Year 2006

Quantization of the modified Dirac action

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

1. to 2-dimensional delta function as the expression for the bosonic Noether charges identified as configuration space Hamiltonians would suggest, or
2. to 1-dimensional delta function along 1-D curve of partonic 2-surfaces conformal field theory picture would suggest.

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 r^Δ, 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 hierarchy for the zeros of Riemann zeta assumed to define a hierarchy of algebraic extensions of rationals,

• the discretization of the partonic 2-surface by replacing it with a subset of the discrete intersection of the real partonic 2-surface and its p-adic counterpart obtained by algebraic continuation of algebraic equations defining the 2-surface, and

• the hierarchy of quantum phases associated with the hierarchy of Jones inclusions related to the generalization of the notion of imbedding space

The mode expansion of the second quantized spinor field has a natural cutoff for angular momentum l and isospin I corresponding to the integers n_a and n_b characterizing the orders of maximal cyclic subgroups of groups G_a and G_b defining the Jones inclusion in M⁴ and CP₂ degrees of freedom and characterizing the Planck constants. More precisely: one has l≤ n_a and I≤ n_b. 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∑_kn_ky_k, where s_k=1/2+iy_kare 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.

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 h^k 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 δh^k of a given extremal

∂_α J^α_k = 0 ,

J^α_k = (∂² L/∂ h^k_α∂ h^l_β) δ h^l_β ,

where h^k_α denotes the partial derivative ∂_α h^k. 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

D_αJ^α,k = 0

where D_α 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 j_A^k of M⁴ translations which means that second variations define the analog of a local gauge algebra in M⁴ degrees of freedom.

∂_αJ^A,α_n = 0 ,

J^A,α_n = J^α,k_n j^A_k .

Conservation for Killing vector fields reduces to the contraction of a symmetric tensor with D_kj_l which vanishes. The reason is that action depends on induced metric and Kähler form only.

Also covariantly constant right handed neutrino spinors Ψ_R define a collection of conserved super currents associated with small deformations at extremum

J^α_n = J^α,k_nγ_kΨ_R .

Second variation gives also a total divergence term which gives contributions at two 3-dimensional ends of the space-time sheet as the difference

Q_n(X³_f)-Q_n(X³) = 0 ,

Q_n(Y³) = ∫_Y³ d³x J_n ,

J_n = J^tk h_klδh^l_n .

The contribution of the fixed end X³ vanishes. For the extremum with respect to the variations of the time derivatives ∂_th^k at X³ the total variation must vanish. This implies that the charges Q_n defined by second variations are identically vanishing

Q_n(X³_f) = ∫_X³fJ_n = 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 Construction of configuration space Kähler geometry from symmetry principles: Part II .

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 r_M which corresponds to a lightlike direction of lightcone boundary δM⁴_+/- and functions of coordinates of r_M constant sphere and CP₂ coordinates. The assumption has been that the functions in question are powers of form (r_M/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.

1. Number theoretic universality of Riemann Zeta states that the factors 1/(1+p^s) appearing in its product representation are algebraic numbers for the zeros s=1/2+iy of Riemann zeta, and thus also for their linear combinations. Thus for any prime p, any zero s, and any p-adic number field, the number p^iy belongs to some finite-dimensional algebraic extension of the p-adic number field in question.

2. If the radial conformal weights are linear combinations of zeros of Zeta with integer coefficients, then for rational values of r_M/r₀ the exponents (r_M/r₀)^Δ are in some finite-dimensional algebraic extension of the p-adic number field in question. This is crucial for the p-adicization of quantum TGD implying for instance that S-matrix elements are algebraic numbers.

3. Quantum classical correspondence is realized in the sense that the radial conformal weights Δ are represented as (mapped to) points of CP₂ much like momenta have classical representation as 3-vectors. CP₂ would play a role of heavenly sphere, so to say.

4. The third hypothesis could be called braiding hypothesis.
   • For a given parton surface X² identified as intersection of Δ M⁴_+/-× CP₂ and lightlike partonic orbit X³_l the images of radial conformal weights have interpretation as a braid.

   • The Kac-Moody type conformal algebra associated with X³_l restricted to X² acts on the radial conformal weights like on points of complex plane. Also the Kac-Moody algebra of X³_l acts on the radial conformal weights in a non-trivial manner. There exists a unique braiding operation defined by the dynamics of X² defined by X³_l . This operation is highly relevant for the model of topological quantum computation and TGD based model of anyons and quantum Hall effect.

   • These braids relate closely to the hierarchy of braids providing representation for a Jones inclusion of von Neumann algebra known as hyperfinite factor of type II₁ and emerging naturally as the infinite-dimensional Clifford algebra of the "world of the classical worlds".

   • These braids define the finite sets of points which appear in the construction of universal S-matrix whose elements are algebraic numbers and thus can be interpreted as elements of any number field. This would mean that it is possible to construct S-matrix for say p-adic-to-real transitions representing transformation of intention to action using same formulas as for ordinary S-matrix.

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

Δ= ζ^-1(ξ¹/ξ²),

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.

1. The inverse of zeta has infinitely many branches in one-one correspondence with the zeros of zeta and the branch can change only for certain values of r_M such that imaginary part of Δ changes: this has very interesting physical implications.

2. Accepting the universality of the zeros of Riemann Zeta, one also ends up naturally with the hypothesis that the points of the partonic 2-surface appearing in the construction of the number theoretically universal S-matrix correspond to images ζ(s) of points s=∑ n_ks_k expressible as linear combinations of zeros of zeta with the additional condition that r_M/r₀ is rational. In this manner one indeed obtains representation of allowed conformal weights on the "heavenly sphere" defined by CP₂ and also other hypothesis follow naturally.

3. In this framework braids are actually replaced by tangles for which the strand of braid can turn backwards.

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