Position and momentum observables on R and on R3

“…The sole requirement of boost covariance without further assumptions is not enough to lead to the form (7). It is interesting that there are even commutative boost covariant observables which are not smearings of P. This is illustrated in the following example.…”

Section: Smearingsmentioning

confidence: 94%

“…Compared to (4), here we have the additional requirement of boost invariance. It is proved in [7] that E is a position observable exactly when E = Q ρ for some probability measure ρ.…”

Section: Relativistic Approachmentioning

confidence: 99%

Why Unsharp Observables?

2007

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“…Here by an approximate position measurement we mean an observable F , which is covariant for position shifts, and commutes with momentum, so that F (X − q) = W (q, p)F (X)W * (q, p). These are necessarily of the form F ρ = µ * E Q ρ for some measure µ (see, e.g., 21 ). Suppose that we have such an approximate position measurement and, similarly, an approximate momentum measurement given by a noise measure ν.…”

Section: A Covariant Phase Space Observablesmentioning

confidence: 99%

Measurement uncertainty relations

Busch¹,

Lahti²,

Werner³

2014

Journal of Mathematical Physics

133

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Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order $\alpha$ rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.Comment: This version 2 contains minor corrections and reformulation

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Position and momentum observables on R and on R3

Abstract: Abstract. We characterize all position and momentum observables on R and on R 3 . We study some of their operational properties and discuss their covariant joint observables.

Cited by 20 publications

References 14 publications

On the coexistence of position and momentum observables

On the coexistence of position and momentum observables

Why Unsharp Observables?

Measurement uncertainty relations

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