Symmetries of the Free Schr\"odinger Equation in the Non-Commutative Plane

We study all the symmetries of the free Schr\"odinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schr\"odinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.


Introduction and results
The symmetries of a free massive non-relativistic particle and the associated Schrödinger equation have been investigated. The projective symmetries of the Schrödinger equation induced by the transformation on the coordinates (t, x) are well known. They form the Schrödinger group [12,19,20,23] that, apart from the Galilei symmetries, contains the dilatation and the expansion. Recently Valenzuela [24] (see also [4]) discussed higher-order symmetries of the free Schrödinger equation. These symmetry transformations form an infinite-dimensional Weyl algebra constructed from the generators of space-translation and the ordinary commuting Galilean boost. The extra symmetries that do not belong to the Schrödinger group correspond to higher spin symmetries. These transformations are not induced by the transformations on the coordinates but they map solutions into solutions of the Schrödinger equation.
In the case of 2+1 dimensions, the Galilei group admits two central extensions [2,5,14,15,21], one associated to the exotic non-commuting boost and other appearing in the commutator of the ordinary boost and spatial translations. The non-relativistic particle in the non-commutative plane was introduced in [22] by considering a higher order Galilean invariant Lagrangian for the coordinates (t, x) of the particle. A first order Lagrangian depending on the coordinates (t, x) This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html and extra coordinates v was introduced in [9]. For these Lagrangians there are two possible realizations, one with non-commuting (exotic) boosts, and the other with ordinary commuting boosts [5,16] (see [15] for a review).
In this paper we study all the infinitesimal Noether symmetries of a massive free particle in the (2 + 1)-dimensional non-commutative plane. The Noether symmetries are constructed from the Heisenberg algebra of commuting boosts X i and the generators of translations P i , {X i , P j } = δ ij , i, j = 1, 2, all of which are constants of motion and are written explicitly in terms of the initial conditions. The algebra of these symmetries is the infinite-dimensional Weyl algebra associated with the Heisenberg algebra. A general element of the Weyl algebra is given by G(X i , P j ). The generators given by higher degree polynomials do not form a closed algebra for any finite degree. These infinite symmetries are the non-relativistic counterpart of all the symmetries of the free massless Klein-Gordon equation [10]. There is no known realization of this Weyl algebra for an Schrödinger equation with interaction. These symmetries could be useful to construct a non-relativistic analogue of Vasiliev's higher spin theories [25].
The subset of generators constructed up to quadratic polynomials of (X i , P j ) form a finitedimensional sub-algebra, which in turn contains the 9-dimensional Schrödinger algebra. We study the realization of this algebra in the classical unreduced phase-space, as well as in the reduced one, the later appearing due to the presence of second class constraints. We also study all the symmetries of the free Schrödinger equation in the non-commutative plane. The symmetries are in one to one correspondence with the Noether symmetries of the free particle in the noncommutative plane. This analysis is done in the quantum reduced phase space, as well as in the extended one. In the extended space we impose non-hermitian combinations of the second class constraints. In this case we consider two representations for the physical states, namely a Fock representation [16] and a coordinate representation. We study the Schrödinger subalgebra in detail, and we show the equivalence between the reduced and extended space formulations. We show that, in general, the quadratic (and higher) generators in the extended space contain second order derivatives and hence do not generate point transformations for the coordinates.
The organization of the paper is as follows. In Section 2 we construct all Noether symmetries of the massive particle in the non-commutative plane. Section 3 is devoted to the study of the quantum symmetries of the Schrödinger equation.
2 Classical symmetries of the non-relativistic particle Lagrangian in the non-commutative plane In this section we introduce a first order Lagrangian describing a particle in the non-commutative plane [9], and present the corresponding Hamiltonian formalism. The main result of the section is the construction of all the Noether symmetries of the non-relativistic particle in the non-commutative plane (equations (2.11)-(2.14) and the ensuing discussion). For the sake of completeness, we review the construction of the standard and exotic Galilei algebras and of the Schödinger generators [3,5,15,16,17]. We also perform the reduction of the second class constraints of the system for later use in the quantization in the reduced phase space. The first order Lagrangian of a non-relativistic particle in the non-commutative plane, see for example [9], is given by where the overdot means derivative with respect to "time" t. This Lagrangian can be obtained using the NLR method [7,6] applied to the exotic Galilei group in 2 + 1 dimensions 1 ; see [1] for the case of exotic Newton-Hooke whose flat limit gives (2.1). The coordinates x i 's are the Goldstone bosons of the transverse translations and v i 's are the Goldstone bosons of the broken boost. The v i 's and κ are dimensionless. The Lagrangian (2.1) gives two primary second class constraints where p i and π i are the momenta canonically conjugate to x i and v i . The constraints (2.2) satisfy relations and the Dirac Hamiltonian is up to quadratic terms in the constraints. From the canonical pairs (x, v, p, π) we can get a new set of canonical pairs (x,ṽ,p,π) given by In terms of the new variables the constraints (2.2) become a canonical pair, The position and momentum of the particle are expressed as The phase space is a direct product of two spaces. One is spanned by (ṽ,π) with the con- and thus classically trivial. The other one is spanned by (x,p) with the Hamiltonian (2.7). It is a system of a free non-relativistic particle in 2 + 1 dimensions but with the coordinatesx i . In the classical reduced phase space defined by the second class constraints (2.8) the coordinates x i become non-commutative (see also Subsection 2.1), (2.9) particle proposed in [22]. It can be obtained from (2.1) using the inverse Higgs mechanism [18].
If we consider a point transformation (x, v) → (y, u) which is the Lagrangian of a free non-relativistic particle with the commutative coordinates y i . Although it has a form of free particle we keep x i as the "position coordinates" of this system. Local interactions would be introduced at the position x i rather than y i . The coordinates y i in (2.10) are identified with the commuting coordinatesx i in (2.6), while x i are non-commutative as in (2.9). All the Noether symmetries are generated by constants of motion which are arbitrary functions G(X i , P j ) of When computing the variation of the Lagrangian (2.1) under (2.12), the (p i , π i ) are replaced, using the definition of momenta (2.2), by It follows that the variation of the Lagrangian becomes a total derivative, (2.14) All these Noether symmetries generate an infinite-dimensional Weyl algebra. The Weyl algebra, denoted by [h * 2 ], can be defined [24] as the one generated by (the Weyl ordered) polynomials of the Heisenberg algebra generators, is the infinitedimensional algebra of a particle in the non-commutative plane. These infinite symmetries are the non-relativistic counterpart of the complete set of symmetries of the free massless Klein-Gordon equation [10]. The existence of a realization of this Weyl algebra for an interacting Schrödinger equation is an interesting open question.
There are finite-dimensional subalgebras of the higher spin algebra whose generators are constructed from the product of generators X i , P j up to second order: (2) is the Schrödinger algebra 2 in 2D, whose generators are those of the Galilean algebra X i , P i , H, J, together with the dilatation, D, and the expansion, C.
Let us restrict now to Galilean and Schrödinger symmetries. We start by considering the Galilean symmetries of (2.1). The action is invariant under translations, boosts, rotations, and time translations The corresponding Noether charges of translations and boosts are given by while the angular momentum is Together with the total Hamiltonian (2.3), they generate the exotic Galilei algebra [2,5,14,15,21] {H, J} = 0, From this, it may seem that the Lagrangian (2.1) gives a phase space realization of the (2 + 1)dimensional Galilei group with two central charges m, κ. However, one of the central charges is trivial since, if we modify the generator of the boost as in [5,13], one gets that (H, P,K, J) verifies the standard Galilean algebra without κ. 3 Physically, the result of changing the boost generators is a shift in the parameter of the translations Note that the modified boost generatorsK i are proportional to the coordinates at t = 0, X i =x i (0), that verify {X i , X j } = 0, and we have a realization with only one non-trivial central charge associated to the mass of the particle 4 . The Schrödinger generators are those of the Galilean algebra X i , P i , H, J, and the dilatation, D, and the expansion, C, given by In the same spirit, we also redefine the generator of rotations as which, up to square of constraints, coincides with (2.15).
The new, non-zero Poisson brackets are The transformations of the coordinates x i , v i under dilatation and expansion are obtained from (2.12) as where α and λ are the corresponding infinitesimal parameters.

Reduction of second class constraints
The classical symmetry algebra is also realized in the reduced phase space defined by the second class constraints Π i = V i = 0. The Dirac bracket is and yields In this space, the symmetry transformations are generated using the Dirac bracket and the reduced generators, which can be obtained by substituting v i = p i /m, π i = −κ/(2m) ij p j into the standard ones.
The infinite Weyl symmetries are generated by In particular the Schrödinger generators are given by [3] They generate the Schrödinger algebra with the Dirac bracket, sinceK generate a Heisenberg algebra: Symmetry transformations are generated either using the Poisson brackets in the original phase space or using the Dirac brackets with the reduced generators, (2.17)-(2.23). For example the "exotic Galilei" generators K i satisfy and generate "standard(covariant) Galilei" transformation of (x i , p i ) as The "standard Galilei" generatorsK i satisfy and generate "exotic Galilei" (non-covariant) transformations of x i , p i ,

Quantum symmetries of free Schrödinger equation in the non-commutative plane
In this section we will study the quantization of the model at the level of the Schrödinger equation and their symmetries. We will quantize it in two approaches, one in the reduced phase space and the other in the extended phase space.

Quantization in the reduced phase space
In the classical theory, x i has a nonzero Dirac bracket {x i , x j } * as in (2.16) in the reduced phase space. Since Dirac brackets are replaced by commutators in the canonical quantization, one cannot have a x i -coordinate representation of quantum states 5 . To discuss symmetries of Schrödinger equations we introduce new coordinates such that The coordinate y i is the one introduced in (2.10) and q i is its conjugate. In these coordinates, the Schrödinger equation (i∂ t − H)|Ψ(t) = 0 takes the form corresponding to a free particle for the wave function Ψ(y, t) = y|Ψ(t) ,ŷ i |y = y i |y , y|y = δ 2 (y − y ), i.e.
Note that y i are not covariant under exotic Galilei transformation generated by K i but covariant under the Galilei transformation generated byK i The position operators, covariant under K i , arê They are hermitian sinceŷ i = y i ,q i = −i∂ y i , with appropriate boundary conditions on Ψ (y, t), are hermitian. Although in the free theory we are able to work with both the commutativeŷ i = y i and the non-commutativex i = y i − κ 2m 2 ij (−i∂ y j ) position operators, this may not be the case in an interacting theory. For example, if we consider an interaction with a background electromagnetic field, which introduces couplings with a source at position x i , the non-commutative coordinates are naturally selected (see, for example, [8,9,15,17]). If we denote generically by G (R) (t, x, p) = G(X, P )| Π=V =0 the generators of the Weyl algebra in the reduced classical space, the generators in this quantization are given bŷ withq i = −i∂/∂y i and with the appropriate dealing of operator ordering. The knowledge of all the symmetries of the Schrödinger equation in terms of the coordinates y i ,ŷ i is the non-commutative analog in 2 + 1 dimensions of the high spin symmetries of the relativistic massless Klein Gordon equation [10]. The Vasiliev [25] non-linear theory has these high spin symmetries. In this sense these high spin-nonrelativistic symmetries could be useful in order to construct a non-relativistic Vasiliev theory [24].
We consider next in detail the Schrödinger generators, given bŷ

4)
where a Weyl ordering has been used forD (1) andĈ (1) . These generators are hermitian operators when acting on the wave functions Ψ(t, y). Furthermore, they obey the abstract quantum Schrödinger algebra off shell, with non-zero commutators given by Using these, together with  where the α i are the parameters of the transformations. In particular, for the on-shell Schrödinger transformations one has where the coordinate transformations of (y, t) are those of the N = 1 conformal Galilean transformation, and the multiplicative factor is e A+iB , with A and B real functions of the coordinates and of the parameters of the transformation given by (see, for instance, [11,23]

4) (spatial translations and boost)
The difference with respect to the transformation of the ordinary Schrödinger equation is that in the non-commutative case the coordinates that are transformed by conformal Galilean transformations are the canonical ones y i , and not the physical position of the particle, x i .
The invariance of the solutions of the Schrödinger equation under a general element of the Weyl algebra can be proved using the invariance under the generators of the Heisenberg algebra and the commutators (3.10).

Fock representation
In order to quantize the model in the extended phase space the second class constraints (2.2) are imposed as physical state conditions by taking their non-hermitian combinations as in [1]. We first consider the canonical transformation (2.4) that separates the second class constraints as new coordinates. It is realized at quantum level as a unitary transformatioñ For example, It is useful to introduce the complex combinations of the phase space variablesπ ± =π 1 ± iπ 2 andṽ ± =ṽ 1 ± iṽ 2 , which allow us to introduce two pairs of annihilation and creation operators with nonzero commutators [ã ± ,ã † ± ] = 1. Using the Fock representation for (ṽ,π) and coordinate representation for (x,p), any state of this system is described by where |n + , n − is the eigenstate ofÑ ± =ã † ±ã ± with eigenvalues n ± ∈ N ∪ {0} and |y is the eigenstate of commuting operatorsx i with eigenvalue y i . They are normalized as n + , n − |n + , n − = δ n + n + δ n − n − , y|y = δ 2 (y − y ).
In the quantization in the extended phase space the second class constraints (2.2) are imposed as physical state conditions by taking their non-hermitian combination, a ± |Ψ phys (t) = 0. (3.12) This means that physical states are minimum uncertainty states in (ṽ,π). Condition (3.12) selects out only the n + = n − = 0 state, so that Φ n + n − (y, t) = 0 except for Φ 0,0 (y, t) ≡ Φ 0 (y, t), The Schrödinger equation is and thus The generators of the Weyl algebra are given in the extended space as polynomials G(X, P ) of the operator equivalent of (2.11), and, since they commute withã ± andã † ± , physical states remain physical 6 . They act on the physical states as Ψ phys (t) → |Ψ phys (t) = e iG(X,P ) |Ψ phys (t) and it turns out that the transformation of the wave function Φ 0 (y, t) is Φ 0 (y, t) = e iG(X,P ) Φ 0 (y, t) = e iG(y−t(−i∂y),(−i∂y)) Φ 0 (y, t).
This transformation has the same form as the one in the reduced phase space generated by (3.2)-(3.8). Then the wave function in the reduced space Ψ(y, t) = y|Ψ(t) and Φ 0 (y, t) = y| ⊗ 00|Ψ(t) that appear in the extended space quantization are identified. Note that in the former y| is eigenstate ofŷ i = x i + κ 2m 2 ij p j in (3.1) but y| in the latter is eigenstate ofx i that are commuting in the extended space.
We can see now how the non-commutativity of the position operators appears.x ± = x 1 ± ix 2 are commuting in the extended phase space. Using (2.4) we write In the reduced space quantization procedure, theã ± are effectively put to zero and x ± becomes a non-commutative operator on |Ψ(t) . On the other hand in the quantization in the extended space, expectation values of the position operators between two physical states are given by = dy Φ 0 (y, t) y ± ± i κ 2m 2 (−2i∂ y ± ) Φ 0 (y, t).
It is useful to consider the unitary transformation U in (3.11) on the creation and annihilation operatorsã ± ,ã † ± , a + = U † a + U = a + ,ã − = U † a − U = a − − κ 2m 2 p − . (3.13) The quantization in the extended phase space can be also done by considering the constraint equations (3.12) in terms of the operators a ± , a † ± . The physical state conditions (3.12) are a + |Ψ phys (t) = 0, p − − 2m 2 κ a − |Ψ phys (t) = 0, and |Ψ phys is a coherent state of a − with eigenvalue κ 2m 2 p − [16]. In this representation, the Schrödinger generators are hence do not generate point transformations for the coordinates x, v. This is in agreement with the results obtained in the reduced space quantization and the Fock space representation. In any case, the fact that the linear generators commute withŜ 1 ,Ŝ 2 andŜ 3 allows to prove that the quadratic ones also commute, and thus generate symmetries of the Schrödinger equation of the free particle in the non-commutative plane.