Noether’s Theorem

Table of Contents

1. Introduction
2. Historical Background
3. Symmetry in Physics
4. Continuous Symmetries and Conservation Laws
5. Statement of Noether’s Theorem
6. Mathematical Framework
7. Derivation from the Lagrangian Formalism
8. Examples of Noether Currents
9. Time Translation and Energy Conservation
10. Spatial Translation and Momentum Conservation
11. Rotational Symmetry and Angular Momentum
12. Gauge Symmetry and Charge Conservation
13. Application in Classical Field Theory
14. Application in Quantum Field Theory
15. Local vs Global Symmetries
16. Conserved Currents and Charges
17. Noether’s Second Theorem
18. Anomalies and Broken Symmetries
19. Role in Modern Theoretical Physics
20. Conclusion

1. Introduction

Noether’s theorem is one of the most profound results in theoretical physics and mathematics. It connects symmetries of physical systems to conservation laws, providing deep insights into the fundamental structure of nature.

2. Historical Background

Formulated by Emmy Noether in 1915 and published in 1918, the theorem was a response to challenges in general relativity. It established a systematic connection between symmetries and conserved quantities and has since become foundational in classical and quantum physics.

3. Symmetry in Physics

Symmetry describes invariance under transformations:

• Time translation → energy conservation
• Space translation → momentum conservation
• Rotation → angular momentum conservation

Noether’s theorem formalizes this connection for continuous symmetries.

4. Continuous Symmetries and Conservation Laws

A continuous symmetry involves transformations parameterized by a real number, such as shifting time by \( \epsilon \). Noether’s theorem shows that such a symmetry implies a corresponding conserved current.

5. Statement of Noether’s Theorem

For every continuous differentiable symmetry of the action of a physical system, there exists a corresponding conserved quantity.

This is true in both classical mechanics and field theory.

6. Mathematical Framework

Consider a Lagrangian \( \mathcal{L}(\phi, \partial_\mu \phi, x) \) and a transformation:

\[
x^\mu \rightarrow x’^\mu = x^\mu + \delta x^\mu, \quad \phi \rightarrow \phi’ = \phi + \delta \phi
\]

If the variation of the action vanishes under this transformation, then:

\[
\partial_\mu j^\mu = 0
\]

where \( j^\mu \) is the conserved Noether current.

7. Derivation from the Lagrangian Formalism

Let the variation of the Lagrangian be:

\[
\delta \mathcal{L} = \partial_\mu K^\mu
\]

Then, using the Euler-Lagrange equations:

\[
\partial_\mu j^\mu = 0, \quad \text{where } j^\mu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta \phi – K^\mu
\]

This current is conserved: the associated charge is constant in time.

8. Examples of Noether Currents

Each continuous symmetry gives rise to a Noether current \( j^\mu \) and a conserved charge:

\[
Q = \int d^3x\, j^0(x)
\]

Examples:

• Energy-momentum tensor
• Angular momentum tensor
• Electric current

9. Time Translation and Energy Conservation

If the Lagrangian does not explicitly depend on time, it is invariant under time translation. Noether’s theorem implies energy is conserved:

\[
E = \frac{\partial \mathcal{L}}{\partial \dot{q}} \dot{q} – \mathcal{L}
\]

10. Spatial Translation and Momentum Conservation

Invariance under space translation leads to momentum conservation:

\[
p_i = \frac{\partial \mathcal{L}}{\partial \dot{q}_i}
\]

Each spatial direction gives a conserved momentum component.

11. Rotational Symmetry and Angular Momentum

Rotational invariance yields conservation of angular momentum:

\[
L_i = \epsilon_{ijk} x_j p_k
\]

Associated with invariance under infinitesimal rotations.

12. Gauge Symmetry and Charge Conservation

In gauge theories like QED:

• Local U(1) symmetry leads to charge conservation
• Conserved current: \( j^\mu = \bar{\psi} \gamma^\mu \psi \)
• Conserved charge: electric charge

13. Application in Classical Field Theory

In field theory, Noether’s theorem applies to fields \( \phi(x) \) with Lagrangian density \( \mathcal{L} \). Continuous symmetries yield conserved currents:

\[
j^\mu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta \phi
\]

14. Application in Quantum Field Theory

In QFT, conserved currents play roles in:

• Defining quantum numbers
• Constructing Fock space states
• Deriving Ward identities
• Ensuring renormalizability

15. Local vs Global Symmetries

• Global symmetry: transformation same at all spacetime points → true conservation laws.
• Local symmetry: requires introduction of gauge fields; conservation law becomes a constraint (via equations of motion).

16. Conserved Currents and Charges

Given \( \partial_\mu j^\mu = 0 \), define:

\[
Q = \int d^3x\, j^0
\]

This charge is constant in time:

\[
\frac{dQ}{dt} = 0
\]

17. Noether’s Second Theorem

Applies to local symmetries:

• Results in identities between field equations
• Central in gauge theories
• E.g., Ward identities in QED

18. Anomalies and Broken Symmetries

At quantum level, some classical symmetries are broken:

• Anomalies break conservation laws (e.g., axial anomaly)
• Important in understanding symmetry breaking and consistency

19. Role in Modern Theoretical Physics

Noether’s theorem:

• Forms the basis of gauge theories
• Guides model building in particle physics
• Connects mathematical symmetry to physical observables
• Is embedded in modern approaches like string theory and supersymmetry

20. Conclusion

Noether’s theorem is a foundational result that unveils the deep connection between symmetry and conservation in physics. It applies across classical mechanics, field theory, and quantum mechanics, underpinning our understanding of fundamental forces and particles. Its mathematical elegance and physical power make it indispensable in theoretical physics.