Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. ...

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Interesting identity on $SU(3)$

In arXiv:hep-ph/1307.5414 Grabovsky use an interesting identity which is not derived in the paper: ...
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1answer
176 views

Does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom?

The starting point and notations used here are presented in Two puzzles on the Projective Symmetry Group(PSG)?. As we know, Invariant Gauge Group(IGG) is a normal subgroup of Projective Symmetry ...
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177 views

Conformal group in two dimensions [closed]

How can one show in a group-theoretical way that each of SO(d,2) and SO(d+1,1) is isomorphic to two-copies of Virasoro algebra for d=2?
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1answer
151 views

Action of the Lorentz group on scalar fields

The Lorentz groups act on the scalar fields as: $\phi'(x)=\phi(\Lambda^{-1} x)$ The conditions for an action of a group on a set are that the identity does nothing and that $(g_1g_2)s=g_1(g_2s)$. ...
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1answer
269 views

Proof of Pauli group preservation by Clifford group conjugation?

A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words: $C(P_{1} \otimes P_{2})C^{\dagger} = P_{3} \otimes P_{4}$, with $C \in$ Clifford group and ...
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176 views

Why is the crystallography restriction obeyed?

The crystallography restriction states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of ...
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3answers
272 views

Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?

A lot of known quantum gates are in the Pauli group (I,X,Z,Y) or in the Clifford group (H,P,Cnot). I need examples of the quantum gates that aren't in this groups. Also, are there are matlab functions ...
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3answers
326 views

Difference between $SU(2)$ and $SU(2)$ gauge transformations?

I hear this jargon all the time, so what is the difference? (Of course this is nothing special to $SU(2)$, but rather I just took it as an example)
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1answer
245 views

Why do we use the complexification of the Lorentz group?

I do understand why we are using the double cover, but why exactly do we make the transition to complex Lorentz transformations? Where and why are they needed? To be precise: The double cover of ...
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1answer
191 views

Spin(n) group SO(n) relation

Is it correct to state that the elements of Spin(n) fulfill a Clifford algebra and that the Lie group generators of Spin(n) is given by the commutator of the elements? If not, then what is the ...
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1answer
270 views

Irreducible tensor representations with “covariant” indices

As a follow-up of my question on the "most general" $\mathrm{SU}(2)$-symmetric interaction of two spin 1/2 particles, I ponder the following question: Consider an operator acting just on one particle ...
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1answer
109 views

What is an isoscalar factor?

I need to find a definition for "the isoscalar factors of 3j-symbols for the restriction $SO(n)\supset SO(n-1)$...denoted by brackets with a composite subscript $(n: n-1)$..." They are given as: $$ ...
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1answer
48 views

Read-off particle from (projected) Dynkin labels

In the review of Slanksy "Group theory for unified model building" in chapter 6: How do one relate the projected Dynkin diagrams from for example $\overline{5}+10$ of $su(5)$ to the corresponding ...
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2answers
77 views

Coset space and transitiviy

I have a question regarding coset space or homogeneous space $SO(n+1)/SO(n)$ which is simply $S^n$. I need some intuition regarding this result. As everyone knows that for a simple case of ...
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1answer
139 views

Matrix representation of a triplet state

The $SU(2)$ triplet state is typically given in the fundamental representation as a column vector, e.g. \begin{equation} \vec{\Delta} = \left( \begin{array}{c} \delta^{++} \\ \delta^+ \\ ...
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1answer
59 views

The Euler-Poincare equation

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ...
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1answer
53 views

Irrep decomposition of direct product of stress tensors

I have stress tensors direct product of the form $T^{ab}(x)T^{cd}(y)$. I want to write this in terms of a tensor $I^{abcd}$ in the form. $T^{ab}(x)T^{cd}(y)= I^{abcd}$. This is like decomposing the ...
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1answer
144 views

Question on derivation of Ward identity

I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field ...
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1answer
199 views

$t_1$, $t_2$, $t_3$ Hermitian generators of $SU(2)$

What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & ...
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2answers
183 views

A proof in Howard Georgi's “Lie Algebras in Particle Physics”

I am reading the book referenced above and in the first chapter, in the proof of theorem 1.3 (fifth line of the proof), it says: But because D2 is irreducible, P must project onto the whole space ...
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2answers
345 views

Symmetries & Lie groups in physics

This is not a homework, neither it is any exercise. It is my understanding of $U(1)$ symmetry. I would request if anybody can please correct me on any one of the following understandings: The ...
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1answer
140 views

Commutation relation of $J^2$ and $R(\alpha,\beta,\gamma)$

If $R(\alpha,\beta,\gamma)$ is the Rotation operator and $\alpha,\beta,\gamma$ are Euler angles and $J$ is the total angular momentum then how to get to this: $$[J^2,R]~=~0?$$ This is stated in ...
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2answers
331 views

high spin atoms SU(2) representation

I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry. Why not $SU(N)$?
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0answers
32 views

Lie theory and particle physics [duplicate]

I have recently been reading Intro to Lie algebras and representation theory by Humphreys, and when I am finished I am interested in reading about Lie groups and Lie algebras and their applications to ...
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0answers
16 views

How does the choice of a basis decide how many Goldstone bosons there are under spontaneous symmetry breaking?

I have a question about how the basis you choose in a field theory problem semmingly decides how many Goldstone bosons you get after spontaneous symmetry breaking. For SU(2), if you choose the 3 Pauli ...
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96 views

what kind of system respects $SU(N)$ symmetry?

I read this post, Is the symmetry group of two spin 1/2 particles $SU(2) \times SU(2)$ or $SU(4)$? If the picked answer is correct, can I believe that an $N$-degenerate system respects $SU(N)$ ...
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0answers
58 views

Where do $L_+$ and $L_-$ live, if not in $\mathfrak{so(3)}$?

This question is continuation to the previous post. The lie algebra of $ \mathfrak{so(3)} $ is real Lie-algebra and hence, $ L_{\pm} = L_1 \pm i L_2 $ don't belong to $ \mathfrak{so(3)} $. However, ...
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0answers
77 views

How are symmetries defined mathematically? [duplicate]

I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding ...
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0answers
71 views

Information about fields and superfields [closed]

I want some explanation of fields and superfields (types and components), and what the relationship between them and representation of a group.
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1answer
180 views

Is the spin 1/2 rotation matrix taken to be counterclockwise?

The spin 1/2 rotation matrix around the z-axis I worked out to be $$ e^{i\theta S_z}=\begin{pmatrix} \exp\frac{i\theta}{2}&0\\ 0&\exp\frac{-i\theta}{2}\\ \end{pmatrix} $$ Is this taken to be ...
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0answers
81 views

Finding parity eigenvalues from a character table

The all-electron code Wien2K will optionally calculate the character tables for a specified list of $k$-points. I'd like to know the parity eigenvalue for a given $k$-point and band index. Is there ...
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130 views

Relations between fields transforming by Lorentz and Poincare groups

We can analyze fields transforming by the Lorentz group as $(m, n)$ representations, where $m,n$ are the max eigenvalues of two SU(2) operators, which generate the irreducible representation of the ...
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128 views

Deriving term symbols from electron configuration using Young tableaux

Can somebody explain me how to derive all term symbols using Young tableaux? Our lecturer showed us but I couldn't quite understand it without any background on group theory. I have some vague ...
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0answers
75 views

Wigner $3j$ symbols

I am trying to determine the expansion that requires using $3j$ symbols; however, I am running into some conceptual snags. First, the expansion produces symbols that have m's that do not agree with ...
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0answers
44 views

How to obtain deconfined theory from an s-confined N=1 susy gauge theory?

Is there a systematic procedure for obtaining a deconfined theory from an s-confining theory (as defined in hep-th/9610139 for example)?
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0answers
59 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 ...
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3answers
181 views

On Group Theory: Symmetry Groups and Our Interest

Over the past few years, I've been doing a lot of self education in the Quantum Mechanics and General Relativity, and of course, there are mathematical elements of both doctrines that are matrices. ...
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1answer
57 views

Sign function whose argument is an element of a group?

Let $G$ be the group of the permutation of $N$ particles, $P\in G$. Therefore, there are $N!$ elements in $G$. For its subgroup, e.g., even permutation, we can calculate $\text{sign}(P)$ and get ...
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2answers
3k views

Wyckoff positions and lattice coordinations

Many papers use Wyckoff positions and Space Groups to report atom coordinates, making use of the structure's symmetries to save space in the notation (e.g. diamond = Fd-3m, only). How can I ...
0
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1answer
58 views

Rotating a complex number

Let us begin in a two-dimensional Euclidean plane. The vector is e.g. $\vec{V}(x,y)$ It is often useful ā€“ but in this case, it's just a mathematical trick that doesn't make the complex numbers ...
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1answer
49 views

Connection to spin 1/2 electron system?

In another Physics stack exchange thread here, Spin matrix for various spacetime fields I obtained the generator of rotations of the SO(2) rotation group for an infinitesimal rotation of 2D vectors. ...
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2answers
134 views

Reasons for choosing $SU(3)$ as the color group vs. $SO(4)$

What are the reasons that $SU(3)$ is used for QCD? Why wouldn't the simpler & smaller group $SO(4)$ make a better candidate?
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1answer
158 views

Symmetry groups [closed]

I am quite new to this subject. I am just repeating in a few words, what I have learned so far: There are 4 fundamental forces of nature: strong, weak, electromagnetism and gravity. Physicists are ...
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1answer
58 views

Notation in the book Symmetry by Hermann Weyl

I'm having troubles understanding a notation of the symmetry groups in a book "Symmetry" by Hermann Weyl. On the page 80 of the 1952 Princeton University Press edition of the book, Weyl lists the ...
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1answer
148 views

Plotting a bandstructure along High-symmetry points when kx,ky,kz is known

Suppose you know kx,ky,kz points along with the corresponding energies. Basically, you know about the 4-D E(k) dispersion. How you do then convert that data into the bandstructure plots you commonly ...
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1answer
281 views

How to get Gell-Mann matrices?

How to get Gell-Mann matrices $f_{i}$ (more or less strictly)? What are the requirements for getting them, excluding $||f_{i}|| = 1$, commutational law $[f_{i}, f_{j}] = if_{ijk}f_{k}$ and hermitian ...
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1answer
66 views

Proper notation when working with three Euclidean spatial coordinates in a setting with a time parameter

The How does the Euclidean metric is the symmetry group of Euclidean space. It includes rotations and translations. Say I consider an Euclidean space and a time parameter. How does the Euclidean ...
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12 views

Degeneracy in quater-wave stack

Consider a 1D photonic crystal, the quarter-wave stack, and its band structure. A famous conclusion is that there's no gap at Brillouin zone's center. In other words, successive bands are degenerate ...
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76 views

General definition of vector spinor and spin

I am looking for basic and exact definitions of fundamental physical consepts in graduate level. I reach this following definitions. Could you please help to improve these definitions. Spin: ...
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0answers
38 views

What are differences between Spin(3,1), SL(2,C), SO(3,1) and SU(2) representations? Which one is correct exact representation for spinor fields? [duplicate]

I want to understand which group transformations exactly represent spinor fields. That is, do spinor fields transform under the Lorentz group $\mathrm{SO}(3,1)$ or under $\mathrm{Spin}(3,1)$? What ...