Invariances of generalized uncertainty principles

Yeo Cheng Xun

Catarina Bastos, Alex E. Bernardini, Orfeu Bertolami A +2 more

A more general measurement disturbance uncertainty principle is presented in a Robertson-Schrödinger formulation. It is shown that it is stronger and having nicer properties than Ozawa's uncertainty relations. In particular it is invariant under symplectic transformations. One shows also that there are states of the probe (measuring device) that saturate the matrix formulation of measurement disturbance uncertainty principle.

David Mustard

Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

Burcu Elif Akten

Robert T. Deck

We demonstrate two points: 1) the formalism of quantum mechanics can be understood simply as a structure for the expression of the physical notion that not all observables can have values simultaneously; 2) the specific uncertainty relations can be derived (rigorously) by combination of the invariance principle with a general uncertainty relation based only on the existence of unspecified pairs of conjugate observables. For this purpose, we present a formulation of quantum mechanics based strictly on the invariance principle and a “weak” statement of the uncertainty principle that asserts only the existence of incompatible (conjugate) observables without specifying which observables are incompatible. We go on to show that the invariance principle can be used to develop the equations of motion of the theory, including the Klein-Gordon and Schrodinger equations.

E. U. Condon