Projective representations of the inhomogeneous symplectic group: Quantum symmetry origins of the Heisenberg commutation relations

TL;DR

This paper explores how projective representations of the inhomogeneous symplectic group reveal the quantum symmetry origins of the Heisenberg commutation relations, linking group theory to fundamental quantum mechanics principles.

Contribution

It provides a mathematical framework connecting projective representations of ISp(2n) to the Weyl-Heisenberg group, explaining the origin of Heisenberg relations.

Findings

Central extension of ISp(2n) yields the Weyl-Heisenberg group.

Projective representations relate to quantum symmetries in transition probabilities.

Heisenberg commutation relations originate from the group's topological properties.

Abstract

Quantum symmetries that leave invariant physical transition probabilities are described by projective representations of Lie groups. The mathematical theory of projected representations for topologically connected Lie groups is reviewed and applied to the inhomogeneous symplectic group ISp(2n). The projective representations are given in terms of the ordinary unitary representations of the central extension of the group with respect to its first homotopy group direct product with its second cohomology group. The second cohomology group of ISp(2n) is the one dimensional abelian group and the central extension turns the 2n dimension abelian normal subgroup of translations of ISp(2n) into the Weyl-Heisenberg group. This is the quantum symmetry origin of the Heisenberg commutation relations that are the Hermitian representation of the Lie algebra of the Weyl-Heisenberg group.