The Poincare algebra is defined as \begin{align*} i[J^{\mu\nu},J^{\rho\sigma}]&=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}+\eta^{\sigma\nu}J^{\rho\mu}-\eta^{\sigma\mu}J^{\rho\nu}\\ i[P^{\mu},J^{\rho\sigma}]&=\eta^{\mu\rho}P^{\sigma}-\eta^{\mu\sigma}P^{\rho}\\ [P^{\mu},P^{\rho}]&=0 \end{align*} where $\eta^{\mu\nu}$ is the Minkowski metric, the $J^{0i}$ can be identified with boosts in the direction $i$, and the $J^{ij}$ are related to the rotation operator about the axis $k$ by $J^{ij}=\epsilon^{ijk}R^k$. Physicists give very special preference to the generator $P^0$. In particular the $S$-matrix $\hat{S}=T\big(exp(i\hat{P^0})\big)$ is one of the main objects of study in quantum field theory and obviously has important relevance to the physical world. But, by looking at the algebra alone, there does not appear to be any mathematical reason to single out this element.

Question: Would a mathematician who studied this algebra find any reason to single out $\hat{P^0}$?

  • $\begingroup$ Well, the Hamiltonian formulation with some choice of an evolution parameter breaks manifest Lorentz symmetry. $\endgroup$ – Qmechanic Apr 19 '18 at 13:21
  • 2
    $\begingroup$ Well, the Minkowski metric clearly singles out time as a special coordinate. A mathematician who ran into this algebra in the wild might be interested in knowing what's up with that, and they could discover the light cone structure and realize that the Hamiltonian generates translations inside the light cone. That's one possibility, though the most obvious one. $\endgroup$ – Javier Apr 19 '18 at 13:28

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