Quantum Mechanics and Its Evolving Formulations

In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical bra-ket formalism of Dirac, and the more recent rigged Hilbert space approach.

A second attack on the problem is wave mechanics, introduced by Schrödinger in 1926 [3]. The idea was to introduce a wave equation, inspired by the classical analysis of waves, that takes the form of an eigenvalue problem for the Hamiltonian. The latter is obtained from its classical expression by certain correspondence rules. The task of the physicist is to solve the equation-that is, to find the eigenvalues, which are the allowed values of the energy.
In a third paper, Schrödinger showed, albeit in a somewhat heuristic way, that the two approaches are equivalent [4]. Dirac, in turn, proved in 1930 that the equivalence of the two theories results from his transformation theory [5].
In parallel to these two supposedly equivalent formulations of the theory, another set of controversies emerged concerning its interpretation. Based on a statistical approach, largely due to Born, and the notion of noncommuting "observables", which leads to Heisenberg's uncertainty relations, the so-called Copenhagen interpretation gradually took over, although there are always oppositions (Einstein, for one, never admitted it: "God does not play dice!"). In fact, most physicists use it, more or less explicitly, but there are still exceptions, even today.
Clearly, the situation was rather messy. As Gottfried puts it [6]: "The old quantum theory was a diabolically clever hodge-podge of classical laws and seemingly unrelated ad hoc recipes." The mathematics is not well defined, one uses eigenfunctions of infinite norm, as well as the so-called delta "function", operators are treated in a formal way, and so on. Clearly, the time was ripe for a precise mathematical overhaul of the whole theory, and this was the achievement of John von Neumann [7].

Von Neumann's Axiomatics
The first thing to do was obviously to define rigorously the mathematical framework. To that effect, von Neumann begins his book with a long chapter (170 pages) on the abstract Hilbert space. This is in fact the first complete definition of a Hilbert space (Hilbert himself had only considered a special case, namely the space 2 of square integrable sequences). All the basic ingredients are there: a vector space equipped with an inner product and the corresponding norm, plus the requirement of completion. Then operators, bounded or unbounded, unitary operators, self-adjoint operators (which he calls "hypermaximal"), trace class (also called nuclear) and Hilbert-Schmidt operators, orthogonal projections, and the spectral theorem for self-adjoint operators. This pioneering work may well be von Neumann's essential contribution.
Next, von Neumann turns to quantum mechanics (QM). However, he never gives a precise list of its axioms. Instead, he bases everything on the statistical affirmations of QM; thus, he endorses the probabilistic interpretation of Born-Jordan-Dirac. The starting point is the following postulate.
Take a system in the state φ = φ(q 1 , q 2 , . . . , q k ) ∈ L 2 (R 3k ). Then, the probability that the system is in the volume V of configuration space is given by In other words, |φ(q 1 , q 2 , . . . , q k )| 2 is the probability density.
From this statement, von Neumann derives the whole structure of QM: expectation values, statistical operator, measurability and simultaneous measurability, uncertainty relations, and so on.
At this stage, one may formalize the rules (axioms) of QM as follows. 1.
The space of states of a system is a Hilbert space H.
In particular, the requirement that H be complete offers a certain mathematical "comfort"-that is, powerful theorems that do not hold without it.

3.
A physical observable is represented by a self-adjoint operator in H. Therefore, one may use . The spectral theorem for self-adjoint operators.
. Stone's theorem: e iAt is unitary if and only if A = A * ; this is the key of the time evolution.

4.
Given a physical observable represented by the self-adjoint operator A, its expectation value in the state ρ is given by For a pure state φ, this becomes

5.
Given two states φ, ψ ∈ H, the transition probability φ → ψ is given by Before continuing, we would like to point out a recent book [8] that provides a thorough analysis of the space of density matrices (finite dimensional Hilbert-Schmidt operators) and its geometry. This space is in fact a Hilbert space with the trace inner product A|B = Tr A * B.
This formulation is rigorous, but several problems remain, namely, More generally, the question is how to select the "good" operators. This asks for a precise definition of the concept of observable, containing two aspects: the physical definition-that is, how the quantity can be measured experimentally-and the mathematical definition of the representative operator. • There are problems with unbounded operators: one has to make their domain precise; they cannot be added freely. Similarly, operators with a continuous spectrum have the problem that there are no eigenvectors corresponding to points of the continuous spectrum. Another difficulty is that the transition between the Schrödinger picture and the Heisenberg picture may be problematic if the Hamiltonian is unbounded.
Of course, there are exceptions; some textbooks are more mathematically flavoredfor instance, those of Prugovečki [16] or Isham [17]. An extreme opposite attitude is that of Pauli, who once wrote in a letter to Born (19 July 1925) "You are only going to spoil Heisenberg's physical ideas with your futile mathematics!"

The "Bra-Ket" Formalism
The formalism may be synthesized in four "axioms": 1.
The states of a quantum system constitute a vector space E , equipped with a Hermitian inner product ·|· , which manifests the superposition principle, and there is a bijective correspondence bra φ| ↔ ket |φ .

3.
Each physical observable is represented by a linear Hermitian operator on E , and these operators form a (non-Abelian) algebra: A + B and AB are well defined. 4.
Each physical system has a complete system of commuting observables (CSCO), and the eigenvectors of the elements of this CSCO constitute a basis of E . Thus, every state ψ ∈ E may be expanded into that basis.

What about a Rigorous Dirac Formalism?
All this is practical and used by most physicists, but it is not correct in a Hilbert space. Therefore, E cannot be taken as a Hilbert space H . . . What is it then? How can one reconcile the two approaches? A solution is to work in a Gel'fand triplet (rigged Hilbert space). This approach was proposed independently by Roberts [18,19], the author [20][21][22] and Bohm [23]. We will discuss it in the next section.

The RHS Approach
where Φ × is the conjugate dual of Φ, i.e., the space of continuous conjugate linear functionals on Φ (this ensures that both natural embeddings Φ → H and H → Φ × are linear).

More about the RHS
A natural interpretation of the elements of the triplet (1) runs as follows: . Φ represents those states that are physically preparable.
. H is von Neumann's Hilbert space.
. Φ × contains generalized states associated with measurement operations.
As for mathematical properties, one requires that Φ be . complete, i.e., every Cauchy sequence converges to an element of Φ; otherwise, its completion might fail to be contained in H. . reflexive: Φ ×× = Φ, so that no other space than Φ and Φ × has to be considered, which would ruin the interpretation above. . nuclear, which allows to apply the generalized spectral theorem of Gel'fand-Maurin.
This demands a definition. In the simplest case, assume Φ is the intersection of a decreasing scale of Hilbert spaces In technical terms, Φ is a projective limit Φ = ←− lim H n , with the corresponding locally convex topology.
Then Φ is said to be nuclear if, for each n ∈ N, there is a m ∈ N, m > n, such that the natural embedding H m → H n is a Hilbert-Schmidt operator. The classical example is the Schwartz space S(R) of smooth fast decreasing functions.

Eigenvectors
Let A be a closed operator in H, continuous from Φ to Φ. Then, one defines the adjoint A × : Φ × → Φ × , extension of the Hilbertian adjoint A † , by A vector ξ λ ∈ Φ × is called a generalized eigenvector of A, with eigenvalue λ ∈ C, if one has

Using Dirac's notations, this becomes
Actually, for obtaining a complete parallel with Dirac's bra-ket formalism, one needs to consider a second RHS, namely: where Φ denotes the dual of Φ, i.e., the space of continuous linear functionals on Φ. Then, whereas Φ × contains the ket vectors, Φ contains the bra vectors, and indeed the two are in one-to-one correspondence. For a pedagogical description of this formulation, one may consult [24].

Assume
. A has a self-adjoint extension A 0 in H; thus, A × is an extension of A et A 0 (collectively, A) . Φ is nuclear and complete.

Spectral Projections
According to von Neumann, there is a decomposition into a direct integral that "decomposes" A : The problem is that H(λ) is not a subspace of H if λ is a point of null µ-measure. Thus, there are no true eigenvectors corresponding to points of the continuous spectrum.
However, there is a difficulty here: the spectrum of A in Φ × may contain points that do not belong to the Hilbertian spectrum, a situation to be avoided.
To give an example: Take the momentum operator p = 1 i d dx ; then, its spectrum σ(p) is Fortunately, there are criteria for ensuring that one gets a tight rigging [25], that is, a rigging in which the RHS spectrum coincides with the Hilbertian spectrum.
The standard example of a RHS analysis is that of a particle, either free or in sufficiently regular potential V: The observables are: position Q, momentum P, energy H = −P 2 /2m + V(Q), and the natural RHS is the Schwartz triplet

Conclusions
The conclusion of this analysis is that von Neumann's approach to QM is necessary, but not sufficient. The RHS approach completes the latter and allows a rigorous version of the practical formalism of Dirac. It leads to a good formulation of scattering theory: plane waves, Gamow states, resonances, Lippman-Schwinger equation, etc. In addition, it extends naturally to quantum field theory (Borchers, Wightman,. . . ).
An additional bonus is that the RHS approach provides a good realization of symmetries. Let U be a unitary representation of an invariance group G in H. Then, we obtain two additional representations of G: . U Φ acting in Φ (active point of view) . U × Φ acting in Φ × (passive point of view) and these two representations are contragredient of each other, as consequence of the unitarity of U in H): Before quitting, let me give four more references, which are particularly relevant to the present paper. First, a general discussion of the RHS approach may be found in the so-called Compendium of Quantum Physics [26]. For going further in the philosophical implications of QM and its various interpretations, an invaluable source is the classical book by Jammer [27], who also makes contact with the quantum logic, involving nondistributive lattices.
Next, the textbook by Hannabuss [28] gives a clear and pedagogical treatment of QM, suitable for 2nd or 3rd year mathematics students. Notably, it contains a remarkable collection of original citations.
Finally, one should quote the book by Lacki [29]. He analyzes von Neumann's book as concretization of Hilbert's program on the axiomatization of physics, already undertaken by Jordan and Nordheim. In particular, he analyzes in detail Neumann's "proof" of the impossibility of hidden variables in QM.
However, this is not the end of the story. All we have said concerns the standard approach to QM, valid for a single system, in its successive formulations: von Neumann's, Dirac's, and the RHS version. In the background, this still relies in some form of the Copenhagen interpretation. An extension to a collection of identical systems, which is the realm of statistical physics, is readily available.
First, QM has evolved into what is called the standard model of elementary particles. This is certainly not the final word, although it has passed successfully all comparisons with experiment. Then, new extensions of QM have appeared in the recent years, trying to combine it in a rigorous manner with gravity and also information theory. Of course, this is not the place to go into detail. Instead, we might redirect the reader to the fascinating little book by Al-Khalili [30] for a thorough, but nontechnical panorama of the recent evolution of physics. Of course, all these recent developments go far beyond our considerations in the present paper, but nevertheless, these have a nontrivial place in the theory: mathematical rigor is not a luxury, notwithstanding Pauli's opinion quoted above.
Funding: This research received no external funding.