The concept of a strict deformation quantization provides a mathematical
formalism that describes the transition from a classical theory to a
quantum theory in terms of deformations of (commutative) Poisson
algebras (representing the classical theory) into non-commutative C*
-algebras (characterizing the quantum theory). In the first part of this
talk we introduce the definitions, give several examples and show how
quantization of the closed unit 3-ball $B^3$ in $R^3$ is related to
quantization of its smooth boundary (i.e. the two-sphere $S^2$). In the
second part we apply this formalism to prove the existence of the
'classical limit' of mean-field quantum spin systems and Schrödinger
operators yielding a probability measure on the pertinent phase space.
We moreover discuss the concept of spontaneous symmetry breaking (SSB)
as an emergent phenomenon when passing from the quantum realm to the
classical world by taking a limit in the relevant semiclassical parameter.