Higher-spin symmetries of the free Schrodinger equation
aa r X i v : . [ h e p - t h ] J u l Higher-spin symmetries of the free Schr¨odinger equation
Mauricio Valenzuela ∗ Facultad de Ingeniera y Tecnolog´ıaUniversidad San Sebasti´an, General Lagos 1163, Valdivia 5110693, Chile
Abstract
It is shown that the Schr¨odinger symmetry algebra of a free particle in d spatial dimensionscan be embedded into a representation of the higher-spin algebra. The latter spans an infinitedimensional algebra of higher-order symmetry generators of the free Schr¨odinger equation.An explicit representation of the maximal finite dimensional subalgebra of the higher spinalgebra is given in terms of non-relativistic generators. We show also how to convert Vasiliev’sequations into an explicit non-relativistic covariant form, such that they might apply to non-relativistic systems. Our procedure reveals that the space of solutions of the Schr¨odingerequation can be regarded also as a supersymmetric module. Keywords: Higher-spin theory, Non-relativistic symmetry, Supersymmetry
The Schr¨odinger group was discovered by S. Lie [1] and, even earlier, the conserved quantitiesassociated with the Schr¨odinger invariance were already known to Jacobi [2] (see also [3, 4]). Itsname, however, is taken from quantum mechanics [5, 6, 7, 8, 9], since it extends the Galileansymmetries of the free Schr¨odinger equation with dilatation and expansion transformations. TheSchr¨odinger group is isomorphic to the Newton-Hooke conformal group of the harmonic oscillator[8], and it appears also in several contexts, e.g., magnetic monopoles [10], vortices [11, 12, 13, 14],fluid mechanics [15], and strongly correlated fermions [16, 17]. The Schr¨odinger group has someinfinite dimensional generalizations [3, 16], and it can be also realized geometrically as space-timeisometries [18, 19, 20]. The Schr¨odinger symmetry have attracted also renewed interest in thecontext of non-relativistic
AdS/CF T correspondence [21, 22, 23], which relates an asymptotictheory on a curved background to a non-relativistic quantum system [20, 21, 22, 23]. This isa consequence of the embedding of the Schr¨odinger algebras into relativistic conformal algebras[24].The goal of this letter is to show that Galileo boosts, translations and the mass generators arebuilding blocks for constructing higher-order symmetries of the free Schr¨odinger equation. This isdone by noticing that from Galileo boosts, translations and the mass operator, which satisfy theHeisenberg algebra, we can construct a representation of the Weyl algebra. The latter will span ∗ [email protected]
1n infinite set of conserved charges, containing in particular all Schr¨odinger generators. This canbe regarded as the non-relativistic analog of the Eastwood result [25] on the maximal symmetriesof the massless Klein-Gordon equation, which is spanned by polynomials in conformal symmetrygenerators.Endowing the generators of the Weyl algebra with a (super)commutator product yields theso-called higher spin (super)algebras [26, 27, 28, 29]. These algebras are well-known in the contextof higher spin gauge theory (see also [30, 31, 32, 33]). It follows that the free Schr¨odinger equationexhibits higher spin symmetries, and reciprocally, Schr¨odinger symmetry is naturally contained inhigher spin theory. As we shall see at the end of this paper, Vasiliev equations of higher spin gaugefields can be written in an explicit non-relativistic form, by simple identification of the symplectic-spinor indices of the higher spin fields with spatial indices of a non-relativistic space-time. Indeed,the truncation of the higher spin algebra to its maximal finite dimensional subalgebra containsthe Schr¨odinger algebra. Reciprocally, the correspondence between symplectic-spinor indices andnon-relativistic spatial indices allows to endow the Sch¨odinger generators with a supercommutatorproduct, yielding an orthosymplectic-type supersymmetry of the Schr¨odinger equation.
Consider d dimensional Heisenberg algebra, h d = { , G i , P i ; [ G i , P j ] = i δ ij , i, j = 1 , ..., d } . The Weyl algebra, denoted h ∗ d , can be defined as the algebra of Weyl ordered polynomials ofthe Heisenberg algebra generators .An associative algebra can always be endowed with a commutator product, thus yielding aLie algebra, [ h ∗ d ] = n h ∗ d ; [ A, B ] := AB − BA, A, B ∈ h ∗ d o . (1)Alternatively, as the Weyl algebra is graded over Z , we can endow their generators with asupercommutator product yielding a Lie superalgebra,[ h ∗ d } := n h ∗ d ; [ A, B } := AB − ( − | A || B | BA, A, B ∈ h ∗ d o . (2)Here, | · | denotes the degree of the generators, respectively | · | = 0 and | · | = 1 for even andodd order polynomials in the generators G i and P i of the Heisenberg algebra. The constantsof structure of (1) and (2) are derived from those of h d , and the (super)Jacobi identity followsfrom the associativity of the Weyl algebra. The algebras (1) and (2) are called higher spin(super)algebras [26, 27, 28], since they contain a maximal subalgebra of compact generatorsunder which the remaining generators transform as tensors of arbitrary spin.In our approach the Heisenberg algebra is composed of a mass-central-charge, Galileo boostsand translations generators of a non-relativistic particle, h d = { m = , G i = x i − tP i , P i = − i ∂/∂x i } . (3)Consider now the free Schr¨odinger equation in d spatial dimensions, b S | ψ ( x, t ) i = 0 , b S = i ∂∂t − H, H = 12 −→ P . (4) It is indeed the universal enveloping of the Heisenberg algebra which in suitable basis can be defined as thesymmetrized products of the h d generators, owing the Poincar´e-Birkhoff-Witt theorem (see eg. [34]). O = ∂ O /∂t + i[ H, O ] , which can be written in terms of the Schr¨odinger operator as˙ O = − i h b S, O i . (5)This equation can be also regarded as the first class constraint associated to the time-parametrizationinvariance of the free non-relativistic particle [35].The equations (4) and (5) imply that constants of motion are symmetry generators, since, asthey commutes with the Schr¨odinger operator, they leave invariant the space of solutions of theSchr¨odinger equation.The product of constant of motion is also a constant of motion. This follows from the Leibnizrule satisfied by the derivative with respect to the time and by the adjoint action of b S acting onthe product of constants of motion. For the free Schrodinger equation, from Galileo boosts andtranslations ˙ G i = − i[ b S, G i ] = 0 , ˙ P i = − i[ b S, P i ] = 0 , (6)any polynomial of G i and P i will be also a constant of motion. Therefore the free Schr¨odingerequation admits infinitely many conserved charges, spanned by arbitrary operator functions of G i and P i , and which contains the Weyl algebra as the basis of the polynomial class of functions. Once the generators G i and P i are provided we can form the vector L a = ( G , ..., G d , P , ..., P d ) , a = 1 , ..., d. (7)The commutation relations of G i = L i and P j = L d + j become now,[ L a , L b ] = i C ab , C ab = (cid:16) I d × d − I d × d (cid:17) , (8)which defines the symplectic matrix C ab . The symmetrized second order products of generators(7), M ab = 12 { L a , L b } , (9)commute with themselves as,[ M ab , M cd ] = i( C ac M bd + C bd M ac + C ad M bc + C bc M ad ) , (10)as it is deduced from (8), generating a representation of the sp (2 d ) algebra. This representationis usually referred as to “oscillator representation”, since one of their compact generators can beidentified with a harmonic oscillator Hamiltonian. Here the Hamiltonian is, however, identifiedwith a non-compact generator, the one of the non-relativistic free particle. M ab together with L a yields the commutation relations,[ M ab , L c ] = i( C ac L b + C bc L a ) . (11)The generators M ab , L a and yields the maximal finite dimensional subalgebra of (1) (see (8)-(10)-(11)), h d B sp (2 d ) = n , L a , M ab ; [ · , · ] o ⊂ [ h ∗ d ] . (12)3rom M ab we can define the generators, C = 12 δ ij M ij , D = − δ ij M i d + j , H = 12 δ ij M d + i d + j , J ij = M i d + j − M j d + i , i, j = 1 , ..., d . (13)Now (13) together with (3) yields the Schr¨odinger algebra sch ( d ),[ G i , P j ] = i mδ ij , [ J ij , J kl ] = i( δ ik J jl + δ jl M ik − δ il J jk − δ jk J il ) , [ J ij , P k ] = i( δ ik P j − δ jk P i ) , [ J ij , G k ] = i( δ ik G j − δ jk G i ) , [ H, G i ] = − i P i , [ D, C ] = 2i C, [ D, H ] = − H, [ H, C ] = i D, [ D, P i ] = − i P i , [ D, G i ] = i G i , [ C, P i ] = i G i . Other commutators vanish. We stress that the full Schr¨odinger algebra sch ( d ) is implied by theHeisenberg commutation relation (8) and the definitions (13). Indeed, from (3) and (13), thestandard representation of sch ( d ) is recovered,Hamiltonian H = −→ P / , rotations J ij = x i P j − x j P i , translations P i = − i ∂/∂x i , boosts G i = mx i − tP i , mass m = , expansions C = − t H + tD + −→ x / , dilatations D = 2 tH − −→ x · −→ P + i d/ . The Schr¨odinger algebra, having the structure sch ( d ) = h d B { so ( d ) ⊕ sl (2 , R ) } , is subalgebra of(12) and the higher spin algebra (1), i.e. sch ( d ) ⊂ ( h d B sp (2 d )) ⊂ [ h ∗ d ].Rotations, dilations and expansion generators, are second order operators in Galileo boostsand translations. They generate independent symmetries however, as it is well known. Indeed, thefinite transformations generated by D and C are respectively (see e.g. [7, 4]) ( t, x i ) → ( λ t, λx i )and ( t, x i ) → (1 − κt ) − ( t, x i ) where λ and κ are transformation parameters.The new generators of h d B sp (2 d ) not contained in the Schr¨odinger algebra are of secondorder in spatial derivatives, P ij = M d + i d + j − d δ ij H, G ij = M ij − d δ ij C, Z ij = M i d + j + M j d + i + d δ ij D,P ij = P i P j − d δ ij H, G ij = G i G j − d δ ij C, Z ij = G i P j + P i G j + d δ ij D. (14)These operators are traceless, δ ij P ij = δ ij G ij = δ ij Z ij = 0 . The non-vanishing remaining com-4utation relations read,[ P ij , G k ] = − i2 ( δ ik P j + δ jk P i ) + i d δ ij P k , [ Z ij , G k ] = − i( δ ik G j + δ jk G i ) + d δ ij G k , [ G ij , P k ] = i2 ( δ ik G j + δ jk G i ) + i d δ ij G k , [ Z ij , P k ] = i( δ ik P j + δ jk P i ) − d δ ij P k , [ H, G ij ] = − i2 Z ij , [ H, Z ij ] = − P ij , [ C, P ij ] = i2 Z ij , [ C, Z ij ] = 4i G ij , [ D, G ij ] = 2i G ij , [ D, P ij ] = − P ij , [ G ij , P kl ] = i8 ( δ ik ( Z jl + J jl ) + δ il ( Z jk + J jk ) + δ jk ( Z il + J il ) + δ jl ( Z ik + J ik )) − i2 d ( δ ik δ jl + δ il δ jk − d δ ij δ kl ) D − i2 d ( δ ij Z kl + δ kl Z ij ) , [ G ij , Z kl ] = i( δ ik G jl + δ il G jk + δ jk G il + δ jl G ik )+ d ( δ ik δ jl + δ il δ jk − d δ ij δ kl ) C − d ( δ ij G kl + δ kl G ij ) , [ P ij , Z kl ] = − i( δ ik P jl + δ il P jk + δ jk P il + δ jl G ik ) − d ( δ ik δ jl + δ il δ jk − d δ ij δ kl ) H + d ( δ ij P kl + δ kl P ij ) , [ Z ij , Z kl ] = − δ ik J jl + δ il J jk + δ jk J il + δ jl J ik ) , extending sch ( d ) to h d B sp (2 d ), which in Galileo covariant notation is generated by (cf. (12)), h d B sp (2 d ) = n , G i , P i , J ij , C, D, H, P ij , G ij , Z ij ; [ · , · ] o . (15)Notice that endowing the generators (15) instead with a supercommutator product, which ispossible owing the Z grading (2) , yields the superalgebra osp (1 | d ) ⊕ u (1) = n , G i , P i , J ij , C, D, H, P ij , G ij , Z ij ; [ · , ·} o , (16)with (anti)commutation relations equivalent to (9), (10) and (11) (see the definitions (7), (13)and (14)). It is the maximal finite dimensional subalgebra of (2). Here, the Galileo boosts andtranslations generators, G i and P i , are regarded as supercharges (cf. (14)), { G i , G j } = 2 G ij + 2 d δ ij C, { P i , P j } = 2 P ij + 2 d δ ij H, { G i , P j } = J ij + Z ij − d δ ij D. Indeed, the vanishing commutation relations of the Hamiltonian with the non-trivial reflectiongrading-operator reveals the double degeneracy of the Hamiltonian own to supersymmetry, whichis characteristic in supersymmetric quantum mechanics.The Schr¨odinger algebra extension (12) can be seen hence as the bosonic counterpart ofthe orthosymplectic supersymmetry osp (1 | d ) , using the terminology of [40] where the bosoniccounterpart of super-Poincar´e was studied. Of course, the introduction of other degrees of freedomsuch as Clifford or Grassmann variables as in, e.g., [38, 39], would yield a more standard typeof supersymmetry. The supersymmetry induced by parity under spatial reflections has beenwidely studied by M. Plyushchay and collaborators in diverse interacting quantum mechanicalsystem which do not involve fermionic degrees of freedom, and called for this reason bosonizedsupersymmetry , or hidden supersymmetry [37]. The algebra (15) has been also discussed in a different context [36]. Indeed, it is the reflection operator, R Ψ( x ) = Ψ( − x ), which induces the Z grading, as it anti-commutes withGalileo boosts, translations and all their odd powers in the Weyl algebra, whereas it commutes with their evenorder powers. The symmetric and antisymmetric projections of the wave function Ψ ± ( x, t ) = (Ψ( x, t ) ± Ψ( − x, t )),can be seen as they were “bosonic” and “fermionic”. Observe also that the odd projection satisfies a Pauli exclusionlike principle, Ψ − (0 , t ) = 0. on-relativistic covariance of Vasiliev equations The theory of Vasiliev is a generalization of the Cartan formulation of gravity (see e.g. [28]),determining the dynamics of the higher spin fields forms by means of a Cartan-integrable systemof equations. Topological higher spin gravity admits also a Chern-Simons action principle [31](see also [33] in a more recent context) in three dimensions, and in four dimensions an actionprinciple was proposed in [32]. Here we show that the higher spin gauge theory [28, 30, 31]exhibits also non-relativistic symmetries.Vasiliev’s theory makes use of differential forms valued in the higher spin gauge algebra ofthe type W ( X ) = ∞ X n =0 n ! W a a ··· a n ( X ) Y a Y a · · · Y a n , a , a , ..., a n = 1 , , ..., d. (17) W a a ··· a n ( X ) are completely symmetric in a -indices and X labels local coordinates of the basespace-time manifold. The gauge field (17) may involve also the reflection operator (or Kleinoperator) in its expansion, as well as Clifford or Grassmann variables, which we omit here forsimplicity. The representation of the Weyl algebra can be realized also in terms of commuting Y a -symbols endowed with an associative star product, whereas the higher spin (super)algebrais obtained from the correspondent (anti)symmetrization of the star product. Identifying theHeisenberg algebra of the oscillators Y a with the Galileo-boost/translation generators (7) (seeTable 1), allow us to view the expansion (17) as a one-form valued in the universal enveloping ofthe non-relativistic Schr¨odinger algebra h ∗ d , making explicit its non-relativistic covariance.HS theorydimension notation for Heisenbergalgebra generators dimensionNR theory Correspondence withNR generators2 + 1 Y a , a = 1 , Y → G , Y → P Y a = ( y α , y ˙ α ) α, ˙ α = 1 , Y α → ( G , P ) , Y ˙ α → ( G , P ) D > Y Aa ,a = 1 , , A = 0 , ..., D − , D + 1 Y a → ( P , G ) , Y ia → ( G i +1 , P i +1 ) ,i = 1 , ..., D − Y ’s with the generators of Galilean boost andtranslations.In D > Y Aa , Y Bb ] =i C ab η AB , where η AB is the Lorentz metric in D dimensions. Thus the formulation of higher spingravity in general (relativistic) space-time dimensions D > Y Aa ( A = 0 , ..., D − a = 1 , × D . In three and four dimensions these phase-spaces have respectively dimensions 2 and4, and the generators of the Heisenberg algebras transform as spinors under the relativisticLorentz group. Therefore the latter phase-spaces can be labeled instead in terms of Galilean non-relativistic covariance in dimension D + 1, including the time direction, for higher-spin gravityin D >
4, while the non-relativistic covariance is 1 + 1 (one spatial non-relativistic direction andthe time) for higher-spin gravity in D = 3 dimensions, and it is 2 + 1 (two non-relativistic spatialdirection and the time) for higher-spin gravity in D = 4 relativistic space-time. The expected6orrespondence between higher spin gravity and non-relativistic quantum mechanics is thereforeas given in the table 1. We have shown in a simple way how the free Schr¨odinger equation enjoys infinite symmetriesgenerated by the Weyl algebra. Since the Weyl algebra can be made covariant under non-relativistic (Galilean) transformations and also under relativistic (higher spin) transformations,we observe that both, the Schr¨odinger equation and higher spin gravity, have non-relativistic andrelativistic symmetries, depending on the choice of algebra labels.More generally, since the (unitary) representations of the sp (2 d ) algebra, or the osp (1 | d )superalgebra, admit the embeddings cf ( D − , ≈ so ( D − , ֒ → osp (1 | [ D/ ) . the space of solution of the the free Schr¨odinger equation in d = 2 [ D/ − spatial dimensionsspans a representation of the (super)conformal algebra in D − D dimensions.From our results, we would expect that Vasiliev theory in its complete formulation could applyalso to non-relativistic system. It would be challenging to find these systems since they will beso closely related to gravity. Therefore, extensions of our study could lead to an holographiccorrespondence between higher spin theory and a non-relativistic quantum theory.It is worth to mention here that the spin-statistics theorem does not hold in non-relativisticfield theories [41], i.e. statistics and spin may be unconnected. Hence, there is not a priory areason to discard super-commutator product of the non-relativistic particle symmetry generatorsin a first- or a second-quantized theory (cf. [42]). In that case, one may speculate about the exis-tence of a new type of holographic correspondence, between supersymmetric relativistic theoriesand non-relativistic theories which apparently are non supersymmetric but which exhibit a Z graded structure.The results here presented can be also generalized to the harmonic oscillator Schr¨odingerequation, taking advantage of the isomorphism of the Schr¨odinger algebra and the conformalNewton-Hooke algebra [7]. We would like to thank the referee for pointing out that, moregenerally, there is a local diffeomorphism between solutions of linear second-order differentialequations which in particular may be useful to map solutions of the free particle Schrodingerequation to the solutions of any second order Hamiltonian and vice versa . See for instance [43].Bearing in mind this theorem, we would expect that this theorem might be helpful to extend ourresults to any second order hamiltonian system in higher dimensions, at least locally. Acknowledgments
We thank X. Bekaert, M. Hassaine, P. Horvathy, M. Plyushchay and M.A. Vasiliev for valuablediscussions. This work was supported by
Anillos de Investigaci´on en Ciencia y Tecnolog´ıa , projectACT 56
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