Eisenhart lift of Koopman-von Neumann mechanics

Sen, Abhijit; Parida, Bikram Keshari; Dhasmana, Shailesh; Силагадзе, З. К.

doi:10.1016/j.geomphys.2022.104732

Cited by 11 publications

(3 citation statements)

References 87 publications

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“…In the case of pure states, this difficulty was resolved with the demonstration that the Wigner function should be interpreted as a phase space probability amplitude. This is in direct analogy with the Koopman-von Neumann (KvN) representation of classical dynamics [19,[21][22][23][24][25][26][27][28][29][30][31][32][33] which explicitly admits a wavefunction on phase space, and which the Wigner function of a pure state corresponds to in the classical limit. The extension of this interpretation to mixed states has to date been lacking however, given that such states must be described by densities and therefore lack a direct correspondence to wavefunctions.…”

Section: Introductionmentioning

confidence: 93%

The wave operator representation of quantum and classical dynamics

McCaul¹,

Zhdanov²,

Bondar³

2023

Preprint

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The choice of mathematical representation when describing physical systems is of great consequence, and this choice is usually determined by the properties of the problem at hand. Here we examine the little-known wave operator representation of quantum dynamics, and explore its connection to standard methods of quantum dynamics, such as the Wigner phase space function. This method takes as its central object the square root of the density matrix, and consequently enjoys several unusual advantages over standard representations. By combining this with purification techniques imported from quantum information, we are able to obtain a number of results. Not only is this formalism able to provide a natural bridge between phase and Hilbert space representations of both quantum and classical dynamics, we also find the wave operator representation leads to novel semiclassical approximations of both real and imaginary time dynamics, as well as a transparent correspondence to the classical limit. This is demonstrated via the example of quadratic and quartic Hamiltonians, while the potential extensions of the wave operator and its application to quantum-classical hybrids is discussed. We argue that the wave operator provides a new perspective that links previously unrelated representations, and is a natural candidate model for scenarios (such as hybrids) in which positivity cannot be otherwise guaranteed.

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Section: Introductionmentioning

confidence: 93%

The wave operator representation of quantum and classical dynamics

McCaul¹,

Zhdanov²,

Bondar³

2023

Preprint

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“…In this research, we expand the scope of linearization within differential geometry by incorporating a geometric approach that extends Lie symmetry analysis. Our specific focus is on a subset of autonomous dynamical systems, characterized by a set of geodesic equations derived through the application of the Eisenhart lift [28][29][30][31][32][33]. The investigation delves into the geometric properties and the symmetries inherent in the geodesic space.…”

Section: Introductionmentioning

confidence: 99%

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

Paliathanasis

2024

Axioms

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The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis.

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“…See [16] for an introduction and [17] for a detailed presentation. This elegant method has been applied to a large variety of systems, from time-dependent systems, celestial mechanics, inflation and more exotic systems [18][19][20][21][22][23][24][25][26][27], and even quantum mechanics [28]. An interesting fact about this formalism is that the lift metric is equipped with a covariantly constant null Killing vector.…”

Section: Introductionmentioning

confidence: 99%

Schrödinger Symmetry in Gravitational Mini-Superspaces

Ben Achour,

Livine,

Oriti

et al. 2023

Universe

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We prove that the simplest gravitational symmetry-reduced models describing cosmology and black hole mechanics are invariant under the Schrödinger group. We consider the flat FRW cosmology filled with a massless scalar field and the Schwarzschild black hole mechanics and construct their conserved charges using the Eisenhart–Duval (ED) lift method in order to show that they form a Schrödinger algebra. Our method illustrates how the ED lift and the more standard approach analyzing the geometry of the field space are complementary in revealing different sets of symmetries of these systems. We further identify an infinite-dimensional symmetry for those two models, generated by conserved charges organized in two copies of a Witt algebra. These extended charge algebras provide a new algebraic characterization of these homogeneous gravitational sectors. They guide the path to their quantization and open the road to non-linear extensions of quantum cosmology and quantum black hole models in terms of hydrodynamic equations in field space.

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Eisenhart lift of Koopman-von Neumann mechanics

Cited by 11 publications

References 87 publications

The wave operator representation of quantum and classical dynamics

The wave operator representation of quantum and classical dynamics

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

Schrödinger Symmetry in Gravitational Mini-Superspaces

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