First of all, when considering real Lie algebras it is cleanest to use the mathematical convention, i.e. $su(2)$ consists of traceless anti-hermitian $2\times 2$ matrices. OP has already shown that $sl(2,\mathbb{R})$ (which is the set of traceless real $2\times 2$ matrices) contains 2 ladder operators $\sigma_{\pm}$, such that $[\sigma_3,\sigma_{\pm}]=\pm 2\... 2 I think you missed an important point in the very first line of the question: An electron is confined in a prism-shaped quantum dot of$D_{3h}$symmetry i.e. the quantum dot they're referring to is a triangular prism, or a quantum Toblerone. A point group is not a molecule, it is a description of a finite set of operations (rotations, reflections and ... 1 I cannot decipher your peculiar notes of your understanding, but the paper appears straightforward. Defining$U_L=\exp (i\vec \tau \cdot \vec \theta),\qquad U_R=\exp (i\vec \tau \cdot \vec \phi)$, you may observe $$\mathcal{ H}= \begin{bmatrix} H^+\\ H^0 \\ \end{bmatrix}, ~~~\tilde{\mathcal{ H}}=-i\tau_2 \mathcal{ H}^*= \begin{bmatrix}- H^0\\ H^- \\ \... 2 Consider the book Group Theory in Physics - Problems and Solutions by Michael Aivazis. Here is the table of contents. (1) Basic group theory (2) Group Representations (3) General properties of Irreducible Vectors and Operators (4) Representations of the Symmetry group (5) 1 - D Continuous groups (6) SO(3) (7) SU(2) and SO(3) (8) Euclidean groups (9) Lorentz ... 1 There are two related but different notions of "light cone" (or "null cone"): one in the spacetime [which Minkowski introduced in 1907/1908 as part of his "Space and Time", which introduced the "spacetime viewpoint" and the various terms we use today in relativity], and the other in the tangent space of a spacetime ... 4 Light cones can more generally be defined for a curved spacetime: Consider a 4-dimensional Lorentzian manifold, that is, a 4-dimensional smooth manifold \mathcal{M} together with a metric g with signature (+,-,-,-). Then we can define for each point p\in\mathcal{M} the "light-cone" V_{p}\subset T_{p}\mathcal{M} to be the set of all &... 8 Given a four-vector A^\mu, define the “interval” associated with A as$$ \Delta s_A = \eta_{\mu\nu}A^\mu A^\nu = \left(A^0\right)^2 -\vec A{}^2 $$We say that A is “spacelike” if \Delta s_A < 0. An example is (0, \vec A). “timelike” if \Delta s_A > 0. An example is (A^0, \vec 0). “lightlike” if \Delta s_A = 0. The “light cone” is the ... 2 The light cone is definied to be the set of 4-vectors (ct,x,y,z) satisfying$$c^2t^2 - x^2 - y^2 - z^2 = 0.$$Or written in covariant notation$$\eta_{\mu\nu} x^\mu x^\nu = 0.$$(image from Einstein for Everyone - Spacetime) 0 I do not have that text, and I would rather not shadow-box with its logic. There are better ways to introduce the SU(2) conjugate doublet representation, \tilde \xi\equiv \zeta \xi^*, linked to, e.g. this or this. He is using the special form of U, (3), dictated by unitarity, to demonstrate what he claims, namely that \xi \sim \tilde \xi, (10). However, ... 3 We do the exact same thing in the standard model when we break the electroweak symmetry. There are 3 broken generators \delta_-/2, \sigma_1/2, \sigma_2/2, and a leftover preserved U(1)_{EM} generator, \delta_+/2, where$$ \delta_\pm = \frac{1}{2}(\mathbf{1}\pm \sigma_3) $$are linear combinations of the EW gauge group generators. You can verify that ... 2 It is assumed you have appreciated Inönü, E.; Wigner, E. P. (1953), "On the Contraction of Groups and Their Representations" Proc. Natl. Acad. Sci. 39 (6): 510–24, and the super-helpful Gilmore text in Group contraction. Very crudely, the Poincaré Lie algebra,$$ [J_m,P_n] = i \epsilon_{mnk} P_k ~, \qquad [J_i, P_0] = 0 ~, \\ [K_i,P_k] = i \... 3 As a mathematically careful and reasonably rigorous discussion of the mathematical structures of conformal field theory, I highly recommend Schottenloher's "A Mathematical Introduction to Conformal Field Theory". In particular, its chapters 2 and 5 deal almost exclusively with the correct notions of "conformal group" and "conformal ... 1 Mukhanov is talking about the SVT decomposition, $$h_{ij} = 2 C \delta_{ij} + 2 \left( \partial_i \partial_j - \frac13 \delta_{ij} \nabla^2 \right) E + 2 \partial_{(i} \hat{E}_{j)} + 2 \hat{E}_{ij}$$ where$C$and$E$are scalars,$\hat{E}$is a vector, and$\hat{E}_{ij}$is a tensor. As you can see, all of the terms in the expansion are symmetric tensors, ... 1 Thus it seems we can only reach$|{\uparrow \downarrow}\rangle$and$|\! \downarrow \uparrow \rangle$with real coefficients [...]. How can I rewrite these observation as a span of some basis? You can't. The span of a basis is a vector subspace$V$, which means that if$|{\uparrow \downarrow}\rangle \in V$then you must also have$i|{\uparrow \downarrow}\...