Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space
Using Isham's group-theoretic quantization scheme, we construct the canonical groups of the systems on the two-dimensional sphere and one-dimensional complex projective space, which are homeomorphic. In the first case, we take SO(3) as the natural canonical Lie group of rotations of the two-sph...
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| Format: | Article |
| Language: | English |
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Institute of Physics Publishing
2014
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| Online Access: | http://psasir.upm.edu.my/id/eprint/36698/1/Canonical%20Groups%20for%20Quantization%20on%20the%20Two-Dimensional%20Sphere%20and%20One-Dimensional%20Complex%20Projective%20Space.pdf |
| _version_ | 1825948892565340160 |
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| author | Ahamad Sumadi, Ahmad Hazazi Zainuddin, Hishamuddin |
| author_facet | Ahamad Sumadi, Ahmad Hazazi Zainuddin, Hishamuddin |
| author_sort | Ahamad Sumadi, Ahmad Hazazi |
| collection | UPM |
| description | Using Isham's group-theoretic quantization scheme, we construct the canonical groups of the systems on the two-dimensional sphere and one-dimensional complex projective space, which are homeomorphic. In the first case, we take SO(3) as the natural canonical Lie group of rotations of the two-sphere and find all the possible Hamiltonian vector fields, and followed by verifying the commutator and Poisson bracket algebra correspondences with the Lie algebra of the group. In the second case, the same technique is resumed to define the Lie group, in this case SU (2), of CP'. We show that one can simply use a coordinate transformation from S2 to CP1 to obtain all the Hamiltonian vector fields of CP1. We explicitly show that the Lie algebra structures of both canonical groups are locally homomorphic. On the other hand, globally their corresponding canonical groups are acting on different geometries, the latter of which is almost complex. Thus the canonical group for CP1 is the double-covering group of SO(3), namely SU(2). The relevance of the proposed formalism is to understand the idea of CP1 as a space of where the qubit lives which is known as a Bloch sphere. |
| first_indexed | 2024-03-06T08:36:06Z |
| format | Article |
| id | upm.eprints-36698 |
| institution | Universiti Putra Malaysia |
| language | English |
| last_indexed | 2024-03-06T08:36:06Z |
| publishDate | 2014 |
| publisher | Institute of Physics Publishing |
| record_format | dspace |
| spelling | upm.eprints-366982015-12-02T05:24:58Z http://psasir.upm.edu.my/id/eprint/36698/ Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space Ahamad Sumadi, Ahmad Hazazi Zainuddin, Hishamuddin Using Isham's group-theoretic quantization scheme, we construct the canonical groups of the systems on the two-dimensional sphere and one-dimensional complex projective space, which are homeomorphic. In the first case, we take SO(3) as the natural canonical Lie group of rotations of the two-sphere and find all the possible Hamiltonian vector fields, and followed by verifying the commutator and Poisson bracket algebra correspondences with the Lie algebra of the group. In the second case, the same technique is resumed to define the Lie group, in this case SU (2), of CP'. We show that one can simply use a coordinate transformation from S2 to CP1 to obtain all the Hamiltonian vector fields of CP1. We explicitly show that the Lie algebra structures of both canonical groups are locally homomorphic. On the other hand, globally their corresponding canonical groups are acting on different geometries, the latter of which is almost complex. Thus the canonical group for CP1 is the double-covering group of SO(3), namely SU(2). The relevance of the proposed formalism is to understand the idea of CP1 as a space of where the qubit lives which is known as a Bloch sphere. Institute of Physics Publishing 2014 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/36698/1/Canonical%20Groups%20for%20Quantization%20on%20the%20Two-Dimensional%20Sphere%20and%20One-Dimensional%20Complex%20Projective%20Space.pdf Ahamad Sumadi, Ahmad Hazazi and Zainuddin, Hishamuddin (2014) Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space. Journal of Physics: Conference Series, 553. art. no. 012005. pp. 1-5. ISSN 1742-6588; ESSN: 1742-6596 http://iopscience.iop.org/article/10.1088/1742-6596/553/1/012005/meta;jsessionid=84196562F908B70DDA8727A7709516BF.c1.iopscience.cld.iop.org# 10.1088/1742-6596/553/1/012005 |
| spellingShingle | Ahamad Sumadi, Ahmad Hazazi Zainuddin, Hishamuddin Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space |
| title | Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space |
| title_full | Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space |
| title_fullStr | Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space |
| title_full_unstemmed | Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space |
| title_short | Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space |
| title_sort | canonical groups for quantization on the two dimensional sphere and one dimensional complex projective space |
| url | http://psasir.upm.edu.my/id/eprint/36698/1/Canonical%20Groups%20for%20Quantization%20on%20the%20Two-Dimensional%20Sphere%20and%20One-Dimensional%20Complex%20Projective%20Space.pdf |
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