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I have a problem, I have a bi-vector that define like:

$\omega^{\mu \nu}=a^{\mu}b^{\nu}-a^{\nu}b^{\mu}$

where,

$a^{\mu}=(a^0,a^1,a^2,a^3)$ and $b^{\nu}=(b^0,b^1,b^2,b^3)$

I need show that $\Lambda^{\mu}{}_{\nu}=\exp(\alpha \omega^{\mu}{}_{\nu})$ and the explicit form like a matrix.

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Comment to the question (v2): External Lorentz-indices inside the exponential function

$$\Lambda^{\mu}{}_{\nu}~=~\exp(\alpha \omega^{\mu}{}_{\nu})\qquad\qquad (\leftarrow \text{wrong})$$

does not make sense. Instead the indices should be outside the exponential function

$$\Lambda^{\mu}{}_{\nu}~=~\exp(\alpha \omega)^{\mu}{}_{\nu}~=~\delta^{\mu}_{\nu}+\alpha \omega^{\mu}{}_{\nu}+ \frac{\alpha^2}{2}\omega^{\mu}{}_{\lambda}\omega^{\lambda}{}_{\nu}+{\cal O}(\alpha^3). $$

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  • $\begingroup$ To add to Qmechanic's answer, see Prof. Tong's notes on quantum field theory, specifically the section on the spinor representation. $\endgroup$ – JamalS May 31 '14 at 12:59
  • $\begingroup$ In the exercise the indices are inside the exponential function, I was looking the Tong's notes but isn't clear. Th exercise appear in Relativity Electrodynamics. $\endgroup$ – user47619 May 31 '14 at 20:36
  • $\begingroup$ @user47619: Which link and which page? $\endgroup$ – Qmechanic May 31 '14 at 20:56
  • $\begingroup$ damtp.cam.ac.uk/user/tong/qft/four.pdf the page 81 that named JamalS. This exercise is from my School. $\endgroup$ – user47619 May 31 '14 at 22:06

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