Mathematical Foundations of Quantum Mechanics 137Sol`er^13 , Holland^14 and Aerts–van Steirteghem^15 have found sufficient hypotheses,
in terms of the existence of infinite orthogonal systems, for this to happen. Under
these hypotheses, if the ring isRorC, we obtain precisely the lattice of orthogo-
nal projectors of the separable Hilbert space. In the case ofH, one gets a similar
generalized structure. In all these arguments, the assumption of irreducibility is
not really crucial: if property (v) fails, the lattice can be split into irreducible
sublattices[20; 15]. Physically-speaking this situation is natural in the presence
ofsuperselection rules, of which more will be explained soon.
It is worth stressing that the covering property in Theorem 2.3.13 is a crucial
property. Indeed there are other lattices relevant in physics verifying all the re-
maining properties in the aforementioned theorem. Remarkably the family of the
so-calledcausally closed setsin a general spacetime satisfies all the said properties
but the covering one^16. This obstruction prevents one from endowing a spacetime
with a natural (generalized) Hilbert space structure,while it suggests some ideas
towards a formulation of quantum gravity.
2.3.4 States as measures onL(H): Gleason’s theorem
Let us introduce an important family of operators. This family will play a decisive
rˆole in the issue concerning a possible justification of the fact that quantum states
are elements of the projective spacePH.
2.3.4.1 Trace class operators
Definition 2.3.15. IfHis a complex Hilbert space,B 1 (H)⊂B(H)denotes the
set oftrace classornuclearoperators, i.e. the operatorsT∈B(H)satisfying
∑z∈N〈z,|T|z〉<+∞ (2.75)for some Hilbertian basisN∈Hand where|T|:=
√
T†T defined via functional
calculus.
(^13) Soler, M. P.: Characterization of Hilbert spaces by orthomodular spaces.Communications in Algebra, 23 , 219-243 (1995). (^14) Holland, S.S.: Orthomodularity in infinite dimensions; a theorem of M. Soler.Bulletin of the
American Mathematical Society, 32 , 205-234, (1995).
(^15) Aerts, D., van Steirteghem B.: Quantum Axiomatics and a theorem of M.P. Sol ́er.Interna-
tional Journal of Theoretical Physics 16. 39 , 497-502, (2000).
See H. Casini,The logic of causally closed spacetime subsets, Class. Quant. Grav. 19 , 2002,
6389-6404.