Symmetry Breaking and the Metric

[Digs]

Hello,


im a first year phd student, and I am reading http://arxiv.org/abs/0901.1847 right now in an attempt to figure out something cool to do with lie 3 algebras.

In the paper they mention that turning on a B field in the 1,2 dimensions breaks the SO(4) symmetry the theory has to SO(2)xSO(2), and as well the metric becomes diagonal, with some funny entries I don't recognize.

Why does the metric become diagonal? Also, why does the symmetry break to that?

THanks!

 

[xepma]

I ain't no expert, but I think what they mean is that a diagonal form like that is allowed (as opposed to: the most general expression you can have). So with the residual symmetry group it's always possible to obtain such an expression for the metric.

Think of it as normal GR. In that case it's always possible to (locally) obtain the Minkowski metric.

That would be my best guess.

 

[Entropia]

Hello first year PhD student,

Thank you for sharing your thoughts and questions about the paper you are reading. It sounds like you are interested in exploring the concept of symmetry breaking and its relationship to the metric in a lie 3 algebra setting. This is a fascinating topic and I am happy to discuss it with you.

To start, let's define what symmetry breaking is. In physics, symmetry is a fundamental concept that describes the invariance of a system under certain transformations. In other words, a system is symmetric if it remains unchanged when certain operations are performed on it. However, in some cases, the symmetry of a system can be broken, meaning that the system is no longer invariant under those transformations.

In the paper you mentioned, the authors discuss how turning on a B field in the 1,2 dimensions breaks the SO(4) symmetry of the theory to SO(2)xSO(2). This means that the system is no longer symmetric under rotations in all four dimensions, but only under rotations in two dimensions. This breaking of symmetry is due to the presence of the B field, which introduces an external influence on the system.

Now, let's talk about the metric. In simple terms, the metric is a mathematical tool that describes the distance between points in a space. In this case, the metric refers to the spacetime metric, which is used to describe the geometry of spacetime. When the SO(4) symmetry is broken to SO(2)xSO(2), the metric also changes. This is because the symmetry of a system is closely related to its geometry. In this case, the symmetry breaking leads to a change in the geometry of spacetime, resulting in a diagonal metric with some unfamiliar entries.

I hope this helps to answer your questions. Keep exploring and asking questions, as this is an important step in your research journey. Best of luck with your studies!