Noether's theorem

Noether'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:

To every symmetry, there corresponds a conservation law and vice versa.

The formal statement of the theorem derives an expression for the physical quantity that is conserved (and hence also defines it), from the condition of invariance alone. For example:

• 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.

When it comes to quantum field theory, the invariance with respect to general gauge transformations gives the law of conservation of electric charge and so on. Thus, the result is a very important contribution to physics in general, as it helps to provide powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved, invariant.

Proof

Suppose we have an n-dimensional manifold, M and a target manifold T. Let ${\displaystyle {\mathcal {C}}}$  be the configuration space of smooth functions from M to T.

Before we go on, let's give some examples:

• In classical mechanics, M is the one-dimensional manifold ${\displaystyle \mathbb {R} }$ , 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,...,φ_m, then the target manifold is ${\displaystyle \mathbb {R} ^{m}}$ . If the field is a real vector field, then the target manifold is isomorphic to ${\displaystyle \mathbb {R} ^{n}}$ . 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.

Now suppose there is a functional

${\displaystyle S:{\mathcal {C}}\rightarrow \mathbb {R} }$ ,

called the action. (Note that it takes values in to ${\displaystyle \mathbb {R} }$ , rather than ${\displaystyle \mathbb {C} }$ ; this is for physical reasons, and doesn't really matter for this proof.)

To get to the usual version of Noether's theorem, we need additional restrictions on the action. We assume S(φ) is the integral over M of a function

${\displaystyle {\mathcal {L}}(\phi ,\partial _{\mu }\phi ,x)}$ 

called the Lagrangian, depending on φ, its derivative and the position. In other words, for φ in ${\displaystyle {\mathcal {C}}}$ 

${\displaystyle S[\phi ]\equiv \int _{M}d^{n}x{\mathcal {L}}(\phi (x),\partial _{\mu }\phi (x),x).}$ 

Suppose given boundary conditions, which are basically a specification of the value of φ at the boundary of M is compact, or some limit on φ as x approaches ${\displaystyle \infty }$ ; this will help in doing integration by parts). We can denote by N the subset of ${\displaystyle {\mathcal {C}}}$  consisting of functions, φ such that all functional derivatives of S at φ are zero and φ satisfies the given boundary conditions.

Now, suppose we have an infinitesimal transformation on ${\displaystyle {\mathcal {C}}}$ , given by a functional derivation, Q such that

${\displaystyle Q[\int _{N}d^{n}x{\mathcal {L}}]=\int _{\partial N}ds_{\mu }f^{\mu }(\phi (x),\partial \phi ,\partial \partial \phi ,...)}$ 

for all compact submanifolds N. Then, we say Q is a generator of a 1-parameter symmetry Lie group.

Now, for any N, because of the Euler-Lagrange theorem, we have

${\displaystyle Q[\int _{N}d^{n}x{\mathcal {L}}]=\int _{N}d^{n}x({\frac {\partial {\mathcal {L}}}{\partial \phi }}-\partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}})Q[\phi ]+\int _{\partial N}ds_{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]=\int _{\partial N}ds_{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ].}$ 

Since this is true for any N, we have

${\displaystyle \partial _{\mu }({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu })=0.}$ 

You might immediately recognize this as the continuity equation for the current

${\displaystyle J^{\mu }\equiv {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}Q[\phi ]-f^{\mu }}$ 

which is called the Noether current associated with the symmetry. The continuity equation tells us if we integrate this current over a space-like slice, we get a conserved quantity called the Noether charge (provided, of course, if M is noncompact, the currents fall off sufficiently fast at infinity).

This is not generally well-known, but Noether's theorem is really a reflection of the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that

${\displaystyle \int _{\partial N}ds_{\mu }J^{\mu }=0}$ 

Noether's theorem is an on shell theorem.

An example

OK, that was a general proof. Let's look at a specific case. Let's work with a one dimensional manifold with the topology of R (time) coordinatized by t. Let's assume

${\displaystyle S[x]=\int dt{\mathcal {L}}(x(t),{\dot {x}}(t))=\int dt\left({\frac {m}{2}}g_{ij}{\dot {x}}^{i}(t){\dot {x}}^{j}(t)-V(x(t))\right)}$ 

(i.e. a Newtonian particle of mass m moving in a curved Riemannian space (but not curved spacetime!) of metric g with a potential of V).

For Q, let's consider the generator of time translations. In other words, ${\displaystyle Q[x](t)={\dot {x}}(t)}$ . (Quantum field) physicists would often put a factor of i on the right hand side, but what the heck. Note that ${\displaystyle Q[{\mathcal {L}}]=mg_{ij}{\dot {x}}^{i}{\ddot {x}}^{j}-{\frac {\partial }{\partial x^{i}}}V(x){\dot {x}}^{i}}$ . This has the form of ${\displaystyle {\frac {d}{dt}}\left[{\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right]}$  so we can set ${\displaystyle f={\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)}$ . Then, ${\displaystyle j=({\frac {\partial }{\partial {\dot {x}}^{i}}}{\mathcal {L}})Q[x]-f=mg_{ij}{\dot {x}}^{j}{\dot {x}}^{i}-[{\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)]={\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}+V(x)}$ . You might recognize the right hand side as the energy and Noether's theorem states that ${\displaystyle {\dot {j}}=0}$  (i.e. the conservation of energy is a consequence of invariance under time translations).

External links

• Article on Noether's theorem by John Baez