Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity

Kryukov, Alexey A.

doi:10.1007/s10701-005-9012-1

Cited by 11 publications

(30 citation statements)

References 7 publications

(10 reference statements)

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“…[4]. These results demonstrate that QM can be formulated in terms of geometry of the space of states.…”

Section: 2)mentioning

confidence: 64%

“…Provided h 2 is proportional to the identity operator, the latter metric coincides with the FubiniStudi metric on CP n−1 (see Ref. [4]). …”

Section: The Process Of Measurementmentioning

confidence: 97%

“…The principle is in fact a particular instance of the principle of functional relativity introduced in Ref. [4].…”

Section: Tensor Properties Of Equations In the Modelmentioning

confidence: 99%

“…[4], [6]). Since the counters are also located at these points, one can identify them with sources in the space of states of the particle.…”

Section: The Process Of Measurementmentioning

confidence: 99%

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On the Measurement Problem for a Two-level Quantum System

Kryukov¹

2007

Found Phys

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A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrödinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the perturbed metric, the Born rule for probabilities of collapse is derived. The approach is applied to a two-level quantum system to obtain a simple geometric interpretation of quantum commutators, the uncertainty principle and Planck's constant. In light of this, a lucid analysis of the double-slit experiment with collapse and an experiment on a pair of entangled particles is presented.

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“…[4]. These results demonstrate that QM can be formulated in terms of geometry of the space of states.…”

Section: 2)mentioning

confidence: 64%

“…Provided h 2 is proportional to the identity operator, the latter metric coincides with the FubiniStudi metric on CP n−1 (see Ref. [4]). …”

Section: The Process Of Measurementmentioning

confidence: 97%

“…The principle is in fact a particular instance of the principle of functional relativity introduced in Ref. [4].…”

Section: Tensor Properties Of Equations In the Modelmentioning

confidence: 99%

“…[4], [6]). Since the counters are also located at these points, one can identify them with sources in the space of states of the particle.…”

Section: The Process Of Measurementmentioning

confidence: 99%

See 2 more Smart Citations

On the Measurement Problem for a Two-level Quantum System

Kryukov¹

2007

Found Phys

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“…Following Kryukov work, [14][15][16] we can define a metric g K (X, Y) for any two tangent vectors X = (x, x*), Y = (y, y*), by…”

Section: Let Us Definementioning

confidence: 99%

Geometric approach to non-relativistic quantum dynamics of mixed states

Gimeno

Sotoca

2013

Journal of Mathematical Physics

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In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincaré quantum recurrence in a physical system with finite energy levels. C 2013 AIP Publishing LLC. [http://dx

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$\mathfrak{D}$ -Differentiation in Hilbert Space and the Structure of Quantum Mechanics

Hurley

Vandyck

2009

Found Phys

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An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of C-and D-differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman 'path integral' and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates when space is flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect.

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Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity

Cited by 11 publications

References 7 publications

On the Measurement Problem for a Two-level Quantum System

On the Measurement Problem for a Two-level Quantum System

Geometric approach to non-relativistic quantum dynamics of mixed states

$\mathfrak{D}$ -Differentiation in Hilbert Space and the Structure of Quantum Mechanics

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