The Schrödinger group is the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of the free particle

Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...

. Mathematically, the group

SL(2,R) In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: : \mbox(2,\mathbf) = \left\. It is a connected non-compact simple real Lie group of dimension 3 with ap ...

acts The Acts of the Apostles (, ''Práxeis Apostólōn''; ) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message to the Roman Empire. Acts and the Gospel of Luke make up a two-par ...

on the

Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form : \begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b' ...

outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...

s, and the Schrödinger group is the corresponding

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...

Schrödinger algebra

The Schrödinger algebra is the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

of the Schrödinger group. It is not

semi-simple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

. In one space dimension, it can be obtained as a semi-direct sum of the Lie algebra

sl(2,R) In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: : \mbox(2,\mathbf) = \left\. It is a connected non-compact simple real Lie group of dimension 3 with ap ...

and the

Heisenberg algebra In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form : \begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b'' ...

; similar constructions apply to higher spatial dimensions. It contains a Galilei algebra with central extension. :

i \epsilon_ J_c,\,\!

i \epsilon_ P_c,\,\!

i \epsilon_ K_c,\,\!

i \delta_ M,\,\!

i P_a.\,\!

where

J_a, P_a, K_a, H

are generators of rotations (

angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum pro ...

), spatial translations (

momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...

), Galilean boosts and

time translation Time-translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time-translation symmetry is the law that the laws of physics are unchanged ...

(

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

) respectively. (Notes:

i

is the imaginary unit,

i^2=-1

. The specific form of the commutators of the generators of rotation

J_a

is the one of three-dimensional space, then

a,b,c=1,\ldots,3

.). The central extension ''M'' has an interpretation as non-relativistic

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

and corresponds to the symmetry of

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: :

2i H,\,\!

-iK_a,\,\!

_a,C -iK_a,_a,C 0,\,\!

_a,C_a,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 (infinite) families

X_n, Y_m^, M_n

and

R_n^=-R_n^

, 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) :

_n, X_= (n-n') X_

= \left( -m\right) Y_^

_n, M_

_n, R_^

= \delta_ (m-m') M_

= \delta_ Y_^ - \delta_ Y_^

= \delta_ R_^ +\delta_ R_^ -\delta_ R_^ -\delta_ R_^

The Schrödinger algebra is finite-dimensional and contains the generators

X_, Y_^, M_0, R_0^

. In particular, the three generators

X_=H, X_0=D, X_=C

span the sl(2,R) sub-algebra. Space-translations are generated by

Y_^

and the Galilei-transformations by

Y_^

. 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^\partial_t -t^n\vec\cdot\partial_ -t^ \vec\cdot\vec -(n+1) t^n

Y_m^ = -t^ \partial_ - \left(m+\right)  t^ r_j

M_n = - t^n

R_n^ = -t^n \left( r_j \partial_ - r_k \partial_\right)

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. In addition, and in analogy with either the

Virasoro algebra In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel ...

or the

Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...

, further central extensions are possible. However, a non-vanishing result only exists for the commutator

_n, X_/math>, where it must be of the familiar Virasoro form, namely 
:_n, X_(n-n') X_ + (n^3-n)\delta_or for the commutator between the rotations R_n^, 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

Though the Schrödinger group is defined as symmetry group of the free particle

, it is realized in some interacting non-relativistic systems (for example cold atoms at criticality). The Schrödinger group in spatial dimensions can be embedded into relativistic

conformal group In mathematics, the conformal group of an inner product space is the group (mathematics), group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometr ...

in dimensions . This embedding is connected with the fact that one can get the

from the massless

Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is named after Oskar Klein and Walter Gordon. It is second-order i ...

through Kaluza–Klein compactification along null-like dimensions and Bargmann lift of

Newton–Cartan theory Newton–Cartan theory (or geometrized Newtonian gravitation) is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced by Élie Cartan in 1923 and Kurt Friedrichs and later developed by G. Dautcourt, W. G ...

. This embedding can also be viewed as the extension of the Schrödinger algebra to the maximal parabolic sub-algebra of . The Schrödinger group symmetry can give rise to exotic properties to interacting bosonic and fermionic systems, such as the

superfluids Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two i ...

in bosons , and Fermi liquids and non-Fermi liquids in fermions. They have applications in condensed matter and cold atoms. 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.M. Henkel, ''Eur. Phys. J. Spec. Topics'' 226, 605 (2017) It also describes the kinetics of phase-ordering, after a temperature quench from the disordered to the ordered phase, in magnetic systems.

References

* C. R. Hagen, "Scale and Conformal Transformations in Galilean-Covariant Field Theory", ''Phys. Rev.'' D5, 377–388 (1972) * U. Niederer, "The maximal kinematical invariance group of the free Schroedinger equation", ''Helv. Phys. Acta'' 45, 802 (1972) * G. Burdet, M. Perrin, P. Sorba, "About the non-relativistic structure of the conformal algebra", ''Comm. Math. Phys.'' 34, 85 (1973) * M. Henkel, "Schrödinger-invariance and strongly anisotropic critical systems", ''J. Stat. Phys.'' 75, 1023 (1994) * M. Henkel, J. Unterberger, "Schrödinger-invariance and space-time symmetries", ''Nucl. Phys.'' B660, 407 (2003) * A. Röthlein, F. Baumann, M. Pleimling, "Symmetry-based determination of space-time functions in nonequilibrium growth processes", ''Phys. Rev.'' E74, 061604 (2006) -- erratum E76, 019901 (2007) * D.T. Son, "Towards an AdS/cold atoms correspondence: A geometric realization of the Schrödinger symmetry", ''Phys. Rev.'' D78, 046003 (2008) * A. Bagchi, R. Gopakumar, "Galilean Conformal Algebras and AdS/CFT", ''JHEP'' 0907:037 (2009) * M. Henkel, M. Pleimling, ''Non-equilibrium phase transitions, vol 2: ageing and dynamical scaling far from equilibrium'', (Springer, Heidelberg 2010) * J. Unterberger, C. Roger, ''The Schrödinger-Virasoro algebra'', (Springer, Heidelberg 2012)

Schrödinger algebra

The role of the Schrödinger group in mathematical physics

References

See also