# Reparametrization Invariance and Some of the Key Properties of Physical Systems

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## Abstract

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## 1. Introduction

- 1.
- How do we construct such models?
- 2.
- What is the mathematical framework and what are the implications of such models?
- 3.
- What is the meaning/role of an arbitrarily time-parameter $\lambda $ for a particular process?

## 2. Justifying the Reparametrization Invariance (RI)

#### 2.1. Relativistic Particle Lagrangian

#### 2.2. Equivalence of Homogeneous Lagrangians

#### 2.3. Homogeneous Lagrangians of First Order

#### 2.4. Pros and Cons of Homogeneous Lagrangians of the First Order

- (1)
- First of all, the action $S=\int L(x,\frac{dx}{d\lambda})d\lambda $ is a reparametrization invariant.
- (2)
- For any Lagrangian $L(x,v=\frac{dx}{dt})$ one can construct a reparametrization-invariant Lagrangian by enlarging the configuration space $\left\{x\right\}$ to an extended configuration space—the space-time $\{ct,x\}$ [2,21,22]. However, it is an open question whether there is a full equivalence of the corresponding Euler–Lagrange equations.
- (3)
- Parameterization-independent path-integral quantization could be possible since the action S is reparametrization invariant [41].
- (4)
- The reparametrization invariance may help in dealing with singularities [26].
- (5)

- (1)
- (2)
- It follows that the Legendre transformation ($T\left(M\right)\leftrightarrow {T}^{*}\left(M\right)$), which exchanges velocity and momentum coordinates $(x,v)\leftrightarrow (x,p)$, is problematic [20].
- (3)

## 3. One-Time Physics, Causality, Arrow of Time, and the Maximum Speed of Propagation

#### 3.1. One-Time Physics, Maximum Speed of Propagation, and the Space-Time Metric Signature

- I.
- Gravity-like term $\sqrt{g(\overrightarrow{v},\overrightarrow{v})}$ is always present in the matter Lagrangian.
- II.
- The corresponding matter Lagrangian is a real-valued function.

- (1)
- No time coordinates will contradict $g(\overrightarrow{v},\overrightarrow{v})\ge 0$:$$g(\overrightarrow{v},\overrightarrow{v})=-\sum _{\alpha}{\left({v}^{\alpha}\right)}^{2}<0.$$
- (2)
- Two or more time coordinates—unconstrained spacial velocity ${\overrightarrow{v}}_{space}$:$$g(\overrightarrow{v},\overrightarrow{v})={\left({v}^{0}\right)}^{2}+{\left({v}^{1}\right)}^{2}-\sum _{\alpha =2}^{n}{\left({v}^{\alpha}\right)}^{2}\Rightarrow 1+{w}^{2}\ge {\overrightarrow{v}}_{space}^{2}.$$
- (3)
- Only one time coordinate enforces finite spacial velocity ${\overrightarrow{v}}_{space}$:$$g(\overrightarrow{v},\overrightarrow{v})={\left({v}^{0}\right)}^{2}-\sum _{\alpha =1}^{n}{\left({v}^{\alpha}\right)}^{2}\Rightarrow 1\ge {\overrightarrow{v}}_{space}^{2}.$$

#### 3.2. Causality, the Common Arrow of Time, and the Non-Negativity of the Mass

## 4. From Lagrangian to Hamiltonian Mechanics

#### 4.1. Lagrangian Mechanics

#### 4.2. Hamiltonian Formalism

#### 4.3. Problems with the Hamiltonian Function and the Legendre Transform for RI Systems

## 5. From Classical to Quantum Mechanics

#### 5.1. Canonical Quantization

#### 5.2. Extending the Poisson Bracket

#### 5.3. Implementing the Hamiltonian Constraint

#### 5.4. The Schrödinger Equation

## 6. The Meaning of $\mathit{\lambda}$ and the Role of the Hamiltonian Constraint

#### 6.1. The Picture from Lagrangian Mechanics’ Point of View

#### 6.1.1. The Proper Length Parametrization and the Onset of Quantum Length Scale

#### 6.1.2. The Proper Time Parametrization and the Onset of Quantum Time Scale

#### 6.2. The Picture from Hamiltonian Mechanics Point of View

#### 6.2.1. Hamiltonian Constraint for $\lambda $ in Coordinate-Time Role ($\lambda =t$)

#### 6.2.2. Hamiltonian Constraint for $\lambda $ in the Proper-Time Role ($\lambda =\tau $)

#### 6.2.3. The Quantum Mechanics Picture and the Positivity of the Energy

**the positivity of the norm now requires positivity of the energy**$E={p}_{0}>0$ since $\varphi \left(t\right)\to {p}_{0}$. In the rest frame this should correspond to the rest mass of the particle.

#### 6.2.4. The Rate of Change along a Coordinate and Normalizability of the Wave Function

**Now the normalizability of the wave function is related to the usual spatial localization of the physically relevant states**$\psi $ that was modulated by the factor $\sqrt{\varphi}$.

#### 6.2.5. The Notion of Time Reversal

#### 6.3. The Meaning of $\lambda $ and error in the Extended Phase-Space

## 7. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Relativistic Particle

#### Appendix A.1. Coordinate-Time Parametrization

#### Appendix A.1.1. Lagrangian Formulation

#### Appendix A.1.2. Hamiltonian Formulation

#### Appendix A.2. Proper-Time Parametrization

#### Appendix A.2.1. Lagrangian Formulation

#### Appendix A.2.2. Hamiltonian Formulation

#### Appendix A.3. Extended Hamiltonian Framework

#### Appendix A.3.1. Proper-Time Parametrization

#### Appendix A.3.2. Canonical Quantization

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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

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Gueorguiev VG, Maeder A. Reparametrization Invariance and Some of the Key Properties of Physical Systems. *Symmetry*. 2021; 13(3):522.
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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