Quantization of Circulation in Superfluid Helium

Pauli's theorem and quantum canonical pairs: the consistency of a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with non–empty point spectrum

“…Moreover, Pauli's theorem has led many to believe that the canonical commutation relation (3.6) only admits solutions that form a system of imprimitivities on ℜ, so that any operator canonically conjugate to a semibounded Hamiltonian must necessarily be non-self-adjoint (for example Srinivas & Vijayalakshmi 1981, Giannitrapani 1997, Park 1984, Delgado & Muga 1997, Olhovsky &Recami (1974, Toller 1997,1999, Cohen-Tannoudji 1977, Gottfried 1966. Our examples above and the examples of the much ignored earlier works of Galindo (1984), Garrison and Wong (1970), Segal (1967), and especially Nelson (1959) demonstrate otherwise, that in fact there are numerous solutions to equation (3.6). Pauli's Theorem has been requiring solutions for T and H, regardless of the interpretation for T , that form a system of imprimitivities based on the real line under the guidance of the erronous logic leading to equation (2.3).…”

“…A specific direction to head off is to pursue the investigation of the whole class of solutions to the canonical commutation relation in the standard formulation, the relationship of the different solutions to first physical principles (e.g. homogeneity of free space leads to the solution of the pair of position and momentum operators in ℜ (Jauch 1968; Mackey 1968)), and their relationship to specific problems (Garrison & Wong 1970). Now could it be possible that the reason that there is still so much controversy in the quantum time problem because the entirety of its picture is not yet fully understood?…”

“…Thus imprimitivity based on the real line of a canonical pair is not an esteemed property that has to be imposed on all canonical pairs. That is we can not prefer one solution of (3.6) from other solutions without considering the physical context of the solution being sought, a point earlier indicated by Garrison and Wong (1970).…”

“…An ideal quantum clock requires a one-parameter family of states, Λ τ , with the property e −itH Λ τ = Λ τ +t , and a self-adjoint operator T , whose measurement on Λ τ yields τ with negligible dispersion (Susskind &Glogower 1968). An explicit example of such is the oscillator clock of Garrison and Wong (1970) in which a bounded and self-adjoint time operator (canonically conjugate to the oscillator Hamiltonian) is used for T . Another example may arise from the quantization of classical observables, which are real valued functions of the classical canonical coordinates.…”

“…The reason for this pessimism is a theorem due to Pauli (1926Pauli ( , 1958 which has been believed to assert without exception that there exists no self-adjoint time operator canonically conjugate to a semi-bounded Hamiltonian in single Hilbert space: The existence of a self-adjoint time operator implies that the time operator and the Hamiltonian have completely continuous spectra spanning the entire real line, or, in modern parlance, the time operator and the Hamiltonian form a system of imprimitivities based on the real line ℜ. While some examples of self-adjoint canonical pairs are known that do not possess spectra spanning ℜ-the momentum and the position operators of a particle trapped in a box (Reed & Simon 1975;Segal 1967;Nelson 1959), the angular momentum and the angle operators (Kraus 1965), the harmonic oscillator number and phase operators (Galindo 1984, Garrison &Wong 1970)-Pauli's Theorem remains unquestioned and continues to be a major motivation in shaping much of the present works in incorporating a dynamical theory of time in quantum mechanics (see for example Kuusk & Koiv 2001 for a recent reference to Pauli's theorem). This has led to diverse treatments of time within (Leon et.…”