Friday, March 16, 2007

TGD Universe from the condition that all possible statistics are possible

By simple physical arguments H=M4×CP2 is the unique choice for the imbedding space in TGD Universe. This choice follows also from the number theoretic vision. M4 has interpretation as hyper-quaternions and CP2 as space of space of quaternionic planes at a point of (hyper)-octonion space. Space-time surfaces can be seen as hyper-quaternionic sub-manifolds of hyper-octonionic space M8 and correspond to sub-manifolds of H. The basic conjecture is that hyper-quaternionicity corresponds to the property of being a certain preferred extremal of so called Kähler action with corresponding Kähler action defining Kähler function for the "world of classical worlds". The corresponding isometry group SO(3,1)× SU(3) and vielbein group SL(2,C)× U(2)ew have therefore a distinguished position both in physics and quantum TGD.

A new observation induced by a little discussion with Kea is that also the general pattern for inclusions of hyperfinite factors of type II1 selects these groups as something very special: the condition that all possible statistics are realized is guaranteed by the choice M4× CP2.

  1. Inclusions are partially characterized by quantum phase q=exp(i2π/n). n>2 for the quantum counterparts of the fundamental representation of SU(2) means that braid statistics for Jones inclusions cannot give the usual fermionic statistics. That Fermi statistics cannot "emerge" conforms with the role of infinite-D Clifford algebra as a canonical representation of HFF of type II1. SO(3,1) as isometries of H gives Z2 statistics via the action on spinors of M4 and U(2) holonomies for CP2 realize Z2 statistics in CP2 degrees of freedom.

  2. n>3 for more general inclusions [I learned this from the thesis "Hecke algebras of type An and subfactors" of Hans Wenzl (Inv. Math. 92, 349-383 (1988)] in turn excludes Z3 statistics as braid statistics in the general case. SU(3) as isometries induces a non-trivial Z3 action on quark spinors but trivial action at the imbedding space level so that Z3 statistics becomes possible in quark sector quite generally.

For more details about general inclusions see the appendix of the chapter Was von Neumann Right After All? of "Towards S-matrix".

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