Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation

@article{Deriglazov2011ReparametrizationinvariantFO,
  title={Reparametrization-invariant formulation of classical mechanics and the Schr{\"o}dinger equation},
  author={Alexei A. Deriglazov and Bruno Rizzuti},
  journal={American Journal of Physics},
  year={2011},
  volume={79},
  pages={882-885}
}
Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x=x(τ) and t=t(τ) instead of x=x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H=pt+H, where H is… 

Reparametrization Invariance and Some of the Key Properties of Physical Systems

By using the freedom of parametrization for a process, it is argued that the corresponding causal structure results in the observed common Arrow of Time and nonnegative masses of the particles.

Simple derivation of Schrodinger equation from Newtonian dynamics

The Eherenfest theorem states that Schrodinger representation of quantum mechanics (wave mechanics) reproduces Newton laws of motion in terms of expectation values. Remarkably, the contrary is

Non-Grassmann mechanical model of the Dirac equation

We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both Γμ

Quantum space, quantum time, and relativistic quantum mechanics

  • Ashmeet Singh
  • Physics
    Quantum Studies: Mathematics and Foundations
  • 2021
We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear,

A variational principle, wave-particle duality, and the Schr\"{o}dinger equation

. A principle is proposed according to which the dynamics of a quantum particle in a one-dimensional configuration space (OCS) is determined by a variational problem for two functionals: one is based

Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a

Classical Gauge Principle - From Field Theories to Classical Mechanics

In this paper we discuss how the gauge principle can be applied to classical-mechanics models with finite degrees of freedom. The local invariance of a model is understood as its invariance under the

Gauge Invariance for Classical Massless Particles with Spin

Wigner’s quantum-mechanical classification of particle-types in terms of irreducible representations of the Poincaré group has a classical analogue, which we extend in this paper. We study the

A Perspicuous Description of the Schwarzschild Black Hole Geodesics

Schwarzschild black hole is the simplest black hole that is studied most in detail. Its behavior is best understood by looking at the geodesics of the particles under the influence of its

References

SHOWING 1-10 OF 22 REFERENCES

LOCAL SYMMETRIES AND THE NOETHER IDENTITIES IN THE HAMILTONIAN FRAMEWORK

We study in the Hamiltonian framework the local transformations which leave invariant the Lagrangian action: δeS=div. Manifest form of the symmetry and the corresponding Noether identities is

Local symmetries in the Hamiltonian framework. 1. Hamiltonian form of the symmetries and the Noether identities

We study in the Hamiltonian framework the local transformations δ ǫ q A ( τ ) = P [ k ] k =0 ∂ kτ ǫ a R ( k ) aA ( q B , ˙ q C ) which leave invariant the Lagrangian action: δ ǫ S = div . Manifest

Quantization of Gauge Systems

This is a systematic study of the classical and quantum theories of gauge systems. It starts with Dirac's analysis showing that gauge theories are constrained Hamiltonian systems. The classical

Formal Similarity Between Mathematical Structures of Electrodynamics and Quantum Mechanics

Electromagnetic phenomena can be described by Maxwell equations written for the vectors of electric and magnetic field. Equivalently, electrodynamics can be reformulated in terms of an

Quantization of Fields with Constraints

The quantization of singular field theories, in particular, gauge theories, is one of the key problems in quantum field theory. This book - which addresses the reader acquainted with the foundations

Improved extended Hamiltonian and search for local symmetries

We analyze a structure of the singular Lagrangian $L$ with first and second class constraints of an arbitrary stage. We show that there exist an equivalent Lagrangian (called the extended Lagrangian

Potential motion in a geometric setting: presenting differential geometry methods in a classical mechanics course

The standard classical mechanics textbooks used at graduate level mention geometrization of the potential motion kinematics. We show that the complete problem can also be geometrized, presenting the

Theory of Particles with Variable Mass. I. Formalism

The equivalence principle (through the mechanism of the gravitational red shift) allows one to set up a classical formalism with the proper time as an extra degree of freedom, independent of the

Generalized Hamiltonian dynamics

  • P. Dirac
  • Physics
    Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1958
The author’s procedure for passing from the Lagrangian to the Hamiltonian when the momenta are not independent functions of the velocities is put into a simpler and more practical form, the main