Abstract
During the past 20 years a long series of papers concerning algebras of unbounded operators appeared in the literature, papers which, though being originally motivated by physical arguments, contain almost no physics at all. On the contrary the mathematical aspects of these algebras have been analyzed in many details and this analysis produced, up to now, the monographes [32] and [2]. Some physics appeared first in [28] and [31], in the attempt to describe systems with a very large (1024) number of degrees of freedom, following some general ideas originally proposed in the famous Haag and Kastler’s paper, [27], on QM∞.
After a brief historical introduction on the standard algebraic approach to quantum mechanics of large systems, QM∞, we review some basic mathematical aspects and few physical applications of algebras of unbounded operators.
This paper is dedicated to Gerard Emch for his 70th birthday
Chapter PDF
Similar content being viewed by others
References
G. Alli and G. L. Sewell, New methods and structures in the theory of the multi-mode Dicke laser model, J. Math. Phys. 36, 5598 (1995)
J.-P. Antoine, A. Inoue and C. Trapani Partial *-algebras and Their Operator Realizations, Kluwer, Dordrecht, 2002
J.-P. Antoine and W. Karwowski, Partial *-Algebras of Closed Linear Operators in Hilbert Space, Publ.RIMS, Kyoto Univ. 21, 205–236 (1985).
F. Bagarello, Applications of Topological *-Algebras of Unbounded Operators, J. Math. Phys., 39, 2730 (1998)
F. Bagarello, Fixed Points in Topological *-Algebras of Unbounded Operators, Publ. RIMS, Kyoto Univ. 37, (2001), 397–418.
F. Bagarello, G. Morchio, Dynamics of mean field spin models from basic results in abstract differential equations, J. Stat. Phys. 66, 849 (1992)
F. Bagarello and C. Trapani, ‘Almost’ Mean Field Ising Model: an Algebraic Approach, J.Statistical Phys. 65, (1991), 469–482.
F. Bagarello and C. Trapani, A note on the algebraic approach to the “almost” mean field Heisenberg model, II Nuovo Cimento B 108, (1993), 779–784.
F. Bagarello and C. Trapani, States and representations of CQ*-algebras, Ann. Inst. H. Poincaré 61,(1994), 103–133.
F. Bagarello, C. Trapani, The Heisenberg Dynamics of Spin Sistems: a Quasi*-Algebras Approach, J. Math. Phys. 37, (1996), 4219–4234.
F. Bagarello, C. Trapani, Algebraic dynamics in 0*-algebras: a perturbative approach , J. Math. Phys. 43, (2002), 3280–3292.
F. Bagarello, A. Inoue, C. Trapani, Derivations of quasi *-algebras, Int. Jour. Math. and Math. Sci., 21, 1077–1096 (2004)
F. Bagarello, A. Inoue, C. Trapani, Exponentiating derivations of quasi *-algebras: possible approaches and applications, Int. Jour. Math, and Math. Sci., 17, 2805–2820 (2005)
F. Bagarello, G. L. Sewell, New Structures in the Theory of the Laser Model II: Microscopic Dynamics and a Non-Equilibrim Entropy Principle, J. Math. Phys., 39 (1998), 2730–2747.
F. Bagarello, C. Trapani Morphisms of Certain Banach C*-Algebras, Publ. RIMS, Kyoto Univ., 36, No. 6, 681–705, (2000)
F. Bagarello, A. Inoue, C. Trapani, Some classes of topological quasi *-algebras, Proc. Amer. Math. Soc, 129, 2973–2980 (2001).
O. Bratteli and D.W. Robinson, Operator algebras and Quantum statistical mechanics I, Springer-Verlag, New York, 1987.
O. Bratteli and D.W. Robinson, Operator algebras and Quantum statistical mechanics 2, Springer-Verlag, New York, 1987.
E. Buffet, P.A. Martin, Dynamics of the Open BCS Model, J. Stat. Phys., 18, No. 6, 585–632, (1978)
A.M. Chebotarev, Lectures on quantum probability, Sociedad Matemática Mexicana (2000)
D.A. Dubin, G.L. Sewell, Time-translations in the algebraic formulation of statistical mechanics, J. Math. Phys. 11, (1970), 2990–2998.
G.G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley, New York, (1972)
G.G. Emch, H.J.F. Knops, Pure thermodynamical phases as extremal KMS states, J. Math. Phys. 11, (1970), 3008–3018.
F. Fagnola, Quantum Markov Semigroups and Quantum Flows, Proyecciones, 18, no. 3, 1–144, (1999)
R. Haag, Local Quantum Physics, Springer, Berlin (1992)
R. Haag, N.M. Hugenholtz, M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys., 5, 215–236, (1967)
R. Haag and D. Kastler, An Algebraic Approach to Quantum Field Theory, J.Math.Phys. 5, (1964), 848–861.
G. Lassner, Topological algebras and their applications in Quantum Statistics, Wiss. Z. KMU-Leipzig, Math.-Naturwiss. R. 30, (1981), 572–595.
G. Morchio and F. Strocchi, Mathematical structures for long range dynamics and symmetry breaking J. Math. Phys. 28, (1987), 622–635.
D. Ruelle, Statistical Mechanics, W.A. Benjamin, New York (1969)
M. Schröder and W. Timmermann, Invariance of domains and automorphisms in algebras of unbounded operators, in Proc. Int. Conf. on Operator Algebras and Group Representations, Romania (1980), 134–139.
K. Schmüdgen, Unbounded operator algebras and Representation theory, Birkhäuser, Basel, 1990
G.L. Sewell, Quantum Theory of Collective Phenomena, Oxford University Press, Oxford (1989)
G.L. Sewell, Quantum Mechanics and its Emergent Macrophysics, Princeton University Press, (2002)
F. Strocchi, Elements of Quantum Mechanics of Infinite Systems, World Scientific, (1985)
W. Thirring, A Course in Mathematical Physics 4: Quantum Mechanics of Large Systems, Springer-Verlag, Wien (1980)
W. Thirring and A. Wehrl, On the Mathematical Structure of the B.C.S.-Model, Commun.Math.Phys. 4, (1967), 303–314.
C. Trapani, Bounded elements and spectrum in Banach quasi *-algebras, Studia Matematica, in press
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Hindustan Book Agency
About this chapter
Cite this chapter
Bagarello, F. (2007). Physical Applications of Algebras of Unbounded Operators. In: Ali, S.T., Sinha, K.B. (eds) Contributions in Mathematical Physics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-33-0_4
Download citation
DOI: https://doi.org/10.1007/978-93-86279-33-0_4
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-79-1
Online ISBN: 978-93-86279-33-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)