Schrödinger group

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

Schrödinger algebra[edit]

The Schrödinger algebra is the Lie algebra of the Schrödinger group. It is not semi-simple. In one space dimension, it can be obtained as a semi-direct sum of the Lie algebra sl(2,R) and the Heisenberg algebra; similar constructions apply to higher spatial dimensions.

It contains a Galilei algebra with central extension.

Where are generators of rotations (angular momentum operator), spatial translations (momentum operator), Galilean boosts and time translation (Hamiltonian) correspondingly (Notes: is the imaginary unit, . The specific form of the commutators of the generators of rotation is the one of three-dimensional space, then .). 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 shall 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 (infinite) families and , 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)

The Schrödinger algebra is finite-dimensional and contains the generators . In particular, the three generators span the sl(2,R) sub-algebra. Space-translations are generated by and the Galilei-transformations by .

In the chosen notation, one clearly sees that an infinite-dimensional extension exists, which is called the Schrödinger-Virasoro algebra. Then, the generators 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[1]

This shows how the central extension of the non-semi-simple and finite-dimensional Schrödinger algebra becomes a component of an infinite family in the Schrödinger-Virasoro algebra. In addition, and in analogy with either the Virasoro algebra or the Kac-Moody algebra, further central extensions are possible. However, a non-vanishing result only exists for the commutator , where it must be of the familiar Virasoro form, namely

or for the commutator between the rotations , where it must have a Kac-Moody form. Any other possible central extension can be absorbed into the Lie algebra generators.

The role of the Schrödinger group in mathematical physics[edit]

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.[2] It also describes the kinetics of phase-ordering, after a temperature quench from the disordered to the ordered phase, in magnetic systems.


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See also[edit]