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- TL;DR
- The relationship between the generators of translation and the representation of the Poincare group
I just want to make sure I understand this correctly.
For an infinite-dimensional representation, the generators of translation can be written as ##i \frac{\partial}{\partial_{\mu}}= i \partial_{\mu}##, where the generators of the Lorentz group can be written as ##i (x^{\mu}\partial_{\nu} - x^{\nu}\partial_{\mu})##.
However, for a finite-dimensional case, where the generators of the Lorentz group are formulated as boost and rotation matrices, the generators of translation can be written as matrices with elements ##i \delta x## in the appropriate row/column locations, where ##\delta x## denotes an infinitesimal translation in some basis of ##x##.
In other words, in finite-dimensional representation, the generators of translation are infinitesimal displacements (instead of partial derivates as in the infinite-dimensional representation).
For an infinite-dimensional representation, the generators of translation can be written as ##i \frac{\partial}{\partial_{\mu}}= i \partial_{\mu}##, where the generators of the Lorentz group can be written as ##i (x^{\mu}\partial_{\nu} - x^{\nu}\partial_{\mu})##.
However, for a finite-dimensional case, where the generators of the Lorentz group are formulated as boost and rotation matrices, the generators of translation can be written as matrices with elements ##i \delta x## in the appropriate row/column locations, where ##\delta x## denotes an infinitesimal translation in some basis of ##x##.
In other words, in finite-dimensional representation, the generators of translation are infinitesimal displacements (instead of partial derivates as in the infinite-dimensional representation).