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adjoint admits algebra of operators assigns automorphism bounded operators bracket operation canonical coordinates canonical variables classical and quantum classical mechanics common dense domain commutation relations completely determined conjugation and bracket conjugation operation constraints construct Corollary denote easy to verify elements equations of motion essentially self-adjoint expected values exponential polynomials extended functions extended motions finite follows formulation of mechanics function f group of motions Hamiltonian Heisenberg group hence Hermitian functional Hermitian polynomial Hilbert space induced isometry joint moments Lemma Lie algebra linear combination maximal extension maximal representation measure-preserving transformation measuring process mechanical system mial monomials Moreover non-commutative normal form obtained one-parameter group polyno polynomial f polynomial g(z positive linear functional probability distribution Proof properties quantum mechanics representation of AQ require result satisfy the relations standard representation statistics structure subalgebra subset Theorem 5.1 tion total degree two-sided ideal unbounded unique unitary operators vanish variance vector