This one-dimensional matrix entry is of the form:  in which "θ" is the parameter of the group, and the group is Abelian since one-dimensional matrices always commute under matrix multiplication. Lagrangians in quantum field theory for complex scalar fields are often invariant under U(1) transformations. If there is a quantum number "a" associated with the U(1) symmetry, for example baryon and the three lepton numbers in electromagnetic interactions, we have:  The general form of an element of a U(2) element is parametrized by two complex numbers "a" and "b":  and for SU(2), the determinant is restricted to 1:  In group theoretic language, the Pauli matrices are the generators of the special unitary group in two dimensions, denoted SU(2). Their commutation relation is the same as for orbital angular momentum, aside from a factor of 2:  A group element of SU(2) can be written:  where "σj" is a Pauli matrix, and the group parameters are the angles turned through about an axis. The two-dimensional isotropic quantum harmonic oscillator has symmetry group SU(2), while the symmetry algebra of the rational anisotropic QHO is a nonlinear extension of u(2). The eight Gell-Mann matrices (see article for them and the structure constants) are important for quantum chromodynamics. They originally arose in the theory SU(3) of flavor which is still of practical importance in nuclear physics. They are the generators for the SU(3) group, so an element of SU(3) can be written analogously to an element of SU(2):  where "θn" are eight independent parameters. The matrices satisfy the commutator:  where the indices , , take the values 1, 2, 3... The structure constants "fabc" are totally antisymmetric in all indices analogous to those of SU(2). In the standard colour charge basis ("r" for red, "g" for green, "b" for blue):  the colour states are eigenstates of the and matrices, while the other matrices mix colour states together. The eight gluons states (8-dimensional column vectors) are simultaneous eigenstates of the adjoint representation of , the 8-dimensional representation acting on its own Lie algebra , for the and matrices. By forming tensor products of representations (the standard representation and its dual) and taking appropriate quotients, protons and neutrons, and other hadrons are eigenstates of various representations of of color.