Systems of covariance and subrepresentations of induced representations

In this paper, a new conserved current for Klein-Gordon equation is derived. It is shown, for 1 + 1-dimensions, the first component of this current is non-negative and reduces to |φ| 2 in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from |φ| 2 only when the uncertainty in momentum is greater than m0c.

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“…where 0 ≤ η ≤ 1 and 0 ≤ θ ≤ 1. Using equations (44) and (45), the Eq. (40) reads R n (p) = A n (p) + B n (p) + C n (p) + D n (p),…”

Section: Appendix Bmentioning

confidence: 99%

Probability density of relativistic spinless particles

2018

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“…The problem of finding a general representation for a covariant POV measure has been studied by several authors [51,54,55,56,57]. Here we give, for easier reference, a self-contained treatment of the particular case in which we are interested.…”

Section: Translation Covariant Pov Measuresmentioning

confidence: 99%

“…By taking this formula into account and by adding the new representation indices, the eqs. (43), (49), (51) and (53) take the form…”

Section: Translation Covariant Pov Measuresmentioning

confidence: 99%

Localization of events in space-time

Toller

1999

Phys. Rev. A

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The present paper deals with the quantum coordinates of an event in space-time, individuated by a quantum object. It is known that these observables cannot be described by self-adjoint operators or by the corresponding spectral projection-valued measure. We describe them by means of a positive-operator-valued (POV) measure in the Minkowski space-time, satisfying a suitable covariance condition with respect to the Poincaré group. This POV measure determines the probability that a measurement of the coordinates of the event gives results belonging to a given set in space-time. We show that this measure must vanish on the vacuum and the one-particle states, which cannot define any event. We give a general expression for the Poincaré covariant POV measures. We define the baricentric events, which lie on the world-line of the centre-of-mass, and we find a simple expression for the average values of their coordinates. Finally, we discuss the conditions which permit the determination of the coordinates with an arbitrary accuracy. PACS: 03.65.Bz -quantum theory; 02.20.+b -group theory.

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“…Up to our knowledge, Poulsen [26] first proved it for Lie groups using elliptic regularity theorem. Other proofs, for locally compact groups, were given by Davies [9], Neumann [23], Scutaru [31], Cattaneo [7], and Castrigiano and Henrichs [8]. All these proofs reduce the problem to the projective case, either by the Neumark dilation theorem [28], or by Stinespring's theory of positive functions on C * -algebras [32].…”

Section: Introductionmentioning

confidence: 99%

“…Our proof is independent of both the Neumark theorem and the Mackey theorem, which are corollaries of our result. It is inspired by the proof of the Mackey imprimitivity theorem given by Orsted [24], as suggested by a remark in [8] (compare also with [11, Ch. XXII, Sec.…”

Section: Introductionmentioning

confidence: 99%

Square-integrability modulo a subgroup

Cassinelli

Vito

2002

Trans. Amer. Math. Soc.

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Abstract. We prove a weak form of the Frobenius reciprocity theorem for locally compact groups. As a consequence, we propose a definition of squareintegrable representation modulo a subgroup that clarifies the relations between coherent states, wavelet transforms and covariant localisation observables. A self-contained proof of the imprimitivity theorem for covariant positive operator-valued measures is given.

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Systems of covariance and subrepresentations of induced representations

Cited by 26 publications

References 8 publications

Probability density of relativistic spinless particles

Probability density of relativistic spinless particles

Localization of events in space-time

Square-integrability modulo a subgroup

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