Classical mechanics in reparametrization-invariant formulation and the Schrödinger equation

Abstract

The dynamics of any classical-mechanics system can be formulated in the reparametrization-invariant (RI) form (that is we use the parametric representation for trajectories, $\mathbf{x}=\mathbf{x}(\tau )$, $t=t(\tau )$ instead of $\mathbf{x}=\mathbf{x}(t)$). In this pedagogical note we discuss what the quantization rules look like for the RI formulation of mechanics. We point out that in this case some of the rules acquire an intuitively clearer form. Hence the formulation could be an alternative starting point for teaching the basic principles of quantum mechanics. The advantages can be resumed as follows. a) In RI formulation both the temporal and the spatial coordinates are subject to quantization. b) The canonical Hamiltonian of RI formulation is proportional to the quantity $\tilde{H}=p_t+H$, where $H$ is the Hamiltonian of the initial formulation. Due to the reparametrization invariance, the quantity $\tilde{H}$ vanishes for any solution, $\tilde{H}=0$. So the corresponding quantum-mechanical operator annihilates the wave function, $\hat{\tilde{H}}\Psi =0$, which is precisely the Schrödinger equation $i\hslash {\partial }_t\Psi =\hat{H}\Psi$. As an illustration, we discuss quantum mechanics of the relativistic particle.

5 pages, Two references added, Misprints corrected

References in corpus (3)

• The Time-Energy Uncertainty Relation
• Improved extended Hamiltonian and search for local symmetries
• Formal similarity between mathematical structures of electrodynamics and quantum mechanics

Cited by in corpus (9)

• Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces
• Reparametrization Invariance and Some of the Key Properties of Physical Systems
• Quantum Space, Quantum Time, and Relativistic Quantum Mechanics
• Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase
• Symmetries and Covariant Poisson brackets on pre-symplectic manifolds
• Non-Grassmann mechanical model of the Dirac equation
• Classical gauge principle -- From field theories to classical mechanics
• Gauge Invariance for Classical Massless Particles with Spin
• Particle Dynamics and Lie-algebraic type of Non-commutativity of space-time