Lorentz-invariant membranes and finite matrix approximations

0.14 MB·English

by  Jens Hoppe

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: [e-mail] R
†e-mail: [e-mail]
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