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The quantum state of a spin- 1/2 particle can be written, in the momentum representation, as a two-component spinor,

$$\textit{Ψ}(\textbf{p})=\left(\begin{matrix}a_{1}(\textbf{p})\\a_{2}(\textbf{p})\end{matrix}\right),$$

With the density matrix corresponding to 2-component spinor above is

$$\textit{ρ}(\textbf{p′,p′′})=\left(\begin{matrix}a_{1}(\textbf{p′})a_{1}(\textbf{p′′})^*&a_{1}(\textbf{p′})a_{2}(\textbf{p′′})^*\\a_{2}(\textbf{p′})a_{1}(\textbf{p′′})^*&a_{2}(\textbf{p′})a_{2}(\textbf{p′′})^*\end{matrix}\right),$$

The reduced density for spin, increase irrespective of momentum, is obtained by setting $$\textbf{p} = \textbf{p′} = \textbf{p′′}$$ and integrating over p, with the bloch vector n is given by

$$n_{z} = \int [|a_{1}(\textbf{p})|^2-|a_{2}(\textbf{p})|^2]d\textbf{p}$$

and

$$n_{x} - in_{y} = \int a_{1}(\textbf{p})a_{2}(\textbf{p})^*d\textbf{p}$$

we can write the reduced density matrix for spin as $$\textit{𝜏}=\left(\begin{matrix}1+n_{z}&n_{x}-in_{y}\\n_{x}+in_{y}&1-n_{z}\end{matrix}\right),$$

i still dont understand how integrating over p can get the 𝜏 form of matrix, can someone help me ?

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The Hilbert space is $\mathcal H= L^2(R^3)\times \mathbb C^2$, to find the reduced spin density matrix you have to trace over $L^2(R^3)$. That is why you are integrating $\int d^3 p \; \rho(p,p) $.

Remember that your "wavefunctions" are normalized (or, the density matrix should contain a normalization factor) so that $\int d^3p |a_1(p)|^2 + |a_2(p)|^2=1 $.

Now, take your original $\rho(p,p')$, and write down the integral I mentioned above. Then use the normalization condition to rewrite the diagonal components, and you will get the $\tau$ expression.

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Integrating over p will not get you a matrix but will get you the matrix elements in the matrix.

Your $\rho$ is hermitian with trace=1 as per general properties of the density matrix. Since any hermitian matrix with trace=1 can be written in the form $$\textit{𝜏}=\left(\begin{matrix}1+n_{z}&n_{x}-in_{y}\\n_{x}+in_{y}&1-n_{z}\end{matrix}\right),$$ it's really just a matter matching the various matrix elements in terms of spinors with those in terms of the $n_k$ parametrization.

If you prefer, $n_z$ is defined as the average difference $$n_{z} = \int [|a_{1}(\textbf{p})|^2-|a_{2}(\textbf{p})|^2]d\textbf{p}$$ with the average taken over momenta, etc.

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