On the coexistence of position and momentum observables

Abstract

Abstract. We investigate the problem of coexistence of position and momentum observables. We characterize those pairs of position and momentum observables which have a joint observable.

Why Unsharp Observables?

“…This result leads to the complete characterization of jointly measurable pairs of position and momentum observables as explained in [8]. In particular, combining [7, Proposition 5] and [8, Proposition 7], we conclude that for a position observable Q ρ , the following conditions are equivalent:…”

“…Also, one can show that if Q ρ (X) is a nontrivial projection operator for some X ∈ B(R), then Q ρ is not jointly measurable (or even coexistent) with any momentum observable [8,Proposition 11].…”

Universal joint-measurement uncertainty relation for error bars

“…As shown in [8], observables Q µ , P ν are jointly measurable exactly when there is a covariant phase space observable G m of they are the marginals. In that case the resolution widths are given by the widths of the probability measures µ m , ν m (via Eq.…”

On the Complementarity of the Quadrature Observables

“…As will be seen below, these two approaches are closely related with each other. On the other hand, the unsharp pair (μ * Q, ν * P) has joint observable exactly when the probability measures μ and ν have Fourier related densities, in which case (μ * Q, ν * P) are the (Cartesian) marginal observables of a covariant phase space observable G K generated by a positive trace-1 operator K on H [12]. We recall that G K is defined by the operator density (q, p) → W (q, p)KW (q, p) * , that is,…”