The evolution of quantum mechanics through the 1920s was profoundly messy. Some physicists believed that it was necessary to throw out much of the conceptual baggage that early quantum mechanics tended to carry around with it and re-establish the theory on much firmer ground. It was at this critical stage that the search for deeper insights into the underlying reality was set aside in favour of mathematical expediency. All the conceptual problems appeared to be coming from the wavefunctions. But whatever was to replace them needed to retain all the properties and relationships that had so far been discovered. Dirac and von Neumann chose to derive a new quantum formalism by replacing the wavefunctions with state vectors operating in an abstract Hilbert space, and formally embedding all the most important definitions and relations within a system of axioms.
The evolution of quantum mechanics through the 1920s was profoundly messy. Some physicists believed that it was necessary to throw out much of the conceptual baggage that early quantum mechanics tended to carry around with it and re-establish the theory on much firmer ground. It was at this critical stage that the search for deeper insights into the underlying reality was set aside in favour of mathematical expediency. All the conceptual problems appeared to be coming from the wavefunctions. But whatever was to replace them needed to retain all the properties and relationships that had so far been discovered. Dirac and von Neumann chose to derive a new quantum formalism by replacing the wavefunctions with state vectors operating in an abstract Hilbert space, and formally embedding all the most important definitions and relations within a system of axioms.
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.
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.
The Quantum Cookbook shows that whilst quantum mechanics is mathematically challenging, some basic knowledge and a bit of effort will carry you a long way. It also explains how quantum mechanics was derived from the physics. The abstract formalism based on state vectors in Hilbert space was introduced only when it was deemed desirable to lend the theory greater mathematical consistency, and to reject some of its historical baggage. The best way to come to terms with this formalism is to understand how and why it came about. Debates about interpretation continue to this day and, by providing some historical context, you should get the impression that any lack of comprehension of its meaning on your part is absolutely not your fault. Quantum mechanics challenges our comprehension of what any (and all) scientific theories are meant to be telling us about the nature of reality. It’s okay to have doubts.
I n the early 20th century, the groundbreaking theories of quantum mechanics and special relativity emerged at roughly the same time.At first, both theories appeared to be entirely incompatible with each other.While the quantum equations written out by Erwin Schrödinger and Wolfgang Pauli dealt with the behaviours of electrons, atoms, and molecules moving at speeds encountered in classical physics, the rules of special relativity first imagined by Albert Einstein described the behaviour of matter moving at close to the speed of light.Initially, this apparent clash of theories presented a crisis in physics.However, in the late 1920s, further theoretical work carried out by physicists including Paul Dirac, Oskar Klein, and Walter Gordon culminated in a new set of equations, which proved definitively that quantum mechanics and special relativity are compatible with each other after all. An educational challengeYet with this groundbreaking discovery came an entirely new problem.In adding a new layer of complexity to the equations of non-relativistic quantum mechanics, the more complete theories of relativistic
. We put forward a new take on the logic of quantum mechanics, following Schrödinger's point of view that it is composition which makes quantum theory what it is, rather than its particular propositional structure due to the existence of superpositions, as proposed by Birkhoff and von Neumann. This gives rise to an intrinsically quantitative kind of logic, which truly deserves the name ‘logic’ in that it also models meaning in natural language, the latter being the origin of logic, that it supports automation, the most prominent practical use of logic, and that it supports probabilistic inference. The physics and the logic of quantum-ish logic. In 1932 John von Neumann formalized Quantum Mechanics in his book “Mathematische Grundlagen der Quantenmechanik”. This was effectively the official birth of the quantum mechanical formalism which until now, some 75 years later, has remained the same. Quantum theory underpins so many things in our daily lives including chemical industry, energy production and information technology, which arguably makes it the most technologically successful theory of physics ever. However, in 1935, merely three years after the birth of his brainchild, von Neumann wrote in a letter to American mathematician Garrett Birkhoff: “I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.” (sic)—for more details see [73]. Soon thereafter they published a paper entitled “The Logic of Quantum Mechanics” [13]. Their ‘quantum logic’ was cast in order-theoretic terms, very much in the spirit of the then reigning algebraic view of logic, with the distributive law being replaced with a weaker (ortho)modular law. This resulted in a research community of quantum logicians [68, 71, 47, 30]. However, despite von Neumann's reputation, and the large body of research that has been produced in the area, one does not find a trace of this activity neither in the mainstream physics, mathematics, nor logic literature. Hence, 75 years later one may want to conclude that this activity was a failure. What went wrong? The mathematics of it. Let us consider the raison d'être for the Hilbert space formalism.
Abstract The standard von Neumann-Dirac formulation of quantum mechanics is presented as a set of five basic rules. We discuss each rule is discussed in turn paying particular attention to the conceptual history of the theory. Of central importance is the standard interpretation of states (the eigenvalue-eigenstate link) and the dynamical laws of the theory (the random collapse dynamics and the deterministic linear dynamics) and how the interpretation and dynamics work together to predict and explain the results of basic quantum experiments. While the focus is on the behavior of electrons, we also briefly consider how the theory uses the same mathematical formalism to treat other phenomena like the behavior of neutral K mesons and qbits in a quantum computer.