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arXiv:1509.02442 · Foundations of Physics

Lagrangian Description for Particle Interpretations of Quantum Mechanics: Entangled Many-Particle Case

Roderick I. Sutherland

Published Nov 4, 2016• 17 citations
arXivHTMLPDF
DOI
10.1007/s10701-016-0043-6

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A Lagrangian description is presented here that can be used in conjunction with a particle interpretation of quantum mechanics. The advantages of this description are that the field equations and the particle equations of motion can both be deduced from a single Lagrangian density expression and that conservation of energy and momentum are assured. This Lagrangian formulation is then applied to the special case of the well-known Bohm model as an example. It is thereby demonstrated that such a formulation is compatible with the predictions of quantum mechanics.
Classical mechanics, which is a deterministic theory, was developed largely with the work of Newton. Later, the Lagrangian approach of obtaining stationary paths of an action was introduced to obtain the Newtonian results. Thus a deterministic theory was considered in the interpretation of varying paths in the action with the stationary path being the classical mechanical result. In (1) we have argued that both relativistic and nonrelativistic quantum mechanics follow from varying x and t independently in v= x/t = constant in the classical action for a free particle with constant momentum. In this case, it seems one considers a fluctuating distance or a stochastic approach, yet in this note we argue that for dAction/dx =p for the quantum case is very similar to dL/dv = p for the classical. It is the interpretation which seems to differ. We argue that an energy flow through space i.e. a p in quantum mechanics seems to follow from a certain kind of spatial probability fluctuation i.e. a periodic one. On average the free particle case and also the bound state case yield classical results so the Lagrangian approach of classical mechanics is linked to quantum rms values.
We discuss the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler-Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie-Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie-Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.