Enlarged NH symmetries: Particle Dynamics and Gauge Symmetries

We show how the Newton-Hooke (NH) symmetries, representing a nonrelativistic version of de-Sitter symmetries, can be enlarged by a pair of translation vectors describing in Galilean limit the class of accelerations linear in time. We study the Cartan-Maurer one-forms corresponding to such enlarged NH symmetry group and by using cohomological methods we determine the general 2-parameter (in D=2+1 4-parameter)central extension of the corresponding Lie algebra. We derive by using nonlinear realizations method the most general group - invariant particle dynamics depending on two (in D=2+1 on four) central charges occurring as the Lagrangean parameters. Due to the presence of gauge invariances we show that for the enlarged NH symmetries quasicovariant dynamics reduces to the one following from standard NH symmetries, with one central charge in arbitrary dimension D and with second exotic central charge in D=2+1.


Introduction
In the classification of all kinematical groups in D = 3 + 1 one finds the two nonrelativistic counterparts of dS and AdS symmetries -two Newton-Hooke (NH) cosmological groups generated by two NH nonrelativistic algebras [1,2,3,4]. In comparison with the Galilei algebra, described by the generators J ij (nonrelativistic rotation), P i (space translations), K i (nonrelativistic boots) and H (time translations) in NH algebra one relation is deformed for the finite value R of nonrelativistic (A)dS radius In arbitrary dimension D one can introduce one central charge described by the mass parameter m [P i , K j ] = m δ ij . (1.2) If D = 2 + 1 one can add second "exotic" central charge θ [5,6,7,8,3,9] [ Recently the NH symmetries were enlarged by adding the constant accelerations, described by the generator F i [10]. We denote corresponding algebra by NH and list below the relations satisfied by new generators F i , in arbitrary dimension D with central extensions included: We see that the relation (1.4d) does not provide a finite limit R → ∞; in order to obtain such an Galilean limit we should introduce R-dependent Newton-Hooke mass parameter In such a way the relation (1.2) takes the form and in the Galilean limit we obtain 6) in accordance with the formulae introduced in [11]. Further if D = 2 + 1, one can introduce two "exotic" central charges θ, θ ′ , satisfying besides the relations (1.3) the following relations [10] [ One of the aims of this paper is to enlarge the NH algebra by adding another vectorial Abelian generator R i , and introduce a new extension of NH algebra, which we call doubly enlarged NH algebra. The new algebra describe the symmetries of a class of nonrelativistic reference frames which are characterized also by a relative acceleration linear in time in the flat limit R −→ ∞. We consider a group element of this algebra and we construct the Cartan-Maurer one-forms 1 from which we can calculate the nontrivial Eilenberg-Chevalley cohomology, providind the non-trivial closed invariant two forms which determine the central extensions 2 . In any dimension D the NH Lie algebra has two central charges, and the number of central charges in D = 2 + 1, similarly like for NH Lie algebra [10], is equal to four. 3 We further study the physical consequences of the additional generators F i , R i . We construct the action of a particle with NH symmetries, by the method of non-linear realizations [14]. The action can be written as sum of WZ terms multiplied by central charge parameters. For a generic dimension D the lagrangrean depends on two parameters, but exceptionally if D = 2+1 it depends on four parameters. The parameters appearing in the lagrangean are (besides the dS radius R) in one to one correspondence with the central extensions of the NH algebra. The consistency of the field equations imply the existence of one (two in D = 2 + 1) relations among the two (four) dimensionfull lagrangean parameters. These constraints permit the existence of two local gauge invariances of the lagrangean. If a suitable gauge fixing is chosen the gauge-fixed action can be identified with the one considered in [3,9]. We also consider the restriction to the first order NH quasi lagrangean by imposing the vanishing of the Golstone bosons associated to generators R. If we impose some covariant relations representing inverse Higgs mechanism [15] we recover the NH quasi invariant higher lagrangean of reference [10] The plan of the paper is as follows: in section 2 we will introduce the double enlarged NH algebra and its central extension; the case D = 2 + 1 1 Some of the calculations with forms have being done using the Mathematica code for differential forms EDC [12].
2 See for example [13]. 3 We disregard here in D = 2 + 1 the "trivial" central extension due to the deformation of the commutator [H, J], where J ij = ǫ ij J. is considered separately. In section 3 we construct the most general NH invariant first order lagrangean and its gauge invariances. We also show that the higher order actions considered in [10,11] can be derived from our linear action. Finally in section 4 we provide some conclusions.

Doubly Enlarged Newton-Hooke Symmetries
The nonrelativistic de-Sitter and anti de-Sitter symmetries (see (1.1)) describe respectively the nonrelativistic expanding and nonrelativistic oscillating universes. The cosmological constant κ = 1 R describes the time scale determining the rate of expansion or the period of oscillations of the universe. When κ → 0 we obtain the Galilean group and the standard flat nonrelativistic space-time.
The Newton-Hooke algebra describing nonrelativistic (anti)-de-Sitter symmetries [9,17] was extended in [11] to acceleration -extended Newton-Hooke algebra (see (1.1)-(1.6)). The change of nonrelativistic space-time under the transformations of the acceleration -extended NH group are the following [11] with the following assignements of the parameters a i -spatial translations (generators Let us observe that in the flat limit R → ∞ we obtain from (2.1) the following formulae for acceleration -enlarged Galilean transformations In this paper we propose to extend the formula (2.2) by adding subsequent term The transformations (2.1) and (2.4) leads to the following differential realization of the doubly enlarged NH algebra on nonrelativistic space-time The algebra of these vector fields give the unextended version of the algebra Let us write this Lie algebra in terms of the dual formulation. We introduce the Lie-valued left-invariant Maurer-Cartan (MC) 1-form where g is an general element of the group and L H , L P i , L K i , L F i , L R l , L I ij are 1-forms, whose explicit form depends on the parametrization of the group element. The MC one-form verifies the MC equation dΩ + Ω ∧ Ω = 0, in terms of the 1-forms L's we have (2.8) If D = 2 + 1 the rotations I ij reduce to Abelian rotation I ij = ǫ ij I, and the relations (2.8) take the following form: where L I ij = ǫ ij L I . Now we will construct the extended algebra NH. In order to describe the central charges in arbitrary dimension one should look for the invariant closed 2-forms Ω 2 , with closure property dΩ 2 = 0 following from the relations (2.8), (2.9), and write the 2-form, Ω 2 = dΩ 1 as bilinear exterior product of the MC 1-forms (see e.g. [13]). One obtains in arbitrary dimension D the following two nontrivial cohomology classes of rank 2: In D = 2 + 1 one obtain besides the cohomology classes (2.10a-2.10b) still two "exotic" cohomology classes, namely In order to obtain the Liouville one-forms Θ one should provide a parametrization of the group element. From the forms (2.10a-b) we can construct the centrally extended Lie algebra in arbitrary dimension D. The algebra is given by NH relations (2.6) with the commuting algebra of generators (P i , K i F i , R i ) replaced by the non-vanishing commutation relations: where c and M are two central charges.
If D = 2+1 from (2.11a-b) we obtain two additional central charges θ, θ ′′ : We see therefore that our new enlarged NH algebra in D = 2 + 1 depends on two standard (m and M) and two exotic (θ and θ ′′ ) central charges.

Particle Lagrangean from Nonlinear Realizations
In this section we will construct the most general particle lagrangean (quasi) invariant under the group NH and its gauge invariances. We will use the method of non-linear realizations [14]. As we shall see the most general NH first order invariant action is endowed with gauge symmetries. We also discuss the corresponding NH first order quasi invariant lagrangean as well as the connection with the higher order lagrangean proposed in [10].

General Reparametrization Invariant Action
Let us consider the coset of the unextended algebra d d NH SO(d) . We locally parametrize the coset as follows g = e tH e x P e v K e w F e s R .
( 3 1) The (Goldstone) coordinates of the coset depend on the parameter τ that parametrizes the world line of the particle (see for example [16]). The corresponding Maurer-Cartan one-form In arbitrary dimension D the 1-forms L are Substituting the one-forms in the formulae (2.10a,b) and (2.11a,b) one obtains the following formulae for the corresponding Liouville one-forms, satisfying the relations (2.11c) and In arbitrary dimension D the action is given by and in the D = 2 + 1 we have where * means pullback on the worldline of the particle parametrized by τ . One can fix the diffeomorphism invariance by choosing t = τ . In D dimensions the Lagrangean deduced from(3.6) has the following explicit form The field equations following from (3.8) are

Gauge Invariances and Reduction to Oscillator Dynamics
We see that in field equations (3.11a-b) out of four functions x i (t), v i (t), w i (t), s i (t) two can be chosen arbitrary. Such a property reflects the presence (if condition (3.10) is satisfied) of two-parameter local gauge invariance. Indeed, the action (3.8) depends only on the following two linear combinations of the field variables.
Substituting (3.12a-b) into (3.8) one obtains The dependence of L on linear combinations (3.12a,b) means that the action (3.8) is invariant under the following two local gauge transformations, leaving the variables u i and y i invariant (3.14) By fixing the gauge invariances (3.14) we obtain the following gauge-fixed form of the action (3.8) The field equations derived from (3.16) are (3.17) lead to the hyperbolic oscillator equation with the frequency described by the inverse of dS radius.
In D = 2 + 1 dimensions from (3.5a-b) one obtains the following part of the action depending on two exotic charges θ, θ ′′ Introducing new variable u i (see (3.12a)) one gets the field equations are by consistency (ifṡ i = 0) we have In such a case the action (3.20) depends only on two variables (3.12a-b) The most general action in D = 2 + 1 is described by the sum of the action (3.8) and (3.19). If we write down the field equations it appears that one can derive the following equation The solvability of Eq. (3.24) leads to i.e. we reproduce the constraints (3.10) and (3.22). Using the conditions (3.15a-b) and (3.10), (3.22) one obtains the following general gauge-fixed action in D = 2 + 1 which can be identified with the one presented in [9] (see formula (3.5a,b)) if we pass from dS to AdS case (R → iR), parametrize velocity with opposite sign (v i → −v i ) and put m = − c 2R 2 , θ = 1 2 κ. One can state therefore that the physical content of our model is the same as in the case of standard Newton-Hooke symmetries, with the same number of generators as in Galilean case.

Relation with NH Higher Order Lagrangean
Firstly we derive the NH first order lagrangean from our results presented above. For that purpose we should put equal to zero the Goldstone fields associated with generators R , ( s = 0). For dimension D = 2 + 1 the action density becomes L 1 | s=0 , where the lagrangean L 1 is given by (3.8). Instead for D=2+1 there is an extra term L 2 | s=0 with L 2 described (3.19).
Let us use the variables

28)
We can write the Lagrangeans as If we employ the covariant inverse Higgs mechanism [15], by putting L K i = L P i = 0 and using (3.3), (3.28) we obtain v i = −ẋ i (3.32)

Conclusions
We demonstrated that the action constructed via Cartan-Maurer one-forms for extended NH groups with additional constant and linear acceleration generators (F i , R i ) supplements only a gauge sector of the standard Newton-Hooke dynamics. To achieve such equivalence we have to reduce the number of central charges via suitably chosen relations (3.10) and (3.20). We recall that changing the sign of R 2 (see (1.1)) we obtain from the hyperbolic oscillator (see (3.18)) the standard one, with trigonometric solutions (see [9]). It seems therefore that if consider more general NH algebras with extra Abelian generators, the particle dynamics will be equivalent to the one describing ordinary NH particle. This is due the appearance of gauge symmetries associated the extra generators. It will be interesting to understand better such a property.
We have also seen that if we assume that Goldstone fields associated with R vanish and we impose some covariant relations one recovers the NH higher order lagrangians introduced in reference [10].
Finally we would like to observe that the transformations (2.1)-(2.4), for R −→ ∞, can be extended as follows (4.1) We get in such a way the reparametrization group of Newton's equation of particle in the presence of nonrelativistic gravitational potential V (x, t)(see e.g. [18]).
which is invariant under (4.1) if V ′ = V −ä i x i . It seems interesting to consider the link between the extension of system (4.2) for R finite and the enlarged NH symmetries of arbitrary order.