Abstract
We present the bundle (Aff(3)⊗ℂ⊗Λ)(ℝ3), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (ℂ⊗Λ)(ℝ3) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗ℂ)(ℤ3). This space allows a simple physical interpretation as a phase space of a lattice of cells.
We find the SM SU(3) c ×SU(2) L ×U(1) Y action on (Aff(3)⊗ℂ⊗Λ)(ℝ3) to be a maximal anomaly-free gauge action preserving E(3) symmetry and symplectic structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations.
The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with ℤ2-degenerated vacuum state. We construct anticommuting fermion operators for the resulting ℤ2-valued (spin) field theory.
A metric theory of gravity compatible with this model is presented too.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research. Wiley, New York (1962)
Banks, T., Dothan, Y., Horn, D.: Geometric fermions. Phys. Lett. B 117, 413 (1982)
Banks, T., Kogut, J., Susskind, L.: PRD 13, 1043 (1996)
Becher, P.: Phys. Lett. B 104, 221 (1981)
Becher, P., Joos, H.: The Dirac-Kähler equation and fermions on the lattice. Z. Phys. C Part. Fields 15, 343–365 (1982)
Berezin, F.A.: The Method of Second Quantization. Academic Press, New York (1966)
Bleuer, K.: Helv. Phys. Acta 23(5), 567–586 (1950)
Chadha, S., Nielsen, H.B.: Nucl. Phys. B 217, 125–144 (1983)
Daviau, C.: Dirac equation in the Clifford algebra of space. In: Dietrich, V., Habetha, K., Jank, G. (eds.) Proc. of Clifford Algebra and Their Applications in Mathematical Physics, Aachen, 1996, pp. 67–87. Kluwer, Dordrecht (1998)
Dirac, P.A.M.: Proc. R. Soc. A 114(767), 243–265 (1927)
Fermi, E.: Rev. Mod. Phys. 4(1), 87–132 (1932)
Frampton, P., Glashow, S.: Chiral color: an alternative to the standard model. Phys. Lett. B 190, 157 (1987)
Georgi, H., Glashow, S.: Unity of all elementary-particle forces. Phys. Rev. Lett. 32, 438 (1974)
Gupta, S.: Proc. Phys. Lett. B 521, 429 (1950)
Gupta, R.: Introduction to lattice QCD. arXiv:hep-lat/9807028 (1998)
Hestenes, D.: Space-time structure of weak and electromagnetic interactions. Found. Phys. 12, 153–168 (1982)
Isham, C.: Canonical quantum gravity and the problem of time. arXiv:gr-qc/9210011 (1992)
Kadic, A., Edelen, D.G.B.: A Gauge Theory of Dislocations and Disclinations. Lecture Notes in Physics, vol. 174. Springer, Berlin (1983)
Kähler, E.: Rendiconti Mat. 21(3–4), 425 (1962). See also Ivanenko, D., Landau, L., Z. Phys. 48, 341 (1928)
Kogut, J., Susskind, L.: PRD 11, 395 (1975)
Pati, J.C., Salam, A.: Lepton number as the fourth color. Phys. Rev. D 10, 275 (1974)
Pete, G.: Morse theory. http://www.math.u-szeged.hu/~gpete/morse.ps
Preskill, J.: Do black holes destroy information? arXiv:hep-th/9209058 (1992)
Rabin, J.M.: Nucl. Phys. B 201, 315 (1982)
Schmelzer, I.: A generalization of the Lorentz ether to gravity with general-relativistic limit. arXiv:gr-qc/0205035 (2002)
Susskind, L.: Phys. Rev. D 16, 3031 (1977)
Volovik, G.E.: Induced gravity in superfluid 3He. arXiv:cond-mat/9806010 (1998)
Wheeler, J.A.: Geometrodynamica. Academic Press, New York (1962)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schmelzer, I. A Condensed Matter Interpretation of SM Fermions and Gauge Fields. Found Phys 39, 73–107 (2009). https://doi.org/10.1007/s10701-008-9262-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-008-9262-9