Starting from the formal expressions of the hydrodynamical (or “local”) quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce—in terms of the ordinary tensorial language—a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin- ${\frac{1}{2}}$ particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a (...) second part associated with the so-called Zitterbewegung, which is the spin “internal” motion observed in the center-of-mass frame (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a nonconstant spin function, but also in the Schrödinger case, when the wavefunction is a spin eigenstate. Thus, one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore, that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a nonuniform spin vector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schrödinger theory), also a component which is parallel to the curl of the spin vector. (shrink)

We present an approach to understanding the origin of inertia involving the electromagnetic component of the quantum vacuum and propose this as a step toward an alternative to Mach's principle. Preliminary analysis of the momentum flux of the classical electromagnetic zero-point radiation impinging on accelerated objects as viewed by an inertial observer suggests that the resistance to acceleration attributed to inertia may be at least in part a force of opposition originating in the vacuum. This analysis avoids the ad hoc (...) modeling of particle-field interaction dynamics used previously by Haisch, Rueda, and Puthoff (Phys. Rev. A 49, 678, (1994)) to derive a similar result. This present approach is not dependent upon what happens at the particle point, but on how an external observer assesses the kinematical characteristics of the zero-point radiation impinging on the accelerated object. A relativistic form of the equation of motion results from the present analysis. Its manifestly covariant form yields a simple result that may be interpreted as a contribution to inertial mass. We note that our approach is related by the principle of equivalence to Sakharov's conjecture (Sov. Phys. Dokl. 12, 1040, (1968)) of a connection between Einstein action and the vacuum. The argument presented may thus be construed as a descendant of Sakharov's conjecture by which we attempt to attribute a mass-giving property to the electromagnetic component—and possibly other components—of the vacuum. In this view the physical momentum of an object is related to the radiative momentum flux of the vacuum instantaneously contained in the characteristic proper volume of the object. The interaction process between the accelerated object and the vacuum (akin to absorption or scattering of electromagnetic radiation) appears to generate a physical resistance (reaction force) to acceleration suggestive of what has been historically known as inertia. (shrink)

The idea that the mass m of an elementary particle is explained in the semi-classical approximation by quantum-mechanical zero-point vacuum fluctuations has been applied previously to spin-1/2 fermions to yield a real and positive constant value for m, expressed through the spinorial connection Γ i in the curved-space Dirac equation for the wave function ψ due to Fock. This conjecture is extended here to bosonic particles of spin 0 and spin 1, starting from the basic assumption that all fundamental fields (...) must be conformally invariant. As a result, in curved space-time there is an effective scalar mass-squared term $m_{0}^{2}=-R/6=2\varLambda_{\mathrm{b}}/3$ , where R is the Ricci scalar and Λ b is the cosmological constant, corresponding to the bosonic zero-point energy-density, which is positive, implying a real and positive constant value for m 0, through the positive-energy theorem. The Maxwell Lagrangian density $\mathcal{L} =- \sqrt{-g}F_{ij}F^{ij}/4$ for the Abelian vector field F ij ≡A j,i −A i,j is conformally invariant without modification, however, and the equation of motion for the four-vector potential A i contains no mass-like term in curved space. Therefore, according to our hypothesis, the free photon field A i must be massless, in agreement with both terrestrial experiment and the notion of gauge invariance. (shrink)

The goal of this paper is to introduce the notion of a four-dimensional time in classical mechanics and in quantum mechanics as a natural concept related with the angular momentum. The four-dimensional time is a consequence of the geometrical relation in the particle in a given plane defined by the angular momentum. A quaternion is the mathematical entity that gives the correct direction to the four-dimensional time.Taking into account the four-dimensional time as a vectorial quaternionic idea, we develop a set (...) of generalizations and conclusions over the mechanics.In quantum mechanics, the four-dimensional time appears as an observable. (shrink)

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. The model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh frequency, and the (...) possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment. (shrink)

The generator of electromagnetic gauge transformations in the Dirac equation has a unique geometric interpretation and a unique extension to the generators of the gauge group SU(2) × U(1) for the Weinberg-Salam theory of weak and electromagnetic interactions. It follows that internal symmetries of the weak interactions can be interpreted as space-time symmetries of spinor fields in the Dirac algebra. The possibilities for interpreting strong interaction symmetries in a similar way are highly restricted.

In the context of our recently developed emergent quantum mechanics, and, in particular, based on an assumed sub-quantum thermodynamics, the necessity of energy quantization as originally postulated by Max Planck is explained by means of purely classical physics. Moreover, under the same premises, also the energy spectrum of the quantum mechanical harmonic oscillator is derived. Essentially, Planck’s constant h is shown to be indicative of a particle’s “zitterbewegung” and thus of a fundamental angular momentum. The latter is identified with quantum (...) mechanical spin, a residue of which is thus present even in the non-relativistic Schrödinger theory. (shrink)

The concept of mass is one of the most fundamental notions in physics, comparable in importance only to those of space and time. But in contrast to the latter, which are the subject of innumerable physical and philosophical studies, the concept of mass has been but rarely investigated. Here Max Jammer, a leading philosopher and historian of physics, provides a concise but comprehensive, coherent, and self-contained study of the concept of mass as it is defined, interpreted, and applied in contemporary (...) physics and as it is critically examined in the modern philosophy of science. With its focus on theories proposed after the mid-1950s, the book is the first of its kind, covering the most recent experimental and theoretical investigations into the nature of mass and its role in modern physics, from the realm of elementary particles to the cosmology of galaxies.The book begins with an analysis of the persistent difficulties of defining inertial mass in a noncircular manner and discusses the related question of whether mass is an observational or a theoretical concept. It then studies the notion of mass in special relativity and the delicate problem of whether the relativistic rest mass is the only legitimate notion of mass and whether it is identical with the classical mass. This is followed by a critical analysis of the different derivations of the famous mass-energy relationship E = mc2 and its conflicting interpretations. Jammer then devotes a chapter to the distinction between inertial and gravitational mass and to the various versions of the so-called equivalence principle with which Newton initiated his Principia but which also became the starting point of Einstein's general relativity, which supersedes Newtonian physics. The book concludes with a presentation of recently proposed global and local dynamical theories of the origin and nature of mass.Destined to become a much-consulted reference for philosophers and physicists, this book is also written for the nonprofessional general reader interested in the foundations of physics. (shrink)