The Logic of Quantum Mechanics

Beltrametti, Enrico G.; Cassinelli, Gianni; Shimony, Abner

doi:10.1017/cbo9781107340725

Cited by 262 publications

(239 citation statements)

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“…Ethnography thus asks which transform(s) α have a fixed point that describes certain empirically observable and measurable population properties of a descent sequence following a claimed rule or set of rules. Comparison to "superposition" is thus present, similar to the idea of "mixed pure" systems as discussed by [6]: the combinations tr depict a single system simultaneously using rules with different structural numbers, resulting in physically testable predictions that are proven correct in [2,3] and elsewhere. Social sciences such as demography (which studies eigenstates of age-structured birth and death matrices [13]) nor economics (which uses formalism of thermodynamics [12]) are unable to make such predictions.…”

Section: Commentmentioning

confidence: 96%

Some Properties of Transforms in Cultural Theory

Ballonoff¹

2010

Int J Theor Phys

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It is shown that, in certain circumstances, systems of cultural rules may be represented by doubly stochastic matrices denoted , called "possibility transforms," and by certain real valued "possibility densities" π = (π 1 , π 2 , . . . , π n ) with inner product π, π = i π 2 i = 1. We may characterize a certain problem of ethnographic or ethological description as a problem of prediction, in which observations are predicted by properties of fixed points of transforms of "pure systems", or by properties of convex combinations of such "pure systems". Other relationships to quantum methods are noted.Background 1 This paper follows and adopts background from [4, 5] which we summarize here. We assume a finite non-empty set P whose members are called individuals, and a finite non-empty set R whose members are called rules. An evolutionary structure is a quintuple S := (P, R, D, B, M) where D, B, and M are binary relations on P, satisfying these four axioms: (1) D is totally non-symmetric and transitive; (2) M is symmetric; (3) if bDc and there exists no d ∈ P, d = b, c for which bDd and dDc, then we write cP b, and require bBc iff for b, c, d ∈ P, dP b and dP c; (4) |bM| ≤ 2. A rule R ∈ R, R = ∅, is a statement concerning the relationships between the D, B, and/or M, which does not violate those four axioms. A family of subsets G = {G t |G t ⊆ P, t ∈ T} for t ∈ T a set of consecutive nonnegative integers starting with 0, is called a descent sequence of S. G t is called a generation of S, in case, for all G t ∈ G each cell bB occurs in only one generation, each subset bM occurs in only one generation, and for t > 0 when G t ∈ G, b ∈ G t , and cP b, then c ∈ G t−1 (that is, the set G t contains all of the immediate descendants of individuals in G t−1 ). We assume a "Darwinian Sequences axiom" which says that all descent sequences of a given evolutionary structure can be traced back through a chain of descent in an unbroken series of

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Section: Commentmentioning

confidence: 96%

Some Properties of Transforms in Cultural Theory

Ballonoff¹

2010

Int J Theor Phys

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“…Like all other attempts to do so (cf. [43,44,49,37,14]), the axioms appear to be contingent. This is particularly true of Axiom AHS2 and of our Axiom 2, which lie at the heart of quantum mechanics.…”

Section: Poisson Spaces With a Transition Probabilitymentioning

confidence: 98%

“…Note that the dimension of L(P) as a lattice equals the dimension [31] of P as a transition probability space. It is orthocomplemented by ⊥, and is easily shown to be a complete atomic orthomodular lattice [59,13,14] (cf. [31] for the general theory of orthomodular lattices).…”

Section: Transition Probabilities On Pure State Spacesmentioning

confidence: 99%

“…Let us assume, therefore, that σ lies neither in Q nor in Q ⊥ . Define ϕ Q (σ) by the following procedure: lift σ to a unit vector Σ in H, project Σ onto the subspace defined by Q, normalize the resulting vector to unity, and project back to PH (this is a Sasaki projection in the sense of lattice theory [14,31]). In the Hilbert space case relevant to us, the transition probabilities satisfy…”

Section: Spectral Theoremmentioning

confidence: 99%

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Poisson Spaces with a Transition Probability

Landsman

1997

Rev. Math. Phys.

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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.

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“…Ihe following is the rank > 1 version of[6, Propoaition (3.5)] (compare also with [3, Corollary (1.1)] and [4, Lemnía (3.1)]). Leí £ be a le-jet ample vector buvidie o» X. Leí ---, z~be 1 distincí poi»ts o» X a»d leí a 1,..., at be 1 positive ¿viteyers.…”

mentioning

confidence: 99%

On generation of jets for vector bundles

1999

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Wc introduce aud study the le-jet ampleneas aud the le-jet spannedness for a vector bundle, E, on a projective manifold. We obtain different characterizationa of projective apace in ternís of such positivity properties for E. Wc compare dic 1-jet ampleneas with different notions of very ampleneas in the literature.2. given a point x E IP(S) and a nonzero tangent vector r,, at x, then the differential dt(r~) ja not zero.Let z, y be two distinct pointa of IP(S) with z' : p(z) # u' = and note that by the 1-jet amplenesa of 5 the map H0(£) -> 5,,'~S,,,, ia onto. Thus we can choose sectiona s~, s 2 of H 0(S) auch that, regarding S 1, 82 Rs sectiona of fi <>(¿e) under the isomorphiam H 0(£)~fi 0(p,¿e), one has s,(z) = O~s 2(z) and s2(y) = O~s,(y). To see this recalí

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The Logic of Quantum Mechanics

Cited by 262 publications

References 0 publications

Some Properties of Transforms in Cultural Theory

Some Properties of Transforms in Cultural Theory

Poisson Spaces with a Transition Probability

On generation of jets for vector bundles

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