Galilean Exotic Planar Supersymmetries and Nonrelativistic Supersymmetric Wave Equations

Transcription

1 arxiv:hep-th/060198v1 0 Feb 006 Galilean Exotic Planar Supersymmetries and Nonrelativistic Supersymmetric Wave Equations J. Lukierski 1), I. Próchnicka 1), P.C. Stichel ) and W.J. Zakrzewski 3) 1) Institute for Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, Wroc law, Poland lukier@ift.uni.wroc.pl; ipro@ift.uni.wroc.pl ) An der Krebskuhle 1, D Bielefeld, Germany peter@physik.uni-bielefeld.de 3) Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK W.J.Zakrzewski@durham.ac.uk Abstract We describe the most general N-extended D = ( + 1) Galilean symmetries obtained, respectively, from the N-extended D = 3 Poincare algebras with maximal sets of central charges. We confirm the consistency of supersymmetry with the presence of the exotic second central charge θ. We show further how to introduce a N = Galilean superfield equation describing nonrelativistic spin 0 and spin 1 free particles. 1

2 1 Introduction The Galilean invariance is the fundamental space-time symmetry of nonrelativistic systems. The Galilean projective representations (see e.g. [1]), used in the description of quantum d-dimensional nonrelativistic systems, are generated by a central extension of the D = (d + 1) Galilei algebra called also Bargmann or quantum Galilei algebra, with mass m as a central charge. For example, for d = 3 the Galilei algebra takes the following form (r, s, t = 1,, 3; we assume that the generators are Hermitean): [J r, J s ] = i ǫ rst J t, [J r, K s ] = i ǫ rst K t, [K r, K s ] = 0, [P r, P s ] = 0, [J r, P s ] = i ǫ rst P t, [H, J s ] = 0, [H, P s ] = 0, (1) [H, K r ] = i P r, [P r, K s ] = i δ rs m, where J r describe O(3) rotations, K r - Galilean boosts, P r - momenta and H is the energy operator. It is known that the Galilei algebra (1) can be obtained from the D = 4 Poincaré algebra by the relativistic contraction c ([]). The (+1) dimensional Galilean algebra is a special case as, exceptionally, it allows for the existence of a second central charge θ. This algebra, with central charges m and θ, also called exotic, takes the following form (see e.g. [3]-[5]) (i, j = 1, ) [J, K i ] = i ǫ ij K j, [K i, K j ] = i ǫ ij θ, [P i, P j ] = 0, [J, P i ] = i ǫ ij P j, [H, J] = [H, P i ] = 0, () [H, K i ] = i P i, [P i, K j ] = i δ ij m, with the generator J describing the O() rotations. In [6] it was shown that the second central charge θ can be reinterpreted as introducing noncommutativity in the two-dimensional space [ˆx i, ˆx j ] = i θ m ǫ ij. (3)

3 The distinguished role of planar Galilean algebra follows from the covariance of the relations (3) under space rotations. For d > (d - number of space dimensions) it is not possible to have noncommutativity relation for space coordinates which are covariant under classical O(d) rotations what corresponds to the property that the respective Galilean algebras do not allow for the existence of the second central charge. The first attempt at obtaining supersymmetric extensions of the D = (3 + 1) Galilean algebra (1) was made by Puzalowski ([7]) who considered the N = 1 and N = cases. The supersymmetric extension of the D = ( + 1) Galilean symmetries were considered for N = 1 and N = in [8]- [11]. We should observe that the Galilean symmetries were extended to the so-called Schrödinger symmetries of free quantum mechanical systems (free nonrelativistic particle ([1]) and harmonic oscillator ([13])) by adding two additional generators: D (dilatations) and K (extensions, corresponding to the conformal time transformations). The Galilean superalgebra was then obtained as a subalgebra of supersymmetrically extended Schrödinger algebra ([9], [14]-[16]). However, the N extended supersymmetrization of the D = (+1) Galilean algebras have, so far, not been described in its general form; in particular, the D = ( + 1) Schrödinger super-algebra has never been written down in the presence of the second central charge θ. The aim of this paper is twofold: To describe the most general N extended D = (+1) Galilean superalgebras, which we obtain by the nonrelativistic contraction c (c - velocity of light) of the corresponding D = 3 relativistic Poincare algebras with maximal sets of central charges. We obtain N(N 1) Galilean central charges - i.e. the same number as in the relativistic case 1. In Sect.. we show that the contraction of the central charges sector, which preserves their number, requires a suitable rescaling to obtain finite results in the c limit. We show further that for D = (+1) the two-parameter (m, θ) central extension (see ()) is consistent with supersymmetry. To present the realization of the N = D = ( + 1) Galilean superalgebra describing the supersymmetric nonrelativistic particle multiplet with spin 0 and 1. In particular, we obtain the Levy-Leblond equations for nonrelativistic spin 1 fields ([18]). The model can easily be made invariant under the exotic planar Galilean supersymmetry with the central charge θ 0. 1 By central charges we denote here the Abelian generators which commute with supercharges and Galilean generators. In general they transform as tensors under the internal symmetry group. 3

4 Let us add that recently the Galilean supersymmetries have been applied to the light-cone description of superstrings ([19]), nonrelativistic super-pbranes ([1]) and D-branes ([]). We conjecture that our D = ( + 1) Galilean supersymmetries with central charge θ 0 can find application in the description of nonrelativistic supermembranes with noncommutative world volume geometry. N- extended Galilean D = ( + 1) superalgebras as contraction limits Let us recall that the N-extended D = 3 Poincare superalgebra is given by (α, β = 1, ; µ = 0, 1, ; A, B = 1..N): {Q A α, Q B β } = (σ µ ) αβ P µ δ AB + ǫ αβ Z AB, (4) where P µ = η µν P ν (η µν =diag(-1,1,1)), Z AB = Z BA are N(N 1) real central charges and σ µ = (σ i = γ 0 γ i, σ 0 = γ0 = 1 ). We choose γ 1 = ( ), γ = ( ), γ 0 = ( ) = ǫ. (5) The full superalgebra is described by N real supercharges Q A α, the D = 3 Poincare algebra (P µ = (P 0, P i ), M µν = (M 1 = J, M i0 = N i )), N(N 1) central charges Z AB and the O(N) generators T AB = T BA describing internal symmetries: i) D = 3 Poincare algebra [J, N i ] = i ǫ ij N j [N i, N j ] = i ǫ ij J [J, P i ] = i ǫ ij P j, [J, P 0 ] = 0 (6) [P 0, N i ] = i P i, [P 0, P i ] = [P i, P j ] = 0, [P i, N j ] = i δ ij P 0. ii) Supercovariance relations for N real supercharges Q A α [Q A α, N i ] = i (σ i) αβ Q βa, [Q A α, P i ] = 0, [Q A α, J] = i ǫ αβq βa, [Q A α, P 0 ] = 0. (7) 4

5 iii) The internal index A is rotated by the O(N) generators T AB, where and [T AB, P µ ] = [T AB, M µν ] = 0, [T AB, T CD ] = i ( δ AC T BD δ AD T BC + δ BD T AC δ BC T AD) (8) where (τ ab ) d c O(N) and [T AB, Q C α ] = i (τab ) C D QD α, describe the vectorial N N matrix representation of [T AB, Z CD ] = i ( δ AC Z BD δ AD Z BC + δ BD Z AC δ BC Z AD). (9) iv) Central charges Z AB are Abelian and commute with the supercharges Q A α and with (P µ, M µν )). For a given choice Z(0) AB of the central charges the internal unbroken symmetry is described by the generators T AB T AB which satisfy the relation (c.p. (9)) [ T AB, Z(0) CD ] = 0. (10) The nonrelativistic contraction of the D = 3 Poincaré algebra part of the super-algebra (see (6)) is obtained by the introduction of the Hamiltonian H in the following way: If we now perform the rescaling P 0 = mc + 1 H. (11) c N i = cl i and take the limit c then we obtain from (6) the relations () with θ = 0 and K i replaced by L i. In order to get the algebra () (i.e. the exotic (+1) - dimensional Galilean algebra) one should perform the following linear change of basis K i = L i + θ m ǫ ij P j. (1) In the case of a simple N = 1 D = ( + 1) Galilean superalgebra the relation (4) takes the simple form Next we introduce the rescaled supercharges {Q α, Q β } = δ αβ P 0 + (σ i ) αβ P i. (13) S α = 1 c Q α (14) 5

6 and using (13) find that in the c limit {S α, S β } = δ αβ m. (15) Further, from (7) and (14) we get the following completion of the N = 1 D = 3 Galilean super-algebra: [K i, S α ] = 0 [J, S α ] = i ǫ αβs β. (16) Next we derive the N-extended D = ( + 1)-Galilean superalgebra for N = k (k = 1,,..). We start from the superalgebra (4) where A, B = 1,...k. We define Q ±a α = Q a α ± ǫ αβq k+a β, (17) where in the formula (17) we consider a = 1,...k. Using (4) and (17) we get (a, b = 1...k)) {Q +a α, Q+b β } = δ αβp 0 δ ab + ǫ αβ (Z ab + Zã b) + δ αβ (Y ab Ỹ ab ), (18) {Q +a α, Q b β } = (σ ip i ) αβ δ ab + ǫ αβ (Z ab Zã b) + δ αβ (Y ab + Ỹ ab ), (19) {Q a α, Q b β } = δ αβp 0 δ ab + ǫ αβ (Z ab + Zã b) δ αβ (Y ab Ỹ ab ), (0) where the k k matrix of central charges Z AB is described by the four k k matrices Z ab, Zab = Z a+k b+k, satisfying the symmetry properties: Y ab = Z k+a b, Ỹ ab = Z a k+b, (1) Z ab = Z ba, Zab = Z ba () Y ab = Ỹ ba. Finally, one gets (Y (ab) = 1 (Y ab + Y ba ), Y [ab] = 1 (Y ab Y ba )) {Q +a α, Q+b β } = δ αβ(p 0 δ ab + Y (ab) ) + ǫ αβ (Z ab + Z ab ) (3) {Q +a α, Q b β } = ((σ ip i ) αβ δ ab + δ αβ Y [ab] ) + ǫ αβ (Z ab Z ab ) (4) {Q a α, Q b β } = δ αβ(p 0 δ ab Y (ab) ) + ǫ αβ (Z ab + Z ab ). (5) 6

7 Before taking the nonrelativistic limit we rescale the supercharges as follows: S a α = 1 c Q +a α, Ra α = cq a α. (6) The limit c of the relations (3-5) exists if we assume that Y (ab) = δ ab mc + Ỹ (ab), c Z [ab] + Z [ab] = Ũ[ab] c Z [ab] Z [ab] = U [ab] (7) with central charges Y [ab], U [ab], Ỹ (ab) and Ũ(ab) having finite c limits. Using (6) and (7) we see that in this limit {S a α, Sb β } = 4m δ αβ δ ab, {S a α, Rb β } = (σ ip i ) αβ + δ αβ Y [ab] + ǫ αβ U [ab] (8) {R a α, Rb β } = δ αβ(hδ ab Ỹ (ab) ) + ǫ αβ Ũ [ab]. We see that the N-extended (+1)-dimensional algebra (N = k) (8) contains the following set of central charges, with their number given in the second row, U [ab], Y [ab], Ũ [ab], Ỹ (ab) k(k 1), k(k 1), k(k 1), k(k + 1), (9) As k(k+1) + 3 k(k 1) = N(N 1) (N = k), the maximal number of central charges for the D = 3 Poincare and D = ( + 1) Galilean symmetry is the same. The supercovariance relations (7) take the following form [N i, Q ±a α ] = i (σ i) αβ Q a β (30) and after rescalings (17) and (6) one gets, in the c limit or, after performing the shift (1) [L i, S a α ] = 0, [L i, R a α ] = i (σ i) αβ S a β (31) [K i, S a α ] = 0, [K i, R a α ] = i (σ i) αβ S a β. (3) 7

8 Further, from (7) we get [J, S a α ] = i ǫ αβs a β, [J, Ra α ] = i ǫ αβr a β. (33) It is easy to see that the relations (8) are covariant under the O( N ) = O(k) rotations, with indices a, b describing the k-dimensional vector indices. In comparison with (4) we see that the O(N) covariance of the relations (4) has been reduced in the contraction procedure to the covariance with respect to the diagonal O(k) symmetry obtained by constraining the O(N) generators T AB as follows: T ab = T k+ak+b, T a k+b = 0. (34) In consequence, all the central charges (1) are the second rank O(k) tensors and the Galilean supercharges (Sα a, Ra α ) are the O(k) vectors. A special case corresponds to k = 1 (N = ), when Z AB = ǫ AB Z defines the scalar central charge Z (from () we see that for k = 1, Y = Ỹ = Z). One gets {Q + α, Q + β } = δ αβ(p 0 + Z) {Q + α, Q β } = (σ ip i ) αβ (35) {Q α, Q β } = δ αβ(p 0 Z). Writing the first relation in (8) in the form Z = mc + U and using (11) and c (7) we get, when c {S α, S β } = 4mδ αβ {S α, R β } = (σ i P i ) αβ (36) {R α, R β } = (H U)δ αβ, where U plays the role of the correction to the Hamiltonian originating from the central charge Z. Further, from the relation (8) we get (T AB = ǫ AB T; A, B = 1, ) [T, Q ± α ] = i Q± α (37) and after contraction [T, S α ] = i S α, [T, R α ] = i R α (38) In addition we get, using (1), (7) and the contraction limit [K i, S α ] = 0, [K i, R α ] = i (σ i) αβ S β (39) as well as [J, S α ] = i ǫ αβs β, [J, R α ] = i ǫ αβr β. (40) 8

9 3 Superfield wave equation with standard and exotic N = Galilean supersymmetry In this section we derive a super-galilean covariant wave equation for our superfield Ψ(t, x; θ, η) where θ and η two real valued anticommuting spinors with components θ α and η β. The proposed superfield equation is the following (see also [3]): α Ψ = 0, (41) where α is a supercovariant derivative. Here, we will describe a direct way of obtaining (41) without the need of the Lagrangian etc. In order to determine the explicit form of our supercovariant derivative we require, in addition to (41) that we have and [ α, H] Ψ = 0, [ α, P i ] Ψ = 0, (4) { α, S β } Ψ = 0, { α, R β } Ψ = 0, (43) [ α, J] Ψ = 0, [ α, K i ] Ψ = 0. (44) In addition (41) ought to be the square-root of the Schrödinger equation [15]; i.e. we require that by { α, α } Ψ (H p ) Ψ. (45) m The super-derivative α, satisfying all the requirements (4-45) is given α = 1 (σ ip i ) αβ S β m R α. (46) To prove our claim we use the N = super-galilean algebra with a vanishing central charge U discussed in section. First we note that the invariance of α with respect to space-time translations is evident. Furthermore, 1. from (36) we have { α, α } = 4m (H P m ) (47) Note that the supercovariance of this superderivative holds only in a weak sense, ie as is clear from (50,51), it is valid only on the solutions of the wave equation (41) 9

10 . and { α, S β } = 0 (48) 3. from (39) and (40) 4. from (36) and therefore, due to (41) and (47) [ α, J] = 1 (σ i) αβ β, [ α, K i ] = 0 (49) P { α, R β } = m δ αβ ( H) (50) m { α, R β }Ψ = 0. (51) In order to express the wave equation (41) with α given by (46), in terms of derivatives with respect to the arguments of Ψ, we need a realisation of the N = super-galilei algebra in terms of differential operators. It is easily seen that such a realisation is given by (U = 0): H = i t, P i = i i, (5) K i = tp i mx i + θ m ǫ ijp j + i η α(σ i ) αβ J = ǫ ij x i P j + i ǫ αβ ( θ α θ β + η α η β, (53) θ β ), (54) S α = θ α + 1 (σ ip i ) αβ η β + mθ α, (55) R α = η α + 1 (σ ip i ) αβ θ β + i η α t, (56) where the spinor derivatives are defined as left-derivatives and the x i are commuting variables. Note that the spinor part of K i is necessary so that K i has the correct expressions for its commutators with the supercharges S α and R α. This extra term leads also to a nontrivial behaviour of the spinor θ α with respect to boosts i.e: [K i, θ α ] = i (σ i) αβ η β. (57) From (55) and (56) we can read off the transformation properties of our variables in superspace Y (t, x, θ, η) under infinitesimal supertranslations. Then with δy = [Y, ǫ β S β + ρ β R β ] (58) 10

11 we obtain δη α = ρ α, δθ α = ǫ α, (59) δx i = i (ǫ β(σ i ) βγ η γ + ρ β (σ i ) βγ θ γ ), (60) δt = i ρ βη β. (61) Finally, using (55) and (56) for the supercharges S α, R α we obtain the following decomposition of the super-covariant derivative α where α = D α η α( P mi t ), (6) D α = (σ i P i ) αβ θ β m η α. (63) Thus the wave equation (41) is equivalent to the following pair of differential equations: D α Ψ = 0, (64) with D α given by (63) and ( i t P m As the superfield ψ has the expansion ) Ψ = 0. (65) Ψ(t, x; θ α, η α ) = φ(t, x) + θ α ψ α (t, x) + η α χ α (t, x) +... (66) the equation (64) gives us the following equation for the spinor fields (σ i P i ) αβ ψ β (t, x) mχ α (t, x) = 0 (67) which, when combined with (65) gives us i t ψ α = 1 (σ ip i ) ψ α = (σ i P i ) αβ χ β. (68) The set of eq.(67)-(68) provides the Levy-Leblond equations for the nonrelativistic spin 1/ particles ([18]; see also [8]). 11

12 4 Conclusions In this paper we have restricted ourselves to the case of the D = + 1 nonrelativistic supersymmetries but the discussion of the D = case can be performed in an analogous way. In particular, the basic equations (46), (64) and (67) can be extended to a three-dimensional space by introducing complex two-component spinors θ α, η α and three x complex Pauli matrices. Note that in Section 3 we derived the superfield form of the nonrelativistic Levy-Leblond equations without postulating a classical action in the superspace (x i, t, θ α, η α ). Note also that our superfield equation does not depend on the exotic parameter θ. In Section we have presented the most general nonrelativistic contraction scheme for D = 3 extended Poincare algebra. Had we started from the most general D = 4 N-extended Poincare algebra with N(N 1) complex central charges ([4]), we would have obtained analogous results for the D = 3+1 nonrelativistic supersymmetric theory. Acknowledgments Two of the authors (JL) and (PCS) would like to thank the University of Durham for hospitality and the EPSRC for financial support. Partial financial support by a KBN grant 1 P03B 0188 is also acknowledged (J.L.). References [1] V. Bargmann, Ann. Math. 59, 1 (1954) [] E. Inonu and E.P. Wigner, Proc. Nat. Acad. Sci. (USA) 39, 510 (1953). [3] J.M. Levy-Leblond in E.Loebl Group Theory and Applications, Acad. Press, New York (197). [4] A. Ballesteros, M. Gadella and A.A. del Olmo, J. Math. Phys. 33, 3379 (199). [5] D.R. Grigore, J. Math. Phys. 37, 460 (1996); Forsch. d. Physik 44, 63 (1996). [6] J. Lukierski, P.C. Stichel and W.J. Zakrzewski, Ann. Phys. 60, 4 (1997). [7] R. Puzalowski, Acta Phys. Austr. 50, 45 (1978). 1

13 [8] J.A. de Azcarraga, D. Ginestar, J. Math. Phys. 3, 3500 (1991). [9] M. Leblanc, G. Lozano and H. Min Ann. Phys. (NY), 19, 38 (199). [10] O. Bergman and C.B. Thorn, Phys. Rev. D5, 5980 (1995). [11] O. Bergman, Galilean supersymmetry in (+1) dimensions, hep-th/ [1] C.R. Hagen, Phys. Rev. D5, 377 (197). [13] U. Niederer, Helv. Phys. Acta 45, 80 (197). [14] J. Beckers and V. Hussin, Phys. Lett. A 118, 319 (1986). [15] J.P. Gauntlett, J. Gomis and P.K. Townsend, Phys. Lett. B48, 88 (1990). [16] C. Duval and P.A. Horvathy, J. Math. Phys. 35, 516 (1994). [17] C. Duval and P.A. Horvathy, Phys. Lett. B479, 84 (000); J. Phys. A34, (001). [18] J.M. Levy-Leblond, Comm. Math.Phys 6, 86 (1967). [19] O. Bergman and C.B. Thorn, Phys. Rev. D5, 5997 (1995). [0] J. Brugnes, T. Curtright, J. Gomis and L. Mezinescu, Phys. Lett. B 594, 7 (004). [1] J. Gomis, K. Kamimura an P. Townsend, JHEP, 11, 051 (004). [] J. Gomis, F. Passerini, T. Ramirez and A. Van Proyen, hep-th/ [3] I.A. Batalin and P.H. Damgaard, Phys. Lett. B 578, 3 (004) [4] R. Haag, J.T. Lopuszański, M.F. Sohnius, Nucl. Phys. B 88, 57 (1975) 13