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 group the special property that the exponential map is a one-to-one and onto map from the Lie algebra formula_16 to the group formula_1. More general Heisenberg groups formula_31 may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces. The simplest general case is the real Heisenberg group of dimension formula_32, for any integer formula_33. As a group of matrices, formula_31 (or formula_35 to indicate this is the Heisenberg group over the field formula_36 of real numbers) is defined as the group formula_37 matrices with entries in formula_36 and having the form: where This is indeed