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Questions tagged [group-theory]

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. Groups are used in physics to describe symmetry operations of physical systems.

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1answer
80 views

Symmetries of Wigner $3j$-symbols by exchange

I know that Wigner $3j$-symbols have certain symmetry factors arising by exchange of two columns within one symbol. But what happens if you have two 3j symbols and do an exchange like this: $ \left(\...
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1answer
63 views

An invariant of the gauge group $G$ that is totally symmetric with three indices in the adjoint representation

In Ch.19 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.680 the property of a quantity $$\mathcal{A}^{abc}=\mathrm{tr}\left[t^a\{t^b,t^c\}\right]\tag{19.132}$$ ...
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0answers
48 views

Representation of the Lorentz group using matrices of $SL(2,\mathbb{C})$

There is a correspondence between the Lorentz group and the group $SL(2,\mathbb{C})$. To each Lorentz transformation $\Lambda$ we can associate two matrices $\pm A(\Lambda) \in SL(2,\mathbb{C})$ such ...
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0answers
35 views

Symmetry relation of Wigner-Eckart

I saw a symmetry relation following from the the Wigner-Eckart Theorem looking like this $$(\xi j|| T_L || \xi'j') = (-1)^{j-j'} (\xi' j'|| T_L || \xi j)^*$$ I know that it must come somehow under ...
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0answers
54 views

Quantum representation of a system of identical particles

I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
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1answer
75 views

Representation and Lie algebra of $SO(3)$

Studyng the book Group Theory in Physics of Wu-Ki Tung, I have read: "... every representation of the [$SO(3)$] group is automatically a representation of the corresponding Lie algebra, (...) a ...
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0answers
32 views

Representation of the Lorentz group and correspondence with the $SL(2,\mathbb{C})$ group

We can find a correspondence between the restricted Lorentz group and the group $SL(2,\mathbb{C})$ if to each coordenate $x^{\mu}$ we associate a $2\times 2$ hermitian matrix $X$ given by $$X = x^{\mu}...
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2answers
170 views

$SU(2)$ symmetry of $\mathcal{L}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$

I'm considering a Lagrangian of two complex scalar field: $$\mathcal{L}=\partial_{\mu}\phi_1^{*}\partial^{\mu}\phi_1-m_1^2\phi_1^{*}\phi_1+\partial_{\mu}\phi_2^{*}\partial^{\mu}\phi_2-m_2^2\phi_2^{*}\...
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1answer
63 views

Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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0answers
44 views

Difference between $\tilde{\textrm{Diff}}_+(S^1)$ and ${\textrm{Diff}}_+(S^1)$

In this paper, where Liouville theory is being studied on a strip, after equation 2.3 it is mentioned that the conformal transformations of the strip are given by the same chiral and anti-chiral ...
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2answers
91 views

What is a rotation group and how do we get its unitary representation?

The rotation group is ${\rm SO(3)}$. It is the group of $3\times 3$ orthogonal matrices $\{g(\theta)\}$ with unit determinant. So these are already defined in terms of $3\times 3$ matrices. But we use ...
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1answer
88 views

Parametrizing $SU(2)$ with Hermitian matrices

There is something that is not clear to me Here is what I know: Pauli matrices are $\sigma_1 = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$, $\sigma_2 = \begin{pmatrix}0 & -i \\ i & 0\...
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1answer
80 views

Pin groups in Physics

There is a very interesting paper from "The Pin Groups in Physics: C, P, and T" on improper and antichronous Lorentz transformations, but on page 5 I got quite confused as it states there ${L^\...
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1answer
38 views

Lie Subgroups of $SL(2,\mathbb{R})$

I'm wondering about the Lie subgroups of $SL(2,\mathbb{R})$. It's Lie algebra is the algebra of real traceless matrices and has basis elements $$L_0 = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \...
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1answer
45 views

How do we know the group property of a new particle?

Suppose I have a particle $W'$ which can decay into $\mu$ and $\nu_{\mu}$ and $e$ and $\nu_{e}$. Suppose we know such new gauge bosons come from some additional gauge group added to the Standard ...
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1answer
66 views

Representations of the rotation group

(I have already done a similar question, but I did not express myself very well and the question was a bit confusing, so let me try again. If you find the question repetitive, I apologize and you can ...
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1answer
103 views

Do canonical transformations form a group?

In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
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1answer
94 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
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1answer
96 views

$U(N)$ & $SU(N)$ : What's the conceptual difference in Gauge Theory?

I know the mathematical difference that one means $ absolutevalue(det) = 1$ and one means det = 1 (rotation) and that ones the subgroup of the other and so on. But: has a local/gauged $SU(3)$ ...
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0answers
45 views

Fluxes on a finite group $G$

So I've been studying about topological quantum computation and I have a few questions I haven't been able to solve. The first one is why fluxes take values on a finite group $G$? Does it have to do ...
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2answers
59 views

How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
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1answer
42 views

How to construct invariant forms under the effect of an arbitrary group?

First I would like to mention that I do not know that should I post this question here or in the math community, but since my background is in physics and this kind of question is usually asked by ...
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1answer
84 views

How is possible to see that Maxwell's field have $U\left(1\right)$ symmetry?

As it is well knowing, $U\left(1\right)$ is the group of the unitary matrix of the first order and that this group is connected with rotation operations. Under the complex scalar field perspective $$\...
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0answers
57 views

Why do particle physicists often use $\otimes$ instead of $\times$ to denote the direct product of groups? [closed]

For example, sometimes one sees $SU(3)\otimes SU(2)\otimes U(1)$ instead of $SU(3)\times SU(2)\times U(1)$. My understanding is the the product here is just the usual direct product (aka Cartesian ...
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1answer
43 views

Combining SU(N) multiplets using Young diagrams

I am trying to follow the Particle Data Group's instruction (PDF link) to combine SU(N) multiplets. On page 3, they show an example calculation of SU(3)'s $\textbf 8\otimes \textbf 8$. I understand ...
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1answer
88 views

Berry phase: Spin in a magnetic field parameter space manifold

Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce ...
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1answer
51 views

Gauge group of Electroweak theory

I am doing a question that asks me to identify the gauge groups of a Lagrangian with the field strength tensors $$\bf{F}_{\mu \nu} = \partial_{\mu}\bf{W}_{\nu} - \partial_{\nu} \bf{W}_{\mu} - g\bf{...
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0answers
14 views

Dielcetric Tensor Transforms under Product Representation

How do I show that the Dielectric tensor in 2D transforms as a product of representations? I am told that the electric displacement and electric field transforms with the representation $D^v(g)$, and ...
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2answers
83 views

How to unify the rotation matrix of $SU(2)$ operator and $(z_1, z_2)$ representation?

I am following the Xiao-Gang Wen's book: Quantum Field Theory of Many-body Systems. In Ch. 5.6 about non-linear $\sigma$ model, it use a rotation operator $U$ to change the spin quantization from $z$ ...
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1answer
502 views

Why do we require gauge symmetries to commute?

My question arises after reading the 87th page of Elementary particle physics in a nutshell by Tully: which is also given by the following link: https://books.google.se/books?id=vLy2YlkXZuEC&pg=...
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0answers
50 views

Error with generators of Lorentz group (basis of Lorentz Lie algebra) [closed]

Can someone help me figure out why my $J_y$ is incorrect? :/ It's supposed to be \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & -...
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2answers
129 views

What does matrices act on different spaces mean in QFT?

I have a Dirac kinetic term in a Lagrangian. $$ i\bar{\psi}\gamma^\mu D_\mu\psi = i\bar{\psi}\gamma^\mu\partial_\mu\psi + g\bar{\psi}\gamma^\mu\psi A^a_\mu T^a,$$ However, I usually heard that ...
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2answers
97 views

Symplectic group $Sp(2N)$ in Srednicki's book

There is a question in Mark Srednicki's Book (Problem 24.4, p.160) about $Sp(2N)$, but I am not sure I understand the significance (application?) of this group. In that chapter, he talks about $SO(N)$ ...
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0answers
15 views

How to map the symmetry property of the lattice unit cell to the symmetry properties of eigen modes

For example, in $\mathbf{r}$ space, a honeycomb lattice (like graphene) has C6v symmetry about the center of the unit cell. The ground state (singlet) eigen mode has C6v symmetry and the 1st order ...
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1answer
41 views

Basis function of $\Gamma_2$ irrep of point group $T_d$?

From Properties of the Thirty-Two Point Groups (Koster, et. al.), the basis function of the $\Gamma_2$ irrep of the point group $T_d$ is $l_xl_yl_z$, where $l$ is the angular momentum operator. ...
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4answers
249 views

Why in QFT what really matters is $\exp(\mathfrak{so}(1,3))$ instead of $O(1,3)$?

In QFT fields are classified according to representations of the Lorentz group $O(1,3)$. Now, most books when getting into this say that in order to understand the representations of $O(1,3)$ we need ...
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1answer
97 views

Representation of $SO(3)$ acting on 3-index tensors in Zee's Group Theory Book

In page 193, the book Group Theory in a Nutshell for Physicists started to explain inductively how we need to consider only the traceless symmetric (in all indices) tensors to construct $3^j$-...
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3answers
169 views

Three spin states of a spin-1 particle

For a spin-1 particle at rest, it has three spin states(+1, -1, 0, along the z axis). If we rotate the z axis to -z direction, the spin +1 state will become the spin -1 state. Can we transfer the spin ...
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1answer
58 views

Operators of the special orthogonal group $\mathrm{SO}(3)$ in 3 dimensions

My professor taught us that when we want to rotate a 3D vector we need a $3\times 3$ matrix $R$ that is a rotation matrix. The set of all these matrices is the special orthogonal group in three ...
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2answers
47 views

Confusion about Young tableaux composition rule

I'm following the rules in this document to combine irreps of $SU(N)$ using Young tableaux. If I'm not mistaken the product of two irreps should be symmetrical, that is $A \otimes B = B \otimes A$. I'...
2
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0answers
45 views

Why would $B_{1g}$ Raman mode still exist under $D_{4h}$ where $xx = yy$?

I come across some issues about Raman tensors under high symmetry , e.g. $D_{4h}$. A preliminary thought tells me that $x$ and $y$ are supposed to be equivalent in this point group therefore the Raman ...
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0answers
41 views

Particle spectrum in dimensional reduction

First of all, sorry if this question is a bit stupid, but my knowledge of certain aspects of particle physics and group theory is a bit limited. I am compactifying the heterotic $E_{8}\times E_{8}$ ...
2
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1answer
48 views

$SU(5)$ group theory: Contracting three adjoints to make a singlet

There's probably a trivial answer to my question, but I'm having trouble finding it. In an $SU(5)$ GUT, we have a gauge field, $A_{\mu} = A^{a}_{\mu}T^{a}$ which lives in the $24 (\text{Adjoint})$ ...
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1answer
86 views

Structure constants in Lie Algebra

As a nit-picking question, I wanted to clarify a point of confusion. This arises from definitions found in a plethora of books, lectures notes and even the Wikipage on structure constants and Lie ...
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0answers
39 views

How does hexagonal boundary arise in $SU(3)$ representation?

I am having trouble trying to understand the hexagonal boundary in $t_3$ and $y$ representation of $SU (3)$. I tried to work it out like we did in $SU \left( 2 \right)$ but got not luck. Could someone ...
0
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1answer
76 views

Do non-abelian group mathematics have any use in the real world math? [closed]

I know physics uses a lot of non-abelian mathematics (though I cannot wrap my head around ab does not equal ba).. Is there any real world (macro world we live in) uses for non-abelian mathematics? ...
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0answers
35 views

Lie Algebra in Particle Physics

In his book " Lie Algebra in Particle Physics" Georgie directly put the relation $$(1-P)D(g)(1-P)=D(g)(1-P)...(1)$$ This came from the two previous relations: $$PD(g)P=D(g)P$$ $$PD(g)P=PD(g).$$ where ...
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1answer
107 views

Parameter space of $SO(3)$ and $SU(2)$

Is it parameter space of $SO(3)$ and $SU(2)$ are same? can we use quaternions to represent both groups? what about their connectedness?
3
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2answers
199 views

Complex conjugated representation and its Young tableaux

This post is an exact copy of one that I posted in Math's site. I do this copy because people there suggested me to do it since, apparentely, in Mathematics and Physics we use different conventions ...
2
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3answers
271 views

If infinitesimal transformations commute why don't the generators of the Lorentz group commute?

If infinitesimal transformations commute as proved e.g. on this mathworld.wolfram page, why are the commutators for the generators of the Lorentz group nonzero?