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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.
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Keywords:
- canonical quantization /
- boundary conditions /
- Dirac constraints /
- Dirac brackets
[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 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
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[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 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
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