Continuity Equation - Conserved Currents From Noether's Theorem

Conserved Currents From Noether's Theorem

For more detailed explanations and derivations, see Noether's theorem.

If a field φ(r, t), which can be a tensor field of any order describing (say) gravitational, electromagnetic, or even quantum fields, is varied by a small amount;

then Noether's theorem states the Lagrangian L (also Lagrangian density ℒ for fields) is invariant (doesn't change) under a continuous symmetry (the field is a continuous variable):

This leads to conserved current densities in a completely general form:

because they satisfy the continuity equation

Derivation of conserved currents from Lagrangian density

The variation in is:

\begin{align} \delta\mathcal{L} & = \frac{\partial \mathcal{L}}{\partial \phi}\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta (\partial_\mu \phi)\\ & = \frac{\partial \mathcal{L}}{\partial \phi}\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} (\partial_\mu \delta \phi)\\ \end{align}

next using the Euler-Lagrange equations for a field:

the variation is:

\begin{align} \delta\mathcal{L} & = \partial_\mu\left\delta\phi + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} (\partial_\mu \delta \phi)\\ & = \partial_\mu\left \\ & = 0 \end{align}

where the product rule has been used, so we obtain the continuity equation in a different general form:

So the conserved current (density) has to be:

Integrating Jμ over a spacelike region of volume gives the total amount of conserved quantity within that volume:

Read more about this topic:  Continuity Equation

Famous quotes containing the words currents and/or theorem:

    “For it does not follow because many books are written by persons born in America that there exists an American literature. Books which imitate or represent the thoughts and life of Europe do not constitute an American literature. Before such can exist, an original idea must animate this nation and fresh currents of life must call into life fresh thoughts along its shores.”
    —Margaret Fuller (1810–1850)

    “To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
    —Albert Camus (1913–1960)