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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5answers
939 views

Eigenspaces of angular momentum operator and its square (Casimir operator)

The casimir operator $\textbf{L}^2$ commutates with the elements $L_i$ of the angular momentum operator $\textbf{L}$: $$ [\textbf{L}^2, L_i] = 0. $$ However, the $L_i$ do not commute among ...
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3answers
446 views

In physics, what is the importance of distinguishing between a matrix and a group? [closed]

On the topic of Pauli matrices, I have noticed that some authors tend to use the term matrix and group interchangeably. I am asking because I do not see see any profound difference referring to the ...
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1answer
477 views

$(1/2, 0)$ representation of the Lorentz Group $SO(1,3)$

Let us consider the $(j, j') = \left(\frac{1}{2}, 0\right)$ representation of $SO(1, 3)\cong SU(2) \otimes SU(2)$. $j = \frac{1}{2}$ corresponds to $SU(2)$ generated by $$ \tag{1} N_i^+ = \frac{1}{...
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2answers
231 views

Is there any $SU(\infty)$ gauge theory in quantum field theory?

The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there ...
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4answers
103 views

Uniqueness of expression of a Lie group element

Just take the SU(2) group as an example. The three generators are $J_z$, $J_+$, and $J_-$. For an element $ g $, sometimes we want to express it as $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
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2answers
1k views

An identity of Pauli matrices

I am studying spin recently, and textbook gives some identities of Pauli matrices, one said that for any two unit vectors $\bf m$ and $\bf n$, $[\bf m \cdot \bf{\sigma},\bf {n \cdot \sigma}]= 2i(m\...
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2answers
787 views

Lorentz algebra and its generators

I'm reading Maggiore's book A Modern Introduction to Quantum Field Theory and I'm getting a bit confused when he writes about Lorentz algebra: $$K^i = J^{i0},$$ $$J^{i}=\frac{1}{2}\epsilon^{ijk}J^{...
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2answers
677 views

Lie algebra - basis for adjoint matrix products in SU(N)?

In $SU(N)$, the set of matrices in the fundamental representation plus the identity: $$ \left\{ \mathbf{1}, t^a \right\} $$ acts as a basis for generator products under matrix multiplication, such ...
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2answers
498 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: http://www.staff.science.uu.nl/~wit00103/...
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3answers
2k views

Showing a mapping between $SU(2)$ and $SO(3)$

I know this has been done on this site in a different manner but I'm wondering if it's possible to show the 2:1 Lie group homomorphism between $SU(2)$ and $SO(3)$ using exponentials of the generators ...
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1answer
998 views

What is the four-dimensional representation of the $SU(2)$ generators?

Recently, I have been learning about non-Abelian gauge field theory by myself. Thanks @ACuriousMind very much, as with his help, I have made some progress. I am trying to extend the Dirac field ...
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3answers
773 views

Quantum mechanical angular momentum and spin formalism/notation

I am currently stuck on the following notation: $\frac{1}{2}\otimes\frac{1}{2} = 0 \text{ (antisym) } \oplus 1 \text{ (sym) }$ No matter what I tried, I couldn't derive the identity. I am sure that ...
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1answer
91 views

Why isn't $SO(n)/SO(n\!-\!1)$ a symmetric space?

It's my understanding that one way to define a symmetric space $G/H$ is by the commutation relations $$ [T^a, T^b] = f^{abc} T^c, \qquad [T^a, X^{\hat{b}}] = f^{a\hat{b}\hat{c}}X^{\hat{c}}, \qquad [X^{...
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2answers
285 views

A useful identity for Gell-Mann $su(3)$ matrices?

We have the following beautiful result for Pauli $su(2)$ matrices $$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b}) = \mathbb{I} ~\vec{a}\cdot\vec{b} + i (\vec{a} \times \vec{b}) \cdot \vec{\...
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2answers
420 views

What is the relationship of Clebsch-Gordan decomposition with Young tableau?

Until recently, I had the impression that any representation $R_1 \otimes R_2$ for spins $J_1$ and $J_2$ is reducible, for example, into $(2 \min{(J_1,J_2)}+1)$ multiplets. $$ J_1 \otimes J_2 = (J_1 +...
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2answers
137 views

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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2answers
196 views

Updating link variables in lattice $SU(N)$ gauge theory

I'm currently writing a basic program in python to simulate a 1 + 1 dimensional yang mills gauge theory with symmetry group $SU(2)$. On the lattice you work with link variables, which are $SU(N)$ ...
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1answer
229 views

General analysis of internal symmetries in QFT

I am trying to understand as much as I can about internal symmetries in QFT, without using a Lagrangian or the canonical formalism (nor perturbation theory), but I am having a hard time to find good ...
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1answer
68 views

Gluon have colour-anticolour; what about weak bosons?

Gluons can be red-antiblue, or green-antired, etc. What about weak interaction bosons? (Say before symmetry breaking, to make matters simpler.) Is there a similar "weak charge" structure of charge-...
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1answer
325 views

Symmetry transformation on Quantum Field

I stumbled upon this point several times, the latest beeing this question: Connection between conserved charge and the generator of a symmetry I want to understand, why Quantum fields transform under ...
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2answers
121 views

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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2answers
434 views

Clebsch-Gordan coefficents necessary and sufficent condition to be non-zero

I know that the Clebsch-Gordan coefficient, $$\left<J_1, m_1, J_2, m_2|J,M\right>,$$ is zero if the following conditions are not satisfied: $$|J_1-J_2| \le J \le J_1+J_2,$$ $$m_1+m_2=M,$$ $$|M| \...
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1answer
238 views

Anticommutative Sets of SU(N) Generators? Anticommutative Analogue to Cartan subalgebra?

I am currently studying SU(N) generators in order to find bases that may suit a problem at hand. I am especially interested in getting as large anticommuting sets within a basis as possible. In SU(2) ...
3
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1answer
156 views

What's the significance of the difference between the quantum numbers, $\ell$ and $m_{\ell}$?

I know that $m_{\ell}$ is associated with the projection of the angular momentum vector onto the $z$ axis and $\ell$ is associated with the length of the angular momentum vector. To me this implies ...
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1answer
249 views

Real representation is physically real?

In Peskin & Schroder, Introduction to Quantum Field Theory equation (15.82) states that $$ t^a_{\bar{r}} = -(t^a_{r})^* = (t^a_{r})^T $$ Why is the representation which satisfies $$ t_{\...
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1answer
1k 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 & ...
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1answer
204 views

Mathematically, how do we deduce that angular momentum is bounded?

So, how do we know $J_{+}|j,(m=j)\rangle =|0\rangle$? I.e. that m is bounded by j. We know that $J_{+}|j,(m=j)\rangle =C|j, j+1\rangle$, but how do I know that gives zero? Is it by looking at its ...
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1answer
598 views

How do I find the tensor components of all weights of a representation of $SU(3)$, e.g. the six dimensional representation $(2,0)$?

How do I find the corresponding tensor component $v^{ij}$ of the six dimensional representation of $SU(3)$ with Dynkin label $(2,0)$?
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1answer
1k views

Weinberg's way of deriving Lie algebra related to a Lie group

I was reading the second chapter of the first volume of Weinberg's books on QFT. I am quite confused by the way he derives the Lie algebra of a connected Lie group. He starts with a connected Lie ...
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2answers
96 views

Kac-Moody algebra from WZW model via Poisson brackets

In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
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2answers
332 views

Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
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2answers
252 views

Adjoint representation in Liouville-von Neumann equation

I am having trouble understanding the adjoint representation of a Lie algebra in the scope of a very specific example, so I thought physics.SE would be the best place to ask. Background: A $N \times ...
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1answer
163 views

Building $\mathfrak{so}(1,3)$ reps using $\mathfrak{so}(1,3)\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$

I'm going through the representation theory of $\mathfrak{so}(1,3)$, building Dirac/Weyl spinors and vectors, and I'm a bit confused on the mathematical definitions involved. We have $\mathfrak{so}(1,...
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3answers
114 views

Wrong sign in Conformal Casimir

The quadratic conformal Casimir in $d$-dimensional Euclidean space is given by \begin{equation} C = \frac{1}{2}L_{\mu \nu}L^{\mu \nu} - D^2 -\frac{1}{2}\left(P^\mu K_\mu + K^\mu P_\mu \right) \end{...
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2answers
750 views

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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1answer
364 views

The adjoint representation and the gauge boson of $O(n)$

I'm learning "Gauge theory of Elementary Particle Physics Problems and Solutions" by Cheng and Li. In problem 8.4 "$O(n)$ gauge theory" on page 165, Under infinitesimal $O(n)$ representations a ...
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2answers
900 views

Angular momentum - proof for integer or half-integer eigenvalues

I am confused about a proof my Quantum Mechanics textbook has left "as an exercise for the reader". So, we've got the angular momentum operator $\hat{L}$. We've also got the generalized angular ...
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1answer
902 views

Why $SU(3)$ has eight generators?

The generators of $SU(3)$ group are Gell-Mann matrices and one can construct these generators from Pauli spin matrices, basically expanding in 3d and rotating about each axis. Take $\sigma_3$, assume ...
3
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1answer
246 views

How does the Lorentz group act on a 4-vector in the spinor-helicity formalism $p_{\alpha\dot{\alpha}}$?

Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I ...
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1answer
578 views

Commutator of SU(2) Casimir operators in su(3)

I have a $\mathfrak{su}(3)$ Lie algebra spanned by 8 generators, $$\left\{J_1,J_2,J_3,J_4,J_5,J_6,J_7,J_8 \right\}$$ Now, I can choose infinitely many $\mathfrak{su}(2)$ sub-algebras composed of ...
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1answer
639 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 $\...
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1answer
444 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 ...
3
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2answers
94 views

Why are all transformations of quantum operators inner automorphisms?

Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator $\times \frac{1}{i\hbar}$. This is then connected to the state space i.e. the Hilbert ...
3
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1answer
247 views

Levi-Civita tensor and the Lorentz group generators in the vector representation

In the vector representation of the Lorentz group its generators are given by - $$(J^{\mu\nu})_{\alpha\beta} = i(\delta^\mu_\alpha\delta^\nu_\beta-\delta^\mu_\beta\delta^\nu_\alpha)$$ It can be ...
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2answers
1k views

Commutation relations of the generators of the Lorentz group

$$ J^{\mu\nu} = i(x^\mu\partial^\nu-x^\nu\partial^\mu). \tag{3.16}$$ We will soon see that these six operators generate the three boosts and three rotations of the Lorentz group. To determine ...
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1answer
245 views

Quantization of angular momentum in $SO(3)$

When hermitian operators $L_1, L_2, L_3$ follow the commutation relations: $$ [L_1,L_2]=i\;L_3 \\ [L_2,L_3]=i\;L_1 \\ [L_3,L_1]=i\;L_2 $$ one can show that, assuming they are in finite number, their ...
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1answer
230 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 $SO(1,3)$ ...
3
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1answer
297 views

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 ...
3
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1answer
544 views

The states of the adjoint representation correspond to the generators

From section 2.4 of von Steinkirk's Introduction to Group Theory for Physicists [PDF] Defining a set of matrices $T_a$ as $$[T_a]_{bc} \equiv -if_{abc}$$ it is possible to recover (2.1.2): $$[...
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1answer
2k views

Angular momentum coupling-calculation of Clebsch–Gordan coefficients

I am facing problem in calculating the value of given Clebsch–Gordan coefficients representing the coupled angular momenta of two-particle system. For example $$\begin{pmatrix}2 & 1 & 2 \\ 1 &...