# zbMATH — the first resource for mathematics

Automorphisms and symmetries of quantum logics. (English) Zbl 0697.03034
Let L be a quantum logic and let A(L) (S(L),C(L), resp.) denote the group of all automorphisms of L (the group of all symmetries of L, the group of all automorphisms of L which preserve the convex combinations of states, resp.). Then we have natural embeddings i: S(L)$$\to A(L)$$ and e: S(L)$$\to C(L)$$. Thus, with these embeddings, the triple S(L), A(L) and C(L) forms an amalgam of groups. The author raises the following interesting and highly nontrivial question: Given an amalgam of groups, can we always find its “quantum logic” representation as described above? She proves that we can. She also ensures that the representation enjoys properties significant within quantum theories. The proof she provides uses an advanced graph-theory technique.
Reviewer: P.Ptak

##### MSC:
 03G12 Quantum logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 20B27 Infinite automorphism groups
Full Text:
##### References:
 [1] Cook, T. A., and Rüttiman, G. T. (1985). Symmetries on quantum logics,Reports on Mathematical Physics,21, 121–126. · Zbl 0591.03047 [2] Greechie, R. J. (1970). On generating pathological orthomodular structures, Technical Report No. 13, Kansas State University. [3] Kallus, M., and Trnková, V. (1987). Symmetries and retracts of quantum logics,International Journal of Theoretical Physics,26, 1–9. · Zbl 0626.06013 [4] Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, London. · Zbl 0512.06011 [5] Kalmbach, G. (1984). Automorphism groups of orthomodular lattices,Bulletin of the Australian Mathematical Society,29, 309–313. · Zbl 0538.06009 [6] Kurosh, A. G. (1957).Group Theory, State Publishing House of Technical Literature, Moscow. · Zbl 0061.02101 [7] Mackey, G. W. (1963).The Mathematical Foundation of Quantum Mechanics, Benjamin, New York. · Zbl 0114.44002 [8] Pulmannová, S. (1977). Symmetries in quantum logics,International Journal of Theoretical Physics,16, 681–688. · Zbl 0388.06007 [9] Pultr, A., and Trnková, V. (1980).Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, North-Holland, Amsterdam. · Zbl 0418.18004 [10] Sabidussi, G. (1957). Graphs with given group and given graph theoretical properties,Canadian Journal of Mathematics,9, 515–525. · Zbl 0079.39202 [11] Trnková, V. (1986). Simultaneous representation in discrete structures,Commentationes Mathematicae Universitatis Carolinae,27, 633–649. · Zbl 0645.05041 [12] Trnková, V. (1988). Symmetries and state automorphisms of quantum logics, inProceedings of the First Winter School on Measure Theory, Liptovský Jan, January 10–15, 1988, A. Dvurečenskij and S. Pulmannová, eds., Czechoslovakia.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.