Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. ...

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How to directly calculate the infinitesimal generator of SU(2)

We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\frac{\partial U}{\...
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613 views

Wigner-Eckart theorem of SU(3)

I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann ...
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1answer
56 views

Eigenvalues of spherical harmonics in $d$ dimensions

I'm working on the Schrodinger equation for a hydrogen atom in a $d$-dimensional space, so I'm interested in the possible eigenvalues of the angular momentum part of the $d$-dimensional Laplace ...
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1answer
318 views

Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
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1k views

Why helicity is proportional to the spin of particle and has two values?

How can it be shown without using the little group formalism? Let's have the Wigner's classification for the irreducible represetation of the Poincare group. For the massless case the eigenvalues of ...
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2answers
641 views

Equivalent Rotation using Baker-Campbell-Hausdorff relation

Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering ...
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138 views

Proving Lemma 4 in Georgi's Lie Algebra in Particle Physics 2nd p 251

The lemma 4 is given in the above picture. My question is, how to verify linear dependence (20.15) for diagram (a)? I tried to extend the matrix for the simple root in wikipedia $$ \left [\begin{...
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64 views

$(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$ [duplicate]

I am confused about the notation. What's the differences between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$, or maybe $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ? (...
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355 views

How to calculate $3\otimes 3$ and $3\otimes 3\otimes 3$ in $SU(3)$? [closed]

EDIT: I have boiled my question down to How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? A derivation would be nice too. OP: I know that I can ...
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108 views

From $U(3)$ to $SU(3)\times U(1)$ Color symmetry. There is a “gluon” photon-like?

Suppose that $U(3)$ was the gauge group. We can decompose this as $U(3)=U(1)\times SU(3)$, which implies that in addition to the $SU(3)$ that has eight generators corresponding to eight gluons, there ...
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235 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
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111 views

Finding symmetry of a part of an equation, given the group transformation property of another part

I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
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115 views

Calabi Yau compactification based on U(1) charges

In Green-Schwarz-Witten Volume 2, chapter 15, it is argued (roughly) that we need 6-dimensional manifolds of $SU(3)$ holonomy in order to receive 1 covariantly constant spinor field. And it turns out ...
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147 views

Does reversal of one spatial direction count as a discrete Lorentz transformation?

A transformation $\Lambda$ is a Lorentz transformation if it satisfies $\Lambda^T g \Lambda = g$, for the flat metric $g = \left( \begin{array}{cccc} 1 &&& \\ & -1 &&& \\ &...
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243 views

A whole lot of doubts on Lorentz representation

Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on ...
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244 views

What is the axial transformation of a group, i.e. $SU(3)$?

The Gell-Mann matrices $\lambda^\alpha$ are the generators of $SU(3)$. Applying an SU(3) - transformation on the triple $q = ( u , d, s )$ of 4-spinors looks like this: $$ q \rightarrow q' = e^{i \...
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representation of SU(2)

The question is regarding SU(2) group and SU(2) algebra. The SU(2) group can be generated by exponentiating the generators of SU(2) algebra $X_a$ as $exp(i t_a X_a )$ with $t_a$ being three parameters....
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235 views

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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2answers
159 views

Where does the $50^*$ in $SU(5): 10\otimes10= 5^*\oplus45^*\oplus 50^*$ in A. Zee QFT?

See A. Zee, QFT in a nutshell, Appendix B, eq. (24) (p. 469 in first edition with a typo $55^*\to50^*$, cf. Zee errata; p. 530 in second edition.) Where does the $50^*$ in $SU(5)$: $$10\otimes10= 5^*\...
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Are there any known potentially useful nontrivial irreducible representations of the Lorentz Group $O(3,1)$ of dimension bigger than 4? Examples?

Are there any known potentially useful, nontrivial, irreducible representations of the Lorentz Group $O(3,1)$ of dimension more than $4$? Examples? A $5$-dimensional representation? EDIT: Is there ...
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126 views

Convexity — reference request

I've been reading a few papers on generalized probabilistic theories, and have been struggling through proofs of some results that involve use of convexity and group theory, e.g. this paper on bit ...
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2answers
85 views

Is there a relation between spin and the spin group?

In Quantum Mechanics spin appears as one type of angular momentum. Indeed, in Quantum Mechanics one angular momentum on the state space $\mathcal{E}$ is a triplet of observables $\mathbf{J}=(J_1,J_2,...
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149 views

What's the relationship between uncertainty principle and symplectic groups?

What's the relationship between uncertainty principle and symplectic groups? Does the symplectic groups mathematically capture anything fundamental about uncertainty principle?
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312 views

Algebra, commutators and test functions

I am trying to make sense out of the algebra of the generators of the conformal group and I am running into some issues regarding how to calculate commutators. For instance, for translations of a "...
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1answer
125 views

$SO(4,2)$ symmetry of the hydrogen atom

The hydrogen atom with Hamiltonian obviously has $SO(3)$ symmetry since it just depends on the radius. $$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r}$$ This is generated by angular momentum $\mathbf{L}...
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1answer
139 views

Why are one-particle states called representations of Poincaré group?

The one-particle states in the Hilbert space of a quantized relativistic field theory are said to form representations of the Poincaré group. Why is that? I mean, popular texts in QFT do not ...
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315 views

Lorentz Algebra Representation and QFT

I just have a trouble making a full analogy between Lorentz Algebra Representation in Quantum Field Theory (QFT) and SU(2) representation in Quantum Mechanics (QM). To make my point, I will write few ...
4
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2answers
245 views

Traces in different representation

I am actually working with Green-Schwarz anomaly cancellation mechanism in which I have came across a strange formula which relates trace in the adjoint representation (Tr) to trace in fundamental ...
4
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1answer
217 views

Fractionalization and the structure of spin rotation group?

As we know, the phenomena of fractionalizations in condensed matter physics is fantastic, like fractional spin, fractional charge , fractional statistics, .... And one key point is that the ...
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2answers
91 views

What does U(N)xU(N)/U(1) mean?

I am studying a model in which SSB occurs and the original symmetry group is U(N)xU(N)/U(1) it acts as: $M'=AMB^{-1}$ (M is an NxN matrix containing the fields) The groud state i have found is ...
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1answer
76 views

What is the $10$ in the $\mathbf{4}\otimes\mathbf{4}$ tensor product of $SO(6)$?

This is the question 22.D in Howard Georgi's Lie Algebras book, I thought about for a minute, but couldn't come up with a plausible answer. It's a fact that the SO(6) and SU(4) algebras are ...
4
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1answer
104 views

Integrating elements of a Lie group with respect to parameters of the corresponding Lie algebra

I am working with an operator $\textbf{M}$ that is represented by the Lie group SO(1,3), thus it can be written as, $$ \textbf{M} = \exp{\textbf{L}} $$ where, $$ \textbf{L} = \begin{bmatrix} 0&a&...
4
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2answers
282 views

SU(3) antiquark triplet transformation

I'm reading a rather elementary particle physics text, Modern Particle Physics by Thomson. He is staying away from the heavy group theoretic stuff. He derives the transformation law for an SU(2) ...
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1answer
74 views

What does “action of a gauge group on a particle” mean?

I have come across this phrase left handed fermions transform under $SU(3)\times{}SU(2)\times{}U(1)$ differently from the way right handed fermions do. I am just beginning to learn about how the ...
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1answer
161 views

Spinor representation restricted under subgroup, a formula from Polchinski

The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested ...
4
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1answer
295 views

Question about the Killing form in Georgi

I'm trying to understand the Killing form described on page 49 in the book by Howard Georgi. He starts by saying that one defines the inner product between two generators $T_a$ and $T_b$ in the ...
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3answers
428 views

Must all symmetries have consequences?

Must all symmetries have consequences? We know that transnational invariance, for example, leads to momentum conservation, etc, cf. Noether's Theorem. Is it possible for a theory or a model to have ...
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1answer
53 views

Non coherence of Fermions and Bosons through $U(1)$

I "know" the textbook answer why we cannot write, $$ |\psi\rangle = a|j=\tfrac{1}{2}\rangle + b|j=1\rangle $$ as "each term in the quantum superposition transforms differently under $U(1)$", $$ U(2\...
4
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3answers
79 views

In most physical cases, the elements of a group can be represented by unitary matrices. Why no time-reversal?

In Dresselhaus's group theory page 19, a theorem writes: Every representation (of a Hamitonian's group) with matrices having non-vanishing determinants can be brought into unitary form by an ...
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1answer
143 views

How to interpret spin observables constructed by non-standard phase choices?

If we try to find matrix elements of ladder operators ( $J_{\pm}$) for spin when they act on eigenstates of $J^2$ and $J_z$ ( $\newcommand{ket}[1]{\left|#1\right\rangle} \newcommand{avg}[1]{\left\...
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1answer
352 views

Is General Relativity based on a Symmetry?

In short: Is there any kind of symmetry one can start with to derive general relativity (GR)? Longer: Einstein had the opinion that GR was the generalisation of special relativity, because instead of ...
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3answers
540 views

Can a Wick rotation be performed on the Pauli algebra to get from $+++$ to $+--$ signature?

Wick rotation makes sense for the Dirac algebra: it has $+---$ and $++++$ and $----$ signatures. Just wondering if one can Wick rotate the Pauli algebra from the standard $+++$ Pauli matrices to $+--$....
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2answers
83 views

Bilinears in adjoint representation

Below are two statements from my notes and I am trying to verify them explicitly. In both cases the fields are assumed to transform under the fundamental representation of $O(N)$ - --'The kinetic ...
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1answer
99 views

General construction of equations of motion for free particles

I've got a question regarding the different Symmetrie-Lie-Groups of Newtonian Mechanics and special realtivity. Is there a canonical way to obtain the equations of motion for a free particle only by ...
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1answer
639 views

Is the symmetry group of two spin 1/2 particles $SU(2) \times SU(2)$ or $SU(4)$?

This is a simple question. Please forgive me, as I am a lowly experimentalist. Suppose we have two free spin 1/2 particles, i.e. a 4-fold degenerate system. What is the set of symmetry operations ...
4
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1answer
280 views

Does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom?

The starting point and notations used here are presented in Two puzzles on the Projective Symmetry Group(PSG)?. As we know, Invariant Gauge Group(IGG) is a normal subgroup of Projective Symmetry ...
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1answer
439 views

Lorentz group and classification of fields by their transformation under Lorentz transformations

Let's have Lorentz group with generators of 3-rotations, $\hat {R}_{i}$, and Lorentz boosts, $\hat {L}_{i}$. By introducing operators $\hat {J}_{i} = \frac{1}{2}\left(\hat {R}_{i} + i\hat {L}_{i}\...
4
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1answer
224 views

Linearizing Quantum Operators

I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$ The right hand side ...
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89 views

Completely positive maps and symmetric states

Let $\mathcal{N}$ be a completetely positive trace preserving map (aka a quantum channel) acting on a finite dimensional system $\mathrm{A}$, and let $\pi$ denote the maximally mixed state on $\mathrm{...
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138 views

Decomposing a representation under a subgroup [closed]

I am trying to understand what is the method for decomposing representations of a group under one of its subgroups. I already had a look in Slansky, but I could not extract a concrete set of rules/...