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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Most general proper Lorentz transformations do not form a group?

How to prove that most general Lorentz transformations with relative velocities in arbitrary directions do not form a group? If the Lorentz transformations always satisfy, $$\eta=\Lambda^T\eta\Lambda$$...
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Lorentz elements of inverse transformation [duplicate]

I wanted to find explicit elements of inverse transformation of any Lorentz transformation with matrix form: $$L=\begin{pmatrix}L^0_{0}&L^0_{1}&L^0_{2}&L^0_{3}\\L^1_{0}&L^1_{1}&L^...
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Why is the flavour space for flavour symmetry defined in terms of low mass quarks?

In the textbook, I am following it describes flavour space with basis up, down and strange quarks. I am not sure why we did not choose up, charm and top as the basis and why only three bases can ...
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Most general Gauge Lie group in a Yang-Mills theory

Mathematicians have done a complete classification of all possible Lie groups. Is there a set of conditions that allows us to identify which Lie groups from the classification can possibly act as a ...
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Show reciprocal lattice vectors are normal to direct lattice planes [closed]

Given a direct lattice: $\textbf{R}=m\vec{a}+n\vec{b}+p\vec{c}$ where $m,\ n,\ p$ are integers and a reciprocal lattice: $\textbf{G}=h\vec{a^{*}}+k\vec{b^{*}}+l\vec{c^{*}}$ where $h,\ k,\ l$ are ...
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Detail on seeing the double cover of $SO^{+}(1, 3)$ as $SL(2, \mathbb{C})$

We can identify Minkowski space-time $M^4$, of metric signature $(1, -1, -1, -1)$, with the (real) space of $2 \times 2$ (complex) Hermitian matrices under the map $(v_0, v_1, v_2, v_3) \mapsto v_0 I +...
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Showing invariance of tensor trace under $\rm SO(N)$

If $O$ is an element of $SO(N)$, then $O$ is an $N\times N$ matrix satisfying $O^TO=1$ and det$(O)=1$. Let tensor $T^{ij}$ be a representation of the group and let the trace be Tr$(T^{ij})=\delta^{ij}...
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What happens to the space group of a crystal when introducing a non-trivial basis?

I am trying to understand crystallography and the space groups of crystals, but I have one major question bugging me. The book I am using adresses different lattice symmetries and applications of ...
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Derivation of Raising operator of $\rm SU(2)$

I'm reading a paper called: "A Simple Introduction to Particle Physics Part I - Foundations and the Standard Model" and i have some questions regarding the derivation of the raising and the ...
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Relations between the spin of representations of Lorentz group and Poincare group

It is known that Finite dimensional irreducible representations of Lorentz group can be indexed by two half integers $(s_1,s_2)$ and the sum $s_1+s_2$ is called the spin. Infinite dimensional unitary ...
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What does it mean for particles to “be” the irreducible unitary representations of the Poincare group?

I am studying QFT. My question is as the title says. I have read Weinberg and Schwartz about this topic and I am still confused. I do understand the meanings of the words "Poincaré group", &...
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Complex Lie algebra vs Real Lie algebra in Physics [closed]

A Lie algebra is a vector space $\mathfrak{g}$ over some field $F$ together with a binary operation $$\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$$ called the Lie bracket satisfying the following ...
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VEV implying symmetry breaking, but unable to pick out specific subgroup?

Let's say we have a scalar theory with an $O(N)$ symmetry, for which the scalar fields $\phi_{nm}$ transform as a rank $2$ tensor. I can write down an action which spontaneously breaks the symmetry $$...
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How do you observe “silent” quantum vibrations?

In the theory of quantum vibrations (aka phonons) it is useful to divide up the vibrational normal modes of a crystal based on their representation within the symmetry group of the crystal. The ...
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Representing a rotation around an arbitrary axis using Wigner $D$-matrix

It is known that an arbitrary rotation can be expressed in terms of three consecutive rotations called the Euler rotations. So instead of expressing the rotation operator as $\hat{R}(\hat{n},\phi) = \...
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Book on Representations and Group Theory in Particle Physics for Mathematicians

Is there a book for someone who already knows some group theory and theory of group representations on the mathematical side, and just wants something which explains the applications in particle ...
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The largest possible global symmetry of a 2-dimensional Hilbert space?

Suppose we have a quantum system of a 2-dimensional Hilbert space $\mathcal{H}$ and a Hamiltonian $\hat H$. My puzzle: What is the largest possible global symmetry for the Hilbert space $\mathcal{H}$ ...
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Murray Gell-Mann's independent discovery of group theory [closed]

In this article I found an interesting remark on how group (representation) theory was introduced into the physical sciences : Murray Gell-Mann developed the “eight-fold way” to explain the spectrum ...
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How to prove the equivalence of two definitions of hypercharge?

Before introducing top bottom and charm quarks,Strong Hypercharge is defined in the following two ways--- $1.\,\,\,Y=B+S$ where $Y,B,S$ are the hypercharge, baryon number and strangeness respectively. ...
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How to write quark composition of $\rm SU(3)$ mesons?

In $\rm SU(2)$, taking up quark and down quark as a doublet we can easily apply the isospin ladder operators to write the combination of 2 quark or 3 quark (baryon) systems. In $\rm SU(3)$ quark model,...
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Proof of Rivlin–Ericksen representation theorem relies on arbitrary tensors

I am having philosophical difficulties with the use of arbitrary orthogonal tensors in the proof of the Rivlin–Ericksen representation theorem on page 6 of the these lecture notes (author unknown; ...
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Why is group theory important in physics? [closed]

I have read quite a bit about it and I have studied group theory in more than one math class. I assumed that the idea is basically you show that some set of objects are a group and then you can ...
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Rotation of a Spinor

I have a question about an intuitive approach on spinors as certain mathematical objects which have certain properties that make them similar to vectors but on the other hand there is a property which ...
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1answer
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Time evolution of Galilean boost

I was introduced the generator of Galilean boost $K=mx-pt$. I was given an Hamiltonian with several particles: $H=\sum_i \frac{p_i^2}{2m_i}+V(|x_i-x_j|)$ where the potential only depends on the ...
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How can we go from a 4-dimensional representation of $SO(4)$ to the 3-dimensional one of its proper subgroup $SO(3)$?

In Walter Greiner's book, "Relativistic Quantum Mechanics", when discussing infinitesimal tranformations: $$x^{\prime\nu}=a^{\nu}{}_{\mu}x^{\mu},$$ where the $a^{\nu}{}_{\mu}$ is are ...
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How can we write explicitly the length, in the relativistic formalism, as orthogonality relations?

In the last chapter of W. Greiner's book, Relativistic Quantum Mechanics, orthogonal transformations are defined as follows : $$ x^{\prime\mu}=a^{\mu}{}_{\nu}x^{\nu} $$ When proceeding to find $a^{\mu}...
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Three blocks and the representation of $S_3$

I've been studying chapter 1 of the famous group "Lie Algebras in Particles Physics" by Georgi. I am rather confused by section 1.16. The claim is the following. Consider a system of three ...
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Why are Casimir operators required to be Hermitian?

What is the physical significance of requiring Casimir operators (of e.g. Poincare group or the conformal group) to be Hermitian? What breaks down if we do not impose this condition? EDIT: To be ...
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Mass as a coupling and mass as a Casimir operator

In Poincare group, we consider mass as a Casimir of the group. Hence it is a constant in various frames (I do not mean old fashion Lorentz transformation). But, in the quantum field theory mass is the ...
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Relativistic velocity addition - algebraic structure?

The relativistic velocity addition $\oplus$ for 3-velocities $\bf{u}$ can be constructed from Lorentz boosts $\Lambda$ acting on 4-velocities $u$ $$ u \to u^\prime = \Lambda(v) \, u \implies \bf{u}^\...
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What does projective space $\mathbf{S}^{2 N-1} / U(1) \cong \mathbf{C} \mathbf{P}^{N-1}$ mean?

In David Tong's lecture notes on Gauge Theory, in the section on 'Quantising the Colour Degree of Freedom', the following statement is confusing me, Since we already have the constraint (2.16), this ...
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Infinite dimensional representation of generators of the Lorentz group

I was reading Schwichtenberg's "Physics from Symmetry 2nd Edition". In Section 3.7.11, there is the discussion on the infinite dimensional representations. If we consider a transformation of ...
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Isometries of $AdS_2$ space

So, in many places it is mentioned that isometries of pure $AdS_2$ space is the group $SL(2,R)$, defined by the transformation, $t = (at+b)/(ct+d)$ where $ad-bc=1$. Here, the boundary of $AdS_2$ is ...
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The groups and symmetries of superstrings/SUGRA/M-theory

When giving talks to Laymen, we find out that the M-theory paradigm says that there are 5 (only 5) superstring theory types, 11d SUGRA and M-theory. Curiously, we read regularly the symmetry groups of ...
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Why a group that describe rotations always have $su(2)$ Lie algebra?

I'm reading the book Physics from Symmetry by Jakob Schwichtenberg. In part II the author explain the Lie group theory and in particular he treat the $SU(2)$ group. At a certain point the author tells ...
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Change of sign of angular momentum operator in Weinberg's QFT

In Chapter 2 of Weinberg's QFT, in order to represent a "boost" $L(p)$ , $$ L(p)=R(\hat{\mathbf{p}}) B(|\mathbf{p}| / \kappa), \quad(2.5 .44) $$ he tries to define a rotation $R(\hat{\mathbf{...
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Physical significance of the reality of an ${\bf N}$ representation: how the nature of interactions is affected?

Background The fundamental representation of ${\rm SU(N)}$ is denoted by ${\bf N}$ and the conjugate of the fundamental is denoted by ${\bar{\bf N}}$. If the representation ${\bf N}$ is related to ${\...
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Lorentz algebra structure in Hilbert space vs Vector space representation “must have the same commutation relations”

I'm following the Srednicki book, Ch. 2 The definitions are: Small Lorentz transformation on Minkowski space, so $S^{\mu \nu}$ forms the vector representation of the Lorentz group $$ \Lambda^\rho_{\;\;...
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1answer
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What does a broken symmetry mean for the Lagrangian?

I am a little confused about symmetry breaking - in particular, what I see to be too different interpretations of it. First, what I have seen taken to be the definition of a broken symmetry - we start ...
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Do the spinor transformation matrices form a matrix representation of the corresponding Lorentz group?

Suppose $\Psi$ is a Dirac spinor, then let the transformation matrix $S$ be defined as usual: $\Psi'=S(\Lambda)\Psi$, where $\Lambda$ is the Lorentz transformation matrix. Then the questions is: for ...
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1answer
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What is the overlap between different spin component eigenstates?

I am trying to find an expression for the overlap between the eigenstates of different spin component operators in a spin-S system. Say I have operators $\hat{S}_i,~i=x,y,z$ with eigenvalue equations $...
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Origin of the form of symmetry operation

It is well known that the symmetry operations $U$acting on the operators could be written as $$U AU^{-1}$$ Now I want to know the logical origin or motivation of this form of operation, my thought ...
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1answer
39 views

Representation of homogeneous Lorentz transformation

In Page 63, Section 2.5 of Weinberg's QFT Volume 1, on "One-particle states", he considers the representation of homogeneous Lorentz transformation, $U(\Lambda, 0) \equiv U(\Lambda)$ $$ U(\...
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When constructing a general Lorentz boost using an $x$-axis boost, what is the second rotation in relation ro the first rotation?

As is discussed in this question and this other question, it is possible to construct Lorentz boosts along an arbitrary direction using only the Lorentz boost along the $x$-axis by performing the ...
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1answer
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Commutation relations of angular momentum operators

On page 54 of Weinberg's QFT I, he says that an element $T(\theta)$ of a connected Lie group can be represented by a unitary operator $U(T(\theta))$ acting on the physical Hilbert space. Near the ...
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1answer
47 views

Hermitian operators in the expansion of symmetry operators in Weinberg's QFT

This is related to Taylor series for unitary operator in Weinberg and Weinberg derivation of Lie Algebra. $\textbf{The first question}$ On page 54 of Weinberg's QFT I, he says that an element $T(\...
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Question in Derivation of Lie algebra

In Weinberg's QFT Volume 1 Chapter2, he "derives" the Lie algebra from the Lie group as follows [...] a connected Lie group [...is a...] group of transformations 𝑇($\theta$) that are ...
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Why don't $\rm SO(10)/Spin(10)$ or larger GUTs permit electron-pair decay?

$\rm Spin(10)$ unifies all left-handed fermions and anti-fermions, and all right-handed fermions and anti-fermions. And its Pati-Salam subgroup unifies quarks with leptons (SU(4)) and complements SU(...
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Index Placement Conventions in Group Theory

In his book on group theory, Wu-Ki Tung seems to utilize a few peculiar conventions regarding the placement of indices on vector and matrix symbols. For instance, if $|x\rangle=\sum_{i=1}^n|e_i\rangle ...
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Elements of Lorentz transformation matrix

I just began to study relativity and got introduced to tensor algebra few months ago, so I'm confused with this question that wants me to show that the elements of the Lorentz transformation matrix (...

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