The major developments in twentieth-century physics, grouped under the general headings of quantum theory and relativity, have provoked much discussion and re-evaluation of the subject-object relationship in natural science and, to a lesser extent, the role of models in the development and articulation of physical theory. This chapter attempts to approach the subject-object problem from a standpoint developed from an analysis of the nature and function of models as a basis of explanation in physics, in different historical periods and areas of study within the field. A distinction between several major types of model observable at work in classical physics relates to the amount of detail contained in the pictorial description of the model as opposed to the amount of detail in the observational situation to which it relates. The quantum-mechanical model, employs in general a plurality of pictorial representations within a single model, which separates in simple situations into the several classical models mathematically related and restricted.
The major developments in twentieth-century physics, grouped under the general headings of quantum theory and relativity, have provoked much discussion and re-evaluation of the subject-object relationship in natural science and, to a lesser extent, the role of models in the development and articulation of physical theory. This chapter attempts to approach the subject-object problem from a standpoint developed from an analysis of the nature and function of models as a basis of explanation in physics, in different historical periods and areas of study within the field. A distinction between several major types of model observable at work in classical physics relates to the amount of detail contained in the pictorial description of the model as opposed to the amount of detail in the observational situation to which it relates. The quantum-mechanical model, employs in general a plurality of pictorial representations within a single model, which separates in simple situations into the several classical models mathematically related and restricted.
The world of physical systems at the most fundamental levels is replete with efficient, interesting models possessing sufficient ability to represent the reality to a considerable extent. So far, quantum mechanics (QM) forming the basis of almost all natural phenomena, has found beyond doubt its intrinsic ingenuity, capacity and robustness to stand the rigorous tests of validity from and through appropriate calculations and experiments. No serious failures of quantum mechanical predictions have been reported, yet. However, Albert Einstein, the greatest theoretical physicist of the twentieth century and some other eminent men of science have stated firmly and categorically that QM, though successful by and large, is incomplete. There are classical and quantum reality models including those based on consciousness. Relativistic quantum theoretical approaches to clearly understand the ultimate nature of matter as well as radiation have still much to accomplish in order to qualify for a final theory of everything (TOE). Mathematical models of better, suitable character as also strength are needed to achieve satisfactory explanation of natural processes and phenomena. We, in this paper, discuss some of these matters with certain apt illustrations as well.
The relationship between physical notions and the structures expressing them is an issue of primary importance the history and philosophy of pre-modern natural philosophy and of modern science. Historical evidence suggests that, when investigating the relationship between physics and mathematics, one should pay attention to the material, sensually perceivable forms mediating it: spoken or written words, drawings, artefacts, descriptions of experimental set-ups or, as the case discussed the present paper, symbolic notation. The reason for this is that, historically, physical notions and their expressions have very often been developed as one, and only distinguished from each other at a later stage. This was for example the case with what we today regard as mathematical and classical mechanics, which were developed the course of the eighteenth century as the discipline of the analysis of motion. When considered from a later point of view, both the physical notions and the concepts involved historical developments appear as being in flux, vague, and, as far as the side is concerned, non-rigorous. In such cases, verbal description, mechanical artefacts or symbols do not simply mediate between physical notions and structures, as might be the case at a later stage, but represent both of them one, allowing the historical actors to construct and manipulate physical-mathematical meaning by following, at times, their physical reasoning and, at times, their criteria of rigour. The history of quantum mechanics offers many examples of how new physics and new mathematics can emerge as one, with symbolic notation playing a central role shaping both. Among the many creators of quantum theory, Paul Dirac was particularly sensitive to questions of notation:
As we have seen in Section 8.4, the basis of our perception of an objective reality is statistical mechanics and field theory. These are routinely used in equilibrium and nonequilibrium statistical thermodynamics (see, for example, Calzetta & Hu [59]), which govern the macroscopic laws of everyday life. Thus the quantum theory of fields must be the fundamental description of Nature. Therefore, the simpler quantum mechanics of particles is necessarily a derived description. This chapter discusses the extent to which a particle picture of matter and radiation is appropriate. In relativistic quantum field theory, particles appear only as asymptotic excitations of the fields. The quantum particle concept therefore makes mathematical (and intuitive) sense only under very special circumstances, namely in those where a system actually behaves like particles do: - As quasiparticles, they are useful in the domain, where the geometric optics perspective applies. - Before and after a scattering event in a dilute gas or a particle collider ring, particles can be considered as being essentially free. During the very short interaction time itself, the scattering process cannot be described by a particle picture, but needs a hydromechanical description or a kinetic description via Kadanoff-Baym equations. Already at the very beginning of modern quantum mechanics, when Born found his probability interpretation, the scattering process of single particles in an external classical field was understood (see Section 14.1 of the Appendix) to be a wave process relating initially and finally stationary particles, freely moving in their center of mass frame. It is a historical accident that one continues to use the name particle in the many microscopic situations, where it is grossly inappropriate to think of it with the classical meaning of a tiny bullet moving through space. When we restrict the use of the particle concept to where it is appropriate, or when we do not think of particles as “objects”, in both cases all mystery is gone, and the foundations become fully rational. The inappropriate focus on the particle aspect of quantum mechanics created the appearance of mystery; common sense is restored by focussing instead on the field aspect. To substantiate the above summary view, we give in Section 12.1 an analysis of the photoelectric effect and in Section 12.2 a review of the origin of particle tracks. This leads in Section 12.3 to a discussion about the reality of particles. In the remainder of the chapter, we discuss the relation between fields and particles in more detail. Section 12.4 poses the problem how the nonrelativistic multiparticle description arises from relativistic quantum field theory. Section 12.5 considers the case of free quantum fields, and Section 12.6 the case of interacting quantum fields. Section 12.7, the final section, discusses the semiclassical view of quasiparticles in the approximation familiar from geometric optics.
A simplified scheme is used to discuss the idea of the mathematical formalism of physical theory. The physical theory is associated with a certain mathematical structure which makes it possible to define objects, i.e., the subject of investigation, and some of their characteristics. The task of the theory is to find the characteristics of an object from its input data that implicitly define it, on the basis of the laws from the nature of the basic structure. It stands to reason that the task includes the finding of the indicated laws dictated by the structure, the elucidation of the general properties of the objects, the establishment of the relationship between the characteristics, etc. The finiteness of the models used to illustrate the given scheme is brought about by the replacement of the space or the space-time continuum of "real" physics by a finite set.
This study offers a comprehensive analysis of David Bohm’s article “A New Theory of the Relationship of Mind and Matter” (Philosophical Psychology, 1990) — one of the most original attempts of the twentieth century to overcome Cartesian dualism on a physical foundation. The paper proceeds in sequence through the historical context of the mind–body problem, the physical basis of the causal interpretation of quantum mechanics, the central concept of active information, and the implicate order. Bohm’s framework is then compared with several major contemporary approaches: Orchestrated Objective Reduction (Hameroff & Penrose, 2014), the holographic brain model (Pribram, 1991), Integrated Information Theory (Tononi, 2004), and the Global Workspace model (Baars, 1988; Dehaene, 2014). A dedicated section examines the relationship between Bohm’s ideas and Rovelli’s Relational Quantum Mechanics (Rovelli, 1996), as well as Loop Quantum Gravity (Rovelli, 2004; Rovelli & Smolin, 1995), and Qbism, developed by Christopher Fuchs, Rüdiger Schack, and Carlton Caves (Fuchs, 2002; Fuchs et al., 2014). The concluding section evaluates the compatibility of Bohm’s ideas with quantum field theory and modern quantum information science. The principal conclusion is that Bohm’s 1990 article does not constitute a complete theory of consciousness, yet it anticipates with remarkable prescience the information-ontological tendencies of the twenty-first century and may best be understood as a philosophical-physical metatheory whose relevance has only grown since its publication.
The argument of this chapter is governed by the double meaning of its title: (a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over ℂ and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity, and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck, following B. Riemann's comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.