homework and exercises - Inonu-Wigner Group Contraction

I am trying to understand how one obtains the Galilean algebra from the Poincare algebra, specifically through the method of central extension. I'm doing this by imposing that the generators of the Poincare group scale with the velocity in certain ways, and then taking the small velocity limit. Also I was told I should make the definition $$H = M+W,\tag{1}$$ where $H$ is the Hamiltonian, and $M$ will turn out to be a central element of the algebra. Now I have been able to fine many of the commutation relations and they make sense (i.e. match the Galilean algebra) but I don't really understand how to show $M$ commutes with everything. Also, is the physical interpretation that $M$ is the potential energy and $W$ is the kinetic energy?

Answer

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....