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Local Well-Posedness for Membranes in the Light Cone Gauge

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Abstract

In this paper we consider the classical initial value problem for the bosonic membrane in light cone gauge. A Hamiltonian reduction gives a system with one constraint, the area preserving constraint. The Hamiltonian evolution equations corresponding to this system, however, fail to be hyperbolic. Making use of the area preserving constraint, an equivalent system of evolution equations is found, which is hyperbolic and has a well-posed initial value problem. We are thus able to solve the initial value problem for the Hamiltonian evolution equations by means of this equivalent system. We furthermore obtain a blowup criterion for the membrane evolution equations, and show, making use of the constraint, that one may achieve improved regularity estimates.

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References

  1. Abrahams A., Anderson A., Choquet-Bruhat Y., York J.W. Jr: Geometrical hyperbolic systems for general relativity and gauge theories. Class. Quant. Grav. 14, A9–A22 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Acharya B.S., Gukov S.: M theory and singularities of exceptional holonomy manifolds. Phys. Rep. 392(3), 121–189 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  3. Andersson L., Moncrief V.: Elliptic-hyperbolic systems and the Einstein equations. Ann. Henri Poincaré 4(1), 1–34 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Aragone C., Restuccia A.: Signal-front gravidynamics of vector fields in the ray gauge. Phys. Rev. D (3) 13(2), 207–217 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  5. Arnlind J., Bordemann M., Hofer L., Hoppe J., Shimada H.: Noncommutative Riemann surfaces by embeddings in \({\mathbb R^3}\) . Commun. Math. Phys. 288(2), 403–429 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Atiyah M., Witten E.: M-theory dynamics on a manifold of G 2 holonomy. Adv. Theor. Math. Phys. 6(1), 1–106 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Aurilia A., Christodoulou D.: Theory of strings and membranes in an external field. I. General formulation. J. Math. Phys. 20(7), 1446–1452 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  8. Batalin I.A., Fradkin E.S.: Operatorial quantization of dynamical systems subject to constraints. A further study of the construction. Ann. Inst. H. Poincaré Phys. Théor. 49(2), 145–214 (1988)

    MathSciNet  Google Scholar 

  9. Batalin I.A., Vilkovisky G.A.: Quantization of gauge theories with linearly dependent generators. Phys. Rev. D (3) 28(10), 2567–2582 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  10. Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Physics 111(1), 61–110 (1978)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Bergshoeff E.A., Sezgin E., Townsend P.K.: Supermembranes and eleven-dimensional supergravity. Phys. Lett. B 189(1–2), 75–78 (1987)

    MathSciNet  ADS  Google Scholar 

  12. Bergshoeff E.A., Sezgin E., Townsend P.K.: Properties of the eleven-dimensional supermembrane theory. Ann. Phys. 185(2), 330–368 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  13. Bordemann M., Hoppe J., Schaller P., Schlichenmaier M.: gl(∞) and geometric quantization. Commun. Math. Phys. 138(2), 209–244 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Boulton L.S., García del Moral M.P., Restuccia A.: Discreteness of the spectrum of the compactified D = 11 supermembrane with nontrivial winding. Nucl. Phys. B 671(1–3), 343– (2003)

    Article  MATH  ADS  Google Scholar 

  15. Claudson M., Halpern M.B.: Supersymmetric ground state wave functions. Nucl. Phys. B 250(4), 689–715 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  16. de Wit, B., Hoppe, J., Nicolai, H.: On the quantum mechanics of supermembranes, Nucl. Phys. B 305 (1988), no. 4, FS23, 545–581

  17. de Wit B., Lüscher M., Nicolai H.: The supermembrane is unstable. Nucl. Phys. B 320(1), 135–159 (1989)

    Article  ADS  Google Scholar 

  18. de Wit B., Marquard U., Nicolai H.: Area-preserving diffeomorphisms and supermembrane Lorentz invariance. Commun. Math. Phys. 128(1), 39–62 (1990)

    Article  MATH  ADS  Google Scholar 

  19. Dirac, P.A.M.: Lectures on quantum mechanics. Belfer Graduate School of Science Monographs Series, Vol. 2, New York: Belfer Graduate School of Science, 1967, Second printing of the 1964 original

  20. Farkas, H.M., Kra, I.: Riemann surfaces. Second ed., Graduate Texts in Mathematics, Vol. 71, New York: Springer-Verlag, 1992

  21. Fedosov B.V.: A simple geometrical construction of deformation quantization. J. Diff. Geom. 40(2), 213–238 (1994)

    MATH  MathSciNet  Google Scholar 

  22. Fischer A.E., Marsden J.E.: The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I. Commun. Math. Phys. 28, 1–38 (1972)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Fourès-Bruhat Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fradkin E.S., Vilkovisky G.A.: Quantization of relativistic systems with constraints. Phys. Lett. B 55(2), 224–226 (1975)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Garciadel Moral M.P., Mendez Navarro L.J., Pérez A.J., Restuccia A.: Intrinsic moment of inertia of membranes as bounds for the mass gap of Yang-Mills theories. Nucl. Phys. B 765(3), 287–298 (2007)

    Article  ADS  Google Scholar 

  26. Hoppe, J.: Canonical 3+1 description of relativistic membranes, http://arxiv.org/abs/hep-th/9407103v2, 1994

  27. Hoppe, J.: Quantum theory of a massless relativistic surface. Ph.D. thesis, MIT, 1982, available at http://www.aei.mpg.de/~hoppe

  28. Hoppe J., Nicolai H.: Relativistic minimal surfaces. Phys. Lett. B 196(4), 451–455 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  29. Hoppe J., Yau S.-T.: Some properties of matrix harmonics on S 2. Commun. Math. Phys. 195(1), 67–77 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Isenberg, J., Nester, J.: Canonical gravity. In: General relativity and gravitation, Vol. 1, New York: Plenum, 1980, pp. 23–97

  31. Kaku M.: Ghost-free formulation of quantum gravity in the light-cone gauge. Nucl. Phys. B 91(1), 99–108 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  32. Koch H.: Mixed problems for fully nonlinear hyperbolic equations. Math. Z. 214(1), 9–42 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Leibbrandt G.: Introduction to noncovariant gauges. Rev. Mod. Phys. 59(4), 1067–1119 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  35. Lesky P. Jr: Local existence for solutions of fully nonlinear wave equations. Math. Methods Appl. Sci. 14(7), 483–508 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. Lüscher M.: Some analytic results concerning the mass spectrum of Yang-Mills gauge theories on a torus. Nucl. Phys. B 219(1), 233–261 (1983)

    Article  MATH  ADS  Google Scholar 

  37. Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Sciences, Vol. 53, New York: Springer-Verlag, 1984

  38. Martín, I., Ovalle, J., Restuccia, A.: Compactified D = 11 supermembranes and symplectic noncommutative gauge theories. Phys. Rev. D 64(3) (2001), no. 4, 046001, 6

    Google Scholar 

  39. Maz′ya V., Shubin M.: Discreteness of spectrum and positivity criteria for Schrödinger operators. Ann. of Math. (2) 162(2), 919–942 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  40. Milbredt, O.: The Cauchy problem for membranes. http://arxiv.org/abs/0807.3465v1 [gr-qc], 2008, adopted from the author’s dissertation http://arxiv.org/abs/0807.2539v1 [gr-qc], 2008

  41. Molčanov A.M.: On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order. Trudy Moskov. Mat. Obšč. 2, 169–199 (1953)

    Google Scholar 

  42. Moncrief V.: Can one ADM quantize relativistic bosonic strings and membranes?. Gen. Rel. Grav. 38(4), 561–575 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. Scherk J., Schwarz J.H.: Gravitation in the light cone gauge. Gen. Rel. and Grav. 6(6), 537–550 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  44. Schmidhuber C.: D-brane actions. Nucl. Phys. B 467(1–2), 146–158 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. Senjanovic P.: Path integral quantization of field theories with second-class constraints. Ann. Phys. 100(1–2), 227–261 (1976)

    ADS  Google Scholar 

  46. Shatah, J., Struwe, M.: Geometric wave equations. Courant Lecture Notes in Mathematics, Vol. 2, New York: New York University Courant Institute of Mathematical Sciences, 1998

  47. Simon B.: Some quantum operators with discrete spectrum but classically continuous spectrum. Ann. Phys. 146(1), 209–220 (1983)

    Article  MATH  ADS  Google Scholar 

  48. Taylor, M.E.: Partial differential equations. III. Applied Mathematical Sciences, Vol. 117, New York: Springer-Verlag, 1997, corrected reprint of the 1996 original

  49. Townsend P.K.: D-branes from M-branes. Phys. Lett. B 373(1–3), 68–75 (1996)

    MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Albert Einstein Institute, Am Mühlenberg 1, 14476, Potsdam, Germany

    Paul T. Allen & Lars Andersson

  2. Department of Mathematical Sciences, Lewis& Clark College, 0615 SW Palatine Hill Road, Portland, OR, 97219, USA

    Paul T. Allen

  3. Department of Physics, Simon Bolivar University, Caracas, Venezuela

    Alvaro Restuccia

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  1. Paul T. Allen
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  2. Lars Andersson
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  3. Alvaro Restuccia
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Correspondence to Lars Andersson.

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Communicated by p.T. Chruściel

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Allen, P.T., Andersson, L. & Restuccia, A. Local Well-Posedness for Membranes in the Light Cone Gauge. Commun. Math. Phys. 301, 383–410 (2011). https://doi.org/10.1007/s00220-010-1141-5

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