ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY

Transcription

1 ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY CIPRIAN ACATRINEI Natonal Insttute of Nuclear Physcs and Engneerng P.O. Box MG-6, Bucharest, Romana E-mal: Receved March 6, 2008 We dscuss the formulaton of mechancs wth nhomogeneous noncommutatvty at the classcal and quantum level. Mechancs wth noncommutng coordnates has recently been a subect of consderable nterest []. Classcal extensons to nhomogeneous noncommutatvty have also been dscussed, n partcular for the case n whch dmensonal reducton taes place [2]. Our purpose n ths wor s manly to extend to varable commutators a path ntegral formulaton for noncommutatve mechancs, ntally proposed n [3] for constant commutators. As customary n the feld, we wor n 2+ dmensons, for smplcty n notaton. Coordnates are denoted by q and q 2, momenta by p and p 2, and both types of phase space coordnates by x a, x,2,3,4 = q, q 2, p, p 2. Assume the Posson bracets of the theory to be { q q } θ ( q, p), { p, p } = σ ( q, p), {, } = δ,, 2 2 q p or equvalently {x a, x b } = Θ ab, Θ ab = ( ω ), 0 θ 0 θ 0 0 Θ = 0 0 σ 0 σ 0 = () ab where 0 σ 0 σ 0 0.e. ω =. 0 0 (2) θσ θ 0 0 θ The form of σ(q, p) and θ(q, p) above s restrcted only by the Jacob denttes p 2 σ θ qσ = 0, qθ σ p2θ = 0, (3) p σ + θ q2σ = 0, q2θ + pθ = 0. (4) Rom. Journ. Phys., Vol. 54, Nos. 2, P. 9 3, Bucharest, 2009

2 0 Cpran Acatrne 2 The smplest approach to the dynamcs engendered by the bracets () s the varatonal one: we wrte down the acton whch generates equatons of moton of the type H x& a = Θab, (5) xb.e x& a = { xa, H} but wth { xa, x b } = Θab. For constant ω(or Θ) ths acton s [2] S = dt ω ab xax& b H ( x). 2 The quantum theory s descrbed by the Hamltonan H and the commutaton relatons [ xa, xb ] = Θab, x,2,3,4 = q, q2, p, p2. (7) For constant Θ the relevant path ntegral s [3] 4 4 dt ω x x& H ( x) S 2 Z Dx e Dx e. (8) = = To prove that (8) enforces the commutaton relatons (7) one only needs to now that Z represents a transton ampltude between two states of a gven Hlbert space, and that tme-orderng of operators s enforced, as usual, by the path ntegral, { DxOO 2e T O O2 } S = [3]. It s possble to nfer the path ntegral formulaton for nhomogeneous Θ(x) provded one starts from the symplectc form,.e. from the acton S = T dt[ Aa ( x) x& a H ( x) ], (9) 0 wth a Ab ( x) b Aa ( x) = ω ab( x) Θab, (0) whch leads to the equatons of moton (5) for nonconstant Θ as well. The path ntegral wll be taen to be the contnuum lmt of the dscretzed partton functon N 4 ( n+ ) ( n ) ( n ) A ( a( x n )( xa xa ) ε H( x )) Z dx e, () = a= n where ε s the tme ncrement, ε = T/N, and xa s the value of the phase space varable x a at tme t 0 + nε, n = 0,, 2,, N. Let us consder the expectaton value O of, where O ( x ) x n s an operator dependng on the x s. Integratng by parts under the path ntegral (), one gets

3 3 On mechancs wth varable noncommutatvty O x S% = T O x,. x n T{,} represents the tme-orderng of operators, whch means (n)-orderng n the dscrete case. S % s the dscretzed form of the acton sttng n the exponent of S% ( n+ ) ( n ) H (), and = ω ( x x x ) ε. Choosng O = x n, convertng the x x (n)-orderng nto a commutator, and tang then the contnuum lmt ε 0, (2) becomes ω, x x x = δ, (3) whch mples x x =Θ ( x) = ( ω ),,.e. the commutaton relatons (7). To derve the Hesenberg equatons of moton as well, we choose Ô proportonal to the dentty operator. Then, one gets S% x n = 0, leadng to d H d ω x =. Solvng for x, one obtans dt x dt d H x, =Θ = x H, dt x (4) whch are the extended Hesenberg equatons of moton (the quantum form of (5), wth {,} [,] ) PB We derved the commutaton relatons and the operator equatons of moton from the path ntegral (), for nhomogeneous noncommutatvty. One notes that the tme orderng appearng n the prevous dervaton allowed us to gnore at a frst glance the orderng of the operators appearng smultaneously n S %. We now swtch to the dervaton of the path ntegral from the operatoral formulaton. Here orderng problems wll become relevant. Snce p, ε q = 0, we can consder the bass spanned by the set of egenvectors of q and p 2, { q, p2 }, or alternatvely by { q2, p }. In order to calculate the transton ampltude HT ( 0) ( 0) 2 2 (2) A = q, p e q, p, (5) between two states wth prescrbed poston along the frst axs of coordnates, and well defned momentum along the second axs, we now from the constant

4 2 Cpran Acatrne 4 noncommutatvty case [3] that t s suffcent to evaluate q, p2 q2, p, whch n that case was q, p2 q2, p = exp ( qp q2p2 + θpp2 σqq 2). 2 θσ (6) Consder frst the case n whch θ = 0. Then we can represent r p, p = σ q, q, q, p = δ q q, q = q q, q, p = + A q [ 2] ( 2) by 2 2 q provded A A σ ( q, q ). q, p q, p = However, we wll need later on 2 2 q 2 q2 2 to obtan the expresson for the path ntegral. In the q2, p bass the nondagonal operators can be representd by q q, p = q, p, p q, p = + σ q, p. (7) 2 p q2 p 2 Usng the notaton q, p2 q2, p f e α, one obtans the followng lnear partal dfferental equatons for f qf = pf, pf 2 = ( q2 + σ p) f. (8) Integrablty s ensured by the Jacob denttes, and n the end one obtans α = qp qp q σ q, q (9) 2 2 q 2 2 where an addtonal arbtrary functon ( q, p ) φ was dropped. It s a standard thng now to construct the path ntegral, and t s easly seen that Eqs. (0, ) are correctly reproduced when θ = 0: The gauge feld connecton emerges n the path ntegral under the expected form ( Θ( x) ) x&. If θ 0 but σ = 0 one can proceed x smlarly; one smply dualzes q and p, n fact. If however Fθ 0, the dervaton becomes more complcated, and orderng prescrptons become mportant too; one prefers to retreat to partcular forms for θ and σ. There s however an nterestng stuaton whch can stll be treated for generc Θ. The Jacob denttes (3, 4) mply that the phase space varables enter θ and σ n pars: f they depend on q, they must also depend on p 2, and vceversa. (q 2, p ) also enter as a par. Ths s a fortunate stuaton, snce the members of a par commute wth each other, hence no orderng problems wll appear. We consder the case n whch Θ depends on only one par, say (q 2, p ),.e. [ q, q2] = θ ( q2, p), [ p, p2] = σ( q 2, p), q, p = δ. (20) Then q, p2can be represented va q q, p = + θ q, p, p q, p = + σ q, p. (2) ([ ] 2 ( p q ) ( q p ) q, p 2 = 0 s ensured by the Jacob dentty.) Ths representaton leads to the followng dfferental equatons for q, p2 q2, p f e α :

5 5 On mechancs wth varable noncommutatvty 3 α + θ α = q, α + σ α = p. (22) p q2 q2 p 2 The Jacob dentty (4) ensures the ntegrablty condton, and permts to put the soluton ether n the form wth an arbtrary addtve functon q + θ p α = θσ φ q, q, p dropped, or n the form α = 2, (23) p 2 2 wth an arbtrary functon φ ( q, p, p ) ntegral (, 0) s reconstructed. 2 2 p σ q θσ dropped out. In ether case, the path 2, (24) q2 REFERENCES. F. Delduc, Q. Duret, F. Geres, M. Lefrancos, arxv: C.S. Acatrne, J. Math. Phys. 49 (2008) ; see also hep-th/ C.S. Acatrne, JHEP 009 (200) 007.

6 4 Cpran Acatrne 6