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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Physical consequences of the choice of the Lagrangian

Nivaldo Agostinho Lemos

Published Aug 15, 1981• 33 citations
Publisher

Abstract

The physical consequences of the existence of inequivalent Lagrangians associated with a given equation of motion are examined in quantum mechanics. It is shown that in the case of conservative systems one additional condition is sufficient to select the physically correct Lagrangian. It is remarked that in the case of dissipative systems the situation is quite different, and no conditions are known to single out a unique Lagrangian.

DOI
10.1103/physrevd.24.1036

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The physical consequences of the existence of inequivalent Lagrangians associated with a given equation of motion are examined in quantum mechanics. It is shown that in the case of conservative systems one additional condition is sufficient to select the physically correct Lagrangian. It is remarked that in the case of dissipative systems the situation is quite different, and no conditions are known to single out a unique Lagrangian.
A Lagrangian that yields equations of motion for a damped simple harmonic oscillator is shown not to describe this system, but a completely different physical system. Thus, we have a simple counterexample to the statement: ’’If a Lagrangian gives the correct equations of motion for a given system then the Lagrangian describes the system.’’ We also construct a physical system that the Lagrangian describes and derive some of its properties. The quantum theory based on the above Lagrangian has been used to attempt to discuss the quantum theory of dissipation. This is, however, incorrect since the system the Lagrangian describes has no dissipation.
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.
One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
In this article, I use a number of remarks made by Eugene Wigner to defend the claim that the nature of the connection between symmetries and conservation laws is different in quantum and in classical mechanics. In particular, I provide a list of three differences that obtain between the Hilbert space formulation of quantum mechanics and the Lagrangian formulation of classical mechanics. I also show that these differences are due to the fact that conservation laws are not the only consequence that symmetries have in quantum mechanics and to the fact that, in classical mechanics, the connection between symmetries and conservation laws does not always obtain.
This work starts with classical equations of motion and sets very general quantization conditions (commutation relations). It is proved that these conditions imply that the equations of motion are equivalent to the Euler–Lagrange equations of a Lagrangian L. The result is a generalization of work by Feynman, recently reported by Dyson [Am. J. Phys. 58, 209–211 (1990)]. The Lagrangian L need not be unique. Examples are given, including classical equations that do not come from a Lagrangian and therefore cannot be quantized consistently.
Abstract Refusal to use inertial reference frames in favor of non-inertial reference frames means significant changes in the axiomatics of both classical and quantum physics. Taking into account the influences of random small forces and fields makes the equations of classical physics, expanding, subject not only to deterministic laws, but also expands the equations of classical physics with additional variables that have an indeterministic, probabilistic behavior. Extending the scope of consideration of physical systems to non-inertial frames of reference leads to the already well-known new field of study known as quantum correction of Newton’s laws of motion. In this case, not only stable deterministic trajectories are considered, but also unstable, random ones, which usually constitute the measurement error in experiments in cases of applying classical physics and the probability distribution of quantum mechanics. Such a quantum correction can extend the action of Newton’s laws to the microworld by abandoning the basic idea of classical physics that physical systems should be described by differential equations of no higher than second order in favor of supplementing them with higher derivative variables that describe unstable random trajectories.