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advanced_tools:group_theory:representation_theory:tensor_product_representation [2023/02/25 18:42] edi created |
advanced_tools:group_theory:representation_theory:tensor_product_representation [2023/03/19 21:46] (current) edi [Concrete] |
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<tabbox Intuitive> | <tabbox Intuitive> | ||
+ | Given an $n$- and $m$-dimensional representations, we can construct an $nm$-dimensional tensor-product representation by letting it act on the tensor-product space. In particular, we can construct an $n^2$-dimensional representation from two copies of an $n$-dimensional representation. | ||
- | <note tip> | + | Tensor-product representations are useful for constructing new representations from old ones. Take the product of two old representations and break the result up into irreducible representations. |
- | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
- | </note> | + | In quantum mechanics, tensor-product representations are useful for describing multi-particle systems and study entanglement among the particles. |
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<tabbox Concrete> | <tabbox Concrete> | ||
+ | **Example** | ||
+ | |||
+ | The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the tensor-product representation from two copies of the defining representation, we let it act on the tensor-product space, $\mathbb{C}^2 \otimes \mathbb{C}^2$, as shown in the lower branch. | ||
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+ | The resulting 4-dimensional representation is reducible, breaking up into a 1- and 3-dimensional irreducible representation. This is usually written as $\bf 2 \otimes 2 = 1 \oplus 3$. | ||
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+ | This tensor-product representation is useful for describing a system of two spin-1/2 particles, in particular, to analyze the spin states of the combined system. | ||
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+ | [{{ :advanced_tools:group_theory:representation_theory:su2_tensor_rep.jpg?nolink }}] | ||
- | <note tip> | + | For a more detailed explanation of this diagram see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#su2_tensor_rep|Fun with Symmetry]]. |
- | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | |
- | </note> | + | |
<tabbox Abstract> | <tabbox Abstract> |