Conserved Noether current under SO(3) symmetry of some Lagrangian

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 $\phi_{a}$.

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 $(\vec{n} \times \vec{\phi})_{a}$?

Answers and Replies

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Orodruin
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1. Yes.
2. It is an arbitrary vector, the vector you rotate about.
3. There is no scalar product here. $\theta$ is an angle, not a vector.

3. There is no scalar product here. θθ\theta is an angle, not a vector.
I meant the scalar multiplication of the scalar $\theta$ and the vector $\vec{n} \times \phi$. Sorry for the confusion. It is, indeed, a scalar multiplication, is it not?

I think I'm now beginning to understand the form of the $SO(3)$ rotation of the vector $\vec{\phi}$ of fields about the arbitrary unit vector $\vec{n}$ by an infinitesimal angle $\theta$.

My difficulty lay with the visualisation of the space within which the rotation takes place. I think I have now cleared my confusion. Let me explain:

Unlike transformations such as translations and Lorentz transformations, which transform spacetime coordinates and thereby induce a transformation of the field configuration(s), this example does not consider a transformation of the spacetime coordinates, but rather directly considers an internal transformation of the field configurations themselves. To understand the transformation, we need to visualise a right-handed coordinate system where the internal points are labelled by a triplet of real fields $\phi_{a} (a=1,2,3)$. Each of the internal points corresponds to a triplet of real fields $\phi_{a} (a=1,2,3)$, and therefore each of the internal points can be labelled by the vector $\vec{\phi} = (\phi_{a},\phi_{b},\phi_{c})$. The state of a system governed by the Lagrangian density $\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi_{a}\partial^{\mu}\phi_{a}-\frac{1}{2}m^{2}\phi_{a}\phi_{a}$ is given by some internal point $(\phi_{a},\phi_{b},\phi_{c})$ in this internal space. We can imagine various transformations of this internal point. (The Lagrangian density may or may be invariant under any of these transformations.) One particular transformation is the $SO(3)$ rotation of the internal point $(\phi_{a},\phi_{b},\phi_{c})$, or equivalently, of the vector $\vec{\phi}=(\phi_{a},\phi_{b},\phi_{c})$ of fields, about an arbitrary unit vector $\vec{n}$ by an infinitesimal angle $\theta$. The form of the rotations is therefore given by $\delta \phi_{a} = \theta (\vec{n}\times\vec{\phi})_{a} = \theta\epsilon_{abc}n_{b}\phi_{c}$.

Am I correct?

If someone could please tell me if my understanding of the internal space of fields (within which the infinitesimal $SO(3)$ rotation is taking place) is correct, then I could proceed with solving the problem.