Abstract
A convex scheme of quantum theory is outlined where the states are not necessarily the density matrices in a Hilbert space. The physical interpretation of the scheme is given in terms of generalized “impossibility principles”. The geometry of the convex set of all pure and mixed states (called a statistical figure) is conditioned by the dynamics of the system. This provides a method of constructing the statistical figures for non-linear variants of quantum mechanics where the superposition principle is no longer valid. Examples of that construction are given and its possible significance for the interrelation between quantum theory and general relativity is discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Segal, I. E.: Ann. Math.48, 930 (1947)
Birkhoff, G., von Neumann, J.: Ann. Math.37, 823 (1936)
Dähn, G.: Commun. math. Phys.9, 192 (1968);28, 109 (1972);28, 123 (1972)
Davies, E. B., Lewis, J. T.: Commun. math. Phys.17, 239 (1970)
Davies, E. B.: Commun. math. Phys.15, 277 (1969);19, 83 (1970);22, 51 (1971)
Edwards, C. M.: Commun. math. Phys.16, 207 (1970);20, 26 (1971);24, 260 (1972)
Flato, M., Sternheimer, D.: Commun. math. Phys.14, 5 (1969)
Giles, R.: J. Math. Phys.11, 2139 (1970)
Gunson, J.: Commun. math. Phys.6, 262 (1967)
Haag, R., Kastler, D.: J. Math. Phys.5, 848 (1964)
Hellwig, K. E., Kraus, K.: Commun. math. Phys.11, 214 (1969);16, 142 (1970)
Jauch, J. M., Piron, C.: Helv. Phys. Acta42, 842 (1969)
Leiter, D.: Intern. J. Theor. Phys.4, 83 (1971)
Ludwig, G.: Z. Naturforsch.22a, 1303 (1967);22a, 1324 (1967)
Ludwig, G.: Commun. math. Phys.4, 331 (1967);9, 1 (1968);26, 78 (1972)
Mielnik, B.: Commun. math. Phys.15, 1 (1969)
Neumann, H.: Commun. math. Phys.23, 100 (1971)
Piron, C.: Helv. Phys. Acta37, 439 (1964)
Pool, J. C. T.: Commun, math. Phys.9, 118 (1968);9, 212 (1968)
Sachs, M.: Intern. J. Theor. Phys.4, 83 (1971)
Stolz, P.: Commun. math. Phys.11, 303 (1969);23, 117 (1971)
Gudder, S.: Commun. math. Phys.29, 249 (1973)
Author information
Authors and Affiliations
Additional information
Communicated by R. Haag
Rights and permissions
About this article
Cite this article
Mielnik, B. Generalized quantum mechanics. Commun.Math. Phys. 37, 221–256 (1974). https://doi.org/10.1007/BF01646346
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01646346