Spin Algebra and Naimark’s Extension: A Tutorial Approach with Examples
Abstract
In analyzing two-electron systems, the interactions of interest often include the spin-spin operator S→1×S→2) and the spin-orbit operator L→·S→. When these operators act on entangled or indistinguishable particles, their measurement and physical interpretation may extend beyond the standard projective framework. This tutorial introduces the algebraic structure of spin interactions in two electron quantum systems and establishes its conceptual and mathematical connection with \emph{Naimark's Extension Theorem}. Through explicit examples for two-electron systems, we demonstrate how spin operators arise in reduced Hilbert spaces, and how \emph{Naimark's theorem} provides a formal framework for extending these to projective measurements in enlarged spaces. The application of \emph{Naimark's Extension Theorem} in deriving their matrix elements opens up a window into the structure of quantum measurements in such composite systems.
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References
M.A. Naimark, Izv. Acad. Nauk SSSR Ser. Mat. 4, 277 (1940).
Spin Algebra and Naimark’s Extension: A Tutorial Approach with Examples
F. Riesz, and B.S. Nagy, Functional Analysis, (Dover, New York, 1990)
N.I. Akhizer, and I.M. Glazman, Theory of Linear Operators in Hilbert Space, (Ungar, New York, 1963).
V. Paulsen, Completely Bounded Maps and Operator Algebras, (Cambridge University Press, Cambridge, 2020).
G. Araki. Proc. Phys. Math. Soc. Japan, 19, 128 (1937). https://doi.org/10.11429/ppmsj1919.19.0 128
G. Araki, Phys.Rev. 101, 1410 (1956). https://doi.org/10.1103/PhysRev.101.1410
G. Araki, M. Ohta and K. Mano, Phys. Rev. 116, 651 (1959). https://doi.org/10.1103/PhysRev.116.651
P.A.M. Dirac, Priciples of Quantum Mechanics, (Oxford U.P., Oxford, 1958).
H. Bethe, and E.E. Saltpeter, Quantum Mechanics of one or two electron atoms, (Platinum Publishing Corporation, New York, 1977).
R.P. Feynman, Int. J. Th. Phys. 21, 467 (1982). https://doi.org/10.1007/BF02650179
S.D. Kenny, G. Rajagopal, and R.J. Needs, Phys. Rev. A, 51, 1898 (1982). https://doi.org/10.1103/PhysRevA.51.1898
S.A. Alexander, Sumita Datta, and R L. Coldwell, Phys. Rev. A, 81, 032519 (2010). https://doi.org/10.1103/PhysRevA.81.032519
Sumita Datta, S.A. Alexander, and R.L. Coldwell, Int. J. Qu. Chem. 112, 731 (2012). https://doi.org/10.1002/qua.23039
C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 75, 4714 (1995). https://doi.org/10.1103/PhysRevLett.75.4714
A. Steane, Rep. Prog. Phys. 61, 117-173 (1998). https://doi.org/10.1088/0034-4885/61/2/002
E.C.R. da Rosa, and C. Lima, arXiv:2210.15506 v1[quant-ph] 27 Oct. 2022. https://doi.org/10.48550/arXiv.2210.15506
R.V. Kadison, Contemporary Mathematics, 167, 21 (1994).
R. Beneduci, J. Phys.:Conference Series,1638,012006(2020) https://doi.org/10.1088/1742-6596/1638/1/012006
B. Daribaye, A. Mukhanbet, N. Azatbekuly and T. Imankulov, Algoritms,17,327(2024) URL
M.A. Naimark, and S.V. Fomin, Continuous direct sums of Hilbert spaces and some of their applications. Ann. Math. Soc. Transl. Ser. 2, 5, 35 (1957).
M.G.A. Paris, The Eur. Phys. J. Special Topics, 203, 61 (2012). https://doi.org/10.1140/epjst/e2012-01535-1
K. Erdmann, and M.J. Wildon, Introduction to Lie Algebra. Mathematical Institute University, (Oxford, UK, 2006).
Casper VAN HAL, An introduction into Lie Group, Lie Algebra, Representations and Spin, Double Bachelor’s Thesis, (Mathematics and Physics, Utrecht University, 2021.
R. Clinton, J. Bub, and H. Halvorson, Foundations of Physics, 33, 1561 (2003). https://doi.org/10.1023/A:1026056716397
J.H. Yoo, Lie Groups, Lie Algebras, and Applications in Physics, (2015). https://api.semanticscholar.org/CorpusID:43265836
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