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We consider gauge theories in a strong external magnetic like field. Thissituation can appear either in conventional four-dimensional theories, but alsonaturally in extra-dimensional theories and especially in brane world models.We show that in the lowest Landau level approximation, some of the coordinatesbecome non-commutative. We find physical reasons to formal problems withnon-commutative gauge theories such as the issue with SU(N) gauge symmetries.Our construction is applied to a minimal extension of the standard model. It isshown that the Higgs sector might be non-commutative whereas the remainingsectors of the standard model remain commutative. Signatures of this model atthe LHC are discussed. We then discuss an application to a dark matter sectorcoupled to the Higgs sector of the standard model and show that here again,dark matter could be non-commutative, the standard model fields remainingcommutative.

Lowest Landau Level of Relativistic Field Theories in a Strong Background Field