Crítica, Revista Hispanoamericana de Filosofía, Volume 11, number 31, April 1979
Posibilidades de generalización de las lógicas cuánticas
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Carlos Lungarzo
Universidade Estadual de Campinas, Brasil

Abstract: The complete and atomic orthomodular lattice structure which is formed by the projectors (or equivalently, by the closed subspaces) of a complex and separable Hilbert space will be indicated in this paper by LG (H), where H is the Hilbert space. (We take operations and distinct elements for granted.)
LG (H) is, classically considered, the “typical” logic of quantum mechanics, in the following sense: if H is the space associated with physical system S, then H’s rays, the ultramaximal operators, etc., represent typical properties of S, such as observable states and magnitudes.
LG (H) is simply an element in model sets of more general theories such as the orthomodular lattice set, which we formalize in § 2, using a second-order logic. This generalization, which we call “formal” because it consists merely of giving more permissive axioms that facilitate the apparition of new structures other than LG (H), can yet be extended. In § 2 we use infinitarian logics, where a numerable number of conjunctions and quantificators is permitted, to express the theory of orthomodular partially ordered sets (posets). That formulation uses essentially the logic called (omega-1, omega-1), and a conjecture is raised about proving it cannot be realized within a weaker logic. The way to prove this, it is suggested, should be analogous to the way the indefinability of well-ordering is proved.
Such generalizations are analogous to those obtained when we pass, e.g., from the field of real numbers to any commutative, Archimedian, etc., field; then to a commutative ring with and unity, and so on. They are, therefore, formal, in the sense that (1) they do not “say” much about the initial structure which motivated the generalization and that (2) in generalizing, we allow the “entrance” of new structures that have little to do with the original motivation. For instance, in the theory of orthomodular lattices, besides Hilbert-space logics (or those based on other topological vectorial spaces), new models appear, such as finite or orthomodular lattices, for which the name “quantum logic” has little sense.
In § 3 we study the modifications, and eventual generalizations, of the basic structure that generates quantum logic, i.e., the infinite-dimensional separable complex Hilbert space structure.
A first way seems to be the generalizing of the field, and it is analyzed in § 3.1. In principle, it seems natural to admit that the field must contain some copy of the field of real numbers, or have at least an expressive force as great as theirs. This is why the use of real fields —and for a greater certainty, of complex fields— in the H spaces of quantum mechanics seems to be fully justified. They study of quaternions, carried on by Emch and others, but not clearly completed yet, could lead to an equivalent quantum mechanics, notwithstanding the non-commutativity of this field. Anyway, real, complex and quaternions seem to be only fields with a “natural” right to sustentate these spaces. The incompleteness of rational fields, and their undesirable properties as to dimension and measurability, cause them to be rejected.
Other proposals are studied. Leaving aside Hensel’s bodies, that seem to be wholly different in nature, we are left with p-adics Qp and Galois’ fields as the only reasonable candidates.
Know algebraic results are then invoked, according to which an extension or completion of these bodies would not lead to the desired results. Quite simply, we would not, save on trivial instances, have any logic at all.
It is a well-known fact that by completing p-adic fields (let K be such a completion) we arrive at space (V, K) upon that field, which does not allow to define hermitian forms for dimensions greater than 4. Actually, no involutory anti-automorphism of the field (such as complex conjugation) permits to associate any quadratic form for a dimension greater than or equal to 5.
Things are even worse in the case of Galois’ fields where limitative results begin to appear already in dimensions greater than 2.
In § 3.2 we choose to accept the complex numbers body C as a canonic body, and see whether it is possible to generalize space into a class that need not be Hilbert’s. There are various possibilities: to consider Banach spaces, or else more or less general local convex spaces, and particularly, to focus analysis on Mackey spaces.
For Banach spaces there appear Maczynski’s limitative results; for Mackey spaces, “analogous” results from Wilbur. (Cf. references at the end of the essay.)
As a matter of fact, only infinite-dimensional spaces can be of any interest. In this case, if a Mackey space has an “acceptable” topology, i.e., a stronger one than the so-called weak convergence topology, and its LG (M) is really a logic, then M is once again a Hilbert space.
Several of these results to the conclusion that we can hardly obtain a different logic (a more general or anyway more informative one) by taking the class LG (E) of some non-Hilbert space E. Rayski’s proposition about substituting Schwartz space for Hilbert space is also considered, but although such a substitution would modify actual work “within” the space in the interests of formalization, it is natural to expect the apparition of a logic LG (S), equivalent to former logics.
Maybe the part special logics play in quantum mechanics will be entirely unnecessary when the axiomatizing of a physical system can be done in terms of some kind of spaces that leaves no doubt about its usefulness.

[C.L.]
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