Theory of filters

Mielnik, Bogdan

doi:10.1007/bf01645423

Cited by 138 publications

(106 citation statements)

References 15 publications

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“…One of the most important goals of OQL is to impose operationally motivated axioms on a lattice structure in order that it can be made isomorphic to a projection lattice on a Hilbert space. There are different positions in the literature about the question of whether this goal has been achieved or not [14], and also, of course, alternative operational approaches to physics, as the convex operational one [47,48,49,50,51,4]. In this work, we are interested in the great generality of the OQL approach.…”

Section: Introductionmentioning

confidence: 99%

A discussion on the origin of quantum probabilities

2014

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We study the origin of quantum probabilities as arising from non-boolean propositionaloperational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorvian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).

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Section: Introductionmentioning

confidence: 99%

A discussion on the origin of quantum probabilities

2014

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“…Clearly, the extreme boundary ∂ e K of a given compact convex set K as a topological space does not contain enough information to reconstruct K. However, one can equip ∂ e K with the additional structure of a so-called transition probability, as first indicated by Mielnik [41] (also cf. [50]).…”

Section: Transition Probability Spacesmentioning

confidence: 99%

Poisson Spaces with a Transition Probability

Landsman

1997

Rev. Math. Phys.

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The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p : P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context).In classical mechanics, where p(ρ, σ) = δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of P as a transition probability space coincide with the symplectic leaves of P as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant).Motivated by E.M. Alfsen, H.

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“…It seems it was Mielnik who noticed [19,20,21,22] that since there exist many different and inequivalent probability models they may be related to some nonlinear version of quantum maechanics. Such a perspective seems quite natural and suggests that one should try to investigate theories where obsevables, like in classical mechanics, belong to a more general set of functionals.…”

Section: B Observablesmentioning

confidence: 99%

Nonlinear Schrödinger equation and two-level atoms

Czachor¹

1996

Phys. Rev. A

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Starting with the same form of atomic nonlinearity Weinberg [1] and Wódkiewicz and Scully [2] obtained contradictory results concerning an evolution of the atomic inversion w in a two-level atom in Weinberg's nonlinear quantum mechanics: If the atom is initially in a ground state then either (1) the evolution of w can be linear if one uses a nonlinear generalization of the Jaynes-Cummings Hamiltonian, or (2) is always nonlinear if one uses the nonlinear Bloch equations derived from the nonlinear atomic Hamiltonian function. It is shown that the difference is rooted in inequivalent descriptions of the composite "atom+field" system. The linear evolution of w results from a "fasterthan-light communication" between the atom and the field. If one applies a description without the "faster-than-light telegraph" then the calculations based on a suitably modified Jaynes-Cummings Hamiltonian lead to the same dynamics of w as this found in semiclassical calculations based on Bloch equations. A physical background of the two approaches is discussed in detail, and two, inequivalent kinds of such "telegraphs" are described. It is shown also that a nonlinear quantum mechanics based on a nonlinear Schrödinger equation does not possess a natural probability interpretation. Various definitions of nonlinear eigenvalues and probabilities are discussed and illustrated with examples. Finally, I discuss problems that appear if we apply the above results to the "object+observer" composite system, where the "observer" can evolve in a nonlinear way.

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Theory of filters

Cited by 138 publications

References 15 publications

A discussion on the origin of quantum probabilities

A discussion on the origin of quantum probabilities

Poisson Spaces with a Transition Probability

Nonlinear Schrödinger equation and two-level atoms

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