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有限空间中经典场的正则量子化

刘波 王青 李永明 隆正文

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有限空间中经典场的正则量子化

刘波, 王青, 李永明, 隆正文

Canonical quantization of classical fields in finite volume

Liu Bo, Wang Qing, Li Yong-Ming, Long Zheng-Wen
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  • 从离散的角度研究带边界的1+1维经典标量场和Dirac场的正则量子化问题. 与以往不同的是, 这里将时间和空间两个变量同时进行变步长的离散, 应用变步长离散的变分原理, 得到离散形式的运动方程、边界条件和能量守恒的表达式. 然后, 根据Dirac理论, 将边界条件当作初级约束, 将边界条件和内在约束统一处理. 研究表明, 采用此方法, 不仅在每个离散的时空格点上能够建立起Dirac括号, 从而可以完成该模型的正则量子化;而且, 该方法还保持了离散情况下的能量守恒.
    We study the problem of canonical quantization of classical scalar and Dirac field theories in the finite volumes respectively in this paper. Unlike previous studies, we work in a completely discrete version. We discretize both the space and time variables in variable steps and use the difference discrete variational principle with variable steps to obtain the equations of motion and boundary conditions as well as the conservation of energy in discrete form. For the case of classical scalar field, the quantization procedure is simpler since it does not contain any intrinsic constraint. We take the boundary conditions as primary Dirac constraints and use the Dirac theory to construct Dirac brackets directly. However, for the case of classical Dirac field in a finite volume, things are complex since, besides boundary conditions, it contains intrinsic constraints which are introduced by the singularity of the Lagrangian. Furthermore, these two kinds of constraints are entangled at the spatial boundaries. In order to simplify the process of calculation, we calculate the final Dirac brackets in two steps. We calculate the intermediate Dirac brackets by using intrinsic constraints. And then, we obtain the final Dirac brackets by bracketing the boundary conditions. Our studies show that we can not only construct well-defined Dirac brackets at each discrete space-time lattice but also keep the conservation of energy discretely at the same time.
    • 基金项目: 国家自然科学基金(批准号:10865003)资助的课题.
    • Funds: Project supported by National Natural Science Foundation of China (Grant No. 10865003).
    [1]

    Sheikh-Jabbari M M, Shirzad A 2001 Eur. Phys. J. C 19 383

    [2]

    Jing J 2005 Eur. Phys. J. C 39 123

    [3]

    Jing J, Long Z W 2005 Phys. Rev. D 72 126002

    [4]

    Long Z W, Chen L 2007 High Energy Phys. and Nucl. Phys. 31 14 (in Chinese) [隆正文, 陈琳 2007 高能物理与核物理 31 14]

    [5]

    Wang Q, Long Z W, Luo C B 2013 Acta Phys. Sin. 62 100305 (in Chinese) [王青, 隆正文, 罗翠柏 2013 物理学报 62 100305]

    [6]

    Dirac P A M 1964 Lecture Notes on Quantum Mechanics (1st Ed.) (New York: Yeshiva University) p8

    [7]

    Faddeev L D, Jackiw R 1988 Phys. Rev. Lett. 60 1692

    [8]

    Long Z W, Jing J 2003 Phys. Lett. B 560 128

    [9]

    Jing J, Long Z W, Tian L J, Jin S 2003 Euro. Phys. J. C 29 447

    [10]

    Lee T D 1983 Phys. Lett. B 122 217

    [11]

    Ruth R D 1983 IEEE Trans. Nucl. Sci. 30 1669

    [12]

    Feng K 1985 Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations—Computation of Partial Differential Equations (edited by Feng Keng) (Beijing: Science Press)

    [13]

    Guo H Y, Wu K 2003 J. Math. Phys. 44 5978

    [14]

    Guo H Y, Wu K, Wang S K, Wang S H, Wang S K, Wei J M 2000 Commu. Theor. Phys. 34 307

    [15]

    Guo H Y, Li Y Q, Wu K 2001 Commu. Theor. Phys. 35 703

    [16]

    Xia L L, Chen L Q, Fu J L, Wu J H 2014 Chin. Phys. B. 23 070201

    [17]

    Gitman D M, Tyutin I V 1990 Quantization of Fields with Constraints (1st Ed.) (New York: Springer-Verlag) p276

  • [1]

    Sheikh-Jabbari M M, Shirzad A 2001 Eur. Phys. J. C 19 383

    [2]

    Jing J 2005 Eur. Phys. J. C 39 123

    [3]

    Jing J, Long Z W 2005 Phys. Rev. D 72 126002

    [4]

    Long Z W, Chen L 2007 High Energy Phys. and Nucl. Phys. 31 14 (in Chinese) [隆正文, 陈琳 2007 高能物理与核物理 31 14]

    [5]

    Wang Q, Long Z W, Luo C B 2013 Acta Phys. Sin. 62 100305 (in Chinese) [王青, 隆正文, 罗翠柏 2013 物理学报 62 100305]

    [6]

    Dirac P A M 1964 Lecture Notes on Quantum Mechanics (1st Ed.) (New York: Yeshiva University) p8

    [7]

    Faddeev L D, Jackiw R 1988 Phys. Rev. Lett. 60 1692

    [8]

    Long Z W, Jing J 2003 Phys. Lett. B 560 128

    [9]

    Jing J, Long Z W, Tian L J, Jin S 2003 Euro. Phys. J. C 29 447

    [10]

    Lee T D 1983 Phys. Lett. B 122 217

    [11]

    Ruth R D 1983 IEEE Trans. Nucl. Sci. 30 1669

    [12]

    Feng K 1985 Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations—Computation of Partial Differential Equations (edited by Feng Keng) (Beijing: Science Press)

    [13]

    Guo H Y, Wu K 2003 J. Math. Phys. 44 5978

    [14]

    Guo H Y, Wu K, Wang S K, Wang S H, Wang S K, Wei J M 2000 Commu. Theor. Phys. 34 307

    [15]

    Guo H Y, Li Y Q, Wu K 2001 Commu. Theor. Phys. 35 703

    [16]

    Xia L L, Chen L Q, Fu J L, Wu J H 2014 Chin. Phys. B. 23 070201

    [17]

    Gitman D M, Tyutin I V 1990 Quantization of Fields with Constraints (1st Ed.) (New York: Springer-Verlag) p276

计量
  • 文章访问数:  2596
  • PDF下载量:  837
  • 被引次数: 0
出版历程
  • 收稿日期:  2014-10-16
  • 修回日期:  2014-12-04
  • 刊出日期:  2015-05-05

有限空间中经典场的正则量子化

  • 1. 北京化工大学理学院物理与电子科学技术系, 北京 100029;
  • 2. 新疆大学物理科学与技术学院, 乌鲁木齐 830046;
  • 3. 新疆大学信息科学与工程学院, 乌鲁木齐 830046;
  • 4. 贵州大学物理系光电子技术和应用实验室, 贵阳 550025
    基金项目: 国家自然科学基金(批准号:10865003)资助的课题.

摘要: 从离散的角度研究带边界的1+1维经典标量场和Dirac场的正则量子化问题. 与以往不同的是, 这里将时间和空间两个变量同时进行变步长的离散, 应用变步长离散的变分原理, 得到离散形式的运动方程、边界条件和能量守恒的表达式. 然后, 根据Dirac理论, 将边界条件当作初级约束, 将边界条件和内在约束统一处理. 研究表明, 采用此方法, 不仅在每个离散的时空格点上能够建立起Dirac括号, 从而可以完成该模型的正则量子化;而且, 该方法还保持了离散情况下的能量守恒.

English Abstract

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