation of the imbedding space.

  3. Since TGD should have a close relationship to the ordinary quantum field theories it would be highly desirable that the second quantized free induced spinor field would somehow appear in the definition of the configuration space geometry. This is indeed true if the complexified configuration space gamma matrices are linearly related to the oscillator operators associated with the second quantized induced spinor field on the space-time surface and/or its boundaries. There is actually no deep reason forbidding the gamma matrices of the configuration space to be spin half odd-integer objects whereas in the finite-dimensional case this is not possible in general. In fact, in the finite-dimensional case the equivalence of the spinorial and vectorial vielbeins forces the spinor and vector representations of the vielbein group SO(D) to have same dimension and this is possible for D=8-dimensional Euclidian space only. This coincidence might explain the success of 10-dimensional super string models for which the physical degrees of freedom effectively correspond to an 8-dimensional Euclidian space.

  4. It took a long time to realize that the ordinary definition of the gamma matrix algebra in terms of the anti-commutators {gA,gB} = 2gAB must in TGD context be replaced with {gAf,gB} = iJAB\per, where JAB denotes the matrix elements of the Kähler form of the configuration space. The presence of the Hermitian conjugation is necessary because configuration space gamma matrices carry fermion number. This definition is numerically equivalent with the standard one in the complex coordinates. The realization of this delicacy is necessary in order to understand how the square of the configuration space Dirac operator comes out correctly.

  5. The only possible option is that second quantized induced spinor fields are defined at 3-D light-like causal determinants associated with 4-D space-time sheet. The unique partonic dynamics is almost topological QFT defined by Chern-Simons action for the induced Kähler gauge potential and by the modified Dirac action constructed from it by requiring super-conformal symmetry. The resulting theory has all the desired super-conformal symmetries and is exactly solvable at parton level. It is 3-dimensional lightlike 3-surfaces rather than generic 3-surfaces which are the fundamental dynamical objects in this approach.

    The classical dynamics of the interior of space-time surface defines a classical correlate for the partonic quantum dynamics and provides a realization of quantum measurement theory. It is determined by the vacuum functional identified as the Dirac determinant. There are good arguments suggesting that it reduces to an exponent of absolute extremum of Kähler action in each region of the space-time sheet where the Kähler action density has a definite sign.

2. Modified Dirac equation for induced classical spinor fields

The identification of the light-like partonic 3-surfaces as carriers of elementary particle quantum numbers inspired by the TGD based quantum measurement theory forces the identification of the modified Dirac action as that associated with the Chern-Simons action for the induced Kähler gauge potential. At the fundamental level TGD would be almost-topological super-conformal QFT in the sense that only the light-likeness condition for the partonic 3-surfaces would involve the induced metric. Chern-Simons dynamics would thus involve the induced metric only via the generalized eigenvalue equation for the modified Dirac operator involving the light-like normal of X3l subset X4. N=4 super-conformal symmetry emerges as a maximal Super-Kac Moody symmetry for this option. The application of D to any generalized eigen-mode gives a zero mode and zero modes and generalized eigen-modes define a cohomology.

The basic idea is that Dirac determinant defined by eigenvalues of DC-S can be identified as the exponent of Kähler action for a preferred extremal. There are however two problems. Without further conditions the eigenvalues of DC-S are functions of the transversal coordinates of X3l and the standard definition of Dirac determinant fails. Second problem is how to feed the information about preferred extremal to the eigenvalue spectrum. The solution of these problems is discussed below.

The eigen modes of the modified Dirac equation are interpreted as generators of exact N=4 super-conformal symmetries in both quark and lepton sectors. These super-symmetries correspond to pure super gauge transformations and no spartners of ordinary particles are predicted: in particular N=2 space-time super-symmetry is generated by the righthanded neutrino is absent contrary to the earliest beliefs. There is no need to emphasize the experimental implications of this finding.

An essential difference with respect to standard super-conformal symmetries is that Majorana condition is not satisfied, the super generators carry quark or lepton number, and the usual super-space formalism does not apply. The situation is saved by the fact that super generators of super-conformal algebras anticommute to Hamiltonians of symplectic transformations rather than vector fields representing the transformations.

Configuration space gamma matrices identified as super generators of super-symplectic or super Kac-Moody algebras (depending on CH coordinates used) are expressible in terms of the oscillator operators associated with the eigen modes of the modified Dirac operator. The number of generalized eigen modes turns out to be finite so that standard canonical quantization does not work unless one restricts the set of points involved defined as intersection of number theoretic braid with the partonic 2-surface. The interpretation is in terms of finite measurement resolution and the surprising thing is that this notion is implied by the vacuum degeneracy of Kähler action.

3. The exponent of Kähler function as Dirac determinant for the modified Dirac action

Although quantum criticality in principle predicts the possible values of Kähler coupling strength, one might hope that there exists even more fundamental approach involving no coupling constants and predicting even quantum criticality and realizing quantum gravitational holography.

  1. The Dirac determinant defined by the product of Dirac determinants associated with the light-like partonic 3-surfaces X3l associated with a given space-time sheet X4 is the simplest candidate for vacuum functional identifiable as the exponent of the Kähler function. One can of course worry about the finiteness of the Dirac determinant. p-Adicization requires that the eigenvalues belong to a given algebraic extension of rationals. This restriction would imply a hierarchy of physics corresponding to different extensions and could automatically imply the finiteness and algebraic number property of the Dirac determinants if only finite number of eigenvalues would contribute. The regularization would be performed by physics itself if this were the case.

  2. The basic problem has been how to feed in the information about the preferred extremal of Kähler action to the eigenvalue spectrum of C-S Dirac operator DC-S at light-like 3-surface X3l. The identification of the preferred extremal came possible via boundary conditions at X3l dictated by number theoretical compactification. The basic observation is that the Dirac equation associated with the 4-D Dirac operator DK defined by Kähler action can be seen as a conservation law for a super current. By restricting the super current to flow along X3l by requiring that its normal component vanishes, one obtains a singular solution of 4-D modified Dirac equation restricted to X3l. The ënergy" spectrum for the solutions of DK corresponds to the spectrum of eigenvalues for DC-S and the product of the eigenvalues defines the Dirac determinant in standard manner. Since the eigenmodes are restricted to those localized to regions of non-vanishing induced Kähler form, the number of eigen modes is finite and therefore also Dirac determinant is finite. The eigenvalues can be also algebraic numbers.

  3. It remains to be proven that the product of eigenvalues gives rise to the exponent of Kähler action for the preferred extremal of Kähler action. At this moment the only justification for the conjecture is that this the only thing that one can imagine. The identification of super-symplectic conformal weights as zeros of zeta function defined by the eigenvalues of modified Dirac operator would couple them with the dynamics defined by the Kähler action.

  4. A long-standing conjecture has been that the zeros of Riemann Zeta are somehow relevant for quantum TGD. Rieman zeta is however naturally replaced Dirac zeta defined by the eigenvalues of DC-S and closely related to Riemann Zeta since the spectrum consists essentially for the cyclotron energy spectra for localized solutions region of non-vanishing induced Kähler magnetic field and hence is in good approximation integer valued up to some cutoff integer. In zero energy ontology the Dirac zeta function associated with these eigenvalues defines"square root" of thermodynamics assuming that the energy levels of the system in question are expressible as logarithms of the eigenvalues of the modified Dirac operator defining kind of fundamental constants. Critical points correspond to approximate zeros of Dirac zeta and if Kähler function vanishes at criticality as it ineed should, the thermal energies at critical points are in first order approximation proportional to zeros themselves so that a connection between quantum criticality and approximate zeros of Dirac zeta emerges.

  5. The discretization induced by the number theoretic braids reduces the world of classical worlds to effectively finite-dimensional space and configuration space Clifford algebra reduces to a finite-dimensional algebra. The interpretation is in terms of finite measurement resolution represented in terms of Jones inclusion M subset N of HFFs with M taking the role of complex numbers. The finite-D quantum Clifford algebra spanned by fermionic oscillator operators is identified as a representation for the coset space N/M describing physical states modulo measurement resolution. In the sectors of generalized imbedding space corresponding to non-standard values of Planck constant quantum version of Clifford algebra is in question.

4. Super-conformal symmetries

The almost topological QFT property of partonic formulation based on Chern-Simons action and corresponding modified Dirac action allows a rich structure of N=4 super-conformal symmetries. In particular, the generalized Kac-Moody symmetries leave corresponding X3-local isometries respecting the light-likeness condition. A rather detailed view about various aspects of super-conformal symmetries emerge leading to identification of fermionic anti-commutation relations and explicit expressions for configuration space gamma matrices and Kähler metric. This picture is consistent with the conditions posed by p-adic mass calculations.

Number theoretical considerations play a key role and lead to the picture in which effective discretization occurs so that partonic two-surface is effectively replaced by a discrete set of algebraic points belonging to the intersection of the real partonic 2-surface and its p-adic counterpart obeying the same algebraic equations. This implies effective discretization of super-conformal field theory giving N-point functions defining vertices via discrete versions of stringy formulas.

For the updated version of the chapter see Configuration Space Spinor Structure of "Physics as Infinite-Dimensional Geometry".

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