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NORDITA-2011-7 Lorentz-invariant membranes and finite matrix approximations 1 1 Jens Hoppea and Maciej Trzetrzelewskia,b 0 ∗ † 2 n a a Department of Mathematics, J Royal Institute of Technology, 3 2 KTH, 100 44 Stockholm, ] Sweden h t - p b NORDITA, e h Roslagstullsbacken 23, 106 91 Stockholm, [ 1 Sweden v 3 0 4 4 Abstract . 1 0 The question of Lorentz invariance for finite N approximations of 1 relativistic membranes is addressed. We find that one of the classical 1 : manifestationsofLorentz-invarianceisnot possibleforN N matrices v × (at least whenN = 2or 3). How thesymmetryis restored in thelarge i X N limit is studied numerically. r a 1 Motivation Acrucialmanifestation ofLorentzinvarianceinthelight-cone descrip- tion of the relativistic dynamics of M-dimensional extended objects is the existence of a field ζ (the longitudinal embedding coordinate) constructed out of the purely transverse fields ~x and ~p (and two ad- ditional discrete degrees of freedom, ζ = ζdMϕ and η = p ) that 0 + ∗e-mail: [email protected] R †e-mail: [email protected] 1 satisfies the same non-linear wave equation then the components of ~x, namely 1 ggab η2ζ¨= ∆ζ := ∂ ∂ ζ , (1) a a ρ ρ (cid:18) (cid:19) where g is the determinant of g = ∂ ~x∂ ~x, ρ is a certain non- ab a b dynamical density of unit weight ( ρdMϕ = 1), and the Hamiltonian for the phase-space variables (~x,p~;η,ζ ), constrained by 0 R fap~∂ ~x = 0 whenever ∂ (ρfa) = 0, (2) a a Z is given by (cp. [1, 2]) 1 p~2+g H = dMϕ. (3) 2η ρ Z The equations of motion, p~ p~˙ 1 ggab ˙ η~x = , η = ∂ ∂ ~x = ∆~x a a ρ ρ ρ ρ (cid:18) (cid:19) H η˙ = 0, ζ˙ = , (4) 0 η and (2), imply that ζ can, via p~2+g p~ η2ζ˙ = , η∂ ζ = ∂ ~x, (5) 2ρ2 a ρ a consistently be reconstructed; and (1) easily follows from (5): ˙ p~p~ g ~p g p~ η2ζ¨= + gab∂ ~x∂ ~x˙ = ∆~x+ gab∂ ~x∂ ρρ ρ2 a b ηρ ρ2 a bρη 1 ggab p~ g ~p ∆ζ = ∂ ∂ ζ = ∆~x+ ggab∂ ~x∂ . (6) ρ a ρ b ηρ ρ2 a bηρ (cid:18) (cid:19) Notethattheoriginal,manifestlyLorentz-invariant,formulation,namely ∆xµ = 0 (xµ, µ = 0,1,...,D 1 being the embedding coordinates − of the M +1 dimensional manifold swept out in space-time), di- rectly implies (1), as ζ = x0 xD 1M(and time τ = x0+xD−1), while − − 2 the chosen light-cone gauge (cp. [1, 2]) with G = 0 (a = 1,...,M), 0a G = g reduces ∆ = 1 ∂ √GGαβ∂ , the Laplacian on , to a ab − ab √G α β M 2 non-linear wave operator proportional to η2∂2 ∆. Also note that an t − explicit formula for ζ was given by Goldstone in the mid-eighties [2], 1 ~p ζ = ζ + G(ϕ,ϕ˜)˜a ∂˜ ~x ρ(ϕ˜)dMϕ˜ (7) 0 a η ∇ ρ Z (cid:18) (cid:19) δ(ϕ,ϕ˜) G(ϕ,ϕ˜)ρ(ϕ)dMϕ = 0, ∆ G(ϕ,ϕ˜) = 1, ϕ˜ ρ(ϕ) − Z and recently [3] rewritten as ~x p~ ~x ~p ζ + · · ρdMϕ 0 2ηρ − 2ηρ Z 1 p~ p~ + G(ϕ,ϕ˜) ∆~x ~x∆ (ϕ˜)ρ(ϕ˜)dMϕ˜ (8) 2 ηρ − ηρ Z (cid:18) (cid:19) -implying that ζ˜, defined as the part of 2η(ζ ζ ) not containing 0 − P~ = ~pdMϕ, can be rewritten as R ζ˜= (d +e )~x p~ Y (ϕ) (9) αβγ αβγ β γ α · where µ µ d = Y Y Y ρdMϕ, e := β − γd , (10) αβγ α β γ αβγ αβγ µ α Z withY and µ beingthe(non-constant)eigenfunctions,resp. eigen- α α − values, of a Laplacian on the parameter space (with a metric whose determinant is ρ2). Note that (7) satisfies the first equation in (5) without having to use (2) (i.e. ”strongly”) p~ ~p 2η2ζ˙ = 2ηH +2 G(ϕ,ϕ˜)˜a ∂ +∆~x∂ ~x (ϕ˜)ρ˜dMϕ a a ∇ ρ ρ Z (cid:18) (cid:19) p~2 1 ggbc = 2ηH+ G∆˜ ρdMϕ+2 G(ϕ,ϕ˜)˜a ∂ ∂ ~x ∂ ~x ρ˜dMϕ˜ ρ2 ∇ ρ b ρ c a Z (cid:18) (cid:19) Z (cid:20) (cid:18) (cid:19) (cid:21) p~2 g p~2+g = 2ηH + (∆˜G) + (ϕ˜)ρdMϕ˜= (11) ρ2 ρ2 ρ2 Z (cid:18) (cid:19) - having used that 1 g 1 g 2 a ∂ δb ∂ g = ∆ . (12) ∇ ρ b ρ a− 2ρ2 a ρ2 (cid:20) (cid:18) (cid:19) (cid:21) (cid:18) (cid:19) 3 On the other hand, p~ δ(ϕ,ϕ˜) η∂ ζ ∂ ~x = ∂ ∂˜bG(ϕ,ϕ˜)+ δb p~∂ ~x(ρ˜)dMϕ˜ (13) a − ρ a − a ρ a b Z (cid:18) (cid:19) is of the form (2) with δ(ϕ,ϕ˜) ∂˜ ρ˜ ∂ ∂˜bG(ϕ,ϕ˜)+ δb = ρ˜∂ ∆˜G(ϕ,ϕ˜) ∂ δ(ϕ,ϕ˜) = 0, b a ρ a a − a (cid:20) (cid:18) (cid:19)(cid:21) (14) implying that (13) vanishes on the constrained phase space, hence the second part of (5) holds (weakly) too. These considerations will later when we have to guess/know which part of η2ζ¨ ∆ζ is only weakly − zero, of some relevance. (1) (together with the first equation in (5)), immediately implies that the Lorentz-generator M = (x p ζ)dMϕ (15) i i i − H− Z Poisson-commutes with H, as ˙η ηM˙ = p H ηζ˙ dMϕ+ x H ∆ζ ρdMϕ. (16) i i i − ρ − ρ − ! Z (cid:18) (cid:19) Z 2 Matrix approximation, M = 2 In [4] the question of Lorentz-invariance of Matrix Membranes was discussed and a discrete analogue of ζ˜proposed, N2 1 ζ := − µa+µb−µcd(N)~x p~ T(N) (17) N µ abc b· c a a a,b,c=1 X (N) (N) wheretheT arehermiteanN N matrices andd isproportional a × abc (N) (N) (N) to Tr(T T ,T ) (the normalizations suited for N will a { b c } → ∞ be discussed below). In this paper we would like to address the question of Lorentz- invariance for the Matrix theory, focusingon numerical computations, that will tell us that at least for low N (probably all finite N) there are no Matrix • solutions for the natural analogue of (1) 4 show how exactly (and how not) the finite N analogue (17) will • approach ”solving (1)”. Before going into the details, let us note that, on general grounds (cp. [5,6])oneisguaranteedthatforM = 2andanygenus,asequence T , α = 1,2,... of linear maps (from real functions to hermitean α matrices) T :f F(Nα) = T (f) (18) α α → exists, as well as a sequence of increasing positive integers N , and α decreasing positive real numbers ~ (with lim ~ N finite), with α α α α →∞ the following properties (for arbitrary smooth functions f,g,...): lim T (f) < , α α || || ∞ →∞ T (f)T (g) T (f g) 0, α α α || − · || → 1 [T (f),T (g)] T ( f,g ) 0, ||i~ α α − α { } || → α 2π~ Tr(T (f)) fρd2ϕ. (19) α α → Z Here T (f) can be taken as the largest eigenvalue of the hermitean α || || matrix T (f), and α ǫrs f,h := ∂ f∂ h. (20) r s { } ρ For functions on a spherethis map exists for all integers N >1 (hence one can drop the index α and simply write N and ~ instead of N N α and ~ ) and, up to normalisation is given [1] via replacing in α (lm) Y (θ,ϕ) = c x ...x , (21) lm A1...Al A1 Al|~x2=1 AkX=1,2,3 the commuting variables x = rsinθcosϕ, x = rsinθsinϕ, x = rcosθ 1 2 3 by three N N matrices X , X , X satisfying 1 2 3 × 2 (N) (N) (N) [X ,X ] = i ǫ X , A B √N2 1 ABC C − X~2 := X2+X2+X2 = 1 . (22) 1 2 3 N N × 5 Theresulting”Matrix harmonics”T = T (Y )(linear independent lm N lm for l = 0,1,...,N 1, m = l,...,+l, and identically zero for l N) − − ≥ areknown[1,2,7]tohavemanyspecialproperties. Inparticular, they are eigenfunctions of the discrete Laplacian with eigenvalues µ lm − being equal to the infinite N eigenvalues l(l+1) of the parameter − space Laplace 1 1 ∆ = ∂ sinθ∂ + ∂2. (23) sinθ θ θ sin2θ ϕ (N) Apart from the fact that those T are not hermitean (because of lm the Y being complex), they are ideally suited for testing (17). Note lm that for an arbitrary function f the map T can be explicitly given as N N 1 f = ∞ f Y (θ,ϕ) F(N) := − f T(N). (24) lm lm → lm lm l=0,m l l=0 X| |≤ X To get thenormalisation factors right is more difficultfor a variety of reasons: from a practical/computational point of view it is easiest to take Tˆ(N) as given in [7], satisfying Tr(Tˆ(N)Tˆ(N)) = δ δ , and lm lm l′m′ ll′ mm′ in particular equation (6) in [7] as well as (N2 1)l(N 1 l)! Tˆ(N) = √4π − − − c(lm) X(N)...X(N). lm s (N +l)! A1...Al A1 Al AkX=1,2,3 (25) Apart from further ”hermiteanization” note that the usual spherical harmonicsarenormalizedaccordingto Y Y sinθdθdϕ= δ δ l∗m l′m′ ll′ mm′ whereas ρ should satisfy ρd2ϕ = 1. Hence with ρ ρ , Y R → 4π lm → √4πY , explaining the factor √4π in (25). lm R To conform with (19) we need N2π~ 1; we choose1 N → 1 ~ = . (26) N 2π√N2 1 − In order to have 2π~NTr(Tl†m(N)Tl(′Nm)′)→ δll′δmm′ (27) 1Onecouldalsotake2πN~=1(forallN)and(or)multiplyTˆ by√N,ratherthen(cp. (28)) by (N2 1)41. This would have the advantage of having T(1)=1 and T(xA)=XA − hold exactly, for any finite N, and also simplify (29). 6 we multiply (see previous footnote) (25) by (N2 1)14 i.e. take − T˜l(mN) = (N2−1)14Tˆl(mN) = √4πγNl c(Alm1..).AlXA1...XAl (28) AkX=1,2,3 with γ 1 as N . Nl → → ∞ Y D=ue( to1)µmlYm =)oµnl−emca,naantdthTel†men∼de(−as1il)ymgTelt−rmid(oafstahecnononse-qhueremnciecitoyf l∗m − lm (N) problemandconsiderhermiteanmatricesT . Forminglinearcombi- a nations Tl(mN)+Tl†m(N) and Tl(mN)−Tl†m(N) oneobtains thedesiredhermitean √2 √2i basis T(N) N2 1 satisfying { a }a=0− 1 Tr T(N)T(N) = δ . (29) √N2 1 a b ab − (cid:16) (cid:17) The matrix approximation of (3) is then given by (leaving out η from now on, which - just as is done in string theory - can, for most pur- poses, be absorbed in a redefinition of ”time”) 1 1 H = Tr P~2 (2π)2(N2 1)[X ,X ]2 N i j 2√N2 1 − 2 − − (cid:18) (cid:19) 1 1 (N) (N) = p p + f f ~x ~x ~x ~x (30) 2 ia ia 2 abc ab′c′ b· b′ c· c′ (cid:18) (cid:19) and the normalisations, and conventions, 1 d(N) := Tr T(N) T(N)T(N) (31) abc 2√N2 1 a { b c } − (cid:16) (cid:17) (the symbol , here denotes the anticommutator of matrices) and {· ·} 2π f(N) := Tr T(N)[T(N),T(N)] , (32) abc i a b c (cid:16) (cid:17) are such that, as N , (31) and (32) approach, respectively, → ∞ sinθ d = Y Y Y ρdθdϕ, g = Y Y ,Y ρdθdϕ, ρ = . abc a b c abc a b c { } 4π Z Z (33) 7 N 3 Lorentz symmetry at finite ? We now focus on calculating ζ¨ ∆(N)ζ where we take ζ to be N N N − given by N2 1 ζ = − L(N)~x p~ T(N) N abc b c a a,b,c=1 X (N) with for the moment arbitrary coefficients L . Using the discrete abc (N) (N) (N) equations of motion x˙ = p , p˙ = f f ~x ~x x = ∆ x ia ia ia abc c′ba′ c · c′ ia′ aa′ ia′ we find that ζ¨ ∆(N)ζ =~x ~x ~x ~p R(N) T(N) (34) N N m n µ ν amnµν a − · · where R(N) = L(N)f(N)f(N)+L(N)f(N)f(N)+2L(N)f(N)f(N) amnµν acν cdm ndµ aµc cdm ndν aνc cdm ndµ +L(N)(f(N)f(N)+f(N)f(N)) L(N)f(N)f(N). (35) amc cdµ νdn cdν µdn − cµν adm ndc The question that now arises is whether there exist nontrivial coeffi- (N) (N) (N) cients L (i.e. which cannot be written as M f ) such that the abc ak kbc r.h.s of equation (34) is weakly zero. The corresponding equation for (N) R is amnµν R(N) +R(N) = G(N) f(N) (36) amnµν anmµν anmk kµν (N) (N) (N) where G are unknown coefficients. Note that M f is a so- anmk ak kbc lution of (36) for arbitrary M , i.e. satisfies (36) with ab (N) (N) (N) (N) (N) (N) G = M (f f +f f ). anmk − ck amd dnc and dmc In order to see that there are no other solutions it is best to first symmetrize (36) over µ and ν R(N) +R(N) +R(N) +R(N) = 0 amnµν anmµν amnνµ anmνµ (N) and solve the resulting equation with respect to L . By explicit abc calculation for N = 2 and N = 3 we found the general solution to (N) (N) (N) be of the form L = M f , proving that nontrivial solutions do abc ak kbc not exist (for N = 2,3, possibly for all N). Ontheotherhand,inthelargeN limitthetheoryisrelativistically (N) invariant[2];thereforethereshouldexistachoiceofL suchthatthe abc 8 r.h.s. of (34) converges to zero when N for ~x and p~ satisfying a a → ∞ (N) the Gauss constraint G := f ~x ~p = 0. In the following we will a abc b c consider ζ given by (17) i.e. we take N µ +µ µ (N) a b c (N) L = − d . (37) abc µ abc a Inserting (37) into (34) however does not immediately yield a r.h.s. converging to zero (as we found numerically); hence it is necessary to (N) explicitly determine G . To derive the exact form of the subtrac- anmk tion that one has to make to renderconvergence (to zero), as N , → ∞ is non-trivial: due to 1 ggab ∆ζ := ∂ ∂ ζ , b a ρ ρ (cid:18) (cid:19) the term involving constraints is (leaving out the ρ-factors and η for simplicity), cp. (13): ∂ ggab Fc(ϕ,ϕ˜)(p~∂ ~x)(ϕ˜)dMϕ ; b a c − (cid:18) Z (cid:19) for M = 2, the α-component of that is + ∂ Y ǫbb′ǫaa′∂ ~x∂ ~x (ϕ)Fc(ϕ,ϕ˜)(p~∂ ~x)(ϕ˜)d2ϕd2ϕ˜ b α a′ b′ a c Z Z (cid:16) (cid:17) ∂aY 1 γ = Y ,~x ∂ ~x ~x,p~ = g g L ~x ~x ~x p~ α a γ αmβ γµν γnβ m n µ ν − { } µ { } −2 · · γ γ Z X as, using the completeness of vector spherical harmonics on S2 Fc := ∞ 1 ∂ Y (ϕ)∂˜cY (ϕ˜) +δaδ(ϕ,ϕ˜) = ∞ ǫaa′′∂a′′Y (ϕ)ǫcc′∂ Y (ϕ˜). a −µ a γ γ c µ γ c′ γ γ γ Xγ=1 (cid:16) (cid:17) Xγ=1 Hence (note the factor of 2 involved in the relation between ζ and ζ˜) the matrix U :=~x ~x ~x ~p (R(N) S(N) )T(N) (38) m n µ ν amnµν amnµν a · · − with S(N) := L(N)f(N)f(N) amnµν − cnd adm cµν should not contain any terms proportional to the G ’s and therefore a should converge strongly to 0 in the large N limit. 9 Numerical investigation In order to verify that the matrix U indeed converges to 0 we per- formed a numerical analysis for matrices with N = 3,...,11 using the conventions described in section 2 (N = 2 is trivial, U = 0). The elements of the matrix U are polynomials of the form U(N) :=~x ~x ~x p~ R˜(N) [T(N)] ij m· n µ· ν amnµν a ij where R˜(N) := R(N) S(N) . We restrict the analysis to i,j = amnµν amnµν amnµν − 1,2,3, i.e. we analyze what is the N dependence of the SU(3) corner of matrix U, and to 1 a,m,n,µ,ν 8, i.e. we consider only the ≤ ≤ range of the SU(3) adjoint index. (N) A typical polynomial U consists of about 700 terms satisfying ij theserestrictions. Wefoundnumerically thattheyall behavelike 1/N (see Fig. 1). 600 400 s 200 t n e i c 0 i f f e o -200 C - 400 2 4 6 8 10 N Figure 1: N dependence of coefficients from U(N) polynomials ij We would like to make several comments concerning this result. First, the fact that we subtracted the Gauss constraint by consider- ing R˜ instead of R is necessary to see the convergence. If the Gauss 10

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