# 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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### What is “broken symmetry”?

For reference, I come from a mathematics background (mostly differential geometry). I have a very limited understanding of upper-level physics (I'm currently trying to fix this). This is my current ...
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### Where in fundamental physics are Lie groups actually important (and not just Lie algebras)?

I was wondering where in fundamental physics the global structure of a Lie group actually makes a difference. Most of the time physicists are sloppy and don't distinguish groups and algebras ...
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### How is the invariant speed of light encoded in $SL(2, \mathbb C)$?

In quantum field theory, we use the universal cover of the Lorentz group: $SL(2, \mathbb C)$, instead of $SO(3,1)$. The reason for this is, of course, that $SO(3,1)$ representations aren't able to ...
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### Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$?

The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \...
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### Explicit calculation of $J_2$ induced rotation by $-\pi$ reversing the sign of $J_3$ operator

Let $\textbf{J}=(J_1,J_2,J_3)$ be the angular momentum QM operators in the x, y, z spacial directions respectively. These are also the generators of rotations around the x, y, z axes respectively. ...
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### 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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### Generators of $SU(2)$ and $SU(3)$ Symmetry Groups

I've been reading about gauge field theories, and I keep coming across the generators of the $SU(3)$ and $SU(2)$ Symmetry Groups. I read that each generator corresponds to a gauge boson, but I'm ...
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### Is the adjoint representation of $SU(2)$ the same as the triplet representation?

Is the triplet representation of $SU(2)$ the same as its adjoint representation? Where the convention for the adjoint representation used is the one used in particle physics, where the structure ...
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### How do gauge fields transform with an extra inhomogeneous term even though they are Lie Algebra valued in Non-Abelian gauge theories?

I am trying to work out Non-Abelian gauge theories but I couldn't get my head around the fact that gauge fields transform with an extra inhomogeneous term under the adjoint action of a group $G$, that ...
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### Lorentz group $SO(3,1)$ and $SU(2)\times SU(2)$

One way to classify Lorentz representations is to consider the Lie algebra isomorphic of Lorentz group to $SU(2)\times SU(2)$. So that we can classify it by two integers $(j_1,j_2)$. In this way I can ...
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### Generalized Pauli matrix for spin larger than 1/2 [duplicate]

for Spin 1/2, we have Pauli matrix as in wiki. So what's the generalized 3-by-3 Pauli matrix for spin 1 or even larger spin? Is there a generalization method?
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### Degree of Freedom of an $SU(n)$ Group

I've been thinking about the DOF of the $SU(n)$ group. I first consider the DOF of a unitary matrix. See if I get this right: Any unitary matrix can be written in the form of $e^{iH}$, where $H$ is a ...
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### Lie Algebra for fermion fields

A key identity (e.g. when deriving BRST symmetry for gauge fields) is that: $$[c,d]_a =f_{abc}c_b d_c$$ where $c$ and $d$ are both Fermion Fields. How do I derive this from the lie algebra ...
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### Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

The projective unitary representations of a multiply-connected group $G$ is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of ...
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### Generators of $SU(2)\times U(1)$

I'm currently reading about spontanous symmetry breaking. In particular about a Lagrangian that is invariant under $SU(2)\times U(1)$, in other words pretty standard QFT stuff. I know that the ...
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### How to construct a space which is translation invariant but not rotation invariant

I am just confused by the following idea. Consider a 3-dimensional translation invariant space, we now have 3 translation generators. Then let us start with a point, the full 3-dimensional space ...
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### 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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