M Theory Lesson 102

In the 2005 lecture 6 Alain Connes points out that although his framework predicts physical couplings that match $SU(5)$ unification, it achieves this without the addition of extra fields or supersymmetry arguments. Towards the end of the lecture he summarises the situation: the problem is to combine (1) the renormalisation theory (Hopf algebras, motivic Galois group) and (2) the geometric setup from NCG operator theory, in such a way that running geometries (on different scales) are possible. Connes proposes a functional integral over geometries which is spectral in nature, and in fact looks like a matrix model. The question is, what sort of constraints should be applied to geometries? It is made clear that this is a challenge to physicists: his spectral action principle is a statement about the nature of observables, but insufficient in itself to guide, as Connes puts it, the merging of motives and NCG.

These lectures are highly recommended to physicists. Don’t expect to understand all the mathematical gobbledygook, but try to take in the big picture, which is fantastically conveyed. Note that more recent work removes much of the arbitrariness of the original NCG formulation of the SM, but still puts the number of generations in by hand.

2 Responses so far »

  1. 1

    Matti Pitkanen said,

    Thank you for the link. I must find time to read the lecture.

    Finding the proper physical interpretation is essential in order to make progess since mere mathematics leads to the combinatorially exploding garden of branching paths. Some comments.

    a) To me the extension of standard model gauge group is quite too simplistic approach and has led via GUTs and strings to the recent disaster. To my best knowledge TGD is the only approach explaining standard model gauge group and relating it to classical number fields. It predicts a lot of new physics and deviations from standard model but no contradictions with experimental facts as I know them.

    b) The notion of running geometry corresponds in TGD framework to a hierarchy of p-adic geometries and their effective variants in the real context. Coupling constant evolution is discretized to p-adic coupling constant evolution and occurs at the level of free induced spinor fields. Best support comes from the success of p-adic mass calculations but it seems that there is no manner to communicate this to the colleagues (tried for about 13 years now!).

    c) I tend to believe that commutative geometry is not something emerging in Planck scale but results naturally from an appropriate chacterization of finite measurement resolution in quantum measurement theory using inclusions of factors. Included algebra replaces complex ray of Hilbert space in this approach and one obtains non-commutative Hilbert space.

    Again it is really strange that colleagues refuse to consider seriously the fact that quantum measurements have finite resolution: perhaps they see this as “philosophy”!

    d) More general non-commutative geometries could make sense for simple submanifolds of generalized imbedding space. In TGD framework M^2xS^2 appearing as the intersection of variants of H corresponding to different Planck constants and fixed choice of quantization axes is assignable to fully quantum critical systems and in a unique candidate. Quantum M^2 and quantum S^2 are indeed simple things.

    This would imply that string like sub-aminifods of partonic 2-surface for which induced spinor fields commute would be replaced by discrete number theoretic braids when coordinates become non-commutative. Discretization would emerge at the fundamental level from dynamics rather than as ad hoc assumption and would reflect finite resolution of measurement and cognition. But perhaps also this is “philosophy”.

    e) I do not know whether Connes refers to path integral or to genuine functional integral. In any case the existence of this integral in commutative sense is extremely powerful requirement.

    *Cancellation of metric and Gaussian determinants is implied by Kahler geometry. The existence of Kahler metric with Riemann connection implies symmetric space structure, etc.. This in turn implies that imbedding space is highly unique if not completely fixed since configuration space inherits its symmetries from imbedding space. I know: this is “philosophy”: better to accept landscape and antropic principle;-).

    *Cancellation of infinities due to local vertices is guaranteed if Kahler function as functional of 3-surface is non-local. Kahler function as Kahler action for a preferred 4-D extremal of Kahler action containing the 3-surface guarantees this. General Coordinate Invariance is automatic consequence so that a connection with another very deep principle emerges. Yes, yes: a piece of this disgusting “philosophy” again!;-)

  2. 2

    L. Riofrio said,

    Great that Connes believes this can be done without extra fields or supersymnetry. Those old approaches have led to a divergence of theories but no solution. I look forward to hearing more from this work.

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