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Register a new userQuantum Set Theory: Transfer Principle and De Morgans Laws
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Authors
Masanao Ozawa
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In quantum logic, introduced by Birkhoff and von Neumann, De Morgans Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeutis quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeutis quantum set theory has a problem in that De Morgans Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for bounded quantifiers that satisfies De Morgans Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgans Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgans Laws.
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In 1981, Takeuti introduced set theory based on quantum logic by constructing a model analogous to Boolean-valued models for Boolean logic. He defined the quantum logical truth value for every sentence of set theory. He showed that equality axioms do not hold, while axioms of ZFC set theory hold if appropriately modified with the notion of commutators. Here, we consider the problem in Takeutis quantum set theory that De Morgans laws do not hold for bounded quantifiers. We construct a counter-example to De Morgans laws for bounded quantifiers in Takeutis quantum set theory. We redefine the truth value for the membership relation and bounded existential quantification to ensure that De Morgans laws hold. Then, we show that the truth value of every theorem of ZFC set theory is lower bounded by the commutator of constants therein as quantum transfer principle.
We show that the classifying topos for the theory of fields does not satisfy De Morgans law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory based on quantum logic, as the first step to construct quantum mathematics, a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeutis mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.
The $omega$-power of a finitary language L over a finite alphabet $Sigma$ is the language of infinite words over $Sigma$ defined by L $infty$ := {w 0 w 1. .. $in$ $Sigma$ $omega$ | $forall$i $in$ $omega$ w i $in$ L}. The $omega$-powers appear very naturally in Theoretical Computer Science in the characterization of several classes of languages of infinite words accepted by various kinds of automata, like B{u}chi automata or B{u}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $omega$-powers.
The transfer of a quantum state between distant nodes in two-dimensional networks, is considered. The fidelity of state transfer is calculated as a function of the number of interactions in networks that are described by regular graphs. It is shown that perfect state transfer is achieved in a network of size N, whose structure is that of a N/2-cross polytope graph, if N is a multiple of 4. The result is reminiscent of the Babinet principle of classical optics. A quantum Babinet principle is derived, which allows for the identification of complementary graphs leading to the same fidelity of state transfer, in analogy with complementary screens providing identical diffraction patterns.
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