Schrodinger Equations for Higher Order Non-relativistic Particles and N-Galilean Conformal Symmetry

We consider Schrodinger equations for a non-relativistic particle obeying N+1-th order higher derivative classical equation of motion. These equations are invariant under N(odd)-extended Galilean conformal (NGC) algebras in general d+1 dimensions. In 2+1 dimensions, the exotic Schrodinger equations are invariant under N(even)-GCA.


I. INTRODUCTION
The Schrödinger equation for a non-relativistic free particle is invariant under the scalar projective representation of the Galilei group [1] [2]. In addition the maximal symmetry of (1) is the Schrödinger algebra [3][4] that includes extra generators associated with dilatation, expansion (special conformal transformations in one dimension) and a central charge. The Schrödinger equation is obtained by the canonical quantization of the non-relativistic free particle whose action S = dt M 2 ( dX dt ) 2 is also invariant under the Schrödinger group.
In this paper we will generalize above result (N=1 case) and show that the higher order non-relativistic particle model given by the Lagrangian 1 has the N-Galilean conformal (NGC ) symmetry [7] It has subalgebras, (H, D, C), one dimensional conformal algebra and (H, P P 0 , P P 1 , J), the unextended Galilean algebra and the acceleration extended Galilean algebra [6] with higher 1 The N =3 case was considered in [5] as an example of a realization of the Galilei algebra with zero mass.
The model was also considered in [6]. 2 In [9] it was conjectured that N +1-th order free equations of motion are NGC invariant for any N, odd and even. order accelerations P P j , (j = 2, ..., N). N is interpreted in terms of the dynamical exponent z under the dilation D of the coordinates as Recently it gets much attention on the Galilean conformal symmetry and its extensions especially in condensed matter physics and gravity [10] [11] [12] [9]. It is interesting to see how such symmetries are realized in a simple particle models.
In sect.2 we introduce a particle action invariant under central extension of the NGC algebra (3). In sect.3 we discuss it in quantum theory that how the symmetry is realized in the Schrödinger equation. Even N cases in 2 + 1 dimensions are briefly commented in sect.4 and summary and discussions are in section 5. In appendix we add how the invariant actions are derived using the method of non-linear realization.

II. NGC INVARIANT PARTICLE MODEL
In order to quantize the particle model described by the Lagrangian (2) we construct the Hamiltonian associated with it. Since the theory is higher order in time derivative we introduce auxiliary coordinates X j =Ẋ j−1 , (1 ≤ j ≤ N −1 2 ), X 0 ≡ X and the Lagrange multipliers Y j to get the Lagrangian as The Ostrogradsky momenta [13] are If we use the second class constraint (Y j , P j Y ) = (P j , 0), the independent canonical pairs are (X j , P j ), (0 ≤ j ≤ N −1 2 ) and the Hamiltonian becomes Using the canonical commutators the Heisenberg equationṡ reproduce the Euler-Lagrangian equation of (2), d N+1 dt N+1 X 0 = 0. The Lagrangian (5) is invariant under the NGC symmetry whose hermitian canonical generators are where It proves the transformations generated by G's are symmetry of the Lagrangian (5). Transformations of Y j are given by those of P j . Their commutators are those of the NGC algebra (3). In addition P P j 's are no longer commuting but appears a central charge Z = M [14], In appendix we show that in a non-linear realization method [15] the Lagrangian (5) appears as the left invariant one-form associated with the central charge Z in (11).

III. SCHRODINGER EQUATION
Now we consider the Schrödinger equation associated with the Hamiltonian (7). The canonical pairs (X j , P j ) satisfying (8) are realized as the hermitian linear operators ( 2 ) with the inner product where ψ is the complex conjugate function of ψ. The Schrödinger equation i∂ t ψ = Hψ for where Φ S is the Schrödinger differential operator defined by The Schrödinger equation can be deduced from the action where d is the spatial dimensions. They are satisfying the NGC algebra (3) and (11). These generators commute with the Φ S , showing that they are constant of motion. Therefore the Schrödinger equation (13)   The H transformation is time translation, For the finite scale transformation, under which the action (15) is invariant. For the finite conformal transformation, αG = κC, with For the finite P P j transformations, The non-trivial projective phase [1] ω 1 associated with the P P j transformations is given by where the transformed coordinates are given by We have a projective representation of the NGC group.
Under successive P P j transformations we get the non-trivial 2-cocycle ω 2 (β, β ′ ), 2π The projective invariance of the Schrödinger equation is one to one correspondence with the fact that the higher order Lagrangian (2), or the corresponding Lagrangian (5) is invariant up to a total derivative under the finite P P j transformations [16], where transformations of X j and Y j are , These two properties are related to the fact that NGC algebra has a central extension (11) [17].

IV. EVEN N MODEL IN 2+1 DIMENSIONS
In 2+1 dimensions we can construct a local higher order Lagrangian given by where N is any positive even integer 3 . The Lagrangian equivalent to (29) is The Ostrogradsky momenta [13] are Using the second class constraints the variables X satisfying Note that the model has built in a noncommutative structure in phase space. The Hamiltonian is and the equations of motion for X ≡ X 0 gives d N+1 dt N+1 X 0 = 0. Note that in this case the order of derivatives, (N + 1), is odd. The canonical generators of NGC algebra are where the central charge Z appears in [P P a j , P P b k ] = −i ǫ ab δ N,j+k (−1) The Hamiltonians (7) and (34) of present models are not positive semi-definite as is known in general for higher time derivative Lagrangian systems 4 . In quantum theory generically the system will contain ghost degrees of freedom. One possible procedure is to change the scalar product that has been applied to higher order harmonic oscillator [20] in reference [21].
Another possibility is to eliminate the ghost spectrum by imposing a BRST like operator on the physical states [22].
Possible extensions of the work is to consider the case of the NGC algebra for even N in any dimensions.We could also study the symmetry properties of the fourth order derivative harmonic oscillator [20], its generalizations. We expect in this case we will have a realization of the Newton-Hooke NGC algebra [14]. There are also possible higher order extensions of the Levy-Leblond equation [23] and the associated superconformal algebra [24].

Appendix A: Particle Model action from Non-linear Realization
Here we show how the NGC invariant action (5), for odd N, is derived using the nonlinear realization of the group G/H [15] (see also [26] and references there in), where G is the centrally extended algebra (3) with (11). The left invariant MC form Ω = −ig −1 dg is expanded as and is satisfying the MC equation dΩ + iΩ ∧ Ω = 0. Using the NGC algebra (3) and (11) their components satisfy the MC equations, They are closed under "d" that is equivalent with that the Jacobi identities of the algebra are satisfied. The right hand side of dL Z is the WZ two form closed and invariant in the non-extended algebra.
We parametrize the coset element as g = e iHt e iP jX j e iCσ e iDρ e iZc .
Here and hereafter when there appearX −1 andX N +1 , they are understood to be zero by definition. Note in the present parametrization of the coset L Z does not depend on neither ρ nor σ.
In the method of NLR the particle action is constructed from J invariant one forms. They are L H , L D , L C , L Z and we can use their linear combination as the invariant action.
where * means pullback to the particle world line parametrized by τ . The first term L c depends only on the su(1,1) variables, (t, σ, ρ), and giving one dimensional conformal mechanics Lagrangian [25].
We take the second term L X as the particle Lagrangian now depending on t andX j in the present parametrization of the coset (A3). Using (A5) and subtracting a surface term it becomes HereX N −j in the first term runs overX N+3 2 , ...,X N and they play roles of Lagrange multipliers giving their equations of motion , Using it back into the Lagrangian (A8) the second term can be dropped.X N+1 2 , ...,X 1 are solved iteratively in terms ofX 0 and its derivatives, If we use them back in the Lagrangian we obtain In a static gaugeṫ = 1 the Lagrangians (A8) and (A12) become (5) and (2)  Applying the similar discussions we arrive at the actions (29) and (30).