Poincare wave equations as Fourier transforms of Galilei wave equations

It is well known that the Galilei algebra is a sub algebra of Poincare algebra in one space dimension more. 1 This fact allows us to relate relativistic Poincare and Galilean theories. An interesting point is that Galilei transformations in two space dimensions are contained in the usual Poincare transformations? This enables us to present Poincare spin zero wavefunctions as Fourier transforms of Galilean ones. In the same way it is possible to see the Klein-Gordon equation as the Fourier transform of the Schrodinger equation in one space dimension less. On the other hand, due to the fact that the Poincare algebra is a subalgebra of the complex Galilei algebra in one space dimension more,3 it is possible to do a similar analysis as in the preceding case, i.e., the Schrodinger equation can be obtained as a Fourier transform of the Klein-Gordon equation. c The aim of this paper is to extend the results above quoted to the arbitrary spin case and study the possible relations between the Lagrangian formulations of Poincare and Galilei theories. The organization of this paper is as follows: In Sec. 2 we give a summary of the results of Ref. 2, in Sec. 3 we extend these results to the arbitrary spin case; in Sec. 4 we study some aspects of the Lagrangian formulation; Sec. 5 is devoted to conclusions.


I. INTRODUCTION
It is well known that the Galilei algebra is a sub algebra of Poincare algebra in one space dimension more. 1 This fact allows us to relate relativistic Poincare and Galilean theories.An interesting point is that Galilei transformations in two space dimensions are contained in the usual Poincare transformations?This enables us to present Poincare spin zero wavefunctions as Fourier transforms of Galilean ones.
In the same way it is possible to see the Klein-Gordon equation as the Fourier transform of the Schrodinger equation in one space dimension less.
On the other hand, due to the fact that the Poincare algebra is a subalgebra of the complex Galilei algebra in one space dimension more,3 it is possible to do a similar analysis as in the preceding case, i.e., the Schrodinger equation can be obtained as a Fourier transform of the Klein-Gordon equation.c  The aim of this paper is to extend the results above quoted to the arbitrary spin case and study the possible relations between the Lagrangian formulations of Poincare and Galilei theories.
The organization of this paper is as follows: In Sec. 2 we give a summary of the results of Ref. 2, in Sec. 3 we extend these results to the arbitrary spin case; in Sec. 4 we study some aspects of the Lagrangian formulation; Sec. 5 is devoted to conclusions.

II. POINCARE AND GAll LEI SPIN-O WAVE EQUATIONS
The light cone transformation where PI!' J, and K are the generators of the (3 + 1) Poincare algebra.
If we take the linearized natural representation of the Poincare group acting on (XO, Xl, x 2 , x-\ 1) and subduce it to the Galilei subgroup generated by (2.2), we obtain, in the coordinates (2.1), the transformation which is the natural representation of the (2 + 1) extended Galilei group.
Notice that the s transformation is related to the phase that appears in the projective representations of the GaIilei group.4 Let us now relate the spin-O Poincare and Galilei wave functions by means of a Fourier transformation.A wavefunction 1/J(t,x 1 ,s) scalar under Poincare group can be expressed as t/!(t,x1,S) = f dTje j "'cp,,(t, Xl) , (2.4) where cp,,(t, Xl) is a scalar wave function under the (2 +1) extended Galilei group.Now we can write the Klein-Gordon equation as 0+ m 2 )",(t, Xl'S) = f dTje•j" '(SCP,,) (t,x[), (2.5) where (Scp,,) is the Schrodinger equation for a (2 +1) Galilean particle of mass Tj.
As in the preceeding case, if we take the linearized natural representation of the extended Galilei group acting on (x, y, z, t, S ,I) and subduce it to the Poincare subgroup generated by (2.7), we can obtain, using (2.6), the natural representaiton of the (2 + I) Poincare group.
Let us remark that the invariance of the Klein-Gordon equation implies, by (2.5), the invariance of the Schrodinger equation.Due to (2.9), the inverse is also true.

III. POINCARE AND GAll LEI SPIN S WAVE EQUATIONS
We shall first concentrate ourselves in the Dirac equation for a spin ~ particle of mass m.This equation is invariant If this representation is subduced using (2.2), to the (2 + 1) Galilei algebra, we obtain an equivalent representation of that of Levy-Leblond 4 for spin !.
Using this matrix, we obtain ~J, which is the usual representation of the Galilei algebra.
If we used another description for a spin s particle, we can obtain simialar results.For example, if we use the reformulate (6s +1) Hurley theory8 as a modified BW set,9 i.e., where D is the Dirac operator and r is the projector r = (~ g).t/JBW is a symmetric multispinor of rank 2s.Due to the fact that r commutes with the 9) representation, of SL (2.C), these equations are invariant under the tensor product of the 9) representation.
Therefore, the Pincare (6s +1) theory is the Fourier transform of the Galilei (6s +1) theory.Summing up, we can say that the possible Poincare multispinor wave equations can be written as Fourier transforms of the Galilean ones.On the other hand, we can span a symmetric multispinor in a basis ofSO(3) tensors.Using this fact it is easy to see that the Poincare tensorial wave equa-tions can be written as Fourier transforms of Galilei tensorial wave equations.
We want to remark, that in this section we have only studied the Poincare wavefunctions as Fourier transforms of the Galilei ones, a related analysis can be performed in the inverse situation.

IV. POINCARE AND GAll LEI LAGRANGIANS
The results of the preceeding sections suggest that it must exist a relation between the Poincare and Galilei Lagrangians.This is due firstly to the fact that Poincare wavefunctions are a Fourier transform of the Galilei ones and secondly that the Lagrangians are bilinear functions of the fields.
Let us begin this analysis with the scalar case.The Lagrangian for the Klein-Gordon fields is given by X KG (XO , x) = m 2 t/J*t/J + J" t/J* Jf't/J , (4.1) if we use the coordinate transformation (2.1), (4.1) can be written as X KG (t, Xl'S) = -m 2 t/J*t/J + as t/J*a, t/J + a, t/J*as t/J -aj t/J*aj t/J .where .Y Soh" is the usual Schrodinger Lagrangian with an additive term which is physically interpreted as an energy shift.Moreover we can write being W KG the action for the Klein-Gordon field, and WSChll the action for the Schrodinger field with mass rt in (2 + 1) dimensions.
We want to remark that the relation (4.6) between the actions is not given by a Fourier transform.This means that if we make a variation 8W KG = 0 it does not imply 8W Sch1 , = 0, therefore we can not take (4.6) as a starting point in order to relate Poincare and Galilean theories.Now let us consider the relativistic Lagrangian for a spin-!particle .7 D(XO, x) = t/Jt'l(y"P" -m)t/J.In order to cancel the m-dependence in the rhs of (4.10) and also to recover the dimensionality of the Levy-Leblond field we define ~"(t.x,) "" Yme<""'" ( _ I;;, ;)~"(t.x,).(4.11)Therefore, we have found J !.t' D(t, Xl' s) ds = (\121m) J d1J!.t' LL'r/(t, Xl), (4.12)   where !.t'LL'r/ is the well-known Levy-Leblond Lagrangian for a particle of spin ~.
In the same way as in the spin zero case, we also have being W D and WLL'r/' the actions for the Dirac and Levy-Leblond fields, respectively.The remarks we have done for the spin-O case can serve here unaltered.

IV. CONCLUSIONS
Due to the relation between the Poincare and Galilei groups, the Poincare and Galilei transformations are seen to be contained one each other in one space dimension more.This fact allows us to write the Poincare wavefunctions as Fourier transforms of Galilean ones, and also to find the transformation properties of the Galilei wavefunctions under the (2 + I) Galilei group from the transformation properties of Poincare wavefunctions under the (3 + 1) Poincare group.In particular it is easy to see the projective character of the representations of the (2 + 1) dimensional Galilei group.Taking into account these last properties we can write the Poincare wave equations for arbitrary spin as Fourier transforms of the Galilean ones, so the Dirac equation can be seen as the Fourier transform of the Levy-Leblond equation.
The relation between Poincare and Galilei wavefunctions allows us to the relate the Lagrangians of the two theories.
The generalization to the higher spin cases, the relation between the energy-momentum tensor of the two theories and the possible introduction of external fields in this framework is under investigation.

mass 1 ]
, and now we can write the Dirac equation for t/r6 asA (y"p'l -m)A -1t/r6(t, xl> s) (3.5)where is the Levy-Leblond equation in (2 +1) dimensions for a particle of mass 1] and spin!.Therefore, the Dirac equation is the Fourier transform of Levy-Leblond equations.Let us search the transformations properties of the Ga-that induces a transformation of the operator (3obtained the well-known classical form of the Levy-Leblond operator.It is important to observe that while the operator of (3,9) is invariant under the (2 + 1) Galilei group due to the invariance of the Dirac operator and the relation (3.5), the Levy-Leblond operator (3.10) is not an invariant one. 5,6 If we write (4.7) in terms of the new Dirac spinor ~, we haveX D(XO, x) = ~ tA + --I'I(YI'Pf' -m)A -I~ , (4.8) but under the coordinate transformation (2.1), (4.8) becomes .Y D(t, Xl! s) being E =ia / at and M =ia / as.Using (3.4) and (4.9), we have f !.t' D(t, Xl' s) ds = V2Jd1J ifJ ' lilean spin or ¢>7]LL from those of the corresponding Dirac spinor.Under the Lorentz group, the Dirac spin or transforms as tP~(x') = fiJ(A )tPn(x) .we restrict ourselves to transformations belonging to the (2 + I) homogeneous Galilei group, A = A o , the new Dirac spinor t/r6 transforms as Now let us note that the operator (3.6) can be put in a more standard form, In order to eliminate the m-dependence we can define a new transformation on the spinors ¢> ;ILL(t', x~) = eifLl (AG)¢>'ILL(t, Xl)'(3.8)where!= [V1Xl + !V l 2t ]1] is the known phase which appears in the projective representations of the Galilei group.4 Xl! s)ds = f drt[ -m2rp~rp,,-irta,(rp~rpT,)