Contractions of the Euclidean Group ISO(3) = E(3)

[Azure Ace]

Homework Statement

Consider the contractions of the 3D Euclidean symmetry while preserving the SO(2) subgroup. In the physics point of view, explain the resulting symmetries G(2) (Galilean symmetry group) and H(3) (Heisenberg-Weyl group for quantum mechanics) and give their Lie algebras explicitly.

Homework Equations

$$ISO(3): J_{x], J_{y}, J_{z}, P_{x}, P_{y}, P_{z}$$ $$[P_{i}, P_{j}] = 0, [J_{i}, J_{j}] = i\epsilon_{ijk} J_{k}, [P_{i}, J_{j}] = i\epsilon_{ijk} P_{k}$$
$$SO(2): J_{z}$$
$$G(2): K_{x], K_{y}, J_{z}, P_{x}, P_{y}, H$$ $$[P_{i}, P_{j}] = [K_{i}, K_{j}] = [P_{i}, H] = [J_{i}, H] = [K_{i}, P_{j}] = 0, [K_{i}, H] = iP_{i}$$ $$[J_{i}, J_{j}] = i\epsilon_{ijk} J_{k}, [P_{i}, J_{j}] = i\epsilon_{ijk} P_{k}, [J_{i}, K_{j}] = i\epsilon_{ijk} K_{k}$$
$$H(2): X, Y, J_{z}, P_{x}, P_{y}, I$$ $$[X_{i}, P_{j}] = i\hbar\delta_{ij} I, [P_{i}, I] = [X_{i}, I] = 0$$

The Attempt at a Solution

For ISO(3) cotract to G(2), I tried $K_{x} = \frac{J_{x}}{c}, K_{y} = \frac{J_{y}}{c}, H = cP_{z}$ and was able to get most of the commutation relations for G(2) except that I get something which doesn't match ($[K_{x}, H] = -iP_{y}, [K_{y}, H] = iP_{x}$.

 

[mighty2000]

I also don't see how to get the Heisenberg algebra from this contraction.First, let's clarify the notation used in the problem. The square brackets in the subscripts of the generators indicate antisymmetrization, so $J_{x]$ means $J_x$ and $J_y$, and similarly for $K_{x]$ and $K_{y]$. Also, $i,j,k$ are indices running from 1 to 3, not complex numbers.

To contract ISO(3) to G(2), we can use the following scaling: $K_x = \frac{J_x}{\epsilon}$, $K_y = \frac{J_y}{\epsilon}$, $H = cP_z$, where $\epsilon$ is a small parameter and $c$ is the speed of light. This gives us the following commutation relations:
$$[P_i,P_j]=0, [K_i,K_j]=0, [P_i,H]=0, [K_i,H]=iP_i, [J_i,J_j]=i\epsilon_{ijk}J_k, [P_i,J_j]=i\epsilon_{ijk}P_k, [J_i,K_j]=i\epsilon_{ijk}K_k$$
Note that we have assumed that $[P_z,P_x]=[P_z,P_y]=0$ and $[J_z,P_x]=[J_z,P_y]=0$, which is consistent with the original commutation relations for ISO(3).

To get the Heisenberg algebra from this contraction, we can take the limit as $\epsilon \rightarrow 0$. In this limit, the generators $K_x$ and $K_y$ become infinitely large, and therefore $[K_x,K_y]=0$. This gives us the Heisenberg algebra:
$$[P_i,P_j]=0, [X_i,X_j]=0, [P_i,X_j]=i\hbar\delta_{ij}I$$
where $X_i=\lim_{\epsilon \rightarrow 0} K_i$. Note that we have also taken the limit $c\rightarrow \infty$ to get rid of the $H$ generator, since it becomes infinitely large in this limit.

Finally, to get the H(2) algebra, we need to contract ISO(3) to H(3). This can be done by taking the following scaling: $X_i=\frac{J_i}{