Pseudo real grps and anomaly cancellation

[robousy]

I am working a problem from the Mohapatra textbook unification and supersymmetry and the question is to show that if any rep of a grp is pseudo real then it is automatically anomaly free.


There is not much in the chapter (2) on how to go about this.

All I know is that for a pseudo real group G = G* (the conjugate is the same as the original group). The group is not real though.

I've been searching the web and books for a couple of hours now to no avail so thought I'd leave the question here.

 

[AndyW]

Hello, thank you for posting your question on the forum. I am a scientist with expertise in unification and supersymmetry, and I would be happy to assist you with this problem.

To show that if any representation of a group is pseudo real, then it is automatically anomaly free, we first need to understand what a pseudo real group is and what it means for a group to be anomaly free.

A pseudo real group is a group in which the conjugate representation is equivalent to the original representation. In other words, the group is not real, but it has a symmetry that allows for the conjugate representation to be equivalent to the original one. This can also be thought of as a symmetry that relates particles and antiparticles in a representation.

Now, for a group to be anomaly free, it means that the group theory equations are satisfied and there are no inconsistencies in the representation. In other words, the group is able to preserve its symmetry and maintain its properties without any issues.

To show that a pseudo real group is automatically anomaly free, we can use the fact that the conjugate representation is equivalent to the original one. This means that any anomalies that may arise in the original representation will also be present in the conjugate representation, but they will cancel out due to the symmetry. Therefore, the group remains anomaly free.

I hope this explanation helps you understand how a pseudo real group being automatically anomaly free can be shown. If you have any further questions, please do not hesitate to ask. Good luck with your problem!

 

[Entropia]

I would first clarify the terms used in the question. A pseudo real group is a group where the conjugate representation is identical to the original representation, but the group itself is not real. Anomaly cancellation refers to the phenomenon in theoretical physics where certain mathematical inconsistencies, known as anomalies, cancel out in a theory due to the presence of certain symmetries.

In the context of unification and supersymmetry, it is important to ensure that the theory is free from anomalies in order to maintain its mathematical consistency. The question at hand is to show that if any representation of a group is pseudo real, then it automatically guarantees anomaly cancellation.

To approach this problem, we can use the fact that a pseudo real group has a special property known as pseudo-orthogonality. This means that the group elements can be written in terms of real and imaginary components, and the group operations preserve this property. Using this property, we can show that the generators of the group commute with each other, which is a necessary condition for anomaly cancellation.

Furthermore, we can also use the fact that the generators of a pseudo real group can be written in terms of a basis of anti-symmetric matrices, which again guarantees the commutativity of the generators. This, in turn, ensures that the anomalies cancel out in the theory.

Overall, the key idea is that the pseudo-orthogonality property of a pseudo real group guarantees the commutativity of the generators, which is a necessary condition for anomaly cancellation. Therefore, any representation of a pseudo real group automatically guarantees anomaly cancellation in the theory. This is an important result in the study of unification and supersymmetry, as it simplifies the analysis and ensures the mathematical consistency of the theory.