On euclidean systems of covariance for massless particles

Position and momentum observables on R and on R3

“…Let E be covariant under dilations. In a similar way as in [8,Lemma 3] it can be shown that E(U) = 1 for all nonempty open sets U, and so, (a) implies (b). Assume then that (b) holds.…”

“…and comparing (7) to (8) we note that E ρ is obtained when the sharply concentrated Dirac measure δ 0 is replaced by the probability measure ρ.…”

A note on the relationship between localization and the norm-1 property

“…The covariance assures that the results of a localization measurement do not depend on the choice of the origin and the orientation of the reference frame. As we have just remarked, in the case of the photon, sharp localization is impossible [2,4,5,3]. Conversely, localization of the photon can be described by means of POVMs F :…”

“…Now, we specialize to the case of relativistic localization in R 3 . In the relativistic case, the relevant group is the Poincare group and sharp localization is defined as follows [4,10]: let W be a continuous unitary representation of the universal covering of the Poincare group. Let U be the restriction of W to the universal covering group…”

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

“…So far, the general technique used to study systems of covariance has been to extend the POV measure to a projector-valued (PV) measure on an enlarged Hilbert space, using in the process a theorem by Naimark [12,13], and then to apply Mackey's theory to this space [14][15][16][17][18]. The use of these general results to study representations of specific kinematic groups, which arise in the context of investigating the problem of particle localization -such as the Euclidean, Galilei, and Poincar6 groups -has received a significant amount of attention [19][20][21][22], and is of central interest to the stOchastic quantum mechanics programme.…”