# Tagged Questions

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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### Working towards finding Clebsch-Gordan coefficients for a single electron

I'm really confused about a problem involving a single electron which eventually wants me to calculate Clebsch-Gordan coefficients. I think this is probably because, I've only ever seen examples done ...
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### What is a Chiral Algebra for a group?

What do we mean by the Chiral Algebra for a group G (SO(3) etc )? Do you know a reference suitable for physicists? Thank you
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### LHC data and mathematics of QFT

I'm reading Frederic's Paugam Towards the Mathematics of Quantum Field Theory, an advanced theoretical physics book. I would like to know how I could apply the theories in this book. For example, ...
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### What is the fundamental representation in field theory?

In field theory we associate to each Gauge theory a continuous group of local transformations (a Gauge group), and then we require\define fermion fields to be irreducible representations belonging to ...
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### General definition of vector, spinor, and spin

I am looking for basic and exact definitions of fundamental physical concepts in graduate level. I reach this following definitions. Could you please help to improve these definitions. Spin: ...
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### The states of the adjoint representation correspond to the generators

I understand the definition of the adjoint representation. It uses structure constants as matrix components of generators, but I can't understand meaning of the states $|X_{a}\rangle$. What does "...
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### What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
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### The role of SO(3) and SU(2) in quantum mechanics [duplicate]

When studying the irreducible representations of SO(3) one usually looks at the irreps of the infinitesimal rotations instead, i.e. the ones of so(3), the Lie Algebra of SO(3). The Irreps of so(3) can ...
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### From Symmetry Group to Physics Equations

To the extent that I know: There are symmetry groups like the rotation groups SO(3), the Groups of Poincare Transformations,... If the physics of a system has a symmetry group G, then it can be ...
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### What is different in representation?

I'm sorry if this is somewhat a dumb question. First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear ...
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### Formalism and representation in Quantum Mechanics

I am just curious about the formalism of basic Quantum Mechanics. Lets take for instance the system of a spin-$\frac{1}{2}$ particle. The state of the particle is described by a vector in an abstract ...
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### Young Tableau Projectors: Does the order of symmetric and anti-symmetric projectors matter?

Given a Young Tableau we find the irreducible basis of an arbitrary tensor by projecting, The projectors are usually defined as first symmetrise over the row entries and then anti-symmetrise over the ...
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### The 6-j symbol and intersecting Wilson loops, redux

This is a quite specific question continuing the problems I have with computing the expectation value of intersecting Wilson loops I laid out here. Using the tools from the answer there, I quite ...
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### Subgroups of the Clifford Group

We recall the definition of a Clifford group (over $n$ qubits) is the set of unitary transformations: $$\{U: UPU^\dagger\in\mathcal{P}\}$$ where $\mathcal{P}$ denotes the corresponding Pauli group (...
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### Intersecting Wilson loops in 2D Yang-Mills

I am currently trying to understand 2D Yang-Mills theory, and I cannot seem to find an explanation for calculation of the expectation value of intersecting Wilson loops. In his On Quantum Gauge ...
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### What exactly is a coherent state and why is it interesting?

Please note that I do not have a background in physics, so if possible please refrain from a bunch of $|x\rangle$ notations, unless clearly specifying what it symbolically means. So I have been ...
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### Spinor reps in $\mathbb{R}^{1,3}\times{}B$ space-times

I am considering spinors in a space-time which is $\mathbb{R}^{1,3}\times{}B$ being $B$ a compact manifold of $D$ dimensions. I know that in ordinary 4 dimensional space-time spinors are ...
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I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.
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### What's a lepto-diquark?

This questions refers to Slansky's Group theory for unified model building, page 106 of chapter 7. He assigns the weight $(1)(01)$, which is stepwise projected from $E_6$ to $SU(2)\times SU(3)$, to a ...
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### Triality and charge

I have a few questions about triality for the representations of $SU(3)$. (I have seen the wikipedia page, but it does not make the connection with physics.) What is triality, how can you compute ...
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### Reducibility of tensor products of Lorentz group representations

Consider the statement: (34.29 in Srednicki's QFT text) $$\tag{34.29} (2,1)\otimes(1,2)\otimes(2,2)~=~(1,1)\oplus\ldots$$ Where of course, $(a,b)$ label representations of Lorentz group in the usual ...
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### How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group? [duplicate]

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
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### What guarantees the existence of unitary operators implementing Lorentz Transformations?

This should be a very basic question. In introductory QFT books, often one of the first things we see is the following claim: for every Lorentz transformation $\Lambda$, we can associate an unitary ...
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### How can one (formally) determine the particle content of a free field theory?

Here's my question: Suppose I'm given a free field theory, where my fields are functions $\phi:\mathbb{R}^4 \rightarrow V$, and the equations of motion are a system of linear Lorentz-invariant ...
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### What are particle multiplets in the Standard Model?

The particles of the standard model are often displayed in groupings known as multiplets. I know that this somehow relates to the underlying symmetries of the standard model, which can be viewed as ...
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### Representation of SU(3) generators

Let's discuss about $SU(3)$. I understand that the most important representations (relevant to physics) are the defining and the adjoint. In the defining representation of $SU(3)$; namely $\mathbf{3}$...
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### Do generalized Pauli Operators generate SU(n)?

A commonly used generalization of Pauli Operators is the "clock" and "shift" operators summarized here: http://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices Pauli Operators are generators ...
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### What is the Physical Significance of Tr(A) w.r.t. Matrix Representations in Group Theory

I've seen the post on mathoverflow.SE asking almost the same question, and I have indeed flipped through said answers, but most are in a more general context ie quantum mechanics and do not provide a ...
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### Why the generators of boosts transform like a vector under rotation?

$$\left[J_i,J_j \right]=i\epsilon_{ijk}J_k$$ $$\left[J_i,M_j \right]=i\epsilon_{ijk}M_k$$ $$\left[M_i,M_j \right]=-i\epsilon_{ijk}J_k$$ where $J_i$ is the generator of rotation of Lorentz group, $M_i$ ...
### Show the Lie algebra is the same for $SU(2) \times SU(2)$ and Lorentz group [duplicate]
So I know: $$[\sigma_{I},\sigma{j}] = 2i \epsilon_{ijk} \sigma_{k}$$ So two products of this should give us the Lorentz group: $SO(4) = SU(2) \times SU(2)$ Where $SO(4)$ has 3 Lie algebra which can ...
### Representations and transformations under an $SU(n)$ Lie groups?
I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the \$\...