Generators for Lorentz transformations

[Wox]

Consider Minkowski spacetime with signature (-+++) and coordinates (ct,x,y,z) with respect to the standard orthogonal basis. I'm looking for the smallest set of matrices that can generate any Lorentz transformation with respect to this basis. I came up with 8 matrices (see below). Am I missing something?

Three boosts (in x, y and z direction)
$$\text{Boost}_{x}(\tilde{\alpha})=\left[\begin{matrix}<br /> \cosh\tilde{\alpha}&\sinh\tilde{\alpha}&0&0\\<br /> \sinh\tilde{\alpha}&\cosh\tilde{\alpha}&0&0\\<br /> 0&0&1&0\\<br /> 0&0&0&1<br /> \end{matrix}\right]\quad<br /> \text{Boost}_{y}(\tilde{\beta})=\left[\begin{matrix}<br /> \cosh\tilde{\beta}&0&\sinh\tilde{\beta}&0\\<br /> 0&1&0&0\\<br /> \sinh\tilde{\beta}&0&\cosh\tilde{\beta}&0\\<br /> 0&0&0&1<br /> \end{matrix}\right]\quad<br /> \text{Boost}_{z}(\tilde{\gamma})=\left[\begin{matrix}<br /> \cosh\tilde{\gamma}&0&0&\sinh\tilde{\gamma}\\<br /> 0&1&0&0\\<br /> 0&0&1&0\\<br /> \sinh\tilde{\gamma}&0&0&\cosh\tilde{\gamma}\\<br /> \end{matrix}\right]\quad$$
Three spatial rotations
$$\text{Rot}_{x}(\alpha)=\left[\begin{matrix}<br /> 1&0&0&0\\<br /> 0&1&0&0\\<br /> 0&0&\cos\alpha&\sin\alpha\\<br /> 0&0&-\sin\alpha&\cos\alpha\\<br /> \end{matrix}\right]\quad<br /> \text{Rot}_{y}(\beta)=\left[\begin{matrix}<br /> 1&0&0&0\\<br /> 0&\cos\beta&0&-\sin\beta\\<br /> 0&0&1&0\\<br /> 0&\sin\beta&0&\cos\beta\\<br /> \end{matrix}\right]\quad<br /> \text{Rot}_{z}(\gamma)=\left[\begin{matrix}<br /> 1&0&0&0\\<br /> 0&\cos\gamma&\sin\gamma&0\\<br /> 0&-\sin\gamma&\cos\gamma&0\\<br /> 0&0&0&1\\<br /> \end{matrix}\right]$$
Two inversions
$$I=\left[\begin{matrix}<br /> 1&0&0&0\\<br /> 0&-1&0&0\\<br /> 0&0&-1&0\\<br /> 0&0&0&-1<br /> \end{matrix}\right]\quad<br /> \tilde{I}=\left[\begin{matrix}<br /> -1&0&0&0\\<br /> 0&1&0&0\\<br /> 0&0&1&0\\<br /> 0&0&0&1<br /> \end{matrix}\right]$$

 

[George Jones]

What do you mean by "generator" and "generate"? I suspect that you take your meanings from the math community. For the physics community, "generator" often means "element of the Lorentz Lie algebra".

Do you really need rotations about all three axes? For example, what does a rotation about the x-axis followed by a rotation about the y-axis give?

If you have all the rotations, do you need boosts in three linearly independent directions?

 

[Wox]

Do you really need rotations about all three axes? For example, what does a rotation about the x-axis followed by a rotation about the y-axis give?

To describe an arbitrary rotation I need all three, no?

As for generators: I mean in the mathematical sense. Every Lorentz transformation can be written as a combination of these generators. I was thinking about the generators of a group, but this is different because $\text{Rot}_{x}(\alpha)$ is not one element of the Lorentz group but many elements, so I'm not sure how this is mathematically formalized.

 

[George Jones]

For example, what does a rotation about the x-axis followed by a rotation about the y-axis give?

I should have written

For example, what does a rotation of pi/2 about they y-axis followed by an arbitrary rotation about the x-axis followed by a rotation of -pi/2 about the y-axis give?

 

[Wox]

Hmmm, I see what you mean. So I can skip one rotation and two boosts. But more importantly, are there Lorentz transformations that can't be written as a combinations of the transformations given?

 

[Ben Niehoff]

Your set of transformations covers everything. However, they are not the "generators". The generators of a Lie group are the elements of the Lie algebra. The boosts and rotations you gave are exponentials of Lie algebra elements, so you're close.

 

[George Jones]

What do you mean by "generator" and "generate"? I suspect that you take your meanings from the math community. For the physics community, "generator" often means "element of the Lorentz Lie algebra".

Your set of transformations covers everything. However, they are not the "generators". The generators of a Lie group are the elements of the Lie algebra. The boosts and rotations you gave are exponentials of Lie algebra elements, so you're close.

From Wikipedia (which I think is common usage in the math community)

Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.