The Schwinger action principle for classical systems*

Manjarres, A. D. Bermúdez

doi:10.1088/1751-8121/ac2321

Cited by 6 publications

(7 citation statements)

References 70 publications

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“…In this section, we show two methods to obtain the Heisenberg equation of motion for the position and velocity operators from the variation of an action operator. The results given here are an expansion of the ones given in [35].…”

Section: Action Principlesmentioning

confidence: 92%

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Operational classical mechanics: holonomic systems

Manjarres¹

2022

J. Phys. A: Math. Theor.

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We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, we rediscover several results from analytical mechanics from an entirely new perspective. We start by expressing the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of Holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D'Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position-velocity to a position-momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman-von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized to include holonomic constraints.

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Section: Action Principlesmentioning

confidence: 92%

“…In section 6 we derive the Lagrange equation of motion from the variation of the so-called Schwinger action, extending previous work on the Schwinger action principle [35] to systems with holonomic constraints. We comment on the possible relation of the Schwinger action principle with the Gauss principle of least constraints.…”

Section: Introductionmentioning

confidence: 93%

Operational classical mechanics: holonomic systems

Manjarres¹

2022

J. Phys. A: Math. Theor.

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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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“…However, its utility is limited in comparison to the classical counterpart because the phase space variables are merely formal parameters, half of which are implicitly defined and are hard to compute. In contrast, the Feynman's path integral approach [7], which is closely related to the Schwinger quantum action principle [8][9][10], can be specialized in terms of conventional phase space variables and is useful in a plethora of applications, from chemistry to quantum gravitation [11][12][13][14][15][16]. However, this approach involves the trajectory-selecting variational analysis only at the level of the semiclassical stationary phase approximation.…”

Section: Introductionmentioning

confidence: 99%

Joint quantum–classical Hamilton variational principle in the phase space*

Zhdanov¹,

Bondar²

2022

J. Phys. A: Math. Theor.

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We show that the dynamics of a closed quantum system obeys the Hamilton variational principle. Even though quantum particles lack well-deﬁned trajectories, their evolution in the Husimi representation can be treated as a ﬂow of multidimensional probability ﬂuid in the phase space. By introducing the classical counterpart of the Husimi representation in a close analogy to the Koopman-von Neumann theory, one can largely unify the formulations of classical and quantum dynamics. We prove that the motions of elementary parcels of both classical and quantum Husimi ﬂuid obey the Hamilton variational principle, and the diﬀerences between associated action functionals stem from the diﬀerences between classical and quantum pure states. The Husimi action functionals are not unique and deﬁned up to the Skodje ﬂux gauge ﬁxing [Skodje R. T. et al. Phys. Rev. A 40, 2894 (1989)]. We demonstrate that the gauge choice can dramatically alter ﬂux trajectories. Applications of the presented theory for constructing semiclassical approximations and hybrid classical-quantum theories are discussed.

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The Schwinger action principle for classical systems*

Cited by 6 publications

References 70 publications

Operational classical mechanics: holonomic systems

Operational classical mechanics: holonomic systems

The wave operator representation of quantum and classical dynamics

Joint quantum–classical Hamilton variational principle in the phase space*

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