Schrödinger group

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

Schrödinger algebra

The Schrödinger algebra is the Lie algebra of the Schrödinger group.

It contains Galilei algebra with central extension.

$[J_i,J_j]=i \epsilon_{ijk} J_k,\,\!$

$[J_i,P_j]=i \epsilon_{ijk} P_k,\,\!$

$[J_i,K_j]=i \epsilon_{ijk} K_k,\,\!$

$[P_i,P_j]=0, [K_i,K_j]=0, [K_i,P_j]=i \delta_{ij} M,\,\!$

$[H,J_i]=0, [H,P_i]=0, [H,K_i]=i P_i.\,\!$

Where 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. Central extension M has 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:

$[H,C]=i D, [C,D]=-2i C, [H,D]=2i H,\,\!$

$[P_i,D]=i P_i, [K_i,D]=-iK_i,\,\!$

$[P_i,C]=-iK_i,[K_i,C]=0,\,\!$

$[J_i,C]=[J_i,D]=0.\,\!$

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

The role of the Schrödinger group in mathematical physics

Though the Schrödinger group is defined as symmetry group of the free particle Schrödinger equation, it is realised in some interacting non-relativistic systems (for example cold atoms at criticality).

The Schrödinger group in d spatial dimensions can be embedded into relativistic conformal group in d+1 dimensions SO(2,d+2). This embedding is connected with the fact that one can get the Schrödinger equation from the massless Klein–Gordon equation through Kaluza–Klein compactification along null-like dimensions and Bargmann lift of Newton–Cartan theory.

References

• C. R. Hagen, "Scale and Conformal Transformations in Galilean-Covariant Field Theory", Phys. Rev. D 5, 377–388 (1972)
• Arjun Bagchi, Rajesh Gopakumar, "Galilean Conformal Algebras and AdS/CFT", JHEP 0907:037,2009
• D.T.Son, "Toward an AdS/cold atoms correspondence: A geometric realization of the Schrödinger symmetry", Phys. Rev. D 78, 046003 (2008)

See also

• Schrödinger equation
• Galilean transformation
• Poincaré group