Tensorial description of quantum mechanics

Clemente-Gallardo, Jesús; Marmo, Giuseppe

doi:10.1088/0031-8949/2013/t153/014012

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…systems whose evolution is influenced by an external environment whose properties can only, at most, be averaged. Following the description of Hilbert and projective complex spaces in geometrical terms, this work aims to characterise geometrically the pure and mixed states of quantum systems, by extending some previous work done by some of us [7,[15][16][17], where a tensorial description of the set of observables O (and its dual space O * ) was considered. Density matrices can be identified as C-linear functionals acting on a complex unital C*-algebra A, whose real elements represent the physical observables in the Heisenberg picture.…”

Section: Introductionmentioning

confidence: 99%

Tensorial dynamics on the space of quantum states

Cariñena¹,

Clemente-Gallardo²,

Jover-Galtier³

et al. 2017

J. Phys. A: Math. Theor.

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A geometric description of the space of states of a finite-dimensional quantum system and of the Markovian evolution associated with the Kossakowski-Lindblad operator is presented. This geometric setting is based on two composition laws on the space of observables defined by a pair of contravariant tensor fields. The first one is a Poisson tensor field that encodes the commutator product and allows us to develop a Hamiltonian mechanics. The other tensor field is symmetric, encodes the Jordan product and provides the variances and covariances of measures associated with the observables. This tensorial formulation of quantum systems is able to describe, in a natural way, the Markovian dynamical evolution as a vector field on the space of states. Therefore, it is possible to consider dynamical effects on non-linear physical quantities, such as entropies, purity and concurrence. In particular, in this work the tensorial formulation is used to consider the dynamical evolution of the symmetric and skewsymmetric tensors and to read off the corresponding limits as giving rise to a contraction of the initial Jordan and Lie products.

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Section: Introductionmentioning

confidence: 99%

Tensorial dynamics on the space of quantum states

Cariñena¹,

Clemente-Gallardo²,

Jover-Galtier³

et al. 2017

J. Phys. A: Math. Theor.

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“…It is reasonable to expect that what will eventually emerge will entail a substantial revision of both theories, which we may expect, will emerge merely as approximations to some underlying new theory which encompasses them. [1] Other motivation for our geometrical formulation is the following. There is no physical principle that ensures that the metric tensor has to be symmetric, g µν = g νµ ; this condition is assumed a priori without a physical justification.…”

Section: Introductionmentioning

confidence: 99%

Implications of nonsymmetric metric theories for particle physics: New interpretation of the Pauli coupling

Teruel

2014

Mod. Phys. Lett. A

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In this work we provide a possible geometrical interpretation of the spin of elementary particles. In particular, it is investigated how the wave equations of matter are altered by the addition of an antisymmetric contribution to the metric tensor. In this scenario the explicit form of the matter wave equations is investigated in a general curved space-time, and then the equations are particularized to the flat case. Unlike traditional approaches of NGT, in which the gravitational field is responsible for breaking the symmetry of the flat Minkowski metric, we find more natural to consider that, in general, the metric of the space-time could be nonsymmetric even in the flat case. The physical consequences of this assumption are explored in detail. Interestingly enough, it is found that the metric tensor splits into a bosonic and a fermionic; the antisymmetric part of the metric is very sensitive to the spin and turns out to be undetectable for spinless scalar particles. However, fermions couple to it in a non-trivial way (only when there are interactions). In addition, the Pauli coupling is derived automatically as a consequence of the nonsymmetric nature of the metric

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Hybrid Koopman C∗ –formalism and the hybrid quantum–classical master equation ^*

Bouthelier-Madre,

Clemente-Gallardo,

González-Bravo

et al. 2023

J. Phys. A: Math. Theor.

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Based on Koopman formalism for classical statistical mechanics, we propose a formalism to define hybrid quantum-classical dynamical systems by defining (outer) automorphisms of the C ∗ algebra of hybrid operators and realizing them as linear transformations on the space of hybrid states. These hybrid states are represented as density matrices on the Hilbert space obtained from the hybrid C*-algebra by the GNS theorem. We also classify all possible dynamical systems which are unitary and obtain the possible hybrid Hamiltonian operators.

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Tensorial description of quantum mechanics

Cited by 3 publications

References 11 publications

Tensorial dynamics on the space of quantum states

Tensorial dynamics on the space of quantum states

Implications of nonsymmetric metric theories for particle physics: New interpretation of the Pauli coupling

Hybrid Koopman C∗ –formalism and the hybrid quantum–classical master equation ^*

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Tensorial description of quantum mechanics

Cited by 3 publications

References 11 publications

Tensorial dynamics on the space of quantum states

Tensorial dynamics on the space of quantum states

Implications of nonsymmetric metric theories for particle physics: New interpretation of the Pauli coupling

Hybrid Koopman C∗ –formalism and the hybrid quantum–classical master equation *

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Product

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Hybrid Koopman C∗ –formalism and the hybrid quantum–classical master equation ^*