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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49 views

Which is correct definition between $J^i\equiv \frac{1}{2}\epsilon^i_{~jk}J^{jk}$ and $J^i\equiv \frac{1}{2}\epsilon^{ijk}J^{jk}$?

The generators of the Lorentz group are denoted by $J^{\mu\nu}$ (suppose they are defined in terms of raised indices, as shown). From this, in my opinion, the angular momentum generators $J^i$'s and $...
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1answer
65 views

Representing $su(2)$ Lie algebra on a torus

I've recently taken up the study of QFT (as a post retirement hobby), based on texts by David Tong and Anthony Zee. My question is based on the Lie Algebra of the $SU(2)$ group, and how this may ...
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1answer
87 views

Lie algebra/group/basis of the four gamma matrices along with the identity?

Do the four gamma matrices along with the identity element constitute a lie algebra? With real coefficients we have $$ \mathbf{v}_{\mathbb{R}}=aI+t\gamma_0+x\gamma_1+y\gamma_2+z\gamma_3 \tag{real ...
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1answer
212 views

How are unitary representations different from other representations?

I understand that unitary representations arise naturally in quantum mechanics when groups act on the Hilbert space in a way that preserves probability. I don't understand what details make unitary ...
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43 views

Why do we use affine groups in gauge theory? What is the purpose?

When we study General Relativity in the frame of gauge theory, what's the importance of affine group?
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2answers
244 views

Are all representations of a finite group unitary?

I am reading Zee's Group theory in a nutshell for physicists and came across the following theorem (Page 96): Unitary representations The all-important unitarity theorem states that finite ...
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51 views

How do I show that the tensor product of $\mathbf{3} \otimes \mathbf{\bar{3}}=\mathbf{1} \oplus \mathbf{8}$? [duplicate]

It's often stated that the tensor product of the representations of $SU(3)$ satisfies $\mathbf{3} \otimes \mathbf{\bar{3}}=\mathbf{1} \oplus \mathbf{8}$, and that this implies that if flavour $SU(3)$ ...
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1answer
39 views

Different definitions of commutator in operator theory/quantum mechanics vs. in group theory

In group theory, the commutator of two elements $g$ and $h$ in a group is defined as $$[g,h]=ghg^{-1}h^{-1}$$ However, in quantum mechanics, we always see commutator relation between two operators $A$...
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308 views

Does a gauge group $G$ determine the Principal $G$-bundle?

I'm trying to understand the mathematical underpinnings of gauge theories in the language of principal $G$-bundles and associated vector bundles. Not long ago, I had assumed that the physical choice ...
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3answers
74 views

How am I to interpret $\text{Tr}(\text{ad}_X\text{ad}_Y)$?

I'm trying to show that the $(2,0)$ Killing tensor is invariant under the $\text{Ad}$ homomorphism: $K(\text{Ad}_A(X),\text{Ad}_A(Y))=K(X,Y),$ with $X,Y\in \mathfrak{g},\hspace{1mm}A\in G,$ and $K(X,Y)...
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Comprehensive book on group theory for physicists?

I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that ...
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Notation of basis functions for irreducible representations

In character tables for symmetry groups, there are typically basis functions for each irreducible representation given. There are basis functions given like $xy$, $S_x$ or $R$. Could someone explain ...
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40 views

What is the Eigenvalue of $T^2$ ($SU(3)$ Casimir)?

For example, in $SU(2)$, $\hat{S}^2|s,m_s>=\bar{h}^2 s(s+1)|s,m_s>$. What about in $SU(3)$, $\hat{T}^2|T,m_3,m_8>=?|T,m_3,m_8>$ where $\hat{T}^2=\sum_i^8 T_iT_i $, $T_i = \frac{\lambda_i}...
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1answer
527 views

Why do Lorentz boosts not form a group?

An excerpt from my lecture notes on relativity (translated from Dutch): "Special (special in the notes indicates that the determinant of the representation matrix equals +1) Lorentz transformations ...
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2answers
164 views

Operator-valued vectors and representation theory

Let $G$ be a Lie group and $\pi : G\to GL(V)$ a finite-dimensional representation of $G$ in the vector space $V$. For every $g\in G$ we have a linear transformation $\pi(g) : V\to V$. Being linear, if ...
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0answers
62 views

If gravity is a gauge theory, what is the Lie group? [duplicate]

Here I asked a question. In one curious comment, I see a statement that gravity is a gauge theory. However, my definition (based on what I read till date) of a gauge theory is a field theory which is ...
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1answer
871 views

How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
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1answer
31 views

Equation for the multiplicity of a set of planes

The multiplicity (m) of lattice planes counts the number of planes related to (hkl) by symmetry. For example, the multiplicity of the {100} planes would be 6 because the following planes are all ...
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1answer
78 views

How does the adjoint of $SO(10)$ branch under $SU(5)$

We can split up $SU(5)$ into real and imaginary parts as $U=U_R+iU_I$ and in doing so embed this in $SO(10)$ as $\begin{pmatrix} U_R & -U_I \\ U_I &U_R\end{pmatrix}$. Hence we know that $SU(5)...
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1answer
42 views

Simultaneous diagonalization of Cartan generators of $SO(6)$

This question is naive but for some reason I'm not getting the expected result. The generators of $SO(6)$ can be written in this way: $$(J_{ab})_{cd}=i(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}),...
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2answers
104 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 ...
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1answer
89 views

What are the $A_{\mu}{}^a$ fields in Yang-Mills theory?

At some point of the demonstration of Yang Mills theory we assume an ansatz that $A_{\mu}=t^a A_{\mu}{}^a$ where $a=1, \ldots,n^2-1$ and the $t^a$ are the generators of the $SU(n)$ symmetry in order ...
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1answer
144 views

Commutator of rotation matrices

How do you compute the commutator of rotation matrices in two different directions by different angles? Let $R_{x}(\alpha)$ be the rotation matrix about the $x$-axis and $R_{z}(\beta)$ be the ...
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1answer
72 views

Peskin Schroeder Higgs mechanism for an $SU(3)$ gauge theory with a scalar field $\varphi$ in the adjoint representation

In Peskin Schroeder pag.696 a Higgs mechanism for an $SU(3)$ gauge theory with a scalar field $\phi$ in the adjoint representation is presented. The covariant derivative of $\phi$: $$ D_{\mu}\phi_{a} =...
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43 views

Decompose $SU(4)$ into $SU(3) \times U(1)$

I'm solving these problems concerning the $SU(4)$ group and I've reached the point where I have determined the Cartan matrix of $SU(4)$, its inverse and the weight schemes for $(1 0 0)$ and $(0 1 0)$ ...
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1answer
110 views

Angular momentum and rotation group representations

In Sakurai's book it's written that the operator $D_{m',m}^{(j)}=\left\langle{j,m'}\Big|\exp{\frac{-i \mathbf{ J\cdot \hat{n} } \phi}{\hbar}}\Big|{j,m}\right\rangle$ is the "$2j+1$-dimensional ...
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0answers
22 views

Group theory and representation theory reference [duplicate]

Could anyone suggest some reference(s) on group theory and representation theory geared to physicist? The reference should be rigorous and not for a novice (but not for an expert either) it should ...
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1answer
72 views

Georgi - decomposition of representations into subgroups

I have long been unable to follow section 12.3 of Georgi - Lie algebras in particle physics. This section deals with how irreps of $SU(3)$ decompose as irreps of subgroups $H \subset SU(3)$ and is ...
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1answer
337 views

Where does in GUT symmetry breaking $U(1)$ come from?

In GUTs one starts with some larger group, like $SU(5)$, which is then broken into smaller groups, for example $$SU(5) ~\longrightarrow~ SU(3) \times SU(2) \times U(1)$$ This can be seen, for ...
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1answer
58 views

Is every unitary operator induced by a Hamiltonian?

Diving deeper into the mathematical inner workings of quantum mechanics: The set of unitary operators on the Hilbert space $\mathcal{H}$ forms a group. While for finite-dimensional Hilbert spaces, ...
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1answer
152 views

What is the weight system for these ${\rm SU}(5)$ representations?

I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The $\mathbf{5}$ ...
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56 views

Surjective homomorphism between ${\rm SL}(2,\mathbb{C})$ and the restricted Lorentz group ${\rm L}_0$

I am reading "Group theory and physics" by Sternberg. Ch. 1.2 deals with homomorphism between ${\rm SL}(2,\mathbb{C})$ and the Lorentz group ${\rm L}$, respectively ${\rm L}_0$, the restricted Lorentz ...
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1answer
211 views

Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
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0answers
37 views

Fidelity of unitary operators: is $||U-\tilde U|| < \delta$ a *necessary* and *sufficient* condition?

There's a notion of fidelity of quantum states. However, is there a standard notion of the fidelity of unitary operators? Say, I wish to approximate a unitary operator $U$ acting on $n$ qubits with a ...
2
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1answer
90 views

What is the Lie group of gravity?

If the lie group of the three gauge forces are $SU(3)×SU(2)×U (1)$, then what is the symmetry group of gravity? $SL(2,C)$? Just a newbie in Lie groups.
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Group action on $2$x$2$?

Group action of group $C_{3v}$ is defined for all $2$x$2$ matrices over field of complex numbers, for all $g$ from $C_{3v}$ $$D(g)A=E(g)AE(g^{-1})$$ where $E$ is $2D$ representation of $C_{3v}$. How ...
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2answers
80 views

How to formulate with a tensor multiplication method $2\otimes 2\otimes 2$ in $SU(2)$ group?

In $SU(2)$ we can write $2\otimes 2=3\oplus 1$ or \begin{equation} q_iq^j=\left(q_iq^j-\frac{1}{2}\delta^i_jq^kq_k\right)+\frac{1}{2}\delta^i_jq^kq_k, \end{equation} where $q_i$ is a $SU(2)$ doublet, ...
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1answer
52 views

$\mathfrak{so}(2n+1)$ Cartan subalgebra

For the Lie algebra $\mathfrak{so}(n)$, $n^2$ $n \times n$ real and antisymmetric matrices can be introduced as $$(M_{pq})_{jk} = \delta_{pj}\delta_{qk}-\delta_{pk}\delta_{qj}, \qquad j,k=1, ..., n $$...
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34 views

How to find standard basis? [migrated]

How to find standard (symmetry adaptated) basis in representation of group $C_{3v}$ in representation $$D(C_{3v})=A_2\otimes E\otimes E=$$ $$ =A_2\otimes (A_1\oplus A_2 \oplus E)=$$ $$=A_2\oplus ...
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1answer
188 views

Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
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1answer
86 views

Why do we need conformal compactification to define the global conformal group?

First I have the definition of a conformal map. Let $(M,g)$ and $(M',g')$ be two pseudo-Riemannian manifolds of same dimension. Let $U\subset M$ and $V\subset M'$, we say that a smooth map of maximal ...
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1answer
158 views

Why there's a Lorentz inner product in the unitary representations of the translation group?

Consider Minkowski spacetime. Its translation group is just the additive group $\mathbb{R}^4$. This is an abelian locally compact group. Next, consider one unitary representation $T : \mathbb{R}^4\to ...
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2answers
363 views

Reconstructing unitary representation of Lie group from its generators

This question is about the motivation for Weinberg's approach in "The Quantum Theory of Fields" to obtain unitary representations of Lie groups out of its generators. One is dealing with a Lie group $...
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2answers
83 views

Why are covering groups more 'fundamental'?

So I understand that the Lie algebra of $SO(1,3)$ is isomorphic to the Lie algebra of $SU(2)\oplus SU(2)$, and the Lie algebra of $SO(3)$ is isomorphic to one copy of $SU(2)$ (at the group level we ...
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2answers
66 views

Branching of $SU(3)$ under $D_8$

The question is to work out the branching of $SU(3)$ representations of $\mathbf{3}, \mathbf{\overline{3}}, \mathbf{8}$ under the dihedral group $D_8 = \langle r,s \mid r^4 = s^2 = e, rs=sr^{-1} \...
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0answers
65 views

$SU(2)$ vs $SO(3)$ in Quantum Mechancs

When we're talking about spatial rotations is quantum mechanics, why do we need to resort to $SU(2)$? Why isn't $SO(3)$ enough? I've read that $SO(3)$ isn't simply connected, and I've read about ...
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1answer
80 views

Symmetry group of quantum optical interactions

Some quantum optical interactions such as the beamsplitter and two-mode squeezing are unitaries that belong to certain continuous groups of transformations. For example, the beamsplitter is an $SU(2)...
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3answers
2k views

How are symmetries precisely defined?

How are symmetries precisely defined? In basic physics courses it is usual to see arguments on symmetry to derive some equations. This, however, is done in a kind of sloppy way: "we are calculating ...
2
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0answers
125 views

Bosonic commutation relations for force carriers?

Why are force carriers bosons? The easiest answer that I can give myself is that the gauge field $A_\mu$ is introduced like this: $$ \partial_\mu \rightarrow D_\mu = \partial_\mu+ieA_\mu, $$ so it ...
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2answers
156 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 ...