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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Translation and Dilation transformations within the conformal group

I am using Di Francesco's book P.39. The equation that the generators of the transformations satisfy is given by: $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta w_a} \partial_{\mu} \Phi - \frac{\delta F}{...
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909 views

How to theoretically determine the angular momentum of an atom?

To determine if an atom is a boson or a fermion I have to count the fermions that constitute the atom (protons, neutrons and electrons). My question is: How to theoretically (as opposed to ...
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362 views

Lie algebra of lorentz group

I'm stuck in following calcualtion from sredniki's QFT book.(Its actually in the solution manual) How can i get from $$\delta\omega_{\rho\sigma}(g^{\sigma\mu}M^{\rho\nu} - g^{\rho\nu}M^{\mu\sigma}) $...
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842 views

high spin atoms SU(2) representation

I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry. Why not $SU(N)$?
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2k views

Wigner-Eckart projection theorem

I'm following the proof of Wigner-Eckart projection theorem which states that: $$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} \...
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47 views

Selfstudy Conformal Field Theory and Lie algebra from scratch roadmap [closed]

A couple of months ago, I stumbled upon Conformal Quantum Mechanics (CQM) which was really interesting yet confusing. I didn't have the prerequisites so I couldn't follow the paper. I want to first ...
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17 views

Inner product on group theoretic coherent states and anti-commutator of Lie algebra generators

This question is related to group theoretic coherent states (Gilmore, Perelomov etc.). I consider a semi-simple Lie group $G$ with Lie algebra $\mathfrak{g}$ and a unitary representation $U(g)$ acting ...
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34 views

Lorentz algebra and group question with regards to operator representaion of $M^{\mu\nu}$

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: I aim to consider the product $L^0{}_0(\Lambda_1\Lambda_2).$ Consider the following ...
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32 views

Why does the Weyl vector $\rho = \sum_{\lambda_i\in\Lambda_W} \lambda_i = \dfrac 12 \sum_{\alpha_i \in\Lambda_R^+}\alpha_i$ represents vacuum?

I was reading https://arxiv.org/pdf/1003.2861.pdf, and in p.5, just below equation (3.1), it was written that The state $|\rho\rangle$ associated with the Weyl vector $\rho$ corresponds to the ...
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36 views

Vanishing Poisson bracket with non-vanishing Moyal bracket

Let $M=\mathbb{R}^{2n}$ be the phase space with standard Poisson bracket on smooth functions on $M$. Fix a classical hamiltonian $h$ (function on $M$) and function $f$ generating symmetry of $h$ i.e. $...
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There is a classification of matrix representations of Euclidean group $\mathrm{E}(n)$ (also called $\mathrm{ISO}(n)$)?

I'm wondering if someone has already done a classification of matrix representations of the Euclidean group $\mathrm{E}(n)$, more specifically I'm trying to classify $\mathrm{E}(2)$. I watched a ...
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50 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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93 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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124 views

Commutation relation Lorentz Algebra

Related question, which I don't understand either. I think is easier to get the Lorentz group algebra as defined by Maggiore, $$ [J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\nu\rho}J^{\mu\sigma} - \eta^{\...
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66 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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Is it always possible to move to the “Cartan Gauge”?

Forgive me for potentially coming up with a new name for what I am about to describe. Let's say we have a scalar field $\phi^a$ which transforms with respect to the adjoint representation of some Lie ...
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1answer
54 views

Gauge group of Electroweak theory

I am doing a question that asks me to identify the gauge groups of a Lagrangian with the field strength tensors $$\bf{F}_{\mu \nu} = \partial_{\mu}\bf{W}_{\nu} - \partial_{\nu} \bf{W}_{\mu} - g\bf{...
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52 views

Error with generators of Lorentz group (basis of Lorentz Lie algebra) [closed]

Can someone help me figure out why my $J_y$ is incorrect? :/ It's supposed to be \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & -...
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149 views

Hypercharge normalization for $SU(5)$ GUT

Reading about $SU(5)$ unification, texts says that they use the renormalization factor $\sqrt{3/5}$ for weak hypercharges in order to embed SM into a $SU(5)$ group. This implies a new $U(1)_Y$ ...
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29 views

Connection between Classical and Quantum symmetries

I am an advanced undergraduate student.I would like to know about the importance of symmetry in classical and quantum mechanics.Also a good book concerning the connection between symmetries of ...
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1answer
61 views

How are Dunkl operators used in Hamiltonian mechanics?

I am currently doing a math research project on the representation theory of Cherednik (double affine Hecke) algebras, specifically the algebra $\mathcal{H}_{t,c}(\mathfrak{S}_n,\mathfrak{h})$, which ...
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98 views

Lie algebra: Proof that the commutator of infinitesimal motions is an infinitesimal motion

I am following Classical and Quantum Mechanics via Lie Algebras by Neumaier and Westra. Setup I am stuck at part of Thm 2.3.1. Consider the matrix group $\mathbb{G}$. The set of $\mathbb{G}$-motions ...
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43 views

Physically, why don't we care about representations that differ only by a similarity transformation?

I was looking at how to derive the (1/2, 0) representation of the Lorentz group when acting on fields. Specifically, I'm interested in understanding the logic behind replacing the "symbols" $A,B$ with ...
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64 views

Symmetry of the Batalin-Vilkovisky (BV) antibracket operation

Batalin and Vilkovisky define $^1$ an operation they call antibracket which is $$(F,H) = \Big(\frac{\partial_r F}{\partial \Phi^A}\Big) \Big(\frac{\partial_l H}{\partial \Phi^* _A} \Big) - \Big(\frac{...
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113 views

Books, papers, etc on Lorentz and Poincare groups/algebras/etc

I'm currently trying to learn more about the Lorentz- and Poincare Lie-algebras and the representation theory about them. But I'm really struggling with the material that we were given and I'm also ...
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1answer
190 views

How much information about a quantum operator is determined by its Poisson bracket Lie algebra?

Hamiltonian quantum mechanics is often built using many ideas from Hamiltonian classical mechanics like the Poisson bracket to determine the commutator between quantum operators, which is appropriate ...
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178 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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69 views

Is there a difference of sign conventions of Dirac Index between mathematics and physics?

In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by $$\mathrm{Ind}(D\!\!\!\!/_{A})=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}(F\wedge F)+\frac{...
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46 views

$SU\left(N\right)$ Dynkin labels, how to compute

Let $V$ be somecomplex irreducible representation of $SU\left(N\right)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of ...
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47 views

Books about Group Theory [duplicate]

I was looking for a book to complement the lecture notes of the course for a more intuitive approach to the subject and full of examples (mathematical), because the handouts seem only a bunch of ...
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130 views

Relation between Lie algebra of conformal Killing vector fields and conformal algebra

I'm new to conformal transformations and I have a question. Following the book of Barrett O'Neill "Semi-Riemannian geometry with applications to relativity", there is a Lie anti-isomorphism between ...
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89 views

How to prove a set of matrices form a representation of Lie algebra?

When reading Paul Langacker's The Standard Model and Beyond, I am quite confused on equation 3.29, which says with a set of fields $\Phi _a$, where $a$ goes from 1 to $n$, are chosen to be transformed ...
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70 views

Time dependent Hamiltonian operator and $SU(1,1)$ generator method

In this screenshot of a paper I am reading, I have the following question: 1.What is a $SU(1,1)$ group and how do we find its generators? 2.From the expression for the Hamiltonian $\hat{H}$, how do ...
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49 views

Integrating over a symmetry vector field by exponential map

Take some constant-of-motion, $H$, and the Poisson bracket, $\{\cdot, \cdot\}$. Then, we recover the symmetry vector field of $H$ by \begin{equation} S_H = \{\cdot, H\} \end{equation} so take for ...
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44 views

Representations of a symmetry group: what is actually being represented? [duplicate]

For definiteness, consider the group $SO(3)$. There is a Lie algebra $so(3)$ given by $$ [T_a, T_b] = if_{abc}T_c $$ The generators of this algebra can be exponentiated to form the elements of $SO(3)...
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109 views

Explicit Quadratic Casimir for $sp(2N)$

We know that $so(3)$ has the explicit quadratic Casimir $$L^2=\sum L_{i}^2.$$ Are there analogs to this in other simple lie algebras? I know that for a simple lie algebra I can always use the ...
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93 views

Vector spaces in second quantization

Studying about fermionic commutation relations, the convention I'm following is to consider a set of creation (destruction) operators $\hat{a}_{i}^{\dagger}\left(\hat{a}_{i}\right)$ with $i=1,...,n$ ...
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79 views

What do quantum spin hamiltonians describe?

I've learned all particles are either fermions or bosons, obeying their respective operator algebras, and then I've seen Hamiltonians describing models carrying one of these two types of particles. So ...
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44 views

How to prove that given operators form an algebra? [closed]

I am new to Group theory and representations and I'm having trouble with this problem in an exercise: Given the two oscillator algebra $$[a, a^†] = 1$$ $$[b, b^†] = 1$$ $$[a, b] = 0$$ show that ...
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142 views

Difficulty understanding Srednicki's derivation of the Lie Algebra Commutators of the Lorentz Group

I have just started reading through Srednicki's QFT in preparation for a few courses I am about to take. I have taken courses that have covered the basics of special relativity including ...
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54 views

Conformal Dimension and Highest Weight States of Coset CFT

I am trying to understand the vertex operator algebras of the following form: $$\frac{U(M|N)_{k_1}}{U(L)_{k_2}}$$ Where $U(M|N)$ is the unitary supergroup, $U(L)$ is the usual unitary group, and $...
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1answer
107 views

On the homomorphism of the Lorentz algebra representation (1/2, 0)

I was reading this answer and I don't quite understand how the $\rho$ homomorphism works. The generators of the two copies of $\mathfrak{su}(2)$ in $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ are given ...
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Representation of the Lorentz group from representation of its Lie algebra [duplicate]

In many books on particle physics, it is first shown that the Lie algebra of the Lorentz group is isomorphic to $$ su(2) \oplus su(2) , $$ then by this fact, it is implicitly assumed that for each ...
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136 views

Representation Theory in a Nutshell for Physicists? [duplicate]

Are there good references and introduction of Representation Theory as "Representation Theory in a Nutshell for Physicists"? For example, we hope that the book/ref contains the "introduction to ...
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68 views

Gauge group properties VS Particle properties

Let us say that, for simplicity, we are studying gauge theory over 4-dimensional Minkowski spacetime, and that the gauge group is $SU(3)$, which is probably the simplest non-abelian gauge group after $...
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113 views

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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2answers
78 views

“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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186 views

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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119 views

Why do we study the projective representations of SO(3) in the context of the theory of angular momentum? [duplicate]

I have been studying the group theoretic formalism of quantum mechanics and I have yet to find a satisfying explanation for the need for projective representations in the theory of angular momentum. I ...
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165 views

How to find generators of a transformation?

We have a 4-vector $V^\mu =(V,0,0,V)$ with sametime and $ z $ components. then we are asked to find the transformations that are not pure rotations and leave $ V $ invariant. Sone one such matrix is ...