Universality and Clustering in 1+1 Dimensional Superstring-Bit Models

UFIFT-HET-95-31 hep-th/9512107

arXiv:hep-th/9512107v1 14 Dec 1995

Universality and Clustering in 1+1 Dimensional Superstring-Bit Models?
Oren Bergman? and Charles B. Thorn?
Institute for Fundamental Theory Department of Physics, University of Florida, Gainesville, FL, 32611, USA

Abstract We construct a 1+1 dimensional superstring-bit model for D=3 Type IIB superstring. This low dimension model escapes the problems encountered in higher dimension models: (1) It possesses full Galilean supersymmetry; (2) For noninteracting polymers of bits, the exactly soluble linear superpotential describing bit interactions is in a large universality class of superpotentials which includes ones bounded at spatial in?nity; (3) The latter are used to construct a superstring-bit model with the clustering properties needed to de?ne an S-matrix for closed polymers of superstring-bits.

?

Supported in part by the Department of Energy under grant DE-FG05-86ER-40272, and by the E-mail address: oren@phys.u?.edu E-mail address: thorn@phys.u?.edu

Institute for Fundamental Theory.
? ?

dimensions as ?eld theories of point-like string-bits moving in d ≡ D ? 2 space bind together to form long closed polymer chains, which in the continuum limit

String-bit models are attempts to reformulate string theories in D ? 1 space

dimensions, in which string is composite, not fundamental [1, 2, 3, 4, 5]. The bits have precisely the properties of closed relativistic string. String-bits must carry an internal “color” degree of freedom which de?nes ordering around the chain. This can be achieved if the bits transform in the adjoint representation of the unitary group U(Nc ), with Nc ≥ 2. A crucial feature of our light-cone approach to string-bit invariance in d + 1 space dimensions: the bits enjoy a non-relativistic dynamics. Second-quantization of string-bits employs a string-bit creation operator φ? (x)β , α models is that they possess Galilean invariance in d space dimensions, not Poincar? e

where α and β run over the Nc colors and x denotes the d space coordinates. Denote the zero-bit state by |0 . A bare closed chain of M bits is then described by: |Ψ( x1 , . . . , xM ) = dx1 · · · dxM (1)

× Tr[φ? (x1 ) · · · φ? (xM )]|0 Ψ(x1 , . . . , xM ) ,

implying that Ψ(x1 , . . . , xM ) is cyclically symmetric. To describe closed chains and their interactions, the Hamiltonian governing string-bit dynamics must allow for their formation and assure their stability. Chain formation requires that string-bits have an attractive interaction between nearest neighbors on a chain, with non-nearest neighbors interacting much more weakly. It is well-known [6, 3, 5] that this pattern of interactions arises in a many body system of particles described by Nc × Nc matrix creation operators using ’t Hooft’s Nc → ∞ limit [7]. For an interaction Hamiltonian of the form Hint = 1 Nc dxdyV (y ? x)Tr[φ? (x)φ? (y)φ(y)φ(x)],

(2)

string-bits. For Nc ?nite but large, O(1/Nc ) e?ects allow a single bare closed chain to break into two bare closed chains. Thus 1/Nc serves as a chain coupling constant, and ’t Hooft’s 1/Nc expansion produces chain perturbation theory. 2

the limit Nc → ∞ leads to nearest-neighbor interactions in a bare closed chain of

M → ∞, m → 0, with mM kept ?xed, or equivalently in the low energy limit given

Free light-cone string is recovered for Nc → ∞ in the continuum limit given by

by E ? T0 /m, where T0 is the string tension and m is the Newtonian mass of a bit. The total Newtonian mass of a chain becomes an e?ectively continuous P + of a string. The x? coordinate of string thus emerges dynamically in string-bit models as

the conjugate to Newtonian mass. The other light-cone coordinate x+ is identi?ed as time, and its conjugate P ? as the bit Hamiltonian. The O(1/Nc) chain interactions become string interactions in the continuum limit. Stability of string depends on the ground state energy of a long closed chain, generically given by: E0,M = 1 b 1 aM + + O( 2 ) . m M M (3)

The ?rst term is the same for a single chain of M bits and two chains of M1 and M2 bits, with M1 + M2 = M. For long chains, the nature of the true ground state then depends on the second term. If b > 0 then E0,M < E0,M1 + E0,M2 , and a long chain is stable. If b < 0 then E0,M > E0,M1 + E0,M2 , the chain is unstable to decay into two smaller chains, and it will through the O(1/Nc) terms alluded to before. Consider a chain of bosonic bits interacting via the nearest-neighbor harmonic potential V (x) = (ω 2 /2m)x2 . The ground state energy for an M-bit chain in d space dimensions is found to be [1, 5] E0,M = nπ 1 πωd 1 ωd M ?1 sin 2M ? = + O( 2 ) , 2m n=1 M m 6M M (4)

and so a long chain of bosonic bits is unstable against decay into two smaller chains. This is just the string-bit manifestation of the tachyonic instability of bosonic string. The negative coe?cient of 1/2mM is the mass-squared of the tachyon. This instability is absent in superstring theory which requires the addition of fermionic modes on string in a supersymmetric fashion. For string-bit models the bits are in supermultiplets with a “statistics” degree of freedom distinguishing bosons from fermions. This degree of freedom gives rise to “statistics waves” on long chains, similar to spin waves. Supersymmetrizing the harmonic string-bit model leads to a cancellation of the contribution to the ground state energy of the coordinate “phonon” 3

waves with that of the “statistics” waves. In fact, the ground state energy is exactly zero for any M[5]. We shall see later that this is a universal property of supersymmetric string-bit models, and not special to the harmonic interaction. Note that in this model all the terms in (3) vanish. If this were not so, the sign of the ?rst nonvanishing term would determine the stability of ?nite long chains. If these were for superstring theory, and at best would only make sense in the continuum limit. unstable except when M = ∞, the bit model could not provide a fundamental basis We have constructed superstring-bit models in 2 + 1 and 8 + 1 dimensions that underlie D = 4 and D = 10 type IIB superstring theory respectively [5]. In the D = 4 case we could include extra degrees of freedom either as real compacti?ed dimensions, or as additional internal bit degrees of freedom [8] which, on long chains, would produce “?avor waves” playing the role of extra dimensions. The N = 2 spacetime Poincar? supersymmetry of the D-dimensional type IIB superstring ree an N = 1 Galilean supersymmetry. The symmetry is Galilean because light-cone variables break the manifest SO(D ? 1, 1) to SO(D ? 2) × SO(1, 1). Discretization of P + breaks this SO(1, 1) and also mixes the right- and left-moving supercharges, leaving only an N = 1 supersymmetry generated by the right + left combinations. subgroup of SO(D ? 1, 1), and have opposite chirality in the SO(1, 1) subgroup. ToSO(D ? 1, 1), and generating the super-Poincar? algebra. e
˙

recovering the free superstring mass spectrum at M = ∞ requires only b = 0, whereas

quires the corresponding (D ? 2) + 1-dimensional superstring-bit model to possess

The Galilean supercharges Q and R transform as spinors of the transverse SO(D ?2) gether they build a single supercharge transforming as a spinor of the Lorentz group For general d the Galilean supercharges QA , RA must each have d components

for a satisfactory superstring limit of the string-bit model. The corresponding superGalilei algebra then reads: {QA , QB } = mMδ AB 1 ˙ ˙ {QA , RB } = αAB · P , 2 ˙ ˙ ˙ ˙ AB B A {R , R } = δ H/2 , ,

(5)

where M is the total number of bits, and P is their total momentum. The superstring-

4

bit models of [5] implement all but the last of these relations: There are additional terms not proportional to δ AB . For the supersymmetric harmonic model these terms are sub-leading in the 1/Nc expansion, so the R-supersymmetry is broken by chain interactions. For other superstring-bit models this happens already at the level of
˙ ˙

the free chain spectrum. In either case, it might still be that the full Poincar? e supersymmetry can be recovered in the continuum stringy physics. only one component each. A 1 + 1-dimensional superstring-bit model would underlie D = 3 superstring, the lowest dimensional superstring possible [9]. Specializing the supercharges of [5] to this case gives Q = m 2 dxTr[eiπ/4 φ? (x)ψ(x) + h.c.] For d = 1, the full superalgebra closes by default, since R and Q then have

1 R = ? √ dxTr[e?iπ/4 φ? (x)ψ ′ (x) + h.c.] 2 2m 1 √ + dxdyW (y ? x) 2Nc 2m × Tr[e?iπ/4 φ? (x)ρ(y)ψ(x) + h.c.] ,

(6)

where φ? (x)β is the bosonic creation operator, ψ ? (x)β is the fermionic creation operα α ator, and ρβ = [φ? φ + ψ ? ψ]β . The superalgebra is given by: α α {Q, Q} = mM , {Q, R} = ?P/2, {R, R} = H/2. (7)

The last equation can be taken as the de?nition of the Hamiltonian for this system, which in turn implies that the model is invariant under both Q and R. If the function W (x) is taken to be odd, the two-bit sector is equivalent to Witten’s

supersymmetric quantum mechanics [10], where W (x) is the superpotential. In that case the ground state energy of a two-bit closed chain vanishes. For understanding superstring theory, we are interested in a class of superpotentials for which the ground state energy of any length chain vanishes, and for which the gap to excite the chain is ?nite. Exploring this issue for noninteracting chains, we consider the ?rst-quantized system obtained by acting with R on a bare chain and taking the limit Nc → ∞.

5

The ?rst-quantized supercharge is then given by R =
M 1 √ (eiπ/4 θk + e?iπ/4 πk )pk 2 2m k=1

? (e?iπ/4 θk + eiπ/4 πk )W (xk+1 ? xk ) ,

(8)

where pk = ?i?/?xk and πk = ?/?θk . The summation is understood to be cyclic, given by HM = 1 M p2 + W 2 (xk+1 ? xk ) 2m k=1 k

i.e. k = M + 1 is equivalent to k = 1. The ?rst-quantized M-bit Hamiltonian is then

+ W ′(xk+1 ? xk )[θk πk ? πk θk + πk+1 θk ? θk+1 πk ? i(θk θk+1 + πk πk+1 )] . (9)

We denote states in the ?rst-quantized Hilbert space of an M-bit chain by | · · ·) to The ground state of the chain is then denoted by |0). Consider the linear superpotential W (x) = T0 x ,

distinguish them from states in the second-quantized bit Fock space, denoted | · · · .

(10)

which de?nes the supersymmetric harmonic model. The ground state is annihilated by R implying it belongs to a “small” representation of the Galilei superalgebra and has zero energy. In addition its spectrum approaches that of a relativistic superstring with tension T0 in the continuum limit. Consider a deformation of the superpotential W (x) → W (x) + δW (x) , (11)

can be treated perturbatively, since for |x| > L the unperturbed wavefunctions are exponentially small. Due to the Galilei superalgebra (7), the exact Hamiltonian can of the ground state to ?rst order is then: δE0,M = 4(0|{R, δR}|0) , 6 (12) be written as the square of the new supercharge R + δR. The change in the energy

√ where δW (x) is small in the interval |x| < L. If L ? 1/ T0 this deformation

which vanishes since R|0) = (0|R = 0. We stress that this holds for any length chain. The spectrum of excitations of the chain is generated by acting on the above ground state with mode raising operators. For the 2 + 1-dimensional harmonic model these were derived in [5]. Dropping a dimension and with it the spinor indices gives: A? = n (?n + iωn xn ) p ? (?n ? iωn xn ) p ? √ √ , An = , 2ωn 2ωn (13)

where ωn = 2T0 sin nπ/M, for the coordinate modes raising and lowering operators, and
? ? ? Bn = ξn θn + ηn πn , Bn = ηn θn + ξn πn , ? ?

(14)

cos nπ/2M ), for the “statistics” modes raising and lowering operators. In the above xn , θn , pn , πn are the Fourier transforms of xk , θk , pk , πk , respectively. Consider an ? ? ? ? excitation with a single raising operator A? |0). The zero’th order energy of this state n is given by nπ 2T0 sin . m M The shift in the energy due to the deformation (11) is given to ?rst order by En,M = (0|An HA? |0) = n
(0)

√ √ where ξn = (1/ 2)(sin nπ/2M + cos nπ/2M ) and ηn = (1/ 2)(sin nπ/2M ?

(15)

δEn,M = 4(0|An {R, δR}A? |0) . n Using the commutation relations i ωn [An , R] = ? eiπn/2M Bn 2 m ? sin nπ/M √ [An , δR] = (e?iπ/4 θk + eiπ/4 πk ) 2 ωn mM k × eiπn(2k+1)/M δW ′ (xk+1 ? xk ) , and the fact that (0|δW ′(xk+1 ? xk )|0) is cyclically invariant we ?nd δEn,M = ?2ieiπn/2M ωn (0|Bn δRA? |0) + h.c. n m 2 nπ = (0|δW ′(x2 ? x1 )|0) sin . m M

(16)

(17)

(18)

The net e?ect is then just to shift the string tension T0 → T0 + (0|δW ′(x2 ? x1 )|0) , 7 (19)

so the gap remains ?nite for all M. For excitations of the form than one raising operator, we have δE = 4(0|
i

i

A? i |0), with more n

Ani {R, δR} eiπnj /2M

i

A? i |0) n A? k |0) n (20)

= ?2i

j

ω nj (0|Bnj Ani δR m i=j

k

+h.c. .

As we commute the A? k ’s to the left we pick up an additional derivative of δW n and a factor of ωnk /M = O(1/M) for each A? k that contracts with δR. The rest n contract with some of the Ani ’s. The remaining Ani ’s are then commuted to the right, all contracting with δR to produce additional derivatives and additional powers of 1/M. Finally, Bnj is contracted with what’s left. The result is a sum of terms of increasing odd number of derivatives of δW multiplied by increasing powers of 1/M, the the l = 0 term contributes, so the analogue of (18) holds for any excitation of the form
i

the coe?cient of δW (2l+1) being proportional to M ?2l?1 . In the limit M → ∞ only
? A? i |0). The argument for excitations created by products of Bn ’s or products n

? of both A? ’s and Bn ’s is similar. Thus the only e?ect of the deformation (11) on long n

chains is to renormalize the string tension as in Eq.(19). We now argue that these results from ?rst order perturbation theory hold to all orders, and indeed should extend to a large universality class of superpotentials. The ground state energy must remain zero as long as the ground state is in a “small” representation of the superalgebra. Clearly the representation can’t change in perturbation theory, but more generally it will remain “small” unless the ?rst excited state becomes degenerate with the ground state, i.e. unless the gap closes. Moreover, the properties of the O(T0 /mM) excitations of long chains have a universal character determined by phonon and statistics waves, which are inevitable collective excitations of stable long chains [4]. Formation of long bare chains is ensured by a large bond-breaking energy and does not require the in?nite range harmonic force. Finally we turn to the issue of interactions between closed chains. By exploiting the Nc → ∞ limit, we have been able to de?ne a bare closed chain, whose interactions 8

with other closed chains is negligible. Taking Nc to be ?nite will “dress” the bare chains, and will give rise to interactions between di?erent chains. The ionization energy required to break a bond in the closed chain is O(T0/m). As long as the scattering energy is below threshold for such ionization, the only way for interactions to occur is through bond rearrangement. A necessary condition for de?ning a closed chain S-matrix is that the model satisfy a clustering property: An initial state of two spatially separated closed chains must evolve as two noninteracting chains until enough time has elapsed for them to get close to one another. With a linearly growing superpotential as in (10), this is impossible at ?nite Nc . The supersymmetric harmonic model is thus unsatisfactory for describing chain scattering. Asymptotically free chains require a bounded superpotential. In order to keep the ionization energy The restricted universality established above indeed allows us to deform the linear superpotential into W = T0 [x + (L ? x)θ(x ? L) ? (L + x)θ(?L ? x)], with no change in the continuum properties of free chains. With the bounded superpotential (21) there can still be large (O(T0 /m)) correlation energies between spatially separated clumps of bits. If the clumps correspond to closed chains, or more generally to singlet states, these correlation energies must vanish. This can be achieved by ?rst noting that the generators for color rotations given by Gβ = α dx[ρ(x) ? σ(x)]β , α (22) (21) O(T0 /m), however, we must still insist that W → ±W∞ = 0 at spatial in?nity.

β where σα =: [φφ? ? ψψ ? ]β :, annihilate singlet states. This motivates replacing ρ(y) α

with ρ(y) ? σ(y) in Eq. (6) for R. It is easy to check that such a change does not disturb the superalgebra. To see that it gives the desired clustering property, consider the action of the new R on a state we denote by S1 S2 |0 , where S1 and S2 are any color singlet functions of creation operators, such that the locations of all creation operators in S1 are more than a distance L from all the locations in S2 . Let (rS)1

9

and (rS)2 be the singlet functions of creation operators de?ned by RS1 |0 = (rS)1 |0 , RS2 |0 = (rS)2 |0 . Then RS1 S2 |0 = (rS)1 S2 |0 + (?)S1 S1 (rS)2 |0 + RI S1 S2 |0 (24) (23)

in the two-body term of R to contract with a creation operator in S1 , and the other annihilation operator to contract with a creation operator in S2 . Because of the spatial separation of the coordinates in S1 and the coordinates in S2 , the superpotential W (x) can be replaced with its asymptotic value and taken out of the S1 or S2 , either of which gives zero. Thus we conclude that

where the action of RI is de?ned by requiring one of the two annihilation operators

integral. Consequently one is left with the color rotation operator (ρ ? σ) acting on

RS1 S2 |0 = (rS)1S2 |0 + (?)S1 S1 (rS)2|0 .

(25)

supercharge is odd, we infer:

Applying R once again, using the supersymmetry algebra, and remembering that the

HS1 S2 |0 = 4[(r 2 S)1 S2 |0 + S1 (r 2 S)2 |0 ],

(26)

implying that the hamiltonian acts independently on the two singlets. Therefore as long as they remain spatially well separated the two singlets propagate without mutual interaction.

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References
[1] R. Giles and C. B. Thorn, Phys. Rev. D16 (1977) 366. [2] I. Klebanov and L. Susskind, Nuc. Phys. B309 (1988) 175. [3] C. B. Thorn, in Sakharov Memorial Lectures in Physics, Ed. L. V. Keldysh and V. Ya. Fainberg, Nova Science Publishers Inc., Commack, NY, 1992; hepth/9405069. [4] C. B. Thorn, Phys. Rev. D51 (1995) 647. [5] O. Bergman and C. B. Thorn, Phys. Rev. D52 (1995) 5980. [6] C. B. Thorn, Phys. Rev. D20 (1979) 1435. [7] G. ’t Hooft, Nucl. Phys. B72 (1974) 461. [8] R. Giles, L. McLerran, and C. B. Thorn, Phys. Rev. D17 (1978) 2058. [9] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Cambridge University Press (1987). [10] E. Witten, Nuc. Phys. B188 (1981) 513.

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