Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical Physics

On an Extension of the Mathematical Framework of the Quantum Theory

Hitoshi Wakita

Published May 1, 1961• 0 citations
Publisher

Abstract

Extending the usual framework of the quantum theory, we have succeeded in gettig a new mathematical framework. The most essential character of the usual theory is preserved in the new theory, that is, the set of physical quantities can be represented by an operator algebra in a separable Hilbert space. In the new theory, on the other hand, we can rigorously treat the interaction Hamiltonians, which cannot be treated in the usual framework. In our new framework there are always solutions of the Schrödinger equation for any Hamiltonian. However, the most serious defect of our theory is in the fact that we have no guarantee of the uniqueness of solutions.

DOI
10.1143/ptp.25.743

Related papers

On an Extension of the Mathematical Framework of the Quantum Theory. II

Hitoshi Wakita

Renormalizing the state vectors which represent the clouds of the proper fields of particles, we can define a new Hilbert space where the interaction Hamiltonian has a definite mathematical meaning. The success of the renormalization theory seems to show that the quantum theory of fields can be treated properly by using this new Hilbert space. Though this Hilbert space is not unique, states will be uniquely determined as linear functionals on some set of physical quantities. The clouds of the proper fields really exist, but they are not observed at all.

Progress of Theoretical Physics•Aug 1, 1962•0 citations

On the Representations of Field Quantities, II

Hitoshi Wakita

In this series of papers we study whether the framework of a Hilbert space is adequate to the quantum theory of fields. Especially in the present paper we inquire what are the characteristic features of field theory, and study how these features can be described in this framework. For this purpose we examine the character of additive field quantities, which are used in this paper in place of field operators, and the relation between a system extending over the whole space and one restricted within a finite volume. After these considerations, we may conclude that there is a sufficient reason to treat the quantum theory of fields within the framework of a Hilbert space.

Progress of Theoretical Physics•Feb 1, 1959•0 citations

On Algebraic Methods in Quantum Theories

Benjamin H. Feintzeig

The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observables—measurable quantities of the system—given by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.

•Jan 1, 2016•0 citations

The mathematical structure of quantum theory

Keith Hannabuss

Abstract We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.

•Mar 20, 1997•0 citations

Rigorous Differential Algebraic Closure Framework for Quantum Dynamical Systems: A Constructive Approach with Certified Computations

shifa liu

This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to quantum dynamical systems, including closed quantum systems, open quantum systems, and quantum field theories. We define thequantum dynamical closure KQDS and quantum field theoretic closure KQFT asdifferentially closed field extensions constructed through recursive adjunction processes that systematically incorporate quantum states, observables, Hamiltonians,evolution operators, measurement processes, and renormalization procedures.Within these closures, we prove that solutions to broad classes of quantum dynamical equations—including the Schr¨odinger equation, Heisenberg equation,Lindblad master equation, and Wheeler-DeWitt equation—admit unified representations that respect the underlying algebraic and geometric structures. The framework rigorously addresses quantum nonlinearity, measurement back-action,decoherence, and renormalization while preserving operator algebraic structures and quantum constraints.We provide detailed constructive proofs with complete mathematical rigor, deriving explicit solution formulas with certified error bounds and establishing convergence criteria in appropriate quantum function spaces. Comprehensive algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with mathematically certified error bounds. A rigorous validation framework employing interval arithmetic and operator-theoretic methods demonstrates the practical effectiveness of our approach.The work demonstrates that explicit analytic solutions exist within appropriately constructed differential algebraic closures, providing new algebraic perspectives on quantum solvability while maintaining consistency with established quantum theory. Extensions to quantum computation, quantum thermodynamics, and quantum gravity establish connections across physical disciplines while maintaining mathematical rigor throughout.

•Dec 31, 2025•0 citations

An alternative foundation of quantum theory

Inge Svein Helland

The final version of a new approach to quantum theory is formulated in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the finite-dimensional case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is indicated here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. In case, this simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. What is new in this version of the article, is that the assumptions given behind the main theorems are simplified considerably, and that all proofs are given explicitly or refered to in other recent papers.

arXiv:2305.06727•May 11, 2023•3 citations
Extending the usual framework of the quantum theory, we have succeeded in gettig a new mathematical framework. The most essential character of the usual theory is preserved in the new theory, that is, the set of physical quantities can be represented by an operator algebra in a separable Hilbert space. In the new theory, on the other hand, we can rigorously treat the interaction Hamiltonians, which cannot be treated in the usual framework. In our new framework there are always solutions of the Schrödinger equation for any Hamiltonian. However, the most serious defect of our theory is in the fact that we have no guarantee of the uniqueness of solutions.
Renormalizing the state vectors which represent the clouds of the proper fields of particles, we can define a new Hilbert space where the interaction Hamiltonian has a definite mathematical meaning. The success of the renormalization theory seems to show that the quantum theory of fields can be treated properly by using this new Hilbert space. Though this Hilbert space is not unique, states will be uniquely determined as linear functionals on some set of physical quantities. The clouds of the proper fields really exist, but they are not observed at all.
In this series of papers we study whether the framework of a Hilbert space is adequate to the quantum theory of fields. Especially in the present paper we inquire what are the characteristic features of field theory, and study how these features can be described in this framework. For this purpose we examine the character of additive field quantities, which are used in this paper in place of field operators, and the relation between a system extending over the whole space and one restricted within a finite volume. After these considerations, we may conclude that there is a sufficient reason to treat the quantum theory of fields within the framework of a Hilbert space.
The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observables—measurable quantities of the system—given by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.
Abstract We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.
This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to quantum dynamical systems, including closed quantum systems, open quantum systems, and quantum field theories. We define thequantum dynamical closure KQDS and quantum field theoretic closure KQFT asdifferentially closed field extensions constructed through recursive adjunction processes that systematically incorporate quantum states, observables, Hamiltonians,evolution operators, measurement processes, and renormalization procedures.Within these closures, we prove that solutions to broad classes of quantum dynamical equations—including the Schr¨odinger equation, Heisenberg equation,Lindblad master equation, and Wheeler-DeWitt equation—admit unified representations that respect the underlying algebraic and geometric structures. The framework rigorously addresses quantum nonlinearity, measurement back-action,decoherence, and renormalization while preserving operator algebraic structures and quantum constraints.We provide detailed constructive proofs with complete mathematical rigor, deriving explicit solution formulas with certified error bounds and establishing convergence criteria in appropriate quantum function spaces. Comprehensive algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with mathematically certified error bounds. A rigorous validation framework employing interval arithmetic and operator-theoretic methods demonstrates the practical effectiveness of our approach.The work demonstrates that explicit analytic solutions exist within appropriately constructed differential algebraic closures, providing new algebraic perspectives on quantum solvability while maintaining consistency with established quantum theory. Extensions to quantum computation, quantum thermodynamics, and quantum gravity establish connections across physical disciplines while maintaining mathematical rigor throughout.
The final version of a new approach to quantum theory is formulated in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the finite-dimensional case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is indicated here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. In case, this simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. What is new in this version of the article, is that the assumptions given behind the main theorems are simplified considerably, and that all proofs are given explicitly or refered to in other recent papers.