# Convex structures and operational quantum mechanics

- 129 Downloads
- 30 Citations

## Abstract

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.

Convex structures are shown to give an axiomatization of the states of a physical system and the metric is physically motivated. We demonstrate how convex structures give a generalizing and unifying formalism for convex set and operational methods in axiomatic quantum mechanics.

## Keywords

Quantum Mechanic Linear Operator Linear Space Quantum Computing Operational Method## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Cornette, W., Ph.D.: Dissertation, University of Denver (in preparation).Google Scholar
- 2.Davies, E. B., Lewis, J. T.: An operational approach to quantum probability. Commun. math. Phys.
**17**, 239–260 (1970).Google Scholar - 3.Edwards, C. M.: Classes of operations in quantum theory. Commun. math. Phys.
**20**, 26–56 (1971).Google Scholar - 4.Gudder, S.: Axiomatic quantum mechanics and generalized probability theory in A. Bharucha-Reid (editor), Probabilistic Methods in Applied Mathematics, Vol. II. New York: Academic Press 1970.Google Scholar
- 5.Gunson, J.: Structure of quantum mechanics. Commun. math. Phys.
**6**, 262–285 (1967).Google Scholar - 6.Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.
**5**, 848–861 (1964).Google Scholar - 7.Jauch, J.: Foundations of quantum mechanics. Reading, Mass.: Addison-Wesley 1968.Google Scholar
- 8.Jauch, J., Misra, B., Gibson, A.: On the asymptotic condition of scattering theory. Helv. Phys. Acta
**41**, 513–527 (1968).Google Scholar - 9.Mielnik, B.: Geometry of quantum states. Commun. math. Phys.
**9**, 55–80 (1968).Google Scholar - 10.Mielnik, B.: Theory of filters. Commun. math. Phys.
**15**, 1–46 (1969).Google Scholar - 11.Neumann, H.: Classical systems and observables in quantum mechanics. Commun. math. Phys.
**23**, 100–116 (1971).Google Scholar - 12.Peressini, A.: Ordered Topological Vector Spaces. New York: Harper and Row 1967.Google Scholar
- 13.Stolz, P.: Attempt of an axiomatic foundation of quantum mechanics and more general theories, VI. Commun. math. Phys.
**23**, 117–126 (1971).Google Scholar - 14.Segal, I.: Mathematical problems of relativistic physics. American Math. Soc. Lectures in Applied Mathematics. Providence, R.I. (1963).Google Scholar
- 15.Varadarajan, V.: Geometry of Quantum Theory, Vol. I. Princeton, N.J.: Van Nostrand 1968.Google Scholar