Abstract
The language of this first section is quite heuristic, as it only aims at a broad presentation of the problems we want to turn our attention to. A rigorous formulation of some of these questions, and of their answers, is reviewed in section 2. A few illustrative models are mentioned briefly in Section 3. Section 4 is entirely devoted to the question raised by the very existence of crystals in a world where the fundamental interactions are believed to be covariant at least under the group of rigid motions in IR3.
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
Poincaré’, H., Leçons sur les Hypotheses Cosmogoniques, 2nd édition, Hermann, Paris, 1913; see in particular pp. LX-LXI, 53–58, and 184–189.
G. Bertin and L. A. Radicati, Astrophys. Journ. 206 (1976) 815–821.
Both of these works contain useful cross-references.
We should emphasize here that in all this section the notation is very loose; H is not necessarily meant to denote an Hamiltonian, but is only to be taken as a symbol for a prescription which defines the model and the solutions of interest.
J. L. Lebowitz, in Rome International Conference on the Mathematical Problems in Theoretical Physics, June 6–15, 1977.
Emch, G. G., Algebraic Methods in Statistical Mechanics and Quantum Field Theory, WileyInterscience, New York, 1972.
See for instance the quantum generalization of the transfer-matrix idea as elaborated for the Heisenberg model by H. Araki (see [61); other models will be mentioned as well in section III.
Araki, H., Commun. math. Phys. 14 (1969) 120–157.
Fröhlich, J., in Elementary Particles and Mathematical Physics, P. Urban, Ed., Springer-Verlag, 1976, pp. 133–269.
Rieckers, A., Lecture Notes, in Tübingen International Summer School on Groups and Many-Body Physics, July 11–16, 1977; his list of references also provides a good background for the present lectures.
von Neumann, J., Grundlagen der Quantenmechanik, Springer, Berlin, 1932.
Kubo, R., J. Phys. Soc. Japan 12 (1957) 570–586
Martin, P. C. and Schwinger, J., Phys. Rev. 115 (1959) 1342–1373
Haag, R., Hugenholtz, N. and Winnink, M., Commun. math. Phys. 5 (1967) 215–236.
Takeda, Z., Tohoku Math. Journ. 7 (1955) 68–86.
Gelfand, I. and Naimark, M. A., Mat. Sborn, N.S. 12 54 (1943) 197–217
Segal, I. E., Ann. Math. 48 (1947) 930–948.
See [8] in this volume.
For a discussion of this point, see [4] pp. 282–3 and references quoted therein; in particular: Hugenholtz, N. and Wieringa, J. D., Commun. math. Phys. 11 (1969) 183–197
Wieringa, J. D., Thesis, Groningen, 1970
Haag, R., Kadison, R. V. and Kastler, D., Commun. math. Phys. 16 (1970) 81–104
Takesaki, M. and Winnink, M., Commun. math. Phys. 30 (1973) 129–152
Ruelle, D., Commun. math. Phys. 3 (1966) 133–150
Lanford, O. E. and Ruelle, D., Journ. Math. Phys. 8 (1967) 1460–1463.
We already mentioned [6]. See also Robinson, D. W., Commun. math. Phys. 7 (1968) 337–348
Ruelle, D., Statistical Mechanics, Benjamin, N.Y., 1969
Manuceau, H. and Trottin, J. C., Ann. Inst. Poincaré, Henri A10 (1969) 359–380
Kishimoto, A., Commun. math. Phys. 47 (1976) 25–32
Robinson, D. W., preprint, Bielefeld, 1977.
For two early counterexamples, see: Dubin, D. A. and Sewell, G., bourn. Math. Phys. 11 (1970) 2990–2998
Emch, G. G. and Knops, J. F., Journ. Math. Phys. 11 (1970) 3008–3018
Radin, Ch., Commun. math. Phys. 54 (1977) 69–79
Battle, G., Thesis, Ph. D., Duke University, 1977; Journ. Math. Phys. (to appear); Commun. math. Phys. (to appear).
See the last two entries in [14] above, and [15].
See the second entry in [15] above.
Roepstorff, G., Commun. math. Phys. 46 (1976) 253–262
and preprint, Aachen, 1976
Sewell, G., Commun. math. Phys. (to appear);Araki, H., in Japan — US Conference on C* algebras and their Applications in Physics, Los Angeles, April 18–22, 1977; for a general review, see Verbeure, A., in Marseille International Colloquium on Algebras of Operators and their Applications to Mathematical Physics, June 20–24,1977; the latter covers, amongst other things, several original contributions on the subject from Verbeure and coworkers.
Pusz, W. and Woronowicz, S. L., in Rome International Conference on the Mathematical Problems in Theoretical Physics, June 6–15, 1977; and Marseille International Colloquium on Algebras of Operators and their Applications to Mathematical Physics, June 20–24, 1977. Their approach can actually be related to the approach via correlation-inequalities, as pointed out in the last entry of [18] above.
q-clustering will bedefined in subsection 6 below.
Haag, R., Kastler, D, and Trych Pohlmeyer, E., Commun. math. Phys. 38 (1974) 173–193
the cluster property, as given in this lecture, is an improvement of the Original version, and is due to
Bratteli, O. and Kastler, D., Commun. math. Phys. 46 (1976) 37–42
this approach has been further extended to a theory of the chemical potential by Araki, H., Haag, R., Kastler, D. and Takesaki, M., Commun. math. Phys. (to appear); see also Kastler, D., Lecture Notes, UCLA 1977.
Gallavotti, G. and Verboven, E. J., Il Nuovo Cimento 28 (1975) 274–286
Aizenmann, M., Goldstein, S., Gruber, C., Lebowitz, J. L. and Martin, Ph., preprint, Lausanne, 1976
Pulvirenti, M., Journ. Math. Phys. (to appear); see also Andersson, S. I., Fortschritte der Physik 24 (1976) 55–83.
Kossakowski, A., Frigerio, A., Gorini, V. and Verri, M., preprint, Milano, 1977.
Takesaki, M., Springer Lecture Notes in Mathematics No. 128, Berlin, 1970
Takesaki, M., Springer Lecture Notes in Physics No. 20, pp. 205–246, Berlin, 1973.
van Daele, A., Journ. Funct. Analysis 15 (1974) 378–393
van Daele, A., JournProceedings of the Varenna Conference (1973) on C*algebras and their Applications to Statistical Mechanics and Quantum Field Theory, Kastler, D., ed.
Rieffel, M. A. and Daele van, A., Pac. J. Math. 69 (1977) 187–221.
Narnhofer, H., Acta Phys. Austriaca 31 (1970) 349–353.
Emch, G. G. and Radin, Ch., Journ. Math. Phys. 12 (1971) 2043–2046.
Robinson, D. W., Commun. math. Phys. 31 (1973) 171–189.
Araki, H., Commun. math. Phys. 28 (1972) 267–277.
Emch, G. G., Commun. math. Phys. 49 (1976) 191–215.
Davies, E. B., Commun. math. Phys. 19 (1970) 83–105
Frigerio, A., preprint, Milano, 1977; Krümmerer, B., in Vlth International Colloquium on Group Theoretical Methods in Physics, Tübingen, July 18–22, 1977.
Emch, G.,G., Knops, H. J. F. and Verboven, E. J., Commun. math. Phys. 8 (1968) 300–314.
Onsager, L., Phys. Rev. 65 (1944) 117–149
Yang, C. N. and Lee, T. D., Phys. Rev. 87 (1952) 404–409 and 410–419
Montroll, E. W., Potts, R. B. and Ward, J. C., Journ. Math. Phys. 4 (1963) 308–322
Schultz, T. D., Mattis, D. C. and Lieb, E. H., Revs. Mod. Phys. 36 (1964) 856–871
recall also ref [3] above.
Lanford, O. E. and Ruelle, D., Commun. math. Phys. 13 (1969) 194–215
Ruelle, D., in Cargése Lecture Notes, Kastler, D., Ed., Gordon and Breach, New York, 1969
Ruelle, D., Journ. Funct. Analysis 6 (1970) 116–151
Robinson, D. W, Commun. math. Phys. 7 (1968) 337–348
Araki, H., ref [6] above.
Sakai, S., C*-Algebras and W*-Algebras, Springer, Berlin, 1971
Wils, W., C. R. Acad. Sc., Paris 267 (1968) 810–812
Wils, W., C. R. Lecture Notes, Aarhus, 1969. See also ref. [24] above.
See for instance the papers by Ruelle, D. in [35] above, by Takesaki, M. and Winnink, M. in
above, and by Andersson, S. I. in [22] above.
Fröhlich, J., Simon, B. and Spencer, T., Phys. Rev. Letters 36 (1976) 804–806
and preprint Princeton, 1976; see also [7] above.
Bardeen, J. L., Cooper, N. and Schrieffer, J. R., Phys. Rev. 108 (1957) 1175–1204
Schrieffer, J. R., Theory of Superconductivity, Benjamin, New York, 1964.
Haag, R., Nuovo Cimento 25 (1962) 287–298.
Araki, H. and Woods, E. J., Journ. Math. Phys. 4 (1963) 637–662
see also the first reference in [15] above.
Ezawa, H., Journ. Math. Phys. 5 (1964) 1078–1090
Emch, G. G. and Guenin, M., Journ. Math. Phys. 7 (1966) 915–921
Thirring, W. and Wehrl, A., Commun. math. Phys. 4 (1967) 303–314
Thirring, W., Commun. math. Phys. 7 (1968) 181–189
Thirring, W., in The Many-Body Problem, Garrido, L. M. et al., Eds., Plenum Press, London, 1969
Jelinek, F., Commun. math. Phys. 9 (1968) 169–175
see also pp. 28–32 in [4] above.
See the second reference in [15] above.
See for instance Kac, M., in Brandeis Summer Institute, 1966 Lecture Notes, Chretien, M., Gross, E. P. and Deser, S., Eds., Gordon and Breach, New York 1968
Kac, M., Fundamental Problems in Statistical Mechanics II, Cohen, E. G. D., Ed., North-Holland, Amsterdam, 1968.
See the papers by Battle in [15] above.
Martin, Ph. A., Journ. Stat. Phys. 16 (1977) 149–168
Giuffre, M. S., Ph. D. dissertation, University of Rochester, 1977, for a review, see Emch, G. G., in Proceedings of the VIth International Colloquium on Group Theoretical Methods in Physics, Tübingen, July 18–22, 1977.
Mermin, D. and Wagner, H., Phys. Rev. Letters 17 (1966) 1133–1136.
See in particular Lieb, E. H., in Rome International Conference on the Mathematical Problems in Theoretical Physics, June 6–15, 1977.
With the exception of the “critical slowing-down” obtained in the first reference of [46] above.
Emch, G. G., Knops, H. J. F. and Verboven, E. J., Journ. Math. Phys. 11 (1970) 1655–1668.
See Takesaki, M. and Winnink, M., in [13] above.
See for instance: Uhlenbeck, G. E., in Fundamental Problems in Statistical Mechanics II, Cohen, E. G. D., Ed., North-Holland, Amsterdam, 1968.
Kastler, D., preprint, Marseille, 1967: this is a seminar talk in which the author reported on work jointly done with Haag, R. and Michel, L.
Kastler, D., Mebkhout, M., Loupias, G. and Michel, L., Commun. math. Phys. 27 (1972) 195222.
Emch, G. G., Knops, H. J. F. and Verboven, E. J., Journ. Phys. Soc. Japan 26 (Suppl.) (1969) 301–303
Knops, H. J. F., Dissertation, University of Nijmegen, 1969.
Gates, D. J., Journ. Math. Phys. 12 (1971) 766–778.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1980 Springer Fachmedien Wiesbaden
About this chapter
Cite this chapter
Emch, G.G. (1980). An Algebraic Approach for Spontaneous Symmetry Breaking in Quantum Statistical Mechanics. In: Kramer, P., Dal Cin, M. (eds) Groups, Systems and Many-Body Physics. Vieweg Tracts in Pure and Applied Physics, vol 4. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-06825-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-663-06825-9_5
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-08444-8
Online ISBN: 978-3-663-06825-9
eBook Packages: Springer Book Archive