The purpose of this paper is to elucidate the concept of an elementary particle in the first-quantized Standard Model. The emphasis is upon the mathematical structures involved, rather than numerical computations. After the general concepts and philosophical outlook are introduced in the opening section, Section 2 addresses the question of what a free elementary particle is, paying particular attention to the relationships between the configuration representation and momentum representation. Section 3 deals with gauge fields, Section 4 deals with interactions between particles and gauge fields, Section 5 deals with composite systems, and Section 6 deals with the representation of Baryons, Mesons, and Hadrons. Section 7 addresses the interpretational question of whether an elementary particle has only one intrinsic state, and Section 8 attempts to elucidate what an interacting elementary particle is in the Standard Model.
The purpose of this paper is to elucidate the concept of an elementary particle in the first-quantized Standard Model. The emphasis is upon the mathematical structures involved, rather than numerical computations. After the general concepts and philosophical outlook are introduced in the opening section, Section 2 addresses the question of what a free elementary particle is, paying particular attention to the relationships between the configuration representation and momentum representation. Section 3 deals with gauge fields, Section 4 deals with interactions between particles and gauge fields, Section 5 deals with composite systems, and Section 6 deals with the representation of Baryons, Mesons, and Hadrons. Section 7 addresses the interpretational question of whether an elementary particle has only one intrinsic state, and Section 8 attempts to elucidate what an interacting elementary particle is in the Standard Model.
Elementary Cycles Theory is a self-consistent, unified formulation of quantum and relativistic physics. Here we introduce its basic quantum aspects. On one hand, Newton's law of inertia states that every isolated particle has persistent motion, i.e. constant energy and momentum. On the other hand, the wave-particle duality associates a space-time recurrence to the elementary particle energy-momentum. Paraphrasing these two fundamental principles, Elementary Cycles Theory postulates that every isolated elementary constituent of nature (every elementary particle) must be characterized by persistent intrinsic space-time periodicity. Elementary particles are the elementary reference clocks of Nature. The space-time periodicity is determined by the kinematical state (energy and momentum), so that interactions imply modulations, and every system is decomposable in terms of modulated elementary cycles. Undulatory mechanics is imposed as constraint "overdetermining" relativistic mechanics, similarly to Einstein's proposal of unification. Surprisingly this mathematically proves that the unification of quantum and relativistic physics is fully achieved by imposing an intrinsically cyclic (or compact) nature for relativistic space-time coordinates. In particular the Minkowskian time must be cyclic. The resulting classical mechanics are in fact fully consistent with relativity and reproduces all the fundamental aspects of quantum-relativistic mechanics without explicit quantization. This "overdetermination" just enforces both the local nature of relativistic space-time and the wave-particle duality. It also implies a fully geometrodynamical formulation of gauge interactions which, similarly to gravity and general relativity, is inferred as modulations of the elementary space-time clocks. This brings novel elements to address most of the fundamental open problems of modern physics.
The basic concepts of space and time have undergone radical changes in the course of history, which render the Newtonian notions of elementary particles obsolete. This was inherent in the formulations of quantum mechanics, as the zitterbewegung of quantum-mechanical free particles are not the same as that of classical ones. The quantum field theories do not fare any better as particles and the corresponding fields are inseparable concepts. As physical concepts are becoming fuzzy, the definition of what constitutes an experimental "discovery" of an elementary particle is being revised. In these attempts, the quasi-particles, which are simple mathematical conveniences, tend to be reclassified as physical entities. The current model depictions of physical vacuum as superconducting medium do not accommodate a picture of elementary particles as individual entities. It is a situation where interactions and interactants become entangled and description of either one independent of the other is unattainable.
The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.
Quantum field theory (QFT) can be analyzed in terms of its mathematical structure, its conceptual system for physical descriptions, or its basic ontology. The analysis can be done logically or historically. In this chapter, only the genesis of the conceptual foundations of QFT relevant to its basic ontology will be treated; no discussion of its mathematical structures or its epistemological underpinnings will be given. Some conceptual problems, such as those related to probability and measurement, will be discussed, but only because of their relevance to the basic ontology of QFT, rather than their intrinsic philosophical interest. Here, by basic ontology I mean the irreducible entities that QFT is invented to describe. The often mentioned candidates for the basic ontology of QFT, in fact of the physical world, are the discrete particle and the continuous field. Another possible candidate (the spacetime point) has also been suggested recently (Redhead, 1983). Since the aim of this chapter is to analyze the historical process in which the conceptual foundations of QFT were laid down, rather than the logical structure of QFT which philosophers of the present day treat, no discussion of the last possibility will be given. The content of this chapter involves the formation and interpretation, in a roughly chronological order, of the concepts of the wave function, quantization, quantization of the field, the vacuum, interactions between fields, and renormalization. The first two topics will be discussed in relation to the quantization of the field, with their role being taken as the starting point of the conceptual development of QFT.
Philosophical reflection on quantum field theory has tended to focus on how it revises our conception of what a particle is. However, there has been relatively little discussion of the threat to the "reality" of particles posed by the possibility of inequivalent quantizations of a classical field theory, i.e., inequivalent representations of the algebra of observables of the field in terms of operators on a Hilbert space. The threat is that each representation embodies its own distinctive conception of what a particle is, and how a "particle" will respond to a suitably operated detector. Our main goal is to clarify the subtle relationship between inequivalent representations of a field theory and their associated particle concepts. We also have a particular interest in the Minkowski versus Rindler quantizations of a free Boson field, because they respectively entail two radically different descriptions of the particle content of the field in the very same region of spacetime. We shall defend the idea that these representations provide complementary descriptions of the same state of the field against the claim that they embody completely incommensurable theories of the field.
Well over a century after the discovery of the electron, we are still faced with serious conceptual issues regarding precisely what an electron is. Since the development of particle physics and the Standard Model, we have accumulated a great deal of knowledge about the relationships among various subatomic particles. However, this knowledge has not significantly aided in our understanding of the fundamental nature of any particular elementary subatomic particle. The fact that many particle properties, such as position, time, speed, energy, momentum, and component of spin, are observer-dependent suggests that these relevant variables do not represent properties per se, but rather the relationship between the observer and the observed. That is, they reflect details about how the electron influences the observer, and vice versa. Here we attempt to understand this by considering a simple model where particles influence one another in a discrete and direct fashion. The resulting framework, referred to as Influence Theory, is shown to faithfully reproduce a surprising amount of physics. While it would be naive to assume that the ideas presented here comprise anything resembling the final word on the matter, it is hoped that this work will demonstrate that a simple and understandable picture of particles, such as the electron, is indeed feasible and should be actively sought after.