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Pertti Lounesto
My research focuses on algebras emerging from problems in geometry and physics,
called Clifford algebras.
In physics, the concept of Clifford algebra, as such or in a disguise,
is a necessity in the description of electron spin,
because spinors cannot be constructed by tensorial methods,
in terms of exterior powers of the vector space.
In geometry, information about orientation of subspaces can be encoded
in simple multivectors, which can be added and multiplied.
Physicists are familiar with this tool in the special case of one-dimensional
subspaces, which they manipulate by vectors
(not by projection operators, which lose information about orientations).
I am also interested in
misconceptions of research mathematicians,
while they enter unexplored domains.
This Spring 2002, I delivered
lectures on Clifford algebras.
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I have authored/co-edited the following
books:
Comparison to
other books on Clifford algebras and spinors:
I.R. Porteous:
Clifford algebras and the classical groups. Cambridge UP, 1995.
J. Snygg:
Clifford algebra, a computational tool for physicists.
Oxford UP, 1997.
D. Hestenes, G. Sobczyk:
Clifford algebra to geometric calculus. Reidel, 1984, 1987.
D. Hestenes:
New foundations for classical mechanics.
Reidel, 1986, 1987, (2nd ed.) 1999.
B. Jancewicz:
Multivectors and Clifford algebra in electrodynamics.
World Scientific, 1988.
W.E. Baylis:
Electrodynamics: a modern geometric approach.
Birkhäuser, 1999.
W.E. Baylis (ed.):
Clifford (geometric) algebras with applications to physics, mathematics,
and engineering.
Springer, 1996.
I.M. Benn, R.W. Tucker:
An introduction to spinors and geometry with applications in physics.
Adam Hilger, 1987.
F.R. Harvey:
Spinors and calibrations.
Academic Press, 1990.
J. Gilbert, M. Murray:
Clifford algebras and Dirac operators in harmonic analysis, CUP, 1991.
G. Sommer:
Geometric computing with Clifford algebras; theoretical foundations and applications in computer vision and robotics. Springer, 2001.
See errata of the books.
See commentary of a
paper,
where the authors prove a conjecture of one of the above authors,
although the conjecture was a corollory of Ado's theorem.
- Tony Smith:
What are Clifford algebras and spinors?
- Bill Pezzaglia:
International Clifford Algebra Society.
- Flow chart on the
history of Clifford's geometric algebras.
- Perttu Puska:
Electromagnetism formulated in Clifford algebra.
- Kirby Urner:
Clifford Algebra for high schoolers?
- Tennessee TU:
6th Conference on Clifford Algebras,
May 18-25, 2002.
- Courses on Clifford
algebras applied to
analysis and
physics.
- Octonions
multiplied,
explained and
applied.
Clifford algebra might make its curriculum debut
in high schools by
replacing the cross product,
in undergraduate courses by
replacing rotation matrices,
and in graduate courses by
condensing the Maxwell equations in a vacuum
into a single equation in terms of Cl3.
Hamilton's
quaternions
are used to represent
3D rotations;
quaternions enable smooth interpolation of rotation matrices
in
computer graphics/
games,
quaternions are used to avoid a singularity, the
gimbal lock
in flight simulation and spacecraft navigation.
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Home-pages of researchers on Clifford's geometric algebra:
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Algebra:
Rafal Ablamowicz,
Bertfried Fauser,
Alexander J. Hahn,
Jacques Helmstetter,
Ian R. Porteous.
Analysis:
Swanhild Bernstein,
Freddy Brackx,
Richard Delanghe,
Sirkka-Liisa Eriksson-Bique,
Guy Laville,
Heinz Leutwiler,
Enrique Ramírez de Arellano,
John Ryan,
Baruch Schneider,
Frank Sommen,
Wolfgang Sprößig.
Celestial Mechanics:
Jan Vrbik.
Chemistry:
Janne Pesonen.
Computer Science:
Adrian Lawrence,
Stephen Mann,
Gerik Scheuermann,
Dongming Wang.
Computer Vision:
Christian Perwass,
Bodo Rosenhahn.
Electromagnetism:
William E. Baylis,
Bernard Jancewicz,
Perttu Puska.
Geometry:
Christian Bär,
Helga Baum,
Sven Buchholz,
Jan Cnops,
Oliver Conradt,
Larry Grove,
Garret Sobczyk,
Andrzej Trautman,
Charles H.T. Wang.
Physics:
Timaeus Bouma,
Philip Charlton,
Robert Coquereaux,
Chris
Doran,
Ramon González Calvet,
Stephen Gull,
David Hestenes,
Heinz Krüger,
Jaime Keller,
Anthony Lasenby,
Antony Lewis,
Garrett Lisi,
Nikolai Marchuk,
Dennis Marks,
James M. Nester,
Josep Parra,
William M. Pezzaglia,
Charles Poole,
Patrick Reany,
Waldyr Rodrigues,
Nikos A.
Salingaros,
Thalanayar S. Santhanam,
John Schutz,
Greg Trayling,
Jose Vargas,
Jayme Vaz,
José Ricardo Zeni.
Robotics:
Eduardo
Bayro-Corrochano,
Curtis Collins,
Leo Dorst,
Seamus Garvey,
Hongbo Li,
Michael McCarthy,
Allan McRobie,
Jon Selig,
Gerald Sommer.
Signal Processing:
Thomas Bülow,
Michael Felsberg,
Joan Lasenby.
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Ten most influential mathematicians in Clifford algebras:
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William R. Hamilton (1805-1865)
invented
quaternions, on Monday Oct 16, 1843.
- Hermann Grassmann (1809-1877) introduced the
exterior algebra, 1844/1862.
- William K. Clifford (1845-1879) classified his
geometric algebras (4 cases), 1878/1882.
- Rudolf Lipschitz (1832-1903) represented rotations by
spin groups, 1880/1886.
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K. Theodor Vahlen (1869-1945):
multiplication rule by binary index sets, 1897,
.
Möbius transformations of
Rn
by 2x2-matrices with entries in Cln, 1902.
- Elie Cartan (1869-1951):
periodicity of 8, 1908,
and triality of Spin(8), 1925.
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Ernst Witt (1911-1991): Clifford algebra Cl(Q) of a
quadratic form
Q, 1937.
- Claude Chevalley (1909-1984) extended the ground field to characteristic 2, 1954.
- Marcel Riesz (1886-1969):
exterior product via the Clifford product, 1958,
.
x^u
= ½(xu +
(-1)kux)
for
x in Rn and
u in /\kRn
\subset Cln+ or
Cln-.
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Ian R. Porteous
(1930- ) classified the scalar products of spinors (32 cases), 1969.
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Jacques Helmstetter
(1943- ) developed symplectic Clifford algebras, 1982/2002.
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W.R. Hamilton |
H. Grassmann |
W.K. Clifford |
R. Lipschitz |
E. Cartan |
E. Witt |
C. Chevalley |
M. Riesz |
J. Helmstetter |
