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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Generators of the (1,2,2) of Pati-Salam

I am working on a project involving breaking SO(10) to its Pati-Salam sub group. In one of the path ways you can use, the broken generators fit in the (6,2,2) of Pati-Salam (recall the Pati-Salam ...
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Can we do better than “a spinor is something that transforms like a spinor”?

It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do ...
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Fundamental representation in Group Theory

I am struggling to understand fundamental representations in Group Theory. I know that the fundamental representation of $SU(N)$, as assigned to a matrix $U=U^i_j \in SU(N)$ can be shown through the ...
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Weinberg derivation of Lie Algebra [duplicate]

In the derivation of the Lie algebra in the first volume of Quantum Theory of Fileds by Weinberg, it is assumed that the operator $U(T(\theta)))$ in equation (2.2.17) is unitary, and the rhs of the ...
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Lorentz transformation of the spinor fields

I have been reading the Srednicki's QFT textbook (available online at https://web.physics.ucsb.edu/~mark/qft.html) and in Chapter 34 the left and right-handed spinors are discussed. There is a step in ...
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Complex conjugate Young Tableaux representation [duplicate]

I have been studying Young Tableaux representation from youtube to represent $2\times 2$ and other examples to in $SU(n)$ symmetry. But i am unable to understand nor able to find relevant answers of ...
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Inönü-Wigner contraction of Poincaré $\oplus$ $\mathfrak{u}$(1)

Metric = (-+++), complex $i$'s are ignored. Using the following decompositions of the Poincaré generators, I can write the Poincaré algebra as I can get the Galilei algebra using the following ...
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What are the most general symmetries that a Hamiltonian of the form $H=\vec{k}\cdot\vec{\sigma}$ can have?

Hamiltonians of the form $H=\vec{k}\cdot\vec{\sigma}$ with $\vec{k}$ being the crystal momentum and $\sigma_i$ being the $i$-th Pauli matrix (an $su(2)$ generator), are pretty common in the study of ...
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Problems book on Lie groups and Lie algebras for physicists

I am trying to get a better gras of Lie groups and Lie algebras and I was wondering if there's any book or general resource with problems and solutions in order to learn through practice.
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Definition of matrix Lie group as closed subset of $GL(n;\mathbb{C})$

I am trying to understand the following definition of matrix Lie group: A matrix Lie group is a subgroup $G$ of a $GL(n;\mathbb{C})$ such that if $A_m$ is any sequence of matrices in $G$ and $A_m$ ...
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How to prove $Q_u = -2Q_d$ from $SU(5)$?

I'm a beginner, and I'm trying to figure out how to prove that charge of Up quark is equal to 2 times the charge of down quark from the 10 representation of $SU(5)$. Please help.
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Understanding the definitions of vector and scalar [migrated]

So I am preparing now to start studying Lagrangian and Hamiltonian mechanics with Marion's book on classical dynamics. It is the first time I encounter the formal definition of vector and scalar, and ...
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Normalization of Generators of $SU(N)$

I have given a finite-dimensional matrix-representation of $SU(N)$. In this representation, the generators are denoted by $G^{a}$ for $a=1,\dots N^{2}-1$. I have to show that I can choose the ...
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Rotating particles of arbitrary spin [closed]

Is there a way to figure out what is the result of a rotation about an axis of any state in arbitrary spin. Just for example, what is $\hat{R}(\pi/2,\mathbf{y})|2,1\rangle_z$ without their ...
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1answer
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Isomorphisms between isometry groups of maximally symmetric spaces

For maximally symmetric spaces in a given dimension, $d$, and a given signature, it is my understanding that there are always three distinctive cases, according to whether the scalar curvature is ...
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There are infinitely many equivalent irreducible representations of $SO(3)$ on $\mathbb R^3$ [migrated]

The irreducible representation of $SO(3)$ on $\mathbb R^3$ is the set of the matrices $M$ such that $MM^T=I$ and $\det(M)=1$. But this is not the only one, indeed if $A$ is an invertible matrix then ...
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Lagrangian in $D$-dimensions in maximal supergravity

In the paper Dualisation of Dualities. I. By Cremmer,Julia, Lu and Pope they derive the $D$-dimensional lagrangian in terms of the generators $E_{i}^j$, $E_{ijk}$ and $D$, I believe $E_{i}^j$ is the ...
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1answer
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$SU(2) \to I$ symmetry breaking

In "Cosmic strings and other topological defects" by A. Vilenkin and E. P. S. Shellard, section 2.1.3, they detail some examples of spontaneous symmetry breaking. In the first example, they discuss ...
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Commutativity and Associativity of Poincare Transformations

Commutativity and Associativity of Poincare Transformations: For commutativity I showed that $2$ successive transformations does not commute with the same transformations reversed. $$(Λ_2 Λ_1, Λ_2 ...
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1answer
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Representation of the Poincaré group by means of exponential

Let $(\Lambda,a)$ be a Poincaré coordinate transformation. Let $U$ be an unitary representation of the Poincaré group on some vector space. Is it always possible to express $U(\Lambda,a)$ in the ...
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1answer
66 views

A particular representation of $SU(2)$ on $\mathbb R^3$

Studying physics I encounter group theory and it has told me that: the matrices that rotates $\mathbb R^3$ vectors in the Euclidean space are the representation of $SU(2)$. Namely, $SO(3)$ matrices ...
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1answer
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What are the physical manifestations of the finite-dimensional irreducible non-unitary representations of the inhomogeneous Lorentz group?

According to https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group: "The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-...
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2answers
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Permitted bound states in extended QCD color group $SU(4)$

I'm very much curious about the possible quark combinations to form bound states in $SU(4)$ QCD. We already know Wilson has proven that only colorless combinations can arise in $SU(3)$ of these ...
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1answer
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Wigner's classification in curved space

Wigner classfied elementary particle as unitary irreducible representations (UIR) of the Poincaré group. Suppose the spacetime is curved with symmetries $G$. Should the elementary particles in this ...
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1answer
85 views

Spin state rotations and spinors rotations

I've tried to do the calculations to derive the SU(2) matrices that rotates spinors from the rotation of the spin eigenstates. The following is the procedure that I followed but at the end I didn't ...
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1answer
72 views

Set of zeroes as coset space

I am currently studying Chapter 6 of Coleman S. - Aspects of Symmetry. We study a spontaneously broken gauge theory in two spatial dimensions where the Lagrangian reads: $$ \mathcal{L} = -\frac{1}{4}...
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1answer
80 views

The word “representation” in the context of Lie groups

I'm still very new to learning about Lie groups, something I find particularly confusing is the use of the word representation in the context of Lie groups. Sources I've checked online go quite far ...
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3answers
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Why is representation theory important in physics?

Given a certain group we can find many representations of it. And If I'm not wrong a representation is a group itself. For example, given the group of the unitary 2x2 matrices with determinant 1 $SU(2)...
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1answer
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Some clarifications about the ideas of representation of a group

I started to study group theory but I have many doubts about the topic, so I'd like to share my current understanding together with some questions, my aim is to understand the general ideas and ...
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How do I prove that all symmetries have an inverse?

I am currently studying discrete symmetries in quantum mechanics and have trouble proving that the set of discrete symmetry operators is a group. An operator, $\hat S$, is called a symmetry operator ...
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1answer
40 views

Terminology about the tensorial represention of a group

This could be extremely trivial but i need to be sure I am not wrong. I am encountering many times statements where the author says "Tensors are examples of representations for the Lorentz group". ...
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1answer
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Baryons in flavor $SU(N)$ (in ChPT)

For flavor $SU(2)$ (Isospin) we have two $\frac{1}{2}^+$ baryons, the nucleons. For flavor $SU(3)$ we have the eight baryons in the octet. In a world with $N$ light quarks we would see a baryon ...
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2answers
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Lorentz representation of totally symmetric tensors?

Pretty much the title. A rank 1 tensor is $(1/2,1/2)$, a symmetric rank 2 tensor is $(0,0)+(1,1)$. I'm curious how this generalizes to rank $n$ totally symmetric tensors.
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Why is the worldvolume theory a $U(1)$ gauge theory?

So in string theory a Dp-brane can be described by a $(p+1)$ dimensional QFT living in its worldvolume. For a string, this is similar to the description of the string as a twodimensional QFT living ...
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Why does having a representation of the Poincaré algebra imply conservation of energy, momentum and angular momentum?

Considering that Poincaré algebra is given by the following relations $$i[J^{\mu\nu},J^{\rho\sigma}]=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}-\eta^{\sigma\mu}J^{\rho\nu}+\eta^{\sigma\...
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References for listings of point group operations

I am looking for a listing of the operations of point groups (preferrably as matrices). Nearly every textbook lists the irreducible representations and characters of the groups, but I have not found ...
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1answer
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Regarding the signs in the Clebsch-Gordan coefficients

Let's take, for example, the $\frac{1}{2}$ $\frac{1}{2}$ spin case. We have, for $J = 1, M = 0$ $$|1,0\rangle=\frac{1}{\sqrt{2}}(|-1 / 2,1 / 2\rangle+|1 / 2,-1 / 2\rangle),$$ and, if we follow the ...
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Are all familiar symmetry transformations, when they act on fields, linear? [duplicate]

Consider symmetry transformation acting on a field or a set of fields. For example, a gauge transformation of the form $$\phi^\prime_a(x)=U_{ab}(x)\phi_b(x)$$ where $U(x)$ is a matrix with elements ...
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1answer
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What's up with $\mathrm{U}(1)$ regarding the spin homomorphism?

Let $\mathcal H(2)$ be the space of hermitian matrices of size $2\times 2$, and let $\sigma:\mathbb R^{4}\rightarrow\mathcal H(2)$, $$ \sigma(x)=x^\mu\sigma_\mu=\left(\begin{matrix} x^0+x^3 & x^1-...
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1answer
48 views

Product of representations of Lorentz group

How represent angular momentum in representation $(a,b)\otimes (c,d)=(a\otimes c,b\otimes d)$ ? I also post Tensor product of representations of Lorentz group Agreements: $I_a$ is $(2a+1)$-...
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Quartic Casimir of the 3D conformal group

I am studying the conformal group in 3 dimensions. The generators of this group are isomorphic to the generators of $SO(1,4)$. Hence two of the Casimir operators are, $$C_1=-\tfrac12J_{AB}J^{AB}$$ $$...
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1answer
119 views

Tensor product of representations of Lorentz group

Where is the rules for tensor product of representations of Lorentz group $(a,b)\otimes (c,d)$ without decomposition of one of these in orthogonal sum?
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Is this extended super-symmetry possible?

The usual extended super-algebras have generators $Q_N^{\alpha},\overline{Q}_N^{\alpha}$ which can have $4^{N}$ components of the algebra. (Which can be arranged on the vertices of a hypercube). e.g. ...
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How to realize explicitly an $SO^+(1,3)$ Lorentz transformation of $(x-y)$ to $(y-x)$ for space-like separation?

Consider two spacetime points $x, y$ which are space-like separated. How can one realize explicitly a proper, orthochronous Lorentz transformation between two frames such that $x-y$ becomes $y-x$? ...
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4answers
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$U(n)$ and $SU(n)$ Symmetry

I am a first year post graduate student and studying symmetries in quantum mechanics. I am not getting the point why we need $U(n)$ and $SU(n)$ symmetry in general and how does groups come into role?
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Irrep of stress energy tensor

We have 4-tensor of second rank. For example energy-momentum tensor $T_{\mu\nu}$, which is symmetric and traceless. Then $T_{\mu\nu}=x_{\mu}x_{\nu}+x_{\nu}x_{\mu}$ where $x_{\mu}$ is 4-vector. Every ...
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1answer
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Internal flavor symmetry of the $N$ left-handed complex Weyl spinors v.s. $N$ real Majorana spinors: ${\rm U}(N)$ vs. ${\rm O}(2N)$ or ${\rm O}(N)$

Consider 4d spacetime, it seems that for massless particles, we can easily change the left-handed complex Weyl spinor basis (2 component in complex $\mathbb{C}$ for Euclidean spacetime Spin(4)) to ...
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3answers
88 views

Lie group and corresponding smooth manifold, and also why $SO(3)$ have a 3-dimensional manifold embedded in 4-dimensional Real space?

I think I have some loop holes on a connecting smooth manifold to a lie group. I state what my concepts are, Lie groups are expressed as manifold because the parameters in corresponding metric form a ...
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0answers
19 views

Bloch theorem for a semi-infinite crystal

How could we formulate the Bloch theorem for a semi-infinite crystal? For simplicity I suggest assuming that the crystal boundary is along one of its crystallographic planes. One could also assume a ...
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29 views

Irreducible representation and character number

I am doing a course on "Symmetry of molecules and crystals". We talked about character tables and irreducible representation, as well as the notation for the representation. I got the following ...

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