# 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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### 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 ...
230 views

### Charge of a field under the action of a group

What does it mean for a field (say, $\phi$) to have a charge (say, $Q$) under the action of a group (say, $U(1)$)?
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### 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: $$Q = I_3 + \frac{Y}{2}$$ where $Q$ is the electric charge of the quarks, $I_3$ is the ...
350 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 ...
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### Why is supersymmetry a continuous symmetry?

Supersymmetry feels like a discrete symmetry to me, since the fermions are turning into bosons, and vice versa. I understand there is an infinitesimal parameter involved in the transformations, but I ...
998 views

### Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
230 views

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### Is there an anticommutator relation for orbital angular momentum?

So I know that there are commutator relations for $L$ such as $[L_x,L_y] = i\hbar L_z$, but is there a relation for the anticommutator? For example, $L_xL_y + L_yL_x$?
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### Euler-Arnold equation

In https://arxiv.org/abs/1905.05765, I came across the Euler-Arnold equation which is equation (2.8) reproduced here for convenience: G_{ab}\frac{dv^b}{ds}-f_{ca}\,^d\,G_{de}\,v^cv^e=...
In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the central extension of the Witt algebra. They give the central extension $$\... 2answers 165 views ### Isometry group on a coset manifold In ''Einstein Gravity in a Nutshell'' Zee says ''On a coset manifold G/H, the isometry group is evidently just G'' when discussing the relation between the Killing vector fields and Lie ... 1answer 110 views ### \mathrm{SU}(2) as a representation of the rotation group I have read in a book that the group \mathrm{SU}(2) is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators J_{1}, J_{2} and ... 2answers 651 views ### Why are 3 colors used in QCD? The mapping of strong charge to RGB left me believing that there are only 3 conserved quantities in QCD. I recently came to the understanding that there are in fact 8 conserved quantities, as ... 1answer 62 views ### How do you subtract colors and divide them by irrational numbers? (Gluons) [closed] There is a gluon that is$$\frac{1}{\sqrt{3}} (red \cdot\overline{red} + blue\cdot\overline{blue} - 2\cdot green \cdot\overline{green})$$This confuses me because I do not understand how adding and ... 1answer 957 views ### Invariant tensors in a general representation and their physical meaning I'm trying to use tensor methods to find invariant elements of representations. Specifically I'm looking at representations of SU(5). I can show that the invariant element in 5\otimes\bar{5} (or ... 0answers 157 views ### What Lie supergroup does the super-Poincare algebra generate? Every Lie supergroup has an associated Lie superalgebra of generators (in general, some of which are bosonic and some fermionic). Which Lie supergroup(s) are generated by the Super-Poincare algebra ... 1answer 968 views ### Commutator of Gauge Covariant derivatives What is the physical meaning of$$ [D_{\mu}, D_{\nu}] ~\propto~ F_{\mu, \nu}, $$where D_{\mu} is the gauge covariant derivative and F_{\mu,\nu} is the field strength? Is it just a definition? ... 2answers 145 views ### Heuristic derivation of W^\mu=\frac{1}{2}\epsilon^{\mu\nu\sigma\rho}P_\nu J_{\sigma\rho} using combination of physical and mathematical arguments If a simple systematic way to derive or guess (either mathematically or by a combination of physical arguments and mathematics) that one of the Casimir operator of Poincare group is W^2\equiv W_\mu W^... 3answers 231 views ### Why do we get fake half-spin values for orbital angular momentum if we solve it algebraically? It is a well-known fact that the values for the square of the orbital angular momentum of a particle L^2 and it's projection in the z-direction L_z are m\hbar and l(l+1)\hbar and that l ... 1answer 47 views ### Hermitian operators in the expansion of symmetry operators in Weinberg's QFT This is related to Taylor series for unitary operator in Weinberg and Weinberg derivation of Lie Algebra. \textbf{The first question} On page 54 of Weinberg's QFT I, he says that an element T(\... 2answers 347 views ### Multiplicity of eigenvalues of angular momentum Reading Dirac's Principles of Quantum Mechanics, I encounter in § 36 (Properties of angular momentum) this fragment: This is for a dynamical system with two angular momenta \mathbf{m}_1 and \... 1answer 537 views ### Anticommutator of spin-1 matrices We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is \{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I. Does a similar relation hold for the spin-1 representation? 1answer 308 views ### Symmetric tensor product decomposition of su(2) Taking the tensor product of two spin 1 representations of su(2) yields$$1 \otimes 1 = 0 \oplus 1 \oplus 2.$$What changes if instead we take the symmetric tensor product 1 \odot 1 of these ... 2answers 157 views ### Half Witt algebra I have the following Lie algebra which is generated by \{L_n|n\geq 0\}. It satisfies the following commutation rule$$ \Big[ L_i ,L_j \Big]=\frac18 \frac{(2i+2j-1)(2j-2i)}{(2j+1)(2i+1)}L_{i+j-1}-\...
The eigenvalue equation of the $L^2$ operator is given by $$L^2f_l^m = \hbar ^2l(l+1)f_l^m$$ Side: So a determinate state for some observable $Q$ is a state where every measurement of $Q$ returns ...