This paper surveys recent explorations into David Carf`ı’s Schwartz Linear Algebra, a framework unifying algebra, analysis, and quantum mechanics through Schwartz families and tempered distributions. We examine the Fourier kernel as a Schwartz basis, critique the Hilbert space foundation of quantum mechanics, and lay the groundwork for the generalized Dirac scalar product of transposable Schwartz families with tempered distributions. These developments address limitations in traditional quantum theory, offering a rigorous, distribution-based alternative.
This paper surveys recent explorations into David Carf`ı’s Schwartz Linear Algebra, a framework unifying algebra, analysis, and quantum mechanics through Schwartz families and tempered distributions. We examine the Fourier kernel as a Schwartz basis, critique the Hilbert space foundation of quantum mechanics, and lay the groundwork for the generalized Dirac scalar product of transposable Schwartz families with tempered distributions. These developments address limitations in traditional quantum theory, offering a rigorous, distribution-based alternative.
In this paper we start to construct some fundamental features of Dirac Calculus, specifically, we go inside the theory of Heisenberg continuous matrices, which, in our Schwartz Linear Algebra, are represented by Schwartz families. We distinguish the important subclass of transposable continuous matrices and give some basic and very important examples in Quantum Mechanics. So we define transposable Schwartz families and their transpose families, we prove the transposability of Dirac families and Fourier families. We find the transpose of regular-distribution families in a much general case. We define symmetric families, the analogous of symmetric ma- trices in the continuous case. We prove the symmetry of Dirac families and of Fourier families. We define Hermitian families, the analogous of Hermitian matrices in the continuous case. We prove the Hermitianity of Dirac families and of Fourier families. We define unitary families, the analogous of unitary matrices in the continuous case. We prove the unitarity of Dirac families and of the fundamental normalized de Broglie family. Then, we use the transpose of a family to find the components of the superpositions of transposable families, we give a general result and we apply this result to the Dirac families and the eigenfamilies of the vector-wave operator. We shall use the transposable families in next chapters to define the Dirac product in distribution spaces, basic product for the entire foundation of Dirac Calculus and Quantum Mechanics formalism.
We propose, by Schwartz Linear Algebra, a significant development of Laurent Schwartz Distribution Theory. The study is conducted by following the (natu- ral and straightforward) way of Weak Duality among topological vector spaces, aiming at the construction of a feasible, rigorous, quite elementary and manageable frame- work for Quantum Mechanics. It turns out that distribution spaces reveal themselves an environment more capable to help in Quantum Mechanics than previously thought. The goal of the research, introduced here, consists in showing that the most natural state-spaces of a quantum system, in the infinite dimensional case, are indeed dis- tribution spaces. Moreover, we show new, natural and straightforward mathematical structures that reproduce very closely several physical objects and many operational procedures required in Quantum Mechanics, systematizing the algorithms and nota- tions of Dirac Calculus, in such a way that it becomes a more versatile and more powerful tool, than we are used to think of.
In this work we show that standard results in quantum tomography, which are typically proven in the context of Hilbert spaces, can analogously be established in the theory of Schwartz spaces. In particular, we will see that quantum states are equivalently characterized by their Wigner function, the characteristic of their Wigner function, and the probability distributions for a family of linear combinations of position and momentum. Physically, the latter can be understood as the evolution of its probability distribution in position over time under, for example, free or harmonic evolution. The result is obtained with techniques that are more accessible to a physicist audience, and shows that, at least for tomography results, the full structure of Hilbert spaces is not necessary.
As is well-known, Dirac [1 gave a very elegant foundation of quantum mechanics, which contained however some self-contradictory concepts from the mathematical point of view.J. von Neumann gave another foundation of the theory based upon his spectral theory of self-adjoint operators of Hilbert space.However the whole spec- tral theory is in fact not necessary and what Dirac actually needs 1 d is only that the self-adjoint operators t. and could be put in i dt diagonal forms.There is also another justification of "improper" iunctions introduced by Dirac to put these operators in the diagonal forms by means of the theory of distributions of L. Schwartz.But
In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.
This study investigates the foundational and applied roles of linear algebra and functional analysis in quantum mechanics, employing a mixed-methods design that integrates qualitative conceptual analysis with quantitative simulations. The methodology combined literature synthesis of recent works (2018–2022), mathematical modeling of eigenvalue problems, and computational simulations within Hilbert spaces to validate theoretical constructs. The results demonstrated that linear algebra provides the structural basis for representing quantum states, operator norms, and matrix formulations, while functional analysis extends these principles to infinite-dimensional spaces, spectral decompositions, and unbounded operators. Tables presented comprehensive datasets, including eigenvalue distributions, operator convergence, and density matrix approximations, whereas figures offered multidimensional visualizations through line, bar, scatter, and pie charts. Together, these outputs revealed that functional analytic methods are indispensable in understanding spectral behavior and operator stability, while linear algebra underpins computational frameworks for quantum algorithms and quantum information processing. The discussion established that this synergy not only consolidates the mathematical rigor of quantum mechanics but also advances practical innovations such as quantum error correction, quantum machine learning, and operator-based quantum algorithms. The study concludes that linear algebra and functional analysis are complementary frameworks that enable both deeper theoretical insights and application-driven progress in quantum mechanics, underscoring their continuing importance in research, pedagogy, and technology development.