# 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 $\lambda$.