Uncertainty relations for a non-canonical phase-space noncommutative algebra

Abstract

We consider a non-canonical phase-space deformation of the Heisenberg-Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg-Pauli-Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of timefrequency analysis.Date: May 7, 2019.where K iν are modified Bessel functions. These solutions are highly oscillatory and not square-integrable. This poses severe interpretational problems. This is a familiar feature in this type of mini superspace models. One faces the problem of determining a "time" variable and a measure, such that on constant "time" hypersurfaces, the wave-function is normalizable and the square of its modulus is a bona fide probability density.

Uncertainty Relations: Curiosities and Inconsistencies

“…(where β = β * = 0), one obtains that λ 5 ψ = 0 for any a and b but λ 5 ψ > 0 for these a and c, which do not fulfil the condition (18) and in this case the uncertainty relation (14) has the standard form. Similar examples can be found for self-adjoint matrices or operators acting in any Hilbert space (see, e.g., Section 2 in [20]).…”

“…The detailed and rigorous mathematical analysis of the Heisenberg's relation (2) together with (4) shows that, e.g., for observables A def = X n and B def = P m , (where P = −ih d dx and m, n ∈ N), using the so-called unitary dilation operator one can build from a normalized state |ψ(x) ∈ L 2 (R) such a function that the product of standard deviations of X n and P m calculated for this function can vanish (for details see, e.g., [20]). This suggest that relations (2) and (4) may not be good relations, strictly speaking that the product ∆ ψ A • ∆ ψ B may not be a good measure of the uncertainty.…”

“…As it is seen, the inconsistencies of this type and others discussed in previous Sections are integrated into inequality (4). For this reason, attempts are being made to improve and refine the Heisenberg as well as Robertson and Schrodinger uncertainty relations (see, e.g., [14][15][16][17]20]). From the analysis presented in Section 5 it follows that a status and role of the uncertainty relations (1), (2) and (4) in P T -symmetric quantum theory seems to be unclear.…”

“…As it was mentioned, there is still a discussion on how to interpret inequalities (4) and (2) and how to improve them (see, e.g., [10] and references therein, [11,12,[14][15][16][17][18][19][20] and many other papers). From the derivation of the formula (4) it follows that the standard deviations ∆ φ A and ∆ φ B characterize the statistical distribution of the most probable values of A and B in the state |φ .…”