In this chapter we present some of the basic concepts needed to describe noncommutative spaces and their topological and geometrical features. We therefore complement the previous chapters where noncommutative spaces have been described by the commutation relations of their coordinates. The full algebraic description of ordinary (commutative) spaces requires the completion of the algebra of coordinates into a \(C^\star\)-algebra, this encodes the Hausdorff topology of the space. The smooth manifold structure is next encoded in a subalgebra (of “smooth” functions). Relaxing the requirement of commutativity of the algebra opens the way to the definition of noncommutative spaces, which in some cases can be a deformation of an ordinary space. A powerful method to study these noncommutative algebras is to represent them as operators on a Hilbert space. We discuss the noncommutative space generated by two noncommuting variables with a constant commutator. This is the space of the noncommutative field theories described in this book, as well as the elementary phase space of quantum mechanics. The Weyl map from operators to functions is introduced in order to produce a \(\star\)-product description of this noncommutative space.
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
D. Hilbert, Grundlagen der Geometrie, Teubner (1899), English translation The Foundations of Geometry, available at http://www.gutenberg.org
J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton (1955).
A. Connes, Noncommutative Geometry, Academic Press, San Diego (1994).
G. Landi, An Introduction to Noncommutative Spaces and Their Geometries, Springer Lect. Notes Phys. 51, Springer Verlag (Berlin Heidelberg) (1997), [hep-th/9701078].
J. M. Gracia-Bondia, J. C. Varilly and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, Boston, MA (2000).
J. Madore, An introduction to noncommutative differential geometry and physical applications, Lond. Math. Soc. Lect. Note Ser. 257.
G. Landi, F. Lizzi and R. J. Szabo, Physical Applications of Noncommutative Geometry, Birkhäuser, Boston to appear.
G. Murphy, \(C^*\) -algebras and Operator Theory, Academic Press, San Diego (1990).
J. M. G. Fell and R. S. Doran, Representations of \(^*\) -Algebras, Locally Compact Groups and Banach \(^*\) -Algebraic Bundles, Academic Press, San Diego (1988).
A. Connes, On the spectral characterization of manifolds, arXiv:0810.2088.
J. Dixmier, Les \(C^*\) -algébres et leurs Représentations, Gauthier-Villars, Paris (1964).
M. A. Rieffel, Induced representation of \(C^*\) -algebras, Adv. Math. 13, 176 (1974).
A. P. Balachandran, G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, Finite quantum physics and noncommutative geometry, Nucl. Phys. Proc. Suppl. 37C, 20 (1995), [hep-th/9403067].
M. A Rieffel, \(C^{*}\) -algebras associated with irrational rotations, Pacific J. Math. 93, 415–429 (1981).
H. Weyl, The theory of Groups and Quantum Mechanics, Dover, New York (1931), translation of Gruppentheorie und Quantemmechanik, Hirzel Verlag, Leipzig (1928).
E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
J. C. Pool, Mathematical aspects of the Weyl correspondence, J. Math. Phys. 7, 66 (1996), [hep-th/0512169].
G. S. Agarwal and E. Wolf, Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators, Phys. Rev. D 2, 2161 (1970).
J. M. Gracia-Bondia, Generalized moyal quantization on homogeneous symplectic spaces, Contemp. Math., 134, 93 (1992).
H. Grönewold, On principles of quantum mechanics, Physica 12, 405 (1946).
J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45, 99 (1949).
J. M. Gracia-Bondia and J. C. Varilly, Algebras of distributions suitable for phase space quantum mechanics. (I), J. Math. Phys. 29, 869 (1988).
J. C. Varilly and J. M. Gracia-Bondia, Algebras of distributions suitable for phase space quantum mechanics. II. Topologies on the moyal algebra, J. Math. Phys. 29, 880 (1988).
R. Estrada, J. M. Gracia-Bondía and J. C. Várilly, On asymptotic expansions of twisted products, J. Math. Phys. 30, 2789 (1989).
M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math. 79, 59 (1964).
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. 1. Deformations of symplectic structures, Annals Phys. 111, 61 (1978).
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization. 2. Physical applications, Annals Phys. 111, 111 (1978).
M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66, 157 (2003), [q-alg/9709040].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Springer
About this chapter
Cite this chapter
Lizzi, F. (2009). Noncommutative Spaces. In: Noncommutative Spacetimes. Lecture Notes in Physics, vol 774. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89793-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-89793-4_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89792-7
Online ISBN: 978-3-540-89793-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)