# 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

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
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.