NONLINEAR
PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal
2010, Vol.13, No.3, pp.249-266
The Lorentz Group, Noncommutative Space-Time, and Nonlinear
Electrodynamics in Majorana - Oppenheimer Formalism.
V. Red'kov and E. Tolkachev
Non-linear electrodynamics arising in the frames of field
theories in non-commutative space-time is examined on the base
of the Riemann - Zilberstein - Majorana - Oppenheimer formalism. The
problem of form-invariance of the non-linear constitutive
relations governed by six non-commutative parameters 
K = n + im is explored in detail on the
base of the complex orthogonal group theory SO(3.C). Two Abelian
2-parametric small groups, isomorphic to each other in abstract
sense, and leaving unchangeable the extended constitutive
relations at arbitrary six parameters 
of
effective media have been found, their realization depends
explicitly on invariant length K2. In the case of
non-vanishing length a special reference frame in which the small
group has the structure SO(2)
SO(1,1) has been found. In
isotropic case no such reference frame exists. The way to
interpret both Abelian small groups in physical terms consists in
factorizing corresponding Lorentz transformations into Euclidean
rotations and boosts.
In the context of general study of various dual symmetries in
non-commutative field theory,
it is demonstrated explicitly that
the non-linear constitutive equations in non-commutative
electro-dynamics are not invariant under continuous dual
rotations, instead the invariance under discrete dual
transformation exists only.
Key words:
non-commutative electrodynamics, Majorana -
Oppenheimer approach, non-linear constitutive relations, Lorentz symmetry
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Last updated: September 15, 2010