  The algebraic approach to quantum mechanics has been vital to the development
of quantum theory since its inception, and it has evolved into a mathematically
rigorous $C^\ast$-algebraic formulation of the theory's axioms. Conversely, the
functional approach in the form of Feynman path integrals is far from
mathematically rigorous: Nevertheless, path integrals provide an equally valid
and useful formulation of the axioms of quantum mechanics. The two approaches
can be merged by employing a notion of functional integration based on
topological groups that allows to construct functional integral representations
of $C^\ast$-algebras. The merger achieves a hybrid formulation of the axioms of
quantum mechanics in which topological groups play a leading role. To
illustrate the formalism, we apply the framework to non-relativistic and
relativistic quantum mechanics via the Heisenberg and Poincar\'{e} groups.
