Note on properties of spin-1 matrices

C. Leubner

Daniel Grucker

For a two-dimensional complex vector space, the spin matrices can be calculated directly from the angular momentum commutator definition. The 3 Pauli matrices are retrieved and 23 other triplet solutions are found. In the three-dimensional space, we show that no matrix fulfills the spin equations and preserves the norm of the vectors. By using a Clifford geometric algebra it is possible in the four-dimensional spacetime (STA) to retrieve the 24 different spins 1/2. In this framework, spins 1/2 are rotations characterized by multivectors composed of 3 vectors and 3 bivectors. Spins 1 can be defined as rotations characterized by 4 vectors, 6 bivectors and 4 trivectors which result in unit multivectors which preserve the norm. Let us note that this simple derivation retrieves the main spin properties of particle physics.

Vishnu K. Agrawala

Richard J. Blume

ELENA M. GEORGIEVA, Y. S. Kim

It is noted that 2x2 "S" matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two spacelike and one timelike dimensions. It is also noted that Wigner's little groups have a slide-rule-like property that allows us to perform multiplications by additions. It is shown that these two mathematical properties lead to a cyclic representation of the S matrix for multilayer optics, as in the case of ABCD matrices for laser cavities. It is therefore possible to write the N-layer S matrix as a multiplication of the N single-layer S matrices resulting in the same mathematical expression with one of the parameters multiplied by N. In addition, it is noted, as in the case of lens optics, that multilayer optics can serve as an analog computer for the contraction of Wigner's little groups for internal space-time symmetries of relativistic particles.

Brian R. Judd

1 Basic properties.- 1.1 Reduced Density Matrices.- 1.2 N-representability.- 1.3 Norbs, Nags and Least Squares.- 1.4 Fock Space.- 1.5 Reduced Density Matrices on Fock Space.- 1.6 Unitary Invariance.- 1.7 Notes.- 1.8 Exercises.- 2 N-Representability.- 2.1 The p-rank of a State.- 2.2 Convexity.- 2.3 Ensemble N-representability.- 2.4 A Family of Convex Surfaces.- 2.5 Parameterizing.- 2.6 The Total Energy.- 2.7 The One-matrix.- 2.8 Notes.- 2.9 Exercises.- 2.10 Appendix - Figures for Chapter 2.- 3 Correlation.- 3.1 Sasaki's Formula.- 3.2 Bounds on for fermions.- 3.3 Necessary Conditions.- 3.4 A Universal Fermion Model.- 3.5 Strong Orthogonality.- 3.6 Unitary Decomposition.- 3.7 Reduced Hamiltonian Orbitals.- 3.7.1 For LiH.- 3.7.2 For H2O.- 3.8 Intracules, extracules and Statistics.- 3.9 Notes for Chapter 3.- 3.10 Exercises for Chapter 3.- 3.11 Appendix - The Pfaffian.- 4 Generalized AGP.- 4.1 Canonical form for g(12).- 4.2 Properties of GAGP.- 4.3 Extreme AGP.- 4.4 Scope of GAGP.- 4.5 Error Control.- 4.6 The GAGP Energy.- 4.7 Critical values of E.- 4.8 Hartree-Fock-Bogolubov Equations.- 4.9 Notes.- 4.10 Exercises.- 5 Boson Systems.- 5.1 Yang's Paper.- 5.2 Order Indices.- 5.3 Green's Functions and Propagators.- 5.4 Bose Condensation.- 5.5 Mesoscopic Condensation.- 5.6 Bose Liquid.- 5.7 Collective Excitations.- 5.8 Notes.- 6 Fermion Systems.- 6.1 The Superconducting Transition.- 6.2 Heterophase Superconductor.- 6.3 Notes.- 6.4 Figures.- 7 Coulson's Challenge.- 7.1 The Reduced Hamiltonian.- 7.2 Pairon Distribution function.- 7.3 Facing the Challenge!.- 7.4 Notes.