You can find here a little pdf file where I tried to describe the scenario of quantum geometry of space, offered by the Loop Quantisation of Gravity (LQG), well discussed in my master's thesis, as user-friendly as possible.
Thereafter, the quantum geometry of space is modeled by the theory in a discrete lattice made of spin tetrahedra and one is able to describe the quantum geometric properties of space as observables of the theory: they reduce to some self-adjoint operators among finite dimensional vector spaces with real spectra containing the possible outcomes of the measurements, in a perfect quantum flavor.
Everything lies within the framework of multilinear algebra and here you can find Python code computing such matricial geometric observables of the gravitational field.
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Quantum Theory: Gravitational field with gauge symmetries
$\text{SU}(2)$ and diffeomorphisms. -
Dynamic Variable: Discretized connection on a lattice
$\Gamma$ with$N$ nodes and$L$ links, i.e. a map in$\text{SU}(2)^L$ . -
Quantum States: Functionals of discrete connections, expressed in the Peter-Weyl product basis in terms of the fundamental representation
$\rho^{1/2}$ of$\text{SU}(2)$ .
- Product representation
$\rho = \bigotimes_{i=1}^{k} (\rho^{j_i})$ identifying the lattice$\Gamma\leftrightarrow(j_1,...,j_k)$ with$1$ node and$k$ links. - Use of the theorem
$\rho^{j} = \text{Sym}^{2j}$ , being$\text{Sym}^k:=\bigodot_{i=1}^k\rho^{1/2}: \text{SU}(2)\to\text{GL}(\mathbb{C}^{k+1})$ .
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Construction: Utilizing the Pauli matrices
$\sigma_a$ and defining$T\rho(\tau_a) = -\frac{i}{2} T\rho(\sigma_a)$ ,$T$ being the tangent map. -
Form: Lie operators
${L_{(node, link)}}_a$ as tensor products of identities and$\tau_a$ in the$T\rho$ representation.
- Spinnets are elements of a Hilbert space
$H_{(j_1,...,j_k)}$ corresponding to an isotropic subspace$\text{Inv}(\rho) = {v \in V |\ \rho(U)v = v,\ \forall U \in \text{SU}(2)}$ , being$V\cong\bigotimes_{i=1}^k\mathbb{C^{2j_i+1}}$ the support vector space of$\rho$ . - Dimension of
$\text{Inv}\left({\rho^{1/2}}^{\otimes4}\right)$ for the ground state quantum tetrahedron is 2. - Dimension of
$\text{Inv}\left({\rho^1}^{\otimes4}\right)$ is instead 3. - In general, dimension of
$\text{Inv}\left({\rho^j}^{\otimes4}\right)$ turns out to be$2j+1$ .
- Also called geometric operators, are defined as
$D_{\alpha\beta} = (L_\alpha)^a\delta_{ab}(L_\beta)^b$ . - Restriction to the subspace
$\text{Inv}(\rho)$ .
- Generalize to tetrahedra with an arbitrary number of legs.
- Develop a program to calculate gauge invariant operators and Lie operators in a multilinear algebra context.
You can mainly refer to the folder lqg, that's where you can find a comprehensible workflow of the theory of quantum spin