Unified description of classical and quantum behaviours in a variational principle

2020

Annals of Physics

We revisit the problem of the uncertainty relation for angle by using quantum hydrodynamics formulated in the stochastic variational method (SVM), where we need not define the angle operator. We derive both the Kennard and Robertson-Schrödinger inequalities for canonical variables in polar coordinates. The inequalities have state-dependent minimum values which can be smaller than /2 and then permit a finite uncertainty of angle for the eigenstate of the angular momentum. The present approach provides a useful methodology to study quantum behaviors in arbitrary canonical coordinates.

Section: Discussionmentioning

confidence: 99%

Uncertainty relation for angle from a quantum-hydrodynamical perspective

Gazeau

2020

Annals of Physics

“…For the σ-algebra of all measurable events of x t , P t and F t represent an increasing and a decreasing family of sub-σ-algebras, respectively. These derivatives are connected through the stochastic partial integration [45],…”

Section: A Forward and Backward Stochastic Differential Equationsmentioning

confidence: 99%

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

J. Phys. A: Math. Theor.

Kodama

2020

Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, differently from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier-Stokes-Fourier (NSF) equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gaugeinvariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.

“…In the formulation of the variational principle, we should fix not only an initial condition but also a final condition. This implies that the forward SDE alone is not sufficient [2,3]. We further introduce the backward SDE for dt < 0 as…”

Section: Formulationmentioning

confidence: 99%

“…For the σ-algebra of all measurable events ofx t , P t and F t represent an increasing and a decreasing family of subσ-algebras, respectively. These derivatives are connected through the stochastic partial integration [3],…”

Section: Formulationmentioning

confidence: 99%

See 1 more Smart Citation

Novel effect induced by spacetime curvature in quantum hydrodynamics

Kodama

2019

Physics Letters A

The interplay between quantum fluctuation and spacetime curvature is shown to induce an additional quantum-curvature (QC) term in the energy-momentum tensor of fluid using the generalized framework of the stochastic variational method (SVM). The QC term is necessary to satisfy the momentum conservation but the corresponding quantum hydrodynamics is not necessarily cast into the form of the Schrödinger equation, differently from the case of the Euclidean spacetime. This seems to suggest that the existence of the Hilbert space is not a priori requirement in the quantization of curved spacetime systems. As an example, we apply the Friedmann-Robertson-Walker (FRW) metric and show that this term contributes to the cosmological acceleration although it is too small in the present non-relativistic toy model.