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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3
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1answer
69 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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2answers
105 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
352 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
964 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}\...
1
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1answer
36 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 $\phi_b$, ...
0
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0answers
62 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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0answers
33 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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0answers
51 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 ...
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33 views

Generators of 2D global conformal group in terms of differential operators?

I'm looking for a reference that lists generators of two dimensional global conformal group on a complex sphere in terms of differential operators that may act on quasi primary fields $\phi(z,\bar z)$....
1
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1answer
140 views

Orthochronous indefinite orthogonal group $O^+(m, n)$ forms a group

My question is based on Qmechanic's answer here which proves that $O^+(m, 1)$ forms a group -- that if two Lorentz transformations have positive time-time co-ordinate, so does their product. The key ...
2
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0answers
40 views

Space group for vector fields with fractional periods in subspaces

Consider a periodic system of 3-dimensional vector fields, the corresponding Hamiltonian is a 3-by-3 hermitian matrix: $$ H(\mathbf{r})=\begin{pmatrix} a(\mathbf{r}) & 0 & d(\mathbf{r})\\ 0 &...
14
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4answers
2k views

Why Lie algebras if what we care about in physics are groups?

In physics, we want irreducible representations of the symmetry group of our system. However, one frequently sees representations of the corresponding Lie algebra being studied instead. Is it that in ...
6
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2answers
247 views

What is a Borel subalgebra?

Borel subalgebra appears here https://arxiv.org/abs/hep-th/9508170 in the context of quantum double of $SU(2)$. I request a layman explanation of what a Borel subalgebra is.
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0answers
51 views

Has this group something to do with the cone of light?

Consider the group $V=(-1,1)$ with addition $+_{rel}:V\times V\to V$ defined as: $$v+_{rel}w=\frac{v+w}{1+vw}$$ This group is analogous to the relativistic velocities where the speed of light equals ...
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2answers
63 views

Emergence of rotational symmetry on 2D square lattice

On page 74 of David Tong's Statistical Field Theory lecture notes, it is said that $(\partial_1\phi)^2 + (\partial_2\phi)^2 $ respects both $D_8$ (that includes discrete four-dimensional rotation ...
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0answers
68 views

What does it mean to take the tensor product of two reps of the Lorentz group? [duplicate]

If I reduce the Lorentz group to the representation $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$, I can write left and right-handed Weyl spinors respectively as $\left( \frac{1}{2},0 \right)$ and $\left(...
4
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2answers
154 views

Why would a spinor transform under Lorentz transformations?

From my understanding of spinors, they arise as projective representations of $SO_0(1,3)$ that do not correspond to representations of $SO_0(1,3)$. But still one says here - and virtually everywhere - ...
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0answers
192 views

Significance of the Little group

My current understanding of the Little group is that it is the symmetry of a given state in the Fock space. This means that given a massive or massless particle in n dimensions, I can tell the number ...
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0answers
51 views

Why use $SU(3)$ and not $SL(3, \mathbb{R})$ for color charge? [duplicate]

Why do we use the group $SU(3)$ and not $SL(3, \mathbb{R})$ for color charge? As far as I can tell, the $SL(3, \mathbb{R})$ is volume and orientation preserving, by the fact that it has unit ...
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0answers
11 views

What does “representation spanned by the antisymmetrized product of a degenerate representation” mean in direct product tables

In some tables for direct products (e.g. the one for $C_2$ here), some of the components for direct products of irreps (seemingly only for degenerate irreps) are "the representation spanned by the ...
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0answers
49 views

Regarding notation used for infintesimal parameters of the Lorentz algebra and generators of the Lorentz group

I have a confusion regarding the notation that is used for infintesimal Lorentz transformations and the parameters that define the Lorentz transformation (used in various books such as Srednicki's and ...
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0answers
55 views

Spinor transformations as representations of $\mathrm{SL}(2, \mathbb{C})$

Background In the Weyl representation of the Dirac $\gamma$-matrices, the spinor transformations $S=e^{\frac{1}{2} \omega_{\alpha \beta} \Sigma^{\alpha \beta}} \in \,\mathrm{G}_{\mathrm{L}} \leq \...
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0answers
23 views

Physics-y resource request on Killing-Cartan forms

Most books that treat nonabelian gauge theory do not contain detailed discussion on Killing-Cartan forms, they'll usually just say that in $\text{SU}(N)$ Yang-Mills theory, one can choose generators $...
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0answers
23 views

Resources on Physical Baryon States as mixings of multiplet states

I've been told, and have seen in papers (search for "mixing") that baryon resonances are not plucked from pure group theoretic multiplets, but are most often found to be mixings of states in different ...
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3answers
97 views

Why the full conformal symmetry is $Vir\otimes \overline{Vir}$ instead of $Vir\oplus \overline{Vir}$

In 2D CFT, we have the Virasoro generators $L_m$ and the generators $\bar L_m$, which are such that $[L_m,\bar L_n]=0$. Hence I thought that the full conformal algebra was $Vir\oplus \overline{Vir}$. ...
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2answers
67 views

$3+3$ representation of $SO(4)$

In Zee's Group Theory in a Nutshell book, he says that the antisymmetric tensor $A^{ij}$ furnishes a 6 dimensional representation of $SO(4)$. He further argues that this 6 dimensional representation ...
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0answers
65 views

Representation of Poincaré group

Let's consider the most general Lorentz transformation: $x'^{\mu} = \Lambda^{\mu}_{\ \ \nu} x^{\nu} + a^{\mu}$. These transformations form the Poincaré group. The generators of translations of this ...
2
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1answer
106 views

Why is $\rm{Conf}(\mathbb{R}^{1,1}) = \rm{Diff}(S^1) \times \rm{Diff}(S^1)$ and not $ \rm{Diff}(\mathbb{R}) \times \rm{Diff}(\mathbb{R})$?

The Minkowski metric for $\mathbb{R}^{1,1}$ is $$ ds^2 = dt^2 - dx^2 = du dv $$ for coordinates $$ u = t + x \hspace{1cm} v = t - x $$ If you do any coordinate transformation that acts independently ...
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0answers
41 views

Connection between $2n$ real fermions and $SO(2n)$

In section 11.4 of "Basic Concepts of String Theory" by Blumenhagen et al, they say: Consider a system of $2n$ two-dimensional real fermion (...) transforming as a vector of $SO(2n)$. I guess they ...
0
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2answers
46 views

Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels. For a simple example if we take $SU(2)$, with $[R]$ indicating the highest ...
0
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0answers
36 views

What's the relation between the Lorenztz group and spin of particles?

I know that particles are defined in terms of irreducible representations of the Poincaré group, and that the state of a massive particle is defined by its mass and spin, which are the eigenvalues of ...
2
votes
1answer
43 views

Wicks contractions of stress-energy tensor and plane partitions

I am working out the number of wick contraction of a number $n$ of stress-energy tensor in 4D CFT. The strategy is as follows: For 1 stress energy tensor $T_{\alpha\beta}$, you have only one ...
1
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1answer
60 views

Some Clebsch-Gordan coefficients for $j_{1}=1$ and $j_{2}=1$

I've successfully derived every coefficient, but not the one that has $j=0$. Starting from $|J=2,M=2⟩$ and applying $J_{-}$ we derive $|2,1⟩$ and $|2,0⟩$ and using orthonormality (and the Condon-...
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0answers
50 views

$SU(3)$ and flavor symmetry technical question

In the HW of a particle physics class I was asked about a global $SU(3)_G$ symmetry of $N$ complex scalar fields that transform as $\phi_i(3)$ with $i=1\dots N$, $i$ is the flavor index. The ...
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0answers
21 views

What is the representation of the different classes of nontrivial textures in an ordered field of biaxial nematics in terms of $SU(2)$-like rotations?

I was reading Mermin's classic review on Topological Defects in Ordered Media, in which he describes ordered media with non-Abelian Fundamental Groups by taking the example of biaxial nematics with ...
0
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1answer
87 views

Conventional unit cells and Bravais lattices

Conventional unit cell is defined in the following: A definition of a conventional unit cell of a lattice is one that contains the same point group symmetries as the overall lattice and is the ...
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0answers
54 views

Computation of the Faddeev-Popov determinant

I am studying the Faddeev-Popov method from Pokorski's Gauge Theory book, and I am puzzled by what happens in the step below. He is writing the group element $g = 1 - i T_a\ \Theta^a(x)$ in a ...
2
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1answer
86 views

Symmetries of Wigner $3j$-symbols by exchange

I know that Wigner $3j$-symbols have certain symmetry factors arising by exchange of two columns within one symbol. But what happens if you have two 3j symbols and do an exchange like this: $ \left(\...
1
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1answer
66 views

An invariant of the gauge group $G$ that is totally symmetric with three indices in the adjoint representation

In Ch.19 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.680 the property of a quantity $$\mathcal{A}^{abc}=\mathrm{tr}\left[t^a\{t^b,t^c\}\right]\tag{19.132}$$ ...
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0answers
65 views

Representation of the Lorentz group using matrices of $SL(2,\mathbb{C})$

There is a correspondence between the Lorentz group and the group $SL(2,\mathbb{C})$. To each Lorentz transformation $\Lambda$ we can associate two matrices $\pm A(\Lambda) \in SL(2,\mathbb{C})$ such ...
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0answers
38 views

Symmetry relation of Wigner-Eckart

I saw a symmetry relation following from the the Wigner-Eckart Theorem looking like this $$(\xi j|| T_L || \xi'j') = (-1)^{j-j'} (\xi' j'|| T_L || \xi j)^*$$ I know that it must come somehow under ...
3
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0answers
56 views

Quantum representation of a system of identical particles

I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
1
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1answer
86 views

Representation and Lie algebra of $SO(3)$

Studyng the book Group Theory in Physics of Wu-Ki Tung, I have read: "... every representation of the [$SO(3)$] group is automatically a representation of the corresponding Lie algebra, (...) a ...
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0answers
39 views

Representation of the Lorentz group and correspondence with the $SL(2,\mathbb{C})$ group

We can find a correspondence between the restricted Lorentz group and the group $SL(2,\mathbb{C})$ if to each coordenate $x^{\mu}$ we associate a $2\times 2$ hermitian matrix $X$ given by $$X = x^{\mu}...
2
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2answers
198 views

$SU(2)$ symmetry of $\mathcal{L}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$

I'm considering a Lagrangian of two complex scalar field: $$\mathcal{L}=\partial_{\mu}\phi_1^{*}\partial^{\mu}\phi_1-m_1^2\phi_1^{*}\phi_1+\partial_{\mu}\phi_2^{*}\partial^{\mu}\phi_2-m_2^2\phi_2^{*}\...
1
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1answer
84 views

Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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0answers
46 views

Difference between $\tilde{\textrm{Diff}}_+(S^1)$ and ${\textrm{Diff}}_+(S^1)$

In this paper, where Liouville theory is being studied on a strip, after equation 2.3 it is mentioned that the conformal transformations of the strip are given by the same chiral and anti-chiral ...
0
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2answers
173 views

What is a rotation group and how do we get its unitary representation?

The rotation group is ${\rm SO(3)}$. It is the group of $3\times 3$ orthogonal matrices $\{g(\theta)\}$ with unit determinant. So these are already defined in terms of $3\times 3$ matrices. But we use ...
1
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1answer
103 views

Parametrizing $SU(2)$ with Hermitian matrices

There is something that is not clear to me Here is what I know: Pauli matrices are $\sigma_1 = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$, $\sigma_2 = \begin{pmatrix}0 & -i \\ i & 0\...
1
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1answer
85 views

Pin groups in Physics

There is a very interesting paper from "The Pin Groups in Physics: C, P, and T" on improper and antichronous Lorentz transformations, but on page 5 I got quite confused as it states there ${L^\...

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