Quaternions| 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].
> Online resources:
see Wikipedia page.
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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 24 may 2021