Sunday, October 08, 2006

About the identification of Kac Moody algebra and corresponding Virasoro algebra

It is relatively straightforward to deduce the detailed form of the TGD countetpart of Kac-Moody algebra identified as X3-local infinitesimal transformations of H_+/-=M4+/-× CP2 respecting the lightlikeness of partonic 3-surface X3. This involves the identification of Kac-Moody transformations and corresponding super-generators carrying now fermion numbers and anticommuting to a multiple of unit matrix. Also the generalization the notions of Ramond and N-S algebra is needed. Especially interesting is the relationship with the super-canonical algebra consisting of canonical transformations of δ M4+/-× CP2.

1. Bosonic part of the algebra

The bosonic part of Kac-Moody algebra can be identified as symmetries respecting the light-likeness of the partonic 3-surface X3 in H=M4+/-× CP2. The educated guess is that a subset of X3-local diffeomorphisms of is in question. The allowed infinitesimeal transformation of this kind must reduce to a conformal transformation of the induced metric plus diffeomorphism of X3. The explicit study of the conditions allows to conclude that conformal transformations of M4+/- and isometries of CP2 made local with respect to X3 satisfy the defining conditions. Choosing special coordinates for X3 one finds that the vector fields defining the transformations must be orthogonal of the light-like direction of X3. The resulting partial differential equations fix the infinitesimal diffeomorphism of X3 once the functions appearing in Kac-Moody generator are fixed. The functions appearing in generators can be chosen to proportional to powers of the radial coordinate multiplied by functions of transversal coordinates whose dynamics is dictated by consistency conditions

The resulting algebra is essentially 3-dimensional and therefore much larger than ordinary Kac-Moody algebra. One can identify the counterpart of ordinary Kac-Moody algebra as a sub-algebra for which generators are in one-one correspondence with the powers of the light-like coordinate assignable to X3. This algebra corresponds to the stringy sub-algebra E2× SO(2)×SU(3) if one selects the preferred coordinate of M4 as a lightlike coordinate assignable to the lightlike ray of δ M4+/- defining orbifold structure in M4+/- ("massless" case) and E3× SO(3)×SU(3) if the preferred coordinate is M4 time coordinate (massive case).

The local transformation in the preferred direction is not free but fixed by the condition that Kac-Moody transformation does not affect the value of the light-like coordinate of X3. This is completely analogous to the non-dynamical character of longitudinal degrees of freedom of Kac-Moody algebra in string models.

The algebra decomposes into a direct sum of sub-spaces left invariant by Kac-Moody algebra and one has a structure analogous to that defining coset space structure (say SU(3)/U(2)). This feature means that the space of physical states is much larger than in string models and Kac Moody algebra of string models takes the role of the little algebra in the representations of Poincare group. Mackey's construction of induced representations should generalized to this situation.

Just as in the case of super-canonical algebra, the Noether charges assignable to the Kac-Moody transformations define Hamiltonians in the world of classical worlds as integrals over the partonic two surface and reducible to one-dimensional integrals if the SO(2)× SU(3) quantum numbers of the generator vanish. The intepretation is that this algebra leaves invariant various quantization axes and acts as symmetries of quantum measurement situation.

2. Fermionic sector

The zero modes and generalized eigen spinors of the modified Dirac equation define the counterparts of Ramond and N-S type super generators.

The hypothesis inspired by number theoretical conjectures related to Riemann Zeta is that the eigenvalues of the generalized eigen modes associated with ground states correspond to non-trivial zeros of zeta. Also non-trivial eigenvalues must be considered.

  1. Neveu-Schwartz type eigenvalues which are expressible as λ=1/2+i∑knkyk, where sk=1/2+iyk is zero of Riemann zeta. Higher Virasoro excitations would correspond to conformal weights λ=n+1/2+i∑knkyk.

  2. Zero modes correspond naturally Ramond type representations for which the ground state conformal weight vanishes so that a zero mode (solution of the modified Dirac equation is in question) and higher conformal weights would be integer valued.

  3. If one accepts non-trivial zeros as generalized eigenvalues one would have additional Ramond type representations with a tachyonic ground state conformal weight lambda= -2n, n>0.

Thus N-S type ground state conformal weights would involve also imaginary part and this has an interpretation in terms of an inherent arrow of time associated with particles distinguishing positive energy particle propagating to the geometric future from negative energy particle propagating to geometric past. p-Adic mass calculations suggest that y could characterize the decay width of the particle. The action of Kac-Moody generator to the state can be defined and affects the conformal weight in the expected manner.

3. Super Virasoro algebras

The identification of SKM Virasoro algebra as that associated with radial diffeomorphisms is obvious and this algebra replaces the usual Virasoro algebra associated with the complex coordinate of partonic 2-surface X2 in the construction of states and mass calculations. This algebra does not not annihilate physical states and this gives justification for p-adic thermodynamics. The commutators of super canonical and super Kac-Moody (and corresponding super Virasoro) algebras would however annihilate naturally the physical states. Four-momentum does not appear in the expressions for the Virasoro generators and mass squared is identified as p-adic thermal expectation value of conformal weight. There are no problems with Lorentz invariance.

One can wonder about the role of ordinary conformal transformations assignable to the partonic 2-surface X2. The stringy quantization implies the reduction of this part of algebra to algebraically 1-D form and the corresponding conformal weight labels different radial SKM representations. Conformal weights are not constants of motion along X3 unlike radial conformal weights. TGD analog of super-conformal symmetries of condensed matter physics rather than stringy super-conformal symmetry would be in question.

The last sections of the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix" and the chapter The Evolution of Quantum TGD of "TGD: an Overall View" give detailed summary of the recent picture.

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