Abstract
Quantum algebra was created by Dirac. Its evolution also bears the imprint of the genius of many great mathematicians and physicists such as Weyl, von Neumann, Schwinger, Moyal, Flato, and others. It has inspired developments in deformation theory, representation theory, quantum groups, and many other mathematical themes.
This essay and the next are based on lectures given at Howard University, Washington D.C., sponsored by my friend D. Sundararaman, in the 1990s.
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
P. A. M. Dirac, Proc. Roy. Soc. A 109 (1925), 642–653.
B. L. van der Waerden, Sources of Quantum Mechanics, Dover, 1967, 307–320; this source book will be referred to as SQM.
W. Heisenberg, Zeit. Phys. 33 (1925), 879–893; see also SQM, 261–276.
M. Born and P. Jordan, Zeit. Phys. 34 (1925), 858–888; SQM, 277–306.
M. Born, W. Heisenberg, and P. Jordan, Zeit. Phys. 35 (1926), 557–615.
SQM, 53–54, Letter to Pauli.
H. Weyl, Theory of Groups and Quantum Mechanics, Dover, 1931. Weyl’s ideas were first published in H. Weyl, Zeit. Phys. 46 (1927), 1–46.
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
V. S. Varadarajan, Geometry of Quantum Theory, Springer-Verlag, 1985.
V. S. Varadarajan, Int. Jour. Theor. Phys. 32 (1993), 1815–1834.
E. G. Beltrametti, and G. Cassinelli, The Logic of Quantum Mechanics, Encyclopedia of Mathematics and its Applications, Vol 15, Addison-Wesley, 1981.
B. C. Van Fraassen, Quantum Mechanics, Oxford, 1991.
G. Birkhoff and J. Von Neumann, Ann. Math. 37 (1936), 823–843.
G. Cassinelli, E. De Vito, P. Lahti, and A. Levrero, Rev. Math. Phys. 9 (1997), 921–941.
J. Schwinger, Proc. Nat. Acad. Sci. USA 45 (1959), 1542–1553; see also Selected papers (1937–1976) of Julian Schwinger, M. Flato, C. Fronsdal, K. A. Milton (eds), D. Reidel Publishing Company, Boston, 127–138. See also the beautiful book of Schwinger, especially the Prologue: J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, Ed. Berthold-Georg Englert, Springer, 2001.
J. Schwinger, Quantum Kinematics and Dynamics, W. A. Benjamin, New York, 1970.
J. Schwinger, Hermann Weyl and quantum kinematics, in Exact Sciences and Their Philosophical Foundations, Verlag Peter Lang, Frankfurt, 1988, 107–129.
L. Accardi, Nuovo Cimento, 110 (1995), 685–721.
G. W. Mackey, Unitary group Representations in Physics, Probability, and Number theory, Benjamin, 1978.
J. Schwinger, Proc. Nat. Acad. Sci. USA, 46 (1960), 570–579.
E. Husstad, Endeligdimensjonale approksimasjoner til kvantesystemer, Thesis, University of Trondheim, 1991/92.
T. Digernes, V. S. Varadarajan, and S. R. S. Varadhan, Rev. Math. Phys. 6 (1994), 621–648.
M. H. Stone, Linear transformations in Hilbert Space. III. Operational methods and group theory, Proc. Nat. Acad. Sci. USA, 16, (1930), 172–175.
J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann., 104 (1931), 570–578.
V. S. Varadarajan, in Analysis, Geometry, and probability, Rajendra Bhatia (ed), Hindustan Book Agency, New Delhi, 1996, 362–396. For Mackey’s paper see Duke Math. Jour. 16 (1949), 313–326, and also the book Tata Lectures on Theta III by D. Mumford with Madhav Nori and Peter Norman, Birkhäuser,Basal, 1991, pp. 1–14.
V. S. Varadarajan, Lett. Math. Phys. 34 (1995), 319–326.
J. E. Moyal, Proc. Camb. Phil. Soc. 45 (1949), 99–124.
F. Bayen, M. Flato, M, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer. Quantum mechanics as a deformation of classical mechanics, Lett. Math. Phys. 1 (1975/77), no. 6, 521–530.
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1,61–110.
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), no. 1, 111–151.
H. Basart, M. Flato, A. Lichnerowicz, and D. Sternheimer, Deformation theory applied to quantization and statistical physics, Lett. Math. Phys. 8(1984), 483–494.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Varadarajan, V.S. (2011). Quantum Algebra. In: Reflections on Quanta, Symmetries, and Supersymmetries. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0667-0_2
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0667-0_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0666-3
Online ISBN: 978-1-4419-0667-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)