Finite spin-half tensor chains

Let $N\ge 1$ be an integer. The finite spin-half tensor chain is the complex inner-product space ${\mathcal{H}}_N:=({\mathbb{C}}^2{)}^{\otimes N}$. On each factor use the inner product $⟨(z_0,z_1),(w_0,w_1){⟩}_{{\mathbb{C}}^2}:=z_0\overline{w_0}+z_1\overline{w_1},$ which is linear in its first argument, and let $\mathrm{\mid }0⟩=(1,0)$ and $\mathrm{\mid }1⟩=(0,1)$. By the finite tensor-product basis theorem, for $\alpha =(a_1,\dots ,a_N)\in \{0,1{\}}^N$ the words $\mathrm{\mid }\alpha ⟩:=\mathrm{\mid }{a}_1⟩\otimes \cdots \otimes \mathrm{\mid }{a}_N⟩$ form a basis of ${\mathcal{H}}_N$. For $u={\sum }_{\alpha \in \{0,1{\}}^N}{c}_{\alpha }\mathrm{\mid }\alpha ⟩$ and $v={\sum }_{\alpha \in \{0,1{\}}^N}{d}_{\alpha }\mathrm{\mid }\alpha ⟩$, define $⟨u,v{⟩}_{{\mathcal{H}}_N}:={\sum }_{\alpha \in \{0,1{\}}^N}{c}_{\alpha }\overline{d_{\alpha }}\mathrm{.}$ This is a positive-definite inner product because $⟨u,u⟩={\sum }_{\alpha }\mathrm{\mid }{c}_{\alpha }{\mathrm{\mid }}^2$, and it makes the tensor words orthonormal. Expanding each factor in $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$ gives its elementary-tensor product formula $⟨v_1\otimes \cdots \otimes v_N,w_1\otimes \cdots \otimes w_N{⟩}_{{\mathcal{H}}_N}={\prod }_{j=1}^N⟨v_j,w_j{⟩}_{{\mathbb{C}}^2}\mathrm{.}$ Let ${\sigma }^x,{\sigma }^y,{\sigma }^z$ be the Pauli matrices of the finite-spin matrix supplier, and for $1\le k\le N$ define ${\sigma }_k^a:=I^{\otimes (k-1)}\otimes {\sigma }^a\otimes I^{\otimes (N-k)},J_k^a:=\frac{1}{2}{\sigma }_k^a(a\in \{x,y,z\})\mathrm{.}$ For $1\le i<N$, define the adjacent swap $s_i$ on elementary tensors by $s_i(v_1\otimes \cdots \otimes v_N):=v_1\otimes \cdots \otimes v_{i+1}\otimes v_i\otimes \cdots \otimes v_N\mathrm{.}$ Write $\mathrm{\mid }\omega ⟩:=\frac{1}{\sqrt{2}}(\mathrm{\mid }0⟩\otimes \mathrm{\mid }1⟩-\mathrm{\mid }1⟩\otimes \mathrm{\mid }0⟩),P_i:=I^{\otimes (i-1)}\otimes P_{\omega }\otimes I^{\otimes (N-i-1)},$ where $P_{\omega }v:=⟨v,\omega ⟩\omega$ on the adjacent two factors. Also put $e_i:=I_{{\mathcal{H}}_N}-s_i$ for $1\le i<N$. If $N=1$, these bond-indexed families are empty.

This is a mathematical definition of finite-dimensional tensor spaces and operators; it makes no physical claim about a material, a particle species, or an adopted statistics sector.

Real and complex inner product spaces, with the inner product linear in the first argument

The elementary tensors of two bases form the product basis of the tensor product

Finite spin matrices

The convention $J_k^a={\sigma }_k^a\mathrm{/}2$ is dimensionless. If a later model interprets angular momentum in SI units, it uses $S_k^a=\mathrm{\hslash }{J}_k^a$ with $\mathrm{\hslash }$ in joule-seconds; that physical interpretation is not part of this mathematical definition.