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arXiv:1801.06520 · Journal of Statistical Physics

Nonrelativistic Hydrodynamics from Quantum Field Theory: (I) Normal Fluid Composed of Spinless Schrödinger Fields

Masaru Hongo

Published Jan 28, 2019• 11 citations
arXivHTMLPDF
DOI
10.1007/s10955-019-02224-4

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We derive the quantum potential directly from the material derivative of the osmotic velocity and formulate a two-fluid model that reproduces the Madelung equations. Interactions between the two fluids are included but remain secondary. The framework is generalized to incorporate electromagnetic fields, yielding a self-consistent description of both the Schrödinger and Klein-Gordon equations. Extension to the relativistic Klein-Gordon case demonstrates the model's flexibility and applicability to spinless relativistic quantum systems. This approach unifies hydrodynamic, kinematic, and electromagnetic perspectives, providing a clear physical interpretation of quantum potentials and forces and offering a versatile platform for modeling complex quantum systems in both nonrelativistic and relativistic regimes.
We consider a Finslerian-type geometrization of the nonrelativistic quantum mechanics in its hydrodynamical (Madelung) formulation, by also taking into account the effects of the presence of the electromagnetic fields on the particle motion. In the Madelung representation, the Schr\"odinger equation can be reformulated as the classical continuity and Euler equations of classical fluid mechanics in the presence of a quantum potential, representing the quantum hydrodynamical evolution equations. The equation of particle motion can then be obtained from a Lagrangian similarly to its classical counterpart. After the reparametrization of the Lagrangian, it turns out that the Finsler metric describing the geometric properties of quantum hydrodynamics is a Kropina metric. We present and discuss in detail the metric and the geodesic equations describing the geometric properties of the quantum motion in the presence of electromagnetic fields. As an application of the obtained formalism, we consider the Zermelo navigation problem in a quantum hydrodynamical system, whose solution is given by a Kropina metric. The case of the Finsler geometrization of the quantum hydrodynamical motion of spinless particles in the absence of electromagnetic interactions is also considered in detail, and the Zermelo navigation problem for this case is also discussed.
We propose new ideal hydrodynamics in the function space which describes a fluid composed of the 1+1 dimensional real scalar field in the framework of the stochastic variational method (SVM). In the derivation, the thermal equilibrium is assumed to the internal state of fluid elements in the function space of the scalar-field configuration. The deterministic trajectory of the functional fluid element is related to the functional generalization of the Bohmian trajectory in relativistic quantum field theory. To find the correspondence relation to standard hydrodynamics, a further coarse-graining should be introduced. Thus functional hydrodynamics is regarded as a mesoscopic theory such as the Boltzmann equation in the dynamical hierarchy of many-body systems. Functional hydrodynamics reproduces the exact behaviors of relativistic quantum field theory in a certain limit. We thus expect that our theory is applicable to study the influence of quantum-field theoretical fluctuation in collective flows of produced particles in relativistic heavy-ion collisions.