scholarly journals Transfer principle in quantum set theory

2007 ◽  
Vol 72 (2) ◽  
pp. 625-648 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
Von Neumann Algebra ◽  
Transfer Principle ◽  
Real Numbers ◽  
Von Neumann ◽  
Equality Relation ◽  
Adjoint Operators ◽  
Transfer Theorems

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

scholarly journals From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 397
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
World View ◽  
Open Problems ◽  
Boolean Valued Analysis ◽  
Mathematical World ◽  
Distributive Law ◽  
And Algebra ◽  
Adjoint Operators

Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the 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 that is 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 Takeuti’s 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.

scholarly journals On spectra and Brown's spectral measures of elements in free products of matrix algebras

2008 ◽  
Vol 103 (1) ◽  
pp. 77
Author(s):  
Junsheng Fang ◽  
Don Hadwin ◽  
Xiujuan Ma
Keyword(s):  
Free Product ◽  
Von Neumann Algebra ◽  
Single Point ◽  
Diagonal Operator ◽  
Spectral Measures ◽  
Free Products ◽  
Hyperinvariant Subspace ◽  
Von Neumann ◽  
Matrix Algebras ◽  
Adjoint Operators

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$, the reduced free product von Neumann algebra of $M_2(\mathsf {C})$ with $M_2(\mathsf {C})$. Examples include $AB$ and $A+B$, where $A$ and $B$ are matrices in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*1$ and $1*(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})$, respectively. We prove that $AB$ is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if $\mathrm{Tr}(A)=\mathrm{Tr}(B)=0$. We show that if $X=AB$ or $X=A+B$ and $A,B$ are not scalar matrices, then the Brown measure of $X$ is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if $X=AB$ or $X=A+B$ and $X\neq \lambda 1$, then $X$ has a nontrivial hyperinvariant subspace affiliated with $(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$.

Popper and the Quantum Theory

1995 ◽  
Vol 39 ◽  
pp. 163-176
Author(s):  
Michael Redhead
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Scientific Discovery ◽  
Von Neumann ◽  
Interpretation Of Probability ◽  
Propensity Interpretation

Popper wrote extensively on the quantum theory. In Logic der Forschung (LSD) he devoted a whole chapter to the topic, while the whole of Volume 3 of the Postscript to the Logic of Scientific Discovery is devoted to the quantum theory. This volume entitled Quantum Theory and the Schism in Physics (QTSP) incorporated a famous earlier essay, ‘Quantum Mechanics without “the Observer”’ (QM). In addition Popper's development of the propensity interpretation of probability was much influenced by his views on the role of probability in quantum theory, and he also wrote an insightful critique of the 1936 paper of Birkhoff and von Neumann on nondistributive quantum logic (BNIQM).

The center of an orthologic

1972 ◽  
Vol 37 (4) ◽  
pp. 641-645 ◽  
Author(s):  
Barbara Jeffcott
Keyword(s):  
Quantum Mechanics ◽  
Boolean Algebra ◽  
Quantum Logic ◽  
Decisive Role ◽  
Orthomodular Poset ◽  
Fundamental Relation ◽  
Von Neumann ◽  
Modular Ortholattice ◽  
Probability And Statistics ◽  
Kolmogorov Theory

Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable.Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.

TRACE JENSEN INEQUALITY FOR SELF-ADJOINT OPERATORS IN SEMI-FINITE VON NEUMANN ALGEBRAS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350075
Author(s):  
HIDEKI KOSAKI
Keyword(s):  
Convex Function ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Jensen Inequality ◽  
Measurable Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Measurable Operators ◽  
Adjoint Operators ◽  
Finite Von Neumann Algebras

Let [Formula: see text] be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace τ, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality in our previous work states τ(f(a*xa)) ≤ τ(a*f(x)a) (as long as the both sides are well-defined) for a contraction [Formula: see text] and a semi-bounded τ-measurable operator x. Validity of this inequality for (not necessarily semi-bounded) self-adjoint τ-measurable operators is investigated.

scholarly journals A quantum route to the classical Lagrangian formalism

2021 ◽  
pp. 2150091
Author(s):  
F. M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort ◽  
G. Marmo ◽  
L. Schiavone ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Smooth Manifold ◽  
Von Neumann Algebra ◽  
Fundamental Question ◽  
Lagrangian Formalism ◽  
Classical Dynamics ◽  
New Approach ◽  
Von Neumann ◽  
Dynamics Of Particles

Using the recently developed groupoidal description of Schwinger’s picture of Quantum Mechanics, a new approach to Dirac’s fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function [Formula: see text] on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.

scholarly journals Causality in Schwinger’s Picture of Quantum Mechanics

Entropy ◽  
2022 ◽  
Vol 24 (1) ◽  
pp. 75
Author(s):  
Florio M. Ciaglia ◽  
Fabio Di Di Cosmo ◽  
Alberto Ibort ◽  
Giuseppe Marmo ◽  
Luca Schiavone ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Quantum System ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Quantum Mechanical ◽  
Von Neumann ◽  
Triangular Operator ◽  
Quantum Mechanical Systems ◽  
Causal Structures ◽  
Incidence Theorem

This paper begins the study of the relation between causality and quantum mechanics, taking advantage of the groupoidal description of quantum mechanical systems inspired by Schwinger’s picture of quantum mechanics. After identifying causal structures on groupoids with a particular class of subcategories, called causal categories accordingly, it will be shown that causal structures can be recovered from a particular class of non-selfadjoint class of algebras, known as triangular operator algebras, contained in the von Neumann algebra of the groupoid of the quantum system. As a consequence of this, Sorkin’s incidence theorem will be proved and some illustrative examples will be discussed.

Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers

2001 ◽  
Vol 66 (3) ◽  
pp. 1231-1258 ◽  
Author(s):  
Philip Ehrlich
Keyword(s):  
Set Theory ◽  
Number Fields ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Ordered Group ◽  
Von Neumann ◽  
Ordered Field ◽  
Number Systems ◽  
Ordered Fields

Introduction. In his monograph On Numbers and Games [7], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including ω, ω, /2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered “number” fields—be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice, henceforth NBG [cf. 21, Ch. 4], it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique homogeneous universal Archimedean ordered field, No is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [10], [12], [13].However, in addition to its distinguished structure as an ordered field, No has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or simplicity hierarchy, as we have called it [15], depends upon No's (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with No's structure as an ordered group and an ordered field, respectively, it being understood that x is simpler than y just in case x is a predecessor of y in the tree.

scholarly journals Quantum logic as motivated by quantum computing

2005 ◽  
Vol 70 (2) ◽  
pp. 353-359 ◽  
Author(s):  
J. Michael Dunn ◽  
Tobias J. Hagge ◽  
Lawrence S. Moss ◽  
Zhenghan Wang
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Computer Science ◽  
Quantum Logic ◽  
Theoretical Computer Science ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Definition Of

§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection.The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.

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