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Question about principal chiral field, why conserved currents arise

+ 2 like - 0 dislike
142 views

See here for this 1999 preprint of MacKay, The $SO(N)$ principal chiral field on a half-line.

The principal chiral model may be defined by the Lagrangian$$\mathcal{L} = \text{Tr}(\partial_\mu g^{-1}\partial^\mu g),$$where the field $g(x^\mu)$ takes values in a compact Lie group $\mathcal{G}$, here chosen to be $SO(N)$. It has a global $\mathcal{G}_L \times \mathcal{G}_R$ symmetry with conserved currents$$j(x, t)_\mu^L = \partial_\mu gg^{-1},\text{ }j(x, t)_\mu^R = -g^{-1}\partial_\mu g$$which take values in the Lie algebra $\mathbf{g}$ of $\mathcal{G}$; that is, $j = j^at^a$ (for $g^L$ or $g^R$: henceforth we drop this superscript) where $t^a$ are the generators of $\mathbf{g}$.

Can anyone expand on the derivation of these conserved currents? It is not very clear to me where they come from. Thanks.

This post imported from StackExchange MathOverflow at 2015-11-17 15:56 (UTC), posted by SE-user Antonio
asked Nov 16, 2015 in Theoretical Physics by Antonio (10 points) [ no revision ]
retagged Nov 17, 2015
Cross-posted to Physics Stack Exchange.

This post imported from StackExchange MathOverflow at 2015-11-17 15:56 (UTC), posted by SE-user HDE 226868

1 Answer

+ 1 like - 0 dislike

This comes from an application of Noether's theorem, which associates with every symmetry of a relativistic system a 4-current. The following Wikipedia link shows the derivation and the final formula for a general Lagrangian.

answered Nov 17, 2015 by Arnold Neumaier (12,355 points) [ no revision ]

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