Quantum mechanics comes in a variety of different forms and formalisms. There is wave mechanics, matrix mechanics, Dirac's version of quantum mechanics, von Neumann's version of quantum mechanics. The philosopher of physics has to ask not only what quantum mechanics means, but also what it is. Orthodoxy in the philosophy of physics treats quantum mechanics as the general theory of microphysics which takes Hilbert space as its state space and which associates observables with certain special operators on that space. This amounts to a decision as to what quantum mechanics is. However it is a far from arbitrary decision. For the Hilbert space, or von Neumann, formalism summarizes and captures everything that is in the other versions of quantum mechanics, without any of the dubious mathematics of, for example, Dirac's version. Contemporary philosophers of physics take a very formal view of what quantum mechanics is, a view which has many advantages and some disadvantages. Whatever the disadvantages of focussing on the formalism of quantum mechanics - and it may lead to losing sight of the ‘physics' and possibly to a bias against Bohrian Copenhagenism - it is certainly true that the literate philosopher of physics must have some acquaintance with Hilbert space. There are in fact several topics we have to deal with, though we deal with them only very cursorily. (This chapter is no substitute for Jauch's classic text Foundations of Quantum Mechanics .)
Quantum mechanics comes in a variety of different forms and formalisms. There is wave mechanics, matrix mechanics, Dirac's version of quantum mechanics, von Neumann's version of quantum mechanics. The philosopher of physics has to ask not only what quantum mechanics means, but also what it is. Orthodoxy in the philosophy of physics treats quantum mechanics as the general theory of microphysics which takes Hilbert space as its state space and which associates observables with certain special operators on that space. This amounts to a decision as to what quantum mechanics is. However it is a far from arbitrary decision. For the Hilbert space, or von Neumann, formalism summarizes and captures everything that is in the other versions of quantum mechanics, without any of the dubious mathematics of, for example, Dirac's version. Contemporary philosophers of physics take a very formal view of what quantum mechanics is, a view which has many advantages and some disadvantages. Whatever the disadvantages of focussing on the formalism of quantum mechanics - and it may lead to losing sight of the ‘physics' and possibly to a bias against Bohrian Copenhagenism - it is certainly true that the literate philosopher of physics must have some acquaintance with Hilbert space. There are in fact several topics we have to deal with, though we deal with them only very cursorily. (This chapter is no substitute for Jauch's classic text Foundations of Quantum Mechanics .)
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation: There are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be the subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. This latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1] . Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2] , Chapter 3 of [3] , or Chapter 2 of [4] . We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section. General physical system The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc. Quantum space of states The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ ψ 〈 – or ∣ ψ(t) 〈 if one wishes to make the time dependence explicit.
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.