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Advances in High Energy Physics

Volume 2011 (2011), Article ID 493514, 30 pages

http://dx.doi.org/10.1155/2011/493514

Review Article

## On the Minimal Length Uncertainty Relation and the Foundations of String Theory

Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA

Received 1 June 2011; Accepted 9 August 2011

Academic Editor: Yang-Hui He

Copyright © 2011 Lay Nam Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- M. B. Green, J. H. Schwarz, and E. Witten,
*Superstring Theory*, vol. I,II, Cambridge University Press, New York, NY, USA, 1988. - J. Polchinski,
*String Theory*, vol. I,II, Cambridge University Press, New York, NY, USA, 1998. - K. Becker, M. Becker, and J. H. Schwarz,
*String Theory and M-Theory: A Modern Introduction*, Cambridge University Press, New York, NY, USA, 2007. - J. A. Wheeler, “On the nature of quantum geometrodynamics,”
*Annals of Physics*, vol. 2, no. 6, pp. 604–614, 1957. - C. A. Mead, “Possible connection between gravitation and fundamental length,”
*Physical Review*, vol. 135, no. 3B, pp. B849–B862, 1964. View at Publisher · View at Google Scholar - M. Maggiore, “A generalized uncertainty principle in quantum gravity,”
*Physics Letters Section B*, vol. 304, no. 1-2, pp. 65–69, 1993. View at Scopus - L. J. Garay, “Quantum gravity and minimum length,”
*International Journal of Modern Physics A*, vol. 10, no. 2, pp. 145–165, 1995. - S. Weinberger, “The cosmological constant problem,”
*Reviews of Modern Physics*, vol. 61, no. 1, pp. 1–23, 1989. - S. M. Carroll, “The cosmological constant,”
*Living Reviews in Relativity*, vol. 4, article 1, 2001. - E. Witten, “The cosmological constant from the viewpoint of string theory,” . In press, http://arxiv.org/abs/hep-ph/0002297.
- N. Straumann, “The history of the cosmological constant problem,” . In press, http://arxiv.org/abs/gr-qc/0208027.
- S. Nobbenhuis, “Categorizing different approaches to the cosmological constant problem,”
*Foundations of Physics*, vol. 36, no. 5, pp. 613–680, 2006. View at Publisher · View at Google Scholar · View at Scopus - J. Polchinski, “What is string theory?” . In press, http://arxiv.org/abs/hep-th/9411028.
- L. N. Chang, Z. Lewis, D. Minic, T. Takeuchi, and C. H. Tze, “Bell's inequalities, superquantum correlations, and string theory,” . In press, http://arxiv.org/abs/1104.3359.
- H. Kragh, “Arthur March, Werner Heisenberg, and the search for a smallest length,”
*Revue d’Histoire des Sciences*, vol. 8, no. 4, pp. 401–434, 1995. - H. Kragh, “Heisenberg's lattice world: the 1930 theory sketch,”
*American Journal of Physics*, vol. 63, pp. 595–605, 1995. - W. Heisenberg and W. Pauli, “Zur Quantendynamik der Wellenfelder,”
*Zeitschrift für Physik*, vol. 56, no. 1-2, pp. 1–61, 1929. View at Publisher · View at Google Scholar · View at Scopus - M. Born, “Modified field equations with a finite radius of the electron,”
*Nature*, vol. 132, no. 3329, p. 282, 1933. View at Scopus - H. S. Snyder, “Quantized space-time,”
*Physical Review*, vol. 71, no. 1, pp. 38–41, 1947. - H. S. Snyder, “The electromagnetic field in quantized space-time,”
*Physical Review*, vol. 72, no. 9, pp. 68–71, 1947. View at Publisher · View at Google Scholar · View at Scopus - C. N. Yang, “On quantized space-time,”
*Physical Review*, vol. 72, no. 1, p. 874, 1947. - C. A. Mead, “Observable consequences of fundamental-length hypotheses,”
*Physical Review*, vol. 143, no. 4, pp. 990–1005, 1966. View at Publisher · View at Google Scholar · View at Scopus - T. G. Pavlopoulos, “Breakdown of Lorentz invariance,”
*Physical Review*, vol. 159, no. 5, pp. 1106–1110, 1967. View at Publisher · View at Google Scholar · View at Scopus - T. Padmanabhan, “Physical significance of planck length,”
*Annals of Physics*, vol. 165, no. 1, pp. 38–58, 1985. - T. Padmanabhan, “Limitations on the operational definition of spacetime events and quantum gravity,”
*Classical and Quantum Gravity*, vol. 4, article L107, 1987. - T. Padmanabhan, “Duality and zero-point length of spacetime,”
*Physical Review Letters*, vol. 78, no. 10, pp. 1854–1857, 1997. - S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer, and H. Stöcker, “Signatures in the Planck regime,”
*Physics Letters Section B*, vol. 575, no. 1-2, pp. 85–99, 2003. View at Publisher · View at Google Scholar · View at Scopus - U. Harbach, S. Hossenfelder, M. Bleicher, and H. Stöcker, “Probing the minimal length scale by precision tests of the muon g - 2,”
*Physics Letters Section B*, vol. 584, no. 1-2, pp. 109–113, 2004. View at Publisher · View at Google Scholar · View at Scopus - U. Harbach and S. Hossenfelder, “The Casimir effect in the presence of a minimal length,”
*Physics Letters Section B*, vol. 632, no. 2-3, pp. 379–383, 2006. View at Publisher · View at Google Scholar · View at Scopus - S. Hossenfelder, “Running coupling with minimal length,”
*Physical Review D*, vol. 70, no. 10, Article ID 105003, 2004. View at Publisher · View at Google Scholar - S. Hossenfelder, “A note on theories with a minimal length,”
*Classical and Quantum Gravity*, vol. 23, no. 5, pp. 1815–1821, 2006. View at Publisher · View at Google Scholar - S. Hossenfelder, “Interpretation of quantum field theories with a minimal length scale,”
*Physical Review D*, vol. 73, no. 10, Article ID 105013, 2006. View at Publisher · View at Google Scholar - S. Das and E. C. Vagenas, “Universality of quantum gravity corrections,”
*Physical Review Letters*, vol. 101, no. 22, Article ID 221301, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Das and E. C. Vagenas, “Phenomenological implications of the generalized uncertainty principle,”
*Canadian Journal of Physics*, vol. 87, no. 3, pp. 233–240, 2009. View at Publisher · View at Google Scholar - S. Das and E. C. Vagenas, “Das and Vagenas reply:,”
*Physical Review Letters*, vol. 104, no. 11, Article ID 119002, 2010. View at Publisher · View at Google Scholar - A. F. Ali, S. Das, and E. C. Vagenas, “Discreteness of space from the generalized uncertainty principle,”
*Physics Letters Section B*, vol. 678, no. 5, pp. 497–499, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Basilakos, S. Das, and E. C. Vagenas, “Quantum gravity corrections and entropy at the planck time,”
*Journal of Cosmology and Astroparticle Physics*, vol. 2010, no. 9, article 027, 2010. View at Publisher · View at Google Scholar - S. Das, E. C. Vagenas, and A. F. Ali, “Discreteness of space from GUP II: relativistic wave equations,”
*Physics Letters Section B*, vol. 690, no. 4, pp. 407–412, 2010. View at Publisher · View at Google Scholar - S. Das, E. C. Vagenas, and A. Farag Ali, “Erratum to Discreteness of space from GUP II: relativistic wave equations,”
*Physics Letters Section B*, vol. 692, no. 5, p. 342, 2010. View at Publisher · View at Google Scholar · View at Scopus - B. Bagchi and A. Fring, “Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems,”
*Physics Letters Section A*, vol. 373, no. 47, pp. 4307–4310, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. Fring, L. Gouba, and F. G. Scholtz, “Strings from position-dependent noncommutativity,”
*Journal of Physics A*, vol. 43, no. 34, Article ID 345401, 2010. View at Publisher · View at Google Scholar · View at Scopus - A. Fring, L. Gouba, and B. Bagchi, “Minimal areas from q-deformed oscillator algebras,”
*Journal of Physics A*, vol. 43, no. 42, Article ID 425202, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Weinberg, “Precision tests of quantum mechanics,”
*Physical Review Letters*, vol. 62, no. 5, pp. 485–488, 1989. View at Publisher · View at Google Scholar · View at Scopus - S. Weinberg, “Testing quantum mechanics,”
*Annals of Physics*, vol. 194, no. 2, pp. 336–386, 1989. View at Scopus - C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,”
*eConf*, vol. 617, Article ID C0306234, 2003. - C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,”
*Physical Review Letters*, vol. 89, Article ID 270401, 2002. - C. M. Bender, D. C. Brody, and H. F. Jones, “Erratum: complex extension of quantum mechanics,”
*Physical Review Letters*, vol. 92, no. 11, Article ID 119902, 2004. View at Publisher · View at Google Scholar · View at Scopus - M. Maggiore, “Quantum groups, gravity, and the generalized uncertainty principle,”
*Physical Review D*, vol. 49, no. 10, pp. 5182–5187, 1994. View at Publisher · View at Google Scholar · View at Scopus - M. Maggiore, “The algebraic structure of the generalized uncertainty principle,”
*Physics Letters Section B*, vol. 319, no. 1–3, pp. 83–86, 1993. - A. Kempf, G. Mangano, and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,”
*Physical Review D*, vol. 52, no. 2, pp. 1108–1118, 1995. View at Publisher · View at Google Scholar · View at Scopus - D. Amati, M. Ciafaloni, and G. Veneziano, “Can spacetime be probed below the string size?”
*Physics Letters B*, vol. 216, no. 1-2, pp. 41–47, 1989. View at Scopus - E. Witten, “Reflections on the fate of spacetime,”
*Physics Today*, vol. 49, no. 4, pp. 24–30, 1996. View at Scopus - D. J. Gross and P. F. Mende, “The high-energy behavior of string scattering amplitudes,”
*Physics Letters B*, vol. 197, no. 1-2, pp. 129–134, 1987. View at Scopus - D. J. Gross and P. F. Mende, “String theory beyond the Planck scale,”
*Nuclear Physics Section B*, vol. 303, no. 3, pp. 407–454, 1988. View at Scopus - D. Amati, M. Ciafaloni, and G. Veneziano, “Superstring collisions at planckian energies,”
*Physics Letters B*, vol. 197, no. 1-2, pp. 81–88, 1987. View at Scopus - D. Amati, M. Ciafaloni, and G. Veneziano, “Classical and quantum gravity effects from Planckian energy superstring collisions,”
*International Journal of Modern Physics A*, vol. 3, no. 7, pp. 1615–1661, 1988. View at Scopus - W. Heisenberg,
*The Physical Principles of the Quantum Theory*, University of Chicago Press, Dover Publications, 1930. - J. A. Wheeler, “On the mathematical description of light nuclei by the method of resonating group structure,”
*Physical Review*, vol. 52, no. 11, pp. 1107–1122, 1937. View at Publisher · View at Google Scholar · View at Scopus - W. Heisenberg, “Die beobachtbaren Größen in der theorie der elementarteilchen,”
*Zeitschrift für Physik*, vol. 120, no. 7–10, pp. 513–538, 1943. View at Publisher · View at Google Scholar · View at Scopus - W. Heisenberg, “Die beobachtbaren Größen in der theorie der elementarteilchen. II,”
*Zeitschrift für Physik*, vol. 120, no. 11-12, pp. 673–702, 1943. - W. Heisenberg, “Die beobachtbaren Größen in der theorie der elementarteilchen. III,”
*Zeitschrift für Physik*, vol. 123, no. 1-2, pp. 93–112, 1944. - L. Susskind and E. Witten, “The holographic bound in anti-de sitter space,” . In press, http://arxiv.org/abs/hep-th/9805114.
- A. W. Peet and J. Polchinski, “UV-IR relations in AdS dynamics,”
*Physical Review D*, vol. 59, no. 6, pp. 1–5, 1999. - L. I. Mandelshtam and I. E. Tamm, “The uncertainty relation between energy and time
in nonrelativistic quantum mechanics,”
*Journal of Physics-USSR*, vol. 9, pp. 249–254, 1945. - E. P. Wigner, “On the time–energy uncertainty relation,” in
*Aspects of Quantum Theory*, A. Salam and E. P. Wigner, Eds., pp. 237–247, Cambridge University Press, New York, NY, USA, 1972. - M. Bauer and P. A. Mello, “The time-energy uncertainty relation,”
*Annals of Physics*, vol. 111, no. 1, pp. 38–60, 1978. View at Scopus - T. Yoneya, “On the interpretation of minimal length in string theories,”
*Modern Physics Letters A*, vol. 4, no. 16, pp. 1587–1595, 1989. - T. Yoneya, “Schild action and space-time uncertainty principle in string theory,”
*Progress of Theoretical Physics*, vol. 97, no. 6, pp. 949–961, 1997. - T. Yoneya, “String theory and the space-time uncertainty principle,”
*Progress of Theoretical Physics*, vol. 103, no. 6, pp. 1081–1125, 2000. View at Scopus - M. Li and T. Yoneya, “Short-distance space-time structure and black holes in string theory: a short review of the present status,”
*Chaos, Solitons and Fractals*, vol. 10, no. 2, pp. 423–443, 1999. View at Scopus - A. Jevicki and T. Yoneya, “Space-time uncertainty principle and conformal symmetry in D-particle dynamics,”
*Nuclear Physics B*, vol. 535, no. 1-2, pp. 335–348, 1998. View at Scopus - H. Awata, M. Li, D. Minic, and T. Yoneya, “On the quantization of Nambu brackets,”
*Journal of High Energy Physics*, vol. 5, no. 2, article 013, 2001. View at Scopus - D. Minic, “On the space-time uncertainty principle and holography,”
*Physics Letters Section B*, vol. 442, no. 1–4, pp. 102–108, 1998. View at Scopus - F. Brau, “Minimal length uncertainty relation and the hydrogen atom,”
*Journal of Physics A*, vol. 32, no. 44, pp. 7691–7696, 1999. View at Publisher · View at Google Scholar - F. Brau and F. Buisseret, “Minimal length uncertainty relation and gravitational quantum well,”
*Physical Review D*, vol. 74, no. 3, Article ID 036002, 2006. View at Publisher · View at Google Scholar - S. Z. Benczik, , Ph.D. thesis, Virginia Tech, 2007.
- A. Kempf, “Non-pointlike particles in harmonic oscillators,”
*Journal of Physics A*, vol. 30, no. 6, pp. 2093–2101, 1997. View at Scopus - L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, “Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations,”
*Physical Review D*, vol. 65, no. 12, Article ID 125027, 2002. View at Publisher · View at Google Scholar · View at Scopus - S. Benczik, L. N. Chang, D. Minic, and T. Takeuchi, “Hydrogen-atom spectrum under a minimal-length hypothesis,”
*Physical Review A*, vol. 72, no. 1, Article ID 012104, p. 4, 2005. View at Publisher · View at Google Scholar · View at Scopus - A. Chodos and E. Myers, “Gravitational contribution to the Casimir energy in Kaluza-Klein theories,”
*Annals of Physics*, vol. 156, no. 2, pp. 412–441, 1984. View at Scopus - A. Higuchi, “Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1),”
*Journal of Mathematical Physics*, vol. 28, no. 7, pp. 1553–1566, 1987. View at Scopus - N. A. Vilenkin,
*Special Functions and the Theory of Group Representations*, American Mathematical Society, 1968. - I. S. Gradshteyn and I. M. Ryzhik,
*Table of Integrals, Series and Products*, Acedemic Press, 2000. - M. Abramowitz and I. A. Stegun,
*Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Table*, Dover Publications, 1965. - E. W. Weisstein, “Confluent Hypergeometric Function of the Second Kind,” http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html.
- N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “The hierarchy problem and new dimensions at a millimeter,”
*Physics Letters Section B*, vol. 429, no. 3-4, pp. 263–272, 1998. View at Scopus - K. R. Dienes, E. Dudas, and T. Gherghetta, “Extra spacetime dimensions and unification,”
*Physics Letters Section B*, vol. 436, no. 1-2, pp. 55–65, 1998. View at Scopus - T. Han, J. D. Lykken, and R. J. Zhang, “Kaluza-Klein states from large extra dimensions,”
*Physical Review D*, vol. 59, no. 10, pp. 1–14, 1999. View at Scopus - T. Appelquist, H.-C. Cheng, and B. A. Dobrescu, “Bounds on universal extra dimensions,”
*Physical Review D*, vol. 64, no. 3, Article ID 035002, 2001. - C. Schwob, L. Jozefowski, B. De Beauvoir et al., “Optical frequency measurement of the 2S-12D transitions in hydrogen and deuterium: Rydberg constant and lamb shift determinations,”
*Physical Review Letters*, vol. 82, no. 25, pp. 4960–4963, 1999. View at Scopus - S. Mallampalli and J. Sapirstein, “Perturbed orbital contribution to the two-loop lamb shift in hydrogen,”
*Physical Review Letters*, vol. 80, no. 24, pp. 5297–5300, 1998. View at Scopus - M. I. Eides, H. Grotch, and V. A. Shelyuto, “Theory of light hydrogenlike atoms,”
*Physics Report*, vol. 342, no. 2-3, pp. 63–261, 2001. View at Scopus - G. G. Simon, C. Schmitt, F. Borkowski, and V. H. Walther, “Absolute electron-proton cross sections at low momentum transfer measured with a high pressure gas target system,”
*Nuclear Physics Section A*, vol. 333, no. 3, pp. 381–391, 1980. View at Scopus - V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov et al., “Quantum states of neutrons in the Earth's gravitational field,”
*Nature*, vol. 415, no. 6869, pp. 297–299, 2002. View at Publisher · View at Google Scholar · View at Scopus - V. V. Nesvizhevsky, et al., “Measurement of quantum states of neutrons in the Earth’s gravitational field,”
*Physical Review D*, vol. 67, no. 10, Article ID 102002, 2003. - V. V. Nesvizhevsky, A. K. Petukhov, H. G. Börner et al., “Study of the neutron quantum states in the gravity field,”
*European Physical Journal C*, vol. 40, no. 4, pp. 479–491, 2005. View at Publisher · View at Google Scholar · View at Scopus - L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, “Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem,”
*Physical Review D*, vol. 65, no. 12, Article ID 125028, 2002. View at Publisher · View at Google Scholar · View at Scopus - S. Benczik, L. N. Chang, D. Minic, N. Okamura, S. Rayyan, and T. Takeuchi, “Short distance versus long distance physics: the classical limit of the minimal length uncertainty relation,”
*Physical Review D*, vol. 66, no. 2, Article ID 026003, 2002. View at Publisher · View at Google Scholar · View at Scopus - T. Banks, “The Cosmological Constant Problem,”
*Physics Today*, vol. 57, no. 3, pp. 46–51, 2004. View at Scopus - J. Polchinski, “The cosmological constant and the string landscape,” . In press, http://arxiv.org/abs/hep-th/0603249.
- E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,”
*International Journal of Modern Physics D*, vol. 15, no. 11, pp. 1753–1935, 2006. View at Publisher · View at Google Scholar · View at Scopus - E. Komatsu, et al., “Seven-year Wilkinson microwave anisotropy probe (WMAP) observations: cosmological interpretation,”
*The Astrophysical Journal Supplement*, vol. 192, no. 2, 2011. - E. P. Wigner, “Relativistic invariance and quantum phenomena,”
*Reviews of Modern Physics*, vol. 29, no. 3, pp. 255–268, 1957. View at Publisher · View at Google Scholar · View at Scopus - H. Salecker and E. P. Wigner, “Quantum limitations of the measurement of space-time distances,”
*Physical Review*, vol. 109, no. 2, pp. 571–577, 1958. View at Publisher · View at Google Scholar · View at Scopus - F. Karolyhazy, “Gravitation and quantum mechanics of macroscopic objects,”
*Nuovo Cimento A*, vol. 42, no. 2, pp. 390–402, 1966. View at Publisher · View at Google Scholar · View at Scopus - Y. J. Ng and H. Van Dam, “Limit to space-time measurement,”
*Modern Physics Letters A*, vol. 9, no. 4, pp. 335–340, 1994. - Y. J. Ng and H. Van Dam, “Remarks on gravitational sources,”
*Modern Physics Letters A*, vol. 10, no. 36, pp. 2801–2808, 1995. - Y. J. Ng and H. Van Dam, “Measuring the foaminess of space-time with gravity-wave interferometers,”
*Foundations of Physics*, vol. 30, no. 5, pp. 795–805, 2000. - Y. J. Ng, “Spacetime foam: from entropy and holography to infinite statistics and nonlocality,”
*Entropy*, vol. 10, no. 4, pp. 441–461, 2008. - Non-relativistic space-time foam has more general “turbulent” scaling.
- G. Amelino-Camelia, “Limits on the measurability of space-time distances in (the semiclassical approximation of) quantum gravity,”
*Modern Physics Letters A*, vol. 9, no. 37, pp. 3415–3422, 1994. - L. Diósi and B. Lukács, “On the minimum uncertainty of space-time geodesics,”
*Physics Letters A*, vol. 142, no. 6-7, pp. 331–334, 1989. - L. Diósi and B. Lukács, “Critique of proposed limit to space-time measurement, based on Wigner's clocks and mirrors,”
*Europhysics Letters*, vol. 34, no. 7, pp. 479–481, 1996. View at Publisher · View at Google Scholar - V. Jejjala, D. Minic, Y. J. Ng, and C. H. Tze, “Turbulence and holography,”
*Classical and Quantum Gravity*, vol. 25, no. 22, Article ID 225012, 2008. View at Publisher · View at Google Scholar - T. Banks, “Cosmological breaking of supersymmetry?”
*International Journal of Modern Physics A*, vol. 16, no. 5, pp. 910–921, 2001. - M. Gell-Mann, P. Ramond, and R. Slansky, “Complex spinors and unified theories,” in
*Supergravity*, P. van Nieuwenhuizen and D. Z. Freedman, Eds., North Holland, 1979. - T. Yanagida, “Horizontal symmetry and masses of neutrinos,”
*Progress of Theoretical Physics*, vol. 64, no. 3, pp. 1103–1105, 1980. - R. N. Mohapatra and G. Senjanović, “Neutrino mass and spontaneous parity nonconservation,”
*Physical Review Letters*, vol. 44, no. 14, pp. 912–915, 1980. View at Publisher · View at Google Scholar · View at Scopus - S. Weinberg, “Varieties of baryon and lepton nonconservation,”
*Physical Review D*, vol. 22, no. 7, pp. 1694–1700, 1980. View at Publisher · View at Google Scholar · View at Scopus - B. Schmittmann, K. Hwang, and R. K. P. Zia, “Onset of spatial structures in biased diffusion of two species,”
*Europhysics Letters*, vol. 19, no. 1, article 19, 1992. - D. Helbing, I. J. Farkas, and T. Vicsek, “Freezing by heating in a driven mesoscopic system,”
*Physical Review Letters*, vol. 84, no. 6, pp. 1240–1243, 2000. View at Scopus - D. Helbing, “Traffic and related self-driven many-particle systems,”
*Reviews of Modern Physics*, vol. 73, no. 4, pp. 1067–1141, 2001. View at Publisher · View at Google Scholar · View at Scopus - R. K. P. Zia, E. L. Praestgaard, and O. G. Mouritsen, “Getting more from pushing less: negative specific heat and conductivity in nonequilibrium steady states,”
*American Journal of Physics*, vol. 70, no. 4, pp. 384–392, 2002. View at Publisher · View at Google Scholar · View at Scopus - D. Minic and C. H. Tze, “Background independent quantum mechanics and gravity,”
*Physical Review D*, vol. 68, no. 6, Article ID 061501, 2003. View at Publisher · View at Google Scholar - D. Minic and C. H. Tze, “A general theory of quantum relativity,”
*Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics*, vol. 581, no. 1-2, pp. 111–118, 2004. View at Publisher · View at Google Scholar · View at Scopus - V. Jejjala and D. Mimic, “Why there is something so close to nothing: towards a fundamental theory of the cosmological constant,”
*International Journal of Modern Physics A*, vol. 22, no. 10, pp. 1797–1818, 2007. View at Publisher · View at Google Scholar - V. Jejjala, M. Kavic, D. Minic, and C. H. Tze, “On the origin of time and the universe,”
*International Journal of Modern Physics A*, vol. 25, no. 12, pp. 2515–2523, 2010. View at Publisher · View at Google Scholar · View at Scopus - V. Jejjala, M. Kavic, D. Minic, and C. -H. Tze, “The big bang as the ultimate traffic jam,”
*International Journal of Modern Physics D*, vol. 18, no. 14, pp. 2257–2263, 2009. View at Publisher · View at Google Scholar - Y. A. Gol’fand,
*Soviet Physics JETP*, vol. 10, p. 842, 1960. - Y. A. Gol’fand, “Quantum field theory in constant curvature p-space,”
*Soviet Physics JETP*, vol. 16, p. 184, 1963. - Y. A. Gol’fand,
*Soviet Physics JETP*, vol. 37, p. 365, 1966. - L. Freidel and E. R. Livine, “3D quantum gravity and effective noncommutative quantum field theory,”
*Physical Review Letters*, vol. 96, no. 22, Article ID 221301, p. 2947, 2006. - G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, and L. Smolin, “The principle of relative locality,” . In press, http://arxiv.org/abs/1101.0931.
- L. N. Chang, D. Minic, and T. Takeuchi, “Quantum gravity, dynamical energymomentum space and vacuum energy,”
*Modern Physics Letters A*, vol. 25, no. 35, pp. 2947–2954, 2010. View at Publisher · View at Google Scholar · View at Scopus - I. Bars, “Gauge symmetry in phase space consequences for physics and space-time,”
*International Journal of Modern Physics A*, vol. 25, no. 9, pp. 5235–5252, 2010. - A. Einstein, “Investigations on the Theory of the Brownian Movement,” 2011, http://www.bnpublishing.net/.
- J. Magueijo and L. Smolin, “Lorentz invariance with an invariant energy scale,”
*Physical Review Letters*, vol. 88, no. 19, Article ID 190403, 2002. - J. Magueijo and L. Smolin, “Generalized Lorentz invariance with an invariant energy scale,”
*Physical Review D*, vol. 67, no. 4, Article ID 044017, 2003. - J. Kowalski-Glikman, “Introduction to doubly special relativity,”
*Lecture Notes in Physics*, vol. 669, pp. 131–159, 2005. - F. Girelli and E. R. Livine, “Special relativity as a noncommutative geometry: lessons for deformed special relativity,”
*Physical Review D*, vol. 81, no. 8, Article ID 085041, 2010. View at Publisher · View at Google Scholar · View at Scopus - E. Witten, “Noncommutative geometry and string field theory,”
*Nuclear Physics B*, vol. 268, p. 253, 1986. - G. T. Horowitz, J. Lykken, R. Rohm, and A. Strominger, “Purely cubic action for string field theory,”
*Physical Review Letters*, vol. 57, no. 3, pp. 283–286, 1986. View at Publisher · View at Google Scholar · View at Scopus - A. Strominger, “Closed string field theory,”
*Nuclear Physics B*, vol. 294, pp. 93–112, 1987. - S. Okubo,
*Introduction to Octonion and Other Non-Associative Algebras in Physics*, Cambridge University Press, New York, NY, USA, 1995. - H. Awata, M. Li, D. Minic, and T. Yoneya, “On the quantization of Nambu brackets,”
*Journal of High Energy Physics*, vol. 5, no. 2, pp. 13–16, 2001. - Y. Nambu, “Generalized hamiltonian dynamics,”
*Physical Review D*, vol. 7, no. 8, pp. 2405–2412, 1973. View at Publisher · View at Google Scholar · View at Scopus - L. Takhtajan, “On foundation of the generalized Nambu mechanics,”
*Communications in Mathematical Physics*, vol. 160, no. 2, pp. 295–315, 1994. View at Publisher · View at Google Scholar · View at Scopus - R. Chatterjee and L. Takhtajan, “Aspects of Classical and Quantum Nambu Mechanics,”
*Letters in Mathematical Physics*, vol. 37, no. 4, pp. 475–482, 1996. View at Scopus - G. Dito, M. Flato, D. Sternheimer, and L. Takhtajan, “Deformation quantization and Nambu mechanics,”
*Communications in Mathematical Physics*, vol. 183, no. 1, pp. 1–22, 1997. View at Scopus - R. G. Leigh, A. Mauri, D. Minic, and A. C. Petkou, “Gauge fields, membranes, and subdeterminant vector models,”
*Physical Review Letters*, vol. 104, no. 22, Article ID 221801, 2010. - R. Jackiw, “Topological investigations of quantized gauge theories,” in
*Current Algebra and Anomalies*, S. Trieman, R. Jackiw, B. Zumino, and E. Witten, Eds., Princeton, 1985. - J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,”
*Advances in Theoretical and Mathematical Physics*, vol. 2, no. 2, pp. 231–252, 1998. - J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,”
*International Journal of Theoretical Physics*, vol. 38, no. 4, pp. 1113–1133, 1999. - S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,”
*Physics Letters Section B*, vol. 428, no. 1-2, pp. 105–114, 1998. View at Scopus - E. Witten, “Anti-de Sitter space and holography,”
*Advances in Theoretical and Mathematical Physics*, vol. 2, pp. 253–291, 1998. - R. Blumenhagen and E. Plauschinn, “Nonassociative gravity in string theory?”
*Journal of Physics A*, vol. 44, no. 1, Article ID 015401, 2011.