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

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1. Dec 20, 2017

### Azure Ace

1. The problem statement, all variables and given/known data
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.

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

3. 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 ($[K_{x}, H] = -iP_{y}, [K_{y}, H] = iP_{x}$.

Last edited by a moderator: Dec 20, 2017
2. Dec 25, 2017

### PF_Help_Bot

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