It is explicitly shown the weak points (being not rigorous) in the reasoning of Dirac, of Landau, of Von Neumann, of Weyl, of Fadeev and of others, deriving many fundamental relations, which later became classical and are being commonly used in Quantum Theory up to now. It is because of their derivations were made when functional analysis was not yet sufficiently developed. Those weak points are of two following reasons:1. Unbounded property of the physical operators is not yet paid by attention.2. The “Dirac delta function” defined by Dirac is not always distinguished clearly with the Dirac distribution defined by Schwartz and with the generalized functions defined by Gel'fand.Particularly, the formulation of the closure relation by using equality of the functionals is not rigorous. Consequently, it is necessary to improve Quantum Theory.
It is explicitly shown the weak points (being not rigorous) in the reasoning of Dirac, of Landau, of Von Neumann, of Weyl, of Fadeev and of others, deriving many fundamental relations, which later became classical and are being commonly used in Quantum Theory up to now. It is because of their derivations were made when functional analysis was not yet sufficiently developed. Those weak points are of two following reasons:1. Unbounded property of the physical operators is not yet paid by attention.2. The “Dirac delta function” defined by Dirac is not always distinguished clearly with the Dirac distribution defined by Schwartz and with the generalized functions defined by Gel'fand.Particularly, the formulation of the closure relation by using equality of the functionals is not rigorous. Consequently, it is necessary to improve Quantum Theory.
The aim of this chapter is to present (in a compact way) the fundamentals of conventional quantum theory. We select the for- mulation of QM as an axiomatic theory in the complete accordance with the Hilbert’s program on axiomatization of physical theories. We remark that the first step toward the present formulation of QM was done in the seminal work of Hilbert, Nordheim, and von Neumann [112]. This work was substantially extended, completed, and clarified in the book of von Neumann [296]. Another version of the rigorous mathematical approach to QM was presented in the book of Dirac [74]. From the mathematical viewpoint, the main difference between the approaches of von Neumann and Dirac is that the first one is based on the theory of self-adjoint operators in complex Hilbert space and the second one is based on the theory of self-adjoint operators acting (by using the modern mathematical terminology) in rigged Hilbert spaces. This theory operates with generalized eigenvectors belonging to the spaces of distributions. Although the von Neumann’s formalization of QM became dominant in quantum mathematical physics, the Dirac’s formalization is widely used in the works on the physical level of rigorousness. Moreover, the Dirac’s terminology, bra-and ket- vectors, became the standard for quantum information theory. However, since quantum information is based on the castrated version of QM, namely, QM in finite-dimensional Hilbert spaces, the so-called n-qubit spaces, the basic tool of the Dirac’s formalism, usage of generalized eigenvectors, is useless (since in the finite- dimensional case everything is reduced to manipulation with matrices). During the last 80 years, there were written hundreds of books on QM enlightening its foundational aspects. Unfortunately, the majority of these writings advertise exotic (and some of them really esoteric) views on QM and, in particular, on interpretation of the wave function. They are full of such terms as the collapse of the wave function, quantumnonlocality, manyworlds, thewave function of universe, the preparation of Schro¨dinger cats, influence of the future on the past, quantum teleportation. Of course, manipulations with such terminology create an atmosphere of mystery and a feeling that something unusual and impossible can happen. And this atmosphere plays a positive role in attracting young people to QM and especially to quantum foundations.
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
Dirac's bra and ket formalism is investigated and incorporated into a complete mathematical theory. First the axiomatic foundations of quantum mechanics and von Neumann's spectral theory of observables are reviewed and several inadequacies are pointed out. These defects then are remedied by extending the usual Hilbert space to a rigged Hilbert space as introduced by Gel'fand, i.e., a triplet Φ⊂H⊂Φ′ where H is a Hilbert space, Φ a dense subspace of H provided with a new (finer) topology, Φ′ the dual of Φ. It is shown that this mathematical structure, together with the Schwartz nuclear theorem, allows us to reproduce Dirac's formalism in a completely rigorous way, without losing its transparency; this makes the theory easier to handle. The temporal evolution of the system and the wave equation are considered. Finally the probabilistic interpretation and the physical aspects of the theory are discussed; Φ is identified with the set of all physically accessible states of the system, Φ′ with the set of all possible experiments (apparatus) to which it can be subjected; this provides a direct connection with Feynman's formulation of quantum mechanics.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.
Quantum theory is the soul of contemporary physics. It was discovered in an adventurous way, under the urge to solve the puzzles posed by atomic spectra and blackbody radiation. But after its invention, it immediately became clear that it was not just a theory of specific physical systems: it was rather a new language of universal applicability. Already in 1928, the theory had received solid mathematical foundations by Hilbert, von Neumann, and Nordheim, and this work was brought to completion in the monumental work of von Neumann, in the form that we still study nowadays. The theory is extraordinarily successful, and its predictions have been confirmed to an astonishing level of precision in a large spectrum of experiments. However, almost 90 years after von Neumann's book, quantum theory remains mysterious. Its mathematical formulation – based on Hilbert spaces and self-adjoint operators – is far from having an intuitive interpretation. The association of physical systems to Hilbert spaces whose unit vectors represent pure states, the representation of transformations by unitary operators and of observables by self-adjoint operators – all such postulates look artificial and ad hoc . A slightly more operational approach is provided by the C * -algebraic formulation of quantum theory – still, this formulation relies on the assumption that observables form an algebra , where the physical meanings of the multiplication and the sum are far from clear. In short, the postulates of quantum theory impose mathematical structures without providing any simple reason for this choice: the mathematics of Hilbert spaces is adopted as a magic blackbox that “works well” at producing experimental predictions. However, in a satisfactory axiomatization of a physical theory the mathematical structures should emerge as a consequence of postulates that have a direct physical interpretation. By this we mean postulates referring, e.g., to primitive notions like physical system, measurement, or process, rather than notions like, e.g., Hilbert space, C * -algebra, unit vector, or selfadjoint operator. The crucial question thus remains unanswered: why quantum theory? Which are the principles at the basis of the theory? A case that is often invoked in contrast is that of Special Relativity theory, which directly follows from the simple understandable principle of relativity.