The theory of special relativity (TSR) exhibits an unquestionable success, at the expense of an unresolved axiomatic conflict with functional analysis and operator theory, as this paper demonstrates. These mathematical disciplines —amongst the newest developments in the field— could not possibly be incorporated into the original formulation of the TSR because they were in an incipient state at the turn of the 20th century when the TSR was being formulated, maturing only decades later.
The theory of special relativity (TSR) exhibits an unquestionable success, at the expense of an unresolved axiomatic conflict with functional analysis and operator theory, as this paper demonstrates. These mathematical disciplines —amongst the newest developments in the field— could not possibly be incorporated into the original formulation of the TSR because they were in an incipient state at the turn of the 20th century when the TSR was being formulated, maturing only decades later.
Special relativity is a fascinating and challenging branch of physics. It describes the physics of the high velocity/high energy regime, frequently turning up phenomena that appear paradoxical in view of our everyday experience. In this book we will be quite selective in our presentation of the theory of special relativity: we will concentrate on those features that we consider necessary for the later applications to relativistic quantum chemistry. We do this in good conscience, knowing that there is a vast literature on the subject, catering to a wide range of audiences—from the quite elementary to the very sophisticated. A few examples are listed in the reference list, but a visit to any nearby physics library will provide an ample selection of reading material for those wishing to delve deeper into the matter. In the present chapter we adopt a minimalist approach. We develop some of the basic concepts and formulas of special relativity, building on a rather elementary level of basic physics. The aim is to provide a sufficient foundation for those who want to proceed as quickly as possible to the more quantum chemical parts of the text. In later chapters we will introduce more advanced tools of physics and revisit some of the subjects treated here. The theory of special relativity deals with the description of physical phenomena in frames that move at constant velocity relative to each other. The classroom is one such frame, the car passing at constant speed outside the classroom is another. The trajectory of a ball being thrown up vertically in the car will look quite different whether we describe it relative to the interior of the car or relative to the interior of the classroom. In particular we will be concerned with inertial frames. We define an inertial frame as a frame where spatial relations are Euclidean and where there is a universal time such that free particles move with constant velocities. In classical Newtonian mechanics, relations between the spatial parameters and time in two inertial frames S and S’ are expressed in terms of the Galilean transformations.
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.
Classical physics has enabled the acquisition of significant knowledge of the physical properties of nature on a standard macroscopic scale. These achievements were driven by use of the causal ontological approach (proposed originally by Aristotle) to formulate models of physical reality. At the beginning of the 20th century, however, the physics community began to prefer models based on a fundamentally different approach to human knowledge. Copenhagen quantum mechanics (CQM) was used to describe the micro-world. The special theory of relativity was used to describe the kinematics of objects moving at high velocity values in both the macroscopic and microscopic regions. This phenomenological approach to knowledge has been more focused on how things appear - instead of their actual properties and causal sequence. In the middle of the 20th century, the causal ontological approach was used to develop a significant scientific advance: the systematization of fundamental strongly interacting particles on the basis of unified algebra in three-dimensional isotopic spin space for spin values 1, 1/2, and 0. However, it was later strongly deformed under the influence of the phenomenological approach and the quark model. This paper will show that practically all contemporary theoretical models of physical reality contain mistakes or unresolved problems. Further scientific progress can be obtained if (and only if) scientists return to the successful causal ontological approach and falsification. Classical physics may be slightly generalized to enable the description of inertia mass increase in dependence on velocity, replacing the relativity theory and CQM. New assumptions may then be employed using generalized classical physics (GCP) to formulate new descriptions of observed phenomena that were previously inaccurately interpreted and used to promote fundamentally inadequate theories.
The origin of the idea of rationality as a fundamental model of human behaviour (the concept of homo oeconomicus) that was introduced during the neoclassical revolution in economics and that implied the mathematization of economics thereby adopting the mathematical scheme of and borrowing some fundamental conception from classical mechanics, as well as the crisis of this rationality model that followed in the latest decades are here discussed. The origin of the least action principle in 18th century is first described. The least action principle was the basis that lead to the development of Lagrangean dynamics in classical mechanics that, on its turn, entered economics in 19th century as a fundamental method to describe the behaviour of a postulated rational agent and was the basis for the development of mainstream economics in 20th century. The epistemologically unjustified transfer of concepts from mechanics, in particular in Walras? work, is stressed. In the second half of 20th century experimental psychology slowly entered economics and several paradoxes in the real agents? choices were shown giving strong evidence of rationality?s axioms violation in the real behaviour of people. By mid-20th century, quantum mechanics had already developed a highly effective mathematical apparatus to describe the wave-particle behaviour of quantum objects and in particular the phenomenon of probability interference that is clearly shown in the double-slit experiment with single electrons, as well as with photons. Probability interference is peculiar to quantum mechanics and is extraneous to the Kolmogorov axiomatic theory of classical probability. The recent idea of applying the quantum mechanics concepts and formalism, including quantum probability interference and superposition of states, to the description of mental processes is discussed. The aim of this new field of research, which is developing along the lines of the centuries-old question of the mind-body relation, is to create a general framework where cognition might be modeled. Quantum cognition is therefore introduced as a possible answer to the rationality violation paradoxes shown by experiments on ambiguous figures recognition and on the effects brought about by conjunction and disjunction of concepts.
This paper constitutes a background to the paper 'Quantum mechanics as space-time statistical mechanics?', arXiv:quant-ph/0501133, presented previously by the author. But it is also a free-standing and self-contained paper. The purpose of this paper is to give the reader an increased and a deeper understanding of the special theory of relativity, and the ideas lying behind the above mentioned paper. We will here consider, discuss, define, analyse, and explain things such as, e.g., the constancy of the speed of light, synchronization, simultaneity, absolute simultaneity, absolute space and time, the ether, and spacetime. Albert Einstein's original version of the special theory of relativity is fundamentally an operational theory, free from interpretation. But the old Lorentzian interpretation and the standard spacetime interpretation of the special theory of relativity will also be considered. This paper also discusses and analyses aspects of the philosophy of science that in my opinion are relevant for an understanding of the special theory of relativity.