The existence of any elementary particle in universe requires the existence of some region of universe occupied by it. By taking the volume of this occupied region, the author will reformulate the relativistic quantum field theory using new 3-dimensional region-like idealization of elementary particles and hereinafter will call the total volume of all regions occupied by the elementary constituent particles of the quantum system the occupied volume. Also the author will call the set of all regions of universe filled by elementary constituent particles of the quantum system the occupied path. Always any quantum system is existed at a head of its occupied path. This path is growing by mutual filling and leaving regions of universe by its elementary constituent particles. The conservation of this elementary constituent particle requires the conservation of its occupied volume during this process. This requirement could be summarized by the following conditions: 1) the total volume of all regions of universe filled by the elementary constituent particles of the quantum system minus the total volume of all regions of universe left by these elementary constituent particles must be equal to the occupied volume of the quantum system; 2) the total increase in the occupied volume of the quantum system due to the absorption of another elementary particles from outside its occupied regions minus the total decreasing in its occupied volume due to the emission of another elementary particles outside its occupied regions must be equal to the occupied volume of it. The wave-particle duality of the elementary constituent particles implied accumulation of them as the finite set of interfered waves. This accumulation of elementary constituent particles causes the absolute probabilistic nature of event of finding the elementary consistent particle in specified interfered wave, and hence the mathematical representation of this interfered wave should take into account the value of probability amplitude of finding an elementary particle inside the region occupied specified interfered wave. In quantum theory this probability amplitude corresponds to complex amplitude of the wave function of interfered wave. Also in Hilbert’s representation of the quantum theory these wave functions are representing the components of the quantum state vector. In this paper the author will develop the transformation theory of the region-like quantum state of the quantum system.
The existence of any elementary particle in universe requires the existence of some region of universe occupied by it. By taking the volume of this occupied region, the author will reformulate the relativistic quantum field theory using new 3-dimensional region-like idealization of elementary particles and hereinafter will call the total volume of all regions occupied by the elementary constituent particles of the quantum system the occupied volume. Also the author will call the set of all regions of universe filled by elementary constituent particles of the quantum system the occupied path. Always any quantum system is existed at a head of its occupied path. This path is growing by mutual filling and leaving regions of universe by its elementary constituent particles. The conservation of this elementary constituent particle requires the conservation of its occupied volume during this process. This requirement could be summarized by the following conditions: 1) the total volume of all regions of universe filled by the elementary constituent particles of the quantum system minus the total volume of all regions of universe left by these elementary constituent particles must be equal to the occupied volume of the quantum system; 2) the total increase in the occupied volume of the quantum system due to the absorption of another elementary particles from outside its occupied regions minus the total decreasing in its occupied volume due to the emission of another elementary particles outside its occupied regions must be equal to the occupied volume of it. The wave-particle duality of the elementary constituent particles implied accumulation of them as the finite set of interfered waves. This accumulation of elementary constituent particles causes the absolute probabilistic nature of event of finding the elementary consistent particle in specified interfered wave, and hence the mathematical representation of this interfered wave should take into account the value of probability amplitude of finding an elementary particle inside the region occupied specified interfered wave. In quantum theory this probability amplitude corresponds to complex amplitude of the wave function of interfered wave. Also in Hilbert’s representation of the quantum theory these wave functions are representing the components of the quantum state vector. In this paper the author will develop the transformation theory of the region-like quantum state of the quantum system.
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.
This chapter considers a more realistic variation of the measuring procedure, taking into account the fact that the measured particles become entangled with the measuring apparatus at some finite time. The results are also generalized to quantum field theory, where quantum states are represented by wave functions depending on an infinite number of space-time coordinates. Thus, to make Bohmian mechanics covariant, one needs first to reformulate the standard quantum mechanics (QM) in a covariant way such that time is treated on an equal footing with space. Another physically interesting situation is when the entanglement with the environment takes the form, where are coherent states. In general, in the Bohmian interpretation all the other quantities may have a distribution totally different from those predicted by purely probabilistic QM. The space probability density should be generalized to the space-time probability density.
In this paper, a Unitary Quantum Relativistic Theory was developed. The wavy nature of a quantum particle involves the relativistic dynamics of the wave packet of this particle, and a field of interaction described by Lorentz’s force and three of the four Maxwell equations: the electromagnetic induction law, and the flow laws of the electric and magnetic fields. These equations arise from the group velocity, which is of the form of the Lagrange equation, and from a relativistic principle for wave functions, which asserts that in any system of reference, a wave function has a bounded spectrum with a velocity limit c. When a magnetic circuit law is considered for a field interacting with a quantum particle, this is an electromagnetic field, propagating with the velocity c. This means that the limit velocity c of a quantum particle according to the theory of relativity is also obtained as a necessary condition of quantum dynamics. The relativistic particle wavefunctions considered in this framework, describing finite distributions of densities in the two conjugated spaces of the coordinates and momentum, provide a better description of a collisional phenomenon, in a finite volume occupied by these particles and the photon mediating this collision according to Quantum Electrodynamics.
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.
The existence of precise particle trajectories in any quantum state is accounted for in a consistent way by allowing delocalization of the particle charge. The relativistic mass of the particle remains within a small volume surrounding a singularity moving along the particle trajectory. The singularity is the source of an electric displacement field. The field induces a polarization charge in the vacuum and this charge is equated with the charge of the particle. Under dynamic conditions a distributed charge density rho(x,t) is induced in the vacuum. The volume integral of the charge density is equal to the charge of the particle and is rigorously conserved. The charge density is derived from a complex-valued physical field psi(x,t) such that rho(x,t) = |psi(x,t)|^2. The position probability density is equated with the mean charge density. The mean field psi(x,t) for many sample realizations with a given energy E and potential V(x) is the sum of the individual fields. In order for the sum to be non-zero, the components in the spectral decompositions of the individual fields must be spatially coherent. The particle has a spin frequency given by Planck's relation hv = T - V + mc^2, where T is the kinetic energy, determined from the momentum p and V is a quantum potential such that E = T + V is conserved. The instantaneous phase of the spin is given by the phase of exp(ikx) in the spectral decomposition a(k) of psi(x,t). It is spatially coherent, due to the dependence on x. The momentum probability distribution is given by the squared magnitude of the coefficients a(k). The Schrodinger equation is derived by requiring local conservation of mean energy.
In this paper, the quantum dynamics was obtained in the framework of the general theory of relativity, where a quantum particle is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spatial coordinates and of the momentum, called wave functions. For a free particle, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time-dependent phases proportional to the relativistic Lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the particle wave functions, Lorentz’s force and the Maxwell equations were obtained. From the wave/group equation in the momentum space describing the Lorentz force, the expressions of the electric and magnetic fields as functions of the electric potential conjugated to time and of the vector potential conjugated to the coordinates in the particle-field Lagrangian were obtained. With these expressions, the electric and magnetic fields that satisfy the Faraday-Maxwell law of electromagnetic induction and the two Gauss-Maxwell laws of these fields were obtained. The Ampère-Maxwell law is obtained only by taking into account the physical consistency of the matter-field interaction of the equality of the propagation field velocity with the maximum relativistic velocity c. For a quantum particle in the electromagnetic field, dynamic equations in the coordinate and momentum spaces and the particle and antiparticle wave functions were obtained. It was shown that the electromagnetic potentials as functions of the coordinates describing the matter distribution of the quantum particle do not alter this distribution – under the action of an electromagnetic a quantum particle moves as a whole. The scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion, were obtained. This description of a quantum particle as a distribution of matter with a density amplitude/wavefunction of the form of a wave packet, with the time-dependent phase proportional to the relativistic Lagrangian as a function of the metric tensor including also the gravitational field, enables the application of this theory in quantum gravity and quantum field theory in agreement with general relativity.