Thursday, October 15, 2009

Multiverse as space of quaternionic sub-algebras of local octonionic Clifford algebra?

Multiverses as quantum superpositions of geometric objects are unavoidable in any theory of quantum gravitation starting from a geometric description of gravitation.

The notion of multiverse in M-theory context is however extremely poorly defined. Should one introduce probability amplitudes in all possible 11-D space-times and try to geometrize this space and show that Calabi-Yau times circle times M4:s appears as preferred ones? Should one also introduce probability amplitudes for all possible configurations of all possible branes inside particular 11-D manifold? Should one introduce at classical level decomposition of 11-D space-time to regions in good approximation of the desired form?

To me this is a hopeless mess both mathematically and physically. Like thermodynamics before Boltzman whose work colleagues stubbornly refused to recognize with tragic consequences (it seems that the situation is equally difficult with the "complex square root" of thermodynamics;-)).

My own modest proposal is following. Let us start by asking whether the higher-D space-time could be selected uniquely, say by starting from the idea that associativity fixes physics completely.

  1. 8-D space-times with Minkowski signature allow octonionic representation of gamma matrices as products of octonions and Pauli's sigma matrices. Consider local Clifford algebra in M8, which is the simplest possible choice.

  2. Ask what are the local associative sub-algebras of this algebra (one could and must also consider co-associative sub-algebras). Associativity corresponds to a restriction of local Clifford algebra elements to 4-D (hyper-)quaternionic surface Quaternionicity means that one can assign quaternionic plane, not necessarily tangent plane, to each of its points by some rule. If the 4-D quaternionic planes form an integrable distribution in some sense, we have got 4-D space-time.

  3. Do these quaternionic local Clifford sub-algebras allow commutative local sub-algebras? They do. This leads to a slicing of given hyper-quaternionic space-time surface by 2-D stringy surfaces (they are commutative) with slices parametrized by what I call partonic 2-surfaces (Euclidian string world sheets). In finite measurement implying discretization you get a collection of strings. Could M8 should allow slicings by quaternionic local Clifford sug-algebras with slicings parametrized by coquaternionic sub-algebras? This proposal is not a new one but appears naturally in this context.

  4. These properties imply M8-M4×CP2 duality that is mapping of these surfaces in M8 to M4×CP2 giving standard model symmetries and TGD in its basic form.

  5. The meaning of (hyper-)quaternionicity depends on the criteria assigning to given point of space-time surface quaternionic plane. Classical variational principle provides this criterion. For volume as action (non-physical choice) one obtains standard induced gamma matrices spanning tangent space. For Kahler action one obtains modified gammas and quaternionic sub-algebra does not span tangent space. This option is physical and besides producing standard model gauge field dynamics it provides the richests structure (quantum criticality, inclusion hierarchy of super-conformal algebras corresponding to that for HFFS of type II_1, etc..).

The world of classical worlds (WCW) is the multiverse of TGD and can be identified as the space of these quaternionic sub-algebras of the octonionic local Clifford and entire quantum TGD follows from mere algebra. Quantum states are spinor fields in WCW formed by quaternionic local Clifford sub-algebras. No landscape is obtained in this multiverse. Standard model symmetries are always the fundamental symmetries having purely number theoretical meaning. This picture is mathematically precisely defined with well-developed connections with existing physics. Mathematicians could immediately start to apply their methodology and intuition to develop TGD as a purely mathematical discipline.

But first something should be done. Maybe Nobel committee should follow their strategy when it gave peace price for Kissinger: Nobels to the leading string gurus! String wars would cease, landscape nightmare -the Vietnam of physics- would be soon forgotten, and theoreticians would be eagerly studying physics again;-).


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