Noether’s Theorem

Remco Bloemen


To every differentiable symmetry generated by local actions, there corresponds a conserved current.

Discrete Langrangian Mechanics

In Langrangian mechanics a system is described by a set of generalized parameters 𝐱(t)\mathbf x(t), a time depended vector in the state space of the system.

The physics of the system is contained in a Lagrangian function:

$$ L(t, \mathbf x, \mathbf v): ℝ \times \mathrm{T}\mkern2mu X \rightarrow ℝ $$

where 𝐱X\mathbf x ∈ X and 𝐯T𝐱X\mathbf v ∈ \mathrm{T}_{\mathbf x} X.

Hamilton’s principle than states that:

δ𝐱(t)tL(𝐱(t),(dt𝐱)(t),t)=0 \delta \mathbf x(t) \mathrm{\int}^{t}_{} L(\mathbf x(t), (\operatorname*{d}\nolimits_{t}^{}\; {\mathbf x}) (t), t) = 0

this can be restated as the Euler-Lagrange equations:

dtvkL=xkL \operatorname*{d}\nolimits_{t}^{}\; \operatorname*{∂}\nolimits_{v_k}^{}\; L = \operatorname*{∂}\nolimits_{x_k}^{}\; L

Noether’s theorem

Given a symmetry TrT_r, 𝐐r\mathbf Q_r, the quantity

(𝐯L𝐯L)Tr𝐯L𝐐r \left({ \operatorname*{∂}\nolimits_{\mathbf v}^{}\; L \cdot \mathbf v - L }\right) T_r - \operatorname*{∂}\nolimits_{\mathbf v}^{}\; L \cdot \mathbf Q_r

is conserved.

In reality

The standard model

Symmetry Generators Conserved quantity
Time shift 1 Energy
Translation 3 Momentum
Rotation 3 Angular Momentum
Boosts 3 ??
Gauge 6 Color charge, Weak isospin, Electric charge, Weak hypercharge
Phase 4 Baryon, electron, muon and tau numbers

General Relativity

The Einstein-Hilbert Langrangian:

L=c416πGRjjdetgμν L = \frac{c^4}{16 \pi G} R_j^j \sqrt{-\det g_{\mu \nu }}

Killing vector fields are symmetries of the metric tensor.