  The statistical mechanics of quantum-classical systems with holonomic
constraints is formulated rigorously by unifying the classical Dirac bracket
and the quantum-classical bracket in matrix form.
  The resulting Dirac quantum-classical theory, which conserves the holonomic
constraints exactly, is then used to formulate time evolution and statistical
mechanics. The correct momentum-jump approximation for constrained system
arises naturally from this formalism. Finally, in analogy with what was found
in the classical case, it is shown that the rigorous linear response function
of constrained quantum-classical systems contains non-trivial additional terms
which are absent in the response of unconstrained systems.
