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.
2,348 questions
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Nonvanishing matrix elements of electric dipole transition in $O_h$ symmetry from $E_g$-symmetric initial state to $T_{1u}$-symmetric final state
I am trying to work out Q6.3(b) of Dresselhaus's Group Theory: Application to the Physics of Condensed Matter. The question is to find the non-vanishing matrix elements for an electric dipole ...
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Deriving the normalization factors of $SU(2)$
In Georgi's book on Lie Algebras in Particle Physics, he makes the following argument to
derive the normalization factors of $SU(2)$. Define
the raising/lowering operators by $J^\pm = (J_1 \pm i J_2)/\...
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Degree of degeneracy of energy levels and irreducible representations of Hamiltonian symmetry group
In case the Hamiltonian of a system has some non-trivial symmetry regarding the physical space, let's assume symmetry that can be described by a finite symmetry group (e.g. a point group symmetry) one ...
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Proving gauge transformation of non-abelian field strength
For a general Yang-Mills theory, we have the field strength
$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig [A_\mu, A_\nu]$$ I now want to prove that it transforms as $$F_{\mu\nu} \...
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Analogue of Coleman-Mandula theorem for non-relativistic quantum field theory?
For relativistic quantum field theories, the Coleman-Mandula theorem places very strong restrictions on the possible symmetry groups $G$ of the aforementioned quantum field theory, forcing it to be a ...
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Group theory: The energy splitting of the five-fold $d$ orbitals in a $D_{3h}$ crystal field
I am trying to figure out how the degerenate 5-fold $d$ orbitals is split in a $D_{3h}$ crystal field. A practical case would be a subtitutional $\rm Fe$ impurity in the hexagonal graphene lattice. ...
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Systematic procedure of SSB
Whenever I deal with spontaneous symmetry breaking (SSB), I encounter some confusions. This is my flow of logic and related questions:
We begin with UV gauge group. Find matter contents and their ...
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Which is the Lorentz group representation the Majorana Rarita-Schwinger spinor transforms like?
In the textbooks --- for instance "Freedman & van Proeyen" -- it is said that the Majorana Rarita Schwinger spinor is a combination of a Majorana spinor and a vector. In order to the ...
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Is the Poisson bracket related to the Lie bracket of some Lie group?
A Lie algebra is a vector space $\mathcal{L}$ on which an operation called a Lie bracket, (denoted by $[. ,. ]$) is defined, which associates with a pair $X, Y\in \mathcal{L}$, the element $[X, Y]$ of ...
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Derive magnetic space group from lattice space group
Consider we have a simple cubic lattice, space group P23. At each corner of the cubic lattice there is one atom.
Now, if we assign a spin to each atom, and let the spins align in a ferromagnetic ...
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Why are $SO(N)$ symmetries often associated with bosons, and $SU(N)$ with fermions in particle physics?
In the context of particle physics, I’ve often seen $SO(N)$ symmetries linked with bosonic fields and $SU(N)$ symmetries with fermionic fields. I understand that bosons have integer spins and obey ...
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A good reference for irreducible decomposition like $(2,1)\otimes (1,2)=(1,1)_A \oplus (3,1)_S$?
M. Srednicki says in the book "Quantum Field Theory" on page 211 that:
I would really appreciate if someone could introduce me a good reference for understanding the irreducible ...
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On the distinction between Lorentz group and proper orthochronous group via Jacobian determinants
The group $O(1,3)$ is defined as the set of matrices which are orthogonal with respect to the Minkowski metric $\eta$, i.e. those matrices $\Lambda$ satisfying
$$\Lambda^T \eta \Lambda = \eta.$$
...
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Problem with equivalence between local lorentz gauge field transformation law and lorentz transformation law of the spin connection
Defining spin connection, the transformation law under lorentz transformations is
$\omega_{\mu}{}^{a'}{}_{b'}=\Lambda^{a'}_{a}\Lambda^{b}_{b'}\omega_{\mu}{}^{a}{}_{b}+\Lambda^{a'}_{a}(\partial_{\mu}\...
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Untwisted spectrum of $\text{SU}_k(2)/\mathbb{Z}_2$ WZW model
I am interested in constructing the untwisted spectrum of $\text{SU}_k(2)/\mathbb{Z}_2$ Wess-Zumino-Witten model.
Representation theory of $SU(2)$
Labelling the basis of spin-$j$ representation space ...
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What is the nature of spin-$j$ representations of $SU(2)$ group used in QM? Elements of $SU(2j+1)$?
The $j=1$ representation of $J_x,J_y$ and $J_z$ is QM, are all hermitian, and traceless matrices. Therefore, the corresponding rotation matrices, $$U(\hat{n},\alpha)=e^{-iJ_k\alpha_k/\hbar},$$ are $3\...
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Formal Framework for Time-Reversal Symmetry in Quantum Mechanics
Usually when we think of symmetries in Quantum Mechanics we think of some group $G$ and a representation $\rho$ of that group on the physical Hilbert space $\mathcal{H}$ of the system we are ...
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Relation of $SO(3)$ with $SU(2)$ Rotation
It's known that $SO(3)$ is (locally) isomorphic to $SU(2)$, I am trying to establish a relation from exponentiation of their Lie Algebra, the formula I would like to prove is:
$$e^{-i\theta\hat{n}\...
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Explicit derivation of $SO(1,3)$ and $SL(2,\mathbb{C})$ homomorphism
I know that similiar answers have already been given but i could not find an explaination of this exact passage.
I want to build the usual homomorphism between the lorentz group $SO(1,3)$ and $SL(2,\...
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Where are the two different spinor representations of the Lorentz group in the Poincare representation?
When studying the finite dimensional irreducible representations of the Lorentz group, we come across spinors, which can be thought of as objects in the $(j_1,j_2)$ representation. We have Weyl ...
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What subset of the Lorentz group produces a given Weyl spinor?
Let's say I have a unit Weyl spinor $\psi = [\psi_1\ \psi_2]^T$. This has 3 degrees of freedom. Now suppose that this spinor was produced from a reference unit spinor — let's take $[1\ 0]^T$ to be ...
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What is the expected value of orbital angular momentum $𝐿$ in a octahedral crystal field, where 2 electrons occupy the $𝑡_{2𝑔}$ orbitals?
What is the expected value of the orbital angular momentum 𝐿 for a regular octahedral crystal field, where only 2 electrons occupy the 𝑡_2𝑔 orbitals? Since these orbitals are degenerate and the 𝑡...
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A confusion about $SU(2)$ group
I was reading the basic properties about $SU(2)$ group. As the fundamental representation, $SU(2)$ matrices operate on two-dimensional vector
$$
\eta=\begin{pmatrix}\eta_1 \\\eta_2\end{pmatrix}.
$$
...
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The interval of Hyperbolic Space (Euclidean Anti-deSitter Space)
I am learning the Group Theory notes from G. Moore.
https://www.physics.rutgers.edu/~gmoore/618Spring2022/GTLect1-AbstractGroupTheory-2022.pdf
7.4.3 Orbits Of The Lorentz Group In $d > 2$ ...
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How to wrap one's head around the conformal group $O(p\!+\!1,q\!+\!1)$ in a spacetime $\mathbb{R}^{p,q}$?
By preserving angles obviously, har.. har...
In all seriousness, I can appreciate the Lorentz group for 4D being written $O(1,3)$ the same way we write our spacetime $\mathbb{R}^{1,3}$. The Lorentz ...
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Wigner representation on excitated states in string theory
In his string theory lectures David Tong argues on why the first excitated states in the lightcone quantization of the closed string have to be massless, but I don't understand his reasoning.
When ...
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The Cartan subalgebra of $\mathfrak{so}(1,3)$
Consider the four-dimensional representation of the Lie algebra $\mathfrak{so}(1,3)$, given in terms of the following matrices
\begin{align}
J_1&=\left(
\begin{array}{cccc}
0 & 0 & 0 &...
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Computing the spin of the longitudinal and transverse components of a vector field
Considering the generators of the Lie algebra of $SO(3)$:
\begin{eqnarray}
J_1=\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0 \\
\end{array}
\right),\,...
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What is the relation between irreducible representation of $SO(3)$ and spherical harmonics?
I just studied the rep theory of $SU(2)$ (now I know its the double cover of $SO(3)$ so I guess their reps are highly similar) and I also know spherical harmonics (I will use SH for short) are the ...
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What is a crystal system, compared to a Bravais lattice?
In my condensed matter class I have been told there are 7 crystal systems: Triclinic, Monoclinic, Orthorhombic, Tetragonal, Trigonal, Hexagonal, Cubic.
I am also told there are 14 Bravais lattices. I ...
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Components of a 2nd-order Tensor
If we consider a matrix of the form: $T_{ij}=v_iw_j$, where $\vec v$ and $\vec w$ are 3D vectors, that under 3D rotation transform.
One can show that the above mentioned matrix can be written in the ...
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Do projective representations imply a relationship between spin and normalization of single-particle states?
I was reading about how we take projective representations of the Lorentz group because there is no physical meaning to multiplying a single-particle state globally by any complex number. If you ...
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Applying Slater-Koster tight binding method for Rhombohedral Bi
I am trying to build a tight-binding Hamiltonian for Bi which has the $D_{3d}$ point group and $R\bar{3}m$ space group. It has two inequivalent atoms in the unit-cell.
I understood the basics of the ...
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Does Cauchy stress tensor act on $SO(3)$?
I have some confusions surrounding the nature of the Cauchy stress tensor.
Ostensibly, the stress tensor takes some vector normal to a surface and gives another vector which is a linear transformation ...
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Does the set of evolution operators in QM form any known algebraic structure (e.g., a monoid)?
The evolution operator, $U(t,t_0)$, in quantum mechanics is defined by
$$
|\Psi(t)\rangle = U(t,t_0)|\Psi(t_0)\rangle.
$$
with $t\geq t_0$. Is the set of such evolution operators, specified by two ...
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Projective representations and reduction of half-integer spin representations under $C_{\infty v}$
Suppose we have an orthonormal basis of states $|j,m,p\rangle$ where
$j=\frac{1}{2},\frac{3}{2},\ldots$ is the angular momentum quantum number associated with some angular momentum operator $\mathbf{...
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Vector/axial current and vector/axial transformation
As is well known, in particle physics, the vector current is defined as a current of the form
$$J_V^\mu = \bar{\psi} \gamma^\mu \psi $$
and the axial current as
$$J_A^\mu = \bar{\psi} \gamma^5 \gamma^\...
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On JD Jackson's derivation of Matrix Representations of Lorentz Tranformations
Jackson derives the ordinary rotation matrix for a rotation through angle $ \omega$ about the $z$ axis (eq.11.96) via the exponential map of the Lorentz group:
$$A=e^{-\vec{\omega}\cdot\vec S-\vec \...
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Group of time translations for finite-dimensional quantum systems and incommensurable eigenvalues
I am a bit confused by the group of the time translations in finite-dimensional quantum mechanics. Take a finite-dimensional Hamiltonian H. For instance, for a two-dimensional system
$$
H=\begin{...
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How can one says that a particle IS a representation of some group? [duplicate]
I suppose this question has been asked many times. I have been told that an elementary particle is a (moving) point, or (a section from) some field, or an excitation from some field. But now, I am ...
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Cartesian tensor transformation and representations of $SO(3)$
I have hard times understanding the transformation of tensors, that probably stems from my shaky understanding of representation theory.
A cartesian tensor can be decomposed in three terms:
$$
T_{ij}=...
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Can all solutions from any possible dynamical systems be formulated as combinations of harmonic oscillators?
In my understanding, all physical oscillators are unit vectors of a Hilbert space that is represented from the group $SU(3)\times SU(2)\times U(1)$. Every dynamical system operates under this group (...
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Recovering the electromagnetic field from the $(1,0) \oplus (0,1)$ representation
I'm confused as to how one recovers the electromagnetic field $E,B$ from the standard procedure of building the $(1,0) \oplus (0,1)$ representation of the Lorentz Lie algebra. The reason is the ...
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Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry? [duplicate]
This question is a duplicate of this question asked on Maths S-E
From my notes I have that
The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be ...
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Group actions confusion
I have been using results from this paper in calculations. In sections 2.4 and 3.4 they perform a canonical transformation into new coordinates consisting of constants of motion. They then construct ...
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Group properties of density of state for i.i.d. particles microcanonical ensembles
For a $2N$ particle non-interacting gas, we could arbitrarily divide the gas into 2 sets of $N$ particles.
Then the density of states for each one is,
$$
D(N_A, V, U_A) = h^{-N} \int d^Nq~d^Np ~ \...
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Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry?
From my notes I have that
The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be realised if $A_\mu$ is an element of the Lie algebra. It can ...
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1
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Showing that a generator exponentiates to a $\mathbb{R}$ group
I have a generator $G$ that acts on the phase space of Schwarzschild and maps geodesics into each other.
In order to discuss the corresponding symmetry group, I need to exponentiate this generator and ...
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1
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78
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Goldstone Matrix for $SO(3) \longrightarrow SO(2)$ breaking
In these lecture notes (https://arxiv.org/abs/1506.01961) on composite Higgs models, on page 22, the authors calculate the Goldstone Matrix $U[\Pi]$ for an abelian composite Higgs scenario i.e. $SO(3)$...
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Identity for generators of $SU(N)$ in the adjoint representation
For the generators of $SU(N)$ in the fundamental representation, $T_{i j}^a$, the following identity holds
$$T_{i j}^a T_{k \ell}^a=\frac{1}{2}\left(\delta_{i \ell} \delta_{j k}-\frac{1}{N} \delta_{i ...
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