This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter also reviews the canonical anticommutation relations, which encapsulate the quantum mechanics of spin systems. Having catalogued reasons to regard unitary equivalence as a robust criterion of physical equivalence for quantum theories obtained by finding representations of the CCRs/CARs, the chapter presents a pair of theorems—the Stone-von Neumann and Jordan-Wigner theorems—that establish that, provided only finitely many degrees of freedom are involved, all representations of the CCRs/CARs for a given quantum theory are unitarily (and so presumptively physically) equivalent.
Keywords: Hamiltonian quantization, Stone-von Neumann theorem, Jordan-Wigner theorem, unitary equivalence, Hilbert space, quantum theory, canonical commutation relations, Weyl relations, canonical anticommutation relations, physical equivalence
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