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## Homework Statement

Verify that the Lagrangian density ##\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi_{a}\partial^{\mu}\phi_{a}-\frac{1}{2}m^{2}\phi_{a}\phi_{a}## for a triplet of real fields ##\phi_{a} (a=1,2,3)## is invariant under the infinitesimal ##SO(3)## rotation by ##\theta##, i.e., ##\phi_{a} \rightarrow \phi_{a} + \theta \epsilon_{abc}n_{b}\phi_{c}##, where ##n_a## is a unit vector.

Compute the Noether current ##j^{\mu}##.

Deduce that the three quantities ##Q_{a}=\int d^{3}x\ \epsilon_{abc}\ \dot{\phi}_{b}\phi_{c}## are all conserved and verify this directly using the field equations satisfied by ##\phi_{a}##.

## Homework Equations

## The Attempt at a Solution

Before I verify the invariance of the Lagrangian density ##\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi_{a}\partial^{\mu}\phi_{a}-\frac{1}{2}m^{2}\phi_{a}\phi_{a}## for a triplet of real fields ##\phi_{a} (a=1,2,3)## under the given infinitesimal ##SO(3)## rotation,

I need to understand why an infinitesimal ##SO(3)## rotation of the field ##\phi_{a}## by ##\theta## has the form given by ##\phi_{a} \rightarrow \phi_{a} + \theta \epsilon_{abc}n_{b}\phi_{c}##, where ##n_a## is a unit vector.

I understand that ##\epsilon_{abc}n_{b}\phi_{c} = (\vec{n} \times \vec{\phi})_{a}## denotes the three components of a cross product between a unit vector ##\vec{n} = (n_{a},n_{b},n_{c})## and the vector ##\vec{\phi}=(\phi_{a},\phi_{b},\phi_{c})## of fields.

I have three questions to ask:

1. Is ##\theta## an infinitesimal angle?

2. What determines the components of the unit vector ##\vec{n}##?

3. Why is ##\delta \phi_{a}## given by the scalar product of ##\theta## and ##(\vec{n} \times \vec{\phi})_{a}##?