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 ...

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2
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1answer
226 views

Question on derivation of Ward identity

I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field ...
3
votes
1answer
237 views

Lie Algebra Generators

We know that rotations are performed via real and orthogonal matrices, $O^{T}=O^{-1}$. We can write $O$ as, (The proper rotations have unit determinant) $$O = \exp(A),$$ where $A^{T}=-A$. In three ...
5
votes
1answer
165 views

What is the exact relation between $\mathrm{SU(3)}$ flavour symmetry and the Gell-Mann–Nishijima relation

I'm trying to understand how the Gell-Mann–Nishijima relation has been derived: \begin{equation} Q = I_3 + \frac{Y}{2} \end{equation} where $Q$ is the electric charge of the quarks, $I_3$ is the ...
2
votes
0answers
77 views

Casimir Invariants of the Galilean group

I had studied a couple of things about Galilean and Poincare group. But in the Galilean group, there is not enough clarity on how to calculate generators for boosts ($B_i$), which if I do it seems I ...
1
vote
0answers
81 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 ...
2
votes
2answers
248 views

Why do we require the generators of $\mathrm{SU(N)}$ gauge theories to be $N \times N$ matrices?

I have often read that the generators for $\mathrm{SU(N)}$ gauge theories must be $N \times N$ matrices; see for instance these notes at the top of page 3: ...
7
votes
1answer
122 views

Lie algebra of axial charges

Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\tau_a$ are pauli matrices) $L=\bar Ni\gamma^\mu \partial_\mu N+ \frac{1}{2} ...
6
votes
2answers
524 views

Tensor decomposition under $\mathrm{SU(3)}$

In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning ...
5
votes
1answer
124 views

Proving Lemma 4 in Georgi's Lie Algebra in Particle Physics 2nd p 251

The lemma 4 is given in the above picture. My question is, how to verify linear dependence (20.15) for diagram (a)? I tried to extend the matrix for the simple root in wikipedia $$ \left ...
2
votes
0answers
89 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 ...
3
votes
2answers
160 views

Unitary groups and infinitesimal transformations - Schwingers way of deriving Lie groups

In Schwinger's source theory book, he suggests if $G_a$ are the hermitian generators of the Unitary group, then we have an infinitesimal transformation is given by : $$ G = \sum_{a=1}^n ...
6
votes
2answers
334 views

Number of the Generators of Poincare Group

It is said that the Poincare group, $P(3,1)$ has $10$ generators. $6$ of them are the generators of the Lorentz group, $O(3,1)$ and the other $4$ generators are the generators of $4D$ translational ...
2
votes
3answers
295 views

Number of Parameters of Lorentz Group

We embed the rotation group, $SO(3)$ into the Lorentz group, $O(1,3)$ : $SO(3) \hookrightarrow O(1,3)$ and then determine the six generators of Lorentz group: $J_x, J_y, J_z, K_x, K_y, K_z$ from the ...
5
votes
2answers
321 views

Rotation Group and Lorentz Group

It is often stated that rotations in the 3 spatial dimensions are examples of Lorentz transformations. But Lorentz transformations form a group named the Lorentz Group, $O(1,3)$ which is a group a ...
4
votes
1answer
446 views

Scalar field transformation and generators

When we do a transformation (norm preserving one) for a given quantity, from what I have understood it seems like there is a representation of the group element for each quantity depending how they ...
8
votes
1answer
915 views

Generators of Poincare Groups

How can I determine the generators of the Poincare Group, $P(1,3)$ explicitly? Here $P(1,3)$ means a matrix Lie group.
2
votes
1answer
128 views

Does a Super Noether Theorem exist?

I am wondering if an extension of Noether theorem to supergroups exists. In particular the analogy with the usual case should be that supersymmmetries are in 1 to 1 correspondence to certain ...
2
votes
0answers
64 views

Quantum Field Theory and Lie Theory [duplicate]

I am reading Vol.1 of "The Quantum Theory Of Fields" by S. Weinberg. However I have come to a halt when connected Lie groups were introduced. I have solid knowledge in elementary group theory and ...
2
votes
1answer
75 views

1-dimensional Ring geometry - Group of Translations

I considered a Ring-like one dimensional geometry. In this, if we fix an origin (at some point on the circumference), we can think of set of all displacements along the circumference to form a vector ...
25
votes
3answers
1k views

Idea of Covering Group

$SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence? I heard that this fact is related to the description of bosons and fermions. But how does it ...
2
votes
2answers
130 views

Different definition of SL(2,R) algebra?

I'm looking into $SL(2,\mathbb{R})$ group and it's algebra. I found on line that the $sl(2,\mathbb{R})$ algebra is given by the two by two real matrices of trace zero. This Lie algebra has dimension ...
0
votes
1answer
148 views

Triangle inequality Clebsch-Gordan coeffcients

The Clebsch-Gordan coefficients can only be non-zero if the triangle inequality holds: $$\vert j_1-j_2 \vert \le j \le j_1+j_2$$ In my syllabus they give the following proof: $$-j \le m \le j$$ $$-j_1 ...
2
votes
1answer
394 views

Lorentz group representation and transformation of “vectors”

Let $P$ be the parity operator of the Lorentz group, $$P=\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}$$ the commutation relations ...
1
vote
0answers
82 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 ...
13
votes
3answers
737 views

$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = ...
6
votes
0answers
174 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
0
votes
2answers
271 views

Azimuthal quantum number $\ell$ and magnetic quantum number $m$ are from angular momentum?

Azimuthal quantum number $\ell$, and magnetic quantum number $m$ are defined when we do derivation for $L^2f=\ell(\ell+1){\hbar}^2f$ and $L_zf=\ell{\hbar}f$. This is my own conclusion after studying ...
0
votes
2answers
95 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 ...
2
votes
2answers
363 views

Peskin and Schroeder Equation 3.23

I've been trying (for a while) to prove that $S^{\mu\nu}:=\frac{i}{4}\left[\gamma^\mu,\,\gamma^\nu\right]$ is a representation of the Lorentz Lie algebra, that is, to prove that it satisfies the ...
6
votes
2answers
1k views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
3
votes
0answers
709 views

Killing vectors for 2-sphere as generators of $SO(3)$ symmetry

How to get Killing vectors in a form of generators of $SO(3)$ group symmetry? By using Killing equations for metric $ds^{2} = d\theta^{2} + \sin^{2}(\theta^{2}) d\varphi^{2}$ I got $$ ...
4
votes
1answer
289 views

What are the generators of spherical symmetry?

The title says it all. I think this should be a pretty simple question but I just couldn't find the answer. Ok -- I'll give a bit more context to my question. I'm encountering this in the context of ...
4
votes
1answer
645 views

What exactly is the connection between the Jacobi and Bianchi identities

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0$$ ...
9
votes
1answer
411 views

Why does the eightfold way work?

Last year I attended an introductory particle physics course, in which the Eigthfold Way for classifying hadrons has been discussed. The main idea consists in grouping hadrons in multiplets (i.e ...
4
votes
1answer
247 views

Question about the Killing form in Georgi

I'm trying to understand the Killing form described on page 49 in the book by Howard Georgi. He starts by saying that one defines the inner product between two generators $T_a$ and $T_b$ in the ...
3
votes
1answer
345 views

Deriving Virasoro algebra question

I'm reading a book Lie groups, Lie algebras, cohomology and some applications in physics by Azcarraga and Izquierdo, and on page 347, when deriving the exact form of the central extension term I came ...
4
votes
1answer
234 views

Question on conserved quantities and Noether's theorem

I have a question about Noether's theorem in the context of QM, which I'll state in the context of the weak interaction but the basic point could be generalized. According to Noether's theorem, given ...
1
vote
1answer
310 views

$t_1$, $t_2$, $t_3$ Hermitian generators of $SU(2)$

What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & ...
6
votes
2answers
729 views

What is the significance of Lie groups $SO(3)$ and $SU(2)$ to particle physics?

I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics? The application of Lie groups and their ...
2
votes
2answers
460 views

Symmetries & Lie groups in physics

This is not a homework, neither it is any exercise. It is my understanding of $U(1)$ symmetry. I would request if anybody can please correct me on any one of the following understandings: The ...
1
vote
1answer
591 views

How to get Gell-Mann matrices?

How to get Gell-Mann matrices $f_{i}$ (more or less strictly)? What are the requirements for getting them, excluding $||f_{i}|| = 1$, commutational law $[f_{i}, f_{j}] = if_{ijk}f_{k}$ and hermitian ...
8
votes
4answers
358 views

What does “carry a representation” mean (in SUSY algebra)?

I come from a maths background and am struggling with some of the more physical texts on SUSY. In particular they claim that the fermionic generators $Q_A^i$ carry a representation of the Lorentz ...
5
votes
2answers
663 views

Lie Groups and group extensions?

Is $U(1)$ x $SU(2)$ x $SU(3)$ a vector space over a field? I saw an article here http://en.wikipedia.org/wiki/Group_extension that seemed to me that a similar concept to a field extension was ...
0
votes
0answers
65 views

What is ``thermal" about a thermal quotient of EdS and EAds?

This is in continuation of my previous question and is in reference to this paper. I guess that the authors are interested in $S^n$ and $\mathbb{H}^n$ since these are the Euclideanized versions of ...
7
votes
1answer
294 views

link between real particles, representation of algebra and Young tableau

I know that different representations of this algebra correspond to different spin. One can sort the representation according to the casimir. For any simple Lie algebra, the operator $$ T^2 = ...
1
vote
1answer
401 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 ...
4
votes
1answer
159 views

Is the conjecture about $E(11)$ and M-theory (West's conjecture) generally accepted?

I was reading this paper by West, in which it is argued that: Eleven dimensional supergravity can be described by a non-linear realisation based on the group $E\left(11\right)$ From ...
4
votes
1answer
214 views

Decomposition of representations of the Virasoro algebra under $sl(2)$

The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight ...
4
votes
2answers
391 views

Calculating an expression for the trace of generators of two Lie algebra

Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie ...
0
votes
1answer
113 views

splitting spin representation when reducible

I'm looking at the spin representation of the orthogonal algebra $so(2m,C)$. This representation is $2^m$ dimensional and an explicit construction with gamma matrices is well known (see for example ...