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Particles and Fields

Eur. Phys. J. C 20, 185-191
DOI: 10.1007/s100520100623

Reconstruction of the standard model in a generalized differential geometry based on the real structure

Y. Okumura1, H. Kase2 and K. Morita3

1  Department of Physics, Boston University, Boston, MA 02215, USA
2  Department of Physics, Daido Institute of Technology, Nagoya, 457, Japan
3  Department of Physics, Nagoya University, Nagoya, 464-01, Japan

(Received: 31 January 2000 / Revised version: 28 July 2000 / Published online: 25 April 2001 -© Springer-Verlag / Società Italiana di Fisica 2001)

The standard model is reconstructed in a generalized differential geometry (GDG) based on the idea of a real structure as proposed by Coquereaux et al. and Connes. The GDG considered in this article is a kind of non-commutative geometry (NCG) on the discrete space that successfully reproduces the Higgs mechanism of the spontaneously broken gauge theory. Here, a GDG is a direct generalization of the differential geometry on an ordinary continuous manifold to the product space of this manifold with a discrete manifold. In a GDG, a one-form basis $\chi$ on the discrete space is incorporated in addition to the one-form basis ${\mathrm {d}}x^\mu$ on Minkowski space, rather than $\gamma^5$ as in Connes's original work. Although the Lagrangians obtained in this way are the same as those obtained in our previous formulation of GDG, the basic formalism becomes very simply and clear.

© Società Italiana di Fisica, Springer-Verlag 2001