Transformation of a scalar field

In summary, the general rotation of a scalar field under both SU(2)L and SU(2)R is that δΣ = iεaRTaΣ - iεaLΣTa.
  • #1
Shen712
12
0
I read somewhere that, suppose a scalar field Σ transforms as doublet under both SU(2)L and SU(2)R, its general rotation is

δΣ = iεaRTaΣ - iεaLΣTa.

where εaR and εaL are infinitesimal parameters, and Ta are SU(2) generators.

I don't quite understand this. First, why does the first term have positive sign but the second term has a negative sign? Second, why is Σ after Ta in the first term, while Σ is before Ta in the second term?
 
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  • #2
If the scalar field transforms in the (2,2) representation of the 2 SU(2)'s, then there will be a term in the lagrangian of the form:

[itex] \bar{\psi}_L \Sigma \psi_R[/itex]
where the psiL lives in (1,2) and the psiR lives in (2,1)
Transforming that term:
[itex] \bar{\psi}_L e^{ iT_L^a \theta^a} \Sigma' e^{-iT_R^b \omega^b} \psi_R[/itex]
Demanding gauge invariance:
[itex] \Sigma = e^{ iT_L^a \theta^a} \Sigma' e^{-iT_R^b \omega^b}[/itex]
[itex]\Sigma = (1 + i T_L^a \theta^a) \Sigma' (1 - iT_R^b \omega^b)= \Sigma' + iT_L^a \theta^a \Sigma' - i \Sigma' T_R^b \omega^b [/itex]
[itex] \delta \Sigma = \Sigma'-\Sigma = - iT_L^a \theta^a \Sigma + i \Sigma T_R^b \omega^b[/itex]

This might need a review, eg signs, but I guess the main idea is that (as it also was for the SM) and that there is a sign difference (now how it goes depends on how you define the transfs).
What's your reference?
 
Last edited:
  • #3
What about a term ##\bar{\psi}_R \Sigma \psi_L##?
 
  • #4
CAF123 said:
What about a term ##\bar{\psi}_R \Sigma \psi_L##?
It is the same. As I said, the overall signs can change depending on how you define the transformation, but the relative sign will still be the same.
Of course I would prefer writing that term with a sigma-dagger as that's a c.c. term in the Lagrangian?
 

1. What is a scalar field?

A scalar field is a mathematical function that assigns a scalar value (a single number) to every point in a space. This type of field is used in various fields of science, such as physics, engineering, and computer graphics.

2. What does it mean to transform a scalar field?

Transforming a scalar field means altering the function or values of the field in some way. This could involve changing the shape or magnitude of the field, or applying mathematical operations to the function.

3. Why do we need to transform scalar fields?

Scalar fields are often transformed in order to simplify or visualize complex data. For example, in physics, scalar fields may be transformed to represent different physical quantities, such as temperature or electric potential, in a more meaningful way.

4. What are some common techniques for transforming scalar fields?

Some common techniques for transforming scalar fields include scaling, rotation, translation, and deformation. These techniques can be applied using mathematical functions or algorithms.

5. How are transformations of scalar fields used in real-world applications?

Transformations of scalar fields have many real-world applications in fields such as physics, engineering, and computer graphics. For example, in computer graphics, transformations of scalar fields can be used to create realistic simulations of physical phenomena, such as fluid flow or weather patterns.

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