The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics

The relationship between super-canonical algebra (SC) acting at light-cone boundary and Super Kac-Moody algebra (SKM) acting on light-like 3-surfaces has remained somewhat enigmatic due to the lack of physical insights. This is not the only problem. The question to precisely what extent Equivalence Principle (EP) remains true in TGD framework and what might be the precise mathematical realization of EP is waiting for an answer. Also the justification of p-adic thermodynamics for the scaling generator L₀ of Virasoro algebra -in obvious conflict with the basic wisdom that this generator should annihilate physical states- is lacking. It seems that these three problems could have a common solution.

Before going to describe the proposed solution, some background is necessary. The latest proposal for SC-SKM relationship relies on non-standard and therefore somewhat questionable assumptions.

1. SKM Virasoro algebra (SKMV) and SC Virasoro algebra (SCV) (anti)commute for physical states.
2. SC algebra generates states of negative conformal weight annihilated by SCV generators L_n, n < 0, and serving as ground states from which SKM generators create states with non-negative conformal weight.

This picture could make sense for elementary particles. On other hand, the recent model for hadrons [see this] assumes that SC degrees of freedom contribute about 70 per cent to the mass of hadron but at space-time sheet different from those assignable to quarks. The contribution of SC degrees of freedom to the thermal average of the conformal weight would be positive. A contradiction results unless one assumes that there exists also SCV ground states with positive conformal weight annihilated by SCV elements L_n, n < 0, but also this seems implausible.

1. New vision about the relationship between SCV and SKMV

Consider now the new vision about the relationship between SCV and SKMV.

1. The isometries of H assignable with SKM are also symplectic transformations [see this] (note that I have used the term canonical instead of symplectic previously). Hence might consider the possibility that SKM could be identified as a subalgebra of SC. If this makes sense, a generalization of the coset construction obtained by replacing finite-dimensional Lie group with infinite-dimensional symplectic group suggests itself. The differences of SCV and SKMV elements would annihilate physical states and (anti)commute with SKMV. Also the generators O_n, n > 0, for both algebras would annihilate the physical states so that the differences of the elements would annihilate automatically physical states for n > 0.
2. The super-generator G₀ contains the Dirac operator D of H. If the action of SCV and SKMV Dirac operators on physical states are identical then cm of degrees of freedom disappear from the differences G₀(SCV)-G₀(SKMV) and L₀(SCV)-L₀(SKMV). One could interpret the identical action of the Dirac operators as the long sought-for precise realization of Equivalence Principle (EP) in TGD framework. EP would state that the total inertial four-momentum and color quantum numbers assignable to SC (imbedding space level) are equal to the gravitational four-momentum and color quantum numbers assignable to SKM (space-time level). Note that since super-canonical transformations correspond to the isometries of the "world of classical worlds" the assignment of the attribute "inertial" to them is natural.
3. The analog of coset construction applies also to SKM and SC algebras which means that physical states can be thought of as being created by an operator of SKM carrying the conformal weight and by a genuine SC operator with vanishing conformal weight. Therefore the situation does not reduce to that encountered in super-string models.

This picture provides also a justification for p-adic thermodynamics.

1. In physical states the p-adic thermal expectation value of the SKM and SC conformal weights would be non-vanishing and identical and mass squared could be identified to the expectation value of SKM scaling generator L₀. There would be no need to give up Super Virasoro conditions for SCV-SKMV.
2. There is consistency with p-adic mass calculations for hadrons [see this] since the non-perturbative SC contributions and perturbative SKM contributions to the mass correspond to space-time sheets labeled by different p-adic primes. The earlier statement that SC is responsible for the dominating non-perturbative contributions to the hadron mass transforms to a statement reflecting SC-SKM duality. The perturbative quark contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SKM whereas non-perturbative contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SC. Also the proposal that the exotic analogs of baryons resulting when baryon looses its valence quarks [see this] remains intact in this framework.
3. The results of p-adic mass calculations depend crucially on the number N of tensor factors contributing to the Super-Virasoro algebra. The required number is N=5 and during years I have proposed several explanations for this number. It seems that holonomic contributions that is electro-weak and spin contributions must be regarded as contributions separate from those coming from isometries. SKM algebras in electro-weak degrees and spin degrees of of freedom, would give 2+1=3 tensor factors corresponding to U(2)_ew×SU(2). SU(3) and SO(3) (or SO(2) Ì SO(3) leaving the intersection of light-like ray with S² invariant) would give 2 additional tensor factors. Altogether one would indeed have 5 tensor factors.

2. Can SKM be lifted to a sub-algebra of SC?

A picture introducing only a generalization of coset construction as a new element, realizing mathematically Equivalence Principle, and justifying p-adic thermodynamics is highly attractive but there is a problem. SKM is defined at light-like 3-surfaces X³ whereas SC acts at light-cone boundary dH_±=dM⁴_±×CP₂. One should be able to lift SKM to imbedding space level somehow. Also SC should be lifted to entire H. This problem was the reason why I gave up the idea about coset construction and SC-SKM duality as it appeared for the first time.

A possible solution of the lifting problem comes from the observation making possible a more rigorous formulation of HO-H duality stating that one can regard space-time surfaces either as surfaces in hyper-octonionic space HO=M⁸ or in H=M⁴×CP₂ [see this]. Consider first the formulation of HO-H duality.

1. Associativity also in the number theoretical sense becomes the fundamental dynamical principle if HO-H duality holds true [see this]. For a space-time surface X⁴ Ì HO=M⁸ associativity is satisfied at space-time level if the tangent space at each point of X⁴ is some hyper-quaternionic sub-space HQ=M⁴ Ì M⁸. Also partonic 2-surfaces at the boundaries of causal diamonds formed by pairs of future and past directed light-cones defining the basic imbedding space correlate of zero energy state in zero energy ontology and light-like 3-surfaces are assumed to belong to HQ=M⁴ Ì HO.
2. HO-H duality requires something more. If the tangent spaces contain the same preferred commutative and thus hyper-complex plane HC=M², the tangent spaces of X⁴ are parameterized by the points s of CP₂ and X⁴ Ì HO can be mapped to X⁴ Ì M⁴×CP₂ by assigning to a point of X⁴ regarded as point (m,e) of M⁴₀×E⁴=M⁸ the point (m,s). Note that one must also fix a preferred global hyper-quaternionic subspace M⁴₀ Ì M⁸ containing M² to be not confused with the local tangent planes M⁴.
3. The preferred plane M² can be interpreted as the plane of non-physical polarizations so that the interpretation as a number theoretic analog of gauge conditions posed in both quantum field theories and string models is possible.
4. An open question is whether the resulting surface in H is a preferred extremal of Kähler action. This is possible since the tangent spaces at light-like partonic 3-surfaces are fixed to contain M² so that the boundary values of the normal derivatives of H coordinates are fixed and field equations fix in the ideal case X⁴ uniquely and one obtains space-time surface as the analog of Bohr orbit.
5. The light-like "Higgs term" proportional to O=g_kt^k appearing in the generalized eigenvalue equation for the modified Dirac operator [see this] is an essential element of TGD based description of Higgs mechanism. This term can cause complications unless t is a covariantly constant light-like vector. Covariant constancy is achieved if t is constant light-like vector in M². The interpretation as a space-time correlate for the light-like 4-momentum assignable to the parton might be considered.
6. Associativity requires that the hyper-octonionic arguments of N-point functions in HO description are restricted to a hyperquaternionic plane HQ=M⁴ Ì HO required also by the HO-H correspondence. The intersection M⁴Çint(X⁴) consists of a discrete set of points in the generic case. Partonic 3-surfaces are assumed to be associative and belong to M⁴. The set of commutative points at the partonic 2-surface X² is discrete in the generic case whereas the intersection X³ÇM² consists of 1-D curves so that the notion of number theoretical braid crucial for the p-adicization of the theory as almost topological QFT is uniquely defined.
7. The preferred plane M² Ì M⁴ Ì HO can be assigned also to the definition of N-point functions in HO picture. It is not clear whether it must be same as the preferred planes assigned to the partonic 2-surfaces. If not, the interpretation would be that it corresponds to a plane containing the over all cm four-momentum whereas partonic planes M²_i would contain the partonic four-momenta. M² is expected to change at wormhole contacts having Euclidian signature of the induced metric representing horizons and connecting space-time sheets with Minkowskian signature of the induced metric.

The presence of globally defined plane M² and the flexibility provided by the hyper-complex conformal invariance raise the hopes of achieving the lifting of SC and SKM to H. At the light-cone boundary the light-like radial coordinate can be lifted to a hyper-complex coordinate defining coordinate for M². At X³ one can fix the light-like coordinate varying along the braid strands can be lifted to some hyper-complex coordinate of M² defined in the interior of X⁴. The total four-momenta and color quantum numbers assignable to the SC and SKM degrees of freedom are naturally identical since they can be identified as the four-momentum of the partonic 2-surface X² Ì X³ÇdM⁴_±×CP₂. Equivalence Principle would emerge as an identity.

3. Questions

There are still several open questions.

1. Is it possible to define hyper-quaternionic variants of the superconformal algebras in both H and HO or perhaps only in HO. A positive answer to this question would conform with the conjecture that the geometry of "world of classical worlds" allows Hyper-Kähler property in either or both pictures [see this].
2. How this picture relates to what is known about the extremals of field equations [see this] characterized by generalized Hamilton-Jacobi structure bringing in mind the selection of preferred M²?
3. Is this picture consistent with the views about Equivalence Principle and its possible breaking based on the identification of gravitational four-momentum in terms of Einstein tensor is interesting [see this] ?

For more details see the chapter Massless particles and particle massivation.