# Noether’s Theorem

2015-08-26

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 $\mathbf x(t)$, 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 $\mathbf x ∈ X$ and $\mathbf v ∈ \mathrm{T}_{\mathbf x} X$.

Hamilton’s principle than states that:

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

this can be restated as the Euler-Lagrange equations:

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

## Noether’s theorem

Given a symmetry $T_r$, $\mathbf Q_r$, the quantity

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

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:

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

Killing vector fields are symmetries of the metric tensor.