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Quantum symmetries that leave invariant physical transition probabilities aredescribed by projective representations of Lie groups. The mathematical theoryof projected representations for topologically connected Lie groups is reviewedand applied to the inhomogeneous symplectic group ISp(2n). The projectiverepresentations are given in terms of the ordinary unitary representations ofthe central extension of the group with respect to its first homotopy groupdirect product with its second cohomology group. The second cohomology group ofISp(2n) is the one dimensional abelian group and the central extension turnsthe 2n dimension abelian normal subgroup of translations of ISp(2n) into theWeyl-Heisenberg group. This is the quantum symmetry origin of the Heisenbergcommutation relations that are the Hermitian representation of the Lie algebraof the Weyl-Heisenberg group.

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