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Article

Reparametrization Invariance and Some of the Key Properties of Physical Systems

1
Institute for Advanced Physical Studies, Sofia 1784, Bulgaria
2
Ronin Institute for Independent Scholarship, 127 Haddon Pl., Montclair, NJ 07043, USA
3
Geneva Observatory, University of Geneva, Chemin des Maillettes 51, CH-1290 Sauverny, Switzerland
*
Author to whom correspondence should be addressed.
Academic Editor: Rami Ahmad El-Nabulsi
Symmetry 2021, 13(3), 522; https://doi.org/10.3390/sym13030522
Received: 28 February 2021 / Revised: 17 March 2021 / Accepted: 22 March 2021 / Published: 23 March 2021
In this paper, we argue in favor of first-order homogeneous Lagrangians in the velocities. The relevant form of such Lagrangians is discussed and justified physically and geometrically. Such Lagrangian systems possess Reparametrization Invariance (RI) and explain the observed common Arrow of Time as related to the non-negative mass for physical particles. The extended Hamiltonian formulation, which is generally covariant and applicable to reparametrization-invariant systems, is emphasized. The connection between the explicit form of the extended Hamiltonian H and the meaning of the process parameter λ is illustrated. The corresponding extended Hamiltonian H defines the classical phase space-time of the system via the Hamiltonian constraint H=0 and guarantees that the Classical Hamiltonian H corresponds to p0—the energy of the particle when the coordinate time parametrization is chosen. The Schrödinger’s equation and the principle of superposition of quantum states emerge naturally. A connection is demonstrated between the positivity of the energy E=cp0>0 and the normalizability of the wave function by using the extended Hamiltonian that is relevant for the proper-time parametrization. View Full-Text
Keywords: diffeomorphism invariant systems; reparametrization-invariant systems; Hamiltonian constraint; homogeneous singular Lagrangians; generally covariant theory; equivalence of the Lagrangian and Hamiltonian framework; Canonical Quantization formalism; extended phase-space; extended Hamiltonian framework; proper time and proper length; relativistic Hamiltonian framework; relativistic particle; Minkowski space-time physical reality; common Arrow of Time; non-negativity of the mass of particles; positivity of the rest energy; Schrodinger’s equation; wave-function normalization; superposition principle in Quantum Mechanics diffeomorphism invariant systems; reparametrization-invariant systems; Hamiltonian constraint; homogeneous singular Lagrangians; generally covariant theory; equivalence of the Lagrangian and Hamiltonian framework; Canonical Quantization formalism; extended phase-space; extended Hamiltonian framework; proper time and proper length; relativistic Hamiltonian framework; relativistic particle; Minkowski space-time physical reality; common Arrow of Time; non-negativity of the mass of particles; positivity of the rest energy; Schrodinger’s equation; wave-function normalization; superposition principle in Quantum Mechanics
MDPI and ACS Style

Gueorguiev, V.G.; Maeder, A. Reparametrization Invariance and Some of the Key Properties of Physical Systems. Symmetry 2021, 13, 522. https://doi.org/10.3390/sym13030522

AMA Style

Gueorguiev VG, Maeder A. Reparametrization Invariance and Some of the Key Properties of Physical Systems. Symmetry. 2021; 13(3):522. https://doi.org/10.3390/sym13030522

Chicago/Turabian Style

Gueorguiev, Vesselin G., and Andre Maeder. 2021. "Reparametrization Invariance and Some of the Key Properties of Physical Systems" Symmetry 13, no. 3: 522. https://doi.org/10.3390/sym13030522

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