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## Homework Statement

From the Lagrangian density

[tex] L = \frac{1}2 \partial_\mu \phi_a \partial^\mu \phi_a - \frac{1}2 \phi_a \phi_a,[/tex]

where a = 1,2,3 and the transformation

[tex] \phi_a \to \phi _a + \theta \epsilon_{abc} n_b \phi_c [/tex]

show that one gets the conserved charges

[tex]Q_a = \int d^3x \epsilon_{abc}\dot{\phi}_b \phi_c.[/tex]

## Homework Equations

The transformation is a symmetry of the Lagrangian so by Noethers theorem

we got a conserved current which is given by

[tex]j^\mu = \frac{\partial L}{\partial(\partial_\mu \phi_a)} \delta \phi_a = \partial^\mu \phi_a \epsilon_{abc} n_b \phi_c[/tex]

## The Attempt at a Solution

The obvious conserved charge is just

[tex] Q = \int d^3x j^0 = \int d^3 x \dot \phi_a \epsilon_{abc} nb \phi_c [/tex]

but this is not the 3 different charges in the expression for Q_a. There is no normal vector n in that expresion and the time differentiated field has got b-index instead of an a index.

How can one get from the conserved current to the expression for these charges?

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