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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Center of $SU(3)$

I assumed a 3x3 matrix of the form $$A= \begin{pmatrix} a & b & c\\ d & e & f\\ k & l & m \end{pmatrix}$$ Then, since we know that the center is always an Abelian invariant ...
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Inner product on group theoretic coherent states and anti-commutator of Lie algebra generators

This question is related to group theoretic coherent states (Gilmore, Perelomov etc.). I consider a semi-simple Lie group $G$ with Lie algebra $\mathfrak{g}$ and a unitary representation $U(g)$ acting ...
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62 views

Isospin and $\Delta$ Production

I have seen the relation between cross sections postulated: \begin{align} \sigma(\nu+p\to\mu^- + \Delta^{++})=9\sigma(\nu+n\to\mu^- + \Delta^{+}) \end{align} Motivated by isospin symmetry. I wanted ...
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Co-spinors and contra-spinors

As i was reading my teacher's notes on $SU(2)$ and $SO(3)$, i have had this question. Why do co-spinors transform differently under a rotation than contra-spinors?
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Determining the manifold picture of a Lie group — and thus determining global properties

1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious? Allow me to give the definitions I am working with. A Lie group G is a ...
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SO(3) and SU(2) [closed]

If we define $$X(\textbf{a} )=e^{(ia_iL_i)} ,$$ how can we show that $X(\textbf{α} )$ can be written as a $2×2$ matrix in terms of 2 complex parameters $a$ and $b$ with $|a|^2 + |b|^2 = 1$, and ...
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60 views

Matrix Expression of the Maurer-Cartan Form

I'm looking for clarification re: the 'classical' matrix expression for the Maurer-Cartan form $$g^{-1} dg$$ (I have seen the related posts, they don't answer my specific question.) Specifically I ...
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34 views

Lorentz algebra and group question with regards to operator representaion of $M^{\mu\nu}$

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: I aim to consider the product $L^0{}_0(\Lambda_1\Lambda_2).$ Consider the following ...
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1answer
48 views

(Physics version of) Taylor expansion. In the the context of deriving a Lie groups generators (a Lie algebra from a Lie group)

Statement which I'm confused about: "Consider some n-dimensional Lie group whose elements depend on a set of parameters $\alpha = (\alpha_1 ... \alpha_n)$, such that $g(0) = e$ with e as the identity,...
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Significance of Wigner-Eckart theorem [duplicate]

What is the physical importance of the Wigner-Eckart theorem and are there any examples of its physical application?
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27 views

Direct product spaces of angular momentum

Consider the direct product space of two angular momentum eigenfunctions: $$|j_1, j_2; m_1, m_2⟩ = |j_1, m_1⟩|j_2, m_2⟩$$ for the simple case when $$j_1 = j_2 = 1/2.$$ How can i construct the ...
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Why do physical dimensions form an abelian group?

The Wikipedia article on dimensional analysis says: the dimensions form an abelian group under multiplication This is used to justify the manipulation of ratios of incommensurable quantities. My ...
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59 views

Cyclicity of $\text{Tr}\ T^a T^b T^c T^d$ for the unitary group $U(N)$?

I am trying to calculate the trace of $4$ generators of $U(N)$, i.e. $\text{Tr}\ T^a T^b T^c T^d$? I found a plausible result, but I would also like to show that the result is cyclic with respect to ...
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Connection between Group of Schrodinger equation and energy level degeneracy [duplicate]

I am recently study group theory and its application in quantum mechanics, but got stuck at a very important point that how group theory can be applied to analyze energy level degeneracy. In many ...
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Is there a second/many order form of the infinitessimal unitary operator in quantum mechanics?

Is there a second/many order form of the infinitessimal unitary operator in quantum mechanics? We know that a unitarily transformed system must be invariant, i.e. $\langle\psi|\psi\rangle = (\langle\...
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Do total derivatives have anything to do with central extensions?

I recently got interested in the Galilean group and its central extension and found a paper "Quantization on a Lie group: Higher-order Polarizations" by Aldaya, Guerrero and Marmo. Before asking my ...
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1answer
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Question about the Lie group $SU(3) \times SU(2) \times U(1)$ and the concept of manifold

I don't know if this question is a duplicate, so I'll delete if is. Well, I'm in the very beginning of the study of contemporary topics such as gauge theories, I would say that I'm in a "science ...
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63 views

Differences betwen the conformal group and the Schrödinger group?

Facts: The Maxwell (free) equations (4d) are invariant under the 15 dimensional conformal group. The free Schrödinger equation in 3d is invariant under the 15 dimensional group "called" Schrödinger ...
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Symmetry v.s. isometry of Minkowski and AdS or dS spacetime

We know some nice spacetime have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
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Could the Universe have monster symmetry?

The Monster Group is the largest "sporadic simple group" with order: 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 I read that if you compactify 26D bosonic string theory ...
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Product of structure constants of $U(N)$

In $SU(N)$, one can derive the following identity: $$f^{abe}f^{cde} = \frac{2}{N} \left(\delta_{ac}\delta_{bd} - \delta_{ad}\delta_{bc} \right) + d_{ace}d_{bde} - d_{bce}d_{ade}\tag{1}$$ with $f^{...
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As $SL(2,\mathbb{C})$ is a double cover of the Lorentz group, is $SL(2,\mathbb{Z})$ a discrete subgroup of the Lorentz group?

The group $SL(2,\mathbb{C})$, the group of $2 \times 2$ complex matrices with determinent $1$, is a double cover of the Lorentz group. (These transformations can be understood as Mobius ...
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74 views

Can we parameterise $SU(3)$ in such a way that there are clearly 2 parameters corresponding to the cartan torus?

We can parameterise the lie algebra of $SU(3)$ using the Gell-Mann matrices, so that a general element of LA is $\theta_i T_i$, where $T_i=\lambda_i/2$ and $\lambda_i$ are the Gell-Mann matrices. ...
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120 views

Induced representation in Zee's Group Theory

I am trying to understand the topic of Induced representation of the euclidean Group E(2) in A. Zee's Group theory in a Nutshell in Chapter IV.i3. The Lie algebra of E(2) has three elements $P_1, P_2,...
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1answer
98 views

Symmetry of Maxwell equations for electric-magnetic duality

According to Griffiths's book on electrodynamics, including magnetic charge the Maxwell equations become $$ \begin{align*} \nabla \cdot \vec{E} &= \frac{\rho_e}{\epsilon_0} &&& \nabla ...
2
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1answer
27 views

$O(p,q)$ as transformations that conserve quadratic form

Let us try to define $O(p,q)$ in two different ways, which I want to show their equivalence. Define the symmetric bilinear quadratic form $[\cdot ,\cdot]$ which is given by $$[x,y]=\langle x,gy\...
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2answers
72 views

Finding properties of Poincare Transformation

I have started studying the Poincare group for the first time, in preparation for my first QFT course, and I wish to be able to solve the following problem: A Poincare transformation ($\Lambda,a)$ ...
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1answer
44 views

Which matrices represent unitary projective representations of ${\rm SO(3)}$?

I was reading this post which triggered the following question. The group ${\rm SO(3)}$ is real orthogonal. However, it is possible to consider representations of ${\rm SO(3)}$ on a complex vector ...
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1answer
62 views

Where to learn about Poincaré Group properties?

I am studying my first QFT course, and there seems to be a lot that I was not taught in previous courses. In my first assignment, I have to prove several properties about the Poincare group, but I ...
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2answers
62 views

Single sequence of angular momentum ladder in quantum mechanics? — Why there is only a

How do you prove that there is only one sequence of angular momentum eigenstates connected by the ladder operator, within the subspace where the squared modulus of the angular momentum has a given ...
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1answer
97 views

$SU(2)$ and $SO(3)$ WZW models

It seems that the $SU(2)_1$ and $SO(3)_1$ Wess-Zumino-Witten models are quite different despite the Lie algebras being identical. The $SO(3)_1$ model has central charge 3/2 and is equivalent to 3 ...
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4answers
2k views

Why do all fields in a QFT transform like *irreducible* representations of some group?

Emphasis is on the irreducible. I get what's special about them. But is there some principle that I'm missing, that says it can only be irreducible representations? Or is it just 'more beautiful' and ...
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70 views

Simple/elementary explanation for $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$? [duplicate]

I am preparing a talk on the Eightfold Way, and am attempting to explain the spectra of the light mesons/baryons via representation theory. It will be delivered to students who have never seen ...
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2answers
68 views

How does the underlying symmetry of QCD imply the allowance of a 4-gluon vertex?

Quantum chromodynamics allows for a four-gluon vertex such as this, in a diagram Such a vertex would never be allowed in quantum electrodynamics, which has an underlying U(1) gauge symmetry. I know ...
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1answer
67 views

Maurer-Cartan form in Physics

I am just reading about the Maurer-Cartan form in the context of Lie Groups, although the mathematical definition: $$\Theta(g)({\bf v}) = (L_{g^{-1}})_{*g}({\bf v})$$ for $g\in G$, $G$ a Lie group, ${\...
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1answer
180 views

$z$ component of angular momentum under Lorentz transformation for massless particle

This question is related to this Helicity states. Suppose we have $k=[\omega,0,0,\omega]$. In Weinberg's book The Quantum Theory of Fields: Volume I he defines the state $|k,\sigma\rangle$ as an ...
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1answer
72 views

Helicity states

On page 71 of Weinberg's book The Quantum Theory of Fields: Volume I, he defines the operators $$A=J_2+K_1$$and $$B=-J_1+K_2$$ where ${\mathbf{J }}=(J_1,J_2,J_3)$ are the rotation generators and ${\...
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2answers
160 views

Physical significance of the decomposition $3\otimes\overline{3}=8\oplus 1$ in meson classification

Under the assumption of $SU(3)$ flavour symmetry of early days, proposed by Gell-Mann and Neeman, among other irreducible representations, the meson octet was obtained. This result from the $SU(3)$ ...
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1answer
62 views

Helicity under rotation

Suppose that the state $|p,\sigma\rangle$ (for a massless particle) has 3 momentum ${\bf p}=p_3$ (that is the momentum is in the $z$ direction) and that $J_3|p,\sigma\rangle=\sigma|p,\sigma\rangle$ ...
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1answer
50 views

Angular momentum operator transformation under rotation

The generators of rotations $J_i$ under rotation transform as $$J_i'=R_{ij}J_j\,.$$ Now $J^2=J_1J_1+J_2J_2+J_3J_3$ trasform as $$ \begin{aligned} J'^2&=RJ^2R^{-1}=RJ_1R^{-1}RJ_1R^{-1}+RJ_2R^{-1}...
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0answers
30 views

There is a classification of matrix representations of Euclidean group $\mathrm{E}(n)$ (also called $\mathrm{ISO}(n)$)?

I'm wondering if someone has already done a classification of matrix representations of the Euclidean group $\mathrm{E}(n)$, more specifically I'm trying to classify $\mathrm{E}(2)$. I watched a ...
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1answer
58 views

Condition for Lorentz transformation

Today I had my first class of a QFT course, and there were some things that apparently I am supposed to know, but I don't. One of them is regarding Lorentz transformations. My teacher stated that: $\...
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1answer
66 views

Symmetry of the hamiltonian $H = \frac{1}{2m}p^2 + V(r) + a \, \vec{s} \cdot \vec{l} $

Consider the hamiltonian \begin{align} H& = H_0 + a\, \vec{s} \cdot \vec{l} \\& = \frac{1}{2m}p^2+ V(r) + a\, \vec{s} \cdot \vec{l}, \end{align} where $V(r)$ denotes an arbitrary central ...
3
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1answer
58 views

Fierz identity for symplectic group

For the fundamental representation of $SU(N)$, there is a Fierz identity: $$ \sum_iT^i_{ab}T^i_{cd}=\frac{1}{2}\left(\delta_{ad}\delta_{bc}-\frac{1}{N}\delta_{ab}\delta_{cd}\right) $$ where $T^i$ is ...
4
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2answers
333 views

Why are physicists so interested in irreps if in their non-block-diagonal form they mix all components of a vector?

Consider a group $\{G,\circ\}$, with elements $e,g_1,g_2,...$, represented by the matrices $\{D(e), D(g_1), D(g_2)...\}$. If all the matrices can be brought to block diagonal forms by a similarity ...
12
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3answers
893 views

Generator for parity?

The unitary translation operator, $\hat{T}(\lambda) = e^{i\hat{p}\lambda/\hbar}$, is generated from the Hermitian operator $\hat{p}$. The unitary rotation operator, $\hat{R}_z(\alpha)=e^{-i\hat{L_z}\...
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1answer
35 views

How to make a triplet out of 2 doublets in the $SU(2)$ representation?

In Y.Grossman and Y.Nir "The Standard Model" book in chapter 4 (non abelian symmetrys) they present the law of whom we can have a triplet and singlet out of 2 doublets name them $\phi_a$ and $\phi_b$, ...
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50 views

Lorentz group in 1+1 dimension

Consider the Minkowski 2D metric $\eta = \text{diag}(-1, 1)$. The Lorentz group is the set of matrices such that, for a transformation $\Lambda$, we get $$\eta = \Lambda^T \eta \Lambda$$ This means ...
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28 views

How to know the symmetry (point group) of crystal field in a material?

As an example, Let's consider a material $Ba_{2}YMoO_{6}$,(ref:PRB 81,224409), the space group of this material is Fm3m, the crystal structure is shown below (https://journals.aps.org/prb/abstract/...
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22 views

Fierz identities to eliminate all vector and tensor Dirac matrices in effective operator (Weinberg)

In the paper titled "Baryon- and Lepton- Non-conserving processes" (prl, 1979) S. Weinberg used operator formalism in effective field theory to analyse beyond the standard model processes which ...