Canonical quantization of constants of motion

Abstract

We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process. If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory. Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization.

On the Classical-Quantum Relation of Constants of Motion

“…, where we are identifying T * ξ S n−1 √ λ with the plane tangent to ξ (literally). See Belmonte [16] for the proof of the last claim.…”

“…Even though the importance in quantum mechanics of L 2 is well known, the relation via quantization, of L 2 as a COM, with its classical counterpart has not been clarified. In this article we will study such relation, and thus our analysis adds up a novel example of CQR (using the well known quantum decomposition of L 2 ) to the already known cases [16]. Interestingly, the technique used to prove the main result (theorem 4) suggests that we should expect that we can obtain CQR for a broad class of COM.…”

“…Considering the analogies between the mathematical descriptions of classical and quantum COM, it seems natural that there should be a quantization process that preserves COM. In Belmonte [16], a solution for this problem was given. We shall recall its construction in section 2.…”

“…It is important to note that in some cases the right hand of (1) might be zero, in other words, it can happen that Weyl Calculus does preserve COM for some Hamiltonian h 0 . The following theorem, proved in Belmonte [16], provides interesting examples when the equality holds. THEOREM 1.…”

“…The conservation of momentum has been proven to correspond the rotational invariance by Noether's theorem [14]. In Belmonte [16], it was shown that every function on angular momenta is a constant of motion for h 0 (x, ξ ) = ||x|| 2 or h 0 (x, ξ ) = ||ξ || 2 . Among those COM, an important role is played by the angular momenta coordinates themselves, i.e.…”