What is meant by two vectors transforming in the same way under SU(2)?

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In summary, in section 2.3 of 'Quantum Field Theory' by Lewis Ryder, the Lie Group SU(2) is discussed and it is shown that the spinor vectors \xi and \xi^\dagger transform differently under SU(2). However, by using unitarity of U, it can be shown that \begin{pmatrix} \xi_1 \\ \xi_2 \end{pmatrix} and \begin{pmatrix} - \xi_2^* \\ \xi_1^* \end{pmatrix} transform in the same way under SU(2). This is because \xi^\dagger transforms under a representation equivalent to the fundamental representation, making the fundamental and conjugate
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This question comes from section 2.3 of 'Quantum Field Theory' by Lewis Ryder. The discussion is on the Lie Group SU(2). He discusses the transformations of vectors under SU(2). Here it goes:

consider the basic spinor [itex] \xi = \begin{pmatrix} \xi_1 \\ \xi_2 \end{pmatrix} [/itex];
[tex] \xi \to U \xi, [/tex]
[tex]\xi^\dagger \to \xi^\dagger U^\dagger. [/tex]

Then he says, we see that [itex] \xi [/itex] and [itex] \xi^\dagger [/itex] transform in different ways, but we may use the unitarity of [itex] U [/itex] to show that [itex] \begin{pmatrix} \xi_1 \\ \xi_2 \end{pmatrix} [/itex] and [itex] \begin{pmatrix} - \xi_2^* \\ \xi_1^* \end{pmatrix} [/itex] transform in the same way under SU(2).

My question is, what is meant by 'they transform in the same way'? And what is meant by saying that [itex] \xi [/itex] and [itex] \xi^\dagger [/itex] don't transform in the same way?
 
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Define
[tex]\bar{\xi}=\epsilon \xi^*,[/tex]
where
[tex]\epsilon_{12}=-\epsilon_{21}=1, \quad \epsilon_{11}=\epsilon_{22}=0[/tex]
is the Levi-Civita symbol in two dimensions.

Now let
[tex]\xi'=U \xi[/tex]
with an arbitrary [itex]U \in \mathrm{SU}(2)[/itex]. Then obviously
[tex]\bar{\xi}'=\epsilon \xi'^*=\epsilon U^* \xi^*.[/tex]
But now for SU(2) matrices you have
[tex](U^{\dagger} \epsilon U^*)_{jk}=U_{lj}^* \epsilon_{lm} U^*{mk} = \epsilon_{jk} \det(U^*)=\epsilon_{jk}.[/tex]
Now [itex]\epsilon^{-1}=\epsilon^{t}=\epsilon^{\dagger}=-\epsilon[/itex]. Thus we can write
[tex]\epsilon^{-1} U^{\dagger} \epsilon U^{*}=1 \; \Rightarrow \; U^{*}=\epsilon^{-1} U \epsilon=\epsilon U \epsilon^{-1}.[/tex]
This means that [itex]\bar{\xi}[/itex] transforms under SU(2) by a representation that is equivalent to the fundamental representation, which means that [itex]\bar{\xi}[/itex] does not define a new representation compared to the fundamental representation. Thus, for SU(2), the fundamental and the conjugate complex fundamental representation are equivalent and thus from the point of view of reprsentation theory the same.
 

FAQ: What is meant by two vectors transforming in the same way under SU(2)?

1. What does it mean for two vectors to transform in the same way under SU(2)?

This means that the two vectors will undergo similar transformations when subjected to operations within the SU(2) group. These transformations can include rotations, reflections, and combinations of these operations.

2. What is the significance of SU(2) in vector transformations?

SU(2) is a special unitary group that plays a crucial role in the study of vector transformations. It is a fundamental symmetry group in quantum mechanics and is used to describe the behavior of particles with spin.

3. How does SU(2) differ from other symmetry groups in vector transformations?

SU(2) is a special case of the general SU(n) group, where n represents the dimensionality of the vector space. It is unique in that it preserves the length of vectors, making it useful in describing physical systems where the magnitude of a vector is important.

4. Can you provide an example of vector transformations under SU(2)?

One example is the transformation of spin-1/2 particles, such as electrons, under rotations in three-dimensional space. These rotations can be described using the SU(2) group, and the resulting transformations of the particle's spin state can be predicted using the principles of quantum mechanics.

5. How is SU(2) related to other mathematical concepts?

SU(2) is closely related to other mathematical concepts, such as quaternions and the group of rotations in three-dimensional space (SO(3)). It also has connections to Lie algebras and Lie groups, which are important in the study of symmetries and transformations in mathematics and physics.

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