# Which is the correct way to write a Lorentz group component in exponential form?

I have this issue with the Lorentz transformations. I learned at some lectures that any Lorentz group component can be written as: $$\Lambda = e^{\frac{1}{2}w_{ij}M^{ij}}$$ where the different components of $$M$$ indicate boosts or rotations in the Minkowski space-time. However, this brings me some problems. For example, if someone wanted to perform two rotations, it is not the same to do first a rotation around the $$x$$-axis and then a rotation around the $$y$$-axis as the rotation matrices of $$SO(3)$$ don't commute. The same happens with performing a boost and then a rotation. So, wouldn't Lorentz transformations have the form: $$\Lambda=e^Ae^Be^C...$$ which contains the order of the transformations. Both formulations are not the same as you can not separate the exponents in the first formula.

• While it is true that SO(3) is non-Abelian, it still is a group. You can always find w-ij's corresponding to exp(A)exp(B) and exp(B)exp(A), although they will be different. Jul 28, 2022 at 13:21
• Can you be more explicit as to why non-commutativity would be a problem here? The sum $w_{ij}M^{ij}$ is just a matrix and the exponential of matrices is well-defined, so what's the issue? Jul 28, 2022 at 13:51

You can indeed write a general Lorentz transformation in the form of $$\Lambda = e^A e^B \cdots,$$ where each term corresponds to a "simple" transformation (i.e., a rotation around $$x$$, or a boost along $$y$$, etc). However, you can also collect all of the exponentials into a single one. There is an expression known as the Baker–Campbell–Hausdorff formula, which allows one to compute $$Z$$ in the expression $$e^X e^Y = e^Z,$$ where $$X$$ and $$Y$$ are matrices. More explicitly, as given in the Wikipedia page I linked, one has $$Z = X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \cdots,$$ and hence the non-commutative aspects of the matrices is taken into account.
It might be interesting to mention that the idea of expressing elements of a group in the form $$e^X$$ is way more general than just for the Lorentz group. It is a construction that applies to Lie groups in general (I'm not sure if all of them or whether there is some extra condition, but it definitely applies to, e.g., $$SO(N)$$). The space of the matrices being exponentiated (i.e., the space of the $$X$$'s that go into $$e^X$$) is known as the Lie algebra associated to the group.