# Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

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## Abstract

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## 1. Introduction

## 2. Geometrical Setting

**Hamilton stationary action principle**: a configuration $\sigma $ is critical iff for any compact m-region $D\subset M$ and deformation X with $\mathrm{supp}\left(X\right)\subset D$ one has

#### 2.1. Field Equations

## 3. Noether Theorem

## 4. Applications to Mechanics

- i)
- a symmetry that is a vertical vector field on the configuration bundle, i.e., $\mathsf{\Xi}={\xi}^{i}\left(q\right){\partial}_{i}$ such that its tangent prolongation $\widehat{\mathsf{\Xi}}$ leaves the Lagrangian invariant ($\widehat{\mathsf{\Xi}}\left(L\right)=0$);
- ii)
- a symmetry that is a (non-vertical) vector field $\mathsf{\Xi}$ on the configuration bundle, e.g., if the Lagrangian density is independent of time t then $\mathsf{\Xi}={\partial}_{t}$ is in fact a symmetry;
- iii)
- vector fields that leave essentially invariant the system (e.g., when in the notation introduced above one has $\alpha \ne 0$);
- iv)
- generalized vector fields to obtain first integrals that are not simply linear in the momenta, as for example the Runge-Lenz vector in Kepler’s motion.

#### 4.1. Case (i)

#### 4.2. Case (ii)

#### 4.3. Case (iii)

#### 4.4. Case (iv)

## 5. GR and Natural Theories

- i)
- covariant conservation laws are not conservation laws;
- ii)
- covariant conserved quantities would not depend on the observer as physically expected and as pseudotensorial prescriptions do.

#### 5.1. Covariant Conservation Laws

#### 5.2. Observers

#### 5.3. Extended Theories of Gravitation

## 6. Gauge Theories

#### 6.1. Yang–Mills Theory

#### 6.2. Hole Argument

## 7. Frame-Affine Formalism for GR

## 8. Conclusions and Perspectives

## Acknowledgments

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**MDPI and ACS Style**

Fatibene, L.; Francaviglia, M.; Mercadante, S.
Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics. *Symmetry* **2010**, *2*, 970-998.
https://doi.org/10.3390/sym2020970

**AMA Style**

Fatibene L, Francaviglia M, Mercadante S.
Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics. *Symmetry*. 2010; 2(2):970-998.
https://doi.org/10.3390/sym2020970

**Chicago/Turabian Style**

Fatibene, Lorenzo, Mauro Francaviglia, and Silvio Mercadante.
2010. "Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics" *Symmetry* 2, no. 2: 970-998.
https://doi.org/10.3390/sym2020970