How would I relate $e^{i\omega_{\mu\nu}}J^{\mu\nu}$ with lorentz transformation matrix?

Question

How to go from the given exponential form to given transformation matrix?

Do I need to know the generators of boost and rotation? How will I find $\omega_{\mu\nu}$ and $J^{\mu\nu}$ in that case?

Answer 1

If $J^{\mu \nu}$ are the generators of the Lorentz transformations (so should form a basis for the Lie Algebra of the Lorentz group, in particular they should be antisymmetric) and $\omega_{\mu \nu}$ are the parameters governing the Lorentz transformations, then since the generators $J^{01}$, etc. are matrices (we have a matrix Lie algebra) the exponential map back to the Lie group will just be the matrix exponential.

Hence the procedure is to calculate the matrix $A = \omega_{\mu \nu}J^{\mu \nu}$ that governs the transformation that you're doing then find your Lorentz transformation as

$$\textrm{exp}(A) = \sum_0^{\infty} \frac{1}{n!}A^n$$

which will also be a matrix.