The
Poincaré algebra is the
Lie algebra of the Poincaré group. It is a
Lie algebra extension of the Lie algebra of the Lorentz group. More specifically, the proper (\det\Lambda = 1),
orthochronous ({\Lambda^0}_0 \geq 1) part of the Lorentz subgroup (its
identity component), \mathrm{SO}(1, 3)_+^\uparrow, is connected to the identity and is thus provided by the
exponentiation \exp\left(ia_\mu P^\mu\right)\exp\left(\frac{i}{2}\omega_{\mu\nu} M^{\mu\nu}\right) of this
Lie algebra. In component form, the Poincaré algebra is given by the commutation relations: {{Equation box 1 |cellpadding=6 |border |border colour=#0073CF |bgcolor=#F9FFF7 |indent=: | equation=\begin{align}[] [P_\mu, P_\nu] &= 0\, \\ \frac{1}{i}~[M_{\mu\nu}, P_\rho] &= \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu\, \\ \frac{1}{i}~[M_{\mu\nu}, M_{\rho\sigma}] &= \eta_{\mu\rho} M_{\nu\sigma} - \eta_{\mu\sigma} M_{\nu\rho} - \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\nu\sigma} M_{\mu\rho}\, , \end{align} }} where P is the
generator of translations, M is the generator of Lorentz transformations, and \eta is the (+,-,-,-) Minkowski metric (see
Sign convention). The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, J_i = \frac{1}{2}\epsilon_{imn} M^{mn}, and boosts, K_i = M_{i0}. In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as : \begin{align}[] [J_m, P_n] &= i \epsilon_{mnk} P_k ~, \\[] [J_i, P_0] &= 0 ~, \\[] [K_i, P_k] &= i \eta_{ik} P_0 ~, \\[] [K_i, P_0] &= -i P_i ~, \\[] [J_m, J_n] &= i \epsilon_{mnk} J_k ~, \\[] [J_m, K_n] &= i \epsilon_{mnk} K_k ~, \\[] [K_m, K_n] &= -i \epsilon_{mnk} J_k ~, \end{align} where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification [J_m + iK_m,\, J_n -iK_n] = 0 permits reduction of the Lorentz subalgebra to \mathfrak{su}(2) \oplus \mathfrak{su}(2) and efficient treatment of its associated
representations. In terms of the physical parameters, we have : \begin{align} \left[\mathcal H, p_i\right] &= 0 \\ \left[\mathcal H, L_i\right] &= 0 \\ \left[\mathcal H, K_i\right] &= i\hbar cp_i \\ \left[p_i, p_j\right] &= 0 \\ \left[p_i, L_j\right] &= i\hbar\epsilon_{ijk}p_k \\ \left[p_i, K_j\right] &= \frac{i\hbar}c\mathcal H\delta_{ij} \\ \left[L_i, L_j\right] &= i\hbar\epsilon_{ijk}L_k \\ \left[L_i, K_j\right] &= i\hbar\epsilon_{ijk}K_k \\ \left[K_i, K_j\right] &= -i\hbar\epsilon_{ijk}L_k \end{align} The
Casimir invariants of this algebra are P_\mu P^\mu and W_\mu W^\mu where W_\mu is the
Pauli–Lubanski pseudovector; they serve as labels for the representations of the group. The Poincaré group is the full symmetry group of any
relativistic field theory. As a result, all
elementary particles fall in
representations of this group. These are usually specified by the
four-momentum squared of each particle (i.e. its mass squared) and the intrinsic
quantum numbers J^{PC}, where J is the
spin quantum number, P is the
parity and C is the
charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many
quantum field theories; where this occurs, P and C are forfeited. Since
CPT symmetry is
invariant in quantum field theory, a
time-reversal quantum number may be constructed from those given. As a
topological space, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time-reversed and spatially inverted. == Other dimensions ==