# 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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### Why are $E_{d(d)}$ representations labelled by integers?

Reducing 11d supergravity on a torus $T^d$ or, equivalently, IIB supergravity on $T^{d-1}$, one gets an exceptional $E_{d(d)}$ hidden symmetry, which is, in particular, the non-compact split real form ...
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### Quantization of charge from the path integral

Consider a complex scalar field, with the usual Lagrangian: $$\mathcal{L} = | \partial_{\mu} \phi|^2 - V(|\phi|^2).$$ This theory has a $U(1)$ symmetry, $\phi \to e^{i \alpha} \phi$, and the ...
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### Georgi's "Lie Algebras in Particle Physics" Theorem 1.2 proof

In Georgi's "Lie Algebras in Particle Physics", Theorem 1.2 reads Every representation of a finite group is completely reducible. The proof that follows contains the following lines If it ...
1 vote
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### Character of group elements of the Lorentz group

I am looking for a reference for the character of the element of the Lorentz group. If the generator the Lorentz transformations is given by $$\Lambda = e^{i\alpha\cdot J-\beta\cdot K}$$, then I wish ...
1 vote
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### (Anti-)Fundamental Representation of $SU(5)$ GUT

In many places, it has been mentioned that the sum of electrical charges of the particles present in $\overline{5}$ of $SU(5)$ is zero since the trace of $SU(5)$ generators is zero. I do not ...
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### Are representations of (bosonic) Lie groups over Grassmann variables well understood?

When one studies representations of (bosonic) Lie groups in physics, whether dealing with spacetime symmetries or gauge symmetries, it is often left implicit whether the representations are over real ...
1 vote
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### Why do we look for the representations of $\mathfrak{so}(1,3)$ when looking for projective representations of $SO^+(1,3)$?

In a relativistic quantum field theory, one classifies quantum fields by looking for finite dimensional projective representations of the restricted Lorentz group $SO^+(1,3)$ over the target space $V$ ...
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### Confusion on the helicity formalism

I'm studying group representation theory from a more mathematical point of view but I don't understand the link between helicity formalism and the "classical one". They should be both ...
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### Question about spinor inner products

Let a 2D spinor be given by $$\chi_2(p)=\pmatrix{\xi^1\\\xi^2}+i\pmatrix{\xi^3\\\xi^4}$$ with the $\xi^i$'s being real for $i=\{1,2,3,4\}$. Assume, now, that I want to represent this spinor by a real-...
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### Dimension of singlet subspace of $2N$ identical spin-1/2 particles

Consider a quantum system of $2N$ identical spin 1/2 particles, each having a 2D Hilbert space $V$. The total Hilbert space is the tensor product $V^{\otimes 2N} \equiv V \otimes \cdots \otimes V$. ...
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### Question regarding one-particle states being irreducible representations of the Poincare group

I am a bit confused with the way Physics textbooks use the word 'representation'. I was reading Weinberg's QFT and in section (2.5) titled One-Particle states, he shows that (eq. 2.5.3) \begin{...
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### Generators of ${\rm SU}(n)$ are traceless. Why?

A general element of the Lie group ${\rm SU}(n)$ is written as $$g({\vec{\theta}})=e^{-i\sum_a\theta_a T_a}$$ where $\theta_a$ for $(a=1,2,\ldots,n^2-1)$ denotes $n^2-1$ real parameters. The ...
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### What is meant by dimension of the defining representation (and adjoint representation)?

A linear representation of a classical Lie group $G$ is defined by $\rho:g \to GL(V)$ where $g \in G$ is a group element and $V$ is the representation space. The dimension of the representation space(...
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### About the notation of Lie algebra, upstairs and downstairs indices

In some textbooks, Lie algebra is written as $$[X_a, X_b]=if_{abc}X_c$$ where $X_a$'s are the generators of the Lie group. But sometimes, the Lie algebra is written as $$[X_a,X_b]=if_{ab}^{~~~ c}X_c$$ ...
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### Resources on representations theory and harmonic analysis

I'm studing the representation theory of the group $SL(2,R)$ from the book "Lorentz group and harmonic analysis" of Ruhl, in particular I'm interest in chapter 5 and 6. I'm a master students ...
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### The relation between spin and momentum in particle state

In quantum field theory, it seems that when we consider a massive particle's spin degree of freedom, we usually do in the particle's rest frame. And I know the little group will only change spin DOF ...
1 vote
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### Why are the weight vectors corresponding to the raising and lowering operators given by the formula $\bar{t_{1}}- \bar{t_{2}}$?

I am reading Robinson's particle physics paper part-I which is available in the public domain here, particularly the theory of SU(3) Lie Groups. Page 63 of the text states that, Now, repeating ...
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### Are there any indications that the hat polykite or spectre appear as aperiodic structures in materials?

It is known that certain aperiodic structures appear in quasichrystals. For instance, Daniel Schechtman and Ilan Blech discovered that the aluminium-manganese alloy Al$_{6}$Mn has no translational ...
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### Is the $(\frac{1}{2},\frac{1}{2})$ representation of $SO(3,1)$ reducible or not?

From wikipedia and some other sources, I've read that if $G_1$ and $G_2$ are two irreps of some group, then $G_1\otimes G_2$ can be a representation of both the group $G$ and the new group $G \times G$...
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### Is it possible for a Lorentz scalar to NOT be invariant under another linear transformation?

Lorentz scalars are invariant under Lorentz transformations, which are a subset of linear transformations. I wanted to know if it is possible, for a Lorentz scalar, to NOT be invariant with respect to ...
In my lectures on groups and representations, we write the Clebsch-Gordan decomposition for addition of angular momenta $$r_{j_1}\otimes r_{j_2}=\bigoplus_{j=|j_1-j_2|}^{j_1+j_2}r_j\tag{1}$$ where $... 0 votes 1 answer 93 views ### Irreps of$SU(2)$versus the irreps of$SU(3)$For the group$SU(2)$, the fundamental representation corresponds to$j=1/2$, for which the highest (eigen)value of$J_3$is$+1/2$and the lowest (eigen)value is$-1/2$. For$j=1$representation, the ... 1 vote 0 answers 40 views ### Seeking Spinor Operation Analogous to$M^T \eta M = g$for GL$^+(4,\mathbb{R})$/Spin$^c$(3,1) I'm exploring the spinorial representation of the Spin$^c$(3,1) group, especially in the context of metric preservation in general relativity and quantum field theory. For the group GL$^+(4,\mathbb{R}$... 1 vote 1 answer 96 views ### Confused about two different definitions for the same object$\psi_i$in the representation theory of$\mathrm{SU}(n)$Let the entities$\psi^i$transform as the fundamental representation of$\mathrm{SU}(n)$, denoted by${\bf n}$: $$\psi^{\prime i}=U^{i}_{~j}\psi^j,$$ where, of course,$U$represents$n\times n$... 2 votes 0 answers 45 views ### How to derive the Lorentz Invariant bilinear form in the$(1/2, 1/2)$representation? We can represent the complexified proper Lorentz group Lie algebra as the direct sum$\mathfrak{sl}(2, \mathbb{C}) \oplus \mathfrak{sl}(2, \mathbb{C})$. The representation nomenclature is$(n,m)$, two ... 3 votes 2 answers 121 views ### Why is the Lorentz transformation of fields linear? I know that the coordinate,$x^\mu = (t,\vec x)$is a 4-vector and it transforms as $$x'^\mu={\Lambda^\mu}_\nu x^\nu.$$ The related (classical or quantized) field,$\phi_a(x)$, can be classified into ... 1 vote 1 answer 75 views ### Lorentz invariant tensors of odd rank How to prove that Lorentz invariant tensors of odd rank are all zero? I somehow think that determinant or generator of the Lorentz group should be used. 1 vote 1 answer 84 views ### Why the Double Covering? It is known mathematically that given a bilinear form$Q$with signature$(p,q)$then the group$Spin(p,q)$is the double cover of the group$SO(p,q)$associated to$Q$, and that$Pin(p,q)$is the ... 0 votes 2 answers 92 views ### Topological proof of spin-statistics theorem confusion I am currently studying the spin-statistics theorem. I have found a section on John Baez's website which presents a "proof" of the spin-statistics theorem. He states the theorem as: This is ... 0 votes 0 answers 53 views ### Algebra equation for rank-3 tensor Suppose I work in$4$dimensions. I have an algebraic equation in the following form, which contains a rank-3 tensor$X ^{\alpha \lambda \mu }$\begin{equation} X ^{\alpha \lambda \mu }\eta ^{\beta \... 0 votes 0 answers 61 views ### Why is it not Pin(1,3) or Pin(3,1) in condensed matter physics? In electron systems (or condensed matter physics), it is well known that$T^2=-1$and$M^2=-1$, where$T$and$M$are time reversal and reflection along some axis. But in general, the symmetry of ... 0 votes 0 answers 38 views ### Why 3 mmm and 4mmmm point group symmetry elements do not exist? Why 3 mmm and 4mmmm point group symmetry elements do not exist? How we can say 3 mmm and 4mmmm are rather equivalent to 3m and 4mm respectively? 0 votes 1 answer 59 views ### Why must Hamiltonian of a system be invariant under every operation of the relevant point group? It is always claimed in any group theory books: "Hamiltonian of a system must be invariant under every operation of the relevant point group" or$RHR^{-1}=H$. Suppose I have a Hamiltonian of ... 1 vote 1 answer 98 views ### How is the Wigner little group representation of Poincaré group Unitary? From Weinberg's QFT Vol.1, eq(2.5.11): $$U(\Lambda)\Psi_{p,\sigma}=({N(p)\over N(\Lambda p)})\sum_{\sigma'}D_{\sigma'\sigma}(W(\Lambda,p))\Psi_{\Lambda p ,\sigma '}.\tag{2.5.11}$$ However, this is not ... 3 votes 0 answers 70 views ### How many degrees of freedom does a photon have in 2+1D? Wigner's classification of particles implies that the internal degrees of freedom of a particle transform under unitary representations of the subgroup of the Lorentz group that leaves its momentum ... 2 votes 1 answer 74 views ### What kind of a space is$\mathbb{C}_{-1}$? In the first paragraph on page 24 of this paper: https://arxiv.org/abs/0904.1556, it's written the left-handed leptons$\nu_L$and$e^−_L$both have hypercharge$Y=−1$, so each one spans a copy of$\...
In solid-state physics, I understand that in the case of no spin, the time reversal operator $\hat{T}$ is equal to the complex conjugation operator $\hat{K}$ and in the case with spin, \$\hat{T} = \hat{...