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Questions tagged [representation-theory]

The systematic study of group representations, which describe abstract groups in terms of linear transformations of vector spaces, such that group elements or their generators are represented as matrices, reducing group-theoretic problems to linear-algebraic ones.

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Representation of the $\rm SU(5)$ model in GUT

In Srednicki's textbook Quantum Field Theory, section 97 discusses Grand Unification. On page 606, it states: In terms of $\rm SU(5)$, we have \begin{equation} 5 \otimes 5 = 15_{S} \oplus 10_{A} ...
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What is the spinor representation of other groups beside $SO(p, q)$?

I am studying a lecture about superconformal algebras and it claims that there is a superconformal algebra in $d=5$ where supercharges belongs to spinor representation of $F_4$ (which is an ...
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Inner product on group theoretic coherent states and anti-commutator of Lie algebra generators

This question is related to group theoretic coherent states (Gilmore, Perelomov etc.). I consider a semi-simple Lie group $G$ with Lie algebra $\mathfrak{g}$ and a unitary representation $U(g)$ acting ...
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Observables labelling one-particle states in Quantum Field Theory

I'm studying introductory QFT using the first volume of Weinberg's series, and i'm having problems in understanding how single particle states of the free theory are labelled, i.e. what observables ...
401 views

Position representation of spin states and spin operators

How can we represent a spin states $\lvert S_x:+\rangle, \lvert S_y:+\rangle,\lvert S_z:+\rangle ,\lvert S_x:-\rangle, \lvert S_y:-\rangle$ and $\lvert S_z:-\rangle$ in position representation ...
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Representations of the rotation group

(I have already done a similar question, but I did not express myself very well and the question was a bit confusing, so let me try again. If you find the question repetitive, I apologize and you can ...
92 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
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Combining SU(N) multiplets using Young diagrams

I am trying to follow the Particle Data Group's instruction (PDF link) to combine SU(N) multiplets. On page 3, they show an example calculation of SU(3)'s $\textbf 8\otimes \textbf 8$. I understand ...
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Normal mode decomposition of a triangular hexagonal lattice

I was trying to understand and redo the methods used in a previous question: Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice ...
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Unitarity representations of CFT in arbitrary dimensions

There is a well defined notion of unitarity of representations in Euclidean Conformal field theories that follows from the requiring unitarity in the Lorentzian space. Under this notion, all states ...
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Dielcetric Tensor Transforms under Product Representation

How do I show that the Dielectric tensor in 2D transforms as a product of representations? I am told that the electric displacement and electric field transforms with the representation $D^v(g)$, and ...
127 views

What does matrices act on different spaces mean in QFT?

I have a Dirac kinetic term in a Lagrangian. $$i\bar{\psi}\gamma^\mu D_\mu\psi = i\bar{\psi}\gamma^\mu\partial_\mu\psi + g\bar{\psi}\gamma^\mu\psi A^a_\mu T^a,$$ However, I usually heard that ...
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Product of generators in fundamental representation of $SU(N)$

I'm trying to prove equation 25.20 in Schwartz: $$T^a T^b=\frac{1}{2N}\delta ^{ab}+\frac{1}{2}d^{abc}T^c + \frac{1}{2}if^{abc}T^c,\tag{25.20}$$ where $T^a$ are the fundamental representation ...
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Why in QFT what really matters is $\exp(\mathfrak{so}(1,3))$ instead of $O(1,3)$?

In QFT fields are classified according to representations of the Lorentz group $O(1,3)$. Now, most books when getting into this say that in order to understand the representations of $O(1,3)$ we need ...
Reading Griffiths' Quantum Mechanics. We have the electronic confirmation of Carbon as $$(1s)^2 (2s)^2 (2p)^2$$ in the ground state. He says There are two electrons with orbital angular ...
I'm following the rules in this document to combine irreps of $SU(N)$ using Young tableaux. If I'm not mistaken the product of two irreps should be symmetrical, that is $A \otimes B = B \otimes A$. I'...