Abstract
Since the exactly solvable higher-dimensional quantum systems with certain central potentials are usually related to the real orthogonal group O(N) defined by orthogonal n×n matrices, we shall give a brief review of some basic properties of group O(N) based on the monographs and textbooks. We first outline the development in order to make the reader recognize its importance in physics and then review the tensor and spinor representations of the group SO(N) and Γ matrix group.
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Notes
- 1.
There exist two kinds of different meanings of the terminology “abstract group” during the first half of the 20th century. The first meaning was that of a group defined by four axioms given above, but the second one was that of a group defined by generators and commutation relations.
- 2.
A Young pattern [λ] has n boxes lined up on the top and on the left, where the jth row contains λ j boxes. For instance, the Young pattern [2,1] is
It should be noted that the number of boxes in the upper row is not less than in the lower row, and the number of boxes in the left column is not less than that in the right column. We suggest the reader refer to the permutation group S n in Appendix A for more information.
- 3.
B N is the strong space-time reflection matrix and C N is the charge conjugation matrix, which are usually used in particle physics.
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Dong, SH. (2011). Special Orthogonal Group SO(N). In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_2
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