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Relativistic Dynamics, Electromagnetic Field, and Spin, As All Quantum Effects JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 15, NO. 4, DECEMBER 2017 333 Relativistic Dynamics, Electromagnetic Field, and Spin, as All Quantum Effects Eliade Stefanescu Abstract—We consider a quantum particle as a space-time deformation is considered in the time wave packet in the coordinate space. When the dependent phase of a quantum particle, from the group conjugate wave packet in the momentum space is velocity we get the particle dynamics according to the considered, we find that the group velocities of these two general theory of relativity. In this way, the relativistic wave packets, which describe the particle dynamics, are dynamics, the electromagnetic field, and the spin of a in agreement with the Hamilton equations only if in the quantum particle are obtained only from the invariance time dependent phases one considers the Lagrangian of the time dependent phases of the particle wave instead of the Hamiltonian which leads to the functions. conventional Schrödinger equation. We define a Index Terms—Group velocity, Maxwell equations, relativistic quantum principle asserting that a quantum metric tensor, quantum particle, spin, wave packet. particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of 1. Introduction coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two The whole history of physics has been a strained one, due types of possible interactions: 1) An interaction with an to the fundamental problems raised by the particular behavior external field, by a modification of the time dependent of the physical world, unexpected for the common sense. phase differential with the terms proportional to the Although these problems have been largely discussed during differentials of the space-time coordinates multiplied the last century[1]-[7], recently they returned into very active with the components of this field four-potential, and 2) debates[8]-[16]. The fundamental prerequisites of the physical an interaction by a deformation of the space-time laws, as the principles of quantum mechanics, theory of coordinates, due to a gravitational field. From the relativity, and the mathematical methods of the least action, invariance of the time dependent phase with field symmetry transformations, and path integrals, raise interesting components, we obtain a mechanical force of the form of problems of consistency (non-contradiction), when they are Lorentz’s force, and three Maxwell equations: The simultaneously applied for the description of the various Gauss-Maxwell equations for the electric and magnetic physical properties. fluxes, and the Faraday-Maxwell equation for the An interesting property of a quantum particle is the spin, electromagnetic induction. When the fourth equation, which determines the symmetry of the particle wave function, Ampère-Maxwell, is considered, the interaction field and, accordingly, the particle statistics[17]. Important domains of takes the form of the electromagnetic field. For a low the applied physics and technology have been developed on propagation velocity of the particle waves, we get a this basis[18]-[25]. Recently, a new field, for energy production packet of waves with the time dependent phases from the environmental heat, has been developed[26]-[30]. For an proportional to the relativistic Hamiltonian, as in analytical description of such phenomena, the open quantum Dirac’s famous theory of spin, and a slowly-varying physics[31]-[39] has been approached by a more physical, amplitude with a phase proportional to the momentum microscopic method, of Ford, Lewis, and O’Connell[40], and and this velocity. In the framework of our theory, the has been developed in a new field, of explicit quantum master spin is obtained as an all quantum effect, without any equations for particles, with microscopic coefficients[41]-[44]. The additional assumption to the quantum theory. When a particle spin, obtained by Dirac from two different theories, the [45]-[47] quantum mechanics and the special relativity , here is Manuscript received October 19, 2016; revised June 7, 2017. understood in the framework of a unitary quantum relativistic E. Stefanescu is with the Center of Advanced Studies in Physics, theory[48],[49]. Bucharest 050711, Romania, and also with the Institute of We obtain the spin as a consequence of the wavy nature of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest 010702, Romania (e-mail: [email protected]). the quantum particles, where a particle is described by wave Digital Object Identifier: 10.11989/JEST.1674-862X.70623010 packets in the coordinate and momentum spaces, with the 334 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 15, NO. 4, DECEMBER 2017 group velocities in accordance with the Hamiltonian equations. function ; = + : Although these two wave packets are a different representation H0 (r p) T (p) U (r) (4) from the Schrödinger one, these two representations describe With the above Hamilton equations, the same particle distribution at a low velocity, v << c. In this d ∂H dr ∂H dp representation, the time dependent phase is described by the H (r;p) = 0 + 0 dt 0 ∂r dt ∂p dt particle Lagrangian, and not by the particle Hamiltonian as in (5) ∂H ∂H ∂H ∂H the Schrödinger representation. When a time dependent phase = 0 0 − 0 0 = 0; cutting the wave packet spectrum at a limit velocity c, and the ∂r ∂p ∂p ∂r invariance of this phase at an arbitrary change of the coordinate we obtain this Hamiltonian as a time constant called energy, frame, are considered, we obtain the relativistic dynamics of the H (r;p) = T (p) + U (r) = E: (6) quantum particle. 0 Describing the particle interaction with a field by a This means that the characteristic functions T (p) and U (r) Lagrangian variation with a vector potential A conjugated to of the wave packets (1) and (2) are the kinetic and potential the coordinate variations, and a scalar potential U, conjugated energies of a particle represented by these two wave to the time variation, we obtain the electromagnetic induction packets. We notice that for a constant potential, law (Faraday-Maxwell), and the laws for fluxes of the electric U (r) = const:; from (3b) and (3a) we obtain a constant and magnetic fields (Gauss-Maxwell). Considering a field momentum, p = const:; propagation velocity equal to the cut off velocity c of the d particle spectrum, we obtain also the fourth Maxwell equation, p = 0 (7) dt of the magnetic circuit (Ampère-Maxwell). At a low velocity and a constant velocity v << c, the Hamiltonian of a particle in a magnetic field gets a term –μ B, with a dynamic vector μ conjugated to this field, d ∂ s s r˙ ≡ r = T (p) = const: (8) which is the spin. dt ∂p We show that our representation describes the dynamics of From these relations, we notice that the momentum is a a quantum particle according to the general theory of relativity. function of velocity, p(r˙), which, in agreement with the Thus, for the wave packet of a quantum particle, we obtain Newtonian mechanics, is of the form the group acceleration on the geodesic. In a system of spherical coordinates, from the group velocity we obtain the particle p = M0r˙ (9) acceleration in agreement with Schwarzschild solution of the where M0 is a characteristic of the quantum particle called gravitational field, which, for a sufficiently large distance, takes mass. With this expression, from the Hamilton equation the form of the Newtonian gravitational force. (3b), d ∂ p = F(r)=˙ − U (r); (10) 2. Relativistic Dynamics of a Quantum dt ∂r Particle for a quantum particle as a wave packet under the action of In the framework of this theory, a quantum particle is the force F(r) of the electromagnetic field, we obtain the represented by a wave packet in the coordinate space frg, Newtonian equation Z i ∂ 1 fpr− T(p)−U(r) tg 3 = = − : ψ (r;t) = φ (p;t)e h¯ [ ] d p (1) M0r¨ F(r) ˙ U (r) (11) 0 (2πh¯)3=2 0 ∂r By multiplying this equation with ˙, and integrating with with the reverse Fourier transform in the conjugated r time, we obtain the particle energy, and the Hamiltonian momentum space fpg, Z function, i 1 − fpr− T(p)−U(r) tg 3 ; = ; h¯ [ ] : 2 2 φ0 (p t) 3=2 ψ0 (r t)e d r (2) p M0r˙ (2πh¯) H0 (p;r) = + U (r) = + U (r) = E: (12) 2M0 2 With the characteristic functions T (p) and U (r) of these At the same time, from the Hamilton equations (3a) and wave packets, we obtain the group velocities (3b), we obtain the Hamiltonian differential d ∂ ∂ r = T (p) = H (r;p) (3a) ∂H ∂H dt ∂p ∂p 0 dH = 0 dp + 0 dr = r˙dp − p˙dr: (13) 0 ∂p ∂r d ∂ ∂ p = − U (r) = − H (r;p) (3b) With the identity dt ∂r ∂r 0 d(pr˙) = r˙dp + pdr˙; (14) of the form of the Hamilton equations, for the Hamiltonian STEFANESCU: Relativistic Dynamics, Electromagnetic Field, and Spin, as All Quantum Effects 335 we eliminate the momentum differential, 3. Electromagnetic Dynamics as a − = ; = + : d(pr˙ H0) dL0 (r r˙) pdr˙ p˙dr (15) Characteristic of a Field Interacting We obtain the Lagrangian as a function of coordinates and with a Quantum Particle velocities, We consider the interaction of a field with a quantum ; = − ; ; L0 (r r˙) pr˙ H0 (p r) (16) particle, by a variation of the phase differential L0 (r;r˙)dt of and the Lagrange equation, this particle, with terms proportional to the coordinate differentials, and to the time differential, d ∂L0 ∂L0 = ; (17) L(r;r˙;t)dt = L (r;r˙)dt + eA(r;t)dr − eU (r)dt (24) dt ∂r˙ ∂r 0 for the particle described by the wave packets (1) and (2).
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