Abstract
This lecture consists of two sections. In section 1 we consider the simplest version of a q-deformed Heisenberg algebra as an example of a noncommutative structure. We first derive a calculus entirely based on the algebra and then formulate laws of physics based on this calculus. Then we realize that an interpretation of these laws is only possible if we study representations of the algebra and adopt the quantum mechanical scheme. It turns out that observables like position or momentum have discrete eigenvalues and thus space gets a lattice-like structure.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
BA ss E. Sweedler, Hopf Algebras, W. A. Benjamin, New York, 1969.
BB Elicki, Hopf Algebras, Cambridge Tracts in Mathematics, Cambridge University Press, 1977.
BC Klimyk, K. Schmüdgen, Quantum Groups and Their Representations, Springer, Berlin, 1997.
BD Kassel, Quantum Groups, Springer, Berlin, 1995.
BE Majid, Foudations of Quantum Group Theory, Cambridge University Press, 1995.
BF Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge University Press, 1995.
BG Landi, An Introduction to Noncommutative Spaces and their Geometries, Springer Lecture Notes, Springer, Berlin, 1997.
BH Lusztig, Introduction to Quantum Groups, Birkhaeuser, Boston, 1993.
BI D. Faddeev, N. Y. Reshetikhin, L. A. Takhtajan Quantization of Lie Groups and Lie Algebra, Algebra Analysis 1, 178, 1989; translation: Leningrad Math. 3.1., 193, 1990.
BJ G. Drinfeld, Quantum Groups, Proc. ICM Berkeley, 798, 1987,.
BK Jimbo, A q-Difference Analogue of U(q) and the Yang-Baxter Equation, Lett. Math. Phys.10, 1985.
BL Lorek, A. W. Schmidke, J. Wess, SU q (2) Covariant R-Matrices for Reducible Representations, Lett. Math. Phys. 31: 279, 1994.
BM Snyder, Quantized Space-Time, Phys. Rev. 71: 38, 1947.
BN B. Schmidke, J. Wess, B. Zumino, A q-deformed Lorentz Algebra, Z. Phys. C 52: 471, 1991.
BO Ogievetsky, W. B. Schmidke, J. Wess, B. Zumino, Six Generator q-deformed Lorentz Algebra, Lett. Math. Phys.23: 233, 1991.
BP Ogievetsky, W. B. Schmidke, J. Wess, B. Zumino, q-deformed Poincaré Algebra, Comm. Math. Phys.150: 495, 1992.
BQ Lukierski, H. Ruegg, A. Nowicki, V. N. Tolstoy, q-Deformation of Poincaré Algebra, Phys. Lett. B 264: 331, 1992.
BR L. Woronowicz, Twisted SU(2) Group. An Example of Non-commutative Differential Calculus, Publ. RIMS-Kyoto 23: 117, 1987.
BS Wess, B. Zumino, Covariant Differential Calculus on the Quantum Hyperplane, Nucl. Phys. B (Proc. Suppl.) 18 B: 302, 1990.
BT Dimakis, J. Madore, Differential Calculi and Linear Connection, J. Math.Phys. 37: 4647, 1996.
BU Carow-Watamura, M. Schlieker, S. Watamura, W. Weich, Bicovariant Differential Calculus on Quantum Groups SU q (N) and SO q (N), Comm. Math. Phys. 142: 605, 1991.
BV Carow-Watamura, M. Schlieker, S. Watamura, SO q (N) Covariant Differential Calculus on Quantum Space and Quantum Deformation of Schrödinger Equation,Z. Phys. C 49: 439, 1991.
BW Fiore, J. Madore, Leibniz Rules and Reality Conditions, Preprint 98-13, math/9806071, Napoli, 1998.
BX Fiore, Harmonic Oscillator on the Quantum Euclidean Space R q N-1, Proceedings of the Clausthal Symposion on Nonlinear, Dissipative, Irreversible Quantum Systems, 1994.
BY Ogievetsky, B. Zumino, Reality in the Differential Calculus on q-Euclidean Spaces, Lett. Math. Phys.25: 121, 1992.
BZ A. Azcàrraga, P. P. Kulish, F. Rodenas, On the Physical Contents of q-deformed Minkowski Space, Phys. Lett.B 351: 123, 1995.
CA Fiore, J. Madore, The Geometry of Quantum Euclidean Spaces, Preprint LMU-TPW 97-23, math/9904027, 1997.
CB Pillin, On the Deformability of Heisenberg Algebras, Comm. Math. Phys. 180: 23, 1996.
CD Zippold, Hilbert space Representations of an Algebra of Observables for q-deformed Relativistic Quantum Mechanics, Z. Phys. C 67: 681, 1995.
CE Ocampo, SO q (4) Quantum Mechanics, Z. Phys. C 70: 525, 1996.
CF Lorek, W. Weich, J. Wess, Non-commutative Euclidean and Minkowski Structures, Z.Phys. C 76: 375, 1997.
CG L. Cerchiai, J. Wess, q-deformed Minkowski Space based on a q-Lorentz Algebra, E. Phys. J. C 5: 553, 1998.
CH L. Curtright, C. K. Zachos, Deforming maps for quantum algebras, Phys. Lett. B 243: 237, 1990.
CI Pillin, W. B. Schmidke, J. Wess q-Deformed Relativistic One-Particle States, Nucl. Phys. B 403: 223, 1993.
CJ Rohregger, J. Wess, q-deformed Lorentz Algebra in Minkowski Phase Space, E. Phys. J. C 7: 177, 1999.
CK Kiesler, J. Madore, T. Masson, On Finite Differential Calculus, Contemp. Math. 203: 135, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wess, J. (2000). q-Deformed Heisenberg Algebras. In: Gausterer, H., Pittner, L., Grosse, H. (eds) Geometry and Quantum Physics. Lecture Notes in Physics, vol 543. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46552-9_7
Download citation
DOI: https://doi.org/10.1007/3-540-46552-9_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67112-1
Online ISBN: 978-3-540-46552-2
eBook Packages: Springer Book Archive