Lie groupoids and Lie algebroids in physics and noncommutative geometry

Only a handful of physicists appear to be familiar with Lie groupoids and Lie algebroids, whereas the latter are practically unknown even to mathematicians working in noncommutative geometry: so much the worse for its relationship with symplectic geometry! Thus the aim of this review paper is to explain the relevance of both objects to both audiences. We do so by outlining the road from canonical quantization to Lie groupoids and Lie algebroids via Mackey's imprimitivity theorem and its symplectic counterpart. This will also lead the reader into symplectic groupoids, which define a 'classical' category on which quantization may speculatively be defined as a functor into the category KK defined by Kasparov's bivariant K-theory of C∗-algebras. This functor unifies deformation quantization and geometric quantization, the conjectural functoriality of quantization counting the “quantization commutes with reduction” conjecture of Guillemin and Sternberg among its many consequences.