Abstract
Three definitions of logical independence of two von Neumann latticesPℳ1,Pℳ2 of two sub-von Neumann algebras ℳ1, ℳ2 of a von Neumann algebra ℳ are given and the relations of the definitions clarified. It is shown that under weak assumptions the following notion, called “logical independence” is the strongest:A ∧ B ≠ 0 for any 0 ≠A ∈ Pℳ1, 0 ≠B ∈Pℳ2. Propositions relating logical independence ofPℳ1,Pℳ2 toC *-independence,W * independence, and strict locality of ℳ1, ℳ2 are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Haag, R. (1992).Local Quantum Physics. Fields, Particles, Algebras, Springer-Verlag, Berlin.
Haag, R., and Kastler, D. (1964). An algebraic approach to quantum field theory,Journal of Mathematical Physics,5, 848.
Hardegree, G. (1976). The conditional in quantum logic, inLogic and Probability in Quantum Mechanics, P. Suppes, ed., Reidel, Dordrecht.
Hardegree, G. (1979). The conditional in abstract and concrete quantum logic, inThe Logico-Algebraic Approach to Quantum Mechanics, Vol. II.Contemporary Consolidation, C. A. Hooker, ed., Reidel, Dordrecht.
Kraus, K. (1964). General quantum field theories and strict locality,Zeitschrift für Physik,181, 1.
Rédei, M. (1995). Logical independence in quantum logic,Foundations of Physics,25, 411.
Roos, H. (1970). Independence of local algebras in quantum field theory,Communications in Mathematical Physics,16, 238.
Summers, S. J. (1990). On the independence of local algebras in quantum field theory,Reviews in Mathematical Physics,2, 201.
Takesaki, M. (1979).Theory of Operator Algebras, I, Springer-Verlag, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rédei, M. Logically independent von Neumann lattices. Int J Theor Phys 34, 1711–1718 (1995). https://doi.org/10.1007/BF00676284
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00676284