Symplectic Vector Space - Heisenberg Group

Heisenberg Group

A Heisenberg group can be defined for any symplectic vector space, and this is the general way that Heisenberg groups arise.

A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative Lie algebra, meaning with trivial Lie bracket. The Heisenberg group is a central extension of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the canonical commutation relations (CCR), and a Darboux basis corresponds to canonical coordinates – in physics terms, to momentum operators and position operators.

Indeed, by the Stone–von Neumann theorem, every representation satisfying the CCR (every representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one.

Further, the group algebra of (the dual to) a vector space is the symmetric algebra, and the group algebra of the Heisenberg group (of the dual) is the Weyl algebra: one can think of the central extension as corresponding to quantization or deformation.

Formally, the symmetric algebra of V is the group algebra of the dual, Sym(V) := K, and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = K. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).