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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 ...


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 &...


4

Maybe it's useful to think it through in reverse? If you've got a 4-vector $p$, you can always try to put it in some standard form. For example: If $p^2 \neq 0$, you can boost to zero out the 3-momentum part, and you end up with a vector of the form $k = (m,0,0,0)$. If $\Lambda$ is the boosting Lorentz transform, then $k = \Lambda p$, so $p = \Lambda^{-...


4

Hints: Firstly, $$SO(10) ~\supseteq~ SO(4)\times SO(6),\tag{1}$$ so we get the branching rules $$ {\bf 10}~\stackrel{(1)}{\cong}~({\bf 4},{\bf 1}) \oplus ({\bf 1},{\bf 6}),\tag{2}$$ and $${\bf 10}\wedge{\bf 10}~\stackrel{(2)}{\cong}~ ({\bf 4}\wedge{\bf 4},{\bf 1})\oplus ({\bf 4},{\bf 6})\oplus ({\bf 1},{\bf 6}\wedge{\bf 6}).\tag{3}$$ Here the space of 2-...


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 ...


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

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)


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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 \...


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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, ...


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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}\...


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