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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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Why are Spin 1/2 particles invariant to $4\pi$ rotation loops while Spin 1 particles are invariant to $2\pi$ loops?

Why do Spin 1/2 particles when turning them by 360 deg get a phase factor of -1 and a loop of 720 deg leads to the identity while for spin 1 particles a loop of 360 deg gives already the identity?
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Root weights and states in orbifold compactifications

I have the following question regarding orbifold compactifications of the heterotic string: What is the relation between a certain representation and the weights of the root lattice? I mean: take ...
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1answer
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Negative unity matrix not hermitian? (stabilizer formalism)

I read the section in the attached picture about the stabilizer formalism and was wondering about the last sentence in the pic. It says that all operators of the stabilizer group are hermitian, ...
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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 ...
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1answer
72 views

Bosonic representation of $SU(N)$: what values can $n_b$ take?

In Assa Auerbach's book on page 166, he describes the construction of a bosonic representation of $SU(N)$ where the generators $S^{mn} \rightarrow b^\dagger_m b_n$. I'm a bit confused about the ...
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1answer
47 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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Dimension of Lie algebra and Lie group generators

In physics, we define the Lie group generators as the basis of Lie algebra. E.g. for $SO(3)$, whose generators are $$J=\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & i\\ 0 & -i & 0 \end{...
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Relationship between Eigenvectors of Hamiltonian vs function of the Representations of the Group

I am trying to understand the relationship between the eigenvectors obtained from a diagonalizing a Hamiltonian and the basis functions of the Representations of the Group, $G$, used to build the ...
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1answer
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Why $dS^d \cong SO(d,1)/SO(d-1,1)$?

I have found a similar question, but there they give a seemingly rigorous proof, and what I am looking for is just an intuition. I understand that $S^2 \cong SO(3)/SO(2)$: for every point in $S^2$ ...
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1answer
43 views

What happens to the $U(1)$ factor in the $U(N)$ SYM gauge group of the AdS/CFT correspondence?

I'm learning about the AdS/CFT correspondence. I know that from the open string perspective, the dynamics on a stack of $N$ coincident $D3$-branes is given by a $\mathcal{N} = 4$ Super Yang Mills ...
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1answer
35 views

About $(0,1/2)$ representations

While studying representations of Lorentz group, we get the generators to be $J_{i}$ - rotations and $K_{i}$ - boosts. We define $N_{i}^+$ and $N_{i}^-$ operators and these operators obey the same ...
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1answer
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What is the dimensionality of each part of a covariant derivative?

In the standard model, we have the following covariant derivative: $$D_\mu = \partial_\mu - ig_sG_\mu^a\lambda_a-igW_\mu^a\frac{\sigma^a}{2}-ig'B_\mu\frac{Y}{2}$$ If we let this work in on e.g. the ...
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1answer
29 views

Which are the underlying Lie group and algebra related to the translation invariance in field theories?

I'm new to Physics SE. I've seen a lot of interesting questions and answers, and thought it will be very useful to participate a little. I'm currently stuck in a, probably, very simple matter, ...
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2answers
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Role of the special linear Lie algebra in general relativity (GR)

The Lie derivative measures the difference between two paths in the timespace manifold, and hence the commutator bracket occurs naturally, as explained in the presentation What is a Tensor? Lesson 21: ...
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2answers
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Isometry group on a coset manifold

In ''Einstein Gravity in a Nutshell'' Zee says ''On a coset manifold $G/H$, the isometry group is evidently just $G$'' when discussing the relation between the Killing vector fields and Lie ...
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1answer
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Generators in Field Theory and Derivatives

Let's consider a representation of the multiplicative group $(0,\infty)$ on Minkowski space $\mathbb{R}^4$ by dilations. \begin{align} \rho:(0,\infty)&\rightarrow\text{GL}(\mathbb{R}^4)&\\ a ...
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2answers
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Commutator generating transformations

Lately I am encountering the commutator of variations of the variables and I'm not quite sure about its physical meaning. Some examples. 1) "The composition of two supersymmetries generates a time ...
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2answers
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Clarification on statement in “Unitary Symmetry and Elementary Particles” by Lichtenberg

He says that: The set of values of the parameter or parameters which characterize a group element can be considered to be points in some kind of space. The number of parameters characterizes the ...
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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
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Are particles always represented by 4-volume preserving transformations?

I've been reading about affine gauge gravity, which uses the affine group A(4,R) (for example here ). If I'm getting it right there seems to be an “affine higgs” mechanism that breaks symmetry down ...
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What precisely is a *classical* spin-1/2 particle?

I was recently having a Twitter conversation with a UC Riverside Prof. John Carlos Baez about Geometric Quantization, and he said (about his work) that "Right. For example, you can get the ...
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Decomposition of the complex conjugate of the fundamental representation of $SU(5)$ in $SU(3)\times SU(2)\times U(1)$

I know I can decompose the fundamental representation (denoted as $5$) of $SU(5)$ as: $$ (3,1)_{-2c/3} \oplus (1,2)_{c} $$ But how do I get the decomposition of the complex conjugate of this ...
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Why are systems joined via a tensor product? [duplicate]

This question comes from seeing that the triangle addition rule for quantum mechanics comes out of groups/representation theory; I thought this was odd as we haven't used any group ideas in QM up to ...
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2answers
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$SU(2)$ Invariant Lagrangian

Consider two arbitrary scalar multiplets $\Phi$ and $\Psi$ invariant under $SU(2)\times U(1)$. When writing the potential for this model, in addition to usual terms like $\Phi^\dagger \Phi + (\Phi^\...
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What should I read to understand this question?

I understand the strong force as a Yang-Mills theory with $SU(3)$ color invariance. I understand that the quarks live in the fundamental representation of $SU(3)$ and that gluons live in the adjoint. ...
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1answer
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Wigner proof of the non-existence of finite unitary representation of the Lorentz group

I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
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Wigner-Eckard theorem in 3+1 Minkowski dimensions

From this source, I have: I cannot find much (if not any) information online for Wigner-Eckard in 4D, hyperboiloid harmonics etc. And there are many facts just states in this source that I would ...
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0answers
38 views

Particle statistics in fractal dimensions? [closed]

We know that fermions and bosons are the only two (indistinguishable) particle statistics for $d\geq 3$, and that anyons are for $d=2.$ What if the space were a fractal? Like the Sierpinski gasket, ...
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1answer
57 views

Supercharge transformation rules

Consider ${\cal N}=2$ supersymmetry with $SU(2)$ global symmetry group. Then both supercharges $Q_{ai},\bar{Q}_{\dot{a}\dot{j}}$ transform by 2 dimensional representation of $SU(2)$. Denote $SU(2)_I$ ...
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How to write the Poincare transformation for an arbitary path in Minkowski space?

So lets Say for arguments sake we have some vector $V^{a}$ and we drag it along some path $\gamma_{1}$ in Minkowski space $R^{3,1}$. For a straight path (represented by a vector $\Delta\overrightarrow{...
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Some clarifications about Point Groups rapresentations for molecules

This is what I've understood until now about this subject, please confirm if what I write is true or not. Choosen one molecule, there is a certain number of symmetry operations which this molecule ...
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1answer
50 views

Poincaré and Galilei group - notation

On this slide it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of. What does $\mathbb{R}^{1,3}$ mean? Why does $\...
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2answers
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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: ...
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2answers
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Spin statistics from the fundamental group of $SO(D)$

I read the answer to this question and am very intrigued by its simple and elegant explanation of the emergence of anyon, boson & fermion statistics. @Trimok basically says: In a space-time ...
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1answer
79 views

Doubt in Weinberg's book on Quantum Field Theory

In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as: \begin{equation} U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
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1answer
68 views

Is spin 1 described by $SO(3)$ or $SU(2)$ [duplicate]

What spin is described by which rotation group? I always only find information about spin-1/2
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1answer
72 views

Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
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1answer
59 views

$(1,1)$ representation of $SL(2,\mathbb{C})$

How do you prove that the $(1,1)$ representation of the $SL(2,\mathbb{C})$ group acts on symmetric, traceless tensors of rank 2?
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Can I determine the normalizer of the stabilizer of a quantum code?

We can define a state of n qubits $\left|G\right>$ as the single +1 eigenstate of a set of operators ${ K_i }$, which make up a group $S$ called "the stabilizer" of that state. The n qubits are ...
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2answers
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Unlike rotation, why a $3\times 3$ translation matrix cannot be written in 3D? or can it be?

The effect of rotation in 3d on a vector, $\vec{r}=x\hat{x}=y\hat{y}+z\hat{z}$ is given in the form a matrix product:$$\vec{r}\to O\vec{r}$$ where $O$ is a $3\times3$ proper orthogonal matrix. Can we ...
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1answer
40 views

Relation of the Lorentz group to $O(1,3)$

Let $\Lambda$ be an element of the Lorentz group. It satisfies the identity:$${\Lambda}^T\eta\Lambda=\eta$$ where $\eta$ is the Minkowskii metric. Hence by the usual definition of orthogonality, $\...
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1answer
51 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
52 views

Scattering matrix symmetries and standard model

I am not able to get around the following question (if it make sense): Suppose I can derive the scattering matrix S for any particle scattering process. Suppose that the standard model is actually ...
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2answers
79 views

Operational definition of rotation of particle

The question in brief: what does it mean, operationally, to rotate an electron? Elaboration/background: I am trying to understand how representation theory applies to quantum mechanics. A stumbling ...
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1answer
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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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$SU\left(N\right)$ 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 ...
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2answers
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Why does the pion live in a representation of isospin SU(2) and is the mediator of the strong force generated by color SU(3)?

Why does the pion live in a representation of isospin $\rm SU(2)$ and is the mediator of the strong force generated by color $\rm SU(3)$? I somehow find strange that this is the case. Given that $\rm ...
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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 ...
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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 ...
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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 $\mathbf{5}$ ...