Questions tagged [lie-algebra]

A vector space $\mathfrak{g}$ over some field $F$ and kitted with a bilinear, antisymmetric and Jacobi-identity-fulfilling product ("Lie Bracket" or "commutator"). In physics, most often arises as the Lie algebra (tangent space to the identity) of a Lie group; in gauge theories, basis vectors of the gauge group's Lie algebra correspond to Noether currents and conserved quantities.

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268 views

Infinitesimal Poincare transformations , Taylor expansion

Let $(\Lambda,a)\in\text{ ISO}_o(3,1)$ be a finite (proper) Poincare transformation and Let $U(\Lambda,b)$ be the corresponding unitary operator implementing this transformation on the Hilbert space ...
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234 views

Why is the Galilean group not commutative?

As I understand it, the Galilean transformation is a matrix $$ \left[ {\begin{array}{ccccc} R_{11} & R_{12} & R_{13} & v_x & a_x\\ R_{21} & R_{22} & R_{23} & v_y ...
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156 views

Why must a fundamental particle's spin be a multiple of $\frac 1 2$? [duplicate]

A fermion is a particle whose spin is an odd multiple of $\frac 1 2$, and a boson is a particle whose spin is an integer. From what I've seen, these appear to be the only two possibilities; not only ...
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221 views

Why the Lorentz group has complex generators in QFT treatments? [duplicate]

In Schwartz' and Peskin's QFT books, when trying to deal with representations of the Lorentz group the authors study the representations of the Lie algebra of such group. By definition, if $SO(1,3)$ ...
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What was the first motive of physicists to interpret spin rotations as elements of $SU(2)$?

I have read several articles talking about the meaning of spins and spinor spaces. They only have a mathematical, or quantum mechanical identity. Thus one hardly finds a classical (geometrical) ...
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Gauge covariant derivative

I have seen distinct definitions of gauge covariant derivative (in Yang-Mills theory) $$ D_\mu \phi = (\partial_\mu + igA_\mu) \phi $$ vs $$ D_\mu \phi = \partial_\mu \phi + ig[A_\mu,\phi] .$$ I ...
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Representation of the Poincare algebra on the space of smooth functions

The following representation describes how a field $\varphi$ transforms under the Poincaré group $\mathcal{P}$. $$\mathsf{S} : \left\lbrace \begin{aligned} \mathcal{P} \times C^{\infty}(\mathcal{M})...
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Cartan Killing metric and Casimir operators

I'm a little confused about Casimir operators and Cartan-Killing metric. The Lorentz group is a semi-simple group and its Cartan-Killing metric is non-degenerate, say $g_{ab}$; it is invertible and ...
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Reference recommendation for Projective representation, group cohomology, Schur's multiplier and central extension

Recently I read the chapter 2 of Weinberg's QFT vol1. I learned that in QM we need to study the projective representation of symmetry group instead of representation. It says that a Lie group can have ...
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Derivation of generators of Lorentz group for spinor representation

I want to prove $$S^{\mu \nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu].$$ I started from $$[\gamma^\mu,S^{\alpha\beta}]=(J^{\alpha\beta})^\mu_\nu \gamma^\nu$$ Putting the value of $(J^{\alpha\beta})^\mu_\...
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Why multiply the infinitesimal generator for a rotation $R$ by $i$ when constructing $U(R)$?

I'm sure this is a silly question, but I can't figure out the answer. Current I'm reading chapter 4 in Weinberg's Lectures on Quantum mechanics. Earlier in the book, he asserts that unitary operators ...
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What is the difference between a Pauli spinor, a Weyl spinor, and a Cartan spinor?

I know that a spinor is a complex two components "vector", which is acted on by the $SU(2)$ group under a rotation. In the physics litterature, I often read "Weyl spinors", "Pauli spinors", "Cartan ...
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Do spins add when particles combine symmetrically?

Suppose I have three spin $s$ particles. What are the possible spins of a symmetric combination of these three particles? Will one of the states always have spin $3s$? Perhaps the above question is ...
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457 views

How to Derive Baryon Octet Matrix?

In every textbook on particle physics that I've read, I encounter the following matrix when reading about matrix of wave functions for the baryon octet: $$\left[\begin{array}{ccc} \frac{\Sigma^0}{\...
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How does one write a three dimensional de Sitter space as the quotient $SL(2, C)/SL(2, R)$?

In arXiv:hep-th/0110108 the $(2+1)-$dimensional de Sitter space is represented as a quotient space of $SL(2, C)/SL(2, \mathbb{R})$. I couldn't understand how, both mathematically and intuitively, is ...
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Interpretation of the field strength tensor in Yang-Mills Theory

In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\nu]...
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Is there a Virasoro group?

On page 14 of the survey article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive, the authors show that smooth selfmaps of the circle form a Lie group corresponding ...
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356 views

Meaning of Lorentz Generators

I'm trying to understand infinitesimal Lorentz transformations in quantum field theory. I've studied some Lie theory from mathematicians, but I'm having trouble adjusting conceptually to how Lie ...
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291 views

Infinite dimensional representations of $\text{SO}(3)$

In the theory of angular momentum, we wish to study the projective representations of the rotation group $\text{SO}(3)$, for which we turn to the representation theory of the double cover $\text{SU}(2)...
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386 views

Representation and gauge transformation of Yang-Mills theory

Let us consider a classical field theory with gauge fields $A_{\mu}^{a}$ and a scalar $\phi^{a}$ such that the Lagrangian is gauge-invariant under the transformation of the gauge fields $A_{\mu}^{a}$ ...
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162 views

How to properly set up the relation between Lie algebras and observables in QM?

In quantum mechanics, given a Hermitian operator $A$, it gives rise to a symmetry/unitary operator by exponentiation $e^{i\lambda A}$, which can be properly defining using the eigenvector expansion, i....
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$SU(2)\times U(1) \rightarrow U(1)$ Symmetry Breaking

Consider a doublet of complex scalar fields $\mathbf{\Phi}$. The $SU(2)$ transformation is $\textrm{exp}\left(it^{i}\tau_{i}\right)$ with generators $\tau_{i}=\frac{\sigma_{i}}{2}$, whilst the $U(1)$ ...
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Difference between Lorentz group and Poincare group

I am currently studying the proper orthochronous Lorentz group $\text{SO}^+(1,3)$ and I have run into some confusion. I started by defining the Poincare group to be the set of all transformations ...
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190 views

Representations of the Lorentz Group

The Lorentz group can be divided into two separate $SU(2)$ algebras and thus we label such representations with two spins $(j_{1},j_{2})$. The first and second spins correspond to generators $$J^{\pm}...
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Nilpotent symmetry group in classical mechanics

> Hello everyone, I have two questions concerning symmetries in classical mechanics. 1) I am looking for examples of a lagrangian or hamiltonian model with a symmetry group which is a nilpotent or ...
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Proof that (1/2,1/2) Lorentz group representation is a 4-vector

Taken from Quantum Field Theory in a Nutshell by Zee, problem II.3.1: Show by explicit computation that $(\frac{1}{2},\frac{1}{2})$ is indeed the Lorentz vector. This has been asked here: How do I ...
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Lorentz Group Representations

Consider for example the (trivial) spin-1/2 representation of the $SU(2)$ group. This representation has dimension two, which is clear from a quantum mechanical perspective since we need to specify ...
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Lie Algebra Conventions: Hermitian vs. anti-Hermitian

Consider the Lie algebra of $SU(2)$. To find the infinitesimal generators we linearise about the identity $$U=I+i\alpha T$$ where $\alpha$ is some small parameter. To find the form of $T$ use the ...
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336 views

${\bf su}(2)$ generators in 1, 2 and 3 dimensional matrix representations

The ${\bf su}(2)$ Lie algebra in a representation $\bf R$ is defined by $$[T^{a}_{\bf R},T^{b}_{\bf R}]=i\epsilon^{abc}T^{c}_{\bf R},$$ where $T^{a}_{\bf R}$ are the $3$ generators of the algebra. ...
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Justification of gauge field transformations

I am trying to understand the gauge transformation of gauge fields in a gauge quantum field theory. As an example I considered this wikipedia article, section 'An example: Scalar $O(n)$ gauge theory'....
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Hypercharge and Isospin as additive quantum numbers in $SU(3)$ flavour symmetry

I am studying the $SU(3)$ flavour symmetry and I'm reading that we use the fact that hypercharge and isospin are additive quantum numbers in order to decompose the tensor products of the fundamental ...
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On what kind groups of Lie groups are Yang-Mills theories based? [duplicate]

According to https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory: Yang–Mills theory is a gauge theory based on [...] any compact, semi-simple Lie group. But the symmetry group of Standard ...
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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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434 views

How to show that an $N$-dimensional SHO's dynamics symmetry is $SU(N)$?

From Wikipedia: The dynamical symmetry group of the $n$-dimensional quantum harmonic oscillator is the special unitary group $SU(n)$. As an example, the number of infinitesimal generators of ...
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Interpretation of Dynkin diagrams

I am having trouble in understanding the physics represented by dynkin diagrams. Say I have the following diagram: What is the difference between the square nodes and circular nodes? What does the ...
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725 views

How to decompose the representation of $\rm SU(5)$?

This question comes from Srednicki's textbook "Quantum Field Theory". On pages 514-515, it states: Under the unbroken $\rm SU(3)\times SU(2) \times U(1)$ subgroup, the $5$ representation of $\rm ...
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“Strange” right representation of Lie Algebras

while reading Freedman & van Proeyen's book, I found a very strange claim concerning representation on Lie Algebras: they define a generic transformation spanned by the parameter $ε^A$ and "...
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Is there an elegant proof of the existence of Majorana spinors?

Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly: A priori, we are dealing with ...
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Free field (Wakimoto) representation in 2d CFT

This question is more a request for explanations. I'm reading now the Di Francesco book in attempt to understand how the free field representations of 2d CFTs are constructed. The first steps in ...
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135 views

Equivalence of two adjoint representation definitions

I'm following notes that define the adjoint representation as a map $\rho$ from $\mathfrak{g}$ to $End(\mathfrak{g})$ such that, for $t_1, t_2$ $\in \mathfrak{g}$ , \begin{equation} \rho_{\tiny{adj}}(...
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Source of Ladder Operators

This might be a silly question but I am curious as to where the ladder operators in quantum mechanics come from. For example, in introductory texts on quantum mechanics, they try to solve the ...
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481 views

Inonu-Wigner Group Contraction

I am trying to understand how one obtains the Galilean algebra from the Poincare algebra, specifically through the method of central extension. I'm doing this by imposing that the generators of the ...
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197 views

Lie algebra decomposition of the gluon field

It is commonly written in the literature that due to it transforming in the adjoint representation of the gauge group, a gauge field is lie algebra valued and may be decomposed as $A_{\mu} = A_{\mu}^...
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Invariance under boosts but not rotations?

I am aware that there are 6 independent infinitesimal Lorentz transformations that can be separated into 3 rotations and 3 boosts. Is it possible for a quantum field theory to be invariant under the ...
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610 views

How are the generators of $\mathrm{SU}(3)$ represented on the gluon space?

I was watching some new lectures on QCD from Colorado and I have a few questions about what I heard: The $\lambda^a_{ij}$ are generators of $\mathrm{SU}(3)$ in the fundamental representation so are $...
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Heuristic derivation of $W^\mu=\frac{1}{2}\epsilon^{\mu\nu\sigma\rho}P_\nu J_{\sigma\rho}$ using combination of physical and mathematical arguments

If a simple systematic way to derive or guess (either mathematically or by a combination of physical arguments and mathematics) that one of the Casimir operator of Poincare group is $W^2\equiv W_\mu W^...
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What is “broken symmetry”?

For reference, I come from a mathematics background (mostly differential geometry). I have a very limited understanding of upper-level physics (I'm currently trying to fix this). This is my current ...
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Where in fundamental physics are Lie groups actually important (and not just Lie algebras)?

I was wondering where in fundamental physics the global structure of a Lie group actually makes a difference. Most of the time physicists are sloppy and don't distinguish groups and algebras ...
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229 views

How is the invariant speed of light encoded in $SL(2, \mathbb C)$?

In quantum field theory, we use the universal cover of the Lorentz group: $SL(2, \mathbb C)$, instead of $SO(3,1)$. The reason for this is, of course, that $SO(3,1)$ representations aren't able to ...
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Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$?

The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \...