Exploring the Semisimplicity of Gauge Groups in the Standard Model and Beyond

In summary, the use of semisimple gauge groups in the Standard Model (SM) and beyond is important in ensuring that the norm of quantum states remains positive definite for the probability interpretation in quantum mechanics. This is due to the non-definiteness of the Killing form in nonsemisimple groups, which can lead to negative norm states. However, there are ways to remove these states through the use of constraints, such as gauge invariance. For solvable Lie algebras, the vanishing of the Killing form on the derived subalgebra can also lead to unphysical zero norm states, but it may be possible to find constraints to eliminate them.
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Is there a physical reason why all gauge groups considered in SM and especially beyond are always semisimple? [+ U(1)] What would happen if they were solvable?
 
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The norm of a quantum state must be positive definite in order that the probability interpretation of quantum mechanics makes sense. For a nonsemisimple group, the Killing form is not definite, so we can't guarantee that there won't be any negative norm states. There are ways to use constraints to remove the negative norm states for certain theories. A simple example is the way that gauge invariance can be used to remove the negative norm states from the ##SO(d,1)## metric on a massless vector field.

I'm not sure about solvable Lie algebras other than the abelian ones. Since the Killing form vanishes on the derived subalgebra, there will be zero norm states that would be considered unphysical. Perhaps one could find suitable constraints to project these out.
 
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Thank you.
 

Related to Exploring the Semisimplicity of Gauge Groups in the Standard Model and Beyond

1. What are gauge groups?

Gauge groups are mathematical structures used in theoretical physics to describe the symmetries of a physical system. They are groups of transformations that leave the equations of a physical theory unchanged.

2. Why is simplicity of gauge groups important?

The simplicity of gauge groups is important because it allows for a more elegant and concise description of physical theories. It also leads to more efficient and accurate calculations, making it an essential concept in theoretical physics.

3. How is the simplicity of gauge groups quantified?

The simplicity of gauge groups is quantified using the concept of Lie algebras. These are mathematical structures that measure the complexity of a group, with simpler groups having fewer generators and a smaller number of dimensions.

4. Can the simplicity of gauge groups change?

Yes, the simplicity of gauge groups can change depending on the specific physical theory being described. For example, in the Standard Model of particle physics, the gauge group SU(3) is simpler than the gauge group SU(5).

5. What are some applications of the concept of simplicity of gauge groups?

The concept of simplicity of gauge groups has numerous applications in theoretical physics, including the study of elementary particle interactions, quantum field theory, and the development of unified theories. It is also used in applied fields such as condensed matter physics and materials science.

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