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 picture some sense, the Heisenberg picture is more natural and convenient than the equivalent Schrödinger picture, especially for relativistic theories. Lorentz invariance is manifest in the Heisenberg picture, since the state vectors do not single out the time or space. This approach also has a more direct similarity to classical physics: by simply replacing the commutator above by the Poisson bracket, the Heisenberg equation reduces to an equation in Hamiltonian mechanics. For the sake of pedagogy, the Heisenberg picture is introduced here from the subsequent, but more familiar, Schrödinger picture. The expectation value of an observable "A", which is a Hermitian