Tuesday, July 17, 2012

Constraints on super-conformal invariance from p-adic mass calculations and ZEO

The generalization of super-conformal symmetry to 4-D context is basic element of quantum TGD. Several variants for the realization of supersymmetry has been proposed. Especially problematic has been the question about whether the counterpart of standard SUSY is realized in TGD framework or not. Thanks to the progress made in the understanding of preferred extremals of Kähler action and of solutions of modified Dirac equation, it has become to develop the vision in considerable detail. As a consequence some existing alternative visions have been eliminated from consideration. One of them the proposal that Equivalence Principle might have realization in terms of coset representations. Second one is the idea that right-handed neutrino generating space-time supersymmetry might be in color partial wave so that sparticles would be colored: this would explain why sparticles are not observed at LHC. The new view providing a possible alternative explanation for the absence of sparticles has been discussed in previous posting.

Concerning the general understanding of super-conformal invariance in TGD framework, an important physical constraint comes from the success p-adic thermodynamics: superconformal invariance indeed forms a core element of p-adic mass calculations (see this, especially this).

  1. The first thing that one can get worried about relates to the extension of conformal symmetries. If the conformal symmetries generalize to D=4, how can one take seriously the results of p-adic mass calculations based on 2-D conformal invariance? There is no reason to worry. The reduction of the conformal invariance to 2-D one for the preferred extremals takes care of this problem. This however requires that the fermionic contributions assignable to string world sheets and/or partonic 2-surfaces - Super- Kac-Moody contributions - should dictate the elementary particle masses. For hadrons also symplectic contributions should be present (see this). This is a valuable hint in attempts to identify the mathematical structure in more detail.

  2. Zero Energy Ontology (ZEO) suggests that all particles, even virtual ones correspond to massless wormhole throats carrying fermions. As a consequence, twistor approach would work and the kinematical constraints to vertices would allow th cancellation of both UV and IR divergences. This would suggest that the p-adic thermal expectation value is for the longitudinal M2 momentum squared (the definition of CD selects M1⊂ M2⊂ M4 as also does number theoretic vision). Also propagator would be determined by M2 momentum. Lorentz invariance would be obtained by integration of the moduli for CD including also Lorentz boosts of CD. This is definitely something new from standard physics point of view, but suggested already by the first p-adic mass calculations. The fact that parton distributions in hadrons are functions of longitudinal momentum fraction, and the division of tangent space of M4 to longitudinal and transversal parts for gauge bosons suggests the same. Also number theoretical vision and the need to assign to realize the choice of spin quantization axis and time like direction defining the rest system in quantum measurement theory at the level of WCW geometry also this.

  3. In the original approach one allows states with arbitrary large values of L0 as physical states. Usually one would require that L0 annihilates the states. In the calculations however mass squared was assumed to be proportional L0 apart from vacuum contribution. This is a questionable assumption. ZEO suggests that total mass squared vanishes and that one can decompose mass squared to a sum of longitudinal and transversal parts. If one can do the same decomposition to longitudinal and transverse parts also for the Super Virasoro algebra then one can calculate longitudinal mass squared as a p-adic thermal expectation in the transversal super-Virasoro algebra and only states with L0=0 would contribute and one would have conformal invariance in the standard sense.

  4. The assumption motivated by Lorentz invariance has been that mass squared is replaced with conformal weight in thermodynamics, and that one first calculates the thermal average of the conformal weight and then equates it with mass squared. This assumption is somewhat ad hoc. ZEO however suggests an alternative interpretation in which one has zero energy states for which longitudinal mass squared of positive energy state derive from p-adic thermodynamics. Thermodynamics - or rather, its square root - would become part of quantum theory in ZEO. M-matrix is indeed product of hermitian square root of density matrix multiplied by unitary S-matrix and defines the entanglement coefficients between positive and negative energy parts of zero energy state.

  5. The crucial constraint is that the number of super-conformal tensor factors is N=5: this suggests that thermodynamics applied in Super-Kac-Moody degrees of freedom assignable to string world sheets is enough, when one is interested in the masses of fermions and gauge bosons. Super-symplectic degrees of freedom can also contribute and determine the dominant contribution to baryon masses. Should also this contribution obey p-adic thermodynamics in the case when it is present? Or does the very fact that this contribution need not be present mean that it is not thermal? The symplectic contribution should correspond to hadronic p-adic length prime rather the one assignable to (say) u quark. Hadronic p-adic mass squared and partonic p-adic mass squared cannot be summed since primes are different. If one accepts the basic rules (see this), longitudinal energy and momentum are additive as indeed assumed in perturbative QCD.

  6. Calculations work if the vacuum expectation value of the mass squared must be assumed to be tachyonic. There are two options depending on whether one whether p-adic thermodynamics gives total mass squared or longitudinal mass squared.

    1. One could argue that the total mass squared has naturally tachyonic ground state expectation since for massless extremals longitudinal momentum is light-like and transversal momentum squared is necessary present and non-vanishing by the localization to topological light ray of finite thickness of order p-adic length scale. Transversal degrees of freedom would be modeled with a particle in a box.

    2. If longitudinal mass squared is what is calculated, the condition would require that transversal momentum squared is negative so that instead of plane wave like behavior exponential damping would be required. This would conform with the localization in transversal degrees of freedom.

  7. For preferred extremals Einstein's equations with cosmological term are satisfied as a consistency condition guaranteing algebraization of the equations. Hence Equivalence Principle holds true. But what about possible quantum realization of Equivalence Principle in this framework? A possible quantum counterpart of Equivalence Principle could be that the longitudinal parts of the imbedding space mass squared operator for a given massless state equals to that for d'Alembert operator assignable to the modified Dirac action. The attempts to formulate this in more precise manner however seem to produce only troubles.

To sum up, the basic new element inspired by ZEO is that p-adic mass calculations are for the longitudinal momentum squared and that elementary particles and even hadrons are basically massless. This assumption looks certainly strange at first. Thank so the integration over boosts of CDs it is however consistent with Lorentz invariance and also allows to understand p-adic thermodynamics as thermodynamics for the transversal part of scaling generator L0. ZEO allows indeed superposition of pairs of positive and negative energies with different momenta for positive energy state without a loss of Lorentz invariance. Also p-adic thermodynamics finds a natural interpretation in terms of M-matrix defining square root of hermitian density matrix.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.


At 9:31 AM, Blogger Ulla said...



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