Inonu-Wigner Group Contraction

In physical, non-covariant language, (WP conventions), the Poincaré algebra presents as $$ [P_0,P_i]=0,\\ [P_i,P_j]=0, \\ [J_i,P_0] = 0 ~, \\ [K_i,P_k] =- i \delta_{ik} P_0 ~,\\ [K_i,P_0] = -i P_i ~,\\ [K_m,K_n] = -i \epsilon_{mnk} J_k ~, \\ [J_m,P_n] = i \epsilon_{mnk} P_k ~,\\ [J_m,K_n] = i \epsilon_{mnk} K_k ~, \\ [J_m,J_n] = i \epsilon_{mnk} J_k ~, $$ where one of the Casimir invariants is $P_\mu P^\mu= P_0^2-\vec{P}^2$.

Now redefine the boosts and $P^0$ up and down by the speed of light $c$, so $K^i\equiv c C^i$ and $P_0\equiv \frac {1}{c} E$.

The Wigner-İnönü contraction $c\to\infty$ (slowness!) results in the naive Galilean Lie algebra G(3), $$[E,P_i]=0, \\ [P_i,P_j]=0, \\ [L_i,E]=0 , \\ [C_i,P_j]= 0 ~,\\ [C_i,E]=i P_i , \\ [C_i,C_j]=0, \\ [L_{m},L_{n}]=i \epsilon _{ink} L_{k} ,\\ [L_{m},P_k]=i \epsilon _{mkj}P_j , \\ [L_{m},C_k]=i \epsilon _{mkj}C_j . $$

In effect, the boost has lost its time-translation piece and is but space translations proportional to the time, Galilean boosts, $C^i$; and the timelike momentum is a plain time-translation oblivious of c, namely a "hamiltonian", E. The spacelike $P_i$ are generators of translations as before (momentum operators), and $L^i$ stand for generators of space rotations, having merely changed name from J, to banish any inapposite thoughts of spin.

Observe how this limit has trivialized several right-hand sides to 0. In fact the 10D regular representation (matrix of structure constants, Gilmore p 220) is a very sparse matrix, indeed. It amounts to extreme structural violence.

Note the quadratic invariants $P^iP^i$ and $L^iL^i$.

The Bargmann central extension algebra is the above, but now with $[C_i,P_j]=iM\delta_{ij} $ instead of the above trivial relation ($E/c^2\to M$ as $c\to \infty$), where the central charge M is an invariant, as the name implies, easy to check consistency of. The quadratic momentum invariant now morphs into a new invariant, $ME-P^2/2$, the mass-shell invariant, and since M is invariant, $E-\frac{P^2}{2M}$ is invariant as well, the potential energy.

E is like the Hamiltonian, but it is not an algebra invariant, as it fails to commute with the Galilean boosts. It is merely a time invariant, i.e. it commutes with itself--pfffft....

  1. Cosmas Zachos has already given a correct answer. The main point is that

    The natural non-relativistic Lie algebra in Newtonian mechanics is the Bargmann algebra, not the Galilean algebra!

  2. The Bargmann algebra is an Inonu-Wigner contraction of $$iso(n\!-\!1,1)\oplus u(1).\tag{A} $$
    Here $iso(n\!-\!1,1)$ is the Poincare algebra in $n$ spacetime dimensions, generated by $J^{\mu\nu}$ and $p^{\mu}$; while $u(1)$ is an algebra generated by the mass generator $m$, which belongs to the center.

  3. Concerning OP's eq. (1), the Hamiltonian $$H~=~p^0c-mc^2~=~\sqrt{{\bf p}^2c^2+m^2c^4}-mc^2\quad\longrightarrow\quad\frac{{\bf p}^2}{2m}\quad\text{for}\quad c\to \infty \tag{B}$$ is the kinetic energy, i.e. the energy minus the rest energy.

  4. For further details, see e.g. my Phys.SE answer here.