Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

Abstract

There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states $\mathbf{P}$ -- i.e., the set of one-dimensional projections on a complex Hilbert space $H$ -- and the orthomodular lattice $\mathbf{L}$ of closed subspaces of $H$. These six groups are isomorphic when the dimension of $H$ is $\geq 3$. Despite of the difficulties caused by $M_2(\mathbb{C})$, rank two algebras are used for quantum mechanics description of the spin state of spin-$\frac12$ particles, there is a counterexample for Uhlhorn's version of Wigner's theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, let $M$ and $N$ be two atomic JBW$^*$-triples not containing rank-one Cartan factors, and let $\mathcal{U} (M)$ and $\mathcal{U} (N)$ denote the set of all tripotents in $M$ and $N$, respectively. We show that each bijection $Φ: \mathcal{U} (M)\to \mathcal{U} (N)$, preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Moln{á}r to the wider setting of atomic JBW$^*$-triples not containing rank-one Cartan factors, and provides new models to present quantum behavior.