How can an SU(2) triplet be represented as a 2x2 matrix in the Lagrangian?

In summary, the conversation discusses the transformation of a fundamental scalar SU(2) triplet into a 2x2 matrix representation in the Lagrangian. This can be done by forming the triplet using traceless matrices or transforming under the adjoint representation. The conversation also mentions the transformation of a doublet into a triplet using the sigma matrices and the antisymmetric metric. Finally, the conversation clarifies that the triplet is irreducible.
  • #1
ChrisVer
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Sorry for this "stupid" question... but I am having some problem in understanding how can someone start from let's say an SU(2) triplet and arrive in a 2x2 matrix representation of it in the Lagrangian...
An example is the Higgs-triplet models...I think this happens with the W-gauge bosons too in the EW theory, but I'm currently losing where in the derivation/how this happens. I think for the W's this happens with the complexification?
 
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  • #2
Does it happen by a map [itex] \phi_i \rightarrow \phi_i \sigma^i[/itex]?
 
  • #3
The tensor product of two fundamental representations is a singlet representation (the trace of the tensor) and a triplet which is the rank two symmetric tensor. You should therefore not be surprised that you can realize the triplet representation using traceless matrices.

Alternatively, just see it as transforming under the adjoint representation, which is the triplet (the Lie algebra is three dimensional).
 
  • #4
If you have a doublet ##\phi##, you can make a triplet out of it, ##\phi^{\dagger} \vec{\sigma} \phi##. I'm not sure, whether this is what you mean or need. This combination transforms under the fundamental SO(3).
 
  • #5
vanhees71 said:
If you have a doublet ##\phi##, you can make a triplet out of it, ##\phi^{\dagger} \vec{\sigma} \phi##. I'm not sure, whether this is what you mean or need. This combination transforms under the fundamental SO(3).

I do not think he has a doublet that he is forming a triplet from. I think he has a fundamental scalar SU(2) triplet such as that which appears in the type-II seesaw.
 
  • #6
Ok, then you can make a doublet via ##\tilde{\phi}=\vec{\sigma} \cdot \vec{\phi}##, where ##\vec{\phi} \in \mathbb{R}^3## is the triplet field.
 
  • #7
vanhees71 said:
Ok, then you can make a doublet via ##\tilde{\phi}=\vec{\sigma} \cdot \vec{\phi}##, where ##\vec{\phi} \in \mathbb{R}^3## is the triplet field.

This is not a doublet, it is an element of the Lie algebra and transforms under the adjoint representation, which is the triplet representation. The triplet is irreducible.
 
  • #8
Yup it's like my Post #2...
It's confusing me ...
the [itex]\phi = \begin{pmatrix} \phi_+ \\ \phi_0 \\ \phi_- \end{pmatrix}[/itex] is a triplet of SU(2). That means that it transforms as triplet under SU(2) transformations and so in the adjoint representation (I think with the [itex]\epsilon_{ab}[/itex]'s the antisymmetric metric)...
Then also the [itex] \phi \cdot \sigma[/itex] (which is now a 2x2 matrix) transforms in the same rep with epsilons?
 
  • #9
The vector you have written down is just vector containing the components of a vector in the representation. The basis vectors of the adjoint representation are the basis vectors of the Lie algebra, i.e., the Pauli matrices in the case of SU(2). So all ##\phi\cdot\sigma## tells you is the full expression for the triplet, i.e., the full element of the Lie algebra.
 
  • #10
ChrisVer said:
Yup it's like my Post #2...
It's confusing me ...
the [itex]\phi = \begin{pmatrix} \phi_+ \\ \phi_0 \\ \phi_- \end{pmatrix}[/itex] is a triplet of SU(2). That means that it transforms as triplet under SU(2) transformations and so in the adjoint representation (I think with the [itex]\epsilon_{ab}[/itex]'s the antisymmetric metric)...
Then also the [itex] \phi \cdot \sigma[/itex] (which is now a 2x2 matrix) transforms in the same rep with epsilons?

The triplet transforms by the adjoint map of the Lie algebra [itex]su(2)[/itex] [tex]\mbox{ad}: \ \ \phi_{i} \to \phi_{i} + \epsilon_{i j k} \phi_{j} \alpha_{k} ,[/tex] while the Hermitian [itex]2 \times 2[/itex] matrix [itex]\Phi = \tau^{i}\phi_{i}[/itex] transforms in the adjoint map of the group [itex]SU(2)[/itex]: [tex]\mbox{Ad}: \ \ \Phi \to U \Phi U^{\dagger} , \ \ U \in SU(2) .[/tex] As usual the two maps are related by [tex]\mbox{Ad} ( e^{\alpha^{i} X^{i}} ) = e^{ \alpha^{i} \mbox{ad}(X^{i}) } .[/tex]
 
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  • #11
Orodruin said:
This is not a doublet, it is an element of the Lie algebra and transforms under the adjoint representation, which is the triplet representation. The triplet is irreducible.
True. Sorry for the confusion.
 

Related to How can an SU(2) triplet be represented as a 2x2 matrix in the Lagrangian?

What is the 2x2 representation of triplet?

The 2x2 representation of triplet is a mathematical concept used in physics and quantum mechanics to describe the properties of particles with spin, such as electrons. It represents the spin states of a particle in a two-dimensional vector space, with the x-axis representing the spin up state and the y-axis representing the spin down state.

How is the 2x2 representation of triplet used in physics?

In physics, the 2x2 representation of triplet is used to describe the spin states of particles and their interactions with other particles. It is an important tool in understanding the behavior of particles, particularly in quantum mechanics where the spin of particles plays a crucial role in determining their properties.

What are the components of the 2x2 representation of triplet?

The 2x2 representation of triplet consists of four complex numbers, denoted as a, b, c, and d. These numbers represent the amplitudes of the spin up and spin down states of a particle in a particular direction.

How is the 2x2 representation of triplet related to the Pauli matrices?

The Pauli matrices, which are a set of three 2x2 matrices, are closely related to the 2x2 representation of triplet. In fact, the Pauli matrices can be used to construct the 2x2 representation of triplet and vice versa, making them important tools in quantum mechanics.

Why is the 2x2 representation of triplet important in quantum mechanics?

The 2x2 representation of triplet allows us to describe the spin states of particles and their interactions with other particles in a more precise and comprehensive manner. This is crucial in understanding the behavior of particles at the quantum level, where the spin of particles plays a major role in determining their properties and behavior.

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