In 1972, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory” was published by Gerard Emch [8], giving the axioms for a physical system in an algebraic setting using the language of Irving Segal [19], and then continuing to obtain the C*-algebraic formalism for a physical system. Of these axioms, only the fifth contained an assumption that was questionable in its physical content. Bearing in mind that for each observable A and state φ , one obtains a distribution of values for the observed results of measurement, we have: Axiom 5: For any element A in the set of observables \(\mathfrak{A}\) and any non-negative integer n , there is at least one element, denoted A n , in \(\mathfrak{A}\) such that (i) the set of dispersion-free states for A n is contained in the set of dispersion-free states for A, (ii) < φ ; A n > = < φ ; A > n for all φ in the set of dispersion-free states for A . (Here < φ ; B > is the expectation of B in state φ .)
In 1972, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory” was published by Gerard Emch [8], giving the axioms for a physical system in an algebraic setting using the language of Irving Segal [19], and then continuing to obtain the C*-algebraic formalism for a physical system. Of these axioms, only the fifth contained an assumption that was questionable in its physical content. Bearing in mind that for each observable A and state φ , one obtains a distribution of values for the observed results of measurement, we have: Axiom 5: For any element A in the set of observables (\mathfrak{A}) and any non-negative integer n , there is at least one element, denoted A n , in (\mathfrak{A}) such that (i) the set of dispersion-free states for A n is contained in the set of dispersion-free states for A, (ii) < φ ; A n > = < φ ; A > n for all φ in the set of dispersion-free states for A . (Here < φ ; B > is the expectation of B in state φ .)
Forty years ago, Haag and Kastler [27] introduced the algebraic approach to Quantum Field Theory, and soon their method was extended to statistical mechanics. The rationale was to free the theory from particular realizations, linked to specific models, and to focus on the basic structure, namely, the algebra of observables and states on it. This more abstract language, exploiting the deep mathematical theories of C*-algebras and von Neumann algebras, soon became the standard approach to the mathematically rigorous description of physical systems with infinitely many degrees of freedom. Textbooks flourished and, among many authors, Gérard Emch argued forcefully for the new approach [6].
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
Another axiom has to be introduced to make the formalism given in Part I physically equivalent to the conventional Hilbert-space formalism. Then it is shown that, given a certain fundamental set of observables, a B*-algebra A can be built into which the set 𝒪b of all bounded observables can be mapped injectively. The closure of the algebra generated by the image of 𝒪b into A is A itself. A Hilbert space ℋ exists into which the set 𝒪0 of all pure physical states can be mapped injectively. The closure of the subset of ℋ which is the image of this mapping is ℋ itself. The algebra A can be mapped in such a fashion into the C*-algebra B(ℋ) of all bounded observables that these mappings provide, essentially, just a translation of the original formalism in a Hilbert-space formalism.
Since 1947 a foundation of Quantum Mechanics (QM) on functional analysis was suggested by Segal. By defining the C*-algebra of the observables, then the Gelfand-Naimark-Segal theorem faithfully represents this algebra into Hilbert space. In the 70’s Emch has reiterated this formulation and improved it. Recently Strocchi improved it even more. First, he suggested an axiomatization of the paradigmatic Dirac-von Neumann’s formulation of QM to which he addresses two basic criticisms, i.e. a weak linkage with the experimental basis of theoretical physics and the obscurity about the separation mark between classical mechanics and QM. Afterwards, through an analysis of the experimental basis of a physical theory he suggests an explanation of Segal’s restriction of the operators to be bounded. Eventually, he represents this algebra into Hilbert space and at last, by means of Weyl algebra he obtains the symmetries of the dynamics of a particle theory. In fact, several characteristic features of this formulation correspond to those determined by the two choices which are the alternative ones to the choices of the dominant formulation. It is a problem-based theory, since it starts rather from than axioms a problem (i.e. the indeterminacy); then, it argues through both doubly negated propositions and an ad absurdum proof. Moreover, its theoretical development is similar to that of an alternative classical theory since it put, before the geometry, the algebra; the bounded operators are represented by a polynomial algebra; which pertains to constructive mathematics. Eventually, he obtains the symmetries of the theory. The problems to be overcome in order to accurately re-construct his formulation according to the two alternative choices which are listed. It is concluded that rather an alternative role, it plays a complementary role to the paradigmatic formulation.
Shall I refuse my dinner because I do not fully understand the process of digestion? No, not if I am satisfied with the result. O. Heaviside, Electromagnetic Theory , vol. 2, 1899 Basic notions of quantum mechanics Quantum axioms Let us start with a brief recapitulation of quantum mechanics on the “how to” level. According to the standard lore, the instantaneous state of any quantum system (that is, everything that can be known about it at a given moment of time) is given by its wave function (state vector) – a complex-valued vector in some abstract Hilbert space; the nature of this space is determined by the system. All the observables (i.e., physical quantities defined for the system and determined by its state – e.g., the position or momentum of a free particle, the energy of an oscillator) are described by Hermitian operators defined in the same Hilbert space. All three elements – the Hilbert space, the state vector, and the set of observables – are necessary to describe the outcome of any experiment one could perform with the system. Since humans cannot directly observe the behaviour of quantum objects, these outcomes are also called measurements , being the result of using some classical apparatus in order to translate the state of a quantum system into the state of the apparatus, which can then be read out by the experimentalist. The classical (i.e., non-quantum) nature of the apparatus is essential, as we shall see in a moment.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.