The General Mathematical Framework of Euclidean Quantum Mechanics, an Outline

1991 ◽  
pp. 123-143 ◽  
Author(s):  
Torbjörn Kolsrud ◽  
Jean-Claude Zambrini
Keyword(s):  
Quantum Mechanics ◽  
Mathematical Framework

scholarly journals QUANTUM SINGULARITIES IN STATIC SPACETIMES

2011 ◽  
Vol 20 (05) ◽  
pp. 729-743 ◽  
Author(s):  
JOÃO PAULO M. PITELLI ◽  
PATRICIO S. LETELIER
Keyword(s):  
Quantum Mechanics ◽  
Wave Operator ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Mathematical Framework ◽  
Physical Content ◽  
Dirac Equations ◽  
Quantum Particles ◽  
Klein Gordon ◽  
Quantum Singularities

We review the mathematical framework necessary to understand the physical content of quantum singularities in static spacetimes. We present many examples of classical singular spacetimes and study their singularities by using wave packets satisfying Klein–Gordon and Dirac equations. We show that in many cases the classical singularities are excluded when tested by quantum particles but unfortunately there are other cases where the singularities remain from the quantum mechanical point of view. When it is possible we also find, for spacetimes where quantum mechanics does not exclude the singularities, the boundary conditions necessary to turn the spatial portion of the wave operator to be self-adjoint and emphasize their importance to the interpretation of quantum singularities.

scholarly journals Entropic Information Theory: Formulae and Quantum Gravity, Bits from bit

2021 ◽  
Author(s):  
olivier denis
Keyword(s):  
Information Theory ◽  
General Relativity ◽  
Dark Matter ◽  
Quantum Mechanics ◽  
Dark Energy ◽  
Quantum Gravity ◽  
Planck Scale ◽  
Basic Structure ◽  
Mathematical Framework ◽  
Information Framework

We show here that entropic information is capable of unifying all aspects of the universe at all scales in a coherent and global theoretical mathematical framework materialized by entropic information framework, theory and formulas, where dark matter, dark energy and gravity are truly informationals processes and where information is code and code is what creates the process, it is itself the process. Mass, energy and movement of information are respectively dark matter, dark energy, and gravity. Here, we reconcile general relativity and quantum mechanics by introducing quantum gravity for the Planckian scale. The formulas of entropic information are expressed in natural units, physical units of measurement based only on universal constants, constants, which refer to the basic structure of the laws of physics: C and G are part of the structure of space-time in general relativity, and h captures the relationship between energy and frequency that is the basis of quantum mechanics. Here we show that entropic information formulas are able to present entropic information in various unifying aspects and introduce gravity at the Planck scale. We prove that Entropic information theory is thus building the bridge between general relativity and quantum mechanics

Qubits and the Framework of Quantum Mechanics

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Quantum Information ◽  
State Space ◽  
Quantum Physics ◽  
Complex Hilbert Space ◽  
The State ◽  
Mathematical Framework ◽  
Infinite Dimensional

In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.

scholarly journals Unification of Commutative Relativity and Noncommutative Quantum Dynamics Via Cyclic Groupoids and Spacetime Fusion Categories

2018 ◽  
Author(s):  
Arturo Tozzi ◽  
James F. Peters
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Binary Operation ◽  
Mathematical Framework ◽  
Mathematical Approach ◽  
Fusion Categories ◽  
Microscopic World ◽  
Noncommutative Quantum

The unexploited unification of general relativity and quantum physics is a painstaking issue that prevents physicists to properly understanding the whole of Nature. Here we propose a pure mathematical approach that introduces the problem in terms of group theory.  Indeed, we build a cyclic groupoid (a nonempty set with a binary operation defined on it) that encompasses both the theories as subsets, making it possible to join together two of their most dissimilar experimental results, i.e., the commutativity detectable in our macroscopic relativistic world and the noncommutativity detectable in the quantum, microscopic world.  This approach, combined with the Connes fusion operator, leads to a mathematical framework useful in the investigation of relativity/quantum mechanics relationships. 

scholarly journals Entropic Information Theory: Formulae and Quantum Gravity Bits from Bit

2021 ◽  
pp. 23-30
Author(s):  
Olivier Denis
Keyword(s):  
Information Theory ◽  
General Relativity ◽  
Dark Matter ◽  
Quantum Mechanics ◽  
Dark Energy ◽  
Quantum Gravity ◽  
Planck Scale ◽  
Basic Structure ◽  
Mathematical Framework ◽  
Information Framework

We show here that entropic information is capable of unifying all aspects of the universe at all scales in a coherent and global theoretical mathematical framework materialized by entropic information framework, theory and formulas, where dark matter, dark energy and gravity are truly informationals processes and where information is code and code is what creates the process, it is itself the process. Mass, energy and movement of information are respectively dark matter, dark energy, and gravity. Here, we reconcile general relativity and quantum mechanics by introducing quantum gravity for the Planckian scale. The formulas of entropic information are expressed in natural units, physical units of measurement based only on universal constants, constants, which refer to the basic structure of the laws of physics:  C and G are part of the structure of space-time in general relativity, and h captures the relationship between energy and frequency that is the basis of quantum mechanics. Here we show that entropic information formulas are able to present entropic information in various unifying aspects and introduce gravity at the Planck scale. We prove that Entropic information theory is thus building the bridge between general relativity and quantum mechanics

scholarly journals Towards the Unification of Quantum Dynamics, Relativity and Living Organisms

2018 ◽  
Author(s):  
Arturo Tozzi ◽  
James F. Peters ◽  
John S. Torday
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Group Theory ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Binary Operation ◽  
Relativity Theory ◽  
Mathematical Framework ◽  
Mathematical Approach ◽  
Living Organisms

The unexploited unification of quantum physics, general relativity and biology is a keystone that paves the way towards a better understanding of the whole of Nature.  Here we propose a mathematical approach that introduces the problem in terms of group theory.  We build a cyclic groupoid (a nonempty set with a binary operation defined on it) that encompasses the three frameworks as subsets, representing two of their most dissimilar experimental results, i.e., 1) the commutativity detectable both in our macroscopic relativistic world and in biology; 2) and the noncommutativity detectable both in the microscopic quantum world and in biology.  This approach leads to a mathematical framework useful in the investigation of the three apparently irreconcilable realms.  Also, we show how cyclic groupoids encompassing quantum mechanics, relativity theory and biology might be equipped with dynamics that can be described by paths on the twisted cylinder of a Möbius strip.   

scholarly journals Entropic Information Theory: Formulae and Quantum Gravity, Bits from Bit

2021 ◽  
Author(s):  
olivier denis
Keyword(s):  
Information Theory ◽  
General Relativity ◽  
Dark Matter ◽  
Quantum Mechanics ◽  
Dark Energy ◽  
Quantum Gravity ◽  
Planck Scale ◽  
Basic Structure ◽  
Mathematical Framework ◽  
Information Framework

We show here that entropic information is capable of unifying all aspects of the universe at all scales in a coherent and global theoretical mathematical framework materialized by entropic information framework, theory and formulas, where dark matter, dark energy and gravity are truly informationals processes and where information is code and code is what creates the process, it is itself the process. Mass, energy and movement of information are respectively dark matter, dark energy, and gravity. Here, we reconcile general relativity and quantum mechanics by introducing quantum gravity for the Planckian scale. The formulas of entropic information are expressed in natural units, physical units of measurement based only on universal constants, constants, which refer to the basic structure of the laws of physics: C and G are part of the structure of space-time in general relativity, and h captures the relationship between energy and frequency that is the basis of quantum mechanics. Here we show that entropic information formulas are able to present entropic information in various unifying aspects and introduce gravity at the Planck scale. We prove that Entropic information theory is thus building the bridge between general relativity and quantum mechanics

Mathematical Foundations of Quantum Mechanics

2018 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Foundations Of Quantum Mechanics ◽  
Theoretical Physics ◽  
Mathematical Framework ◽  
Von Neumann ◽  
New Science ◽  
Freeman Dyson ◽  
And Mathematics ◽  
Mathematical Foundations ◽  
First Time

Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics—a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. This new edition of this classic work has been completely reset in TeX, making the text and equations far easier to read. The book has also seen the correction of a handful of typographic errors, revision of some sentences for clarity and readability, provision of an index for the first time, and prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson have been added. The result brings new life to an essential work in theoretical physics and mathematics.

scholarly journals An Information Ontology for the Process Algebra Model of Non-Relativistic Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 136
Author(s):  
William Sulis
Keyword(s):  
Quantum Mechanics ◽  
Process Algebra ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Mathematical Framework ◽  
Information Propagation ◽  
Discrete Events ◽  
Fundamental Level ◽  
High Degree ◽  
Algebra Model

The process algebra model has been suggested as an alternative mathematical framework for non-relativistic quantum mechanics (NRQM). It appears to reproduce the wave functions of non-relativistic quantum mechanics to a high degree of accuracy. It posits a fundamental level of finite, discrete events upon which the usual entities of NRQM supervene. It has been suggested that the process algebra model provides a true completion of NRQM, free of divergences and paradoxes, with causally local information propagation, contextuality, and realism. Arguments in support of these claims have been mathematical. Missing has been an ontology of this fundamental level from which the formalism naturally emerges. In this paper, it is argued that information and information flow provides this ontology. Higher level constructs such as energy, momentum, mass, spacetime, are all emergent from this fundamental level.

scholarly journals Formulation Of Quantum Mechanics On Poincaré Disks

2021 ◽  
Author(s):  
Arturo Tozzi
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Gauge Theories ◽  
Temporal Evolution ◽  
Wave Functions ◽  
Mathematical Framework ◽  
Genus Zero ◽  
Proper Understanding ◽  
Oscillatory Dynamics ◽  
Topological Theorem

The unexploited unification of general relativity and quantum mechanics (QM) prevents the proper understanding of the micro- and macroscopic world. Here we put forward a mathematical approach that introduces the problem in terms of negative curvature manifolds. We suggest that the oscillatory dynamics described by wave functions might take place on hyperbolic continuous manifolds, standing for the counterpart of QM’s Hilbert spaces. We describe how the tenets of QM, such as the observable A, the autostates ψa and the Schrodinger equation for the temporal evolution of states, might work very well on a Poincaré disk equipped with rotational groups. This curvature-based approach to QM, combined with the noncommutativity formulated in the language of gyrovectors, leads to a mathematical framework that might be useful in the investigation of relativity/QM relationships. Furthermore, we introduce a topological theorem, termed the punctured balloon theorem (PBT), which states that an orientable genus-1 surface cannot encompass disjoint points. PBT suggests that hyperbolic QM manifolds must be of genus ≥ 1 before measuring and genus zero after measuring. We discuss the implications of PBT in gauge theories and in the physics of the black holes.

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