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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38 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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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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29 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(...
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Using Lie theory to understand $U=e^{i\mathcal{H}t}$ [duplicate]

Can we use the exponential map (lie theory) to understand how the Hamiltonian $\mathcal{H}$ gives rise to the unitary, and therefore compliments an essential property of the unitary operator (ie to ...
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96 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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33 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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48 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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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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40 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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35 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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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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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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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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$3+3$ 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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55 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 ...
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101 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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33 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 ...
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35 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 ...
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31 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 ...
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1answer
35 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 ...
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1answer
51 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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$SU(3)$ 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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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 ...
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1answer
24 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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40 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 ...
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1answer
50 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(\...
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1answer
58 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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36 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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28 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 ...
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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 ...
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61 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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26 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}...
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147 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^{*}\...
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1answer
48 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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43 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 ...
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83 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 ...
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36 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\...
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1answer
73 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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1answer
34 views

Lie Subgroups of $SL(2,\mathbb{R})$

I'm wondering about the Lie subgroups of $SL(2,\mathbb{R})$. It's Lie algebra is the algebra of real traceless matrices and has basis elements $$L_0 = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \...
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1answer
44 views

How do we know the group property of a new particle?

Suppose I have a particle $W'$ which can decay into $\mu$ and $\nu_{\mu}$ and $e$ and $\nu_{e}$. Suppose we know such new gauge bosons come from some additional gauge group added to the Standard ...
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1answer
64 views

Representations of the rotation group

(I have already done a similar question, but I did not express myself very well and the question was a bit confusing, so let me try again. If you find the question repetitive, I apologize and you can ...
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1answer
68 views

Do canonical transformations form a group?

In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
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1answer
89 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
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1answer
81 views

$U(N)$ & $SU(N)$ : What's the conceptual difference in Gauge Theory?

I know the mathematical difference that one means $ absolutevalue(det) = 1$ and one means det = 1 (rotation) and that ones the subgroup of the other and so on. But: has a local/gauged $SU(3)$ ...
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45 views

Fluxes on a finite group $G$

So I've been studying about topological quantum computation and I have a few questions I haven't been able to solve. The first one is why fluxes take values on a finite group $G$? Does it have to do ...
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2answers
55 views

How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
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1answer
35 views

How to construct invariant forms under the effect of an arbitrary group?

First I would like to mention that I do not know that should I post this question here or in the math community, but since my background is in physics and this kind of question is usually asked by ...
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1answer
80 views

How is possible to see that Maxwell's field have $U\left(1\right)$ symmetry?

As it is well knowing, $U\left(1\right)$ is the group of the unitary matrix of the first order and that this group is connected with rotation operations. Under the complex scalar field perspective $$\...
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57 views

Why do particle physicists often use $\otimes$ instead of $\times$ to denote the direct product of groups? [closed]

For example, sometimes one sees $SU(3)\otimes SU(2)\otimes U(1)$ instead of $SU(3)\times SU(2)\times U(1)$. My understanding is the the product here is just the usual direct product (aka Cartesian ...
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1answer
34 views

Combining SU(N) multiplets using Young diagrams

I am trying to follow the Particle Data Group's instruction (PDF link) to combine SU(N) multiplets. On page 3, they show an example calculation of SU(3)'s $\textbf 8\otimes \textbf 8$. I understand ...