Reparametrization-invariant formulation of classical mechanics and the Schr?dinger equation

摘要

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? = p_t+H, where H is the usual Hamiltonian and p_t is the momentum conjugate to the variable t. Due to reparametrization invariance, H? vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property ?ψ= 0, which is the time-dependent Schr?dinger equation, ih?_tψ=?ψ. We discuss the quantum mechanics of a relativistic particle as an example.