We argue that the classical description of a symplectic manifold endowed with a Hamiltonian action of an abelian Lie group G and the corresponding quantum theory can be understood as different aspects of the unitary representation theory of G. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory. The proposed argumentative line strongly relies on the conjecture proposed by Guillemin and Sternberg according to which ``quantization commutes with reduction''. By using (...) the generalization of this conjecture to non-zero coadjoint orbits, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This fact points towards a gauge-theoretic interpretation of Heisenberg indeterminacy principle. (shrink)