Noether’s Theorem

Overview

Named after the German mathematician Emmy Noether, Noether’s theorem is a fundamental theorem in theoretical physics and the calculus of variations. It relates symmetries in physical systems to conservation laws, providing a deep insight into the structure of physical laws.

Basic Statement of Noether’s Theorem

In its simplest form, Noether’s theorem states that for every differentiable symmetry of the action of a physical system, there corresponds a conserved quantity. The action of a physical system is defined as the integral over time of the Lagrangian , which is a function that encodes the dynamics of the system:

Here, is the time, and the integral extends over the duration of the physical process under consideration.

Conservation Laws from Noether’s Theorem

From Noether’s theorem, it can be concluded that if a system has a continuous symmetry, then there are quantities in the system that are conserved. The most commonly encountered examples are:

1. Time invariance leads to conservation of energy.
2. Spatial invariance leads to conservation of linear momentum.
3. Rotational invariance leads to conservation of angular momentum.

This can be expressed mathematically as follows: if the Lagrangian of the system is invariant under the transformation , where are the generalized coordinates of the system and is a small parameter, then the quantity is conserved, where is the time derivative of the generalized coordinate .

Application to Field Theory

In the context of field theory, Noether’s theorem applies to both global and local symmetries. Global symmetries lead to conservation laws, as in the simpler cases mentioned above. Local symmetries, on the other hand, lead to what are known as gauge theories, which are the basis for the fundamental forces of nature in the Standard Model of particle physics.

Advanced Topics: Noether’s Second Theorem

Noether’s second theorem is a generalization of the first theorem and deals with the case when the transformations depend on arbitrary functions and their derivatives up to a certain order. It has important implications for the theories of gravity, particularly in the context of general relativity.

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