On the analysis of high-resolution nuclear magnetic resonance spectra

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“…Ordinary operators A can be promoted to superoperators in several ways. Presently we only introduce A l (ρ) ≡ Aρ and A r (ρ) ≡ ρA [9,10], so that (1) amounts to…”

“…The Poisson bracket is then verified to satisfy the Jacobi identity, and one can further define the phase-space volume dpdq, which is invariant 9 under any canonical transformation, i.e., (p, q) → (p ′ , q ′ ) preserving the form of (9). Define the inner product f, g ≡ Γ f * g dpdq, and a density ρ as a positive phase-space function.…”

“…11 8 There will be no confusion between the Poisson bracket and the anticommutator, since the former (latter) only arises in classical (quantum) mechanics. 9 Even the orientation of phase-space volume is preserved. 10 Note a difference with the quantum case, where the "position" part of such a mapping would induce an ordinary operator ψ(q) → ψ(A −1 (q)).…”

“…An observable is an L-symmetry iff it is an integral of H; a canonical map generates an L-symmetry iff it is a Hamiltonian symmetry. For a proof, it suffices to write out [L, f ] = {H, f } and [L, A]ρ = {H−AH, Aρ} respectively [in the former case, (9) shows that Poisson brackets satisfy a product rule]. Note furthermore that the commutator [H, S] simply does not occur in the classical theory.…”

“…In other words, the concept of degeneracy is wider for the von Neumann than for the Schrödinger equation, since only energy differences have to be equal [9]-physically appealing, as it is only those differences which are gauge invariant and, hence, observable. The generalization to more than four states will be obvious.…”