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Tensor representations of \(SU(N)\)

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πŸ”— In Section 5.1 we studied representations of S O ( 3 ) 𝑆 𝑂 ( 3 ) by looking at how tensors transform under rotations. In this section we use a similar approach to study tensor representations of S U ( N ) . 𝑆 π‘ˆ ( 𝑁 ) . πŸ”— Let us start with a brief recap of the construction of tensor representations of S O ( 3 ) 𝑆 𝑂 ( 3 ) in Section 5.1, which applies just as well to tensor representations of S O ( N ) . 𝑆 𝑂 ( 𝑁 ) . The idea was to construct representations of S O ( N ) 𝑆 𝑂 ( 𝑁 ) by constructing objects that transform according to these representations. These objects were called β€œtensors”, and denoted by a letter with many indices, for instance: T i j k . 𝑇 𝑖 𝑗 π‘˜ . The number of indices (called the β€œrank” of the tensor) tells us how these objects transform under a rotation R ∈ S O ( 3 ) : 𝑅 ∈ 𝑆 𝑂 ( 3 ) : πŸ”— If we were to put all independent components of T i j k 𝑇 𝑖 𝑗 π‘˜ in a column vector, the transformation rule above would define a matrix repre

Tensor representations of \(SU(N)\) Skip to main content \(\DeclareMathOperator{\Tr}{Tr} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \) Section 5.5 Tensor representations of \(SU(N)\) ΒΆ Objectives You should be able to: Recognize tensors as objects that transform according to representations of \(SU(N)\text{.}\) Determine the dimension of the irreducible representations of \(SU(N)\) by looking at the corresponding tensors. Show that the defining representation of \(SU(2)\) is pseudo-real. List the irreducible representations of \(SU(3)\) as tensor representations. In Section 5

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