Last update: 30th March 2006,
Keywords: Electromagnetics, electromagnetism, algebra, algebras, vector, vectors, dyad, dyads, dyadic, dyadics, polyadic, multivector, multivectors, bivector, bivectors, Clifford algebra, geometric algebra, geometric calculus, quaternion, quaternions, hypercomplex, exterior calculus, exterior algebra, differential forms, exterior forms, spinor, spinors, tensor, tensors, Maxwell, Grassmann, Hamilton, Cayley, Clifford, Gibbs, Heaviside, Cartan, de RhamI have collected here links and references about algebras used in electromagnetics. The contents of the list of course reflect my preferences. However, taste has not been the only criterion, for I have included only those references that I have had possibility to evaluate, which explains the fact that so many of the classical papers are missing.
Contents
Note that classification is quite artificial, for it is more-often-than-not difficult to label piece of writing as belonging to 'exterior algebras' or 'clifford algebras'. So the reader is advised to consult all categories in order to find suitable texts.
- Vector and dyadic 'algebra'
- Clifford algebras and
- related algebras: Quaternions and Octonions
- Exterior algebras; Differential forms
Vector and dyadic 'algebra'
General
Did you know that the traditional vector algebra is included in Clifford algebra Cl(3,0)? See my notes on that.J. W. Gibbs and E.B. Wilson,Vector Analysis, 2. ed., Scribner, New York, 1909, Dover reprint, New York, 1960.
- J. W. Gibbs is the father of vector and dyadic algebra. His works are still some of the best expositions on dyadic algebra.
- Heaviside is the alternative father of vector algebra. Oliver Heaviside is credited (among many other things) for formulating Maxwell's equations in the language of vector algebra. By-the-way, it is shame that so many of the modern textbooks still fail to pay homage to a man who wrote the equations in a form that made it possible to teach the electromagnetic theory to generations of engineers and physicists. His own writing is a bit heavi read, though (pun intended).
Introductory texts and tutorials
Well, it is almost hopeless to try to recommend anything, there are so many texts available.
Intermediate to advanced
H. C. Chen, Theory of electromagnetic waves, McGraw-Hill, New York, 1983.- Intermediate level dyadics. Out of print for years.
I. Lindell, Methods for Electromagnetic Field Analysis, Oxford: Clarendon Press, 1992. Republished by IEEE 1995.
- Dyadics galore.
Clifford algebras
General
Clifford algebra researchers have an own society which publishes a journal called Advances in Applied Clifford Algebras twice a year.The Geometric Algebra Research Group at Cavendish Laboratory has an on-line intro and several down-loadable ps-format research papers ranging from introductory texts to advanced. Several very nice papers. It is probably so that this Cavendish group has become a forerunner in utilisation of Clifford's geometric algebra in physics.
Tutorials and introductory texts
D. Hestenes: "Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics, " Am. J. Phys. vol. 39, no. 9, pp. 1013-1027, 1971.W. E. Baylis, J. Huschilt, Jiansu Wei: "Why i?," Am. J. Phys. vol. 60, no. 9, pp. 788-797, 1992.
E. F. Bolinder: "Clifford Algebra, What is it?," IEEE Antennas and Propagation Society Newsletter, August, pp. 18-23, 1987.
- Bolinder's brief historical review and discussion of Clifford algebra (I thank dr. S.Sensiper for sending a copy of this article). Did you know that M. Riesz' wrote his famous lecture notes originally for Bolinder?
T. G. Vold: "An introduction to geometric algebra with an application to rigid body mechanics" and "An introduction to geometric calculus and its application to electrodynamics" Am. J. Phys. vol. 61, no. 6, pp. 491-513, 1993.
A. Lewis, a ps-format intro can be down-loaded from his webpage. Also links to other intros, such as the one written by R. Harke.
C. Rodriguez has a compact on-line intro about Clifford algebra.
T. Smith, an on-line intro not entirely restricted to the subject of Clifford algebra.
Intermediate to advanced
W. E. Baylis, and G. Jones, ``The Pauli algebra approach to special relativity,'' J. Phys. A, vol. 22, no. 1, pp. 1-15, 1989.- Shows that there is enough structure in Cl(3,0) ( i.e. Pauli algebra ) to represent spacetime. See also the book by W. E. Baylis.
W. E. Baylis, Electrodynamics, A Modern Geometric Approach Birkhäuser, Boston 1999.
-
The book covers
typical material of intermediate electromagnetics (or electrodynamics)
course and some advanced topics, but all these are
developed in Cl(3,0)! This is possible because Cl(3,0) is
rich enough in structure: For instance, a space-time point
(t,x,y,z) is represented as a paravector:
x = t + x e1 + y e2 + z e3. (1)
Note that time is just a scalar parameter and many of the familiar Gibbs-Heaviside vector 'algebra' ideas are carried over to paravector algebra. This approach Baylis believes to be paedagogically better than the more usual Cl(1,3) or Cl(3,1) formulation of space-time. Now how does the Minkowski metric arise from (1)? Consider the (Clifford) product_ x x = (t+xe1+ye2+ze3)(t-xe1-ye2-ze3) = t^2 - x^2 - y^2 -z^2, (2)
where 'bar' operation - denotes Clifford conjugation, that is, combination of reversion and grade involution (we are using Lounesto's notation here). Thus (2) gives the correct signature and is a valid candidate for a scalar product in space-time.
I. M. Benn and R. W. Tucker,An introduction to Spinors and Geometry, Adam Hilger, London 1987.
- Discusses tensors, Clifford algebras ( spinors are elements of minimal left or right ideals of Clifford algebras, which explains why the word spinor appears so often in Clifford algebra literature ) and applications. Chapter on electromagnetism. Modern differential geometry well represented, therefore this book could also be in the exterior algebra section of this list.
- Circuit engineers will find this interesting.
F. Brackx, R. Delanghe, F. Sommen,Clifford analysis, Pitman, London 1982.
- Analysis branch of the research. Hypercomplex analysis explained. Also from the same Gent research group is
- Analysis branch of the research. More into analysis is also
- PDEs and Clifford algebra. Contains also a short section on electromagnetics.
D. Hestenes, Space-time Algebra, Gordon & Breach, New York, 1966.
- Standard reference. Especially noteworthy for its treatment
on energy-momentum (Maxwell's stress-energy) tensor:
It is a pleasant fact that in Clifford algebra
(vacuum) energy-momentum tensor can be simply written as
T_ij = scalar part of (F ei F ej),
where ek are the basis vectors of Cl(1,3) (or Cl(3,1)) (This great discovery was first made by M. Riesz in 1946) These as well as some other EM-related points are discussed in this work. An update of the book is coming?
D. Hestenes, and G. Sobczyk, Clifford Algebra to Geometric Calculus, Reidel, Dordrecht, 1984, reprint with corrections 1992.
- Expands the work started in Space-time Algebra. Advocates the use of geometric algebras instead of differential forms.
P. Hillion, ``Constitutive relations and Clifford algebra in electromagnetism,'' Adv. in Appl. Cliff. Alg. vol. 5, no. 2, pp. 141-158, 1995.
- One of the few to discuss constitutive relations in Clifford algebra context. Note that the author uses a 'wedge' usually reserved for the exterior products in place of the 'cross' in cross products.
D. A. Hurley, M. A. Vandyck, Geometry Spinors and Applications, Springer and Praxis Publishing, Chichester, 2000.
- Modern mathematical apparatus of physics well presented. Contains an especially interesting section on electromagnetism. One very nice thing about the text is that the formulas in the text are written out fully, e.g. vectors are written with their components and basis vectors.
B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics, World Scientific, Singapore, 1988.
- New insights abound in this 1988 classic. Both electromagnetism and algebra are carefully developed side-by-side, the latter in intuitive manner that will not scare away electrical engineers. This work and the recent book by Baylis are the most mature texts available for those who wish to use Clifford algebra in electromagnetics.
B. Jancewicz: "A Hilbert space for the classical electromagnetic field," Found. Phys. vol. 23, no. 11, 1405-1421, 1993.
- Hilbert space of the vacuum electromagnetic field rendered in Cl(3,0).
G. Juvet, A. Schidlof, Sur les nombres hypercomplexes de Clifford et leurs applications à l'analyse vectorielle ordinaire, à l'électromagnetisme de Minkowski et à la théorie de Dirac. Bull. Soc. Neuchat. Sci. Nat., vol. 57, pp. 127-147, 1932.
- To the best of my knowledge, Juvet and Schidlof were the first authors to write Maxwell equations as a single equation in Clifford algebra. However, they worked with C x Cl(4,0) (Minkowski space with x_4=i ct, where i is the imaginary unit of the field C) and thus did not fully utilize the potential of Clifford algebra (a more natural choice would have been Cl(3,1)/Cl(1,3) or Cl(3,0) with paravectors, cf. Baylis).
J. Kot, G. C. James: "Clifford algebra in electromagnetics", Proceedings of the International Symposium on Electromagnetic Theory, URSI International Union of Radio Science, Aristotle University of Thessaloniki, 25-28 May 1998, Thessaloniki, Greece. pp. 822-824.
- Hypercomplex analysis and electromagnetics.
P. Lounesto,
Clifford Algebras and Spinors,
Cambridge University Press, Cambridge, 1997.
P. Puska
"Covariant Isotropic Constitutive Relations in Clifford's
Geometric Algebra," PIER 32, EMW Publishing, Cambridge, 2001.
M. Riesz, Marcel Riesz: Clifford Numbers and Spinors,
Kluwer Academic Publisher, Dordrecht/Boston, 1993.
J. Snygg,Clifford Algebra, A Computational Tool for Physicists,
Oxford University Press, New York, 1997.
Ftp-site
by H. Baker about historical quaternion papers.
Clyde Davenport's
pages
discuss the use and application of quaternion algebra and
commutative hypercomplex algebra in physics. These
pages include
sections on electromagnetism and special relativity.
Geoffrey Dixon's
site about octonions and related physical
applications/implications. Conference announcements.
A concise tutorial
also available.
Alessandro Rosa's website
"Complex field & quaternion space," displays many nice 3D Mandelbrot
and Julia sets computed with quaternions. (not exactly electromagnetics,
but the pictures are worth looking.
For a similar picure, see also the cover of
"Clifford Algebras with Numeric and Symbolic Computations",
R. Ablamowicz, P. Lounesto, Josep M. Parra (Eds.),
Springer, New York, 1996)
G. A. Deschamps: "Electromagnetics and differential forms", Proc. IEEE,
vol. 69, pp. 676-696, June 1981.
G. A. Deschamps: "Exterior differential forms," pp. 112-161,
in
E. Roubine (ed): Mathematics Applied to Physics,
Springer-Verlag, Berlin and UNESCO, Paris, 1970
It might come as a surprise to a modern reader that
J. W. Gibbs wrote an article about Grassmann's algebra:
"On multiple algebra", Proc. Am. Ass. Adv. of Science, vol. XXXV.
pp. 37-66, 1886.
His approach looks very 'dyadescian', an interesting
and relevant critique of his
approach is recorded in the pages 635-652 of Bull. A.M.S.,
vol. 78, 1972,
in F.Dyson's "Missed Opportunities,". See also pp. 14-15 of
preprint
by J.Parra.
B. Jancewicz: ``A Variable Metric Electrodynamics.
The Coulomb and Biot-Savart Laws in Anisotropic Media,''
Ann. Phys. (N.Y.), vol. 245, 227-274, 1996.
D. G. B. Edelen, Applied Exterior Calculus, Wiley, New York 1985.
I. Lindell and P. Lounesto,
Differentiaalimuodot sähkömagnetiikassa (Differential forms in
electromagnetics),
Helsinki University of Technology,
Electromagnetics laboratory report, Espoo 1995.
K. Meetz and W. L. Engl, Elektromagnetische Felder,
Springer-Verlag, Berlin, 1980.
C. Misner, K. Thorne and J. Wheeler, Gravitation, Freeman,
San Francisco 1973.
F. L. Teixeira
and W. C. Chew: "Unified analysis of perfectly
matched layers using differential forms," Microwave and Optical
Technology Letters, vol. 20, no. 2, pp. 124-126, 1999.
J. W. Wheeler, Geometrodynamics, Academic Press, London 1962.
PIER 32 concentrated on
Geometrical Methods for Computational Electromagnetics.
Geometrical methods in this context means algebraic topology with
its many incarnations (Tonti diagrams, de Rham cohomology...)
The material is available on the
Progress In Electromagnetics Research book series website, with the special focus articles under
PIER 32 heading. Several good papers (you might even spot
the contribution of the yours truly in the last section, which is
tentatively entitled as
Geometric Algebra for Electromagnetics.*).
Stanly Steinberg's website provides a
convenient starting point
for those readers who find the topics in PIER32 interesting.
A. Bossavit, L. Kettunen:
"Yee-like schemes on a tetrahedral mesh, with diagonal lumping"
Int. J. of Num. Mod.: ElectronicNetworks,Devices and Fields.
vol.12, no.1-2, Jan.-April 1999, pp.129-142.
T. Tarhasaari, L. Kettunen: "Some realizations of discrete Hodge operator:
A reinterpretation of finite element techniques," IEEE
Tr. Magn., vol. 35, no. 3, 1999, pp. 1494-1497.
C. Mattiussi:
"An Analysis of Finite Volume, Finite Element, and
Difference Methods Using Some Concepts from Algebraic Topology,"
J. Comp. Phys., vol. 133, 1997, pp. 289-309.
W. Schwalm, B. Moritz, M. Giona, M. Schwalm:
"Vector difference calculus for physical lattice models,"
Phys. Rev. E, vol. 59, no. 1, 1999, pp. 1217-1233.
F. L. Teixeira, W. C. Chew: "Differential forms, metrics,
and the reflectionless absorption of electromagnetic waves,"
J. Electrom. Waves Appls., vol.13,
no.5, 1999, pp .665-686.
F. L. Teixeira,W. C. Chew:
"Lattice electromagnetic theory from
a topological viewpoint,"
J. Math. Phys., vol.40, no.1, Jan. 1999, pp.169-187.
[*]This one is mostly about old-fashioned,
pre-computer electromagnetics in Clifford algebra. But at least the
"geometrical methods" is in common with the other articles. By-the-way,
check the errata.
M. Riesz, "Sur certaines notions fondamentales en théorie quantique
relativiste," C.R. 10e Congrès Math. Scandinaves, Copenhagen,
1946. Jul. Gjellerups Forlag, Copenhagen 1947, pp. 123-148 (You can find the
article in the collected papers of M.Riesz, pp. 545-570 ).
Hypercomplex numbers:
the Quaternions ( a Cl(3,0) subalgebra ) and
their bigger cousins, the Octonions.
General
Timeline of
hypercomplex numbers (i.e. quaternions and octonions ).Introductory texts and tutorials
D. Sweetser has several
on-line tutorials concerning applications of quaternions.
Intermediate to advanced
S. Cornbleet, "An electromagnetic theory of rays in a uniform medium,"
in Huygens' Principle 1690-1990: Theory and Applications,
H. Blok, H.A. Ferwerda, H.K. Kuiken (editors), North-Holland, Amsterdam, 1992.
Exterior algebras; Differential forms
General
An excellent
list of references has been collected by
Richard H. Selfridge, David V. Arnold and Karl F. Warnick at
Brigham Young University. Their site is very much worth checking
out, since they have several papers on-line. The level of exposition
in these papers varies from introductory level texts to
advanced research papers. Especially recommendable is
the one entitled "Teaching Electromagnetic Field Theory Using
Differential Forms," available in ps-format.
Many of the references listed in
the link above are of intro level, so we do not
print them here again, with the exception of
And while we are at it, Deschamps' earlier text
Introductory texts and tutorials
T. Frankel: "Maxwell's equations", Am. Math. Monthly, vol. ?,
pp. 343-349, April 1974.
Intermediate to advanced
J. Baez, J. P. Muniain, Gauge Fields, Knots and Gravity,
World Scientific, Singapore, 1994.
D. Baldomir, P. Hammond Geometry of Electromagnetic Systems,
Clarendon Press, Oxford 1996.
Exterior algebras in computational electromagnetics