Use as a synonym to the representation-theory tag

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2
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1answer
159 views

How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
1
vote
3answers
860 views

Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
4
votes
1answer
110 views

Representations of SO(3) and the classification of relativistic massive particles as in Weinberg's “The Quantum Theory of Fields”

I'm reading about the classification of relativistic massive particles in Weinberg's "The Quantum Theory of Fields", and I found something that doesn't convince me. In Chapter 2, paragraph 5, having ...
9
votes
2answers
1k views

Irreducible Representations Of Lorentz Group

In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as ...
2
votes
1answer
131 views

$SU(3)$ Tensor Methods in a Tetraquark

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
4
votes
1answer
158 views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) ...
6
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2answers
195 views

Does the $\bf{1+3}$ representation of $SU(2)$ also represent $SU(2)\times SU(2)$?

I'm a bit confused about this following issue concerning representations of $SU(2)$. Denote by 1 the 1-dimensional representation of the group $SU(2)$ (=the spin 0). Similarly, denote by 2 and 3 the ...
6
votes
1answer
132 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
2
votes
1answer
116 views

Representation of U(1) on fock space

I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making ...
3
votes
2answers
174 views

Why do decompositons like $16 \otimes 16 = 10 \oplus 120 \oplus 126$ tell us which Higgs representations we can use?

EDIT: I found an answer, which I do not understand: In Gürsey - Symmetry breaking patterns in E6 he writes: " Because of Fermi-Dirac statistics of fermions they must occur in the symmetric part of ...
0
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0answers
19 views

Singularity of $\Gamma(g) - I$ [migrated]

Let $\Gamma$ be a non-trivial irreducible unitary representation(also suppose of finite degree $d$) of a nontrivial group $G$. Consider the matrices $\Gamma(g) - I$, where $g\in G$ and $I$ is the ...
4
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0answers
113 views

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 ...
0
votes
2answers
53 views

Transformation of $| JM\rangle$ under the group of rotations

I am following the Quantum Mechanics I, Galindo A., Pascual P. and in page 207 explaining the matrix representations of the Rotation Operators in the angular momentum it appears the next (obvious) ...
3
votes
1answer
63 views

How to build tetra-quark mesons symmetry group?

I was thinking about the issue while reviewing my group theory notes. One can construct mesons with a nonet as an octet and a singlet, $SU(3)\otimes SU(3) = 8\oplus \bar{1}$. In a same way but for ...
3
votes
1answer
104 views

How to write the Clebsch-Gordan decomposition in tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is known that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ may be decomposed through the ...
0
votes
1answer
4k views

How is multiplicity given by 2S+1?

Suppose there are two electrons in an atom with $s_1 = \frac{1}{2}$, $l_1 = 1$ and $s_2 = \frac{1}{2}$, $l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either $+2$, $+1$ ...
1
vote
1answer
75 views

How to manipulate higher spin systems (higher than spin 1/2) using a given operator?

I’ve been reading ¨Halzen, F., and A. D. Martin. Quarks and Leptons. New York: Wiley Text Books, January 1984. ISBN: 9780471887416¨, and I’d like some clarification of a concept, please: I’m ...
2
votes
1answer
695 views

QCD color factors from quark gluon vertices

The color factors in QCD tell us the relative strength of the coupling of a quark emitting a gluon, a gluon emitting a quark-antiquark pair or a gluon emitting two gluons. To calculate let them we ...
0
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0answers
39 views

Addition of $N$ spin halves

If I have two spin-halves, then \begin{align} \frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 1. \end{align} If I have three spin-halves, then \begin{align} \frac{1}{2} \otimes \frac{1}{2} \otimes ...
0
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0answers
44 views

Deriving Pauli Matrices

How does one derive using, say, the operator formula for reflections $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + ...
1
vote
1answer
44 views

Representation of Symmetry group

Suppose the SE $\boldsymbol{H}\psi=E\psi$ describes a closed system and $G$ is a symmetry group of the system. Then any transformation in $G$ leaves the form of the SE invariant. It seems plausible to ...
3
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0answers
76 views

What is the meaning of SU(2) triplet scalar field? [closed]

The following is an about a Left-Right Symmetric model. $SU(2)\otimes SU(2)$ $(2\otimes 2=3\oplus 1)$ will generate a triplet, which in Left-Right Symmetric model is ...
9
votes
1answer
750 views

Why do we identify symmetric 2nd rank tensors with spin-2 particles in string theory?

I am going through Tong's lecture notes on String Theory and came across the following irrep decomposition (Chap 2, p.43) of the bosonic string first excited states: $$\text{traceless symmetric} ...
3
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1answer
104 views

Finding Electronic Energy Levels by Representation Theory

Let $$u=\left( \begin{array}{cccc} c_1&c_2&c_3&c_4 \end{array} \right)^T$$ for $$\psi = c_1\psi_1 + c_2\psi_2 + c_3\psi_3+ c_4\psi_4$$ We assume that $\left<\psi_i|\psi_j\right> = ...
7
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2answers
114 views

Lie groups with same algebra

I had a problem when considering symmetry breaking in an SO(4) gauge theory: $\mathcal{L} = \left| D_\mu\phi \right|^2$ where $D_\mu$ is the SO(4) covariant derivative. Then assuming there is some ...
3
votes
2answers
77 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 ...
1
vote
1answer
22 views

Bifundamental representations [closed]

Can someone give me explicit examples (in matrix form) of bifundamental representations? Illustrative would be for instance: a) SU(3) x SU(2) b) SO(4) x U(1) c) E6 x U(1) but other you may have ready ...
3
votes
1answer
78 views

Does there exist finite dimensional irreducible rep. of Poincare group where translations act nontrivially?

I read several textbooks of QFT and find that there are two ways to classify the particles or fields. The first one is to study the irreducible representation of Lorentz group (or exactly the ...
5
votes
3answers
2k views

Dirac spinor and Weyl spinor

How can it be shown that the Dirac spinor is the direct sum of a right handed Weyl spinor and a left handed Weyl spinor? EDIT:- Let $\psi_L$ and $\psi_R$ be 2 component left-handed and right-handed ...
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0answers
46 views

Example of a symmetry and the group with which it is modelled? [duplicate]

Could you please provide a specific example of a symmetry and the group with which it is modelled? I am beginner to study symmetry in physics, please answer with just an example. This question is ...
5
votes
1answer
152 views

Unitary gauge for non-abelian case

I'm reading Chapter 19 of Mandle and Shaw's Quantum field theory. In the first section it is explained that one can go with a $SU(2)$ followed by a $U(1)$ transformation from ...
0
votes
1answer
37 views

Number of supercharges in $N\ge 1$ SUSY in $d\ge4$ dimensions

I have a couple of questions concerning supercharges and superalgebras: 1, In four-dimensions, the minimal spinor representations are Weyl spinors. These have 4 real degrees of freedom (dof). This ...
0
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0answers
45 views

Calculating Clebsch-Gordan coefficients through Racah's formula

So Clebsch-Gordan coefficients are found in tables, but I need to calculate them using Racah's formula, which reads as following: $c_+ (J,M) f_{m_1}^{M+1}=c_+(j_1,m_1-1) f_{m_1-1}^M + c_+(j_2,M-m_1) ...
2
votes
2answers
106 views

Why representations instead of just groups?

This question is essentially asking for a clarification on what has already been said in this one. What I don't understand is why it is the representations that are important in Quantum Field Theory ...
6
votes
2answers
85 views

Poincare representations for interacting field theory

I was going through Rudolf Haag's memoir http://link.springer.com/article/10.1140%2Fepjh%2Fe2010-10032-4 and came across these lines: '..in quantum field theory (or for any system of interacting ...
0
votes
1answer
24 views

Decomposition of the adjoint representation of a spontaneously broken compact group

Let be $G$ a compact group, symmetry of the theory I am working with. $G$ is broken into one subgroup $H$. I define the generators of G as $T_A = \{T_a,T_\hat{a}\}$, where the first are the unbroken ...
2
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0answers
46 views

Non-physical representations of double group

In group theory, to account for electron spin, double group is introduced. The key difference between an ordinary point group and a double group is an extra element $\bar{E}$ with the meaning of a ...
3
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0answers
64 views

What is physically irreducible representation?

When I use bilbao crystallographic server recently, I noticed a notation called physically irreducible representation. Paper says it is a direct sum of two complex conjugate representations (if ...
1
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2answers
112 views

Symmetry and degeneracy in quantum mechanics

If an operator commutes with the Hamiltonian of a problem, does it always must admit degeneracy? For example, parity operator commutes with the Hamiltonian in case of a free particle and we have two ...
0
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0answers
80 views

Rotations in Bloch Sphere about an arbitrary axis

I am trying to understand the following statement. "Suppose a single qubit has a state represented by the Bloch vector $\vec{\lambda}$. Then the effect of the rotation $R_{\hat{n}}(\theta)$ on the ...
1
vote
1answer
55 views

Correct Yukawa Term with a SU(2) Higgs Triplet?

Given $SU(2)$ doublet fermions $\Psi^1$ and $\Psi^2$ and a $SU(2)$ triplet Higgs $H$, how does the correct Yukawa term look like in tensor notation? Schematically, we have $$ 2 \otimes 2 \otimes 3 ...
2
votes
1answer
85 views

How is the Lorentz group related to spin? [closed]

I've been reading about the agebra of the Lorentz group. It is given by, $$G\equiv SO(1,3) ~\cong~ SU(2)\times SU^*(2)$$ Now, representations of this group $G$ as labelled by $(j,j')$ where $j$ is ...
5
votes
2answers
236 views

What is the exact relation between $\mathrm{SU(3)}$ flavour symmetry and the Gell-Mann–Nishijima relation

I'm trying to understand how the Gell-Mann–Nishijima relation has been derived: \begin{equation} Q = I_3 + \frac{Y}{2} \end{equation} where $Q$ is the electric charge of the quarks, $I_3$ is the ...
4
votes
1answer
169 views

What defines the spin of a certain field? (formally)

Update: see the restatement of the question below! I've seen this question over and over through the archive of questions, but so far the closer to an answer was this. But I still don't understand. ...
11
votes
3answers
282 views

Group representations as vectors and isomorphism between weights and matrix generators

This might be something basic, but it is unclear to me. So I am used to work with representations of groups as matrices. These matrices represent the structure of the Lie algebra by satisfying the ...
2
votes
1answer
438 views

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 ...
4
votes
2answers
239 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 ...
1
vote
3answers
119 views

Calculations with angular momentum

Is the following correct, when adding 3 angular momenta/spins: \begin{align} 1\otimes 1\otimes \frac{1}{2}&=\left(1\otimes 1\right)\otimes \frac{1}{2} \\ &=\left(2\oplus 1\oplus ...
3
votes
1answer
77 views

How to calculate U-spin for baryons

I am trying to calculate U-spin for sigma baryons. I don't know why U-spin for $\Sigma^{+}$ and $\Sigma^{-}$ and $\Sigma^{*+}$ is 1/2, but for $\Sigma^{*-}$ is 3/2? I know that $\Sigma^{+}$ and ...
1
vote
3answers
123 views

Is the local Lorentz transformation a general coordinate transformation?

There is a saying in Nakahara's Geometry, Topology and Physics P371 about principal bundles and associated vector bundles: In general relativity, the right action corresponds to the local Lorentz ...