A Note on Sesquilinear Forms and the Generalized Uncertainty Relations in *-Algebras

The Uncertainty Principle Derived by The Finite Transmission Speed of Light and Information

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v3i3.2059 ◽

2014 ◽

Vol 3 (3) ◽

pp. 257-266 ◽

Cited By ~ 1

Author(s):

Piero Chiarelli

Keyword(s):

Uncertainty Principle ◽

Large Scale ◽

Uncertainty Relations ◽

Speed Of Light ◽

Energy Fluctuation ◽

Hydrodynamic Analogy ◽

Non Local ◽

Minimum Uncertainty ◽

Quantum Hydrodynamic ◽

Transmission Speed

This work shows that in the frame of the stochastic generalization of the quantum hydrodynamic analogy (QHA) the uncertainty principle is fully compatible with the postulate of finite transmission speed of light and information. The theory shows that the measurement process performed in the large scale classical limit in presence of background noise, cannot have a duration smaller than the time need to the light to travel the distance up to which the quantum non-local interaction extend itself. The product of the minimum measuring time multiplied by the variance of energy fluctuation due to presence of stochastic noise shows to lead to the minimum uncertainty principle. The paper also shows that the uncertainty relations can be also derived if applied to the indetermination of position and momentum of a particle of mass m in a quantum fluctuating environment.

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Lebesgue Decomposition of Non-Commutative Measures

International Mathematics Research Notices ◽

10.1093/imrn/rnaa231 ◽

2020 ◽

Author(s):

Michael T Jury ◽

Robert T W Martin

Keyword(s):

Hilbert Space ◽

Reproducing Kernel ◽

Fock Space ◽

Reproducing Kernel Hilbert Space ◽

Semigroup Algebra ◽

Space Theory ◽

Lebesgue Decomposition ◽

Sesquilinear Forms ◽

Positive Linear Functionals ◽

Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

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