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Journal of High Energy Physics

, 2014:110 | Cite as

Space from string bits

  • Charles B. Thorn
Open Access
Regular Article - Theoretical Physics

Abstract

We develop superstring bit models, in which the lightcone transverse coordinates in D spacetime dimensions are replaced with d = D −2 double-valued “flavor” indices x k f k = 1, 2; k = 2, . . . , d + 1. In such models the string bits have no space to move. Let-ting each string bit be an adjoint of a “color” group U(N), we then analyze the physics of ’t Hooft’s limit N → ∞, in which closed chains of many string bits behave like free lightcone IIB superstrings with d compact coordinate bosonic worldsheet fields x k , and s pairs of Grassmann fermionic fields θ L,R a , a = 1, . . . , s. The coordinates x k emerge because, on the long chains, flavor fluctuations enjoy the dynamics of d anisotropic Heisenberg spin chains. It is well-known that the low energy excitations of a many-spin Heisenberg chain are identical to those of a string worldsheet coordinate compactified on a circle of radius R k , which is related to the anisotropy parameter −1 ≤ Δ k ≤ 1 of the corresponding Heisenberg system. Furthermore there is a limit of this parameter, Δ k → ±1, in which R k → ∞. As noted in earlier work [Phys. Rev. D 89 (2014) 105002], these multi-string-bit chains are strictly stable at N = ∞ when d < s and only marginally stable when d = s. (Poincaré supersymmetry requires d = s = 8, which is on the boundary between stability and instability.)

Keywords

M(atrix) Theories Bethe Ansatz 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    C.B. Thorn, Reformulating string theory with the 1/N expansion, in Sakharov memorial lectures in physics, vol. 1, L.V. Keldysh and V.Ya. Fainberg eds., Nova Science Publishers, Commack U.S.A. (1992), pg. 447 [hep-th/9405069] [INSPIRE].
  2. [2]
    O. Bergman and C.B. Thorn, String bit models for superstring, Phys. Rev. D 52 (1995) 5980 [hep-th/9506125] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    C.B. Thorn, Substructure of string, hep-th/9607204 [INSPIRE].
  4. [4]
    G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    C.B. Thorn, ≥Fock space description of the 1/N c expansion of quantum chromodynamics, Phys. Rev. D 20 (1979) 1435 [INSPIRE].ADSGoogle Scholar
  6. [6]
    G. ’t Hooft, Quantization of discrete deterministic theories by Hilbert space extension, Nucl. Phys. B 342 (1990) 471 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    G. ’t Hooft, On the quantization of space and time, in Proc. of the 4th Seminar on Quantum Gravity, Moscow USSR May 25-29 1987, M.A. Markov ed., World Scientific Press, Singapore (1988) [INSPIRE].
  8. [8]
    G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
  9. [9]
    S. Sun and C.B. Thorn, Stable string bit models, Phys. Rev. D 89 (2014) 105002 [arXiv:1402.7362] [INSPIRE].ADSGoogle Scholar
  10. [10]
    S. Sun, Detailed study of the simplest superstring bit model, to appear.Google Scholar
  11. [11]
    G. ’t Hooft, On the foundations of superstring theory, Found. Phys. 43 (2013) 46 [INSPIRE].
  12. [12]
    P. Goddard, C. Rebbi and C.B. Thorn, Lorentz covariance and the physical states in dual resonance models, Nuovo Cim. A 12 (1972) 425 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Quantum dynamics of a massless relativistic string, Nucl. Phys. B 56 (1973) 109 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    S. Mandelstam, Interacting string picture of dual resonance models, Nucl. Phys. B 64 (1973) 205 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    S. Mandelstam, Interacting string picture of the Neveu-Schwarz-Ramond model, Nucl. Phys. B 69 (1974) 77 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    R. Giles and C.B. Thorn, A lattice approach to string theory, Phys. Rev. D 16 (1977) 366 [INSPIRE].ADSGoogle Scholar
  17. [17]
    H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain (in German), Z. Phys. 71 (1931) 205 [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    C.-N. Yang and C.P. Yang, One-dimensional chain of anisotropic spin spin interactions. 1. Proof of Bethes hypothesis for ground state in a finite system, Phys. Rev. 150 (1966) 321 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    C.N. Yang and C.P. Yang, One-dimensional chain of anisotropic spin spin interactions. 2. Properties of the ground state energy per lattice site for an infinite system, Phys. Rev. 150 (1966) 327 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    F. Gliozzi, J. Scherk and D.I. Olive, Supersymmetry, supergravity theories and the dual spinor model, Nucl. Phys. B 122 (1977) 253 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    P. Ramond, Dual theory for free fermions, Phys. Rev. D 3 (1971) 2415 [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    A. Neveu and J.H. Schwarz, Factorizable dual model of pions, Nucl. Phys. B 31 (1971) 86 [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A. Neveu, J.H. Schwarz and C.B. Thorn, Reformulation of the dual pion model, Phys. Lett. B 35 (1971) 529 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    C.B. Thorn, Embryonic dual model for pions and fermions, Phys. Rev. D 4 (1971) 1112 [INSPIRE].ADSGoogle Scholar
  25. [25]
    A. Neveu and J.H. Schwarz, Quark model of dual pions, Phys. Rev. D 4 (1971) 1109 [INSPIRE].ADSGoogle Scholar
  26. [26]
    K. Bardakci and M.B. Halpern, New dual quark models, Phys. Rev. D 3 (1971) 2493 [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    M.B. Green and J.H. Schwarz, Supersymmetrical dual string theory, Nucl. Phys. B 181 (1981) 502 [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    R. Giles, L.D. McLerran and C.B. Thorn, The string representation for a field theory with internal symmetry, Phys. Rev. D 17 (1978) 2058 [INSPIRE].ADSGoogle Scholar
  29. [29]
    L. Brink and H.B. Nielsen, A simple physical interpretation of the critical dimension of space-time in dual models, Phys. Lett. B 45 (1973) 332 [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    M.B. Green, J.H. Schwarz and L. Brink, Superfield theory of type II superstrings, Nucl. Phys. B 219 (1983) 437 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Institute for Fundamental Theory, Department of PhysicsUniversity of FloridaGainesvilleU.S.A.

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