tensor-iso-quant-matter

"particle" states are (unit) vectors $|v\rangle$ in Hilbert Space $\mathcal{H}=\mathbb{C}^d$ e.g. electron spin is a basis of $\mathbb{C}^2$ $|v\rangle \in \{|\uparrow\rangle,|\downarrow\rangle\}$

"waves" are density matrices $|v\rangle\langle v|:\mathcal{H}\to \mathbb{C}\to \mathcal{H}$ $|\uparrow\rangle\langle \uparrow|=\begin{bmatrix}1\\ 0 \end{bmatrix}\begin{bmatrix} 1 & 0\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}$

$|\downarrow\rangle\langle \downarrow|=\begin{bmatrix}0\\ 1 \end{bmatrix}\begin{bmatrix} 0 & 1\end{bmatrix}=\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix}$

(see the particle/wave duality?)

$n$ -body systems $\mathcal{H}_1\otimes\cdots\otimes \mathcal{H}_n$ ,

E.g. Electron orbital in an atom's shell $\mathbb{C}^2\otimes \mathbb{C}^2$ $\begin{aligned}|\psi\rangle & =\frac{\sqrt{2}}{2}(|\downarrow\uparrow\rangle-|\uparrow\downarrow\rangle)\\ & =\frac{1}{2}\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\end{aligned}$

entanglement = $|\psi\rangle$ where $|\psi\rangle$ has rank more than 1.

How stuff changes

(a mostly true version)

Particles come within Planck length,
Heisenberg uncertainty $\Rightarrow$ at that scale particles can burst into existence, so long as gone by sun down (Planck time), so "virtual particles"

How stuff changes

(a mostly true version)

entangled particles have exchanged some virtual particles, surplus/deficit now in common, but its "random" you cannot say who got what
because virtual particle exchange is net zero, i.e reversible, model by unitary $U|\psi\rangle$