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The Schrodinger group is the symmetry group of the free particle Schrodinger equation . Mathematically, the group SL(2,R) acts on the Heisenberg group by outer automorphisms , and the Schrodinger group is the corresponding semidirect product .

Schrodinger algebra [ edit ]

The Schrodinger algebra is the Lie algebra of the Schrodinger 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.

${\displaystyle [J_{a},J_{b}]=i\epsilon _{abc}J_{c},\,\!}$

${\displaystyle [J_{a},P_{b}]=i\epsilon _{abc}P_{c},\,\!}$

${\displaystyle [J_{a},K_{b}]=i\epsilon _{abc}K_{c},\,\!}$

${\displaystyle [P_{a},P_{b}]=0,[K_{a},K_{b}]=0,[K_{a},P_{b}]=i\delta _{ab}M,\,\!}$

${\displaystyle [H,J_{a}]=0,[H,P_{a}]=0,[H,K_{a}]=iP_{a}.\,\!}$

where ${\displaystyle J_{a},P_{a},K_{a},H}$ are generators of rotations ( angular momentum operator ), spatial translations ( momentum operator ), Galilean boosts and time translation ( Hamiltonian ) respectively. (Notes: ${\displaystyle i}$ is the imaginary unit, ${\displaystyle i^{2}=-1}$. The specific form of the commutators of the generators of rotation ${\displaystyle J_{a}}$ is the one of three-dimensional space, then ${\displaystyle a,b,c=1,\ldots ,3}$.). The central extension M has an interpretation as non-relativistic mass and corresponds to the symmetry of Schrodinger 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:

${\displaystyle [H,C]=iD,[C,D]=-2iC,[H,D]=2iH,\,\!}$

${\displaystyle [P_{a},D]=iP_{a},[K_{i},D]=-iK_{a},\,\!}$

${\displaystyle [P_{a},C]=-iK_{a},[K_{a},C]=0,\,\!}$

${\displaystyle [J_{a},C]=[J_{a},D]=0.\,\!}$

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 ${\displaystyle X_{n},Y_{m}^{(j)},M_{n}}$ and ${\displaystyle R_{n}^{(jk)}=-R_{n}^{(kj)}}$, where n ∈ ? is an integer and m ∈ ?+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 Schrodinger algebra become (euclidean form)

${\displaystyle [X_{n},X_{n'}]=(n-n')X_{n+n'}}$

${\displaystyle [X_{n},Y_{m}^{(j)}]=\left({n \over 2}-m\right)Y_{n+m}^{(j)}}$

${\displaystyle [X_{n},M_{n'}]=-n'M_{n+n'}}$

${\displaystyle [X_{n},R_{n'}^{(jk)}]=-n'R_{n'}^{(jk)}}$

${\displaystyle [Y_{m}^{(j)},Y_{m'}^{(k)}]=\delta _{j,k}(m-m')M_{m+m'}}$

${\displaystyle [R_{n}^{(ij)},Y_{m}^{(k)}]=\delta _{i,k}Y_{n+m}^{(j)}-\delta _{j,k}Y_{n+m}^{(i)}}$

${\displaystyle [R_{n}^{(ij)},R_{n'}^{(kl)}]=\delta _{i,k}R_{n+n'}^{(jl)}+\delta _{j,l}R_{n+n'}^{(ik)}-\delta _{i,l}R_{n+n'}^{(jk)}-\delta _{j,k}R_{n+n'}^{(il)}}$

The Schrodinger algebra is finite-dimensional and contains the generators ${\displaystyle X_{-1,0,1},Y_{-1/2,1/2}^{(j)},M_{0},R_{0}^{(jk)}}$. In particular, the three generators ${\displaystyle X_{-1}=H,X_{0}=D,X_{1}=C}$ span the sl(2,R) sub-algebra. Space-translations are generated by ${\displaystyle Y_{-1/2}^{(j)}}$ and the Galilei-transformations by ${\displaystyle Y_{1/2}^{(j)}}$.

In the chosen notation, one clearly sees that an infinite-dimensional extension exists, which is called the Schrodinger?Virasoro algebra . Then, the generators ${\displaystyle X_{n}}$ with n integer span a loop-Virasoro algebra. An explicit representation as time-space transformations is given by, with n ∈ ? and m ∈ ?+1/2 ^[1]

${\displaystyle X_{n}=-t^{n+1}\partial _{t}-{n+1 \over 2}t^{n}{\vec {r}}\cdot \partial _{\vec {r}}-{n(n+1) \over 4}{\cal {M}}t^{n-1}{\vec {r}}\cdot {\vec {r}}-{x \over 2}(n+1)t^{n}}$

${\displaystyle Y_{m}^{(j)}=-t^{m+1/2}\partial _{r_{j}}-\left(m+{1 \over 2}\right){\cal {M}}t^{m-1/2}r_{j}}$

${\displaystyle M_{n}=-t^{n}{\cal {M}}}$

${\displaystyle R_{n}^{(jk)}=-t^{n}\left(r_{j}\partial _{r_{k}}-r_{k}\partial _{r_{j}}\right)}$

This shows how the central extension ${\displaystyle M_{0}}$ of the non-semi-simple and finite-dimensional Schrodinger algebra becomes a component of an infinite family in the Schrodinger?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 ${\displaystyle [X_{n},X_{n'}]}$, where it must be of the familiar Virasoro form, namely

${\displaystyle [X_{n},X_{n'}]=(n-n')X_{n+n'}+{c \over 12}(n^{3}-n)\delta _{n+n',0}}$

or for the commutator between the rotations ${\displaystyle R_{n}^{(jk)}}$, 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 Schrodinger group in mathematical physics [ edit ]

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

The Schrodinger 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 Schrodinger 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 also be viewed as the extension of the Schrodinger algebra to the maximal parabolic sub-algebra of SO(2, d + 2) .

The Schrodinger group symmetry can give rise to exotic properties to interacting bosonic and fermionic systems, such as the superfluids in bosons ^[2] , ^[3] and Fermi liquids and non-Fermi liquids in fermions. ^[4] They have applications in condensed matter and cold atoms.

The Schrodinger group also arises as dynamical symmetry in condensed-matter applications: it is the dynamical symmetry of the Edwards?Wilkinson model of kinetic interface growth. ^[5] It also describes the kinetics of phase-ordering, after a temperature quench from the disordered to the ordered phase, in magnetic systems.

References [ edit ]

• 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, "Schrodinger-invariance and strongly anisotropic critical systems", J. Stat. Phys. 75 , 1023 (1994)
• M. Henkel, J. Unterberger, "Schrodinger-invariance and space-time symmetries", Nucl. Phys. B660 , 407 (2003)
• A. Rothlein, 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 Schrodinger 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 Schrodinger-Virasoro algebra , (Springer, Heidelberg 2012)

See also [ edit ]

• Schrodinger equation
• Galilean transformation
• Poincare group