**Authors:** Matti Pitkänen

Physics as a generalized number theory program involves three threads: various p-adic physics
and their fusion together with real number based physics to a larger structure, the attempt to
understand basic physics in terms of classical number fields discussed in this article, and infinite
primes whose construction is formally analogous to a repeated second quantization of an arithmetic
quantum field theory.
In this article the connection between standard model symmetries and classical number fields
is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of
classical worlds") is unique from its mere existence. This leads to its identification as union of
symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This
fixes space-time dimension and the decomposition M^{4} x S and the idea is that the symmetries
of the Kähler manifold S make it somehow unique. The motivating observations are that the
dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces,
and imbedding space and M^{8} can be identified as hyper-octonions- a sub-space of complexified
octonions obtained by adding a commuting imaginary unit. This stimulates some questions.
Could one understand S = CP_{2} number theoretically in the sense that M^{8} and H = M^{4} x CP_{2}
be in some deep sense equivalent ("number theoretical compactification" or M^{8} - H duality)?
Could associativity define the fundamental dynamical principle so that space-time surfaces could
be regarded as associative or co-associative (defined properly) sub-manifolds of M^{8} or equivalently
of H.
One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation
of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span
an associate (co-associative) sub-space at each point of space-time surface. Also M^{8} - H duality
holds true if one assumes that this associative sub-space at each point contains preferred plane of
M^{8} identifiable as a preferred commutative or co-commutative plane (this condition generalizes
to an integral distribution of commutative planes in M^{8}). These planes are parametrized by CP_{2}
and this leads to M^{8} - H duality.
WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford
algebra of M^{8} or H which are associative or co-associative. An open conjecture is that this
characterization of the space-time surfaces is equivalent with the preferred extremal property of
Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional
algebra of vanishing second variations.

**Comments:** 28 Pages.

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[v1] 16 Jun 2010

[v2] 2012-01-30 21:56:36

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