Doublets Scalar Potential Question

[Magister]

How can I write the most general real scalar potential in terms of the singlets and doublets corresponding to irreducible representations of S3?

 

[masudr]

Did you mean SO(3)?

 

[denian]

To write the most general real scalar potential in terms of the singlets and doublets corresponding to irreducible representations of S3, we first need to understand the irreducible representations of S3. S3 is a group of symmetries that describes the rotation and reflection of three objects in three-dimensional space. It has three irreducible representations: the trivial representation (singlet), the two-dimensional representation (doublet), and the three-dimensional representation (triplet).

The most general real scalar potential can be written as a linear combination of the singlets and doublets, with coefficients that are determined by the symmetries of S3. This potential should also be invariant under the transformations of S3, which means that it should remain unchanged when we rotate or reflect the objects.

One possible way to write this potential is as follows:

V(ϕ) = m1|ϕ1|^2 + m2|ϕ2|^2 + m3|ϕ3|^2 + λ1(|ϕ1|^2)^2 + λ2(|ϕ2|^2)^2 + λ3(|ϕ3|^2)^2 + λ4|ϕ1|^2|ϕ2|^2 + λ5|ϕ1|^2|ϕ3|^2 + λ6|ϕ2|^2|ϕ3|^2

Where ϕ1, ϕ2, and ϕ3 are the singlet and doublet fields, and m1-m3 and λ1-λ6 are coefficients that are determined by the symmetries of S3. This potential is invariant under the transformations of S3, and it includes all possible combinations of singlets and doublets that are allowed by the symmetries.

It is important to note that this is just one possible way to write the most general real scalar potential in terms of the singlets and doublets corresponding to irreducible representations of S3. Other forms may also exist, depending on the specific system being studied and the symmetries involved. it is crucial to carefully examine the symmetries and properties of the system to determine the most appropriate form of the potential.