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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228 views

Representation of Weyl-Heisenberg Lie Algebra [closed]

I'm reading the book: Coherent State in Quantum Physics, by Jean-Pierre Gazeau. In the page no. 35 of this book, the Weyl-Heisenberg Lie Algebra $\mathfrak{w}_m$ has been given that $$\mathfrak{w}_m=...
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804 views

What is the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1) \subset su(2)$?

What is meant by the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1)\subset su(2)$? I have read it above eqn. (10) in this paper http://arxiv.org/abs/0812.3572 but have also heard it mentioned in ...
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209 views

The universal covering group of a symmetry group [duplicate]

In Weinberg QFT Vol.1, it says one can enlarge the symmetry group $H$ to the universal covering group $C$ such that one obtains a trivial cocycle or $C$ is simply connected whereas $H$ is not. I get ...
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150 views

Given a VEV how can I compute which generators remain unbroken using tensor methods?

This is a follow up to this question. A generator $T_a$ of a given gauge group $G$ remains unbroken after some Higgs field $\Phi$ gets a vev if $$ T_a \langle\Phi\rangle =0 $$ I'm trying to ...
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47 views

How to find the generators of a deformed boost?

I'm reading the paper arXiv:gr-qc/0012051 on doubly special relativity. In page 7, the author wants to find the generators of a deformed boost that preserves $$E^2 = p^2 + m^2 - l_p p^2 E$$ ($l_p$ is ...
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53 views

Name for the transformation into an accelerated frame?

A transformation into a frame that looks at an experiment from a rotated perspective is called a rotation. A transformation into a frame that moves with a different constant velocity is called a ...
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58 views

Reference request: Relation between $Sp(N)$, $Spin(N) $, $SU(N)$ groups and physics [duplicate]

I want to understand the relationship of the so common $SU(N)$ and $SO(N)$ groups in physics with the symplectic group which I think is the double cover of the first and the Spin groups $Spin(N)$. Is ...
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79 views

Spin matrix for various spacetime fields

Let $V^{\mu}$ be a vector field defined in a Minkowski spacetime and suppose it transforms under a Lorentz transformation $V'^{\mu} = \Lambda^{\mu}_{\,\,\,\nu}V^{\nu}$. We can write this like $V'^{\...
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94 views

How are symmetries defined mathematically? [duplicate]

I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding ...
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109 views

Group of translations in two dimensions - A weird treatment

Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is $$ G = p_1\delta x_1 +p_2 \delta x_2 $$ Now, we know clearly that this group is an Abelian ...
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139 views

Action of conformal generators on fields

I am calculating the action of the conformal generators on fields, to be more precise on wavefunctions. For now, I'm classical. I will just paste the part of my report on this to show what I am ...
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112 views

Books on representation theory [duplicate]

Possible Duplicate: Best books for mathematical background? I'm looking for a textbook on the group/representation theory for a student-physicist. The main questions of interest are ...
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1answer
87 views

Irreducibility of $SU(N)$ rank-2 tensors [closed]

Given a rank-2 $\mathrm{SU}(N)$ tensor $X^{ab}$, it transforms as $X'^{ab} = U^a{}_c U^b{}_d X^{cd}$, where $U \in \mathrm{SU}(N)$. We can decompose it into a symmetric and an anti-symmetric part $$ X^...
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377 views

Commutator summation notation

I have the relation $ e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?
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64 views

How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
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1answer
122 views

A doubt with Group Generators in group theory and their algebra

My doubt is this: I saw in a paper that the Lie Algebra is the relation between the commutator of the generators and the generators multiplied by structure constants. $$[S_{i},S_{j}]=c_{ij}^{k}S_{k}$...
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70 views

Incorrect proof that all gauge theories are abelian

Consider a gauge field $W_\mu = W_\mu^{a} \tau_a$ where $\tau_a$ are the generators of the Lie algebra and $W_\mu^{a}$ just numbers. Then: $$ W^2 = W_\mu W^\mu = W_\mu^a\tau_a W^{\mu b} \tau_b = W_\...
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171 views

Quark space tensor product Vs Angular momentum space tensor product

For two triplet angular momenta states, say $J=1$ and $I=1$, if we wanna look at it in the coupled basis $F=I+J$, we use the regular Angular Momentum rules: $$|I-J|\leq F\leq I+J,$$ and from that ...
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168 views

Generator of a rotation matrix

$$T(\phi)= \begin{bmatrix} \cos(\theta) &\sin(\theta) & 0 \\ \sin(\theta)\cos(\phi) & -\cos(\theta)\cos(\phi) & \sin(\phi)\\ \sin(\...
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47 views

Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels. For a simple example if we take $SU(2)$, with $[R]$ indicating the highest ...
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226 views

Does the sedenion algebra offer a grand unification theory?

Stephane Bronoff in The Standard Model of Particle Physics from Sedenions claims that studying the left-multiplication map of unit doubly-pure sedenions solves several mysteries of the standard model. ...
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1answer
298 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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453 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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624 views

Why are 3 colors used in QCD?

The mapping of strong charge to RGB left me believing that there are only 3 conserved quantities in QCD. I recently came to the understanding that there are in fact 8 conserved quantities, as ...
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101 views

The order of Pauli matrices

Is there any special reason why Pauli matrices are: $\sigma _1=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right)$, $\sigma _2=\left( \begin{array}{cc} 0 & -i \\ i & 0 \...
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1answer
73 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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3answers
57 views

Townsend's Infinitesimal Operators

I'm currently reading John Townsend's Modern Approach to Quantum Mechanics and the infinitesimal operators have me a bit puzzled: $$\hat R(d\phi \boldsymbol{k})=1-{i \over \hbar}\hat J_zd\phi$$ $$\...
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2answers
62 views

Product of traceless hermitian $N \times N$ matrices

In Srednicki's textbook Quantum Field Theory, section 80 shows that, for a complete set of $N^{2}$ tracelss hermitian $N \times N$ matrices normalized according to $Tr(T^{a}T^{b}) = \delta^{ab}$, \...
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1answer
112 views

Show that when angular momentum $L_x$ and $L_y$ commute with operator $G$, then $L_z$ also commutes with $G$

I want to prove that if Angular momentum $L_x$ and $L_y$ commute with an operator $G$, angular momentum $L_z$ also commutes with $G$. if $[L_x , G] = [L_y, G] = 0$ then $[L_z , G] = 0$ I know that $...
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1answer
107 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
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1answer
342 views

Pauli matrices identity with no repeating indices

I was just wondering if there is a proof of, or an example utilizing the following relation: $\sigma^i_{\alpha\beta}\sigma^j_{\gamma\delta}+\sigma^j_{\alpha\beta}\sigma^i_{\gamma\delta}+\delta^{ij}(\...
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40 views

Consistency of transformation of scalar fields with mathematical definition of a representation of Lie algebra and Lie group

Transformations of scalar fields under a Lorentz group transformation are generated by differential operators $L_{\mu\nu}=x_\mu\partial_\nu-x_\nu\partial_\mu$. On the other hand, a representation of a ...
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83 views

Relation between $\vec{L}^2$, $\vec{r}$ and $\vec{p}$ [closed]

I'm trying to prove equation (1.35) $$\begin{align} (\mathbf{a}\times\mathbf{b})^2 &= \mathbf{a}^2\mathbf{b}^2 - (\mathbf{a}\cdot\mathbf{b})^2 \\ &− a_j[a_j,b_k]b_k + a_j[a_k,b_k]b_j − a_j[...
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1answer
35 views

Derivation of Commutators for Galillean Transformations in Ballentine

I am trying to follow along the derivation of the commutator relations for the generators of the Galilei Group in Ballentine. He states that the product of two infinitesimal generators and their ...
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1answer
82 views

Weinberg's “Derivation” of Lie algebra commutation relations

I have a question regarding the evolution of Lie algebra conditions in Weinberg's The Quantum Theory of Fields vol. 1: Foundations, chapter 2. I will reproduce the text here and state my two ...
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1answer
62 views

How do you subtract colors and divide them by irrational numbers? (Gluons) [closed]

There is a gluon that is $$\frac{1}{\sqrt{3}} (red \cdot\overline{red} + blue\cdot\overline{blue} - 2\cdot green \cdot\overline{green})$$ This confuses me because I do not understand how adding and ...
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1answer
51 views

Highest and Lowest $SU(3)_F$ states

For the finite dimensional $(p,q)$-irreducible representation of $SU(3)_F$, we can label the states as $\mid T_3,Y\rangle$. Where $T_3$ is the third component of isospin and $Y$ is the hypercharge. ...
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1answer
44 views

Lie Subgroups of $SL(2,\mathbb{R})$

I'm wondering about the Lie subgroups of $SL(2,\mathbb{R})$. It's Lie algebra is the algebra of real traceless matrices and has basis elements $$L_0 = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \...
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1answer
90 views

Generators of conformal transformations change of basis

I recently started going through Introduction to Conformal Field Theory by Blumenhagen and Plauschinn ( springer link ). On page 11, they glue together the generators of conformal transformations as ...
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1answer
89 views

Reference request for Lie algebras

My future adviser just published a beautiful paper, https://arxiv.org/abs/1904.08304, and I am looking for some references/textbooks to look into the following concepts: Lie algebra (central) ...
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1answer
53 views

Generator of 3D rotations in $\mathbb{C}^2 \otimes \mathbb{C}^2$

Let us consider a system of two spinors. The 3D rotation operator around the $\vec{n}$ axis in $\mathbb{C}^2$ is clearly $R(\theta) = \exp(i \frac{\theta}{2}\vec{n}\cdot\vec{\sigma})$. If I wish to ...
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1answer
126 views

Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
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1answer
98 views

How to prove commutation relation between charge and current in current algebra?

I am reading Gauge Theory of Elementary Particle Physics by Tapei Cheng and Lingfong Li. Proceeding equation 5.54, there is a statement which says Then from Lorentz covariance, we can include the ...
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1answer
70 views

Lie algebra valued potential vector [closed]

Maybe it is a simple question but I have some difficulty to understand the explicit matrix form of this usual relation: $$A_\mu=A^a_\mu \tau_a$$ where $A^a_\mu $ is the Lie algebra valued potential ...
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1answer
56 views

Lie algebra definition - Maggior'e book

I'm reading Maggiore's book about QFT, and I'm having a trouble understanding the notation in the part about Lie algebras (Section 2.1): The group generators are defined as $T^a_R=-i\frac{\partial ...
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1answer
62 views

Evaluating $n \otimes_A n^*$ in $SU(n)$

In "Quantum Field Theory in a Nutshell" pg424 the author (Zee) writes: $$(n\oplus n^*)\otimes_A(n \oplus n^*)\quad\cong\quad(n^2-1)\oplus 1 \oplus n(n-1)/2 \oplus ((n(n-1))/2)^*$$ From what I ...
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1answer
1k views

On the generators of the Lorentz group

We already know that the generators of Lorentz group are three boosts and three rotations. We have the relation $$ \left[ M_{\mu \nu},M_{\sigma \lambda} \right] = ig_{\mu \sigma} M_{\nu\lambda} - ...
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1answer
394 views

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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1answer
397 views

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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1answer
236 views

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'....