Space-time Schr\"odinger symmetries of a post-Galilean particle

We study the space-time symmetries of the actions obtained by expanding the action for a massive free relativistic particle around the Galilean action. We obtain all the point space-time symmetries of the post-Galilean actions by working in canonical space. We also construct an infinite collection of generalized Schr\"odinger algebras parameterized by an integer $M$, with $M=0$ corresponding to the standard Schr\"odinger algebra. We discuss the Schr\"odinger equations associated to these algebras, their solutions and projective phases.


Introduction
It is well known that the most general point symmetries of the Schrödinger equation and the free non-relativistic particle is the Bargmann group [2] with two extra generators, the dilations and the special conformal transformations in one dimension, also called expansions [3,4]. Recently, the action and symmetries of a post-Galilean particle, which includes corrections of arbitrarily high order in 1/c to the non-relativistic particle in an infinite extended Minkowski space, has been constructed. [1] This study is motivated by the importance of higher-order post-Galilean corrections to the Keplerian two-body motion in current investigations of binary gravitational wave sources [5][6][7][8][9].
The action of a post-Galilean particle in an infinite dimensional Minkowski space with coordinates (t (m) , x a (m) ) is obtained expanding the action for a massive free relativistic particle S = −mc dτ −Ẋ µẊ µ in powers of 1/c 2 where the coordinates X µ are One gets a series S (0) + S (1) + S (2) + . . ., with S (n) corresponding to power c 2−2n . The first contributions are [1] S (0) = −mc 2 dτṫ (0) , (1.3) whereẋ 4 (0) = (ẋ 2 (0) ) 2 . As shown in [1], the symmetries of action S (M +1) , M ≥ 0, realize the algebra [10][11][12] [J n ab , J m cd ] = δ cb J Disregarding the total derivative term −m/c 2Mṫ (M +1) which appears in S (M +1) , M ≥ 0, the canonical analysis of the above actions suggest the presence of M + 1 first class primary constraints [1] and therefore the existence of M + 1 gauge transformations. For instance, for M = 0 one has the single constraint while for M = 1 there are two constraints 2 Here p (0) a and p (1) a are the canonical momenta associated to x (0)a and x (1)a , respectively, where E (0) and E (1) are minus the canonical conjugated momenta of t (0) and t (1) .
The presence of the above M first class constraints means that there are M gauge symmetries. Fixing the gauge does not destroy the space-time invariances of the theory and just introduces a different realization. It should be noticed [1,13] that, if in addition to fixing the gauge, one performs an adequate projection in the space of all the x, each of the actions in this sequence yields all the terms of the action obtained by expanding the relativistic action up to the given order.
As is well known [3,4], for M = 0 the Galilean algebra can be extended to the Schrödinger algebra adding two new generators, D, the generator of space-time dilatations, and C, which generates special conformal transformations. Together with the Hamiltonian H, they form a sl(2, R) subalgebra which, in the normalization for D that we will use, reads In this paper we first construct the explicit form of the M + 1 of the mass-shell constraints associated to the action S (M +1) , M ≥ 0. These constraints are first class and allow us to construct the canonical action S c M +1 . We write then the most general canonical generator G linear in the momenta. The request that G be a constant of motion implies a set of partial differential equations for the unknown functions of the generator G that we solve 3 . This allows to construct the most general point symmetry transformation of the canonical action S c M +1 . The algebra of these transformations is a generalization of the Schrödinger algebra, and does not contain, except for the case M = 0, an sl(2, R) algebra, and therefore they are not truncations of the Schrödinger-Virasoro algebra [16], and they are also different from the conformal Galilean algebras (CGA) with dynamical exponent z = 2/N , with N positive integer, since these contain an sl(2, R) subalgebra [17][18][19][20]. The algebras that we obtain in this paper contain the generators H (n) , B (n) , J (n) ab that close under the algebra (1.6) and the new generators D (n) , which generalize the dilatations, and a single generator C of expansions.
We consider the M +1 Schödinger equations associated to the quantization of the post-Galilean particle described by the action S c M +1 . We also study the projective character of the wave function.
The paper is organized as follows. In Section 2 we write the canonical action S c For each M , eliminating the M + 1 momenta p (k) a , energies E (k) and multipliers e (k) one can recover the corresponding action in configuration space [1].
Prompted by the standard Galilean boost and rotation transformations on momenta and energy we propose the following generalized transformations for momenta and energies (2.12) It can be seen that they realize the algebra (1. (k) is quasi-invariant under transformations (2.12) and the corresponding transformations for t (k) , x a (k) (see [1] and Appendix B).
3 Symmetries of post-Galilean particle for M=1 The action (1.5) for the second order expansion of a post-Galilean particle in d+1 space-time [1,13] without the total derivative − m c 2ṫ(2) becomes The canonical momenta are given by 2) and they obey the primary first-class constraints which agree with (2.2) for M = 1. The Dirac Hamiltonian is given by and yields the equations of motioṅ 10)

Space-time symmetries
The canonical generator of space-time symmetries is given by with the η, ξ and δF are unknown functions of t (0) , t (1) , x (0) and x (1) , so that the space-time symmetries are obtained as 1) , and then δL 2 = d dτ δF . The equationĠ = 0 allows one to write the following Killing equations [14,15] for the space-time symmetries of S 2 , for a, b = 1, . . . , d, and with ∂ 0 = ∂ . These PDE can be integrated starting from the trivial ones (3.14)-(3.17), and one gets the unique solution given by The non-zero (or non-constant) value of δF is associated to the fact that we dropped the total derivative − m c 2ṫ(2) from the action S (2) . Had we kept that term, we would have obtained δt (2) = − c 2 m δF (plus an arbitrary constant corresponding to shifts in t (2) ). The generator of the point transformations is from which the individual generators can be defined,

Extended space-time algebra
Using the Poisson brackets (2.10) for M = 1 one can compute the dilatation weights ∆ X of the generators, {X, D} = ∆ X X given in Table 1. The standard dynamical exponent, given by the quotient of the weights of H (0) and P (0) , is z = 4/3, in contrast to the z = 2 value of the M = 0 case. The remaining non-zero brackets among generators are plus the rotation algebra of J (0) with itself and with all the generators with vector indexes. The central extension H (2) = m/c 2 appears in both mixing-level translation-boost brackets Table 1. Dilatation weights of the symmetry generators for L 2 . Weights of t (0) , t (1) , x (0) and x (1) are the opposites of H (0) , H (1) , P (0) and P (1) , respectively.
Notice that D (1) transforms B into B, P into P and H into H, but changes the level from (0) to (1) in each case. In this sense it acts like a higher order dilatation, and we will refer to it as a generalized dilatation.
The generators H (0) , H (1) , D, C and D (1) form a solvable, indecomposable 5-dimensional subalgebra. This is in contrast with the case of the Galilean particle, where H = H (0) , C and D are a realization of the semisimple algebra sl(2, R).
Non-relativistic systems with higher order derivatives and with an extended phase space x (n) , p (n) with a single dilatation and Hamiltonian have been proposed, see for example [21][22][23][24].

Generalized Schrödinger algebras
We could proceed now to the M = 2 action and follow the same procedure as before. However, for higher M the computations become very involved quite rapidly and, in particular, solving the resulting system of PDE associated to the conservation of G requires the use of computer algebra packages.
Instead, in order to obtain results for arbitrary M , we rely on the knowledge of the M + 1 constraints presented in Section 2 and propose a generalization of the extended generators found for M = 0 and M = 1. Invariance of the constraints under the extended generators, together with the quasi-invariance of the kinetic terms in the canonical action, justify this approach. The form of the generators is further validated by the closure of the Poisson bracket algebra of generators.
Using the Poisson brackets (2.10) one can compute the action of these generators on the canonical variables x k , p (k) , t (k) , E (k) , k = 0, 1, . . . , M , and the results are given in Appendix B. For boosts and rotations the transformations are, after multiplying by the corresponding parameters, those given in (2.12).
That the above generators correspond to symmetries of S c M +1 is proved as follows. First one can check (see Appendix C) that the constraints φ (k) , k = 0, 1, . . . , M are invariant under the above transformations, that is, ab }. Furthermore, using the results in Appendix B and the commutation of the transformations with the derivation with respect to τ , one can prove that is also invariant, except for the transformations corresponding to the boosts and the special conformal transformation, for which 4 This means that the canonical Lagrangian in S c M +1 is not invariant under the full set of transformations, but it is quasi-invariant, with where µ is the parameter of the special conformal transformation and the v a (k) are the boosts parameters. This agrees with the known result for M = 0 and the result for M = 1 obtained in this paper and, as in those cases, δF (M ) is associated with the dropping of the total derivative −m/c 2Mṫ (M +1) in S (M +1) . This concludes the proof that our generators yield symmetry transformations of the canonical action. Furthermore, they form a closed algebra. Indeed, the brackets of the D 4 We use δ B,j a to denote the transformation under B a . This kind of notation will also be employed for the other generators later on.   Although some of the generators can be redefined so that some of the structure constants become independent of the level M , as was done for the D, D (k) , there are some structure constants for which the dependence on M cannot be erased, that is, increasing M not only brings in new generators but it also changes the brackets of some of the old ones.

Generalized Schrödinger equation and projective invariance
The quantization of the systems that we have considered can be performed by imposing the canonical constraints on the physical states of the corresponding Hilbert space. For M = 1 we have two constraints and we obtain a set of two generalized Schrödinger equations, where Ψ(t (0) , t (1) , x (0) , x (1) ) is the wave function of the physical state in coordinate representation. Working in d = 1 for simplicity and looking for solutions of the form equation (5.2) can be solved by separation of variables with separation constant ε to obtain where A ± are arbitrary constants. The separation constant ε characterizes the dependence of the wave function on t (1) , x (1) , and is the common eigenvalue of the operators corresponding to E (1) and c 2 2m P (1)2 , that is, i∂ t(1) and − c 2 2m ∂ 2 x(1) , respectively. Each of the Ψ ± (t (1) , x (1) ) can be substituted into (5.1) and one obtains a first-order PDE for Ψ 0 , Equation (5.4) can be solved by the method of characteristics. Imposing the initial condition Ψ 0 (0, x (0) ) = F (x (0) ) at t (0) = 0, with F an arbitrary smooth function, one obtains, where F + and F − are the arbitrary functions corresponding to the ± signs in (5.4). Finally, the total wave function solution to the system of Schrödinger equations is where we have written a ε sub-index to indicate the dependence on the parameter ε. Alternatively, we can identify the solution using p to refer to the two eigenvalues ±p of the momentum operator −i∂ x (1) for the two components, with ε = c 2 /(2m)p 2 , and write

Projective phase
We will discuss here the transformation properties of the above wave functions and Schrödinger equations and the associated projective phases under the post-Galilean transformations. For simplicity we will work again in d = 1. We will discuss the case of expansions in detail and summarize the results for the other transformations. Since we have d = 1 we do not consider rotations and generalized rotations. Also, it should be taken into account that the finite transformations presented below are those of the individual generators.
Let Ψ be a solution to (5.1) and (5.2), and letΨ be the transformed solution under a finite C transformation. Assume thatΨ and Ψ are related bŷ If we demandΨ to satisfy (5.1) and (5.2) in the transformed coordinates one gets, after some algebra, the following set of PDE for ϕ ∂ϕ whose solution is Since the Jacobian is trivial, an imaginary part for the constant is not needed to compensate for a change in the measure in x (0) , x (1) space, and we can take it equal to zero. Again, this is different from what happens in the M = 0 case, for which the projective phase acquires an imaginary part, One has then so that |Ψ| 2 dx (0) = |Ψ| 2 dx (0) , as desired.
Returning to the M = 1 expansions, we havê Notice that the projective phase can also be obtained by iteration of the infinitesimal δF corresponding to expansions, with δ n F = δ(δ n−1 F ). Since δF = m/(2c 2 )µx 2 (0) and δx (0) = 0 for expansions in the M = 1 case, the contributions are null after the lineal one. For a general discussion about projective phase, central extensions and invariance up to a total derivative of the Lagrangian see [25][26][27].
As a check, let us assume that Ψ is a solution of (5.1), (5.2) given by (5.6), and let us prove that thenΨ has the same form. Using (5.6), the right-hand side of (5.20) is where we have used thatt (0) = t (0) ,x (0) = x (0) . Expressing now t (1) and x (1) in terms of the transformed variables one gets The extra terms can be combined with the one withx 2 (0) and one gets the perfect square whose exponential can then be absorbed into F ± to yield the correspondingF ± for Ψ, giving it the same form as in (5.6) but in transformed variables. Notice that the parameter ε does not change when going to the transformed coordinates, since for a given solution it corresponds to the value of E (1) , and under the expansions one haŝ . This is not, however, true of some of the other transformations and the change of E (1) , or of c 2 2m p (1)2 , must be taken into account.

2.
Boosts B (0) . The finite transformations for parameter v (0) = v are in this casê and the projective phase is 3. Generalized boosts B (1) . The finite transformations corresponding to parameter v (1) = v arex The projective phase is a constant which, however, cannot be taken as zero and in fact must be imaginary ϕ = i 2 3 λ, to compensate for the change in the measure in (1) . The finite transformations corresponding to parameter λ (1) = λ arex

Generalized dilatations D
and the projective phase is trivial. No constant imaginary part for ϕ is needed, since dx (0) dx (1) = dx (0) dx (1) in this case. 6. Time shifts and space translations H (0) , H (1) , P (0) , P (1) . The finite transformations are just shifts in the corresponding variables, the momenta do not transform and the projective phase can be chosen as zero in all the cases.

Schrödinger equation with higher derivatives
The way that we have constructed the Schrödinger equation of our system corresponds to what is known as weak quantization, where the constraints are imposed as operators on the states of the system. One can also consider a reduced space quantization, in which the gauge invariance is broken and a Hamiltonian in then computed. We can do this for our action (1.5), disregarding the total derivative, by imposing the two gauge conditions [1] which is of fourth order. We can also obtain an equation of this order by plugging (5.2) into (5.1), which is also of fourth order and with the same coefficients, but lacks the term with second derivatives in x (1) . One can conclude then that the system described by action (1.5) is one for which the two quantization procedures yield different results. Nevertheless, one should notice that the wave function obtained by using the gauge fixing (5.47) in (5.7), which by construction is a solution of (5.51) with t (0) = t, is also a solution of (5.50).

Conclusions and outlook
In this paper we have constructed the most general point symmetry transformations of the post Galilean actions [1] that can be obtained from the Minkowskian action for a massive particle. The algebras obtained are generalizations of the ordinary Schrödinger algebra [3,4]. Besides the generalized Galilean transformations, they contain dilatations, D, generalized dilatations D (k) and expansions C. The algebras are different from extensions of Galilean conformal algebras with dynamical exponent z = 2/N , with N positive integer, since these contain an sl(2, R) subalgebra [17,18]. Using a weak quantization procedure, we have introduced an Schrödinger equation for the post-Galilean particle that consists of M + 1 partial differential equations, up to second order in derivatives, for a wave function living in a generalized space. Like the case of ordinary Schrödinger equation, the wave function supports a ray representation of the symmetry group, and we have calculated the projective phase for each transformation. The symmetries of generalized Schrödinger equations in this paper are different from the symmetries of the higher order Schrödinger equations [24].
If we consider the reduced space quantization the corresponding Schrödinger equation is a single differential equation of fourth order. The two procedures of quantization do not coincide in general. Further investigation of the difference between the Schrödinger equations obtained from the weak and reduced space quantizations, and the generalization of this fact for the actions S (M +1) , will also be the subject of future work.
It will be interesting to study the relation of the higher order Schrödinger equation with the expansion up to order v 2 /c 2 of the square root of the Klein-Gordon equation, see for example [28,29], and if it is possible to introduce interaction terms in the new Schrödinger equation that we have found.
Under generalized dilatations one has otherwise, and we have weakly invariance.