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--- advanced_tools:group_theory:representation_theory:tensor_product_representation [2023/02/25 19:01]
edi [Concrete]
+++ advanced_tools:group_theory:representation_theory:tensor_product_representation [2023/02/25 19:28]
edi [Concrete]
@@ Line 2: / Line 2: @@
 <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.
 <tabbox Concrete>
 **Example**
-The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the tensor-product representation of two copies of the defining representation, we let it act on the tensor-product space as shown in the lower branch.
+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.
-The resulting 4-dimensional representation is reducible, breaking up into a 1- and 3-dimensional irreducible representation.
+The resulting 4-dimensional representation is reducible, breaking up into a 1- and 3-dimensional irreducible representation. This is usually written as $\bf 2 \otimes 2 = 1 \oplus 3$.
-This representation is useful for describing a system of two spin-1/2 particles, in particular, to analyze its combined spin.
+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.
 [{{ :advanced_tools:group_theory:representation_theory:su2_tensor_rep.jpg?nolink }}]

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