Find Generators of Lorentz Group: Jackson's List & K Matrix

In summary, the conversation discusses finding generators of the Lorentz group and the confusion surrounding the classification of a specific matrix as a covariant, contravariant, or mixed tensor. The conclusion is that the matrix is, in fact, a mixed tensor due to its role in a Lorentz transformation.
  • #1
eoghan
207
7
Hi!
I'm trying to find the generators of the Lorentz group. Jackson lists them all, for example, the generator of a boost along x is:
[tex]
K=\left( \begin{array}{c}
0\;1\;0\;0 \\
1\;0\;0\;0 \\
0\;0\;0\;0 \\
0\;0\;0\;0
\end{array} \right)
[/tex]

Now, what I don't understand is: this matrix is a covariant, contravariant, or mixed tensor? I mean, should I write
[tex]K_{\mu\nu}\:\:,\:\:K_{\mu}\;^{\nu}\:\:,\:\:K^{\mu \nu}...[/tex] or what else?
 
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  • #2
A Lorentz transformation takes a vector to a vector. A generator is the derivative of that with respect to a parameter in the transformation. Wouldn't that make it a mixed tensor? Like [itex]x'^\mu = T_\nu^\mu x^\nu[/itex].
 
Last edited:

1. What is the purpose of finding generators of the Lorentz group?

The Lorentz group is a key mathematical concept in the study of special relativity, which is essential for understanding the behavior of objects moving at high speeds. Finding the generators of this group allows us to mathematically describe and analyze the transformations that occur between different frames of reference in special relativity.

2. What is Jackson's List and how does it help in finding the generators of the Lorentz group?

Jackson's List is a systematic method for finding the generators of the Lorentz group, developed by physicist John David Jackson. It involves breaking down the Lorentz group into smaller subgroups and using a set of rules to determine the generators of these subgroups. This helps in simplifying the process of finding the generators of the Lorentz group as a whole.

3. What is the K matrix in the context of finding generators of the Lorentz group?

The K matrix, also known as the Lorentz transformation matrix, is a mathematical representation of the transformations that occur between different frames of reference in special relativity. It is a key component in the calculation of the generators of the Lorentz group, as it allows us to determine the specific form of the generators using the elements of the K matrix.

4. Can the generators of the Lorentz group be determined using other methods besides Jackson's List and the K matrix?

Yes, there are other methods for finding the generators of the Lorentz group, such as the infinitesimal transformation method and the direct calculation method. However, Jackson's List and the K matrix are commonly used because they provide a systematic and efficient approach for determining the generators.

5. How are the generators of the Lorentz group related to the physical properties of objects in special relativity?

The generators of the Lorentz group are related to the physical properties of objects in special relativity through the concept of symmetry. The generators represent the transformations that preserve the physical laws of special relativity, such as the conservation of energy and momentum. This allows us to understand the physical behavior of objects moving at high speeds and make predictions about their properties.

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