Contact manifolds and dissipation, classical and quantum

Ciaglia, Florio M.; Cruz, Hans; Marmo, Giuseppe

doi:10.1016/j.aop.2018.09.012

Cited by 65 publications

(82 citation statements)

References 27 publications

(54 reference statements)

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“…We will now exploit the Jordan–Lie-algebra structure of

introduced above to obtain geometric tensor fields on

, specifically, we obtain a symmetric, contravariant bivector field

associated with the Jordan product

, and a Poisson bivector field

associated with the Lie product

. This is the generalization to a generic (finite-dimensional)

-algebra

of what is done in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ] for the specific case

for a finite-dimensional Hilbert space

. Then, we will show how the manifolds of positive linear functionals introduced in the previous section may be interpreted as a sort of analogs of symplectic leaves for the symmetric tensor

in a sense that will be specified later.…”

Section: From the Jordan Product To Riemannian Geometriesmentioning

confidence: 99%

“…From a purely theoretical point of view, there is no need to restrict our attention to geometrical structures on quantum states associated with covariant tensor fields as in the case of metric tensors discussed above. Indeed, in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ], the associative product of the algebra

of linear operators on the finite-dimensional Hilbert space

associated with a quantum system has been suitably exploited to define two contravariant tensor fields on the space of self-adjoint operators on

, and these tensor fields have been used to give a geometrical description of the Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation describing the dynamical evolution of open quantum systems (see [ 19 , 21 , 22 , 24 , 26 , 27 , 28 ]). These two tensor fields, named

and

, are associated with the antisymmetric part (the Lie product) and the symmetric part (the Jordan product) of the associative product in

, respectively.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

Ciaglia

Jost

Schwachhöfer

2020

Entropy

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The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher–Rao metric tensor is recovered in the Abelian case, that the Fubini–Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures–Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on B ( H ) , this alternative geometrical description clarifies the analogy between the Fubini–Study and the Bures–Helstrom metric tensor.

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“…We will now exploit the Jordan–Lie-algebra structure of

introduced above to obtain geometric tensor fields on

, specifically, we obtain a symmetric, contravariant bivector field

associated with the Jordan product

, and a Poisson bivector field

associated with the Lie product

. This is the generalization to a generic (finite-dimensional)

-algebra

of what is done in [ 19 , 20 , 21 , 22 , 23 , 24 , 25 ] for the specific case

for a finite-dimensional Hilbert space

. Then, we will show how the manifolds of positive linear functionals introduced in the previous section may be interpreted as a sort of analogs of symplectic leaves for the symmetric tensor

in a sense that will be specified later.…”

Section: From the Jordan Product To Riemannian Geometriesmentioning

confidence: 99%

of linear operators on the finite-dimensional Hilbert space

associated with a quantum system has been suitably exploited to define two contravariant tensor fields on the space of self-adjoint operators on

and

, are associated with the antisymmetric part (the Lie product) and the symmetric part (the Jordan product) of the associative product in

, respectively.…”

Section: Introductionmentioning

confidence: 99%

From the Jordan Product to Riemannian Geometries on Classical and Quantum States

Ciaglia

Jost

Schwachhöfer

2020

Entropy

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“…If we restrict our attention to solutions to the generalised Euler-Lagrange equations, but allow variations of the endpoint, equation (12) reduces to…”

Section: Herglotz' Variational Principlementioning

confidence: 99%

“…have dimension at most n. Here, borrowing the terminology from [12], we restrict to the following important case.…”

Section: Introductionmentioning

confidence: 99%

Numerical integration in Celestial Mechanics: a case for contact geometry

Bravetti

Seri

Vermeeren

et al. 2020

Celest Mech Dyn Astr

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Several dynamical systems of interest in celestial mechanics can be written in the formqFor instance, the modified Kepler problem, the spin-orbit model and the Lane-Emden equation all belong to this class. In this work we start an investigation of these models from the point of view of contact geometry. In particular we focus on the (contact) Hamiltonisation of these models and on the construction of the corresponding geometric integrators.

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“…-Dissipative systems encompass a wide range of physical research areas, such as general relativity (GR), fluid dynamics, statistical mechanics, and quantum mechanics. Depending on the context, dissipation configures as a general mechanism through which a physical system may partially (or even completely) waste during time evolution its initial energy, entropy, entanglement, and so on [1,2]. Dissipative contributions are fundamental expedients to make a model more realistic, even though the mathematical structure becomes more and more muddled.…”

mentioning

confidence: 99%