# The Role of Time in Reparametrization-Invariant Systems. (arXiv:1903.02483v2 [math-ph] UPDATED)

The relativistic particle Lagrangian justifies the importance of

Reparametrization-Invariant Systems - in particular, the first-order

homogeneous Lagrangians in the velocities. Such systems are studied from the

point of view of the Lagrangian and extended Hamiltonian formalism. The

extended Hamiltonian formulation is using an extended Poisson bracket that is

generally co-variant and applicable to reparametrization-invariant systems. The

extended Poisson bracket is defined over the phase-space-time and includes the

coordinate time and the energy in a way consistent with the Canonical

Quantization formalism. The corresponding extended Hamiltonian $\boldsymbol{H}$

defines the classical phase space-time of the system via the Hamiltonian

constraint $\boldsymbol{H}= 0$ and guarantees that the Classical Hamiltonian

$H$ corresponds to $p_0$ - the energy of the particle when the parametrization

$\lambda = t$ is chosen. When the extended Hamiltonian for a classical system

is quantized ($\boldsymbol{H}\rightarrow\boldsymbol{\hat{H}}$) by following the

Canonical Quantization formalism and the corresponding Hilbert space is defined

by the extended quantum Hamiltonian so that $\boldsymbol{\hat{H}}{\Psi} = 0$

then the Schrdinger's equation emerges naturally. The usual gravitational term

and the principle of finite speed justify the Minkowski space-time physical

reality with only one time-like coordinate. The causal structure and the

freedom of time-like parametrization for a process justify the observed common

Arrow of Time and the non-negative masses of particles. A connection has been

demonstrated between the positivity of the energy and the normalizability of

the wave function by using the extended Hamiltonian that is relevant for the

proper-time parametrization. The choice of the extended Hamiltonian

$\boldsymbol{H}$ is closely related to the meaning of the process time

parameter $\lambda$.