Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors
Friedman, Yaakov
Peralta Pereira, Antonio Miguel
Wigner theorem
Uhlhorn theorem
Partial order set of tripotents
Spin factor
Cartan factors
Atomic JBW∗-triple
Triple isomorphisms
Second author was partially supported by MCIN / AEI / 10. 13039 / 501100011033
/ FEDER “Una manera de hacer Europa” project no. PGC2018-093332-B-I00, Junta de Andalucía grants
FQM375, A-FQM-242-UGR18 and PY20 00255, and by the IMAG–María de Maeztu grant CEX2020-
001105-M / AEI / 10.13039 / 501100011033.
We deeply appreciate the useful and detailed suggestions and comments received from the anonymous
referees of this paper. Special thanks are given to one of the reviewers for pointing out a mistake in one of
the statements of an earlier version.
There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states P—i.e., the set of one-dimensional projections on a complex Hilbert space H– and the orthomodular lattice L of closed subspaces of H. These six groups are isomorphic when the dimension of H is ≥3. The latter hypothesis is absolutely necessary in this identification. For example, the automorphisms group of all bijections preserving orthogonality and the order on L identifies with the bijections on P preserving transition probabilities only if dim(H)≥3. Despite of the difficulties caused by M2(C), rank two algebras are used for quantum mechanics description of the spin state of spin-12 particles. However, 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 U(M) and U(N) denote the set of all tripotents in M and N, respectively. We show that each bijection Φ:U(M)→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 Molnár to the wider setting of atomic JBW∗-triples not containing rank–one Cartan factors, and provides new models to present quantum behavior.
2022-02-04T09:42:49Z
2022-02-04T09:42:49Z
2022-01-20
info:eu-repo/semantics/article
Friedman, Y., Peralta, A.M. Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors. Anal.Math.Phys. 12, 37 (2022). [https://doi.org/10.1007/s13324-021-00644-8]
http://hdl.handle.net/10481/72662
10.1007/s13324-021-00644-8
eng
http://creativecommons.org/licenses/by/3.0/es/
info:eu-repo/semantics/openAccess
Atribución 3.0 España
Springer Nature