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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Relation between the commutator of commutators in Dirac algebra

In an attemption to obtain the curvature tensor related to the spin connection of the fermionic fields I came across this expression with the commutator of the gamma matrices commutators. My question ...
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How to prove that all Minkowski spacetime isometries (transformations in Poincare group) are compositions of translation and Lorentz Transformations?

It is said in wikipedia that Minkowski spacetime isometries, i.e. the transformation that preserves $$ (x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-(t_1-t_2)^2 $$ between points, can be represented as $\mathbb{...
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Metric structure on a group

I have a question about metric structure on a group manifold $G$. Imagine we have a sigma model, i.e. a map $g: \Sigma\rightarrow G$ from some 2D source to our group. One can define the left/right ...
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Clebsch-Gordan Series and Rotation Matrices

I am referring to the first inequality in equation 3.390 on page 217 of Sakurai's "Modern Quatum Mechanics" textbook. The quantity $D^{(j)}(R)$ refers to a rotation operator in the ket space ...
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Equivariance and Wigner $D$ matrices for Spherical harmonic rotations

I am trying to understand equivariance in machine learning, specially as discussed in the following paper. Claim is that equivariance is when Group symmetry operation, such as rotation, commutes with ...
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Questions regarding the 'adjoint of $SO(N)$' section in Zee's group theory

I am reading A. Zee's book about group theory and I am confused about several parts of his discussion in chapter 4.1 about the adjoint of $SO(n)$. First question: The first thing mentioned is that an ...
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Apparent elimination of a 't Hooft anomaly in quantum spin system

The simplest system with a 't Hooft anomaly is the spin $\frac{1}{2}$ system with hamiltonian $\hat{H}=0$. The 't Hooft anomaly follows from the fact that such system has a trivial $SO(3)$ symmetry, ...
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Find commutator $[P_\mu,K_\nu]$ in conformal group

We have conformal group with next element of this group: $$U=e^{i(P_\mu\epsilon^\mu-\frac{1}{2}M_{\mu\nu}\omega^{\mu\nu}+\rho D+\epsilon_\mu K^\mu)},$$ where $D$ is dilatation operator $$x^\mu\...
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From a quant-info perspective why are the reals indexing irreps of the Lorentz group less suspect than continuousness in space-time and general QM?

It is an opinion I occasionally hear, and perhaps hold myself, that the resolution to the 'infinities' that crop up in various bits of physics are artefacts of the approximation that space-time is ...
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A relation for adjoint representation of $U(N)$ acting on product of matrices and $SU(2)$ generators

Is the following relation true, and if so, what is the property that makes it so? \begin{align} \sum_{i=1}^3\mathrm{tr}\left([U^{-1}L_iU]\phi[U^{-1}L_iU]\phi\right) \stackrel{!}{=} \sum_{i=1}^3\...
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Can one express the evolution of a particle with a one-parameter group of $SO(3,1)$?

Can one express the evolution of a particles using a sequence of $SO(3,1)$ transformations? If yes, how? Is it sufficient to apply $SO(3,1)$ transformations to a spinor? $$ \psi(t) = e^{t\mathfrak{so}(...
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Combination of 3 spin $\frac{1}{2}$ particles to yield a state of net spin $\frac{1}{2}$

I did some prior research to this question on stackexchange before posting my question. Due to my limited knowledge in this field, I am not sure if my question is unique since there has been ample ...
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Non-invertibility of crystallographic pole figure

In Group Theory by Morton Hamermesh, while explaining the pole figure and its stereographic projection, it is clear to me that we will get same angle between crystal sample faces if we scale all ...
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Proving rotation about non-intersecting axes leads to translation

In Group Theory by Morton Hamermesh, he states on page 32: For a body of the finite extension, a molecule or the macroscopic form of a mineral, only the first two symmetry types [rotation, reflection]...
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Can the $SU(3)$ gauge field be put in geometric algebra terms?

According to this article on the spacetime algebra, we know the Dirac spinor can be thought of as an even element of the Clifford algebra over spacetime, which in turn can be thought of as a general ...
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Do $Y$ bosons gain a little bit of mass from Higgs of $\bf 5$ rep in $SU(5)$ GUT theory?

I know that when the Higgs in the 24 rep takes a v.e.v. of this form $$v_{24}\,\mathrm{diag}\begin{pmatrix} -\frac{2}{\sqrt{15}} & -\frac{2}{\sqrt{15}} & -\frac{2}{\sqrt{15}} & \frac{3}{\...
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Basic Facts about Lie Algebras

I am reading P&S (Peskin's and Schroeder's book An Introduction to Quantum Field Theory) and in particular Chapter 15.4. At some point the authors say that any infinitesimal group element $g$ can ...
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Why does general irreducible representation $(A, A)$ of a quantum field correspond to traceless symmetric tensor of rank $2A$? [duplicate]

I understand that under rotation, we will have components that transform like integer spin $(2A, 2A-1, ...... 0)$ from decomposition of $(A, A)$ representation. The scalar is the trace, therefore ...
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$U(1)$ factor decoupling at low energy

Consider a (super)-Yang-Mills theory with $U(N)$ gauge group. I read several times that the gauge group can actually be taken to be $SU(N)$ instead because the $U(1)$ factor decouples at low energies, ...
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Invariants of inner product in pseudoreal representation of $SU(2)$

I am reading Peskin's and Schroeder (P&S), "An introduction to Quantum Field Theory", specifically the first paragraph on page 499 in section 15.4 "Basic Facts about Lie Algebras&...
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Constructing gauge invariants

Is there an efficient way for constructing gauge invariants given the number of operators one can use is fixed. For example, if I am given some boson in $\mathbf{3}$ of $SU(2)$, and I want to find ...
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Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

Vectors $\vec A=(A_1,A_2)$ and $\vec B=(B_1,B_2)$ are 2-dimensional representations of $SO(2)$. I want to understand the decomposition $$2\otimes 2=1\oplus 1\oplus 2.$$ I can easy identify that the ...
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I tried to disentangle an exponential of a sum of operators of the type below, but it lead to a different result from that mentioned in the article [closed]

could you explain how we get to equations 10 please, or indicate the formula used. help me mathematiciens ! :)
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What were the $r$ and $n$ of $\theta$s (Polchinski String theory section 8.6 page 265)?

In the Polchinski String theory section 8.6 page 265 In generic backgrounds, all the $\theta$s are distinct and the only massless vectors are the diagonal ones, $i = j$. The unbroken gauge group in ...
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5 answers
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Why does group representation theory look linear?

I'm reading first a few chapters of a physicist's group theory book and one naive question comes into my mind. I feel I probably missed something very basic and got bogged down in the details. My ...
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Mathematical characterisation of diffeomorphisms in General Relativity

Considering the diffeomorphism covariance/invariance of General Relativity, is it possible to characterise mathematically the various kinds of possible transformations $x'^{\mu} = f^{\mu}(x)$? All of ...
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Nonlinear symmetry realization: what is it for and caveats?

I have several doubts regarding the nonlinear realization of a spontaneously broken symmetry and hope they are appropriate to be grouped, and I appreciate any insights. Consider the group breaking ...
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Deriving a recursion relation for Clebsch-Gordan coefficients

I refer to Sakurai's Modern Quantum Mechanics. Using the lower signs for equation $3.369$ on page $212$, $$\sqrt{(j+m)(j-m+1)}\langle j_{1}j_{2};m_{1}m_{2} | j_{1}j_{2};j,m-1\rangle = \sqrt{(j_{1}+m_{...
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$SU(3)$ and $SU(3)\times SU(2)\times U(1)$ Symmetry Breaking [duplicate]

For my master's project I was doing spontaneous symmetry breaking in which I covered U(1), SU(2), SU(2)×U(1) symmetry breaking. My supervisor has said that for the project this much is enough. But now ...
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How is the interaction of leptons and quarks with gauge fields organized in the Standard Model?

Well, standard model has $SU(3)\times SU(2)\times U(1)$ gauge group, so this is a direct product of multiple groups embedded in larger set. So how quarks that must interact with every gauge field and ...
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4 votes
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Wilson loops as representations of the Lorentz group

Wilson loops in lattice $4d$ Yang-Mills theory are used to build various glueball states of different spins when they are applied to the vacuum. The spin dependence of such states is related with the ...
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1 vote
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How to comprehend the fact that parity is an improper rotation in the odd dimension, but not in the even dimension, physically?

Some "clarification" To begin with, I'm not even talking about relativity so, in the following, rotations always act on the Euclidean space or only the space subpart of the Minkowski space. ...
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What is the structure group of the $SU(3)$ manifold?

It has been known for many years that the manifold of the Lie group $SU(3)$ is the non-trivial fiber bundle of the spheres $S_5 \times S_3$, in which $S_5$ is the base and $S_3$ the fibers. What ...
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Sources to learn Gauge Theory, Groups, Lie Algebra, etc [duplicate]

As seen in previous questions, I'm interested in gauge theory, although I have no idea how to do any of the mathematics, though i'd like to start. With that in mind, Are there any good sources that ...
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Parameterization of $SU(4)$

I'm looking for a parameterization or general form for a $4\times 4$ special unitary matrix in closed form. I have looked at the answer to Good reference on the parametrization of $SU(3)$ and $SU(N)$ ...
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What does the exponentiated generator of scale transformation do when it acts on a function? [duplicate]

We know that $d/dx$ is the generator of translation in the sense that $$e^{ad/dx}f(x)=f(x+a)\tag{1}$$ which can be easily be proved from the Taylor series of $f(x+a)$. Studying the very basics of ...
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1 vote
1 answer
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How many massless particles are there in closed bosonic string theory?

How many massless particles are there for closed strings in bosonic string theory? Page 53 of String Theory and $M$-Theory by Becker and Schwarz states that there are 576 massless states for the $N = ...
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4 votes
4 answers
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Precise definitions for higher spin operators

I am trying to understand the matrices and vectors presented in this section https://en.wikipedia.org/wiki/Spin_(physics)#Spin_projection_quantum_number_and_multiplicity I am looking for a reference ...
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2 votes
1 answer
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From the point of view of physics, why is it useful to know the irreps of rotation group?

In 3D, the rank two tensorial physical quantities, for example, the electric susceptibility, the conductivity, the stress tensor etc, are in general, not irreducible representations i.e. neither ...
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Yet more gauge group nonsense: $D3$? $Q8$? $Z8$?

This'll probably make me look like a total idgit, but I have a new question in the same vein as mine about $SU(4)$, but this time without any guesses. I've looked a bit into groups, and it looks like ...
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What does it mean for the $\textbf{B}$-field (Hypercharge) to be in the 0 representation within the SM?

I was reading through the wikipedia page for the mathematical formulation of the standard model and I noticed that it listed the representations of the vector bosons under the SM gauge groups as being ...
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Is every unitary representation of the Poincare group a direct sum of Wigner's irreducible representations?

Is Wigner's classification of unitary irreducible representations of the Poincare group [1] sufficient for constructing all unitary representations of the Poincare group by taking direct sums? The ...
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Deriving an identity with rotation generators [closed]

I am trying to justify the following identity on page 68 of Osborn's notes on group theory: $$e^{-i\pi J_{3}}J_{2}e^{i\pi J_{3}} = -J_{2}.$$ Here, the $J_{i}$ are the typical angular momentum ...
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Extracting $\mathbf b$ from $M = a I + \mathbf b \cdot \mathbf S$, when $S_i$ are higher spin matrices?

This is a cross-post of a question that I posted on the Math SE, that did not get any answers there. It is fundamentally a mathematics question, but it pertains to spin matrices, which many Physics SE ...
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Orthogonality relations for matrix elements of irreducible representations

I am reading Howard Georgi's "Lie Algebras in Particle Physics" and have a question concerning the presented orthogonality relations for matrix elements of irreducible representations. To ...
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Surface effect on non-symmorphic symmetries when applied to tensors

I am curious about the effect that a surface has on a non-symmorphic symmetry. To be concrete, assume that we have a crystal with symmetry-group $G$, which contains non-symmorphic elements $\{R|\tau_R\...
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Tensor products in Howard Georgi's "Lie Algebras in Particle Physics"

My question is regarding eq.(3.39) in the second edition of Georgi's book (for those who have the book:)). The section deals with tensor product states where the states comprising the product ...
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Dilation operator acting on $x$-dependent field

I've been studying conformal field theory (CFT) and got the following "apparent" inconsistency. Let's take dilation ($D$) and translation ($P_\mu$, 4-momentum) generators that according to ...
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2 votes
2 answers
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Exponential of an operator shifted by the derivative operator

Let $p(x)$ and $f(x)$ be sufficiently smooth functions and $D=\frac{d}{dx}$. It is easy to show that $$e^{p(x)D}f(x)=f(e^{p(x)D}x).\tag{1}$$ If $p(x)=a \in \mathbb{R}$ , we have the shift operator as $...
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The irreducible representation of rank $n$ spinor in 3D

I was reading Ref. 1, where it is asserted that the irreducible resolution of a rank $n$ spinor can be written as totally symmetric spinors. $$\Psi^{n}=\Psi^{\{n\}}+\zeta \Psi^{\{n-2\}}+\zeta \zeta \...
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