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Questions tagged [group-representations]

Use as a synonym to the representation-theory tag

3
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1answer
52 views

Doubts on representations of poincare group and QFT

I am studying Poincare group and encountered the term massless representations of the Poincare group. I know Poincare group is studied by the studying the little group of various momenta, massless and ...
-3
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0answers
28 views

Is there a relation between translations of poincare group and annihilation-creation operators in Scalar Theory

Can poincare group for scalar field theory depend on annihilation-creation operators ? Explicitly, for example, how can translations be written in terms of annihilation-creation operators of scalar ...
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0answers
107 views

What is the Lorentz group composition of two electrons?

We know that the wavefunction of an electron transforms as Dirac spinor $(1/2,0)⊕(0,1/2)$ under the Lorentz group $SO(3,1) \sim SU(2)\times SU(2).$ Which representations can we form with two ...
-1
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1answer
63 views

How is mass related to the Poincare group?

I have studied about Poincaré group and some QFT, read that the Casimir elements are $p^{u}p_{u} = m^2$ and Pauli-Lubanski vector (or pseudovector). How does the mass come into picture and spin come ...
1
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1answer
48 views

Poincare group and free theories

How exactly is the Poincare group related to the free relativistic theories in quantum field theory? I know Poincare group is the Lorentz group along with translations but don't see any connected why ...
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0answers
11 views

Calculating adjoint representation of Lie group/algebra [duplicate]

How do I calculate adjoint representation of Lie group and Lie algebra? I would be thankful if anyone can give good example or general formula on calculating adjoint of any Lie group
2
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2answers
71 views

Direct Product vs Tensor Product

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product: ...
2
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1answer
52 views

Is there a simple way to explain a fundamental representation of $O(N)$?

Is there a simple way to explain fundamental representation in Physics? For example, a fundamental representation of $O(N)$?
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1answer
40 views

Can fermions and bosons exist in the same representation?

Some people have made theories where they claim fermions and bosons exist in the same representation for example $E_8$. I can't see how this is possible. But say for example it is. This would imply ...
0
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1answer
74 views

Can $E_8 \times E_8$ contain the standard model?

I know $E_8$ by itself can't be gauge group because it has no complex representation and so would not be chiral. But assuming the existence of mirror matter which also would have $E_8$ gauge group ...
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0answers
20 views

$SU\left(N\right)$ Dynkin labels, how to compute

Let $V$ be somecomplex irreducible representation of $SU\left(N\right)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of ...
0
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1answer
33 views

Weights of $SU\left(5\right)$ representation

Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights? Are these the correct Dynkin labels ...
2
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1answer
39 views

Branching of $SU\left(5\right)$

In the context of branching rules, what is a projection matrix for a subgroup. For instance, the projection matrix for the subgroup $SU\left(2\right)\times SU\left(3\right)$ of $SU\left(5\right)$ is ...
1
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1answer
44 views

What is the weight system for these SU(5) representations?

I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The $\mathbf{5}$ ...
1
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2answers
131 views

Is there a simple way to calculate Clebsch-Gordan coefficients?

I was reading angular momenta coupling when I came across these CG coefficients, there is a table in Griffith's but doesn't help much.
0
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1answer
93 views

Plausibility of the Weyl-Relations of position and Momentum - Physical meaning of the Heisenberg group

In this question I asked about the uniqueness of the momentum operator $\hat{p}$ for a given position operator $\hat{x}$, and wether the uniqueness was fixed by the commutation relations that position ...
3
votes
1answer
99 views

How to determine isospin $T$ (not just $T_z$) of a nuclear ground state

I’m trying to work out the total isospin and the $z$-component of the isospin for specific elements like $^{20}O$ and $^{20}F$ in the ground state. I’ve worked out the $z$-component, as I concluded ...
3
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1answer
90 views

Why is $4=3\oplus 1$? What are propagating modes? Etc

In Schwartz's QFT book, he said that the vector representation of the Lorentz group, $V_\mu$ that is four-dimensional, is the direct sum of two irreducible representations of $SO(3)$: a spin-0 ...
2
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1answer
127 views

Explicit construction of $(\frac{1}{2}, \frac{1}{2})$ representation of Lorentz group

For the vector representation of the Lorentz group (actually the algebra), the $J^1$ generator is $$J_1 = i \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &0 \\ 0 & 0 & 0 ...
0
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1answer
45 views

Wigner $D$ matrix

How to derive symmetry relation of Wigner $D$ matrix? I mean this relation $$ D_{m',m}^j (\alpha,\beta,\gamma) = (-1)^{m'-m} D_{-m',-m}^j (\alpha,\beta,\gamma)^*. $$ I want to derive this, but I ...
1
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1answer
47 views

Relation between Spin 1 representation and angular momentum and SO(3)

This is a naive question. It occurred to me while studying in detail the the Spin 1 angular momentum matrices. The generators of SO(3) are $J_x= \begin{pmatrix} 0&0&0 \\ 0&0&-1 \\ 0&...
3
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2answers
108 views

Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
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0answers
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Why do states acted upon by Spherical Operators transform according to a higher dimensional representation?

In the explanation for the Wigner-Eckart Theorem, what is the exact meaning of "$T_{m_q}^{(q)} \ | l , m_l>$ transforms as a vector in $V_q \otimes V_l\;$"? I understand it comes from the fact that ...
1
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3answers
255 views

Given the transformation of $SU(2)$ triplet $\vec{\phi}$ how to find the transformation of ${\Phi}\equiv\vec{\phi}\cdot\vec{\tau}$?

Given the transformation of a $SU(2)$ triplet $\vec\phi$ $$\phi\to \exp{(-i\vec{T}\cdot\vec{\theta})}~\vec\phi\tag{1}$$ (in the question here by @physicslover) how does obtain the transformation of $\...
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1answer
96 views

How is the complexification of the Lorentz Lie algebra related to the need for Dirac's 4-component spinor in QFT?

There have been several questions with good answers in physics.stackexchange about the motivation of the complexification of the Lorentz Lie algebra, basically as a mathematically nice way to deal ...
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0answers
25 views

$6j$ symbol for U(1) (and groups in general)

The paper here offers a method of constructing solvable quantum mechanical lattice models whose ground states are related to fixed points of the renormalization group associated to gauge theories or ...
1
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2answers
88 views

Notation for the Representations of $SO(3,1)$

I was reading Zee's book "Quantum field theory in a nutshell" and came to the chapter regarding representations and algebra. He writes that the representations of $SO(3,1)$ are $(0,0),(0, 0),( \frac{...
1
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1answer
148 views

General Irreducible Representation of Lorentz Group

Background (just for context, you can skip it if you're familiar with Lorentz representations) A Lorentz transformation can be represented by the matrix $M(\Lambda)=exp(\frac{i}{2}\omega_{\mu\nu}J^{\...
0
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1answer
58 views

Arbitrary functions can be expanded by basis functions of invariant subspaces of a group

On page 96 of M. S. Dresselhaus's Applications of Group Theory to the Physics of Solids, it is said that Any arbitrary function $F$ can be written as a linear combination of a complete set of basis ...
0
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1answer
32 views

Fundamental and antifundamental representations, and hilbrt spaces

I am studying quarks $u$, $d$ and $s$. I know they can be represented by 3-vectors: We also have the antiquarks that are represented by the same 3-vectors: The difference between them is that when ...
9
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2answers
157 views

Chirality of the Electromagnetic Field Tensor

I have learned that chirality is a concept, that appears for $(A,B)$ representations of the Lorentz group, where $A\neq B$. An example would be a Dirac spinor, corresponding to the representation $(\...
2
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1answer
81 views

Irrep corresponding to a rotation, what's the definition?

My character tables for point group $T$(Schönflies-notation but easily convertible into other point group notations) tell me that the rotation around the $z$-axis, $R_z$ (the $z$-direction ...
2
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1answer
117 views

How a symmetry transformation acts on quantum fields

I study particle physics and am finally tired of pushing through QFT with annoying doubts which seem to be both very simple and fundamentally important, and to which several professors of mine couldn'...
1
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1answer
59 views

What prohibits fundamental fermions transforming like the $6$ and $6^*$ IRR's of $SU(3)$?

The lowest IRRs of SU(3) are 3,3* (the fundamental reps), 6,6*, and 8 (the adjoint rep). The quark fields are chosen to transform as 3, 3*, and the gluons as 8 under SU(3), but there is no ...
1
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1answer
76 views

How to prove a set of matrices form a representation of Lie algebra?

When reading Paul Langacker's The Standard Model and Beyond, I am quite confused on equation 3.29, which says with a set of fields $\Phi _a$, where $a$ goes from 1 to $n$, are chosen to be transformed ...
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0answers
46 views

Is there a name for symmetry in which fermions and bosons are in identical adjoint representations?

In a Yang-Mills theory, with gauge group $G$, if the Fermions are in an adjoint representation then for every Fermion with "charge" $Q$ there is a boson with charge $Q$. i.e. there is no difference ...
2
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2answers
95 views

Are rotation matrices faithful representations of the rotation group?

I would like to use rotation matrices as representations of the rotation group. I would like to know if these representations are faithful, i.e. isomorphic to the rotational group elements. I read ...
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0answers
22 views

Representations and classes of square group $D4$

(I post this in physics because its about an excercise in the Thinkman book of theory group for quantum physics). The Group of symmetries of the square (D4) has an order of 8. There are 2 classes in ...
0
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1answer
69 views

Transformation of fields in non-abelian gauge theories

Let us consider a gauge group, e.g. $SU(N)$. One usually says that a fermionic field $\psi$ belongs to the fundamental representation of the group. As far as I understand, the fundamental ...
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0answers
78 views

Question regarding Little Group in Schwartz

In his book on Quantum Field Theory (Pg. 125), M. Schwartz talks about using little groups to resolve the conflict between Lorentz invariance and unitarity. The reasoning in the book is as ...
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0answers
51 views

Lorentz transformation of vector of sigma matrices

$\newcommand{\vec}[1]{\mathbf{#1}}$I'm trying to show that the relations \begin{equation} \rho_L(\Lambda)^\dagger\bar{\sigma}^\mu\rho_L(\Lambda)={\Lambda^\mu}_\nu\bar{\sigma}^\nu\\ \rho_R(\Lambda)^\...
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0answers
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Why is $\Gamma_{R_z} \subset E\otimes E$ for point group symmetries?

I observe that for all non-1-dimensional representations (over $\Bbb R$) in all points groups the irrep that indicates how the rotation around the main axis ($R_z$) transforms is contained in the (...
2
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1answer
129 views

Weinberg's classification of one-particle states and representations of the Poincare group

A representation of a group $G$ is a pair $(\rho, V)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a homomorphism. If $V$ is actually a Hilbert space and $\rho : G\to \mathcal{U}(V)$ maps ...
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0answers
95 views

spinor indices and Weyl spinors

This is a follow up to Spinor dotted and undotted indices, to see if I understand things correctly. So a Dirac spinor $\psi_D$ can be written as a direct sum of a left handed Weyl spinor and a right ...
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1answer
26 views

Questions regarding the elements of vector space spin representations act on

Elements of vector space spin-$1/2$ representations act on are spinors. What about half-integers in general? And what about integer spins? Do spin-$0$,$1$ reps always act on vectors?
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2answers
394 views

Are spin-1/4 theories forbidden?

Ok, this question looks a bit ridiculous at the outset. However, I was thinking, and I couldn't actually come up with a reason why there shouldn't exist a representation of the Lorentz group that was, ...
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0answers
42 views

Mapping from spinor to tetrad

I am reading the journel by Patrick l. Nash: mapping from tetrad to Dirac spinor. While reading this ,I came across the term concrete real 4*4 irreducible representation of SO(3,3). I know SO(3) is ...
7
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2answers
192 views

Group representations and active/passive transformations

Suppose we are in Euclidean 3-space with coordinates $x$ and a scalar function $\phi(x)$ defined on it, and consider the group of rotations $SO(3)$ for simplicity. Take a rotation matrix $R \in SO(3)$;...
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1answer
125 views

What do the symbols $⊕$ and $⊗$ mean?

Please have a look at this presentation on Young tableaux, I'm trying to understand the signs I mention there - what do the $\otimes$ and $\oplus$ symbols mean?
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1answer
49 views

Can somebody explain the relation of Young diagrams and particle physics?

I have the mesons octet and there are Young diagrams below them. how are they drawn? what do they represent?