Phase–number uncertainty from Weyl commutation relations

Sergio A Albeverio, Ambar N. Sengupta

J. C. Garrison, Jack Wong

Apparent difficulties that prevent the definition of canonical conjugates for certain observables, e.g., the number operator, are eliminated by distinguishing between the Heisenberg and Weyl forms of the canonical commutation relations (CCR's). Examples are given for which the uncertainty principle does not follow from the CCR's. An operator F is constructed which is canonically conjugate, in the Heisenberg sense, to the number operator; and F is used to define a quantum time operator.

PEKKA J. LAHTI, Maciej Maczynski

Reinhard Honegger

Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.

Daniel A. Dubin, Mark A. Hennings, T. B. Smith

Jan van Neerven

We revisit the quantum phase operator $Φ$ introduced by Garrison and Wong. Denoting by $N$ the number operator, we provide a detailed proof of the Heisenberg commutation relation $ΦN - NΦ= iI$ on the natural maximal domain $D(ΦN) \cap D(N Φ)$ as well as the failure of the Weyl commutation relations, and discuss some further interesting properties of this pair.