Galilei-covariant tensor formulation

A five dimensional Poincaré algebra leaves the metric ${\displaystyle g^{\mu \nu }}$ invariant,

${\displaystyle {\begin{aligned}[][P_{\mu },P_{\nu }]&=0,\\{\frac {1}{i}}~[M_{\mu \nu },P_{\rho }]&=g_{\mu \rho }P_{\nu }-g_{\nu \rho }P_{\mu },\\{\frac {1}{i}}~[M_{\mu \nu },M_{\rho \sigma }]&=g_{\mu \rho }M_{\nu \sigma }-g_{\mu \sigma }M_{\nu \rho }-g_{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho },\end{aligned}}}$

We can write the generators as

${\displaystyle {\begin{aligned}J_{i}&={\frac {1}{2}}\epsilon _{ijk}M_{jk},\\K_{i}&=M_{5i},\\C_{i}&=M_{4i},\\D&=M_{54}.\end{aligned}}}$

The non-vanishing commutation relations will then be rewritten as

${\displaystyle {\begin{aligned}\left[J_{i},J_{j}\right]&=i\epsilon _{ijk}J_{k},\\\left[J_{i},C_{j}\right]&=i\epsilon _{ijk}C_{k},\\\left[D,K_{i}\right]&=iK_{i},\\\left[P_{4},D\right]&=iP_{4},\\\left[P_{i},K_{j}\right]&=i\delta _{ij}P_{5},\\\left[P_{4},K_{i}\right]&=iP_{i},\\\left[P_{5},D\right]&=-iP_{5},\\[4pt]\left[J_{i},K_{j}\right]&=i\epsilon _{ijk}K_{k},\\\left[K_{i},C_{j}\right]&=i\delta _{ij}D+i\epsilon _{ijk}J_{k},\\\left[C_{i},D\right]&=iC_{i},\\\left[J_{i},P_{j}\right]&=i\epsilon _{ijk}P_{k},\\\left[P_{i},C_{j}\right]&=i\delta _{ij}P_{4},\\\left[P_{5},C_{i}\right]&=iP_{i}.\end{aligned}}}$

An important Lie subalgebra is

${\displaystyle {\begin{aligned}[][P_{4},P_{i}]&=0\\[][P_{i},P_{j}]&=0\\[][J_{i},P_{4}]&=0\\[][K_{i},K_{j}]&=0\\\left[J_{i},J_{j}\right]&=i\epsilon _{ijk}J_{k},\\\left[J_{i},P_{j}\right]&=i\epsilon _{ijk}P_{k},\\\left[J_{i},K_{j}\right]&=i\epsilon _{ijk}K_{k},\\\left[P_{4},K_{i}\right]&=iP_{i},\\\left[P_{i},K_{j}\right]&=i\delta _{ij}P_{5},\end{aligned}}}$

${\displaystyle P_{4}}$ is the generator of time translations (Hamiltonian), P_i is the generator of spatial translations (momentum operator), ${\displaystyle K_{i}}$ is the generator of Galilean boosts, and ${\displaystyle J_{i}}$ stands for a generator of rotations (angular momentum operator). The generator ${\displaystyle P_{5}}$ is a Casimir invariant and ${\displaystyle P^{2}-2P_{4}P_{5}}$ is an additional Casimir invariant. This algebra is isomorphic to the extended Galilean Algebra in (3+1) dimensions with ${\displaystyle P_{5}=-M}$, The central charge, interpreted as mass, and ${\displaystyle P_{4}=-H}$.

The third Casimir invariant is given by ${\displaystyle W_{\mu \,5}W^{\mu }{}_{5}}$, where ${\displaystyle W_{\mu \nu }=\epsilon _{\mu \alpha \beta \rho \nu }P^{\alpha }M^{\beta \rho }}$ is a 5-dimensional analog of the Pauli–Lubanski pseudovector.[4]