Noether's Theorem: Symmetry and Conservation Laws

Noether's Theorem was formulated in 1918 by Emmy Noether, and is one of the most fundamental insights in physics: Every continuous symmetry in a physical system corresponds to a conserved quantity. This essay aims to introduce it and discuss its implications. It builds a foundation in Lagrangian mechanics, defines flows and continuous symmetries. Some initial understanding of what Symmetries are and how they are described by representation theory is beneficial.

Below you will first find the theorem, followed by some semi-informal definitions and explanations of individual parts followed by its proof and its implications.

1. Noether's Theorem

Noether's Theorem. If φ_λ is a continuous symmetry of L, then the quantity

⟨p, v(q)⟩ = ∑_α=1^f p_α v_α(q)

where p_α = ∂L/∂q̇_α is the conjugate momentum, is conserved along any solution of the Euler-Lagrange equations:

d/dt ⟨p, v(q)⟩ = 0

In simple terms: If a system has a symmetry, then there is a conserved quantity that is related to that symmetry.

2. Definitions

A Lagrangian is a special kind of function that describes how a physical system behaves. The abstract definition contains abstract coordinates q, their (partial)time derivatives q̇ and time t. The behaviour of the system is described via the Euler-Lagrange equations: d/dt (∂L/∂q̇_α) - ∂L/∂q_α = 0

which is the set of all possible positions/configurations of a system. For a single particle in 3D, it's ℝ³ and for N particles it becomes ℝ^3N - exponentially bigger. In some situations they become constrained such as in a double pendulum - it simplifies down to two angles - a torus T². In general the coordinates can be written as q = (q₁, ..., q_f) , where f is the number of degrees of freedom. And a curve t ↦ q(t) is a path through this configuration space. (A path is a continuous function from the real numbers to the configuration space, e.g. γ(t), mapping a time parameter t to a position in the configuration space. In the following, our paths will be q(t), describing how the system's configuration evolves over time given a starting point q(t=0) = q.)

A Symmetry φ_λ in the sense of Noether's theorem is a transformation that leaves the Lagrangian unchanged: L(φ_λ(q), ∂/∂t φ_λ(q), t) = L(q, q̇, t) In our context a continuous symmetry is a symmetry which holds for all λ ∈ ℝ and for every curve t ↦ q(t).

Now a flow φ_λ is a one-parameter group of transformations where φ₀ = id (the zero parameter gives the identity), and composing two transformations adds the parameters. In the language of representation theory, a flow is a representation of the additive group (ℝ, +) on the configuration space. φ_λ ∘ φ_μ = φ_λ+μ (λ, μ ∈ ℝ) One sees the 0 for the identity flow (id = φ₀) makes intuitive sense because φ_λ ∘ φ₀ = φ_λ+0 = φ_λ.

Every flow has a generating vector field. A generating vector field v(q) is defined by: v(q) = ∂/∂λ φ_λ(q) |_λ=0 It tells you the infinitesimal direction (the direction at any infinitely small change of q) of the flow for any given point q. The group property of the flow extends this to all λ via: ∂/∂λ φ_λ(q) = v(φ_λ(q)) So by construction the curve q(λ) = φ_λ(q) is the solution of the first-order ODE (Ordinary Differential Equation) dq/dλ = v(q(λ)) where the initial condition q(0) = q is given. The vector field determines the flow and vice versa. (Think about it and realize this is just a fancy way of saying that the flow is the solution to the equation of motion dq/dλ = v(q) and we use q(0) = q as our starting position.)

Careful(!): Noether's theorem only considers continuous symmetries, not discrete symmetries like spatial reflections or time reversal. These discrete symmetries produce quantum numbers used to describe observable eigenstates of quantum systems via Wigner's theorem. Discrete symmetries act as (anti)unitary operators on rays, and their eigenvalues (e.g. parity ±1) label those rays. Since the symmetry operator commutes with time evolution, a state in a ray with a given eigenvalue stays there. We will touch on this in section 9.

3. Proof

Now to prove Noether's theorem (as can be found in any textbook on classical mechanics):

d/dt (∑_α (∂L/∂q̇_α) v_α(q))

= ∑_α [d/dt(∂L/∂q̇_α)] v_α(q) + ∑_α (∂L/∂q̇_α) dv_α/dt

Using the Euler-Lagrange equations to replace d/dt(∂L/∂q̇_α) with ∂L/∂q_α, and applying the definition of the generating vector field, this becomes:

= ∑_α (∂L/∂q_α) ∂/∂λ φ_λ(q)|_λ=0 + ∑_α (∂L/∂q̇_α) ∂/∂λ d/dt φ_λ(q)|_λ=0

= d/dλ L(φ_λ(q(t)), d/dt φ_λ(q(t)), t)|_λ=0

= 0

The last line vanishes because φ_λ is a symmetry.

Note that because the proof uses the Euler-Lagrange equations, the conservation law only holds on actual physical trajectories (expressed as curves) - not on arbitrary curves.

4. The Classical Examples

4.1 Spatial Translations → Total Momentum

Take a multi-particle system, its Lagrangian is L = ∑_i ½ m_i ẋ_i² - V(x₁, ..., x_N), if the potential V depends only on relative positions (i.e., V is invariant under joint translations). The flow then becomes:

Flow: φ_λ(x₁, ..., x_N) = (x₁ + λa, ..., x_N + λa) for any direction a.

And our conserved quantity is: ∑_i m_i ẋ_i · a = P · a. And as a was arbitrary (we never specified the direction, thus it works for any direction we choose), the total momentum P is conserved.

4.2 Rotations → Angular Momentum

If L is invariant under rotations, the flow becomes φ_λ(x₁, ..., x_N) = (R(e, λ)x₁, ..., R(e, λ)x_N), with R(e, λ) being the rotation by angle λ around axis e. And our symmetry is the SO(3) group.

The vector field is given by: v(x₁, ..., x_N) = (e × x₁, ..., e × x_N).

And the conserved quantity is: ∑_i m_i ẋ_i · (e × x_i) = e · ∑_i x_i × m_i ẋ_i = e · J (we use J for angular momentum to avoid confusion with the Lagrangian L). Since e is arbitrary, the total angular momentum J is conserved.

4.3 Cyclic Coordinates → Momentum

The simplest case: if q_α is a cyclic coordinate (does not appear in L), then the flow φ_λ(q_β) = q_β + δ_βαλ (shift only the α-th coordinate) is a symmetry. The generating vector field has only one nontrivial component: v_β = δ_αβ. The conserved quantity is simply the conjugate momentum p_α.

5. Generalization

The basic theorem assumes L is strictly invariant under the flow. One can assume a more general situation where the Lagrangian is not invariant but changes only by a total time derivative:

L(φ_λ(q), ∂/∂t φ_λ(q), t) = L(q, q̇, t) + d/dt F(q, t, λ)

Such Lagrangians are called equivalent as they produce the same equations of motion (EOMs) due to the extra term only affecting the boundary conditions. This causes the conserved quantity to have a offset F:

⟨p, v(q)⟩ - δF

where δF = ∂F/∂λ|_λ=0. - The proof is the same as before, just the last line gives d/dt δF instead of zero, this is where we get this offset F to compensate for it.

6. Energy Conservation

Energy conservation requires a further generalization: the flow transforms time as well as the configuration. An extended flow acts on ℝ^f × ℝ ∋ (q, t):

(q, t) ↦ ψ_λ(q, t) = (φ_λ(q, t), τ_λ(q, t))

with generating vector field (v(q, t), δτ(q, t)). The conserved quantity becomes:

K = ⟨p, v(q)⟩ - (⟨p, q̇⟩ - L) δτ - δF

6.1 Time Translation → Energy

For a time-independent Lagrangian (autonomous system), the pure time translation φ_λ = id, τ_λ(t) = t + λ is a symmetry. Here v = 0, δτ = 1, F = 0. Which leads to the conserved quantity:

K = L - ∑_α (∂L/∂q̇_α) q̇_α = -E

(the sign is up to convention). For a system with kinetic energy T = ½ ∑ G_αβ(q) q̇_α q̇_β and potential V = V(q), this gives E = T + V, the total energy.

6.2 Galilei Boosts → Center of Mass

Now briefly brushing over the Galilei boosts and center of mass (COM). A Galilei boost is a transformation that moves the system uniformly through space-time. There are 10 in total:

1 Time translation
3 Spatial translations: φ_λ(x) = x + λa for a ∈ ℝ³
3 Rotations: φ_λ(x) = R(α, λ)x for α ∈ ℝ³
3 Boosts: φ_λ(x) = x + λvt for v ∈ ℝ³

Side note: This corresponds exactly to the 10 infinitesimal generators of the Galilei algebra. (different topic, google for more information ;) The generating vector field for the combined symmetry, is fully described by ε (time), β (translation), α (rotation), γ (boost): v_i = α × x_i + β + γt - ε ẋ_i The Galilei boost transforms into a uniformly moving frame: φ_λ(x₁, ..., x_N, t) = (x₁ + λvt, ..., x_N + λvt), τ_λ(t) = t. The Lagrangian is not strictly invariant but changes by a total time derivative, given δF = ∑_i m_i x_i · v. The conserved quantity is the center-of-mass (COM) integral:

∑_i m_i ẋ_i · vt - ∑_i m_i x_i · v = -(MX - Pt) · v

Since v is arbitrary, this just says that the center of mass moves at constant velocity.

7. Conserved Quantities → Symmetries

Now to another exciting part: Noether's theorem works in both directions: every conserved quantity F(q, q̇, t) also gives rise to a continuous symmetry. If we impose the condition dF/dt = 0 on solutions, the generating vector field is determined implicitly by:

∑_β (∂²L / ∂q̇_α∂q̇_β) v_β = ∂F/∂q̇_α

Note we assume the mass matrix ∂²L/∂q̇² to be invertible. The corresponding compensating term becomes K = ∑_α v_α (∂L/∂q̇_α) - F. Which confirms that continuous symmetries and conservation laws are equivalent descriptions of the same underlying physics up to the dF/dt = 0 and invertible mass matrix constraints.

8. Connection to Representation Theory

Each continuous symmetry is a representation of a Lie group (for now google, I will come back to this in the future) on the configuration space. The conserved quantities live in the dual space (also google :) ) of the Lie algebra: they are the infinitesimal generators of the symmetry expressed as observable quantities with the dimension of the Lie group telling you the amount of conserved quantities:

Translations in 3D (ℝ³, dimension 3) → 3 components of momentum
Rotations in 3D (SO(3), dimension 3) → 3 components of angular momentum
Time translations (ℝ, dimension 1) → 1 conserved energy
Galilei boosts (ℝ³, dimension 3) → 3 components of center-of-mass integral

This is where representation theory comes in useful. (see finite group representations for more information.)

9. What Noether's Theorem Does NOT Cover

As we saw: Noether's theorem only applies to continuous symmetries. Discrete symmetries (like parity, charge conjugation, or the reflections in D₄) however do not produce conserved quantities via Noether's theorem. What they do however is produce quantum numbers instead (understand them as multiplicative conservation laws rather than additive ones). This is a different mechanism.

There is also a second Noether theorem (often called Noether's second theorem) that deals with gauge symmetries, which are symmetries parametrized by arbitrary functions rather than constants which results in constraints rather than conservation laws, and builds the mathematical foundation of gauge field theories (e.g., electromagnetism, QCD of the Standard Model of Particle Physics). (I aim to write about this in the future.)