Heisenberg group of the Heisenberg group is fairly simple – later generalized by Mackey theory – and was the motivation for its introduction in quantum physics, as discussed below. For each nonzero real number formula_27, we can define an irreducible unitary representation formula_48 of formula_31 acting on the Hilbert space formula_50 by the formula: This representation is known as the Schrödinger representation. The motivation for the this representation is the action of the exponentiated position and momentum operators in quantum mechanics. The parameter formula_52 describes translations in position space, the parameter formula_53 describes translations in momentum space, and the parameter formula_54 gives
Heisenberg model (quantum) respectively. Other approaches do so without Bethe ansatz. The physics of the Heisenberg model strongly depends on the sign of the coupling constant formula_3 and the dimension of the space. For positive formula_3 the ground state is always ferromagnetic. At negative formula_3 the ground state is antiferromagnetic in two and three dimensions, it is from this ground state that the Hubbard model is given. In one dimension the nature of correlations in the antiferromagnetic Heisenberg model depends on the spin of the magnetic dipoles. If the spin is integer then only short-range order is present. A system of half-integer spins