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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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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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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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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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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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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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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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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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) ...
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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 ...
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 $$t_{\... 1answer 1k views t_1, t_2, t_3 Hermitian generators of SU(2) What is the exact SU(2) representation to which these Hermitian generators belong? t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & ... 1answer 204 views Mathematically, how do we deduce that angular momentum is bounded? So, how do we know J_{+}|j,(m=j)\rangle =|0\rangle? I.e. that m is bounded by j. We know that J_{+}|j,(m=j)\rangle =C|j, j+1\rangle, but how do I know that gives zero? Is it by looking at its ... 1answer 598 views How do I find the tensor components of all weights of a representation of SU(3), e.g. the six dimensional representation (2,0)? How do I find the corresponding tensor component v^{ij} of the six dimensional representation of SU(3) with Dynkin label (2,0)? 1answer 1k views Weinberg's way of deriving Lie algebra related to a Lie group I was reading the second chapter of the first volume of Weinberg's books on QFT. I am quite confused by the way he derives the Lie algebra of a connected Lie group. He starts with a connected Lie ... 2answers 96 views Kac-Moody algebra from WZW model via Poisson brackets In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the G\times G symmetry of the WZW model give rise to a Kac-Moody algebra upon ... 2answers 332 views Trace of generators of Lie group In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as$$tr[T^{a}T^{b}]$$which is promptly diagonalised (for compact ... 2answers 252 views Adjoint representation in Liouville-von Neumann equation I am having trouble understanding the adjoint representation of a Lie algebra in the scope of a very specific example, so I thought physics.SE would be the best place to ask. Background: A N \times ... 1answer 163 views Building \mathfrak{so}(1,3) reps using \mathfrak{so}(1,3)\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2) I'm going through the representation theory of \mathfrak{so}(1,3), building Dirac/Weyl spinors and vectors, and I'm a bit confused on the mathematical definitions involved. We have \mathfrak{so}(1,... 3answers 114 views Wrong sign in Conformal Casimir The quadratic conformal Casimir in d-dimensional Euclidean space is given by C = \frac{1}{2}L_{\mu \nu}L^{\mu \nu} - D^2 -\frac{1}{2}\left(P^\mu K_\mu + K^\mu P_\mu \right) \end{... 2answers 750 views Lie Algebra Conventions: Hermitian vs. anti-Hermitian Consider the Lie algebra of SU(2). To find the infinitesimal generators we linearise about the identity$$U=I+i\alpha T$$where \alpha is some small parameter. To find the form of T use the ... 1answer 364 views The adjoint representation and the gauge boson of O(n) I'm learning "Gauge theory of Elementary Particle Physics Problems and Solutions" by Cheng and Li. In problem 8.4 "O(n) gauge theory" on page 165, Under infinitesimal O(n) representations a ... 2answers 900 views Angular momentum - proof for integer or half-integer eigenvalues I am confused about a proof my Quantum Mechanics textbook has left "as an exercise for the reader". So, we've got the angular momentum operator \hat{L}. We've also got the generalized angular ... 1answer 902 views Why SU(3) has eight generators? The generators of SU(3) group are Gell-Mann matrices and one can construct these generators from Pauli spin matrices, basically expanding in 3d and rotating about each axis. Take \sigma_3, assume ... 1answer 246 views How does the Lorentz group act on a 4-vector in the spinor-helicity formalism p_{\alpha\dot{\alpha}}? Given a 4-vector p^\mu the Lorentz group acts on it in the vector representation:$$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$However, I ... 1answer 578 views Commutator of SU(2) Casimir operators in su(3) I have a \mathfrak{su}(3) Lie algebra spanned by 8 generators,$$\left\{J_1,J_2,J_3,J_4,J_5,J_6,J_7,J_8 \right\}$$Now, I can choose infinitely many \mathfrak{su}(2) sub-algebras composed of ... 1answer 639 views Question on derivation of Ward identity I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field \... 1answer 444 views Casimir Invariants of the Galilean group I had studied a couple of things about Galilean and Poincare group. But in the Galilean group, there is not enough clarity on how to calculate generators for boosts (B_i), which if I do it seems I ... 2answers 94 views Why are all transformations of quantum operators inner automorphisms? Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator \times \frac{1}{i\hbar}. This is then connected to the state space i.e. the Hilbert ... 1answer 247 views Levi-Civita tensor and the Lorentz group generators in the vector representation In the vector representation of the Lorentz group its generators are given by -$$(J^{\mu\nu})_{\alpha\beta} = i(\delta^\mu_\alpha\delta^\nu_\beta-\delta^\mu_\beta\delta^\nu_\alpha)$$It can be ... 2answers 1k views Commutation relations of the generators of the Lorentz group$$ J^{\mu\nu} = i(x^\mu\partial^\nu-x^\nu\partial^\mu). \tag{3.16}$$We will soon see that these six operators generate the three boosts and three rotations of the Lorentz group. To determine ... 1answer 245 views Quantization of angular momentum in SO(3) When hermitian operators L_1, L_2, L_3 follow the commutation relations:$$ [L_1,L_2]=i\;L_3 \\ [L_2,L_3]=i\;L_1 \\ [L_3,L_1]=i\;L_2 $$one can show that, assuming they are in finite number, their ... 1answer 230 views Why the Lorentz group has complex generators in QFT treatments? [duplicate] In Schwartz' and Peskin's QFT books, when trying to deal with representations of the Lorentz group the authors study the representations of the Lie algebra of such group. By definition, if SO(1,3) ... 1answer 297 views Hypercharge and Isospin as additive quantum numbers in SU(3) flavour symmetry I am studying the SU(3) flavour symmetry and I'm reading that we use the fact that hypercharge and isospin are additive quantum numbers in order to decompose the tensor products of the fundamental ... 1answer 544 views The states of the adjoint representation correspond to the generators From section 2.4 of von Steinkirk's Introduction to Group Theory for Physicists [PDF] Defining a set of matrices T_a as$$[T_a]_{bc} \equiv -if_{abc}$$it is possible to recover (2.1.2):$$[...
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 &...