Quaternions

In General > s.a. examples of lie groups [SO(4)]; SU(2).
* Idea: Elements of an associative but non-commutative algebra.
* Notation: May be written as ordered pairs of complex numbers, or as

$$\mathbb H$$ = {a + i b + j c + k d | a, b, c, d ∈ $$\mathbb R$$} ,

with i2 = j2 = k2 = −1, i · j = − j · i = k and cyclic permutations; i, j, k can be represented using Pauli matrices by iτ1, iτ2, and iτ3, respectively.
* Relationships: The unit quaternions are isomorphic to SU(2), topologically S3.
@ Textbooks: Conway & Smith 03; Morais et al 14 [real quaternionic calculus, IIb].
@ General references: Aslaksen MI(96) [matrix determinants]; Kravchenko et al in(01)mp [quaternionic Riccati equation]; De Leo et al JMP(02)mp [eigenvalue problems]; De Leo & Ducati JMP(03)mp [quaternionic differential equations]; Staley EJP(10) [and the Dirac belt trick for spinors and rotations]; Colombo & Sabadini JGP(10) [functional calculus]; Familton a1504 [history]; Borsten & Marrani CQG(17) [6-algebra Freudenthal–Rosenfeld–Tits magic square].
@ Biquaternions: Liu mp/01; Kassandrov & Rizcallah G&C(16)-a1612 [biquaternion algebra on a curved manifold].
@ Other generalizations: Raptis mp/01 [graded, deformed]; Volkov a1006 [ternary quaternions].

Applications
* Idea: Quaternions are used to describe rigid body rotations.
@ And rotations, dynamics: Graf a0811 [and dynamics, introduction]; Delphenich a1205 [use to representat physical motions].
@ Quaternionic quantum mechanics: Adler 95; Adler JMP(96)ht [projective group representations]; Brumby & Joshi CSF(96)qp; Horwitz FP(96)qp; Maia ht/99 [spin]; De Leo & Ducati JMP(01)mp/00; Maia & Bezerra IJTP(01)ht [geometric phase]; De Leo & Ducati JMP(06)mp [diffusion by potential step], JMP(07)-a0706 [wave packet behavior]; de Melo & Pimentel AACA(10)-a0809-conf [variational formulation]; McKague a0911 [non-local boxes]; De Leo et al JMP(10)-a1012 [barrier transmission coefficients]; Baez FP(12)-a1101; Graydon FP(13)-a1103 [reduced to ordinary quantum formalism]; Muraleetharan & Thirulogasanthar JMP(15)-a1406 [coherent state quantization]; Giardino a1706 [solutions]; Moretti & Oppio RVMP(19)-a1709 [Poincaré symmetry and reduction to ordinary quantum theory]; Giardino a1803 [in real Hilbert space]; Steinberg et al a2001; > s.a. modified quantum theory [supersymmetric]; Squeezed States.
@ Dirac fields, spinors: De Leo & Rodrigues IJTP(98), IJTP(98) [Dirac electrons]; Arbab IJLEO(17)-a1301 [Maxwell-like equations from quaternionic Dirac equation]; Giardino FP(16)-a1504 [in a square box]; Bolokhov IJMPA(19)-a1712 [wave functions with spin].
@ Field theory: De Leo IJTP(96)ht/95 [guts]; Vandoren ht/00-conf [Yang-Mills instantons]; Maia et al FP(09)-a0809 [and quantum gravity]; Sachs 10 [unified field theory and cosmology]; Giardino & Teotônio-Sobrinho MPLA(13)-a1211 [non-associative scalar field theory]; Peña & Bory a2007, Giardino MPLA-a2010 [electrodynamics]; > s.a. klein-gordon fields.
@ Cosmology: Misner in(94) [mixmaster universe]; Brumby et al PLB(97) [dark matter]; Majerník GRG(04)ap/03, GRG(03) [dark energy].
@ Other physics: Robinson JMP(91) [4D conformal structure]; Lambek MI(95); De Leo & Ducati IJTP(99)ht [general]; Gsponer & Hurni mp/02-conf [general], mp/05, mp/05 [bibliography]; Schwartz JMP(06)ht, JMP(07)ht [wave equation]; Konno QSMF-a1412 [quaternion walks].
> In physics: see modified quantum mechanics; modified quantum field theories; quantum oscillators; special relativity; spinors; spin-3/2 field theories.
> Online resources on quaternions and spatial rotations: see Noel Hughes site [and attitude description, kinematics and dynamics]; Wikipedia page.