13 references, page 1 of 2 [1] H. Snyder, Phys. Rev. 71, 38 (1947).

[2] G. Dunne, R. Jackiw and C. Trugenberger, Phys. Rev. D 41, 661 (1990); G. Dunne and R. Jackiw, Nucl. Phys. (Proc. Suppl.) 33C, 114 (1993); related approaches are found in C. Duval and P. Horv´athy, Phys. Lett. B479, 284 (2000) and hep-th/0106089.

[3] J. Lukierski, P. Stichel and W. Zakrzewski, Ann. Phys. 260, 224 (1997); V. P. Nair and A. P. Polychronakos, Phys. Lett. B505, 267 (2001); J. Gamboa, M. Loewe and J. Rojas, hep-th/0010220.

[4] M. Eliashvili and G. Tsitsishvili, Intl. J. Mod. Phys. B14, 1429 (2000); L. Susskind, hep-th/0101029; A. P. Polychronakos, JHEP 0104, 011 (2001), hep-th/0103013; S. Hellerman and M. Van Raamsdonk, hep-th/0103179.

[5] This instance of the map between a non-commuting field theory and an effective commuting field theory [N. Seiberg and E. Witten, JHEP 9909, 032 (1999)] is worked out by A. Bichl, J. Grimstrup, L. Popp, M. Schweda and R. Wulkenhaar, hep-th/0102044.

[6] Note that a Lorentz non-invariant Chern-Simons addition to the Maxwell Lagrange density, ∝ A · B, similarly modifies the velocity of light, but differently for the two polarizations, hence a Faraday-type rotation occurs; S. Carrol, G. B. Field, R. Jackiw, Phys. Rev. D 41, 1231 (1990); see also J. Harvey and S. Naculich, Phys. Lett. B217, 231 (1989).

M. Chaichian, M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); S. Carroll, J. Harvey, V. A. Kostelecky, C. Lane and T. Okamoto, hepth/0105082.

[8] D. Colladay and V. A. Kostelecky, Phys. Rev. D58, 116002 (1998).

[9] This algebra (with b = 0) was given by L. Landau, Zh. Eksp. Teor. Fiz. 11, 592 (1941) [English translation: J. Phys. USSR, 5, 71 (1941)]; for a review see ref. [11].

[10] V. Arnold, Mathematical Methods in Classical Mechanics, (Springer, New York 1978).