Advances in Mathematical Physics
Volume 2016 (2016), Article ID 5739410, 7 pages
http://dx.doi.org/10.1155/2016/5739410

Research Article

Higher Spin Symmetries of the Free Schrödinger Equation

Mauricio Valenzuela

Facultad de Ingeniera y Tecnología, Universidad San Sebastián, General Lagos 1163, 5110693 Valdivia, Chile

Received 20 April 2016; Accepted 26 July 2016

Academic Editor: Angel Ballesteros

Copyright © 2016 Mauricio Valenzuela. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

It is shown that the Schrödinger symmetry algebra of a free particle in spatial dimensions can be embedded into a representation of the higher spin algebra. The latter spans an infinite dimensional algebra of higher-order symmetry generators of the free Schrödinger equation. An explicit representation of the maximal finite dimensional subalgebra of the higher spin algebra is given in terms of nonrelativistic generators. We show also how to convert Vasiliev’s equations into an explicit nonrelativistic covariant form, such that they might apply to nonrelativistic systems. Our procedure reveals that the space of solutions of the Schrödinger equation can be regarded also as a supersymmetric module.

1. Introduction

The Schrödinger group was discovered by Lie [1] and, even earlier, the conserved quantities associated with the Schrödinger invariance were already known to Jacobi [2] (see also [3, 4]). Its name, however, is taken from quantum mechanics [5–9], since it extends the Galilean symmetries of the free Schrödinger equation with dilatation and expansion transformations. The Schrödinger group is isomorphic to the Newton-Hooke conformal group of the harmonic oscillator [8], and it appears also in several contexts, for example, magnetic monopoles [10], vortices [11–14], fluid mechanics [15], and strongly correlated fermions [16, 17]. The Schrödinger group has some infinite dimensional generalizations [3, 16], and it can be also realized geometrically as space-time isometries [18–20]. The Schrödinger symmetry has attracted also renewed interest in the context of nonrelativistic correspondence [21–23], which relates an asymptotic theory on a curved background to a nonrelativistic quantum system [20–23]. This is a consequence of the embedding of the Schrödinger algebras into relativistic conformal algebras [24].

The goal of this paper is to show that Galileo boosts, translations, and mass generators are building blocks for constructing higher-order symmetries of the free Schrödinger equation. This is done by noticing that from Galileo boosts, translations, and mass operator, which satisfy the Heisenberg algebra, we can construct a representation of the Weyl algebra. The latter will span an infinite set of conserved charges, containing in particular all Schrödinger generators. This can be regarded as the nonrelativistic analog of the Eastwood result [25] on the maximal symmetries of the massless Klein-Gordon equation, which is spanned by polynomials in conformal symmetry generators.

Endowing the generators of the Weyl algebra with a (super)commutator product yields the so-called higher spin (super)algebras [26–29]. These algebras are well known in the context of higher spin gauge theory (see also [30–33]). It follows that the free Schrödinger equation exhibits higher spin symmetries, and, reciprocally, Schrödinger symmetry is naturally contained in higher spin theory. As we will see, at the end of this paper, Vasiliev equations of higher spin gauge fields can be written in an explicit nonrelativistic form, by simple identification of the symplectic-spinor indices of the higher spin fields with spatial indices of a nonrelativistic space-time. Indeed, the truncation of the higher spin algebra to its maximal finite-dimensional subalgebra contains the Schrödinger algebra. Reciprocally, the correspondence between symplectic-spinor indices and nonrelativistic spatial indices allows endowing the Schrödinger generators with a supercommutator product, yielding an orthosymplectic-type supersymmetry of the Schrödinger equation.

2. Symmetries of the Schrödinger Equation

Consider -dimensional Heisenberg algebra:

The Weyl algebra, denoted by , can be defined as the algebra of Weyl ordered polynomials of the Heisenberg algebra generators (it is indeed the universal enveloping of the Heisenberg algebra which in suitable basis can be defined as the symmetrized products of the generators, owing to the Poincaré-Birkhoff-Witt theorem (see, e.g., [34])).

An associative algebra can always be endowed with a commutator product, thus yielding a Lie algebra:

Alternatively, as the Weyl algebra is graded over , we can endow their generators with a supercommutator product yielding a Lie superalgebra:

Here, denotes the degree of the generators, and , respectively, for even and odd order polynomials in the generators and of the Heisenberg algebra. The constants of structure of (2) and (3) are derived from those of , and the (super)Jacobi identity follows from the associativity of the Weyl algebra. The algebras (2) and (3) are called higher spin (super)algebras [26–28], since they contain a maximal subalgebra of compact generators under which the remaining generators transform as tensors of arbitrary spin.

In our approach the Heisenberg algebra is composed of a mass-central-charge, Galileo boosts, and translations generators of a nonrelativistic particle:

Consider now the free Schrödinger equation in spatial dimensions:

The dynamics of a quantum operator is given by the Heisenberg equation , which can be written in terms of the Schrödinger operator as

This equation can be also regarded as the first-class constraint associated with the time-parametrization invariance of the free nonrelativistic particle [35].

Equations (5) and (6) imply that constants of motion are symmetry generators, since as they commute with the Schrödinger operator, they leave the space of solutions of the Schrödinger equation invariant.

The product of constant of motion is also a constant of motion. This follows from the Leibniz rule satisfied by the derivative with respect to the time and by the adjoint action of acting on the product of constants of motion. For the free Schrödinger equation, from Galileo boosts and translations,

any polynomial of and will be also a constant of motion. Therefore the free Schrödinger equation admits infinitely many conserved charges, spanned by arbitrary operator functions of and , which contains the Weyl algebra as the basis of the polynomial class of functions.

3. From Galilean to Higher Spin Symmetries

Once the generators and are provided we can form the vector

The commutation relations of and become now

which defines the symplectic matrix . The symmetrized second-order products of generators (8)

commute with themselves as

as it is deduced from (9), generating a representation of the algebra. This representation is usually referred as to “oscillator representation,” since one of their compact generators can be identified with a harmonic oscillator Hamiltonian. Here the Hamiltonian is, however, identified with a noncompact generator, one of the nonrelativistic free particles.

together with yields the commutation relations:

The generators , , and yield the maximal finite-dimensional subalgebra of (2) (see (9)-(11)-(12)):

From we can define the generators:

Now (14) together with (4) yields the Schrödinger algebra :

Other commutators vanish. We stress out that the full Schrödinger algebra is implied by the Heisenberg commutation relation (9) and the definitions (14). Indeed, from (4) and (14), the standard representation of is recovered:

The Schrödinger algebra, having the structure , is subalgebra of (13) and the higher spin algebra (2); that is, .

Rotations, dilations, and expansion generators are second-order operators in Galileo boosts and translations. They generate independent symmetries however, as it is well known. Indeed, the finite transformations generated by and are, respectively (see, e.g., [4, 7]), and , where and are transformation parameters.

The new generators of not contained in the Schrödinger algebra are of second order in spatial derivatives:

These operators are traceless, The nonvanishing remaining commutation relations read

extending to ), which in Galileo covariant notation is generated by (the algebra (19) has been also discussed in a different context [36]) (cf. (13)):

Notice that endowing the generators (19) instead with a supercommutator product, which is possible owing to the grading (3) (indeed, it is the reflection operator, , which induces the grading, as it anticommutes with Galileo boosts, translations, and all their odd powers in the Weyl algebra, whereas it commutes with their even order powers. The symmetric and antisymmetric projections of the wave function can be seen as they were “bosonic” and “fermionic.” Observe also that the odd projection satisfies a Pauli exclusion-like principle: .), yields the superalgebra

with (anti)commutation relations equivalent to (10), (11), and (12) (see the definitions (8), (14), and (17)). It is the maximal finite dimensional subalgebra of (3). Here, the Galileo boosts and translations generators, and , are regarded as supercharges (cf. (17)):

Indeed, the vanishing commutation relations of the Hamiltonian with the nontrivial reflection grading-operator reveal the double degeneracy of the Hamiltonian owing to supersymmetry, which is characteristic in supersymmetric quantum mechanics.

The Schrödinger algebra extension (13) can be seen hence as the bosonic counterpart of the orthosymplectic supersymmetry , using the terminology of [37, 38], where the bosonic counterpart of super-Poincaré was studied. Of course, the introduction of other degrees of freedom such as Clifford or Grassmann variables as in, for example, [39, 40], would yield a more standard type of supersymmetry. The supersymmetry induced by parity under spatial reflections has been widely studied by Plyushchay and collaborators in diverse interacting quantum mechanical system which do not involve fermionic degrees of freedom and called for this reason bosonized supersymmetry or hidden supersymmetry [41–48].

Nonrelativistic 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 system of 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 action principle was proposed in [32]. Here we show that the higher spin gauge theory [28, 30, 31] exhibits also nonrelativistic symmetries.

Vasiliev’s theory makes use of differential forms valued in the higher spin gauge algebra of the type

are completely symmetric in -indices and labels local coordinates of the base space-time manifold. The gauge field (22) may involve also the reflection operator (or Klein operator) in its expansion, as well as Clifford or Grassmann variables, which we omit here for simplicity. The representation of the Weyl algebra can be realized also in terms of commuting -symbols endowed with an associative star product, whereas the higher spin (super)algebra is obtained from the correspondent (anti)symmetrization of the star product. Identifying the Heisenberg algebra of the oscillators with the Galileo-boost/translation generators (8) (see Table 1) allows us to view the expansion (22) as a one-form taking values in the universal enveloping of the nonrelativistic Schrödinger algebra , making explicit its nonrelativistic covariance.

Table 1: The nonrelativistic covariance of Vasiliev’s higher spin theory can be made explicit by identifying the Heisenberg algebra generators ’s with the generators of Galilean boost and translations.

Dimension HS theory
Notation for Heisenberg algebra generators
Dimension NR theory
Correspondence with NR generators

,  

,  

,  

,
,  

,  ,  

In [30] the Heisenberg algebra employed in Vasiliev’s theory is given by , where is the Lorentz metric in dimensions. Thus the formulation of higher spin gravity in general (relativistic) space-time dimensions makes use of Heisenberg algebras generated by vectors (), so the internal phase-spaces have dimension . In three and four dimensions these phase-spaces have, respectively, dimensions and , and the generators of the Heisenberg algebras transform as spinors under the relativistic Lorentz group. Therefore the latter phase-spaces can be labeled instead in terms of Galilean nonrelativistic covariance in dimension , including the time direction, for higher spin gravity in , while the nonrelativistic covariance is (one spatial nonrelativistic direction and the time) for higher spin gravity in dimensions, and it is (two nonrelativistic spatial direction and the time) for higher spin gravity in relativistic space-time. The expected correspondence between higher spin gravity and nonrelativistic quantum mechanics is therefore as given in Table 1.

4. Conclusions

We have shown in a simple way how the free Schrödinger equation enjoys infinite symmetries generated by the Weyl algebra. Since the Weyl algebra can be made covariant under nonrelativistic (Galilean) transformations and also under relativistic (higher spin) transformations, we observe that both the Schrödinger equation and the higher spin gravity have nonrelativistic and relativistic symmetries, depending on the choice of algebra labels.

More generally, since the (unitary) representations of the algebra or the superalgebra admit the embeddings

The space of solution of the the free Schrödinger equation in spatial dimensions spans a representation of the (super)conformal algebra in space-time dimensions or the anti-de-Sitter algebra in dimensions.

From our results, we would expect that Vasiliev theory in its complete formulation could apply also to nonrelativistic system. It would be challenging to find these systems since they will be so closely related to gravity. Therefore, extensions of our study could lead to an holographic correspondence between higher spin theory and a nonrelativistic quantum theory.

It is worth mentioning here that the spin-statistics theorem does not hold in nonrelativistic field theories [49]; that is, statistics and spin may be unconnected. Hence, there is not a priori reason to discard supercommutator product of the nonrelativistic particle symmetry generators in a first- or a second-quantized theory (cf. [50]). In that case, one may speculate about the existence of a new type of holographic correspondence, between supersymmetric relativistic theories and nonrelativistic theories which apparently are nonsupersymmetric but which exhibit a graded structure.

The results here presented can be also generalized to the harmonic oscillator Schrödinger equation, taking advantage of the isomorphism of the Schrödinger algebra and the conformal Newton-Hooke algebra [7]. We would like to thank the referee for pointing out that, more generally, there is a local diffeomorphism between solutions of linear second-order differential equations which in particular may be useful to map solutions of the free particle Schrödinger equation to the solutions of any second-order Hamiltonian and vice versa. See, for instance, [51]. Bearing in mind this theorem, I would expect that this theorem might be helpful to extend my results to any second-order Hamiltonian system in higher dimensions, at least locally.

Competing Interests

The author declares that there are no competing interests.

Acknowledgments

The author thanks X. Bekaert, M. Hassaine, P. Horvathy, M. Plyushchay, and M. A. Vasiliev for valuable discussions. This work was supported by Anillos de Investigación en Ciencia y Tecnología, Project ACT 56 Lattice and Symmetry (Chile), and a CNRS (France) Postdoctoral Grant (Contract no. 87366).

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