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.
[ J i , J j ] = i ϵ i j k J k , [ J i , P j ] = i ϵ i j k P k , [ J i , K j ] = i ϵ i j k K k , [ P i , P j ] = 0 , [ K i , K j ] = 0 , [ K i , P j ] = i δ i j 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. 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:
[ H , C ] = i D , [ C , D ] = − 2 i C , [ H , D ] = 2 i H , [ P i , D ] = i P i , [ K i , D ] = − i K i , [ P i , C ] = − i K i , [ K i , C ] = 0 , [ J i , C ] = [ J i , D ] = 0. 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 X n , Y m ( j ) , M n and R n ( j k ) = − R n ( k j ) , 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)
[ X n , X n ′ ] = ( n − n ′ ) X n + n ′ [ X n , Y m ( j ) ] = ( n 2 − m ) Y n + m ( j ) [ X n , M n ′ ] = − n ′ M n + n ′ [ X n , R n ′ ( j k ) ] = − n ′ R n ′ ( j k ) [ Y m ( j ) , Y m ′ ( k ) ] = δ j , k ( m − m ′ ) M m + m ′ [ R n ( i j ) , Y m ( k ) ] = δ i , k Y n + m ( j ) − δ j , k Y n + m ( i ) [ R n ( i j ) , R n ′ ( k l ) ] = δ i , k R n + n ′ ( j l ) + δ j , l R n + n ′ ( i k ) − δ i , l R n + n ′ ( j k ) − δ j , k R n + n ′ ( i l ) The Schrödinger algebra is finite-dimensional and contains the generators X − 1 , 0 , 1 , Y − 1 / 2 , 1 / 2 ( j ) , M 0 , R 0 ( j k ) . In particular, the three generators X − 1 = H , X 0 = D , X 1 = C span the sl(2,R) sub-algebra. Space-translations are generated by Y − 1 / 2 ( j ) and the Galilei-transformations by Y 1 / 2 ( j ) .
In the chosen notation, one clearly sees that an infinite-dimensional extension exists, which is called the Schrödinger-Virasoro algebra. Then, the generators X n with n integer span a loop-Virasoro algebra. An explicit representation as time-space transformations is given by, with n ∈ ℤ and m ∈ ℤ+1/2
X n = − t n + 1 ∂ t − n + 1 2 t n r → ⋅ ∂ r → − n ( n + 1 ) 4 M t n − 1 r → ⋅ r → − x 2 ( n + 1 ) t n Y m ( j ) = − t m + 1 / 2 ∂ r j − ( m + 1 2 ) M t m − 1 / 2 r j M n = − t n M R n ( j k ) = − t n ( r j ∂ r k − r k ∂ r j ) This shows how the central extension M 0 of the non-semi-simple and finite-dimensional Schrödinger algebra becomes a component of an infinite family in the Schrödinger-Virasoro algebra. Central extensions are possible (and non-trivial) only for the commutator [ X n , X n ′ ] , where it must be of the familiar Virasoro form, or for the commutator between the rotations R n ( j k ) , where it must have a Kac-Moody form.
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. This embedding can be also be viewed as the extension of the Schrödinger algebra to the maximal parabolic sub-algebra of SO(2,d+2).
The Schrödinger group also arises as dynamical symmetry in condensed-matter applications: it is the dynamical symmetry of the Edwards-Wilkinson model of kinetic interface growth. It also describes the kinetics of phase-ordering, after a temperature quench from the disordered to the ordered phase, in magnetic systems.
