1. Quantum angular momentum matrices – eigenvalues and SVs Peter Loly Department of Physics and Astronomy, University of Manitoba, Winnipeg WCLAM 14-15 May 2016

2. Abstract Replace “matrix norms” by “integer measures”.

3. 3 01 000 1000 000 0001 00010

4. Google Search matrix mechanics -> IMAGES Many of these matrices appear there. Contemporary context includes “qubits”.

5. 1925 Werner Heisenberg 91 st anniversary! Niels Bohr, “Copenhagen” – 1912 electron orbits June 1925 Heisenberg’s “matrix mechanics”, Uncertainty Principle. Paul Dirac 1925 November 7; and 16 th : Born, H and Jordan Wolfgang Pauli 1926/27 – spin 1/2 matrices, Exclusion Principle. 1926 Erwin Schrӧdinger’s “wave mechanics”, his “cat”. 1972 B.L. van der Waerden, Notices of the AMS, Nov. 1997, 323-8. Note remarks on C. Lanczos. Heisenberg just one of several students of Arnold Sommerfeld (Munich) - to win Nobel Prize. Fritz London NOT! His student R.K.Eisenshitz, “Matrix Algebra for Physicists”, c. 1972, supervised my undergraduate dissertation on “The Theory of Measurement in Quantum Mechanics” in 1963.

6. Spin ½ 1921 Stern-Gerlach experiment; 1927 Pauli matrices

7. Background Ph.D. – Interacting spin waves in Heisenberg’s model of ferromagnetism using many-body techniques (Green’s functions, quantum field theory at finite temperatures) for boson and spin models. Much later teaching “Introduction to Theoretical Physics” with a chapter on matrices … numerical examples of simple matrices from MATLAB’s magic(n) function.

8. Overview My interest in these fundamental square matrices is in obtaining integer characteristic polynomials from which integer sums of eigenvalue and singular value powers follow. They include imaginary matrices, and matrices with no eigenvalues (j+, j-) where singular value analysis becomes essential. Previous work on integer matrices, especially Latin squares and magic squares.

9. Matrix sums 1629 Albert Girard – powers of roots of polynomials Later Newton’s identities G.A. Miller 1909, and 1916,1927

10. Girard’s identities 1629

11. Vector Model

12. General formulae (ħ =h/2π) (J.J.Sakurai, Modern Quantum Mechanics)

13. 13 01 000 1000 000 0001 00010

14. Sums

15. n=2j+1jrEVsSVs 2½21,122 3122,2,0832 43/243,3,1,120164 5244,4,2,2,040544 65/265,5, 3,3,1,1701414 7366,6, 4,4,2,2,01123136

16. EV2 and SV2 sums

17. 17 02 000 0000 000 00002 00000

18. n=2j+1jrEVsSVs 2½101,011 312048 43/2301034 524020104 65/25035259 736056560

19. OEIS

20. DISCLAIMER! https://en.wikipedia.org/wiki/Magic_number _(physics) https://en.wikipedia.org/wiki/Magic_number _(physics) The seven most widely recognized magic numbers as of 2007 are 2, 8, 20, 28, 50, 82, and 126 (sequence A018226 in OEIS).A018226OEIS

21. End Thanks to Andrew Senchuk, Adam Rogers 12May - Conjugate transpose used for jy… Question: Are there other physics topics where singular values, sums of EVs and SVs, … might be useful?

22. LINSTAT-2012, IWMS-21 at Bedlewo, Poland http://home.cc.umanitoba.ca/~loly/ -> NEWS 2013 in LATEST NEWS… "Signatura of magic and Latin integer squares: isentropic clans and indexing.” by Ian Cameron, Adam Rogers and Peter D. Loly, from 2012 conference published in Discussiones Mathematicae Probability and Statistics, 33(1-2) (2013) 121-149, from http://www.discuss.wmie.uz.zgora.pl/ps" "Signatura of magic and Latin integer squares: isentropic clans and indexing.” “plain version of 2012 conference paper published in Discussiones Mathematicae Probability and Statistics, 33(1-2) (2013) 121-149. [DMPS] “plain version of 2012 conference paper published in Discussiones Mathematicae Probability and Statistics, 33(1-2) (2013) 121-149. [DMPS] “Data Appendix for Signatura”: Ian Cameron, Adam Rogers and Peter D. Loly “Data Appendix for Signatura”: Ian Cameron, Adam Rogers and Peter D. Loly

23. Some talks and papers on “magical squares” LOLY, The Invariance of the Moment of Inertia of Magic Squares, The Mathematical Gazette, 88, March 2004, 151-153.The Invariance of the Moment of Inertia of Magic Squares Daniel Schindel, Matthew Rempel and Peter LOLY, Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS, Proceedings of the Royal Society A: Physical, Mathematical and Engineering, 462 (2006) 2271-2279.Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS LOLY, Franklin Squares: A Chapter in the Scientific Studies of Magical Squares, Complex Systems 17 (2007) 143-161 from NKS2006 (Wolfram-Mathematica).Franklin Squares: A Chapter in the Scientific Studies of Magical Squares Eigenvalues in the Universe of Matrix Elements 1..n-squared. IWMS-16 2007 – Introductory Keynote Talk (with Ian Cameron and Adam Rogers) – SVDs: Eigenvalues in the Universe of Matrix Elements 1..n-squared. Peter LOLY, Ian Cameron, Walter Trump and Daniel Schindel, Magic square spectra, Linear Algebra and its Applications, 430 (2009)Magic square spectra WCLAM2008 Winnipeg – Compound MS - Adam Rogers with Loly and Styan – Discussion - Loly: some are matrices with only the linesum EV of doubly stochastics. Cameron, Adam Rogers & Peter Loly, Signatura of magic and Latin integer squares: isentropic clans and indexing., IWMS 2012 Bedlewo, Poland – VIDEO http://www.physics.umanitoba.ca/~icamern/Poland2012/ http://www.physics.umanitoba.ca/~icamern/Poland2012/ Discussiones Mathematicae Probability and Statistics, 33(1-2) (2013) 121-149. CMS2014 Winnipeg – Multimagic (Knut Vik) Latin Squares with Cameron and Rogers WCLAM2016 - now

25. Shannon Entropy and Magical Squares Newton, P.K. & DeSalvo, S.A. (2010) [“NDS”] “The Shannon entropy of Sudoku matrices”, Proc. R. Soc. A 466:1957-1975 [Online Feb. 2010.] Immediately clear to us that we could extend NDS using our studies of the singular values of magic squares at IWMS16 in 2007, published in Lin. Alg. Appl. 430 (10) 2659-2680, 2009. DMPS http://home.cc.umanitoba.ca/~loly/

26. Shannon Entropy

27. Magic squares, Morse—Hedlund sequence, Hilbert-Kamke and Waring problems Sergeyev, Y.D., “Generation of symmetric exponential sums”, arXiv:1103.2043v1 [math.NT] 10 Mar 2011

28. “Almost All Integer Matrices Have No Integer Eigenvalues”, Greg Martin and Erick B. Wong, UBC: www.math.ubc.ca/~gerg/papers/downloads/ AAIMHNIE.pdf www.math.ubc.ca/~gerg/papers/downloads/ AAIMHNIE.pdf