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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What is the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1) \subset su(2)$?

What is meant by the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1)\subset su(2)$? I have read it above eqn. (10) in this paper http://arxiv.org/abs/0812.3572 but have also heard it mentioned in ...
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69 views

What type of fields are continuous spin representations?

Continuous spin representations (infinite dimensional representations of the Lorentz group) are pretty rarely discussed, and usually not in that much mathematical details. And usually it is done in a ...
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42 views

Fierz identity for chiral fermions [closed]

First of all I define the convention I use. The matrices $\bar{\sigma}^\mu$ I will use are $\{ Id, \sigma^i \}$ where $\sigma^i$ are the Pauli matrices and $Id$ is the 2x2 identity matrix. I will use ...
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40 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 ...
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1answer
69 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 ...
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1answer
71 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 ...
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1answer
79 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 ...
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37 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, ...
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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 ...
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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 ...
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1answer
40 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 ...
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1answer
107 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). ...
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2answers
62 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) \\ ...
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43 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 ...
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66 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 ...
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1answer
60 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: ...
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27 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 ...
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1answer
78 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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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
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1answer
154 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?
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1answer
92 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 $d$ ...
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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 ...
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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 ...
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1answer
74 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 ...
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94 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 ...
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1answer
79 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 ...
3
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1answer
107 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 ...
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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 ...
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1answer
113 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 ...
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1answer
116 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 ...
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43 views

Angular momentum $J_z$, how do we get eigenvalue of $m\hbar$?

If we have angular orbital momentum for $z$-direction, we assume that for state $|j,m>$ that eigenvalue is $m\hbar$. Similarly for $J^2$, we assume $j(j+1)\hbar^2$ Can I get reference of ...
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61 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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73 views

Irreducible representations of $SU(2)$, Tensor-operators under rotations

First of all: this is my first question so please give feedback to the way I'm formulating the question! The question is about an exercise I have to solve, but I simply get nowhere. It is given the ...
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0answers
49 views

The universal covering group of a symmetry group [duplicate]

In Weinberg QFT Vol.1, it says one can enlarge the symmetry group $H$ to the universal covering group $C$ such that one obtains a trivial cocycle or $C$ is simply connected whereas $H$ is not. I get ...
2
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1answer
152 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 ...
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1answer
116 views

Switch from the position representation to the momentum representation

If we use Fourier Transform, we can switch from the position representation to the momentum representation, like the following formula here comes the problem, if we use dirac notation we can see it ...
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1answer
106 views

Matrix represenation of total angular momentum operator

I see that for total ket in QM of hydrogen atom we define a tensor product of kets of spatial and spin spaces, upon which spatial and spin operators, operate respectively. For the total angular ...
4
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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. ...
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26 views

Index Placement for Spinors in Relativity

This may ultimately be a silly question, but a pedantic mind like mine gets tied into knots over differing notation. (Disclaimer: I'm a mathematician.) Let $\mathbb{W}$ be a complex two-dimensional ...
3
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2answers
129 views

What is the meaning of spin of an particle is $1/2$ and $2$ or something? On which factor does these spin no. depend?

I have read a book. The writer had written that if the spin of an particle is $\frac{1}{2}$, then we have to rotate it at $720$ degree. Imagine that there are two balls joined. Then we have to rotate ...
2
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1answer
119 views

How to determine the trace and determinat of a differential operator?

How to determine the trace and determinant of the operator like $\Box$ or $\nabla^2$ etc. But first of all how to find the same for the simpler operator $\frac{d}{dx}$? I proceeded as follows. What ...
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1answer
82 views

Pauli matrices identity with no repeating indices

I was just wondering if there is a proof of, or an example utilizing the following relation: ...
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1answer
90 views

Infinitesimal rotation of classical fields: why are rotation group representations important?

I understand that $SO(3)$ representations are important in quantum physics, because eigenspaces of the Hamiltonian are irreps of $SO(3)$ if it is part of the symmetry group. But I don't see the reason ...
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1answer
480 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 ...
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2answers
96 views

Complex / real representations of the Lorentz group

In Michele Maggiore's book "A Modern Introduction to QFT" he describes the spinorial representations of the Lorentz group as The representations are in general complex. I always thought the ...
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1answer
159 views

Trivial representation in Clebsch-Gordan decomposition

My professor defined the Clebsch-Gordan series as the direct sum decomposition of the tensor product of two representations of the Lie group SU(2): $$ D_{j_1} \otimes D_{j_2} = D_{j_1+j_2} \oplus ...
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1answer
133 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 ...
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5answers
398 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 ...
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1answer
86 views

In field theory, why are some symmetry transformations applied to the field values while other act on the space that the fields are defined on?

My basic understanding is that a field theory consists of symmetry groups, a space $S$ that the symmetry groups act on and of fields defined on that space $S$. In other words, the space $S$ is the ...
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54 views

Mass, Spin, Internal Energy and 1-Particle States in Galilean Quantum Mechanics

I have been reading an article discussing the unitary representation of Galilean group and non-relativistic quantum mechanics. The link to the article is given below. http://arxiv.org/abs/1107.2442 ...