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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3
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1answer
143 views

Is an electron technically a set of two particles?

The electron - described as a four-spinor in the Dirac equation - transforms according to the $(1/2,0)\oplus(0,1/2)$ representation of the Lorentz group, so it is actually a direct sum of a left- and ...
1
vote
1answer
90 views

Spin of a gauge field

I was wondering what is the simplest way to understand the reduction of the Wigner's little group from $SO(d-1)$ to $SO(d-2)$ when one considers massive and massless fields respectively (in a $d$ ...
3
votes
1answer
76 views

Symmetry breaking to a special subalgebra?

This is a follow-up to my question here. For regular subalgebras of some group's Lie algebra the root system of the subalgebra is a subset of the root system of the original's group algebra. In ...
0
votes
1answer
40 views

Raising and lowering operators for a composite isospin $SU(2)$ system

Consider pion states composed of $q \bar q$ pairs where $q \in \left\{u,d \right\}$ transforms under an $SU(2)$ isospin flavour symmetry. These bound states transform in the tensor product $R_1 ...
0
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0answers
32 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 ...
4
votes
1answer
66 views

What is the Lorentz group representation for a general spin?

Setup, as I understand things so far: One way to think about where the spin of a quantum field comes from is that it is a consequence of the ways that different types of fields transform under ...
-1
votes
0answers
102 views

How much analysis do we need to define a (classical) quantum system?

I'm trying to figure out to what extent is analysis involved in the definition of a (classical) quantum system. Classical meaning not QFT. Here's a definition I came up with: Defnition: A quantum ...
0
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1answer
107 views

How many eigenstates for four (non-identical) spin 1/2 particles? [closed]

Question Consider a system of four non-identical spin 1/2 particles. Find the possible values for the total spin and state the number of eigenstates for each of these. Attempt So I coupled S1 and ...
14
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3answers
1k views

Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: ...
-1
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0answers
80 views

Transformation properties of $2 \times 2$ matrix involving Pauli matrices

Suppose the vector $\phi$ transforms under $SU(2)$ as: $$\phi' = (\exp(-i \alpha \cdot t))_{ij}\phi_j,$$ where $(t_j)_{kl} = −i \epsilon_{jkl}$ and $j, k, l \in \left\{1, 2, 3\right\}.$ Based on ...
15
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1answer
593 views

Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice

Happy holidays, everyone! The following is part question, part visual gallery, and part classical mechanics problem. Inspired by snow over the weekend I began simulating the vibrations of the ...
6
votes
1answer
127 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 ...
3
votes
1answer
74 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 ...
1
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0answers
30 views

Ordering of basis elements of a Lie-group representations tensor product [migrated]

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \oplus_{J}\textbf{r}_J $$ where $\textbf{r}_J$ are ...
0
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0answers
33 views

What's the relation between representation theory and mass / electric charge?

This is a follow-up on this answer, where ACuriousMind wrote Formally, both the mass and the charge classify certain irreducible representations of the Poincaré group and the circle group, ...
2
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0answers
42 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
votes
1answer
76 views

How to find the remaining subgroup after some linear combination of Higgs fields gets a VEV?

This is a follow-up question to this question. How can I compute which generators remain unbroken when a linear combination of Higgs fields $a \Phi_1+ b\Phi_2$ get a vev? If I compute the unbroken ...
3
votes
1answer
64 views

Anticommutative Sets of SU(N) Generators? Anticommutative Analogue to Cartan subalgebra?

I am currently studying SU(N) generators in order to find bases that may suit a problem at hand. I am especially interested in getting as large anticommuting sets within a basis as possible. In SU(2) ...
2
votes
1answer
83 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 ...
4
votes
1answer
165 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. ...
0
votes
1answer
39 views

Matrix for Ladder Operators?

I found this website which shows how to derive the matrices for $L_{+}, L_{-}$ and while I understand the derivation of the equation for $<lm|L_{+}|lm'>$ and $<lm|L_{-}|lm'>$ I do not ...
2
votes
1answer
426 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 ...
1
vote
1answer
96 views

Interpretation of vector mesons in QCD

It is well-known that scalar mesons are interpreted as pseudogoldstone bosons which is connected with spontaneous broken $SU(3) \times SU(3)$ symmetry to $SU(3) \times SU(3) / SU(3)_{chiral}$. Is ...
4
votes
1answer
93 views

Spin 3/2 matrices in terms of Pauli matrices

Let $\sigma_i (\frac{3}{2})$ be the three generators of the irreducible spin 3/2 representation of $SU(2)$ (see http://easyspin.org/documentation/spinoperators.html for their explicit forms). ...
3
votes
1answer
170 views

The derivation of the irreducible representations of the Lorentz group

I took the way of classification of Lorentz group representations from Sexl, Urbantke, Relativity, groups and particles (Germ. ed. 1975). But I don't understand it as I outline in the following: In ...
2
votes
2answers
244 views

Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
2
votes
1answer
142 views

What does Addition of Angular Momenta tell us about Group Theory?

I've come across this a lot, but I've never understood it. I do know basic Group Theory including Lie Groups. In Introduction to Quantum Mechanics, Griffiths ends the chapter on spin with the remark ...
0
votes
2answers
59 views

Good reference on the parametrization of $SU(3)$ and $SU(N)$

For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation: $s_2=\begin{bmatrix} e^{i\alpha}\cos(\theta) & -e^{-i\beta}\sin(\theta) \\ ...
3
votes
1answer
227 views

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 $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
0
votes
0answers
41 views

Dirac Spinors as Representation of $SL(2,\mathbb{C})$ over grassmann algebra

Recently, I've learned that the clifford algebra can be regarded as the quantization of grassmann algebra. This is shown from the following two papers by Berezin. 'Classical spin and Grassmann ...
20
votes
1answer
439 views

Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics: The universal cover: For instance, when representing the rotation group ...
4
votes
4answers
649 views

How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
2
votes
0answers
64 views

Use Cartan subalgebra in spinor representation to find weights of vector representation

For $SO(2n)$ we can construct the lie algebra elements by using antisymmetric combinations of $\gamma_\mu$ which obey the Clifford algebra. Up to some prefactor the elements $ S_{\mu \nu} = \alpha ...
2
votes
1answer
77 views

Proof that trace is independent of representation [closed]

$$\begin{align} \sum_{a'} \langle a'|X|a'\rangle &=\sum_{a',b',b''} \langle a'|b'\rangle \langle b'|X|b''\rangle\langle b''|a'\rangle \\ &=\sum_{b',b''} \langle b''|b'\rangle \langle ...
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0answers
71 views

Books on representation theory [duplicate]

Possible Duplicate: Best books for mathematical background? I'm looking for a textbook on the group/representation theory for a student-physicist. The main questions of interest are ...
1
vote
1answer
59 views

Simple concept question about the dimensionality of a representation in point group

Concept question about the dimensionality of a representation in group theory here: Look at 3.1(c) of problem set, from group theory application to the physics of condensed matter of M.S.Dresselhaus: ...
0
votes
0answers
25 views

Do tensor product tables for irreducible representations apply for non-symmorphic space groups?

I'm reading Dresselhaus's book on group theory for solid-state physics, but I'm having trouble understanding how to get irreducible representations for phonons away from $\mathbf{k} = \mathbf{0}$ for ...
0
votes
0answers
39 views

How do I expand the state $| x \rangle$ in terms of another orthogonal basis?

In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by: $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$ ...
3
votes
1answer
124 views

Why is there no 1/3 spin? [duplicate]

Why do no particles have a 1/3 spin? Why are all particles' spin either a half-integer or integer? How would a particle with such a spin behave, as a fermion, boson, or neither?
2
votes
1answer
100 views

Does scale invariance imply massless or continuous mass distribution?

$\newcommand{\ket}[1]{\lvert #1 \rangle}\newcommand{\bra}[1]{\langle #1 \rvert}\newcommand{\scp}[2]{\langle #1 \vert #2 \rangle}$ In his 2008 slides (PDF), Tzu-Chiang Yuan mentions the following on p. ...
3
votes
1answer
59 views

Decomposing massless N=8 SUGRA multiplet into multiplets of massless N=4

The only massless $N=8$ SUGRA multiplet is given by $(g_{\mu\nu},\psi_\mu^{\Sigma},A_\mu^{[\Sigma\Pi]},\chi_{\alpha}^{[\Sigma\Pi\Lambda]} ,\phi^{[\Sigma\Pi\Lambda\Omega]})$ where the greek upper ...
3
votes
5answers
383 views

Eigenspaces of angular momentum operator and its square (Casimir operator)

The casimir operator $\textbf{L}^2$ commutates with the elements $L_i$ of the angular momentum operator $\textbf{L}$: $$ [\textbf{L}^2, L_i] = 0. $$ However, the $L_i$ do not commute among ...
3
votes
0answers
93 views

Characters of extra representations in the double group of O

Looking at the character table for $\overline{O}$ (double group of $O$) in a book, I noticed that two out of three of the additional irreps (with respect to the five irreps from $O$ itself) are ...
1
vote
1answer
69 views

How to construct fields from from unitary representation of the Poincaré group?

I want to construct fields from unitary representation of the Poincaré group but I do not know how. In Weinberg book he proposed that the Hamiltonian should be of certain kind and from that he derived ...
0
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0answers
39 views

Construct fields from from unitary representation of Poincaré group

I am trying to understand how construct fields from unitary representation of Poincaré group and the reasoning that Weinberg give in his book is the cluster decomposition principle and Lorentz ...
4
votes
1answer
75 views

Why particles with certain properties can't exist

This is inspired by a recent post on why a free electron can't absorb a photon, though my question below is about something considerably more general. The argument in the accepted answer goes (in ...
7
votes
3answers
542 views

Why is $\theta \over 2$ used for a Bloch sphere instead of $\theta$?

I'm a beginner in studying quantum info, and I'm a little confused about the representation of a qubit with a Bloch Sphere. Wikipedia says that we can use $$\lvert\Psi\rangle=\cos\frac{\theta}{2} ...
3
votes
1answer
103 views

What is the difference between scalar and vector mesons?

My understanding is that vectors and pseudooscalars change sign under parity operation and pseudovectors and scalars do not. However, I don't understand what the difference between a vector and ...
0
votes
1answer
52 views

spin representations and polynomials

I'm reading Group Theory and General Relativity by Moshe Carmeli and his discussion of spin representations of SU(2) and the isomorphism to the space of homogenous polynomials is confusing me. I'll ...
2
votes
0answers
90 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 ...