Classical number fields and associativity and commutativity as fundamental law of physics

In the previous posting I told about the possibility that string world sheets with area action could be present in TGD at fundamental level with the ratio of hbar G/R² of string tension to the square of CP₂ radius fixed by quantum criticality. I however found that the assumption that gravitational binding has as correlates strings connecting the bound partonic 2-surfaces leads to grave difficulties: the sizes of the gravitationally bound states cannot be much longer than Planck length. This binding mechanism is strongly suggested by AdS/CFT correspondence but perturbative string theory does not allow it.

I proposed that the replacement of h with h_eff = n× h= h_gr= GMm/v₀ could resolve the problem. It does not. I soo noticed that the typical size scale of string world sheet scales as h_gr^1/2, not as h_gr= GMm/v₀ as one might expect. The only reasonable option is that string tension behave as 1/h_gr². In the following I demonstrate that TGD in its basic form and defined by super-symmetrized Kähler action indeed predicts this behavior if string world sheets emerge. They indeed do so number theoretically from the condition of associativity and also from the condition that electromagnetic charge for the spinor modes is well-defined. By the analog of AdS/CFT correspondence the string tension could characterize the action density of magnetic flux tubes associated with the strings and varying string tension would correspond to the effective string tension of the magnetic flux tubes as carriers of magnetic energy (dark energy is identified as magnetic energy in TGD Universe).

Therefore the visit of string theory to TGD Universe remained rather short but it had a purpose: it made completely clear why superstring are not the theory of gravitation and why TGD can be this theory.

Do associativty and commutativity define the laws of physics?

The dimensions of classical number fields appear as dimensions of basic objects in quantum TGD. Imbedding space has dimension 8, space-time has dimension 4, light-like 3-surfaces are orbits of 2-D partonic surfaces. If conformal QFT applies to 2-surfaces (this is questionable), one-dimensional structures would be the basic objects. The lowest level would correspond to discrete sets of points identifiable as intersections of real and p-adic space-time sheets. This suggests that besides p-adic number fields also classical number fields (reals, complex numbers, quaternions, octonions are involved and the notion of geometry generalizes considerably. In the recent view about quantum TGD the dimensional hierarchy defined by classical number field indeed plays a key role. H=M⁴× CP₂ has a number theoretic interpretation and standard model symmetries can be understood number theoretically as symmetries of hyper-quaternionic planes of hyper-octonionic space.

The associativity condition A(BC)= (AB)C suggests itself as a fundamental physical law of both classical and quantum physics. Commutativity can be considered as an additional condition. In conformal field theories associativity condition indeed fixes the n-point functions of the theory. At the level of classical TGD space-time surfaces could be identified as maximal associative (hyper-quaternionic) sub-manifolds of the imbedding space whose points contain a preferred hyper-complex plane M² in their tangent space and the hierarchy finite fields-rationals-reals-complex numbers-quaternions-octonions could have direct quantum physical counterpart. This leads to the notion of number theoretic compactification analogous to the dualities of M-theory: one can interpret space-time surfaces either as hyper-quaternionic 4-surfaces of M⁸ or as 4-surfaces in M⁴× CP₂. As a matter fact, commutativity in number theoretic sense is a further natural condition and leads to the notion of number theoretic braid naturally as also to direct connection with super string models.

At the level of modified Dirac action the identification of space-time surface as a hyper-quaternionic sub-manifold of H means that the modified gamma matrices of the space-time surface defined in terms of canonical momentum currents of Kähler action using octonionic representation for the gamma matrices of H span a hyper-quaternionic sub-space of hyper-octonions at each point of space-time surface (hyper-octonions are the subspace of complexified octonions for which imaginary units are octonionic imaginary units multiplied by commutating imaginary unit). Hyper-octonionic representation leads to a proposal for how to extend twistor program to TGD framework .

How to achieve associativity in the fermionic sector?

In the fermionic sector an additional complication emerges. The associativity of the tangent- or normal space of the space-time surface need not be enough to guarantee the associativity at the level of Kähler-Dirac or Dirac equation. The reason is the presence of spinor connection. A possible cure could be the vanishing of the components of spinor connection for two conjugates of quaternionic coordinates combined with holomorphy of the modes.

1. The induced spinor connection involves sigma matrices in CP₂ degrees of freedom, which for the octonionic representation of gamma matrices are proportional to octonion units in Minkowski degrees of freedom. This corresponds to a reduction of tangent space group SO(1,7) to G₂. Therefore octonionic Dirac equation identifying Dirac spinors as complexified octonions can lead to non-associativity even when space-time surface is associative or co-associative.
   
   
2. The simplest manner to overcome these problems is to assume that spinors are localized at 2-D string world sheets with 1-D CP₂ projection and thus possible only in Minkowskian regions. Induced gauge fields would vanish. String world sheets would be minimal surfaces in M⁴× D¹⊂ M⁴× CP₂ and the theory would simplify enormously. String area would give rise to an additional term in the action assigned to the Minkowskian space-time regions and for vacuum extremals one would have only strings in the first approximation, which conforms with the success of string models and with the intuitive view that vacuum extremals of Kähler action are basic building bricks of many-sheeted space-time. Note that string world sheets would be also symplectic covariants.
   

   Without further conditions gauge potentials would be non-vanishing but one can hope that one can gauge transform them away in associative manner. If not, one can also consider the possibility that CP₂ projection is geodesic circle S¹: symplectic invariance is considerably reduces for this option since symplectic transformations must reduce to rotations in S¹.
   
   

3. The fist heavy objection is that action would contain Newton's constant G as a fundamental dynamical parameter: this is a standard recipe for building a non-renormalizable theory. The very idea of TGD indeed is that there is only single dimensionless parameter analogous to critical temperature. One can of coure argue that the dimensionless parameter is hbarG/R², R CP₂ "radius".
   

   Second heavy objection is that the Euclidian variant of string action exponentially damps out all string world sheets with area larger than hbar G. Note also that the classical energy of Minkowskian string would be gigantic unless the length of string is of order Planck length. For Minkowskian signature the exponent is oscillatory and one can argue that wild oscillations have the same effect.
   

   The hierarchy of Planck constants would allow the replacement hbar→ hbar_eff but this is not enough. The area of typical string world sheet would scale as h_eff and the size of CD and gravitational Compton lengths of gravitationally bound objects would scale (h_eff)^1/2 rather than h_eff = GMm/v₀ which one wants. The only way out of problem is to assume T ∝ (hbar/h_eff)². This is however un-natural for genuine area action. Hence it seems that the visit of the basic assumption of superstring theory to TGD remains very short. In any case, if one assumes that string connect gravitationally bound masses, super string models in perturbative description are definitely wrong as physical theories as has of course become clear already from landscape catastrophe.
   

Is super-symmetrized Kähler-Dirac action enough?

Could one do without string area in the action and use only K-D action, which is in any case forced by the super-conformal symmetry? This option I have indeed considered hitherto. K-D Dirac equation indeed tends to reduce to a lower-dimensional one: for massless extremals the K-D operator is effectively 1-dimensional. For cosmic strings this reduction does not however take place. In any case, this leads to ask whether in some cases the solutions of Kähler-Dirac equation are localized at lower-dimensional surfaces of space-time surface.

1. The proposal has indeed been that string world sheets carry vanishing W and possibly even Z fields: in this manner the electromagnetic charge of spinor mode could be well-defined. The vanishing conditions force in the generic case 2-dimensionality.
   

   Besides this the canonical momentum currents for Kähler action defining 4 imbedding space vector fields must define an integrable distribution of two planes to give string world sheet. The four canonical momentum currents Π_k^α= ∂ L_K/∂_{∂α h^k} identified as imbedding 1-forms can have only two linearly independent components parallel to the string world sheet. Also the Frobenius conditions stating that the two 1-forms are proportional to gradients of two imbedding space coordinates Φ_i defining also coordinates at string world sheet, must be satisfied. These conditions are rather strong and are expected to select some discrete set of string world sheets.
   
   

2. To construct preferred extremal one should fix the partonic 2-surfaces, their light-like orbits defining boundaries of Euclidian and Minkowskian space-time regions, and string world sheets. At string world sheets the boundary condition would be that the normal components of canonical momentum currents for Kähler action vanish. This picture brings in mind strong form of holography and this suggests that might make sense and also solution of Einstein equations with point like sources.
   
   
3. The localization of spinor modes at 2-D surfaces would would follow from the well-definedness of em charge and one could have situation is which the localization does not occur. For instance, covariantly constant right-handed neutrinos spinor modes at cosmic strings are completely de-localized and one can wonder whether one could give up the localization inside wormhole contacts.
   
   
4. String tension is dynamical and physical intuition suggests that induced metric at string world sheet is replaced by the anti-commutator of the K-D gamma matrices and by conformal invariance only the conformal equivalence class of this metric would matter and it could be even equivalent with the induced metric. A possible interpretation is that the energy density of Kähler action has a singularity localized at the string world sheet.
   

   Another interpretation that I proposed for years ago but gave up is that in spirit with the TGD analog of AdS/CFT duality the Noether charges for Kähler action can be reduced to integrals over string world sheet having interpretation as area in effective metric. In the case of magnetic flux tubes carrying monopole fluxes and containing a string connecting partonic 2-surfaces at its ends this interpretation would be very natural, and string tension would characterize the density of Kähler magnetic energy. String model with dynamical string tension would certainly be a good approximation and string tension would depend on scale of CD.
   
   

5. There is also an objection. For M⁴ type vacuum extremals one would not obtain any non-vacuum string world sheets carrying fermions but the successes of string model strongly suggest that string world sheets are there. String world sheets would represent a deformation of the vacuum extremal and far from string world sheets one would have vacuum extremal in an excellent approximation. Situation would be analogous to that in general relativity with point particles.
   
   
6. The hierarchy of conformal symmetry breakings for K-D action should make string tension proportional to 1/h_eff² with h_eff=h_gr giving correct gravitational Compton length Λ_gr= GM/v₀ defining the minimal size of CD associated with the system. Why the effective string tension of string world sheet should behave like (hbar/hbar_eff)²?
   

   The first point to notice is that the effective metric G^αβ defined as h^klΠ_k^αΠ_l^β, where the canonical momentum current Π_kα=∂ L_K/∂_{∂α h^k} has dimension 1/L² as required. Kähler action density must be dimensionless and since the induced Kähler form is dimensionless the canonical momentum currents are proportional to 1/α_K.
   

   Should one assume that α_K is fundamental coupling strength fixed by quantum criticality to α_K≈1/137? Or should one regard g_K² as fundamental parameter so that one would have 1/α_K= hbar_eff/4π g_K² having spectrum coming as integer multiples (recall the analogy with inverse of critical temperature)?
   

   The latter option is the in spirit with the original idea stating that the increase of h_eff reduces the values of the gauge coupling strengths proportional to α_K so that perturbation series converges (Universe is theoretician friendly). The non-perturbative states would be critical states. The non-determinism of Kähler action implying that the 3-surfaces at the boundaries of CD can be connected by large number of space-time sheets forming n conformal equivalence classes. The latter option would give G^αβ ∝ h_eff² and det(G) ∝ 1/h_eff² as required.
   

7. It must be emphasized that the string tension has interpretation in terms of gravitational coupling on only at the GRT limit of TGD involving the replacement of many-sheeted space-time with single sheeted one. It can have also interpretation as hadronic string tension or effective string tension associated with magnetic flux tubes and telling the density of Kähler magnetic energy per unit length.
   

   Superstring models would describe only the perturbative Planck scale dynamics for emission and absorption of h_eff/h=1 on mass shell gravitons whereas the quantum description of bound states would require h_eff/n>1 when the masses. Also the effective gravitational constant associated with the strings would differ from G.
   

   
   The natural condition is that the size scale of string world sheet associated with the flux tube mediating gravitational binding is G(M+m)/v₀, By expressing string tension in the form 1/T=n² hbar G₁, n=h_eff/h, this condition gives hbar G₁= hbar²/M_red², M_red= Mm/(M+m). The effective Planck length defined by the effective Newton's constant G₁ analogous to that appearing in string tension is just the Compton length associated with the reduced mass of the system and string tension equals to T= [v₀/G(M+m)]² apart from a numerical constant (2G(M+m) is Schwartschild radius for the entire system). Hence the macroscopic stringy description of gravitation in terms of string differs dramatically from the perturbative one. Note that one can also understand why in the Bohr orbit model of Nottale for the planetary system and in its TGD version v₀ must be by a factor 1/5 smaller for outer planets rather than inner planets.
   

Are 4-D spinor modes consistent with associativity?

The condition that octonionic spinors are equivalent with ordinary spinors looks rather natural but in the case of Kähler-Dirac action the non-associativity could leak in. One could of course give up the condition that octonionic and ordinary K-D equation are equivalent in 4-D case. If so, one could see K-D action as related to non-commutative and maybe even non-associative fermion dynamics. Suppose that one does not.

1. K-D action vanishes by K-D equation. Could this save from non-associativity? If the spinors are localized to string world sheets, one obtains just the standard stringy construction of conformal modes of spinor field. The induce spinor connection would have only the holomorphic component A_z. Spinor mode would depend only on z but K-D gamma matrix Γ^z would annihilate the spinor mode so that K-D equation would be satisfied. There are good hopes that the octonionic variant of K-D equation is equivalent with that based on ordinary gamma matrices since quaternionic coordinated reduces to complex coordinate, octonionic quaternionic gamma matrices reduce to complex gamma matrices, sigma matrices are effectively absent by holomorphy.
   
   
2. One can consider also 4-D situation (maybe inside wormhole contacts). Could some form of quaternion holomorphy allow to realize the K-D equation just as in the case of super string models by replacing complex coordinate and its conjugate with quaternion and its 3 conjugates. Only two quaternion conjugates would appear in the spinor mode and the corresponding quaternionic gamma matrices would annihilate the spinor mode. It is essential that in a suitable gauge the spinor connection has non-vanishing components only for two quaternion conjugate coordinates. As a special case one would have a situation in which only one quaternion coordinate appears in the solution. Depending on the character of quaternionion holomorphy the modes would be labelled by one or two integers identifiable as conformal weights.
   

   Even if these octonionic 4-D modes exists (as one expects in the case of cosmic strings), it is far from clear whether the description in terms of them is equivalent with the description using K-D equation based ordinary gamma matrices. The algebraic structure however raises hopes about this. The quaternion coordinate can be represented as sum of two complex coordinates as q=z₁+Jz₂ and the dependence on two quaternion conjugates corresponds to the dependence on two complex coordinates z₁,z₂. The condition that two quaternion complexified gammas annihilate the spinors is equivalent with the corresponding condition for Dirac equation formulated using 2 complex coordinates. This for wormhole contacts. The possible generalization of this condition to Minkowskian regions would be in terms Hamilton-Jacobi structure.
   

   Note that for cosmic strings of form X²× Y²⊂ M⁴× CP₂ the associativity condition for S² sigma matrix and without assuming localization demands that the commutator of Y² imaginary units is proportional to the imaginary unit assignable to X² which however depends on point of X². This condition seems to imply correlation between Y² and S² which does not look physical.
   
   

Summary

To summarize, the minimal and mathematically most optimistic conclusion is that Kähler-Dirac action is indeed enough to understand gravitational binding without giving up the associativity of the fermionic dynamics. Conformal spinor dynamics would be associative if the spinor modes are localized at string world sheets with vanishing W (and maybe also Z) fields guaranteeing well-definedness of em charge and carrying canonical momentum currents parallel to them. It is not quite clear whether string world sheets are present also inside wormhole contacts: for CP₂ type vacuum extremals the Dirac equation would give only right-handed neutrino as a solution (could they give rise to N=2 SUSY?).

Associativity does not favor fermionic modes in the interior of space-time surface unless they represent right-handed neutrinos for which mixing with left-handed neutrinos does not occur: hence the idea about interior modes of fermions as giving rise to SUSY is dead whereas the original idea about partonic oscillator operator algebra as SUSY algebra is well and alive. Evolution can be seen as a generation of gravitationally bound states of increasing size demanding the gradual increase of h__eff implying generation of quantum coherence even in astrophysical scales.

The construction of preferred extremals would realize strong form of holography. By conformal symmetry the effective metric at string world sheet could be conformally equivalent with the induced metric at string world sheets. Dynamical string tension would be proportional to hbar/h_eff² due to the proportionality α_K∝ 1/h_eff and predict correctly the size scales of gravitationally bound states for h_gr=h_eff=GMm/v₀. Gravitational constant would be a prediction of the theory and be expressible in terms of α_K and R² and hbar_eff (G∝ R²/g_K²).

In fact, all bound states - elementary particles as pairs of wormhole contacts, hadronic strings, nuclei, molecules, etc. - are described in the same manner quantum mechanically. This is of course nothing new since magnetic flux tubes associated with the strings provide a universal model for interactions in TGD Universe. This also conforms with the TGD counterpart of AdS/CFT duality.

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" of "Physics as infinite-dimensional geometry".