Friday, August 25, 2006

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 hk 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 δhk of a given extremal

α Jαk = 0 ,

Jαk = (∂2 L/∂ hkα∂ hlβ) δ hlβ ,

where hkα denotes the partial derivative ∂α hk. 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

DαJα,k = 0

where Dα 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 jAk of M4 translations which means that second variations define the analog of a local gauge algebra in M4 degrees of freedom.

αJA,αn = 0 ,

JA,αn = Jα,kn jAk .

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

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

Jαn = Jα,knγkΨR .

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

Qn(X3f)-Qn(X3) = 0 ,

Qn(Y3) = ∫Y3 d3x Jn ,

Jn = Jtk hklδhln .

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

Qn(X3f) = ∫X3fJn = 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 of TGD: Physics as Infinite-Dimensional Geometry


At 11:32 PM, Blogger Mahndisa S. Rigmaiden said...

09 02 06

The notion of many-sheeted space-time and quantization of Planck constant predict that there should exist infinite hierarchy of fractal copies of standard model physics. The interpretation is in terms of dark matter.

Hello Matti:
I hope all is well with you. I can only understand a fraction of what you have written, and thus am always loth to comment. However, since your explanation on how in the TGD framework, we see fractal copies of standard model physics, I am now seeing a bit about where the P Adic number systems come in. I was reading a book on fractal geometry and in the triadic Cantor dust set, P Adic numbers were used, we were expressing the attractor sets in terms of numbers in base three. I also read a paper where the Polyakhov string was quantized in terms of the triadic Cantor dust set. Hmmm Matti your use of the P Adic numbers seems to be the best thing to do when describing certain objects with special geometries. As I learn more, I will come back and visit you. For now, know that I am lurking and cannot always comment-due to lack of knowledge which is currently being rectified.

At 12:42 AM, Blogger Matti Pitkanen said...

Dear Mahndisa,

the applications you mentioned are nice and at high technical level mathematically but tend to be quite too specific to be of help for physicist interested on the interpretation of p-adic physics. I think that it is important to try to develop bird's eye of view to what p-adic physics might mean (to something about which one knows practically nothing yet!).

For me the success of p-adic mass calculations meaning among other things the reduction of the magic number 10^38 of particle physics to number theory made it obvious that p-adic topology must be effective topology of real space-time sheets in some length scale range. I do not believe in p-adicity somehow emerging in Planck scale.

Accepting this one sooner or later articulates the following questions.

a) Could p-adic topology be also a genuine topology at space-time level so that one can speak about p-adic space-time sheets in TGD framework?

b) How to fuse real and various p-adic physics to a larger scheme? The only possible manner seems to be by fusion of these number fields along common rationals (also algebraics for algebraic extensions of p-adics) to a larger structure.

c) What is the interpretation of p-adic space-time sheets? Certainly they cannot be direct correlates for ordinary matter. What is left is the mind stuff speculated by Descartes. Some general properties of p-adic numbers such as inherent non-determinism of p-adic differential equations and hierarchical treelike structure of ultrametric topology suggested already by Parisi to relate to cognition, suggest that the only sensible interpretation of p-adic space-time sheets is as correlates of cognition and intention. Transformation of intention to action would mean quantum in which p-adic space-time sheet becomes real. This of course raises critical questions such as "What about conservation laws?" Taking these questions seriously one ends up to a radically new ontology forced by TGD on basis of quite different arguments.

This picture has quite dramatic implications.

a) That p-adic and real worlds intersect at rational (more generally algebraic) points of imbedding space points is natural outcome of this picture and leads to a coherent view about what the fusion of real and p-adic physics might mean and how cognition/ intention and matter relate at space-time level.

b) One fascinating almost-implication is algebraic universality: S-matrix elements should be algebraic numbers so that they would have universal meaning independent of number field when algebraic extensions of p-adics are allowed. This condition is incredibly strong when combined with super-conformal symmetries.

It of course takes time before physicists are mature to add consciousness and cognition to their vocabulary, and are ready to consider wider framework unifying reals and p-adics but I think this is necessary if one wants to abstract some physics from p-adic mathematics.

By the way, I will add within a week or two to my homepage powerpoint representations about lectures relating to TGD and TGD inspired theory of consciousness. They might help in attempts to get some idea about what is involved.

With Best Wishes,


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