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) Assuming that there's only one critical point, we can conclude that the phase transition happens at formula_22. It is common to name the model depending on the values of formula_23, formula_24 and formula_6: if formula_26, the model is called the Heisenberg XYZ model; in the case of formula_27, it is the Heisenberg XXZ model; if formula_28, it is the Heisenberg XXX model. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz. In the algebraic formulation, these are related to particular Quantum affine algebras and Elliptic Quantum Group in the XXZ and XYZ cases