Deriving the commutation relations of the Lie algebra of Lorentz group

In summary, the Lorentz group has two subgroups for rotations and boosts, and the commutation relation [J_m, J_n] (and [J_m, K_n]) is being derived using the Levi-Cevita symbol and its Kronecker delta dependence. The Minkowski metric is mostly minus and the generators are divided into three spatial ones and three temporal ones. The first step involves replacing ##g_{\alpha \gamma}## with ##- \delta_{ac}##. The next step involves simplifying identities with Kronecker deltas and Levi-Civitas. The commutator can be calculated by replacing ##g## with Kronecker deltas and using the defining commutation amongs ##\mathcal
  • #1
bella987
2
0
Homework Statement
Find commutation relations for Lorentz group
Relevant Equations
See below.
This is the defining generator of the Lorentz group
1_Q.png

which is then divided into subgroups for rotations and boosts
2_Q.png

And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps:
Skjermbilde 2023-04-09 130023.png

especially between here and the following step
4_Q.png

Could someone explain to someone just getting familiar with the Levi-Cevita symbol and its Kronecker delta dependence, what exactly is going on from step to step here? I would be so grateful!!
 
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  • #2
Indices with roman letters are spatial right, and greek time + spatial?
What definition of minkowski metric you have? Diag(+,-,-,-) or Diag(-,+,+,+)?
 
  • #3
Hmm, the greek letters are definitely time + spatial, but in the second included equation we divide the 6 generators into three spatial ones (rotations) and three temporal ones (boosts), so the roman ones are used when seperating them like that.
And it's mostly minus metric.
 
  • #4
Ok, great.
Well that is the first step. ##g_{\alpha \gamma}## becomes ##- \delta_{ac}##.
1681042905024.png


Do you know anything about the ##\mathcal{L}## i.e. are they symmetric or anti-symmetric? This will help you simplify identities with kroneckers and levi-civitas.

Use
1681042861128.png

and calculate the commutator
1681042891053.png

show first that this equals
1681042925729.png

just by replacing ##g## with kroeckers and using the defining commutation amongs ##\mathcal{L}##.

As per the forum rules, you must show your attempt at a solution. "I do not understand" is not enough.
 

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FAQ: Deriving the commutation relations of the Lie algebra of Lorentz group

What is the Lie algebra of the Lorentz group?

The Lie algebra of the Lorentz group consists of the infinitesimal generators of the Lorentz transformations. These generators correspond to rotations and boosts in spacetime and satisfy specific commutation relations. The algebra is typically denoted by \(\mathfrak{so}(3,1)\) or \(\mathfrak{sl}(2,\mathbb{C})\).

What are the generators of the Lorentz group and how are they represented?

The generators of the Lorentz group include three rotation generators \(J_i\) (for \(i = 1, 2, 3\)) and three boost generators \(K_i\). These generators can be represented as \(4 \times 4\) matrices acting on the spacetime coordinates. The rotation generators \(J_i\) correspond to rotations in the 3D spatial subspace, while the boost generators \(K_i\) correspond to boosts (transformations mixing space and time coordinates).

What are the fundamental commutation relations for the generators of the Lorentz group?

The fundamental commutation relations for the generators of the Lorentz group are given by:\[ [J_i, J_j] = i \epsilon_{ijk} J_k \]\[ [K_i, K_j] = -i \epsilon_{ijk} J_k \]\[ [J_i, K_j] = i \epsilon_{ijk} K_k \]Here, \(\epsilon_{ijk}\) is the Levi-Civita symbol, which is antisymmetric in its indices.

How do you derive the commutation relations of the Lorentz group's Lie algebra?

To derive the commutation relations, one typically starts from the definition of the generators as differential operators or as matrices. By computing the commutators of these generators, one can use the properties of the Levi-Civita symbol and the structure of the Lorentz group to arrive at the commutation relations. The derivation involves using the properties of the Minkowski metric and the antisymmetric nature of the generators.

Why are the commutation relations of the Lorentz group important?

The commutation relations of the Lorentz group are crucial because they define the algebraic structure of the group, which is essential for understanding the symmetries of spacetime in the theory of relativity. These relations are fundamental in the formulation of quantum field theory and the Standard Model of particle physics, as they dictate how fields and particles transform under Lorentz transformations.

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