  We consider gauge theories in a strong external magnetic like field. This
situation can appear either in conventional four-dimensional theories, but also
naturally in extra-dimensional theories and especially in brane world models.
We show that in the lowest Landau level approximation, some of the coordinates
become non-commutative. We find physical reasons to formal problems with
non-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 is
shown that the Higgs sector might be non-commutative whereas the remaining
sectors of the standard model remain commutative. Signatures of this model at
the LHC are discussed. We then discuss an application to a dark matter sector
coupled to the Higgs sector of the standard model and show that here again,
dark matter could be non-commutative, the standard model fields remaining
commutative.
