On the construction of coordinates for non-desarguesian complemented modular lattices. I

Fryer, K.D.; Halperin, Israel

doi:10.1016/s1385-7258(58)50020-1

Cited by 6 publications

(2 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To make progress in this direction one has to either follow our line of proof and exclude the possibility of non-Desarguesian projective geometries (cf. [23,24] in the present context), or abandon the use of Hilbert lattices and develop a spectral theory of well-behaved transition probability spaces, analogous to the spectral theory of compact convex sets of Alfsen and Shultz [6,7]. Despite considerable efforts in both directions the author has failed to remove the restriction.…”

Section: Poisson Spaces With a Transition Probabilitymentioning

confidence: 99%

“…For this to apply, the dimension of L(P) as a lattice [31] (which is easily seen to coincide with the dimension of P as a transition probability space) must be ≥ 4, so that we must now assume that dim(P) = 3; the case dim(P) = 2 is covered directly by Axiom 2. (The fact that dimension 3 is excluded is caused by the existence of so-called non-Desarguesian projective geometries; see [24] for a certain analogue of the co-ordinatization procedure in that case.) Accordingly, for dim(P) = 3 there exists a vector space V over a division ring D (both unique up to isomorphism), equipped with an anisotropic Hermitian form θ (defined relative to an involution of D, and unique up to scaling), such that L(P) ≃ L(V ) as orthocomplemented lattices.…”

Section: Proof Of Propositionmentioning

confidence: 99%

See 1 more Smart Citation

Poisson Spaces with a Transition Probability

Landsman

1997

Rev. Math. Phys.

View full text Add to dashboard Cite

The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p : P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context).In classical mechanics, where p(ρ, σ) = δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of P as a transition probability space coincide with the symplectic leaves of P as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant).Motivated by E.M. Alfsen, H.

show abstract

Section: Poisson Spaces With a Transition Probabilitymentioning

confidence: 99%

Section: Proof Of Propositionmentioning

confidence: 99%