scholarly journals Random World and Quantum Mechanics

2021 ◽  
Author(s):  
Jerzy Król ◽  
Krzysztof Bielas ◽  
Torsten Asselmeyer-Maluga
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Infinite Sequence ◽  
Quantum Limit ◽  
Random Real ◽  
Practical Applications ◽  
Infinite Dimensional ◽  
Entire Process ◽  
The Universe ◽  
Infinite Sequences

Abstract Quantum mechanics (QM) predicts probabilities on the fundamentallevel which are, via Born probability law, connected to the formal randomnessof infinite sequences of QM outcomes. Recently it has been shown thatQM is algorithmic 1-random in the sense of Martin-L¨of. We extend this resultand demonstrate that QM is algorithmic ω-random and generic, precisely asdescribed by the ’miniaturisation’ of the Solovay forcing to arithmetic. Thisis extended further to the result that QM becomes Zermelo–Fraenkel Solovayrandom on infinite-dimensional Hilbert spaces. Moreover, it is more likely thatthere exists a standard transitive ZFC model M, where QM is expressed in reality,than in the universe V of sets. Then every generic quantum measurementadds to M the infinite sequence, i.e. random real r ∈ 2ω, and the model undergoesrandom forcing extensions M[r]. The entire process of forcing becomesthe structural ingredient of QM and parallels similar constructions applied tospacetime in the quantum limit, therefore showing the structural resemblanceof both in this limit. We discuss several questions regarding measurability andpossible practical applications of the extended Solovay randomness of QM.The method applied is the formalization based on models of ZFC; however,this is particularly well-suited technique to recognising randomness questionsof QM. When one works in a constant model of ZFC or in axiomatic ZFCitself, the issues considered here remain hidden to a great extent.

scholarly journals Random World and Quantum Mechanics

2021 ◽  
Author(s):  
Jerzy Król ◽  
Krzysztof Bielas ◽  
Torsten Asselmeyer-Maluga
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Infinite Sequence ◽  
Quantum Limit ◽  
Random Real ◽  
Practical Applications ◽  
Infinite Dimensional ◽  
Structural Resemblance ◽  
Random Forcing ◽  
The Universe

Abstract Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω, to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

scholarly journals Random World and Quantum Mechanics

2021 ◽  
Author(s):  
Jerzy Król ◽  
Krzysztof Bielas ◽  
Torsten Asselmeyer-Maluga
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Infinite Sequence ◽  
Quantum Limit ◽  
Random Real ◽  
Practical Applications ◽  
Infinite Dimensional ◽  
Structural Resemblance ◽  
Random Forcing ◽  
The Universe

Abstract Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω , to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

scholarly journals AN INFINITE DIMENSIONAL VERSION OF THE SCHUR CONVEXITY PROPERTY AND APPLICATIONS

2007 ◽  
Vol 05 (02) ◽  
pp. 123-136 ◽  
Author(s):  
CLAUDE VALLÉE ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Symmetric Matrix ◽  
Liouville Problem ◽  
Convexity Property ◽  
Infinite Dimensional ◽  
Selfadjoint Operators ◽  
Dimensional Version ◽  
Sturm Liouville ◽  
Schur Convexity

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the present paper are illustrated by several examples from mechanics or quantum mechanics, including the Sturm–Liouville problem, the Schrödinger equation, and the harmonic oscillator.

scholarly journals TOWARDS A DEFINITION OF QUANTUM INTEGRABILITY

2009 ◽  
Vol 06 (01) ◽  
pp. 129-172 ◽  
Author(s):  
JESÚS CLEMENTE-GALLARDO ◽  
GIUSEPPE MARMO
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Complete Integrability ◽  
Infinite Dimensional ◽  
Quantum Integrability ◽  
Classical Systems ◽  
Definition Of ◽  
Geometrical Formulation ◽  
Quantum Framework

We briefly review the most relevant aspects of complete integrability for classical systems and identify those aspects which should be present in a definition of quantum integrability. We show that a naive extension of classical concepts to the quantum framework would not work because all infinite dimensional Hilbert spaces are unitarilly isomorphic and, as a consequence, it would not be easy to define degrees of freedom. We argue that a geometrical formulation of quantum mechanics might provide a way out.

scholarly journals Dagger linear logic for categorical quantum mechanics

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Robin Cockett ◽  
Cole Comfort ◽  
Priyaa Srinivasan
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Linear Logic ◽  
Infinite Dimensional ◽  
Graphical Calculus ◽  
Finite Dimensional ◽  
Categorical Quantum Mechanics ◽  
Definition Of ◽  
Categorical Semantics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

scholarly journals Generalized Probabilistic Theories in a New Light

2021 ◽  
Author(s):  
Raed Shaiia
Keyword(s):  
Quantum Mechanics ◽  
Measurement Problem ◽  
Quantum Mechanical ◽  
Infinite Dimensional ◽  
Classical Systems ◽  
Generalized Probabilistic Theories ◽  
Deterministic Description ◽  
New Formulation ◽  
The Universe ◽  
Probabilistic Theories

Abstract In this paper we will present a modified formulation of generalized probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, and give the guidelines to how to extend this work to infinite dimensional Hilbert spaces. Moreover, this new formulation which we will call extended operational-probabilistic theories, applies not only to quantum systems, but also equally well to classical systems, without violating Bell’s theorem, and at the same time solves the measurement problem. This is why we will see that the question of why our universe is quantum mechanical rather than classical is misplaced. The only difference that exists between a classical universe and a quantum mechanical one lies merely in which observables are compatible and which are not. Besides, this extended probability theory which we present in this paper shows that it is non-determinacy, or to be more precise, the non-deterministic description of the universe, that makes the laws of physics the way they are. In addition, this paper shows us that what used to be considered as purely classical systems and to be treated that way are in fact able to be manipulated according to the rules of quantum mechanics –with this new understanding of these rules- and that there is still a possibility that there might be a deterministic level from which our universe emerges, which if understood correctly, may open the door wide to applications in areas such as quantum computing. In addition to all that, this paper shows that without the use of complex vector spaces, we cannot have any kind of continuous evolution of the states of any system.

scholarly journals QUANTUM MECHANICS AS A GAUGE THEORY OF METAPLECTIC SPINOR FIELDS

1998 ◽  
Vol 13 (22) ◽  
pp. 3835-3883 ◽  
Author(s):  
M. REUTER
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Phase Space ◽  
Hilbert Spaces ◽  
Canonical Quantization ◽  
Spinor Representation ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Spinor Fields ◽  
Matter Fields

A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.

scholarly journals IMPLICATIONS OF A QUANTUM MECHANICAL TREATMENT OF THE UNIVERSE

1998 ◽  
Vol 13 (05) ◽  
pp. 347-351 ◽  
Author(s):  
MURAT ÖZER
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Early Universe ◽  
Mechanical Treatment ◽  
Correspondence Principle ◽  
Scale Factor ◽  
Quantum Mechanical ◽  
Quantum Mechanical Treatment ◽  
The Standard Model ◽  
The Universe

We attempt to treat the very early Universe according to quantum mechanics. Identifying the scale factor of the Universe with the width of the wave packet associated with it, we show that there cannot be an initial singularity and that the Universe expands. Invoking the correspondence principle, we obtain the scale factor of the Universe and demonstrate that the causality problem of the standard model is solved.

Channel Covariance Conversion and Modelling Using Infinite Dimensional Hilbert Spaces

2021 ◽  
pp. 1-1
Author(s):  
Lorenzo Miretti ◽  
Renato Calvacante ◽  
Slawomir Stanczak
Keyword(s):  
Hilbert Spaces ◽  
Infinite Dimensional

scholarly journals From the Universe to Subsystems: Why Quantum Mechanics Appears More Stochastic than Classical Mechanics

2016 ◽  
Vol 15 (03) ◽  
pp. 1640002 ◽  
Author(s):  
Andrea Oldofredi ◽  
Dustin Lazarovici ◽  
Dirk-André Deckert ◽  
Michael Esfeld
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Bohmian Quantum Mechanics ◽  
The Universe

By means of the examples of classical and Bohmian quantum mechanics, we illustrate the well-known ideas of Boltzmann as to how one gets from laws defined for the universe as a whole the dynamical relations describing the evolution of subsystems. We explain how probabilities enter into this process, what quantum and classical probabilities have in common and where exactly their difference lies.

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