边界Poisson结构及量子化
|
摘要
边界问题一直是世界范围内数学和物理科学领域中人们集中关注的重要问题之一。尤其是边界理论的量子化问题，因其在物理学中的重要性而形成了十分活跃的研究方向。尽管如此，对于如何实现带边界理论的量子化，至今仍没有一套系统而完整的解决方案。因为当一个理论中出现边界时，其原来标准的正则Poisson结构往往会与边界条件不自洽，而本文正是考虑到这一点，才试探性地提出了一种处理这种不自洽性问题的新方法。简单来说，我们的新处理方法是根据带边界理论的因果律及局域性分析，通过对该理论原来无边界情形下标准的正则Poisson结构进行适当修正，从而使其最终与边界条件相自治的。
本文共分为五章。由于边界条件通常是被当作约束来处理的，所以我们在第一章首先对处理约束系统量子化问题常用的Dirac方法作了简单的回顾。接着，我们试图利用Dirac方法对带有边界条件的理论进行量子化。结果发现，在用Dirac方法来处理具有这类特殊边界约束的系统的量子化问题时，总是存在着不可避免的闲难或弊端。这样，分析并提出解决这一问题的更为恰当的新方法就成为我们后面章节中的主要研究任务。
在第二章，我们对D+1维带边界的的质量单标量场的Poisson结构进行了修正，并在此基础上详细讨论了D+1维有质量单标量场在边界相互作用势时的量子化，同时还给出了具有边界相互作用势的2+1维有质量单标量在半平面上的自洽的Poisson结构的基本形式。
第三章我们分别讨论了端点附着于D-膜上的开弦在常的Neveu-Schwarz背景场下的量子化及在非常背景场下的自治的Poisson结构。并且，我们通过对结果的分析指出，D-膜上的场论究竟是对易的还是非对易的，主要取决于观测者所选择的新的自洽的Poisson结构的具体形式。
第四章是为了进一步说明我们这一新方法的可行性，又分别对限制于半直线上或附加了可积边界项的O(N)非线性σ模型、经典非线性σ模型和Gross-Neveu模型的自洽的Poisson结构及量子化进行了简单讨论。第五章则是对本文的总结及对下一步研究工作的展望。
The study of boundary problems continues to be one of the subjects of intensive research in the mathematical and physical sciences worldwide. In particular, because of its significance in physical science, the quantization of a theory with boundary has become a quite active research direction. However, a systematical and complete solution on how to accomplish the quantization for a theory with boundary is still missing, since the appearance of a boundary will generally make the standard canonical Poisson structure inconsistent with the boundary condition. In this thesis, we propose a new method to treat the inconsistency problem. The new treatment is a modified definition of the naive Poisson structure according to the analysis of the causality and locality of the theory in the presence of a boundary condition.
This thesis is divided into five chapters. Since the boundary conditions are usually regarded as constraints, we first give a brief review on the usual method for the Hamil-tonian description of physical system with constraints, i.e. the Dirac method. The special class of constraints which appears in the form of boundary conditions is then considered within the framework of Dirac method and some unavoidable problems or shortcommings of the Dirac method in treating boundary constraints are pointed out. This constitutes Chapter One. Now that the Dirac procedure is not quite appreciated to quantize the boundary model under consideration, to search for a new method to solve this problem becomes our main task in the forthcoming chapters.
In Chapter Two, the quantization for D + 1-dimensional massive single scalar field with boundary is considered. Especially, the quantization of D + 1-dimensional massive single scalar field with boundary interaction potential and the proper Poisson structure of 24- 1-dimensional massive single scalar field with boundary interaction potential VB=1/2 on a half plane are discussed in great detail.
In Chapter Three, consistent boundary Poisson structures for an open string ending on D-branes with a constant and non-constant background Neveu-Schwarz B fields are considered respectively, and rhe results indicate that whether field theories living
on D-branes are commutative or noncommutative depends on which consistent Poisson structure the observer chooses.
Furthermore, to show the feasibility of our new approach, we briefly discuss the quantization of O(N) nonlinear sigma model, classical nonlinear sigma model and Gross-Neveu model which are constrained on a half line or supplemented by integrable boundary terms in Chapter Four. Chapter Five contains some concluding remarks and outlines of several problems for the future research.
本文共分为五章。由于边界条件通常是被当作约束来处理的，所以我们在第一章首先对处理约束系统量子化问题常用的Dirac方法作了简单的回顾。接着，我们试图利用Dirac方法对带有边界条件的理论进行量子化。结果发现，在用Dirac方法来处理具有这类特殊边界约束的系统的量子化问题时，总是存在着不可避免的闲难或弊端。这样，分析并提出解决这一问题的更为恰当的新方法就成为我们后面章节中的主要研究任务。
在第二章，我们对D+1维带边界的的质量单标量场的Poisson结构进行了修正，并在此基础上详细讨论了D+1维有质量单标量场在边界相互作用势时的量子化，同时还给出了具有边界相互作用势的2+1维有质量单标量在半平面上的自洽的Poisson结构的基本形式。
第三章我们分别讨论了端点附着于D-膜上的开弦在常的Neveu-Schwarz背景场下的量子化及在非常背景场下的自治的Poisson结构。并且，我们通过对结果的分析指出，D-膜上的场论究竟是对易的还是非对易的，主要取决于观测者所选择的新的自洽的Poisson结构的具体形式。
第四章是为了进一步说明我们这一新方法的可行性，又分别对限制于半直线上或附加了可积边界项的O(N)非线性σ模型、经典非线性σ模型和Gross-Neveu模型的自洽的Poisson结构及量子化进行了简单讨论。第五章则是对本文的总结及对下一步研究工作的展望。
The study of boundary problems continues to be one of the subjects of intensive research in the mathematical and physical sciences worldwide. In particular, because of its significance in physical science, the quantization of a theory with boundary has become a quite active research direction. However, a systematical and complete solution on how to accomplish the quantization for a theory with boundary is still missing, since the appearance of a boundary will generally make the standard canonical Poisson structure inconsistent with the boundary condition. In this thesis, we propose a new method to treat the inconsistency problem. The new treatment is a modified definition of the naive Poisson structure according to the analysis of the causality and locality of the theory in the presence of a boundary condition.
This thesis is divided into five chapters. Since the boundary conditions are usually regarded as constraints, we first give a brief review on the usual method for the Hamil-tonian description of physical system with constraints, i.e. the Dirac method. The special class of constraints which appears in the form of boundary conditions is then considered within the framework of Dirac method and some unavoidable problems or shortcommings of the Dirac method in treating boundary constraints are pointed out. This constitutes Chapter One. Now that the Dirac procedure is not quite appreciated to quantize the boundary model under consideration, to search for a new method to solve this problem becomes our main task in the forthcoming chapters.
In Chapter Two, the quantization for D + 1-dimensional massive single scalar field with boundary is considered. Especially, the quantization of D + 1-dimensional massive single scalar field with boundary interaction potential and the proper Poisson structure of 24- 1-dimensional massive single scalar field with boundary interaction potential VB=1/2 on a half plane are discussed in great detail.
In Chapter Three, consistent boundary Poisson structures for an open string ending on D-branes with a constant and non-constant background Neveu-Schwarz B fields are considered respectively, and rhe results indicate that whether field theories living
on D-branes are commutative or noncommutative depends on which consistent Poisson structure the observer chooses.
Furthermore, to show the feasibility of our new approach, we briefly discuss the quantization of O(N) nonlinear sigma model, classical nonlinear sigma model and Gross-Neveu model which are constrained on a half line or supplemented by integrable boundary terms in Chapter Four. Chapter Five contains some concluding remarks and outlines of several problems for the future research.
引文
[1] M. M. Sheikh-Jabbari and A. Shirzad. Boundary conditions as dirac constraints. Eur. Phys. J., C19:383, 2001． arXiv:hep-th/9907055．
[2] F. Ardalan, H. Arfaei, and M. M. Sheikh-Jabbari. Mixed branes and m(atrix) theory on noncommutative torus. arXiv.hep-th/9803067．
[3] F. Ardalan, H. Arfaei, and M. M. Sheikh-Jabbari. Noncommutative geometry from strings and branes. JHEP, (9902 016) , 1999． arXiv:hep-th/9810072．
[4] W. T. Kim and J. J. Oh. Noncommutative open strings from dirac quantization. Mod. Phys. Lett., A15:1597-1604, 2000． arXiv:hep-th/9911085．
[5] C. S. Chu and P. M. Ho. Noncommutative open string and d-brane. Nucl. Phys., 3550:151． 1999． arXiv:hep-th/9812219．
[6] C. S. Chu and P. M. Ho. Constrained quantization of open string in background b field and noncommutative d-brane. Nucl. Phys., B568:447-456, 2000． arXiv.hep-th/9906192．
[7] M. A. De Andrade, M. A. Santos, and I. V. Vancea. Unconstrained variables of non-commutative open strings. JHEP, (0106 026) , 2001． arXiv:hep-th/0104154．
[8] A. A. Deriglazov. C. Neves, and W. Oliveira. Open string with a background b-field as the first order mechanics and noncommutativity. arXiv:hep-th/0110183．
[9] N. R. F. Braga and C. F. L. Godinho. Symplectic quantization of open strings and noncommutativity in branes. Phys. Rev., D65(085030) , 2002． arXiv:hep-th/0110297．
[10] Juan M. Romero and J. David Vergara. Boundary conditions as constraints. arXiv:hep-th/0212035．
[11] F. Ardalan. H. Arfaei, and M.M. Sheikh-Jabbari. Dirac quantization of open strings and noncommutativity in branes. Nucl. Phys., 6576:578-596, 2000． arXiv:hep-th/9906161．
[12] Chong-Sun Chu and Pei-Ming Ho. Noncommutative d-brane and open string in pp-wave background with b-field. Nucl. Phys., 6636:141-158, 2002． arXiv:hep-th/0203186．
[13] Ken-Ichi Tezuka. Poisson brackets, strings and membranes. Eur. Phys. J., C25:465-468, 2002． arXiv:hep-th/0201171．
[14] Maxim Zabzine. Hamiltonian systems with boundaries. .JHEP, (0010 042) , 2000． arXiv.hep-th/0005142．
[15] Taejin Lee. Canonical quantization of open string and noncommutative geometry. Phys. Rev., D62(024022) . 2000． arXiv:hep-th/9911140．
[16] Wenli He and Liu Zhao. The origin of noncommutativity? Phys. Lett., B532:345-349． 2002． arXiv:hep-th/0111041．
[17] Liu Zhao and Wenli He. Boundary poisson structure and quantization. Commun.
Theor. Phys., 38:429-432, 2002． arXiv:hep-th/0111005．
[18] P. A. M. Dirac. Generalized hamiltonian dynamics. Canad. J. Math., 2:129-148, 1950．
[19] P. A. M. Dirac. Generalized hamiltonian dynamics. Proc. Roy. Soc. London, ser., A246:326, 1958．
[20] P. A. M. Dirac. Lecture Notes on Quantum Mechanics. Belfer Graduates School of Science, Monographs Series. Yeshiva University Press, New York, 1964．
[21] Yeong-Chuan Kao and Yu-Wen Lee. Schwinger model on a half-line. Phys. Rev., D65(067701) , 2002． arXiv:hep-th/0108024．
[22] S. Ghoshal and A. B. Zainolodchikov. Boundary s-matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys., A9:3841-3886, 1994． arXiv:hep-th/9306002; Erratum-ibid. A9 (1994) 4353．
[23] G. W. Delius, N.J. Mackay, and B. J. Short. Boundary remnant of yangian symmetry and the structure of rational reflection matrices. Phys. Lett., B522:335-344, 2001． arXiv:hep-th/0109115; Erratum-ibid. B524 (2002) 401．
[24] E. Corrigan. On duality and reflection factors for the sinh-gordon model. Int. J. Mod. Phys., A13-.2709-2722, 1998． arXiv:hep-th/9707235．
[25] M. Ablikim and E. Corrigan. On the perturbative expansion of boundary reflection factors of the supersymmetric sinh-gordon model. Int. J. Mod. Phys., A16:625, 2001． arXiv:hep-th/0007214．
[26] P. A. Mattsson. Integrable quantum field theories, in the bulk and with a boundary. arXiv:hep-th/0111261．
[27] Gustav W. Delius and Niall J. MacKay. Quantum group symmetry in sine-gordon and affine toda field theories on the half-line. Commun. Math. Phys., 233:173-190, 2003． arXiv:hep-th/0112023．
[28] M. Moriconi. Integrable boundary conditions for the o(n) nonlinear sigma model. arXiv:hep-th/0111195．
[29] M. Moriconi and A. De Maitino. Quantum integrability of certain boundary conditions. Phys. Lett., 3447:292-297, 1999． arXiv:hep-th/9809178．
[30] M. Moriconi. Integrable boundary conditions and reflection matrices for the o(n) non-linear sigma model. Nucl. Phys., B619:396-414, 2001． aiXiv:hep-th/0108039．
[31] M. F. Mourad and R. Sasaki. Non-linear sigma models on a half plane. Int. J. Mod. Phys., A11:3127-3144, 1996． arXiv:hep-th/9509153．
[32] J. A. Nogueira and P. L. Barbieri. Boundary conditions as mass generation mechanism for real scalar fields. Braz. J. Phys.. 32:798-803, 2002． arXiv:hep-th/0108019．
[33] A. P. C. Malbouisson and J. M. C. Malbouisson. Boundary dependence of the coupling
constant and the mass in the vector n-component (λ 4) D theory. J. Phys., A35:2263-2274, 2002． arXiv:cond-mat/0109259．
[34] I. V. Cherednik. Factorizing particles on a half line and root systems. Theor. Math. Phys., 61:977-983, 1984． Teor. Mat. Fiz. 61 (1984) 35．
[35] A. Fring and R. Koerle. Factorized scattering in the presence of reflecting boundaries. Nucl. Phys., 3421:159-172, 1994． arXiv:hep-th/9304141．
[36] R. Sasaki. Reflection bootstrap equations for toda field theory. arXiv:hep-th/9311027, in Interface Between Physics and Mathematics, eds. W. Nahm and J. M. shen, World Scientific (1994) 201．
[37] E. Corrigan. Integrable field theory with boundary conditions. arXiv:hep-th/9612138．
[38] S. Carlip. Brane-world creation and black holes. Nucl. Phys. Proc. Suppl., 88:10-16, 2000． arXiv:hep-th/9912118．
[39] J. D. Brown and Marc Henneaux. Central charges in the canonical realization of asymptotic symmetries: An example from three dimensinal gravity. Commun. Math. Phys., 104:207-226． 1986．
[40] Edward Witten. Multi-trace operators, boundary conditions and ads/eft correspondence. arXiv:hep-th/0112258．
[41] John L. Cardy. Operator content of two-dimensional conformally invariant theories. Nucl. Phys., B270 [FS16] :186-204, 1986．
[42] Juan M. Maldacena. The large n limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys., 2:231-252, 1998． Int. J. Theor. Phys. 38 (1999) 1113-1133, arXiv.hep-th/9711200．
[43] S. Carlip. The statistical mechanics of the (2 + 1) -dimensional black hole. Phys. Rev., D51:632-637, 1995． arXiv:gr-qc/9409052．
[44] A. Strominger. Black hole entropy from near-horizon microstates. J. High Energy Phys., (9802 009) . 1998． arXiv:hep-th/9712251．
[45] G. 't Hooft. Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026．
[46] L. Susskind. The world as a hologram. J. Math. Phys., 36:6377-6396, 1995． arXiv:hep-th/9409089．
[47] E. Witten. Quantum, field theory and the jones polynomial. Commun. Math. Phys., 121:351-399, 1989． ibid 137 (1991) 29．
[48] Xavier Bekaert and Andres Gomberoff. Quantization of the chern-simons coupling constant. JHEP. (0301 054) . 2003． arXiv:hep-th/0212099．
[49] T. Regge and C. Teitelboim. Role of surface integrals in the hamiltonian formulation of general relativity. Annals Phys. (N. Y.). 88:286, 1974．
[50] J. D. Brown and Marc Henneaux. On the poisson brackets of differentiable generators in classical field theory.J. Math. Phys., 27:489-491, 1986．
[51] Josep M. Pons. Boundary conditions from boundary terms, noetber charges and the trace k lagrangian in general relativity. arXiv:gr-qc/0105032．
[52] J. D. Brown, , and J. W. York. The microcanonical functional integral, i. the gravitational field,. Phys. Rev., 047:1420-1431, 1993． arXiv:gr-qc/9209014．
[53] C. M. Chen and J. M. Nester. Quasilocal quantities for gr and other gravity theories. Class. Quant. Grav., 16:1279, 1999． arXiv:gr-qc/9809020．
[54] C. C. Chang, J. M. Nester, and C. M. Chen. Pseudotensors and quasilocal energy-momentum. Phys. Rev. Lett., 83:1897, 1999． arXiv:gr-qc/9809040．
[55] J. D. Brown, S. R. Lau. and J. W. York. Action and energy of the gravitational field. arXiv:gr-qc/0010024．
[56] Vladimir O. Soloviev. Boundary values as hamiltonian variables, i. new poisson brackets.J. Math. Phys., 34:5747-5769, 1993． arXiv:hep-th/9305133．
[57] Vladimir O. Soloviev. Boundary values as hamiltonian mvariables. ii. graded structures. arXiv:q-alg/9501017．
[58] Vladimir O. Soloviev. Boundary terms and theirm hamiltonian dynamics. Nucl. Phys. Proc. Suppl., 49:35-40, 1996． arXiv:hep-th/9601107．
[59] Vladimir O. Soloviev. Bering's proposal for boundary contribution to the poisson bracket. arXiv:hep-th/9901112．
[60] K. Bering. Family of boundary poisson brackets. Phys.Lett., B486:426-430, 2000． arXiv:hep-th/9912017．
[61] Ian I. Kogan and Nuno B.B. Reis. H-branes and chiral strings. Int. J. Mod. Phys., A16:4567-4590． 2001． arXiv:hep-th/0107163．
[62] Nathan Seiberg and Edward Witten. String theory and noncommutative geometry. JHEP, (9909032) , 1999． arXiv:hep-th/9908142．
[63] Richard J. Szabo. Quantum field theory on noncommutative spaces. arXiv:hep-th/0109162．
[64] A. Connes. Noncommutative Geometry. Academic Press, 1994．
[65] Alain Connes, Michael R. Douglas, and Albert Schwarz. Noacommutative geometry and matrix theory: Compactification on tori. JHEP, (9802 003) , 1998． arXiv:hep-th/9711162．
[66] Edward Witteri. Bound states of strings and p-branes. Nucl. Phys., 8460:335-350, 1996． arXiv:hep-th/9510135．
[67] Michael R. Douglas and Chris Hull. D-branes and the noncommutative torus. JHEP,
(9802008) , 1998． arXiv:hep-th/9711165．
[68] N. Ishibashi. A relation between commutative and noncommutative descriptions of d-branes. arXiv:hep-th/9909176．
[69] T. Banks, W. Fischler, S.H. Shenker, and L. Susskind. M theory as a matrix model: A conjecture. Phys. Rev., D55:5112-5128, 1997． arXiv:hep-th/9610043．
[70] H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, aad T. Tada. Noncommutative yang-mills in iib matrix model. Nucl. Phys., B565:176-192, 2000． arXiv:hep-th/9908141．
[71] Miao Li. Strings from iib matrices. Nucl. Phys., 6499:149-158, 1997． arXiv:hep-th/9612222．
[72] Badis Ydri. Noncommutative geometry as a regulator. Phys.Rev., D63(025004) . 2001． arXiv:hep-th/0003232．
[73] Takashi Tamaki. Tomohiro Harada, Umpei Miyamoto, and Takashi Torii. Have we already detected astrophysical symptoms of space-time noncommutativity? Phys. Rev., D65(083G03) , 2002． arXiv:gr-qc/0111056．
[74] Sean M. Carroll, Jeffrey A. Harvey, V. Alan Kostelecky, Charles, D. Lane, and Takemi Okamoto. Noncommutative field theory and lorentz violation. Phys. Rev. Lett., 87(141601) , 2001． arXiv.hep-th/0105082．
[75] J. Lukierski. P. C. Stichel, and W. J. Zakrewski. Galilean-invariant (2+1) -dimensional models with a chern-simons-like term and d=2 noncommutative geometry. Ann. Phys., 260:224, 1997． arXiv:hep-th/9612017．
[76] Y. Brahaye, C. Gonera, S. Giller, and P. Kosinski. Galilean invariance in 2+ 1 dimensions. arXiv:hep-th/9503046．
[77] L. Mezincesku. St;tr operation in quantum mechanics. arXiv:hep-th/0007046．
[78] J. Gamboa. M. Loewe, and J. C. Rojas. Non-commutative quantum mechanics. Phys. Rev., D64(067901) . 2001． arXiv:hep-th/0010220．
[79] J. Gamboa, M. Loewe, F. Mendez, and J. C. Rojas. The landau problem and non-commutative quantum mechanics. Mod. Phys. Lett, A16:2075-2078, 2001． arXiv:hep-th/0104224．
[80] V. P. Nair and A. P. Polychronakos. Quantum mechanics on the noncommutative plane and sphere. Phys. Lett., 6505:267-274． 2001． arXiv:hep-th/0011172．
[81] V. P. Nair and A. P. Polychronakos. On level quantization for the noncommutative chern-simons theory. Phys. Rev. Lett., 87(030403) , 2001． arXiv:hep-th/0102181．
[82] D. Karabali. V. P. Nair. and A. P. Polychronakos. Spectrum of the schrodinger field in a noncommutative magnetic monopole. Nucl. Phys., B627:565-579, 2002． arXiv:hep-
th/0111249．
[83] S. Bellucci, A. Nersessian, and C. Sochichiu. Two phases of the noncommutative quantum mechanics. Phys. Lett., 8522:345-349, 2001． arXiv:hep-th/0106138．
[84] A. A. Deriglazov. Noncommutative quantum mechanics as a constrained system. arXiv:hep-th/0112053．
[85] Michael R. Douglas and Nikita A. Nekrasov. Noncommutative field theory. Rev. Mod. Phys., 73:977-1029, 2001． arXiv:hep-th/0106048．
[86] Nikita A. Nekrasov. Trieste lectures on solitons in noncommutative gauge theories. 2000． arXiv:hep-th/0011095．
[87] Jeffrey A. Harvey, Per Kraus, Finn Larsen, and Emil J. Martinec. D-branes and strings as non-commutative solitons. JHEP. (0007042) . 2000． arXiv:hep-th/0005031．
[88] Keshav Dasgupta. Sunil Mukhi. and Govindan Rajesh. Noncommutative tachyons. JHEP, (0006 022) , 2000． arXiv:hep-th/0005006．
[89] M. M. Sheikh-Jabbari. More on mixed boundary conditions and d-branes bound states. Phys. Lett., B425:48-54, 1998． arXiv:hep-th/9712199．
[90] H. Arfaei and M. M. Sheikh-Jabbur. Mixed boundary conditions and brane-string bound states. Nucl. Phys., B526:278-294, (1998． arXiv:hep-th/9709054．
[91] Rabin Banerjee, Biswajit Chakraborty, and Subir Ghosh. Noncommutativity in open string: New results in a gauge independent analysis. Phys. Lett., 6537:340-350, 2002． arXiv:hep-th/0203199．
[92] N. Seiberg. Talk given in the conference new ideals in particle physics and cosmology. Uni. Penn. , May 19-22, 1999．
[93] K. Bering. Putting an edge to the poisson bracket. J. Math. Phys., 41:7468-7500, 2000． arXiv:hep-th/9806249．
[94] L. Faddeev and R. Jackiw. Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett.. 60:1692-1694, 1988．
[95] Masahiro Maeno. Light-cone quantization without periodic boundary conditions. arXiv:hep-th/0204028．
[96] F. Loran. On the quantization of constraint systems: A lagrangian approach. arXiv:hep-th/0203117．
[97] Chong-Sun Chu and Frederic Zamora. Manifest supersymmetry in non-commutative geometry. JHEP, (0002022) , 2000． arXiv:hep-th/9912153．
[98] Shoichi Kawamoto and Naoki Sasakura. Open membranes in a constant c-field background and noncommutative boundary strings. JHEP, (0007 014) , 2000． arXiv:hep-th/0005123．
[99] E. Bergshoeff, D. S. Berman, J. P. van der Schaar, and P. Sundell. A noncommutative m-theory five-brane. Nucl. Phys., B590:173-197, 2000． arXiv:hep-th/0005026．
[100] Ashok Das, J. Maharana, and A. Melikyan. Open membranes, p-branes and non-commutativity of boundary string coordinates. JHEP, (0104 016) , 2001． arXiv:hep-th/0103229．
[101] F. Loran. Comment on "constraint quantization of open string in background b field and noncotnmutative d-brane". Phys. Lett., 3544:199-201, 2002． arXiv:hep-th/0207025．
[102] A. Liguori, M. Mintchev, and L. Zhao. Boundary exchange algebras and scattering on the half line. Commun. Math. Phys., 194:569-589, 1998． arXiv.hep-th/960708．
[103] Pei-Ming Ho and Yu-Ting Yeh. Noncommutative d-brane in non-constant ns-ns b field background. Phys. Rev. Lett.. 85:5523-5526, 2000． arXiv:hep-th/0005159．
[104] A. Shirzad. Gauge symmetry in lagrangian formulation and schwinger models. Jour. of Phys. A: Math. Gen., 31:2747-2760, 1998．
[105] C. Battle, J. M. Gomis, and N. Romon-Roy. Equivalence between the lagrangian and hamiltonian formalism for constrained system. Jour. Math. Phys., 27(12) :2953-2962, 1986．
[106] S. Weinberg. The Quantum Theory of Fields, volume 1． Cambridge University Press, Cambridge, 1995．
[107] T. Maskawa and H. Nakajima. Singular lagrangian and the dirac-faddeev method. Prog. Theo. Phys., 56:1295． 1976．
[108] M. E. Peskin and D. V. Schroeder. An Introduction to Quantum Field Theory. Addison-Wesley Publishing, 1995．
[109] E. Ragoucy. Vertex operators for boundary algebras. Lett. Math. Phys., 58:249-260, 2001． arXiv:math.QA/0108221．
[110] Riuji Mochizuki. Kenji Ikegami, and Takayuki Suga. Energy radiation from a moving mirror with finite mass. arXiv:hep-th/9709128．
[111] Niels Tuning and Herman Verlinde. Backreaction on moving mirrors and black hole radiation. arXiv:hep-th/9605063．
[112] Marvin Weinstein. Moving mirrors, black holes, hawking radiation and all that. Nucl. Phys. Proc. Snppl. 108:68-73． 2000． arXiv.gr-qc/0111027．
[113] Robert D. Carlitz and Raymond S. Willey. Reflections on moving mirrors. Phys. Rev., D36(8) :2327-2333, 1987．
[114] P. Mattsson and P. Dorey. Boundary spectrum in the sine-gordon model with dirichlet boundary conditions. J. Phys.. A33:9065, 2000． arXiv:hep-th/0008071．
[115] H. S. Snyder. Quantized space-time. Phys. Rev., 71:38, 1947．
[116] H. S. Snyder. The electromagnetic field in quantized space-time. Phys. Rev., 72:68, 1947．
[117] E. Witten. Noncommutative geometry and string field theory. Nucl. Phys., B268:253, 1986．
[118] A. Connes and M. Rieffel. Yang-mills for noncommutative two-tori. Contemp. Math. Oper. Alg. Math. Phys., 62:237, 1987． in Operator Algebras and Mathematical Physics (Iowa City, Iowa, 1985) AMS.
[119] P. M. Ho and Y. S. Wu. Noncommutative geometry and d-branes. Phys. Lett, B398:52-60, 1997． arXiv:hep-th/9611233．
[120] P. M. Ho, Y. Y. Wu. and Y. S. Wu. Towards a noncommutative geometric approach to matrix compactification. Phys. Rev., D58(026006) , 1998． arXiv:hep-th/9712201．
[121] P. M. Ho and Y. S. Wu. Matrix compactification on orientifolds. Phys. Rev., D60(026002) , 1999． arXiv.hep-th/9812143．
[122] Pei-Ming Ho and Yong-Shi Wu. Noncommutative gauge theories in matrix theory. Phys.Rev., D58(066003) , 1998． arXiv:hep-th/9801147．
[123] Washington Taylor. D-brane field theory on compact spaces. Phys. Lett., B394:283-287, 1997． arXiv:hep-th/9611042．
[124] C. Hofman and E. Verlinde. U-duality of born-infeld on the noncommutative two-torus. JHEP, (9812 010) , 1998． arXiv:hep-th/9810116．
[125] Pei-Ming Ho. Twisted bundle on quantum torus and bps states in matrix theory. Phys. Lett, 6434:41-47, 1998． arXiv:hep-th/9803166．
[126] Volker Schomerus. D-branes and deformation quantization. JHEP, (9906 030) , 1999． arXiv:hep-th/9903205．
[127] C. G. Callan, C. Lovelace. C. R. Nappi. and S. A. Yost. String loop corrections to beta functions. Nucl. Phys.. B288:525． 1987．
[128] A. Abouelsaood, C. G. Callan, C. R. Nappi, and S. A. Yost. Open strings in background gauge fields. Nucl. Phys.. 8280:599． 1987．
[129] The literature on noncommutative field theories is by now quite extensive and out of our means to list, a simple search in the hep-th archive with the word "noncommutative" in the title area will give a full browse about it in recent years.
[130] T. Asakawa and I. Kishimoto. Comments on gauge equivalence in noncommutative geometry. JHEP. (9911:024) , 1999． arXiv:hep-th/9909139．
[131] B. Suo. P. Wang, and L. Zhao. Ambiguities of seiberg-witten map in the presence of matter field. Commun. Theore. Phys.. 37:571． 2002． arXiv:hep-th/0111006．
[132] E. Braaten, T. L. Curtright, and C. K. Zachos. Torsion and geometrostasis in nonlinear sigma models. Nucl Phys., B260:630-688, 1978．
[133] H. Eichenherr. su(n) invariant non-linear σ models. Nucl. Phys., 8146:215-223, 1978．
[134] E. Cremmer and J. Sherk. The supersymmetric non-linear σ-model in four dimensions and its coupling to supergravity. Phys. Lett., B74:341-343, 1978．
[135] V. Golo and A. Perelomov. Solution of the duality equations for the two-dimensional su(n)-invariant chiral model. Phys. Lett, B79:112-113, 1978．
[136] A. D'Adda, P. di Vecchia. and M. Luscher. A 1/n expandable series of non-linear σ models with instantons. Nucl. Phys., B146:63-76,1978． Nucl. Phys. B152 (1979) 125．
[137] M. Bordemann, M. Forger. J. Laartz, and U. Schaeper. The lie-poisson structure of integrable classical non-linear sigma models. Commun.Math.Phys., 152:167-190, 1993． arXiv:hep-th/9201051．
[138] A. Barabanschikov. Boundary σ-model and correctionc to d-brane actions. arXiv:hep-th/0301012．
[139] E. Corrigan and Z. Sheng. Classical integrability of the o(n) nonlinear sigma model on a half-line. Int. J. Mod. Phys., A12:2825-2834, 1997． arXiv:hep-th/9612150．
[140] Andre LeClair. Quantum theory of self-induced transparency. Nucl. Phys., B450 [FS]:753-767, 1995． arXiv:hep-th/9505086．
[141] Xiao-Gang Wen. Edge transport properties of the fractional quantum hall states and weak-impurity scattering of a one-dimensional charge-density wave. Phys. Rev., B44:5708-5719． 1991．
[142] M. Forger, J. Laartz, and U. Schaeper. Current algebra of classical non-linear sigma models. Commun. Math. Phys., 146:397-402, 1992． arXiv:hep-th/9201025．
[143] M.Forger, J.Laartz, and U.Schaeper. The algebra of the energy-momentum tensor and the noether currents in classical non-linear sigma models. Commun. Math. Phys., 159:319-328． 1994． arXiv:hep-th/9210130．
[144] L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Berlin, Heidelberg, New York: Springer, 1987．
[145] J. M. Maillet. Hamiltonian structures for integrable classical theories from graded kac-moody algebras. Phys. Lett.. B167:401-405, 1986．
[146] H. J. de Vega. H. Eichenherr, and J. M. Maillet. Classical and quantum algebras of non-local charges in sigma models. Commun. Math. Phys., 92:507-524, 1984．
[147] J. M. Maillet. Kac-moody algebra and extended yang-baxter relations in the. Phys. Lett., 8162:137-142, 1985．
[148] D. Gross and A. Neveu. Dynamical symmetry breaking in asymptotically free field
theories. Phys. Rev.. 010:3235-3253． 1974．
[149] Subir Ghoshal. Boundary s-matrix of the o(n)-symmetric non-linear sigma model. Phys. Lett., 6334:363-368, 1994． arXiv:hep-th/9401008．
[150] K. Pohlmeyer. Integrable hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys.. 46:207-221, 1976．
[151] A. Neveu and N. Papanicolaou. Integrability of the classical [ψiψi]2 and [ψiψi]22-[ψiγ5ψi]2 interactions. Commun. Math,. Phys., 58:31-64, 1978．
[152] A. Polyakov. Hidden symmetry of the two-dimensional chiral fields. Phys. Lett., B72:224-226, 1977．
[153] E. Witten. Some properties of the (ψψ) model in two dimensions. Nucl. Phys., B142:285-300． 1978．
[154] A. De Martino and M. Moriconi. Boundary s-matrix for the gross-neveu model. Phys. Lett., 6451:354-364． 1999． arXiv:hep-th/9812009．
[2] F. Ardalan, H. Arfaei, and M. M. Sheikh-Jabbari. Mixed branes and m(atrix) theory on noncommutative torus. arXiv.hep-th/9803067．
[3] F. Ardalan, H. Arfaei, and M. M. Sheikh-Jabbari. Noncommutative geometry from strings and branes. JHEP, (9902 016) , 1999． arXiv:hep-th/9810072．
[4] W. T. Kim and J. J. Oh. Noncommutative open strings from dirac quantization. Mod. Phys. Lett., A15:1597-1604, 2000． arXiv:hep-th/9911085．
[5] C. S. Chu and P. M. Ho. Noncommutative open string and d-brane. Nucl. Phys., 3550:151． 1999． arXiv:hep-th/9812219．
[6] C. S. Chu and P. M. Ho. Constrained quantization of open string in background b field and noncommutative d-brane. Nucl. Phys., B568:447-456, 2000． arXiv.hep-th/9906192．
[7] M. A. De Andrade, M. A. Santos, and I. V. Vancea. Unconstrained variables of non-commutative open strings. JHEP, (0106 026) , 2001． arXiv:hep-th/0104154．
[8] A. A. Deriglazov. C. Neves, and W. Oliveira. Open string with a background b-field as the first order mechanics and noncommutativity. arXiv:hep-th/0110183．
[9] N. R. F. Braga and C. F. L. Godinho. Symplectic quantization of open strings and noncommutativity in branes. Phys. Rev., D65(085030) , 2002． arXiv:hep-th/0110297．
[10] Juan M. Romero and J. David Vergara. Boundary conditions as constraints. arXiv:hep-th/0212035．
[11] F. Ardalan. H. Arfaei, and M.M. Sheikh-Jabbari. Dirac quantization of open strings and noncommutativity in branes. Nucl. Phys., 6576:578-596, 2000． arXiv:hep-th/9906161．
[12] Chong-Sun Chu and Pei-Ming Ho. Noncommutative d-brane and open string in pp-wave background with b-field. Nucl. Phys., 6636:141-158, 2002． arXiv:hep-th/0203186．
[13] Ken-Ichi Tezuka. Poisson brackets, strings and membranes. Eur. Phys. J., C25:465-468, 2002． arXiv:hep-th/0201171．
[14] Maxim Zabzine. Hamiltonian systems with boundaries. .JHEP, (0010 042) , 2000． arXiv.hep-th/0005142．
[15] Taejin Lee. Canonical quantization of open string and noncommutative geometry. Phys. Rev., D62(024022) . 2000． arXiv:hep-th/9911140．
[16] Wenli He and Liu Zhao. The origin of noncommutativity? Phys. Lett., B532:345-349． 2002． arXiv:hep-th/0111041．
[17] Liu Zhao and Wenli He. Boundary poisson structure and quantization. Commun.
Theor. Phys., 38:429-432, 2002． arXiv:hep-th/0111005．
[18] P. A. M. Dirac. Generalized hamiltonian dynamics. Canad. J. Math., 2:129-148, 1950．
[19] P. A. M. Dirac. Generalized hamiltonian dynamics. Proc. Roy. Soc. London, ser., A246:326, 1958．
[20] P. A. M. Dirac. Lecture Notes on Quantum Mechanics. Belfer Graduates School of Science, Monographs Series. Yeshiva University Press, New York, 1964．
[21] Yeong-Chuan Kao and Yu-Wen Lee. Schwinger model on a half-line. Phys. Rev., D65(067701) , 2002． arXiv:hep-th/0108024．
[22] S. Ghoshal and A. B. Zainolodchikov. Boundary s-matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys., A9:3841-3886, 1994． arXiv:hep-th/9306002; Erratum-ibid. A9 (1994) 4353．
[23] G. W. Delius, N.J. Mackay, and B. J. Short. Boundary remnant of yangian symmetry and the structure of rational reflection matrices. Phys. Lett., B522:335-344, 2001． arXiv:hep-th/0109115; Erratum-ibid. B524 (2002) 401．
[24] E. Corrigan. On duality and reflection factors for the sinh-gordon model. Int. J. Mod. Phys., A13-.2709-2722, 1998． arXiv:hep-th/9707235．
[25] M. Ablikim and E. Corrigan. On the perturbative expansion of boundary reflection factors of the supersymmetric sinh-gordon model. Int. J. Mod. Phys., A16:625, 2001． arXiv:hep-th/0007214．
[26] P. A. Mattsson. Integrable quantum field theories, in the bulk and with a boundary. arXiv:hep-th/0111261．
[27] Gustav W. Delius and Niall J. MacKay. Quantum group symmetry in sine-gordon and affine toda field theories on the half-line. Commun. Math. Phys., 233:173-190, 2003． arXiv:hep-th/0112023．
[28] M. Moriconi. Integrable boundary conditions for the o(n) nonlinear sigma model. arXiv:hep-th/0111195．
[29] M. Moriconi and A. De Maitino. Quantum integrability of certain boundary conditions. Phys. Lett., 3447:292-297, 1999． arXiv:hep-th/9809178．
[30] M. Moriconi. Integrable boundary conditions and reflection matrices for the o(n) non-linear sigma model. Nucl. Phys., B619:396-414, 2001． aiXiv:hep-th/0108039．
[31] M. F. Mourad and R. Sasaki. Non-linear sigma models on a half plane. Int. J. Mod. Phys., A11:3127-3144, 1996． arXiv:hep-th/9509153．
[32] J. A. Nogueira and P. L. Barbieri. Boundary conditions as mass generation mechanism for real scalar fields. Braz. J. Phys.. 32:798-803, 2002． arXiv:hep-th/0108019．
[33] A. P. C. Malbouisson and J. M. C. Malbouisson. Boundary dependence of the coupling
constant and the mass in the vector n-component (λ 4) D theory. J. Phys., A35:2263-2274, 2002． arXiv:cond-mat/0109259．
[34] I. V. Cherednik. Factorizing particles on a half line and root systems. Theor. Math. Phys., 61:977-983, 1984． Teor. Mat. Fiz. 61 (1984) 35．
[35] A. Fring and R. Koerle. Factorized scattering in the presence of reflecting boundaries. Nucl. Phys., 3421:159-172, 1994． arXiv:hep-th/9304141．
[36] R. Sasaki. Reflection bootstrap equations for toda field theory. arXiv:hep-th/9311027, in Interface Between Physics and Mathematics, eds. W. Nahm and J. M. shen, World Scientific (1994) 201．
[37] E. Corrigan. Integrable field theory with boundary conditions. arXiv:hep-th/9612138．
[38] S. Carlip. Brane-world creation and black holes. Nucl. Phys. Proc. Suppl., 88:10-16, 2000． arXiv:hep-th/9912118．
[39] J. D. Brown and Marc Henneaux. Central charges in the canonical realization of asymptotic symmetries: An example from three dimensinal gravity. Commun. Math. Phys., 104:207-226． 1986．
[40] Edward Witten. Multi-trace operators, boundary conditions and ads/eft correspondence. arXiv:hep-th/0112258．
[41] John L. Cardy. Operator content of two-dimensional conformally invariant theories. Nucl. Phys., B270 [FS16] :186-204, 1986．
[42] Juan M. Maldacena. The large n limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys., 2:231-252, 1998． Int. J. Theor. Phys. 38 (1999) 1113-1133, arXiv.hep-th/9711200．
[43] S. Carlip. The statistical mechanics of the (2 + 1) -dimensional black hole. Phys. Rev., D51:632-637, 1995． arXiv:gr-qc/9409052．
[44] A. Strominger. Black hole entropy from near-horizon microstates. J. High Energy Phys., (9802 009) . 1998． arXiv:hep-th/9712251．
[45] G. 't Hooft. Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026．
[46] L. Susskind. The world as a hologram. J. Math. Phys., 36:6377-6396, 1995． arXiv:hep-th/9409089．
[47] E. Witten. Quantum, field theory and the jones polynomial. Commun. Math. Phys., 121:351-399, 1989． ibid 137 (1991) 29．
[48] Xavier Bekaert and Andres Gomberoff. Quantization of the chern-simons coupling constant. JHEP. (0301 054) . 2003． arXiv:hep-th/0212099．
[49] T. Regge and C. Teitelboim. Role of surface integrals in the hamiltonian formulation of general relativity. Annals Phys. (N. Y.). 88:286, 1974．
[50] J. D. Brown and Marc Henneaux. On the poisson brackets of differentiable generators in classical field theory.J. Math. Phys., 27:489-491, 1986．
[51] Josep M. Pons. Boundary conditions from boundary terms, noetber charges and the trace k lagrangian in general relativity. arXiv:gr-qc/0105032．
[52] J. D. Brown, , and J. W. York. The microcanonical functional integral, i. the gravitational field,. Phys. Rev., 047:1420-1431, 1993． arXiv:gr-qc/9209014．
[53] C. M. Chen and J. M. Nester. Quasilocal quantities for gr and other gravity theories. Class. Quant. Grav., 16:1279, 1999． arXiv:gr-qc/9809020．
[54] C. C. Chang, J. M. Nester, and C. M. Chen. Pseudotensors and quasilocal energy-momentum. Phys. Rev. Lett., 83:1897, 1999． arXiv:gr-qc/9809040．
[55] J. D. Brown, S. R. Lau. and J. W. York. Action and energy of the gravitational field. arXiv:gr-qc/0010024．
[56] Vladimir O. Soloviev. Boundary values as hamiltonian variables, i. new poisson brackets.J. Math. Phys., 34:5747-5769, 1993． arXiv:hep-th/9305133．
[57] Vladimir O. Soloviev. Boundary values as hamiltonian mvariables. ii. graded structures. arXiv:q-alg/9501017．
[58] Vladimir O. Soloviev. Boundary terms and theirm hamiltonian dynamics. Nucl. Phys. Proc. Suppl., 49:35-40, 1996． arXiv:hep-th/9601107．
[59] Vladimir O. Soloviev. Bering's proposal for boundary contribution to the poisson bracket. arXiv:hep-th/9901112．
[60] K. Bering. Family of boundary poisson brackets. Phys.Lett., B486:426-430, 2000． arXiv:hep-th/9912017．
[61] Ian I. Kogan and Nuno B.B. Reis. H-branes and chiral strings. Int. J. Mod. Phys., A16:4567-4590． 2001． arXiv:hep-th/0107163．
[62] Nathan Seiberg and Edward Witten. String theory and noncommutative geometry. JHEP, (9909032) , 1999． arXiv:hep-th/9908142．
[63] Richard J. Szabo. Quantum field theory on noncommutative spaces. arXiv:hep-th/0109162．
[64] A. Connes. Noncommutative Geometry. Academic Press, 1994．
[65] Alain Connes, Michael R. Douglas, and Albert Schwarz. Noacommutative geometry and matrix theory: Compactification on tori. JHEP, (9802 003) , 1998． arXiv:hep-th/9711162．
[66] Edward Witteri. Bound states of strings and p-branes. Nucl. Phys., 8460:335-350, 1996． arXiv:hep-th/9510135．
[67] Michael R. Douglas and Chris Hull. D-branes and the noncommutative torus. JHEP,
(9802008) , 1998． arXiv:hep-th/9711165．
[68] N. Ishibashi. A relation between commutative and noncommutative descriptions of d-branes. arXiv:hep-th/9909176．
[69] T. Banks, W. Fischler, S.H. Shenker, and L. Susskind. M theory as a matrix model: A conjecture. Phys. Rev., D55:5112-5128, 1997． arXiv:hep-th/9610043．
[70] H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, aad T. Tada. Noncommutative yang-mills in iib matrix model. Nucl. Phys., B565:176-192, 2000． arXiv:hep-th/9908141．
[71] Miao Li. Strings from iib matrices. Nucl. Phys., 6499:149-158, 1997． arXiv:hep-th/9612222．
[72] Badis Ydri. Noncommutative geometry as a regulator. Phys.Rev., D63(025004) . 2001． arXiv:hep-th/0003232．
[73] Takashi Tamaki. Tomohiro Harada, Umpei Miyamoto, and Takashi Torii. Have we already detected astrophysical symptoms of space-time noncommutativity? Phys. Rev., D65(083G03) , 2002． arXiv:gr-qc/0111056．
[74] Sean M. Carroll, Jeffrey A. Harvey, V. Alan Kostelecky, Charles, D. Lane, and Takemi Okamoto. Noncommutative field theory and lorentz violation. Phys. Rev. Lett., 87(141601) , 2001． arXiv.hep-th/0105082．
[75] J. Lukierski. P. C. Stichel, and W. J. Zakrewski. Galilean-invariant (2+1) -dimensional models with a chern-simons-like term and d=2 noncommutative geometry. Ann. Phys., 260:224, 1997． arXiv:hep-th/9612017．
[76] Y. Brahaye, C. Gonera, S. Giller, and P. Kosinski. Galilean invariance in 2+ 1 dimensions. arXiv:hep-th/9503046．
[77] L. Mezincesku. St;tr operation in quantum mechanics. arXiv:hep-th/0007046．
[78] J. Gamboa. M. Loewe, and J. C. Rojas. Non-commutative quantum mechanics. Phys. Rev., D64(067901) . 2001． arXiv:hep-th/0010220．
[79] J. Gamboa, M. Loewe, F. Mendez, and J. C. Rojas. The landau problem and non-commutative quantum mechanics. Mod. Phys. Lett, A16:2075-2078, 2001． arXiv:hep-th/0104224．
[80] V. P. Nair and A. P. Polychronakos. Quantum mechanics on the noncommutative plane and sphere. Phys. Lett., 6505:267-274． 2001． arXiv:hep-th/0011172．
[81] V. P. Nair and A. P. Polychronakos. On level quantization for the noncommutative chern-simons theory. Phys. Rev. Lett., 87(030403) , 2001． arXiv:hep-th/0102181．
[82] D. Karabali. V. P. Nair. and A. P. Polychronakos. Spectrum of the schrodinger field in a noncommutative magnetic monopole. Nucl. Phys., B627:565-579, 2002． arXiv:hep-
th/0111249．
[83] S. Bellucci, A. Nersessian, and C. Sochichiu. Two phases of the noncommutative quantum mechanics. Phys. Lett., 8522:345-349, 2001． arXiv:hep-th/0106138．
[84] A. A. Deriglazov. Noncommutative quantum mechanics as a constrained system. arXiv:hep-th/0112053．
[85] Michael R. Douglas and Nikita A. Nekrasov. Noncommutative field theory. Rev. Mod. Phys., 73:977-1029, 2001． arXiv:hep-th/0106048．
[86] Nikita A. Nekrasov. Trieste lectures on solitons in noncommutative gauge theories. 2000． arXiv:hep-th/0011095．
[87] Jeffrey A. Harvey, Per Kraus, Finn Larsen, and Emil J. Martinec. D-branes and strings as non-commutative solitons. JHEP. (0007042) . 2000． arXiv:hep-th/0005031．
[88] Keshav Dasgupta. Sunil Mukhi. and Govindan Rajesh. Noncommutative tachyons. JHEP, (0006 022) , 2000． arXiv:hep-th/0005006．
[89] M. M. Sheikh-Jabbari. More on mixed boundary conditions and d-branes bound states. Phys. Lett., B425:48-54, 1998． arXiv:hep-th/9712199．
[90] H. Arfaei and M. M. Sheikh-Jabbur. Mixed boundary conditions and brane-string bound states. Nucl. Phys., B526:278-294, (1998． arXiv:hep-th/9709054．
[91] Rabin Banerjee, Biswajit Chakraborty, and Subir Ghosh. Noncommutativity in open string: New results in a gauge independent analysis. Phys. Lett., 6537:340-350, 2002． arXiv:hep-th/0203199．
[92] N. Seiberg. Talk given in the conference new ideals in particle physics and cosmology. Uni. Penn. , May 19-22, 1999．
[93] K. Bering. Putting an edge to the poisson bracket. J. Math. Phys., 41:7468-7500, 2000． arXiv:hep-th/9806249．
[94] L. Faddeev and R. Jackiw. Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett.. 60:1692-1694, 1988．
[95] Masahiro Maeno. Light-cone quantization without periodic boundary conditions. arXiv:hep-th/0204028．
[96] F. Loran. On the quantization of constraint systems: A lagrangian approach. arXiv:hep-th/0203117．
[97] Chong-Sun Chu and Frederic Zamora. Manifest supersymmetry in non-commutative geometry. JHEP, (0002022) , 2000． arXiv:hep-th/9912153．
[98] Shoichi Kawamoto and Naoki Sasakura. Open membranes in a constant c-field background and noncommutative boundary strings. JHEP, (0007 014) , 2000． arXiv:hep-th/0005123．
[99] E. Bergshoeff, D. S. Berman, J. P. van der Schaar, and P. Sundell. A noncommutative m-theory five-brane. Nucl. Phys., B590:173-197, 2000． arXiv:hep-th/0005026．
[100] Ashok Das, J. Maharana, and A. Melikyan. Open membranes, p-branes and non-commutativity of boundary string coordinates. JHEP, (0104 016) , 2001． arXiv:hep-th/0103229．
[101] F. Loran. Comment on "constraint quantization of open string in background b field and noncotnmutative d-brane". Phys. Lett., 3544:199-201, 2002． arXiv:hep-th/0207025．
[102] A. Liguori, M. Mintchev, and L. Zhao. Boundary exchange algebras and scattering on the half line. Commun. Math. Phys., 194:569-589, 1998． arXiv.hep-th/960708．
[103] Pei-Ming Ho and Yu-Ting Yeh. Noncommutative d-brane in non-constant ns-ns b field background. Phys. Rev. Lett.. 85:5523-5526, 2000． arXiv:hep-th/0005159．
[104] A. Shirzad. Gauge symmetry in lagrangian formulation and schwinger models. Jour. of Phys. A: Math. Gen., 31:2747-2760, 1998．
[105] C. Battle, J. M. Gomis, and N. Romon-Roy. Equivalence between the lagrangian and hamiltonian formalism for constrained system. Jour. Math. Phys., 27(12) :2953-2962, 1986．
[106] S. Weinberg. The Quantum Theory of Fields, volume 1． Cambridge University Press, Cambridge, 1995．
[107] T. Maskawa and H. Nakajima. Singular lagrangian and the dirac-faddeev method. Prog. Theo. Phys., 56:1295． 1976．
[108] M. E. Peskin and D. V. Schroeder. An Introduction to Quantum Field Theory. Addison-Wesley Publishing, 1995．
[109] E. Ragoucy. Vertex operators for boundary algebras. Lett. Math. Phys., 58:249-260, 2001． arXiv:math.QA/0108221．
[110] Riuji Mochizuki. Kenji Ikegami, and Takayuki Suga. Energy radiation from a moving mirror with finite mass. arXiv:hep-th/9709128．
[111] Niels Tuning and Herman Verlinde. Backreaction on moving mirrors and black hole radiation. arXiv:hep-th/9605063．
[112] Marvin Weinstein. Moving mirrors, black holes, hawking radiation and all that. Nucl. Phys. Proc. Snppl. 108:68-73． 2000． arXiv.gr-qc/0111027．
[113] Robert D. Carlitz and Raymond S. Willey. Reflections on moving mirrors. Phys. Rev., D36(8) :2327-2333, 1987．
[114] P. Mattsson and P. Dorey. Boundary spectrum in the sine-gordon model with dirichlet boundary conditions. J. Phys.. A33:9065, 2000． arXiv:hep-th/0008071．
[115] H. S. Snyder. Quantized space-time. Phys. Rev., 71:38, 1947．
[116] H. S. Snyder. The electromagnetic field in quantized space-time. Phys. Rev., 72:68, 1947．
[117] E. Witten. Noncommutative geometry and string field theory. Nucl. Phys., B268:253, 1986．
[118] A. Connes and M. Rieffel. Yang-mills for noncommutative two-tori. Contemp. Math. Oper. Alg. Math. Phys., 62:237, 1987． in Operator Algebras and Mathematical Physics (Iowa City, Iowa, 1985) AMS.
[119] P. M. Ho and Y. S. Wu. Noncommutative geometry and d-branes. Phys. Lett, B398:52-60, 1997． arXiv:hep-th/9611233．
[120] P. M. Ho, Y. Y. Wu. and Y. S. Wu. Towards a noncommutative geometric approach to matrix compactification. Phys. Rev., D58(026006) , 1998． arXiv:hep-th/9712201．
[121] P. M. Ho and Y. S. Wu. Matrix compactification on orientifolds. Phys. Rev., D60(026002) , 1999． arXiv.hep-th/9812143．
[122] Pei-Ming Ho and Yong-Shi Wu. Noncommutative gauge theories in matrix theory. Phys.Rev., D58(066003) , 1998． arXiv:hep-th/9801147．
[123] Washington Taylor. D-brane field theory on compact spaces. Phys. Lett., B394:283-287, 1997． arXiv:hep-th/9611042．
[124] C. Hofman and E. Verlinde. U-duality of born-infeld on the noncommutative two-torus. JHEP, (9812 010) , 1998． arXiv:hep-th/9810116．
[125] Pei-Ming Ho. Twisted bundle on quantum torus and bps states in matrix theory. Phys. Lett, 6434:41-47, 1998． arXiv:hep-th/9803166．
[126] Volker Schomerus. D-branes and deformation quantization. JHEP, (9906 030) , 1999． arXiv:hep-th/9903205．
[127] C. G. Callan, C. Lovelace. C. R. Nappi. and S. A. Yost. String loop corrections to beta functions. Nucl. Phys.. B288:525． 1987．
[128] A. Abouelsaood, C. G. Callan, C. R. Nappi, and S. A. Yost. Open strings in background gauge fields. Nucl. Phys.. 8280:599． 1987．
[129] The literature on noncommutative field theories is by now quite extensive and out of our means to list, a simple search in the hep-th archive with the word "noncommutative" in the title area will give a full browse about it in recent years.
[130] T. Asakawa and I. Kishimoto. Comments on gauge equivalence in noncommutative geometry. JHEP. (9911:024) , 1999． arXiv:hep-th/9909139．
[131] B. Suo. P. Wang, and L. Zhao. Ambiguities of seiberg-witten map in the presence of matter field. Commun. Theore. Phys.. 37:571． 2002． arXiv:hep-th/0111006．
[132] E. Braaten, T. L. Curtright, and C. K. Zachos. Torsion and geometrostasis in nonlinear sigma models. Nucl Phys., B260:630-688, 1978．
[133] H. Eichenherr. su(n) invariant non-linear σ models. Nucl. Phys., 8146:215-223, 1978．
[134] E. Cremmer and J. Sherk. The supersymmetric non-linear σ-model in four dimensions and its coupling to supergravity. Phys. Lett., B74:341-343, 1978．
[135] V. Golo and A. Perelomov. Solution of the duality equations for the two-dimensional su(n)-invariant chiral model. Phys. Lett, B79:112-113, 1978．
[136] A. D'Adda, P. di Vecchia. and M. Luscher. A 1/n expandable series of non-linear σ models with instantons. Nucl. Phys., B146:63-76,1978． Nucl. Phys. B152 (1979) 125．
[137] M. Bordemann, M. Forger. J. Laartz, and U. Schaeper. The lie-poisson structure of integrable classical non-linear sigma models. Commun.Math.Phys., 152:167-190, 1993． arXiv:hep-th/9201051．
[138] A. Barabanschikov. Boundary σ-model and correctionc to d-brane actions. arXiv:hep-th/0301012．
[139] E. Corrigan and Z. Sheng. Classical integrability of the o(n) nonlinear sigma model on a half-line. Int. J. Mod. Phys., A12:2825-2834, 1997． arXiv:hep-th/9612150．
[140] Andre LeClair. Quantum theory of self-induced transparency. Nucl. Phys., B450 [FS]:753-767, 1995． arXiv:hep-th/9505086．
[141] Xiao-Gang Wen. Edge transport properties of the fractional quantum hall states and weak-impurity scattering of a one-dimensional charge-density wave. Phys. Rev., B44:5708-5719． 1991．
[142] M. Forger, J. Laartz, and U. Schaeper. Current algebra of classical non-linear sigma models. Commun. Math. Phys., 146:397-402, 1992． arXiv:hep-th/9201025．
[143] M.Forger, J.Laartz, and U.Schaeper. The algebra of the energy-momentum tensor and the noether currents in classical non-linear sigma models. Commun. Math. Phys., 159:319-328． 1994． arXiv:hep-th/9210130．
[144] L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Berlin, Heidelberg, New York: Springer, 1987．
[145] J. M. Maillet. Hamiltonian structures for integrable classical theories from graded kac-moody algebras. Phys. Lett.. B167:401-405, 1986．
[146] H. J. de Vega. H. Eichenherr, and J. M. Maillet. Classical and quantum algebras of non-local charges in sigma models. Commun. Math. Phys., 92:507-524, 1984．
[147] J. M. Maillet. Kac-moody algebra and extended yang-baxter relations in the. Phys. Lett., 8162:137-142, 1985．
[148] D. Gross and A. Neveu. Dynamical symmetry breaking in asymptotically free field
theories. Phys. Rev.. 010:3235-3253． 1974．
[149] Subir Ghoshal. Boundary s-matrix of the o(n)-symmetric non-linear sigma model. Phys. Lett., 6334:363-368, 1994． arXiv:hep-th/9401008．
[150] K. Pohlmeyer. Integrable hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys.. 46:207-221, 1976．
[151] A. Neveu and N. Papanicolaou. Integrability of the classical [ψiψi]2 and [ψiψi]22-[ψiγ5ψi]2 interactions. Commun. Math,. Phys., 58:31-64, 1978．
[152] A. Polyakov. Hidden symmetry of the two-dimensional chiral fields. Phys. Lett., B72:224-226, 1977．
[153] E. Witten. Some properties of the (ψψ) model in two dimensions. Nucl. Phys., B142:285-300． 1978．
[154] A. De Martino and M. Moriconi. Boundary s-matrix for the gross-neveu model. Phys. Lett., 6451:354-364． 1999． arXiv:hep-th/9812009．