# SU(2) invariance implies isotropy?

• I
Hello guys,
I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here to share the statements hoping to improve the discussion.

Given that SU(2) is homomorphic to SO(3), we were discussing if this statements were True or not:

 A SU(2)-invariant matter lagrangian is also invariant under rotations
 its energy-momentum tensor is isotropic
 it has the same form as the one from a perfect fluid.

The first statement is true without any doubts but, now, in my viewpoint, that would (probably) imply that the spatial part of ##T_{\mu\nu}## is isotropic, in which case, anisotropic stresses would be zero but there would still be chances to have ##T_{i0} \neq 0## meaning that the second statement wouldn't be enough to ensure the third one.

The tough part is the logical relation between the first statement and the second one, my friend claims that the second one is right but he doesn't offer a clear explanation about it. What do you think guys?

Last edited:

PeterDonis
Mentor
2019 Award
Given that SU(2) is homomorphic to SO(3)
It isn't. These two groups have different topologies, so they can't be homeomorphic. A correct statement is that SU(2) and SO(3) have the same Lie algebra, but that's a weaker condition than being homeomorphic. (That weaker condition might still be enough to ground the reasoning you are attempting.)

kimbyd
Gold Member
Hello guys,
I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here to share the statements hoping to improve the discussion.

Given that SU(2) is homomorphic to SO(3), we were discussing if this statements were True or not:

 A SU(2)-invariant matter lagrangian is also invariant under rotations
 its energy-momentum tensor is isotropic
 it has the same form as the one from a perfect fluid.

The first statement is true without any doubts but, now, in my viewpoint, that would (probably) imply that the spatial part of ##T_{\mu\nu}## is isotropic, in which case, anisotropic stresses would be zero but there would still be chances to have ##T_{i0} \neq 0## meaning that the second statement wouldn't be enough to ensure the third one.

The tough part is the logical relation between the first statement and the second one, my friend claims that the second one is right but he doesn't offer a clear explanation about it. What do you think guys?
The thing that confuses me about this is that I don't see how the SU(2) symmetry can apply as a spatial symmetry in three dimensions.

Typically SU(2) is used as a gauge symmetry to describe the symmetries of local interactions, such that fields which obey this symmetry always have a field configuration at every point in space which obeys the symmetry. That statement says nothing about the isotropy or homogeneity of the field, as there are a great many possible field configurations which obey SU(2), and those configurations can vary wildly across space (and time).

How would you write down a field which has an SU(2) spatial symmetry?

fresh_42
Mentor
The thing that confuses me about this is that I don't see how the SU(2) symmetry can apply as a spatial symmetry in three dimensions.
Neither did I, even if I disregard Peter's homeomorphism and return to the original homomorphism, I don't see this mapping. What we have is ##SU(2) \cong SO(4)/SO(3)##. But I haven't checked whether there is some construction I don't know.

Hello guys,
 A SU(2)-invariant matter lagrangian is also invariant under rotations
 its energy-momentum tensor is isotropic
 it has the same form as the one from a perfect fluid.

The first statement is true without any doubts but, now, in my viewpoint, that would (probably) imply that the spatial part of ##T_{\mu\nu}## is isotropic, in which case, anisotropic stresses would be zero but there would still be chances to have ##T_{i0} \neq 0## meaning that the second statement wouldn't be enough to ensure the third one.

The tough part is the logical relation between the first statement and the second one, my friend claims that the second one is right but he doesn't offer a clear explanation about it. What do you think guys?
Actually the first statement is not true, at least not for the reason implied. All Lagrangians in QFT are invariant under spatial rotations since they are Lorentz invariant, but that is separate from their internal symmetries, i.e. gauge structure. SU(2) as an internal symmetry means the Langragian is invariant with respect to rotations in *field* space. That is, consider the SU(2) components of your fields as your coordinates, and rotate in *this* space. This is a completely different sort of symmetry to spatial rotations, it just happens to be the case that they are mathematically similar in terms of both being describable as a sort of rotation operation.

If you are talking about in curved space, well I am not sure but I think the same applies. Sure, the lagrangian is invariant under rotations, because it is invariant under general (space-time) coordinate transformations, but that is totally separate matter to the gauge group structure.

• JuanC97
bapowell
It isn't. These two groups have different topologies, so they can't be homeomorphic. A correct statement is that SU(2) and SO(3) have the same Lie algebra, but that's a weaker condition than being homeomorphic. (That weaker condition might still be enough to ground the reasoning you are attempting.)
The OP is referring to "homomorphism", not "homeomorphism".

haushofer