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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How to prove that a Hamiltonian is $SU(4)$ invariant?

I am studying the following hamiltonian, consisting of a double quantum dot system \begin{align} \hat{H}_{2QD} = \sum_{i=1,2}v_{i}\hat{n}_{i}+\sum_{i=1,2}U_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow} +...
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Lorentz scalar does not commute with lorentz generator?

If an object is a Lorentz scalar, it should commute with the Lorentz generator $M^{\mu\nu}$. However, I don't know why $[W_{\alpha}\bar{Q}\bar{\sigma}^{\alpha}Q,M^{\mu\nu}]\neq 0$, given the (anti)...
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Explanation of the meaning of those singlets, doublets

I am stuck in understanding and following the explanation (?) written within the red square bracket in the notes below. Can someone please explain me what does this mean? I tried to search in other ...
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What do I need and where do I start to understand unitary (projective) representations in QFT? [closed]

Currently I'm studying QFT from Weinberg and also watching the lectures of Prof. Tobias Osborne through his YouTube channel. He started the first lecture by talking about unitary representations and ...
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What are some examples of sporadic groups in physics?

The existence of sporadic groups is strange in the sense that it seems unnatural. As group theory is unavoidable in physics, have we found that there are things in nature that are better described by ...
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Why the hermitian conjugation swaps the two $SU(2)$ Lie algebras that comprise the Lie algebra of the Lorentz group?

I've start reading the part II (spin 1/2) of srednicki's qft book and I met a problem about group theory. In the section 34, the author describes the left and right handed spinor field. He says that ...
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The proof of the Wigner-Eckart theorem for irreducible tensor operators

I am reading through Wu-Ki Tung's Group Theory in Physics and I met a problem when going through the part of the Wigner-Eckart theorem for irreducible tensor operators. In the 4.3 part of the book, ...
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Sym$^2\mathbb{C}^2$ as the unique 3-dimensional irrep of $\operatorname{SU}(2)$

In this script (Link) regarding GUTs it is stated that the unique 3-dimensional complex representation of $\operatorname{SU}(2)$ up to isomorphism is given by Sym$^2\mathbb{C}^2$, the symmetric ...
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Symmetry and Symplectic Group of Hydrogenic Atom

New version of the question: A simmetry needs to be canonical, following the first answer of this post which states: the symmetry requirement is not necessary in the definition of canonical ...
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Addition of Two Elements of Group Representation (Quantum Mechanics angular momentum)

In Sakurai's Modern Quantum Mechanics I saw the author takes commutation of two infinitesimal 3D rotation matrices. He also claims that the Hilbert space rotation operators should satisfy the same ...
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Do Legendre transformation form a group?

In my classical mechanics class, my professor asked if Legendre transformations form a group, and in my little knowledge about groups, I know that a transformation group consists of a set of ...
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How the factor of $1/2$ is purely conventional?

From Zee's book on group theory, he mentioned that factor $1/2$ is conventional due to historical reasons, but I thought that it was risen to match the Lie algebra of $SO(3)$? For $N=2$, the ...
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Geometric symmetry for hydrogenic atom $SU(2)$ vs $U(2)$

I understand $SO(3)$ is the right group for proper rotation of its Hamiltonian. $O(3)$ describes both proper and improper rotations and the system has symmetry even for reflection. $O(3)$ is not ...
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Direct product of a symmetric $SU(3)$ tensor and a vector

I was trying to do some exercises to feel more confident about decomposition of reducible representations into direct sum of irreducible representations. The exercise involves decomposition of direct ...
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Which is the correct way to write a Lorentz group component in exponential form?

I have this issue with the Lorentz transformations. I learned at some lectures that any Lorentz group component can be written as: $$\Lambda = e^{\frac{1}{2}w_{ij}M^{ij}}$$ where the different ...
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How to show $SU(3)$ symmetry of the following hamiltonian?

I have a hamiltonian of the form: $H = \sum_i (\hat{U^+_i}\hat{U^-_{i+1}} + \hat{U^-_i}\hat{U^+_{i+1}}+\hat{V^+_i}\hat{V^-_{i+1}}+\hat{V^-_i}\hat{V^+_{i+1}}+\hat{T^+_i}\hat{T^-_{i+1}}+\hat{T^-_i}\hat{...
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Are the linear Lie groups matrices, tensors, or both?

In some ways, this is a question about notation. In my experience, I have only seen the classical Lie groups — such as $\operatorname{GL}(n,\mathbb{R})$, $\operatorname{SL}(n,\mathbb{R})$, $\...
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Why do we prefer to use $i$ with generators in Lie Algebra [duplicate]

I am reading A. Zee. Group theory in a Nutshell for Physicists and for some reason, he prefers to write the generators with an $i$ near them For example, a rotation can simply be described as: $$e^{\...
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Do Bloch sphere rotations span $SU(2)$ (up to a global phase)? [duplicate]

It is well known that the Pauli group $\{I,X,Y,Z\}$ spans the group of $2\times2$ unitary matrices, $SU(2)$, for example see this link. A general Bloch sphere rotation by an angle $\alpha$ about an ...
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Reflection property in traces of $SU(N)$ generators

Pure gluon amplitudes could be organized on different basis, most common ones are the trace basis and the DDM (Del Duca-Dixon-Maltoni) basis, see DDM's paper, for better comprehension and to see how ...
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$\operatorname{SU}(n)$ Yang-Mills symmetry breaking to $\operatorname{U}(1)^{n-1}$

I want to break $\operatorname{SU}(n)\to\operatorname{U}(1)^{n-1}$ using a vev of an adjoint scalar and find the charges of the now massive vectors under $\operatorname{U}(1)^{n-1}$, but I don't know ...
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In the Poincaré group, what are explicit representations of translations, boosts, and rotations?

Context In [1], cowlicks asks, ``How can the Gallilean transformations form a group?''. In [1] Selene Routely explains that the Galilean transformations form a group of dimension 10. Routely explains ...
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Doubt on $SU(2)_{L} \times U(1)_{Y}$ covariant derivative and its action on a fermion

I) Introduction I.1) The mathematical structure is quite clear: given a spacetime $M$, and a Lie group $G$ (the gauge group), we can construct the Principal bundle $P^{G}_{M}$. The connection $1$-form ...
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What we talk about when we talk about Lorentz transformations?

Context In [1], cowlicks asks the question, ``How can the Gallilean transformations form a group?'' It is clear what a group is. Borrowing liberally from [2], "A group is a set $G$ together with ...
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Are Pauli matrices invariant tensors in the representation of $\frac12 \otimes \frac12 \otimes 1$?

If we raise the index of the Pauli matrices with Levi-Civita symbol $\epsilon$ we obtain the 2-index spinors $(\sigma_i)^{AB} = (\sigma_i)^A{}_C \ \epsilon^{CB}$. The textbook (Ref. 1) argued that ...
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6 votes
1 answer
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Gauge Theory determined by Gauge Group and Representation: What about specifying the bundle?

I have the following question. In physics, when one talks about (Yang-Mills) gauge theories, one often states that it is enough to specify the following data: The gauge group $G$, which is usually a ...
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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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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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2 answers
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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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13 votes
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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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3 votes
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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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2 answers
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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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3 answers
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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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1 answer
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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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3 votes
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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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10 votes
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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