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Physical Review A

Simultaneous sharp measurability of position and momentum in infinite quantum systems

Thomas Breuer

Published Dec 1, 1994• 0 citations
Publisher

Abstract

Joint position-momentum observables are defined to be covariant POV measures on the group ${\mathit{openR}}^{3}$\ifmmode\times\else\texttimes\fi{}${\mathit{openR}}^{3}$^ of space translations and boosts. For elementary or finite quantum systems, it is shown that there is no covariant PV measure on ${\mathit{openR}}^{3}$\ifmmode\times\else\texttimes\fi{}${\mathit{openR}}^{3}$^ into the von Neumann algebra generated by average position and momentum. So these cannot be measured sharply at the same time. For an infinite quantum system, we construct a covariant PV measure on ${\mathit{openR}}^{3}$\ifmmode\times\else\texttimes\fi{}${\mathit{openR}}^{3}$^ into the von Neumann algebra generated by average position and momentum. Therefore average position and momentum of an infinite quantum system can be measured sharply at the same time.

DOI
10.1103/physreva.50.4565

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Joint position-momentum observables are defined to be covariant POV measures on the group openR3{\mathit{openR}}^{3}openR3\ifmmode\times\else\texttimes\fi{}openR3{\mathit{openR}}^{3}openR3^ of space translations and boosts. For elementary or finite quantum systems, it is shown that there is no covariant PV measure on openR3{\mathit{openR}}^{3}openR3\ifmmode\times\else\texttimes\fi{}openR3{\mathit{openR}}^{3}openR3^ into the von Neumann algebra generated by average position and momentum. So these cannot be measured sharply at the same time. For an infinite quantum system, we construct a covariant PV measure on openR3{\mathit{openR}}^{3}openR3\ifmmode\times\else\texttimes\fi{}openR3{\mathit{openR}}^{3}openR3^ into the von Neumann algebra generated by average position and momentum. Therefore average position and momentum of an infinite quantum system can be measured sharply at the same time.
In the quantum measure approach, quantum theory is viewed as a generalisation of classical stochastic theory rather than of classical deterministic theory. An observable or "be-able" is thus a quantum measurable set in an appropriate event algebra. Thus, the observables in quantum cosmology are covariant quantum measurable sets. In a quantum system with a strongly positive quantum measure or decoherence functional one can construct a vector valued measure whose domain is the algebra of events or physical questions. Using results in the literature, this allows us to specify when a vector valued measure on finite time events can be extended to a measure over infinite time, or equivalently, covariant events. For the class of complex percolation dynamics we show that there is a generic obstruction to this process and that covariant observables can be constructed only for a very restricted class of dynamics.
We present a unified treatment of sequential measurements of two conjugate observables. Our approach is to derive a mathematical structure theorem for all the relevant covariant instruments. As a consequence of this result, we show that every Weyl–Heisenberg covariant observable can be implemented as a sequential measurement of two conjugate observables. This method is applicable both in finite- and infinite-dimensional Hilbert spaces, therefore covering sequential spin component measurements as well as position-momentum sequential measurements.
The question of quantifying the sharpness (or unsharpness) of a quantum mechanical effect is investigated. Apart from sharpness, another property, bias, is found to be relevant for the joint measurability or coexistence of two effects. Measures of bias will be defined and examples given. Dedication The impossibility of measuring jointly certain pairs of observables is an intriguing non-classical feature of quantum theory that Pekka Lahti identified as a candidate for a rigorous formulation of the principle of complementarity. While he was investigating this fundamental no-go statement in the early 1980s, he learned from Peter Mittelstaedt that one of his students was aiming to prove the positive possibility of approximate joint measurements of complementary quantities such as position and momentum. Pekka joined our group as an Alexander von Humboldt Fellow, and together we found that a reconciliation between complementarity and (approximate) joint measurability is possible on the basis of the generalized representation of observables as positive operator measures (POMs). Since then we have pursued together our aspirations of understanding quantum mechanics and understanding Nature. I have benefited much from Pekka’s intellectual rigor and have been privileged ever since to enjoy his warm humanity. It is a great pleasure to present this paper to Pekka as a token of thanks and friendship on the occasion of his 60th birthday, with all good wishes for many happy recurrences and productive years to come. 1
We construct integrals of uniformly bounded ultraweakly measurable functions against positive operator-valued measures (POVMs) in von Neumann algebras, assuming that the integrand algebra has separable predual. The integral lies in the spatial tensor product and is uniquely characterized by its evaluations on product normal states. Integration is unital and completely positive and descends faithfully to the quotient by POVM-null functions. When both preduals are separable, this quotient is a von Neumann algebra and integration is normal, agreeing with the amplification of scalar POVM integration. We establish a dilation factorization, a Leibniz rule and a Fubini theorem, and specify the additional joint-normality hypothesis needed for realization in an ambient algebra. As an application, under the stated separability and joint-normality assumptions, we extend operational quantum reference frame relativization to covariant POVMs on arbitrary measurable spaces carrying a group action. Its domain is the fixed-point von Neumann algebra of system-valued functions modulo null equality. For homogeneous spaces of second countable locally compact Hausdorff groups with closed stabilizers, this domain is identified with the stabilizer-fixed system algebra. For sharp frames, relativization is a normal unital ∗*∗-homomorphism, faithful when ambient multiplication is injective. For localizable frames on metrizable spaces, restriction recovers continuous integrands at supported configurations. On trivial principal bundles, the domain is identified with system-valued fields on the base. These results provide integration machinery for forthcoming applications to relativistic gauge theories.