Noether's Theorem
% Remco Bloemen % 2015-08-26
\renewcommand{\g}[1]{\left({#1}\right)} \renewcommand{\vec}[1]{\mathbf #1} \newcommand{\bindingOperator}[3]{\mathrm{#1}^{#3}{#2}} \newcommand{\∫}[2][]{\bindingOperator{\int}{#1}{#2}} \newcommand{\operator}[3]{\operatorname*{#1}\nolimits{#2}^{#3};} \newcommand{\∂}[2][]{\operator{∂}{#2}{#1}} \newcommand{\δ}[2][]{\operator{δ}{#2}{#1}} \newcommand{\D}[2][]{\operator{D}{#2}{#1}} \renewcommand{\d}[2][]{\operator{d}{#2}{#1}}
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 $\vec x(t)$, a time depended vector in the state space of the system.
The physics of the system is contained in a Lagrangian function:
$$ L(t, \vec x, \vec v): ℝ × \mathrm{T}\mkern2mu X → ℝ $$
where $\vec x ∈ X$ and $\vec v ∈ \mathrm{T}_{\vec x} X$.
Hamilton’s principle than states that:
$$ δ \vec x(t) \∫ t L(\vec x(t), (\d t {\vec x}) (t), t) = 0 $$
this can be restated as the Euler-Lagrange equations:
$$ \d t \∂ {v_k} L = \∂{x_k} L $$
Noether’s theorem
Given a symmetry $T_r$, $\vec Q_r$, the quantity
$$ \g{ \∂ {\vec v} L · \vec v - L } T_r - \∂ {\vec v} L · \vec Q_r $$
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:
$$ L = \frac{c^4}{16 π G} R_j^j \sqrt{-\det g_{μν}} $$
Killing vector fields are symmetries of the metric tensor.