Abstract
Clifford algebras are naturally associated with quadratic forms. These algebras are ℤ2 -graded by construction. However, only a ℤn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cℓ(V) ↔ Λ V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the ℤn-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a ℤn -grading which we now call Clifford algebras of multi-vectors or quantum Clifford algebras. These quantum Clifford algebras are in fact Clifford algebras of a bilinear form in a functorial way. It turns out that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonaliz-ability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cℓp,q which can be decomposed in the symmetric case into a tensor product Cℓp-1,q-1 ⊕ Cℓ1,1. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q -deformed structures through nontrivial vacuum states in quantum theories is outlined.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Ablamowicz, P. Lounesto, On Clifford algebras of a bilinear form with an antisymmetric part, in Clifford Algebras with Numeric and Symbolic Computations, Eds. R. Ablamowicz, P. Lounesto, J. M. Parra, Birkhäuser, Boston, 1996, 167–188.
Algebras, Banff, Alberta Canada, 1995, Ed. W. E. Baylis, Birkhäuser, Boston, 1996, 463–501.
R. Ablamowicz, ‘CLIFFORD’ — Maple V package for Clifford algebra computations, ver. 4, http://math.tntech.edu/rafal/cliff4/.
R. Ablamowicz, B. Fauser, Hecke algebra representations in ideals generated by q-Young Clifford idempotents, in this volume and math.QA/9908062.
M. F. Atiyah, R. Bott, A. Shapiro, Clifford modules, Topology 3 (Suppl. 1) (1964), 3–38.
I. M. Benn, R. W. Tucker, Introduction to Spinors and Geometry with Applications in Physics, Adam Hilger, Bristol, 1987.
F. A. Berezin, The Method of Second Quantization, Academic Press, London, 1966.
N. Bourbaki, Algebra 1, Springer Verlag, Berlin, 1989, Chapters 1–3.
L. de Broglie, La méchanique du photon, Une nouvelle théorie de la Lumière tome 1, La Lumière dans le vide, Hermann Paris, 1940, tome 2, Les interactions entre les photons et la metière, Hermann, Paris, 1942.
P. Budinich, A. Trautmann, The Spinorial Chessboard, Trieste Notes in Physics, Springer, 1988.
E. R. Caianiello, Combinatorics and Renormalization in Quantum Field Theory, W. A. Benjamin, Inc., London, 1973.
C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, New York, 1954.
C. Chevalley, The Construction and Study of Certain Important Algebras, The Mathematical Society of Japan 1955, in Collected Works, Vol. 2, Springer, Berlin, 1997.
A. Crumeyrolle, Orthogonal and Symplectic Clifford Algebras, Spinor Structures, Mathematics and Its Applications, Kluwer, Dordrecht, 1990.
A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994, Ch. V. 10.
O. Conradt, The principle of duality in Clifford algebra and projective geometry, this volume.
H. S. M. Coxeter, W. O. J. Moser, Generators and Relations for Discrete Groups, Fourth Edition, Springer Verlag, Berlin, 1980.
C. Daviau, Application à la théorie de la lumiére de Louis de Broglie d’une réécriture de l’équation de Dirac, Ann. de la Fond. Louis de Broglie, Vol. 23 (3–4) (1998), 121–127.
A. Dimakis, Geometrische Behandlung von Cliffordalgebren und spinoren mit Anwendungen auf die Dynamik von Spinteilchen, Thesis, Univ. Göttingen 1983. A new representation of Clifford algebras, J. Phys. A22 (1989), 3171.
C. Doran, A. Lasenby, S. Gull, States and operators in spacetime algebra, Found. Phys. 23 (9), (1993), 1239. S. Somaroo, A. Lasenby, C. Doran, Geometric algebra and the causal approach to multiparticle quantum mechanics, J. Math. Phys. Vol. 40, No. 7 (July 1999), 3327–3340.
F. J. Dyson, The S-matrix in electrodynamics, Phys. Rev. 75 (1949), 1736–1735.
F. J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635–652.
B. Fauser, H. Dehnen, Isospin from spin by compositeness, in Proceedings of the International Workshop Lorentz Group, CPT and Neutrinos, V. Dvoeglazov, ed., Zacatecas, Mexico, 1999.
B. Fauser, Hecke algebra representations within Clifford geometric algebras of multivectors, J. Phys. A: Math. Gen. 32 (1999), 1919–1936.
B. Fauser, Clifford-algebraische Formulierung und Regularität der Quantenfeldtheorie, Thesis, Uni. Tübingen, 1996.
B. Fauser, H. Stumpf, Positronium as an example of algebraic composite calculations, in Proceedings of The Theory of the Electron, J. Keller, Z. Oziewicz, eds., Cuautitlan, Mexico, 1995, Adv. Appl. Clifford Alg. 7 (Suppl.) (1997), 399–418.
B. Fauser, Clifford geometric parameterization of inequivalent vacua, Submitted to J. Phys. A. (hep-th/9710047).
B. Fauser, On the relation of Clifford-Lipschitz groups to q -symmetric groups, Group 22: Proc. XXII Int. Colloquium on Group Theoretical Methods in Physics, Hobart, Tasmania, 1998, eds., S. P. Corney, R. Delbourgo and P. D. Jarvis, International Press, Cambridge, 1999, 413–417.
B. Fauser, Clifford algebraic remark on the Mandelbrot set of twocomponent number systems, Adv. Appl. Clifford Alg. 6 (1) (1996), 1–26.
B. Fauser, Vertex functions and generalized normal-ordering by triple systems in non-linear spinor field models, preprint hep-th/9611069.
B. Fauser, On an easy transition from operator dynamics to generating functional by Clifford algebras, J. Math. Phys. 39 (1998), 4928–4947.
B. Fauser, Vertex normal ordering as a consequence of nonsymmetric bilinear forms in Clifford algebras, J. Math. Phys. 37 (1996), 72–83.
G. Fiore, On q -deformations of Clifford algebras, this volume.
W. Fulton, J. Harris, Representation Theory, Springer, New York, 1991.
W. H. Greub, Multilinear Algebra, Springer, Berlin, 1967.
R. Haag, On quantum field theories, Mat. Fys. Medd. Dann. Vid. Selsk. 29, No. 12 (1955). Local Quantum Physics Springer, Berlin, 1992.
A. J. Hahn, Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups, Springer, New York, 1994.
M. Hamermesh, Group Theory and Its Application to Physical Problems, Addison-Wesley, London, 1962.
D. Hestenes, Spacetime Algebra, Gordon and Beach, 1966.
D. Hestenes, G. Sobczyk, Clifford Algebra to Geometric Calculus, Reidel, Dordrecht, 1984.
D. Hestenes, A unified language for mathematics and physics, in Proceedings of Clifford Algebras and Their Application in Mathematical Physics, Canterbury, 1985, J. S. R. Chisholm, A. K. Common, eds., Kluwer, Dordrecht, 1986.
D. Hestenes, New Foundation for Classical Mechanics, Kluwer, Dordrecht, 1986.
D. Hestenes, R. Ziegler, Projective geometry with Clifford algebra, Acta Appl. Math. 23 (1991), 25–64.
H. Hiller, Geometry of Coxeter Groups, Pitman Books Ltd., London, 1982.
R. Kerschner, Effektive Dynamik zusammengesetzter Quantenfelder, Thesis, Univ. Tübingen, 1994, Chapter 4.
R. Kerschner, On quantum field theories with finitely many degrees of freedom, preprint, hep-th/9505063.
T. Y. Lam, The Algebraic Theory of Quadratic Forms, The Benjamin/Cummings Publishing Company, Reading, 1973, [Theorem 2.6 and Corollary 2.7, p. 113].
H. B. Lawson, M.-L. Michelsohn, Spin Geometry, Universidade Federal de Ceará, Brazil, 1983, Princeton Univ. Press, Princeton, 1989.
P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press, Cambridge, 1997.
P. Lounesto appendix in M. Riesz, Clifford Numbers and Spinors, Lecture Series No. 38, The Institute of Fluid Dynamics and Applied Mathematics, University of Maryland (1958), 1993 Facsimile, Kluwer, Dordrecht, 1993.
P. Lounesto, Counterexamples in Clifford algebras with CLICAL, in Clifford Algebras with Numeric and Symbolic Computations, R. Ablamowicz, P. Lounesto, J. M. Parra, eds., Birkhäuser, Boston 1996.
S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995.
J. Maks, Modulo (1,1) Periodicity of Clifford Algebras and the Generalized (anti-) Möbius Transformations, Thesis, TU Delft, 1989.
A. Micali, Albert Crumeyrolle, La démarche algébrique d’un géomètre, p. xi in Clifford Algebras and Spinor Structures, A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992), R. Ablamowicz, P. Lounesto, eds., Kluwer, Dordrecht 1995 ix-xiv. P. Lounesto, Crumeyrolle’s strange question: “What is a bi-vector?”, Preprint 1993.
R. V. Moody, A. Pianzola, Lie Algebras with Triangular Decompositions, Wiley-Interscience Publ., New York, 1995.
Z. Oziewicz, From Grassmann to Clifford, in Proceedings Clifford Algebras and Their Application in Mathematical Physics, Canterbury, 1985, UK, J.S.R. Chisholm, A.K. Common, eds., Kluwer, Dordrecht, 1986, 245–256.
Z. Oziewicz, Clifford algebra for Hecke braid, in Clifford Algebras and Spinor Structures, Special Volume Dedicated to the Memory of Albert Crumeyrolle, eds. R. Ablamowicz, P. Lounesto, Kluwer, Dordrecht, 1995, 397–412.
J. M. Parra Serra, In what sense is the Dirac-Hestenes electron a representation of the Poincaré group?, in Proceedings of the International Workshop Lorentz Group, CPT and Neutrinos, V. Dvoeglazov, ed., Zacatecas, 1999, Mexico.
R. Penrose, W. Rindler, Spinors and Space-time: Two-spinor Calculus and Relativistic Fields, Cambridge Monographs on Mathematics, Cambridge University Press, Paperback Reprint Edition, Vol. 1, June, 1987.
I. Porteous, Topological Geometry, Van Nostrand, New York, 1969.
M. Riesz, Clifford Numbers and Spinors, Lecture Series No. 38, The Institute of Fluid Dynamics and Applied Mathematics, University of Maryland 1958, Facsimile by Kluwer, 1993.
H. Sailer, Quantum algebras I, Nuovo Cim. 108B (6)( 1993), 603–630. Quantum algebras II, Nuovo Cim. 109B, (3) (1994), 255–280.
G. Scheja, U. Storch, Lehrbuch der Algebra, Teil 2, Teubner, Stuttgart, 1988.
H. Stumpf, Th. Borne, Composite Particle Dynamics in Quantum Field Theory, Vieweg, Braunschweig, 1994.
G. C. Wick, The evaluation of the collision matrix, Phys. Rev. 80 (1950), 268–272.
E. Witt, Theorie der quadratischen formen in beliebigen Körpern, J. Reine. Angew. Math. 176 (1937), 31–44.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this chapter
Cite this chapter
Fauser, B., Abłamowicz, R. (2000). On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form. In: Abłamowicz, R., Fauser, B. (eds) Clifford Algebras and their Applications in Mathematical Physics. Progress in Physics, vol 18. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1368-0_18
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1368-0_18
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7116-1
Online ISBN: 978-1-4612-1368-0
eBook Packages: Springer Book Archive