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 {r}, Newtonian equation ∫ i ∂ 1 {pr− T(p)−U(r) t} 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 {p}, ∫ function, i 1 − {pr− T(p)−U(r) t} 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). where A(r,t) is a vector potential, and U (r) is a scalar From (16), with (9) and (12), we notice that the phase potential, of this field, while e is the particle charge. With functions of the two wave packets (1) and (2), represent the this phase variation, the wave packets (19) and (20) take a Lagrangian, form: ∫ i p2 p2 1 [Pr−L(r,r˙,t)t] 3 = − − = − , ψ(r,t) = φ(P,t)e h¯ d P (25) L0 U (r) T (p) U (r) (18) h 3/2 M0 2M0 (2π¯) ∫ for a Newtonian dynamics of these two wave packets, i 1 − [Pr−L(r,r˙,t)t] 3 , = , h¯ which are of the form φ(P t) 3/2 ψ(r t)e d r (26) ∫ (2πh¯) i 1 [M0rr˙ −L0(r,r˙)t] 3 3 , = , h¯ with the Lagrangian: ψ0 (r t) / φ0 (r˙ t)e M0 d r˙ (19) (2πh¯)3 2 √ , , = − 2 − 2/ 2 + , − ∫ L(r r˙ t) M0c 1 r˙ c eA(r t)r˙ eU (r) (27) i 1 − [M0rr˙−L0(r,r˙)t] 3 φ (r˙,t) = ψ (r,t)e h¯ d r. (20) and the canonic momentum: 0 (2πh¯)3/2 0 ∂ M r˙ However, the two wave packets are still unphysical, because P = L(r,r˙,t) = √ 0 +eA(r,t) = p+eA(r,t). (28) r˙ − 2/ 2 they have an infinite spectrum, while in the real world ∂ 1 r˙ c everything is finite, even the Universe. With this expression and the group velocity of the wave A finite spectrum, for a wave propagation velocity |r˙| packet (26), we obtain the Lagrange equation: smaller than a limit velocity c, is obtained with the Lagrangian d ∂ ∂ √ P˙ = L(r,r˙,t) = L(r,r˙,t), (29) 2 2 2 dt ∂r˙ ∂r L0 (r˙) = −M0c 1 − r˙ /c , (21) of the form considered in the conventional relativistic while the Lagrangian differential is theory[50]. ∂L ∂L ∂L dL(r,r˙,t) = dr + dr˙ + dt With this Lagrangian, the time dependent phase variation is ∂r ∂r˙ ∂t proportional to the interval variation ∂L = P˙ dr + Pdr˙ + dt. (30) ∂t −L0dt = M0cds, (22) From the difference between the differential: which is of the form √ √ d(Pr˙) = r˙dP + Pdr˙ (31) ds = c 1 − r˙2/c2 dt = c2dt2 − dx2 − dy2 − dz2. (23) and the Lagrangian differential (30), we obtain the In this way, the heuristic relativistic hypothesis of the differential of the Hamiltonian: interval invariance is reduced to a quantum relativistic , , = − , , , principle, of the time dependent phase invariance at an arbitrary H (P r t) Pr˙ L(r r˙ t) (32) change of the coordinate frame. ∂H ∂H ∂H Compared with the conventional relativistic theory, based dH (P,r,t) = dP + dr + dt on a particular technique of measuring distances with light ∂P ∂r ∂t ∂ wave packets, of negligible dimensions, with a = r˙dP − P˙ dr − L(r,r˙,t)dt (33) phenomenological characteristic c, the quantum relativistic ∂t principle considered here refers to an invariance of a quantum which leads to the Hamilton equations: particle representation in an arbitrary coordinate frame. Based ∂ on this principle, the relativistic kinematics and dynamics are r˙ = H (P,r,t) (34) ∂P obtained. 336 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 15, NO. 4, DECEMBER 2017 ∂ with a source: P˙ = − H (P,r,t) (35) ∂r ρ(r) ∂2 ∂ ∂ = − U (r)ε, (48) H (P,r,t) = − L(r,r˙,t). (36) 2 ∂t ∂t ε0 ∂r

With the Lagrangian (27) and the momentum (28), from where ρ(r) is the charge density as ε0 is the electric (32) we obtain the Hamiltonian permittivity. At the same time, for the time derivative: ( ) M c2 d ∂ ∂ H (P,r,t) = √ 0 + eU (r) E(r,t) = r˙ E (r,t) + E (r,t), (49) − 2/ 2 dt ∂r ∂t √1 r˙ c = 2 2 + − , 2 + , with the charge density flow in the field direction: c M0 c [P eA(r t)] eU (r) (37) j(r) = r˙ρ(r), (50) as a function of the two potentials A(r,t) and U (r) of interaction with the quantum particle. and the vectorial identity: At the same time, from (28) we obtain the mechanical force ( ) [ ] ∂ ∂ ∂ acting of this particle, r˙ E(r,t) = r˙ E(r,t) − × [r˙ × E(r,t)] (51) ∂r ∂r ∂r | {z } d d d 0 F = p = P − e A(r,t), (38) e dt dt dt we obtain the time variation equation of the E field: while the canonic force in this equation is obtained from the d ∂ ε E(r,t) = j(r) + ε E(r,t). (52) Lagrange equation (29) with the Lagrangian function (27), 0 dt 0 ∂t d ∂ ∂ ∂ In this way, we found that a field interacting with a P = L(r,r˙,t) = e [A(r,t)r˙] − e U (r). (39) dt ∂r ∂r ∂r quantum particle is described by two vector fields E(r,t) and , With the vectorial formula B(r t), acting with a force (41) on this particle, and satisfying [ ] ( ) the Faraday-Maxwell law (44) of the electromagnetic ∂ ∂ ∂ induction, and the Gauss-Maxwell laws of the electric and r˙ × × A(r,t) = [rA˙ (r,t)] − r˙ A(r,t), (40) ∂r ∂r ∂r magnetic fields, (45) and (47). Considering an interaction field we obtain a mechanical force of Lorentz’s form satisfying an equation symmetric to the induction law (44) for the two vectors E and B of the electromagnetic field, we obtain Fe = eE(r,t) + er˙ × B(r,t) (41) the Ampère-Maxwell law: as a function of the field vector: d 1 ∂ ∂ ε0 E(r,t) = × B(r,t) = j(r) + ε0 E(r,t), (53) ∂ ∂ dt μ0 ∂r ∂t E(r,t) = − U (r) − A(r,t), (42) ∂r ∂t which, with the condition: and of induction vector: 1 c = √ (54) ∂ ε μ B(r,t) = × A(r,t). (43) 0 0 ∂r leads to a field propagating with the limit velocity c of the From (42), we find the Faraday-Maxwell law of the quantum particle (Fig. 1). electromagnetic induction,

∂ ∂ × E(r,t) = − B(r,t) , (44) ∂r ∂t while from (43), we obtain the Gauss-Maxwell law of the magnetic induction flow, ∂ B(r,t) = 0. (45) ∂r

With the gouge condition Fig. 1. Wave packet of a quantum particle, with a relativistic group ∂ velocity, and a bound spectrum for a velocity c, which is equal to A(r,t) = 0, (46) ∂r the propagation velocity of a field interacting with this particle. from (42) we find the Gauss-Maxwell law of the E field This means that the considered interaction with this field is flow: conserved during the whole evolution of the quantum particle. ∂ ρ(r) E(r,t) = , (47) Any difference between these two velocities does not make any ∂r ε 0 sense from the physical point of view. STEFANESCU: Relativistic Dynamics, Electromagnetic Field, and Spin, as All Quantum Effects 337 { [( ) ( ) ] }( ) 0 I 0 (r) 4. Spin as a Characteristic of the σ + + ψ1 c p − M0c eU (r) σ 0 0 I ψ2 (r) Quantum Dynamics ( ) ψ (r) For a low velocity, the wave packets (25) and (26) with = E 1 , (64) ψ (r) (32) and the Hamiltonian (37) take a form with a slowly- 2 varying amplitude, depending on the mechanical momentum p, the Pauli spin ∫ vector σ = I σ , and the unity matrix I. For a non-relativistic i i i i 1 − Pr˙t [Pr+H(P,r)t] 3 , = , h¯ h¯ ≈ 2 >> ψ(r t) / φ(P t) |{z}e e d P case, E M0c eU, we get the Schrödinger equations (2πh)3 2 ¯ slowly−varying [ ] ∫ amplitude σ 2 ( p) + = 1 i {Pr− −M c2+p2/2M −U(r) t} 3 eU (r) ψ1 (r) Ecψ1 (r) (65) ≈ , h¯ [ 0 0 ] 2M 3/2 φ(P t)e d P (55) 0 (2πh¯) ∫ i i [ ] 1 Pr˙t − [Pr+H(P,r)t] 3 2 , = , h¯ h¯ (σp) φ(P t) 3/2 ψ(r t) |{z}e e d r (56) + = , (2πh¯) eU (r) ψ2 (r) Ecψ2 (r) (66) slowly−varying 2M0 amplitude and the Hamiltonian: for the classical energy √ 2 Ec = E − M0c . (67) , , = 2 2 + 2 + H (P r˙ t) c M0 c p eU (r) With the commutation relations of the Pauli spin matrices, ≈ 2 + 2/ + M0c p 2M0 eU (r) (57) the term of the mechanical momentum (58) takes the form: as functions of the mechanical momentum p, which, σ 2 = 2 + σ × . according to (28), is ( p) p i (p p) (68) With the mechanical momentum (58) in the operator form: M r˙ p = P − eA(r,t) = √ 0 . (58) 1 − r˙2/c2 p = −ih¯∇ − eA(r,t), (69) A Schrödinger wave function can be considered for this and the expression (43) of the magnetic field, the vectorial particle, product in the expression (68) can be expessed as ∫ i 1 {Pr− M c2+p2/2M +eU(r) t} 3 p × p = (−ih∇ − eA) × (−ih∇ − eA) , = , h¯ [ 0 0 ] , ¯ ¯ ψE (r t) 3/2 ϕ(P t)e d P (70) (2πh¯) = ihe¯ (∇ × A + A × ∇) , (59) that is as an amplitude of the particle wave function (55), [ ] p × p = ihe¯ B. (71) i 2 2 M c + eU (r) t ψ(r,t) = e h¯ 0 ψ (r,t), (60) E With (68) and (71), the Schrödinger equations (65) and (66) which, besides this factor, includes a rapidly varying factor take an explicit form: that does not influence the particle distribution: [ ] 2 p 2 − μ B + eU (r) ψ (r) = Ecψ (r) (72) , = | , |2 = , . s 1 1 P(r t) ψ(r t) ψE (r t) (61) 2M0 However, the Schrödinger function (59), with a [ ] p2 Hamiltonian of a classical form, cannot be directly used for − μ B + eU (r) ψ (r) = E ψ (r) , (73) 2M s 2 c 2 the construction of a Schrödinger equation, the momentum 0 p in this Hamiltonian not being a canonic one, p , −ih¯∇. By with the energy linearization with Dirac’s operators α , α , α , and α , the 1 2 3 4 E = T (p) + eU (r) − μ B, (74) Hamiltonian (57) becomes c s where H = c(α1 p1 + α2 p2 + α3 p3 + α4 M0c) + eU (r), (62) he¯ as the Schrödinger wave function (59) takes a vectorial μ = σ. (75) s 2M form: 0 2 ( ) This energy includes the kinetic energy T (p) = p /2M0 of ψ1 (r) ψE (r,t) = , (63) the particle, the potential energy eU (r) in the electric field, ψ2 (r) − and the potential energy μsB of this particle in the as a solution of a Schrödinger equation of a matrix form, magnetic field, depending on a characteristic of this particle

μs, which, due to the classical interpretation as a current loop interaction with a magnetic field, is called spin. 338 JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 15, NO. 4, DECEMBER 2017 5. Quantum Particle in Gravitational so that these wave functions take a form  ( √ )  i 1 ∫  ( ) M c − g x˙i x j+c g x˙α x˙βt  Field ( ) 3 3  i h 0 2 i j αβ  M c φ x˙ ,t e ¯  i, = 0  , ,  1 2 3 [51] ψ x t 3/2  ∂(x˙ y˙ z˙) dx ˙ dx ˙ dx ˙ In the formalism of the general theory of relativity , the (2πh¯)   1, 2, 3 wave functions (19) and (20) with the Lagrangian (21) of a ∂(x˙ x˙ x˙ ) (83) quantum particle take the form:  ( √ )  i 1 ∫  ( ) − M c − g x˙i x j+c g x˙α x˙βt    ( )  i, h¯ 0 2 i j αβ  ( ) i ∫ 1 ψ x t e  ∫  M c −g x˙i x˙ j+1 ds i   1 2 3 ( )  i, h¯ 0 ( i j )  φ x˙ ,t =  , , dx dx dx 1 φ x˙ t e  3/2  ∂(x y z)  i, =  , ,  1 2 3 (2πh¯) ψ x t 3/2  ∂(x˙ y˙ z˙) dx ˙ dx ˙ dx ˙ (76) 1, 2, 3 (2πh¯) M3c3  ∂(x x x ) 0 ∂(x˙1, x˙2, x˙3) (84)   i ∫ ∫  ( ) − − i j+  ( )  i M0c ( gi j x˙ x˙ 1)ds with a group velocity 1 ψ x ,t e h¯  ⟨ ⟩ i, =  , ,  1 2 3 ⟨ ⟩ j φ x˙ t 3/2  ∂(x y z) dx dx dx (77) dx (2πh¯)   v j = ∂(x1, x2, x3) dt ⟨ ( √ )⟩ where ∂ √ = 2 ( ) c g x˙α x˙β i αβ α β ∂ gi j x˙ ds = gαβdx dx (78) ⟨ ⟩ ⟨ ⟩ x˙ j dx j = 2c √ = c , (85) is the differential of the space-time interval depending on α β 2 gαβ x˙ x˙ ds the contravariant coordinates xα and the covariant metric tensor g , while the dot over a variable designates the we obtain the group acceleration on a geodesic, αβ ⟨ ⟩ derivative of this variable with the space-time interval s. In ⟨ ⟩ d ⟨ ⟩ d2 x j ds these expressions, x, y, and z are the Euclidean coordinates, a j = v j c = c dt ds2 |{z}dt xα with α =0, 1, 2, 3 represent the space-time coordinates, c while x0 = ct is the time coordinate and xi with i = 1, 2, 3 ⟨ ⟩ ⟨ ⟩ d2 x j dxμ dxν are the spatial coordinates. With these definitions, from (78) = c2 = −c2Γ j , (86) ds2 μν ds ds we obtain the identity: where Γ j is the Christoffel symbol of the second kind. For α β μν dx dx 0 α β x gαβ x˙ x˙ = gαβ = 1, (79) a low velocity, in this expression only the term in ds ds matters, while the other terms are negligible. Lowering the j0 = 0 j j which, for a stationary gravitational field, g g , index j of Γμν, and taking into account the Christoffel = g j0 g0 j, is symbol of the first kind Γλμν as a function of the metric ( ) 2 tensor, the particle acceleration takes the form g x˙α x˙β = g x˙0 + g x˙i x˙ j = 1. (80) αβ 00 i j ⟨ ⟩ ( ) j 2 jλ 1 a = −c g g , + g , − g , , (87) Compared with the coordinate deformations induced by a 2 λ0 0 λ0 0 00 λ sufficiently low gravitational field, we consider a narrow which, for a stationary field, is wave packet and a low velocity: ⟨ ⟩ 1 j j = 2 jλ . dx a c g g00,λ (88) v j = << c (81) 2 dt For simplicity, we consider spherical coordinates, √ dx0 g ≈ c and g ≈ 1. (82) x1 = r , x2 = θ, x3 = ϕ, while, from the gravitation equation 00 dt 00 for the Ricci tensor, With these expressions, the phases of the wave functions = , (76) and (77) can be linearized: Rμν 0 (89) ∫ ( ) ∫ ( √ ) the metric tensor takes a diagonal form, depending on the − i j + = − i j + 2 2 + i j gi j x˙ x˙ 1 ds gi j x˙ dx c dt gi jdx dx parameter m of the gravitational interaction according to the ∫ ( √ ) Schwarzschild solution: = −g x˙idx j + cdt 1 + g x˙i x˙ j ( ) ( ) i j i j −1 [ ( )] 2m 2m ∫ ds2 = 1 − dt2 − 1 − dr2− i j 1 i j r r = −gi j x˙ dx + cdt 1 + gi j x˙ x˙ 2 2 2 − 2 2 2 . ∫ ( √ ) r dθ r sin θdϕ (90) 1 = − g x˙idx j + c g x˙α x˙βdt With this solution, the particle acceleration (88) is 2 i j αβ √ ( ) 1 ⟨ ⟩ 1 ∂ 2m mc2 = g x˙i x j + c g x˙α x˙βt, a j = c2g jλ 1 − = g j1 , (91) 2 i j αβ 2 ∂xλ r r2 STEFANESCU: Relativistic Dynamics, Electromagnetic Field, and Spin, as All Quantum Effects 339 which, with the metric tensor elements in (90), leads to the the famous Dirac’s theory of the particle spin, but not as an radial group acceleration, ingredient coming from a different theory, of relativity. For

( )− a low velocity, the wave function of a quantum particle can ⟨ ⟩ 2m 1 mc2 a1 = − 1 − (92) be considered as a packet of waves depending on the r r2 particle Hamiltonian, with a dynamics described by the ⟨ ⟩ well-known model of spin, and slowly varying amplitudes a2 = 0 (93) with phases proportional to momentum and velocity, while the dynamics of a high velocity particle takes a much more ⟨ ⟩ a3 = 0. (94) complicated form, as a packet of waves depending on the particle Lagrangian. In this way, the acceleration of a quantum particle in a We found that the time dependent phase differential is gravitational field can be understood as the acceleration of proportional to the differential of the space-time interval and the particle packet of waves satisfies the relativistic considered this interval as in the general theory of relativity. quantum principle of the invariant phases. From the group velocity of the particle wave packet, we obtained a the particle acceleration proportional to the 6. Conclusions gravitational field, as a deformation of the metric tensor according to the Schwarzschild solution. Essentially, we found We formulated a unitary relativistic quantum theory, based that the invariance of the space-time interval standing at the on a relativistic quantum principle asserting that the wave basis of the general theory of relativity, based on various function of a quantum particle has a finite frequency spectrum, reasonings with light beams, can be understood in the and that any time dependent phase variation is an invariant to framework of the quantum theory, based on the more any change of coordinates. In this formulation, the Hamiltonian fundamental principle of invariance of the time dependent usually considered in the time dependent phase of the wave phase of a quantum particle. function of a quantum particle is replaced by the particle Lagrangian. In this way, the principle of the least action of the References field theory is understood as a relativistic quantum principle of [1] L. Broglie, Théorie de la quantification dans la nouvelle the time dependent phase invariance of a quantum particle of Mécanique, Paris: Hermann et Cie, 1932. such a field. Unlike the conventional theory of relativity, which [2] W. Heisenberg, The Physical Principles of the Quantum refers to a classical particle, our theory refers to the waves of a Theory, New York: Dover Publications, Inc., 1949. quantum particle. These waves always include a rapidly [3] H. R. 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Eliade Stefanescu was born in Nehoiu, discovered that the penetrability of a potential barrier can be Buzau County, Romania in 1947. He increased by coupling to a dissipative system, and described the graduated from the Faculty of Electronics, decay spectra of some cold fission modes. As a senior scientist I, Section of Physicist Engineers in 1970 from 1997 he developed a microscopic theory of open quantum (M.S. degree), followed a one-year course systems, and discovered a physical principle for the heat of specialization in physics and technology conversion into usable energy. In 2014, he produced a unitary of semiconductor devices in the years of relativistic quantum theory. Now he is working with Advanced 1974 to 1975, and obtained his Ph.D. Studies in Physics Centre of the Romanian Academy at “Simion degree in theoretical physics from Institute of Atomic Physics, Stoilow” Institute of Mathematics of the Romanian Academy as a Bucharest, Romania in 1990. As a scientist from 1976, a senior senior scientist I. He is the member of the American Chemical scientist III from 1978, he worked in technology of semiconductor Society, Division of Physical Chemistry, Subdivision of Energy, devices. From 1978, he worked in physics of optoelectronic and of the Academy of Romanian Scientists. His research interests devices. From 1987, and from 1990 as a senior scientist II, he include open quantum physics with applications in theoretical and worked in the field of open quantum physics. In 1991 he condensed matter physics, and nuclear physics.