Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Representation of symmetry transformations on the sets of tripotents of
  spin and Cartan factors   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 $\Phi: \mathcal{U} (M)\to
\mathcal{U} (N)$, 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{\'a}r to the wider setting of atomic JBW$^*$-triples
not containing rank-one Cartan factors, and provides new models to present
quantum behavior.

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The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol