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

  • #1
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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}##?
 

Answers and Replies

  • #2
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
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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?
 
  • #4
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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.
 

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