So far to date, much has been written about E. Wigner’s and U. Uhlhron’s theorems and their importance for physics and mathematics. For the sake of conciseness, let us go straight to some of the starring results. There are six mathematical models employed in quantum mechanics, among them we have: 1. The C-algebra B(H) of bounded operators; 2. The Jordan algebra B(H)sa of bounded self-adjoint operators; 3. The orthomodular lattice L of closed subspaces of H, equivalently, the lattice of all projections in B(H), where H is a complex Hilbert space.
The natural automorphisms of these mathematical models (i.e., the bijections on these sets preserving the corresponding relevant structure: associative product and involution, Jordan product, and orthogonality and order between subspaces or projections) represent the symmetry groups of quantum mechanics and are endowed with natural topologies induced by the probabilistic structure of quantum mechanics. It is known that these symmetry groups are all isomorphic when dim(H) ≥ 3. The last restriction exclude rank two, where there are no more than two orthogonal projections. This equivalence can be seen as the celebrated Wigner unitary-antiunitary theorem.
By replacing the set of projections P(H) by the wider set PI(H) = U(B(H)), of all partial isometries on H, L. Molnár proved the following result: Let be a complex Hilbert space with dim(H) ≥ 3. Suppose that Φ: U(B(H)) → U(B(H)) is a bijective transformation which preserves the natural partial ordering and the orthogonality between partial isometries in both directions. If Φ is continuous (in the operator norm) at a single element of U(B(H)) different from 0, then Φ extends to a real linear triple isomorphism.
During this talk we shall present new results, obtained in collaboration with Y. Friedman, showing that an extension of the previous results is possible in the case of a bijection between the lattices of tripotents of two Cartan factors and atomic JBW-triples non-containing rank-one Cartan factors. These new result provide new models to understand the quantum models. We shall also see how the results provide new alternatives to complement recent studies by J. Hamhalter proving that the set of partial isometries with its partial order and orthogonality relation is a complete Jordan invariant for von Neumann algebras.
This talk is based on this arXiv paper, joint work with Y. Friedman.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
