9 Tensor Products, Entanglement, and Addition of Spin

If one has two independent quantum systems, with state spaces ${\mathcal{H}}_1$ and ${\mathcal{H}}_2.$ the combined quantum system has a description that exploits the mathematical notion of a “tensor product”, with the combined state space the tensor product ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$ Because of the ability to take linear combinations of states, this combined state space will contain much more than just products of independent states, including states that are described as “entangled”, and responsible for some of the most counter-intuitive behavior of quantum physical systems.

This same tensor product construction is a basic one in representation theory, allowing one to construct a new representation $({\pi }_{W_1\otimes W_2},W_1\otimes W_2)$ out of representations $({\pi }_{W_1},W_1)$ and $({\pi }_{W_2},W_2)$ . When we take the tensor product of states corresponding to two irreducible representations of $SU(2)$ of spins $s_1,s_2$ we will get a new representation $\left({\pi }_{V^{2s_1}\otimes V^{2s_2}},V^{2s_1}\otimes V^{2s_2}\right)$ . It will be reducible, a direct sum of representations of various spins, a situation we will analyze in detail.

Starting with a quantum system with state space $H$ that describes a single particle, a system of n particles can be described by taking an n-fold tensor product ${\mathcal{H}}^{\otimes n}=\mathcal{H}\otimes \mathcal{H}\otimes \cdot \cdot \cdot \otimes \mathcal{H}$ . It turns out that for identical particles, we don’t get the full tensor product space, but only the subspaces either symmetric or antisymmetric under the action of the permutation group by permutations of the factors, depending on whether our particles are “bosons” or “fermions”. This is a separate postulate in quantum mechanics, but finds an explanation when particles are treated as quanta of quantum fields.

Digression. When physicists refer to “tensors”, they generally mean the “tensor fields” used in general relativity or other geometry-based parts of physics, not tensor products of state spaces. A tensor field is a function on a manifold, taking values in some tensor product of copies of the tangent space and its dual space. The simplest tensor fields are vector fields, functions taking values in the tangent space. A more non-trivial example is the metric tensor, which takes values in the dual of the tensor product of two copies of the tangent space.

中文目录

Chapter contents

• 9.1 Tensor products
• 9.2 Composite quantum systems and tensor products
• 9.3 Indecomposable vectors and entanglement
• 9.4 Tensor products of representations
  • 9.4.1 Tensor products of $SU(2)$ representations
  • 9.4.2 Characters of representations
  • 9.4.3 Some examples
• 9.5 Bilinear forms and tensor products
• 9.6 Symmetric and antisymmetric multilinear forms
• 9.7 For further reading

Previous chapter: 8 Representations of SU(2) and SO(3)

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 109、110、111、112、113、114、115、116、117、118、119、120

来源版本：2025-10-20

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