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.
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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. – Ajay Shanmuga Sakthivasan Jul 28 '22 at 13:21
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2Can 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? – ACuriousMind Jul 28 '22 at 13:51
1 Answers
You are writing the same thing in different ways.
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.
If you'd like to learn more about this, I particularly recommend the book by Brian Hall (Lie Groups, Lie Algebras, and Representations) for a more mathy treatment, and the book by Anthony Zee (Group Theory in a Nutshell for Physicists) for a more physical approach. There are tons of other books in Lie groups and algebras with different sorts of treatments (see, e.g., this post for a more comprehensive list), but these two are among my personal favorites.
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