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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8 views

Using bold face letters to label representations [duplicate]

In some papers I see the authors labeling representations of groups and algebras using bold face letters. As an example, from here, "There is a global $SU(4)\cong SO(6)$ symmetry, called an R-...
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1answer
42 views

Why does Higgs field have 4 components and written in a doublet?

I'm studying the Higgs field and I encountered some problems. My book says that the Higgs field has four components which are conveniently arranged into a two-component vector as: $$\phi=\binom{\phi^{+...
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21 views

Fierz identity in Spin(5,5) [on hold]

I am trying to understand(derive) equation 3.8 in 1208.5884 which is First, here $\gamma$ is $16\times 16$ chirally projected gamma matrices. which $\alpha=1,2,3,4,...16$ and $a=1,2,3,4,5,6,7,8,9,10$...
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34 views

Quantum groups and $q$-deformed $SU(2)$

I am looking for a self-contained introduction to quantum groups and $SU(2)_q$ in particular, which a person with background in relativity and particle physics would understand. It has to contain the ...
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1answer
55 views

Real irreducible representation of $SU(2)$ [closed]

Consider a real irreducible representation of $SU(2)$ group in $d$-dimensional space-time. How many components do the spinors (eigenvectors) have? For instance, a real irreducible spinor in 10-dim has ...
3
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1answer
169 views

How are Clifford algebras related to Dirac Equation

Given a vector space $V$ and a quadratic form $q$ for the vector space. The tensor algebra is defined as $\mathcal{T}(V)=\sum_{i=1}^{\infty} V^{\otimes i}$. The set $\mathcal{I}=\{x\otimes x-q(x)\cdot ...
3
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0answers
45 views

Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

Projective representations of a multiply-connected group is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of mine, it appears ...
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40 views

Construction of vector bundles of relativistic fields by Mackey's method of induced representation

I recently stumbled on Sternberg's book on group theory and physics. The ideas expressed in the book are really great, but the detailed reasoning is very hard to follow, I find. I am kind of stuck ...
6
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3answers
250 views

Are there Gauge fields that are not 4-vectors?

In my understanding Gauge fields are fields that have some kind of redundancy, i.e. a transformation that does not change the physical state. As far as I can see all the Gauge fields in the Standard ...
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0answers
40 views

What is the irreducible decomposition of the tensor product of left and right weyl spinor representations, i.e. $(1/2,0)⊗(0,1/2)$?

What is the irreducible decomposition of the tensor product of left and right weyl spinor representations of the group $SL(2,\mathbb{C})$, i.e. $(1/2,0)⊗(0,1/2)$? I mean, the tensor product of two ...
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57 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
51 views

Block Diagonal Matrix Shankar Quantum Page 45

On page 45 of Shankar's intro to qm (you can find a pdf of it online if you want) he says that a specific operator has a block diagonal form because when it operates on some element of an eigenspace ...
6
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1answer
79 views

What is the mathematical motivation for complexifying momenta in BCFW?

One of the first steps in obtaining the on-shell BCFW recursion relations is complexifying the momenta of the external particles. Now complexifying things is not unprecedented (the dispersion program ...
2
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1answer
128 views

What is the relation between the metric tensor and the graviton?

In Zee's quantum theory in a nutshell, at the end of chapter I.10, he states that the graviton is of course the particle associated with the field $g_{\mu\nu}$. My understanding of quantum ...
3
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2answers
138 views

How does Schur's Lemma mean that the Dirac representation is reducible?

In chapter 3 of Peskin and Schroeder, when they're talking about "Dirac Matrices and Dirac Field Bilinears," they introduce $\gamma^{5}$ and give some properties of it. One of the properties is $[\...
3
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1answer
57 views

Conceptual interpretation of the left- and right-handed spinor representations of the Lorentz group

I understand mathematically that the Lorentz group's Lie algrebra $\mathfrak{so(3,1)}$ (given by eqns. (33.11)-(33.13) in Srednicki's QFT book) is isomorphic to $\mathfrak{su(2) \times su(2)}$ (given ...
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0answers
68 views

Some subtleties in quantizing a fermi field

Consider the quantization conditions for a complex Fermi field $\Psi=\Phi_1+i\Phi_2$: $$\{\Psi(x),\Psi(y)\}=\{\Psi^\dagger(x)\Psi^\dagger(y)\}=0,~~~~ \{\Psi^\dagger(x),\Psi(y)\}=\delta(x-y)$$ Compare ...
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1answer
53 views

How do I extract the groundstate of a quadratic field theory?

I have successfully solved a field theory quadratic in fermonic creation and annihilation operators via Bogolyubov transformation to a diagonal field theory. I now want to extract the groundstate of ...
1
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1answer
31 views

Anti-commutator relation of supercharges

Reading mutiple references on SUSY (e.g. Baer and Tata's Weak Scale SUSY and A SUSY Primer by S.P. Martin, arXiv:hep-ph/9709356), there seems to be different anti-commutation relation conventions for ...
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1answer
42 views

What is the spin of the Kalb-Ramond field?

In bosonic string theory the massless states of the closed string are given by a rank 2 tensor, which is divided into its three irreducible spherical tensors: symmetric traceless, antisymmetric and ...
3
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1answer
67 views

$SU(N)$ Yang-Mills Theory

Yang-Mills theory is based on the gauge group $G$ which we take to be $SU(N)$. Consider an example; $$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-\sum_{j=1}^N\bar{\psi}_j(i\gamma^\mu D_\mu-m)\...
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2answers
336 views

Why are usually 4x4 gamma matrices used? [duplicate]

As far as I understand gamma matrices are a representation of the Dirac algebra and there is a representation of the Lorentz group that can be expressed as $$S^{\mu \nu} = \frac{1}{4} \left[ \gamma^\...
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27 views

Transformation law for the eigenfunction representation of dilations (scale transformations) in CFT

On page 32 section 2.1.2. of http://arxiv.org/abs/hep-th/9905111 a representation $\phi(x)$ for the conformal group is chosen such that $D\phi(x) = -i \Delta\phi(x) $. After that it is stated that ...
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2answers
53 views

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 ...
5
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81 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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52 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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45 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 \frac{...
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1answer
78 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 \...
4
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1answer
79 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 ...
3
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1answer
96 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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39 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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0answers
52 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 $2\...
2
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1answer
90 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
43 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 ...
4
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1answer
127 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
64 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) \\ e^{i\beta}\sin(\...
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0answers
50 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 ...
2
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0answers
70 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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0answers
28 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 ...
2
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1answer
80 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 b'|X|b''\...
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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
182 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?
1
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1answer
93 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
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1answer
61 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
81 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 ...
3
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0answers
97 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 ...
5
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1answer
81 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
111 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 ...