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Nonrelativistic conformal transformations in 3 Lagrangian formalism 1 0 2 n K. Andrzejewski∗, J. Gonera, A. Kijanka-Dec a J 8 Department of Theoretical Physics and Computer Science, ] University of L o´d´z, h p - Pomorska 149/153, 90-236 L o´d´z, Poland h t a m [ Abstract 1 v The conformal transformations corresponding to N-Galilean con- 1 formal symmetries, previously defined as canonical symmetry trans- 3 formations on phase space, are constructed as point transformations 5 1 in coordinate space. . 1 0 3 1 Introduction 1 : v In the recent paper [1] we have constructed, using the orbit method [2], i X the general Hamiltonian system on which the centrally extended N-Galilean r conformalalgebra[3](group)actstransitivelyassymmetrygroup(thecentral a extension is possible for N - odd in any space-time and for all N in (2+1)- dimensional one [4]-[6]). It appears that any such system consists of ”external” variables forming standard phase space and the ”internal” ones: spin (related to SU(2) subgroup) and pseudospin (related to SL(2,R) subgroup). The dynamics of external and internal variables are decoupled. Neglecting the internal variables one obtains the free dynamics governed by higher- derivative theory [6] (in fact, the necessity of considering higher-derivative theory was revealed in [7, 8]). ∗ e-mail: [email protected] 1 In Ref.[1] the conformal transformations were introduced as canonical symmetry transformations acting on phase manifold. Here we complete the picture by showing that, within the formalism of higher-derivative theories, the conformal transformations can be defined as point transformations acting on configuration space. Using the results of Ref. [1] we derive the form of point transformations and show that the higher-derivative Lagrangian de- scribing free motion is invariant (up to a total derivative) under the action of these transformations. Thepaperisorganizedasfollows. InSection2weremindthemainresults of Ref. [1] and derive the explicit form of canonical symmetry transformations. In Section 3 and 4 the point transformations on configuration space are defined which coincide ”on shell” with these introduced in Section 2 and it is shown that they are Noether symmetries of free higher-derivative La- grangian with integrals of motion corresponding to the ones on Hamiltonian level (in order to do this we construct an integral of motion for an arbitrary higher-order Lagrangian). Section 5 is devoted to concise conclusions. Some technicalities are relegated to the Appendix. 2 Hamiltonian formalism The N-Galilean conformal algebra has the following structure. First, we have the direct sum of su(2) algebra (spanned by J ’s) and sl(2,R) one k (spanned by H,D and K) [Ja,Jb] = iǫ Jc, abc (1) [D,H] = iH, [D,K] = −iK, [K,H] = 2iD. It is supplemented by 3(N + 1) (in general case of d-dimensional space – d(N + 1)) dimensional abelian algebra which carries D(1,N2) representation of SU(2) × SL(2,R) and is spanned by the generators Ca, a = 1,2,3; i = i 0,1,...,N. The relevant commutation rules involving Ca read i [Ja,Cb] = iǫ Cc, [H,Ca] = −ijCa , j abc j j j−1 N (2) [D,Ca] = i( −j)Ca, [K,Ca] = i(N −j)Ca . j 2 j j j+1 For N odd (and also N even in 2+1 dimensions) the algebra defined by eqs. (1) and (2) admits the central extension [5, 6]. [Cja,Ckb] = iδabδN,j+k(−1)k−2j+1j!k!M, (3) 2 with M being additional central generator. The above algebra can be integrated to the group (SU(2) × SL(2,R)) ⋉ R3N+4 where R3N+4 is nilpotent group and the semidirect product is defined by the D(1,N2)⊕D(0,0) representation of SU(2)×SL(2,R). The question arises what are the dynamical systems exhibiting the symmetry described by N-Galilean conformal group. In the case of centrally extended algebra the answer was given in Ref. [6]. The results obtained there have been generalized in Ref. [1] to the space of arbitrary Hamiltonian system on which our group acts transitively. Below we discuss the proper- ties of the system constructed by Gomis and Kamimura [6]; the general case differs by the existence of additional internal degrees of freedom [1]. The phase space is parametrized by the canonical variables qa, pa, a = k k 1,2,3, k = 0,1..., N−1, obeying 2 {qa,pb} = δ δ . (4) i k ab ik Defining N−1 1 2 h(t) = 2m(p~N2−1)2 + ~qk~pk−1, k=1 X N−1 2 N d(t) = ( −k)~q ~p , k k 2 k=0 X (5) N−3 m N +1 2 2 k(t) = (~qN−1)2 − (N −k)(k +1)~qkp~k+1, 2 2 2 (cid:18) (cid:19) k=0 X N−1 2 ~j(t) = ~q ×p~ , k k k=0 X one finds the Noether charges corresponding to the generators of Lie algebra 3 (1)-(3) h = h(t), d = d(t)−th(t), k = k(t)−2td(t)+t2h(t), ~j =~j(t), (6) (−1)j−N2−1 jk=0 (j−j!k)!tj−k~pk, 0 ≤ j ≤ N2−1, ~cj = (−1)j−N2−1 PkN=2−01 (j−j!k)!tj−kp~k+   m jk=NP+1(−1)j−k(j−j!k)!tj−k~qN−k, N2+1 ≤ j ≤ N. 2  P These charges generatethecanonical transformationsrepresenting theN- Galilean conformal group on Hamiltonian level. Computing systematically the infinitesimal action of all generators we find: – for ~c : j N−n (k +n)! δ~qn = { ~xj~cj,~qn} = (−1)k+n−N2+1 k! tk~xk+n, j k=0 X X (7) n (k +N −n)! δ~pn = { ~xj~cj,p~n} = m (−1)k tk~xk+N−n, k! j k=0 X X – for h : 1 δ~qn = {τh,~qn} = −τ mδN2−1,n~pN2−1 +(1−δN2−1,n)~qn+1 , (8) (cid:18) (cid:19) δ~pn = {τh,p~n} = τ(1−δn0)p~n−1, – for d : N 1 δ~qn = {λd,~qn} = λ −( 2 −n)~qn +t(m~pN2−1δN2−1,n +(1−δN2−1,n)~qn+1 , (cid:18) (cid:19) N δ~pn = {λd,~pn} = λ −n p~n −t(1−δn0)p~n−1 , 2 (cid:18)(cid:18) (cid:19) (cid:19) (9) 4 – for k : δ~qn = {ck,~qn} = c (1−δn0)n(N −n+1)~qn−1+ N (cid:16) 1 2t( 2 −n)~qn −t2 mδn,N2−1p~N2−1 +(1−δn,N2−1)~qn+1 , (cid:0) N +1 2 (cid:1)(cid:17) δ~pn = {ck,~pn} = c mδnN2−1 2 ~qN2−1 −(1−δnN2−1)(N −n)(n+1)p~n+1− N (cid:16) (cid:0) (cid:1) 2t( −n)p~n +t2(1−δn0)p~n−1 . 2 (cid:17) (10) These equations can be integrated out to yield the global transformations. 3 Lagrangian formalism We want to find the realization of N-Galilean conformal group as a group of symmetry transformations on coordinate space (point transformation). Let us note that the Hamiltonian h given by first eq. (5) is the Ostrogradski Hamiltonian [10] corresponding to higher derivative free Lagrangian N+1 2 m d 2 ~q L = . (11) 2 dtN2+1 ! TheprocedureofpassingfromLagrangiantoHamiltonianformalismisslightly involved [10]. First, one enlarges the coordinate space by defining: ~q0 = ~q, ~q1 = ~q˙,...,~qN−1 = ~q(N2−1); (12) 2 then one writes out the Lagrangian 2 N−3 m d~qN−1 2 L˜ = 2 + ~λ (~q˙ −~q ); (13) 2 dt k k k+1 ! k=0 X L˜ is singular so the Dirac method has to be applied. It appears that all resulting constraints are of the second class which allows to eliminate the ~ Lagrange multipliers λ and their conjugate momenta. In this way we arrive k at the Ostrogradski Hamiltonian. 5 To find the action of N-Galilean conformal group in coordinate space let us remind that the general canonical transformation describes the point transformation provided the new coordinates are expressible in terms of old ones (with no momenta involved) while the new momenta are the linear functions of old momenta (with coordinate-dependent coefficient); actually they can be more general affine functions if the Lagrangian transforms by a total derivative. However, the above statement is true only provided the time variable remains unchanged. If it changes the momenta can enter the expressions for the variations of coordinate variables provided they appear only in the form of Poisson brackets of Hamiltonian with coordinates. Then the terms containing momenta canbe removed fromtransformationformulae at the expense of admitting the time variation. The resulting modified point transformations coincide ”on shell” with the initial canonical ones. Asimple inspection of the formulaegiven in Sec. 2 shows that the canoni- calactionofN-Galileanconformalgrouphastheabovementionedproperties. Therefore, one can define the action of this group on coordinate space which ”on-shell” coincides with the transformations defined in Sec. 2. It is not difficult to derive the form of this action. First, let us note that the relations (7)-(10) yield immediately the global form of transformations generated by ~c ’s: k ′ t = t, N−n (k +n)! (14) q~n′(t′) = ~qn(t)+ (−1)k+n−N2+1 k! tk~xn+k. k=0 X The global action of h obviously reads ′ t = t+τ, (15) q~′(t′) = ~q (t). n n To find the action of dilatation we rewrite eq. (9) in the form N δ~q = λ(n− )~q +λt~q˙, (16) n n 2 which can be easy integrated to t′ = e−λt, (17) q~n′(t′) = eλ(n−N2)~qn(t). 6 The case of conformal transformation is slightly more involved. First, by extracting the coefficient in front of h(t) in the expression defining k we find δt = ct2, (18) which integrates to t ′ t = ≡ t(c). (19) 1−ct Now, the first eq. (10) can be written in the form N δ~qn = c n(N −n−1)~qn−1 +2t( −n)~qn −t2~q˙n . (20) 2 (cid:18) (cid:19) The last term on the right-hand side is responsible for time variation. One can get rid of this term by replacing the time variable by its ”running” value (19). In this way we arrive at the following equations d~q (t(c),c) 2t N n dc = n(N −n+1)~qn−1(t(c),c)+ 1−ct( 2 −n)~qn(t(c),c). (21) It is not difficult to integrate eq. (21). The result reads n n (N +k −1)! ck q~n′(t′) = k (N −1)! (1−ct)N+k~qn−k(t). (22) k=0(cid:18) (cid:19) X Finally, the action of rotation subgroup is standard. Let us note that in all cases the following important property holds: dq~′(t′) ~q = ~q˙ implies q~′ (t′) = n . (23) n+1 n n+1 dt′ It allows us to reduce the action of the group under consideration to that on the variables t and ~q = ~q . One finds 0 N t′ = t, q~′(t′) = ~q(t)+ (−1)k−N2+1tk~xk; (24) k=0 X t′ = t+τ, q~′(t′) = ~q(t); (25) t′ = e−λt, q~′(t′) = e−λN2 ~q(t); (26) 7 t ~q(t) t′ = , q~′(t′) = ; (27) 1−ct (1−ct)N as the counterparts of eqs. (14) (15) (17) (19) (22), respectively. It is shown in Appendix that in all cases Ldt (with L given by eq. (15)) is invariant, up to an exact differential, under all the above transformations. Eqs. (24)-(27) allow us to write out the differential realization of the algebra (1), (2). It reads ∂ N ∂ ∂ Hˆ = i , Dˆ = −i ~q +t , ∂t 2 ∂~q ∂t (cid:18) (cid:19) (28) Kˆ = i Nt~q ∂ +t2 ∂ , C~ˆk = i(−1)k−N2−1tk ∂ . ∂q ∂t ∂q~ (cid:18) (cid:19) Let us also note that C~ˆ commutate with each other. This is due to the fact k that M, being central element, act trivially in coadjoint representation. In order to complete the picture we will find all integrals of motion corresponding to the transformations (24)-(27) and compare them with the ones defined on the Hamiltonian level (6). First, let us note that for infinitesimal transformations q~′ = ~q +ǫχ~(q,t), t′ = t+ǫg(t), (29) and an arbitrary higher-order Lagrangian L = L(~q,~q˙,...,~q(R)), R > 1, the symmetry condition dq~′ dRq~′ dt′ d~q dR~q d L(q~′(t′), ,..., ) = L(~q(t), ,..., )+ǫ (δf), (30) dt′ dt′R dt dt dtR dt implies d R ∂L dn~q ǫ (δf)− δ( )−ǫg˙L = 0. (31) dt ∂~q(n) dtn n=0 X For n > 0 the following identity holds dn~q n−1 dk δ( ) = −ǫ (g˙~q(n−k))+ǫχ~(n), (32) dtn dtk k=0 X 8 while for n > 1, k > 0 we get dk ∂L d k−1 dk−l−1 dl ∂L (g˙~q(n−k)) = (−1)l (g˙~q(n−k)) + dtk ∂~q(n) dt dtk−l−1 dtl ∂q~(n) ! l=0 (cid:18) (cid:19)(cid:18) (cid:19) X dk ∂L (−1)kg˙~q(n−k) . dtk ∂~q(n) (cid:18) (cid:19) (33) Using eqs. (32) and (33) one finds R ∂L dn~q R n−1 dk ∂L R ∂L δ( ) = −ǫg˙ (−1)k ~q(n−k) +ǫ χ~(n) ∂~q(n) dtn dtk ∂~q(n) ∂q~(n) n=1 n=1 k=0 (cid:18) (cid:19) n=1 X XX X R n−1 d k−1 dk−l−1 dl ∂L −ǫ (−1)l (g˙~q(n−k)) . dt dtk−l−1 dtl ∂~q(n) ! n=2 k=1 l=0 (cid:18) (cid:19) XX X (34) On the other hand the Ostrogradski Hamiltonian for L can be written in the form R−1 H = ~p ~q(l+1) −L, (35) l l=0 X where R−n−1 d j ∂L ~p = − , n = 0,1,...,R−1; (36) n dt ∂~q(n+j+1) j=0 (cid:18) (cid:19) (cid:18) (cid:19) X consequently the first term on the r.h.s. of eq. (34) can be rewritten as follows R n−1 dk ∂L R R−l d j ∂L −ǫg˙ (−1)k ~q(n−k) = −ǫg˙ − q~(l) = dtk ∂~q(n) dt ∂q~(l+j) n=1 k=0 (cid:18) (cid:19) l=1 j=0 (cid:18) (cid:19) (cid:18) (cid:19) XX XX R−1R−l−1 j d ∂L d −ǫg˙ − ~q(l+1) = −ǫ (gH)−ǫg˙L, dt ∂~q(l+j+1) dt l=0 j=0 (cid:18) (cid:19) (cid:18) (cid:19) X X (37) 9 where H is expressed in terms of ~q and their time derivatives. Substituting this result into eq. (34) and using eq. (31) we obtain the following equation d R n−1 k−1 dk−l−1 dl ∂L ǫ δf +Hg˙ + (−1)l (g˙~q(n−k)) − dt dtk−l−1 dtl ∂q~(n) ! n=2 k=1 l=0 (cid:18) (cid:19) XXX (38) R ∂L ǫ χ~(n) = 0. ∂~q(n) n=0 X Moreover, one checks that R R−1 R k ∂L d d ∂L χ~(n) = p~ χ~(k) +χ~ − . (39) k ∂~q(n) dt dt ∂q~(k) ! n=0 k=0 k=0(cid:18) (cid:19) (cid:18) (cid:19) X X X Together with eq. (38) this leads to the following integral of motion R−1 R n−1 k−1 dk−l−1 dl ∂L C = Hg − p~ χ~(k) + (g˙~q(n−k)) − +δf. k dtk−l−1 dtl ∂q~(n) k=0 n=2 k=1 l=0 (cid:18) (cid:19)(cid:18) (cid:19) X XXX (40) Now, let us apply these general formulae to our Lagrangian (11) and symmetry transformations (24)-(27). In this case the generalized momenta (36) and the Hamiltonian H, when written in terms of ~q’s, read ~pn = m(−1)N2−1−n~q(N−n), n = 0,1,..., N −1, (41) 2 N−3 2 m H = (−1)N2−1−n~q(N−n)~q(n+1) + (~q(N2+1))2. (42) 2 n=0 X Let us now find the integrals of motion. For the transformations (24), g = 0 k and χ~k = (−1)k−N2+1tk, k = 0,...,N while the functions δf~k are of the form (see Appendix, eq. (56) for small ~x) N −1 ~ δf = 0, k = 0,..., ; k 2 k (43) k! N +1 δf~ = m (−t)k−n ~q(N−n), k = ,...,N. k (k −n)! 2 n=XN2+1 10