Noether's theoremNoether's theorem is a central result in theoretical physics that expresses the equivalence of two different properties of physical laws. It is named after the early 20th century mathematician Emmy Noether.
Noether's theorem relates pairs of basic ideas of physics, one being the invariance of the form that a physical law takes with respect to any (generalized) transformation that preserves the coordinate system (both spatial and temporal aspects taken into consideration), and the other being a conservation law of a physical quantity.
Informally, Noether's theorem can be stated as:
- the invariance of physical systems with respect to translation (when simply stated, it is just that the laws of physics don't vary with location in space) translates into the law of conservation of linear momentum;
- invariance with respect to rotation gives law of conservation of angular momentum;
- invariance with respect to time gives the well known law of conservation of energy, et cetera.
Before we go on, let's give some examples:
- In classical mechanics, M is the one dimensional manifold , representing time and the target space is the tangent bundle of space of generalized positions.
- In Field Theory, M is the spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are m real-valued scalar fields, φ1,...,φm, then the target manifold is . If the field is a real vector field, then the target manifold is isomorphic to . There's actually a much more elegant way using tangent bundles over M, but for the purposes of this proof, we'd just stick to this version.
To get to the usual version of Noether's theorom, we need additional restrictions on the action. If , we assume S(φ) is the integral over M of a function of φ, its derivative and the position called the Lagrangian, . In other words,
Given boundary conditions, which is basically a specification of the value of φ at the boundary of M is compact or some limit on φ as x approaches (this will help in doing integration by parts), we can denote the subset of consisting of functions, φ such that all functional derivatives of S at φ are zero and φ satisfies the given boundary conditions.
for all compact submanifolds N. Then, we say δ is a generator of a 1-parameter symmetry Lie group.
Now, for any N, because of the Euler-Lagrange theorem, we have
Since this is true for any N, we have
You might immediately recognize this as the continuity equation for the current