Remark on quantum mechanics with conformal Galilean symmetry

Transcription

1 LMP-TPU 9/08 Remark on quantum mechanics with conformal Galilean symmetry arxiv: v3 [hep-th] 26 Sep 2008 A.V. Galajinsky Laboratory of Mathematical Physics, Tomsk Polytechnic University, Tomsk, Lenin Ave. 30, Russian Federation Abstract Conformal Galilei algebra contains so(1, 2) subalgebra which is the conformal algebra in one dimension. In this note we generalize methods previously developed for one dimensional many body systems and construct a unitary map relating a quantum mechanics invariant under the conformal Galilean transformations to a set of decoupled particles for which the same symmetry is realized in a nonlocal way. Possible applications of the map are discussed.

2 Conformal Galilean symmetry has been extensively studied in the past in the context of quantum mechanics, statistical physics and nonrelativistic field theory (see e.g. [1] [8] and references therein). More recently, various proposals for a nonrelativistic version of the AdS/CFT correspondence [9] [16] stimulated a renewed interest in the conformal Galilean invariance. The generators of time translation, dilatation and special conformal transformation entering the Galilei algebra obey the commutation relations of so(1, 2) which is the conformal algebra in one dimension. By this reason it is natural to expect that quantum mechanics invariant under the conformal Galilean transformations may share some interesting features with one dimensional conformal many body systems. The most prominent example of such a one dimensional system seems to be the Calogero model [17]. An interesting peculiarity of the Calogero model with the oscillator potential is that it can be transformed into a set of decoupled oscillators by applying an appropriate similarity transformation [18, 19], the fact anticipated by Calogero in [20]. In particular, this explains why the spectra of the Calogero model and the decoupled oscillators are so alike. When the oscillator potential is absent the Calogero model exhibits conformal invariance. In [19, 21] a unitary transformation to a set of decoupled particles was constructed. The simplification in the dynamics was achieved at the price of a nonlocal realization of the full conformal algebra in the Hilbert space [19]. The similarity transformation to decoupled particles suggested a simple and straightforward method of building the complete set of eigenstates for the Calogero model [18]. It also provided an efficient means for constructing various N = 2 and N = 4 superconformal many body models in one dimension [19, 22, 23, 24]. An elegant geometric interpretation of the similarity transformation as the inversion of the Klein model of the Lobachevsky plane was proposed in [25]. The purpose of this note is to generalize the results previously obtained for one dimensional systems [19] to higher dimensions, i.e. to the case of the conformal Galilei algebra. Our analysis also covers the time dependent parts of the conformal generators which were disregarded in [19]. Below we construct a simple unitary transformation which relates a quantum mechanics with the conformal Galilean symmetry to a set of decoupled particles for which the same symmetry is realized in a nonlocal way. Possible applications of the map are discussed. Consider a representation of the conformal Galilei algebra realized in a non relativistic quantum mechanics of N particles in d dimensional space 1 H = M ij = 1 2m α p i α pi α +V(x 1,...,x N ) := H 0 +V(x), P i = (x i α pj α xj α pi α ), Ki = m α x i α tpi, M = 1 We work in the standard coordinate representation [x i α,p j β ] = iδij δ αβ with p i α = i x i α natural units for which h = 1. 1 p i α, m α, and use the

3 C = t 2 H +2tD m α x i αx i α, D = th 1 4 (x i αp i α +p i αx i α). (1) Here H is the Hamiltonian which generates time translation, P i, M ij and K i are the generators of space translations, rotations and Galilei boosts, respectively. M is the mass operator (the central charge) and C and D are the generators of special conformal transformation and dilatation, respectively. In the equations above i is the spatial index and α labels the distinct particles, with m α being the particle mass. The commutation relations of the conformal Galilei algebra (vanishing commutators are omitted) [M ij,m kl ] = i(δ ik M jl +δ jl M ik δ jk M il δ il M jk ), [M ij,p k ] = i(δ ik P j δ jk P i ), [M ij,k k ] = i(δ ik K j δ jk K i ), [K i,p j ] = iδ ij M, [H,K i ] = ip i, [D,K i ] = i 2 Ki, [C,P i ] = ik i, [D,P i ] = i 2 Pi, [H,D] = ih, [H,C] = 2iD, [D,C] = ic (2) constrain the potential V(x) which enters the Hamiltonian to obey the system of partial differential equations (x i α x j x j α α x i α )V(x) = 0, V(x) x i α = 0, x i V(x) α +2V(x) = 0. x i α Any solution to (3) defines a quantum mechanics with the Galilean conformal symmetry. The simplest solution V(x) = 0 corresponds to a system of decoupled particles which is governed by the free Hamiltonain H 0. Notice that setting t = 0 in (1) one does not spoil the algebra. It is this case which was considered previously for one dimensional systems in [19]. Below we take into account the time dependent parts of the generators as well. In what follows we shall use the notation C 0 = 1 m α x i α 2 xi α, D 0 = 1 (x i α 4 pi α +pi α xi α ). (4) Theoperators(H, D 0, C 0 )obeythecommutationrelationsofso(1,2)provided therightmost equation in (3) holds. As the next step consider an automorphism of the Galilei algebra T T = e ia T e ia = T + generated by a specific combination of the operators H, D 0, C 0 n=1 (3) i n n! [A,[A,...[A,T]...], (5) }{{} n times A = ah + 1 a C 0 2D 0. (6) 2

4 Here a is an arbitrary parameter of the dimension of length and the number coefficients are adjusted so as to terminate the infinite series in (5) at a final step. Double commutators of the generators of the Galilei algebra with the operator A prove to be proportional to A or vanish identically which automatically terminates the series. Under the unitary transformation (5) the operators M ij and M are unchanged. For P i and K i one finds P i = 1 ( a (Ki +tp i ) K i = 2+ t ) ( K i +a 1+ t 2 P a a) i, (7) while for the conformal generators a straightforward computation gives H = 1 a 2C 0, D = D ( 2+ t ) C 0, a a ( C = a 2 (H 0 +V(x)) 2a 2+ t ) ( D t 2 C 0. (8) a a) A remarkable fact is that the Hamiltonian of the original interacting system is mapped to C 0, while the interaction potential V(x) is moved to C. Our next objective is to realize the map C 0 H 0 by applying the second automorphism similar to (5), (6). It turns out that the map found in [19] for one dimensional systems fits the higher dimensional case as well. It is straightforward to verify that under the unitary transformation T T = e ib T e ib (9) with B = ah 0 1 a C 0 +2D 0 (10) the operators M ij and M remain inert, while P i, K i regain their original form P i = P i, K i = K i. (11) For the conformal subalgebra the power series terminates at the third step and after some calculation one gets H = H 0 D = th 0 +D 0, C = t 2 H 0 +2tD +C 0 +a 2( e ib V(x) e ib). (12) But for the last term in C, this is a representation of the conformal Galilei algebra on a set of N decoupled particles in d dimensions. Thus, theunitaryoperatore ib e ia enablesonetoremove thepotentialtermv(x) fromthe Hamiltonian which, however, resurfaces in the generator of special conformal transformation as a nonlocal contribution. Hence, the price paid for the simplification in the dynamics is a nonlocal realization of the full conformal Galilei algebra in the Hilbert space of a quantum 3

5 mechanical system. Notice, that consistency requires e ib e ia to be independent of the dimensionful parameter a. That d da (eib e ia ) = 0 can be verified by a straightforward calculation (see [19] for more details). Finally, let us discuss possible applications of the transformation considered in this work. Firstly, with appropriate modifications the transformation can be used for constructing exactly solvable quantum mechanics models and building the complete set of eigenstates as, for example, in [18, 26]. Secondly, as a set of decoupled particle is straightforward to supersymmetrize, the map provides an efficient means for constructing various supersymmetric many body models in the spirit of [19, 22, 23, 24, 27]. Thirdly, with some modification the mapping can be applied in the context of trapped Fermi gases at unitarity (see [28] and references therein). Forthly, since the construction is purely algebraic, it is likely to be applicable within the framework of a nonrelativistic field theory as well. Acknowledgments We thank P. Horváthy and T. Mehen for useful comments. The author is grateful to the Institut für Theoretische Physik at the Leibniz Universität Hannover for the hospitality extended to him at the final stage of this work. The research was supported by RF Presidential grants MD , NS and RFBR grant References [1] U. Niederer, Helv. Phys. Acta 45 (1972) 802. [2] C.R. Hagen, Phys. Rev. D 5 (1972) 377. [3] V. Hussin, M. Jacques, J. Phys. A 19 (1986) [4] R. Jackiw, S.Y. Pi, Phys. Rev. D 42 (1990) [5] R. Jackiw, Annals of Phys. 201 (1990) 83. [6] C. Duval, G. Gibbons, P. Horváthy, Phys. Rev. D 43 (1991) [7] M. Henkel, J. Stat. Phys. 75 (1994) [8] Y. Nishida, D.T. Son, Phys. Rev. D 76 (2007) [9] D.T. Son, Phys. Rev. D 78 (2008) [10] K. Balasubramanian, J. McGreevy, Phys. Rev. Lett. 101 (2008) [11] W. Goldberger, AdS/CFT duality for non-relativistic field theory, arxiv: [hepth]. 4

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