Schrödinger group

The Schrödinger group is the symmetry group of the free particle Schrödinger equation.

The Schrödinger algebra is the Lie algebra of the Schrödinger group. It is not semi-simple and can be obtained as a semi-direct sum of the Lie algebra sl(2,R) and the Heisenberg algebra.

It contains a Galilei algebra with central extension.

Where ${\displaystyle J_{i},P_{i},K_{i},H}$ are generators of rotations (angular momentum operator), spatial translations (momentum operator), Galilean boosts and time translation (Hamiltonian) correspondingly. The central extension M has an interpretation as non-relativistic mass and corresponds to the symmetry of Schrödinger equation under phase transformation (and to the conservation of probability).

There are two more generators which we will denote by D and C. They have the following commutation relations:

The generators H, C and D form the sl(2,R) algebra.

A more systematic notation allows to cast these generators into the four families ${\displaystyle X_{n},Y_{m}^{(j)},M_{n}}$ and ${\displaystyle R_{n}^{(jk)}=-R_{n}^{(kj)}}$, where n ∈ ℤ is an integer and m ∈ ℤ+1/2 is a half-integer and j,k=1,...,d label the spatial direction, in d spatial dimensions. The non-vanishing commutators of the Schrödinger algebra become (euclidean form)

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