Time-symmetric relativity theory (TSR) based on Zisman-Stueckelberg-Feynman (ZSF) interpretation includes reference frames going backward in time, maximally expanding their symmetry group to general Lorentz group O(1,3) with 4-inversion (Article 1). Similar time-symmetric (TS) extensions of relativistic quantum mechanics (TS RQM) (Article 2) and QFT (TS QFT) (this article) avoid problems of former RQM and QFT. Former QFT introduced unfounded hypotheses about physical quantum fields in entire space for each type of particle, and these hypotheses led to insoluble problems. TS QFT is limited to known external interaction fields, and describes particles using theory of many-particle systems. In second quantization method, single-particle wave functions and operators, which increase or decrease number of particles in system, form operator amplitudes of transition probabilities. These amplitudes are complex, act in their state spaces for given energy sign, and lead to positive state norm. As in quantum mechanics, they, together with conjugate amplitude, form canonical pair. Canonical formalism in relativity theory is consistent at using Hamilton-Jacobi (HJ) formalism, where 3-momentum and Hamiltonian enter into HJ equation in the same degree. Differential form of relativistic HJ equation is Klein-Gordon (KG) equation, expressing the relationship between energy and momentum, and is general equation for physical components of all operator amplitudes. For bosons, KG equation suffices; for fermions, linearization of HJ equation is required, leading to special first-order equations, such as Dirac equation. Operators of observables are normally ordered, and there are no zero-point fluctuations in ground states. Probability current is determined by 4-momentum and has the same sign; interaction currents are also proportional to probability current and change sign with 4-momentum, which corrects the currents in fermion theory. Introducing interactions into HJ equation leads to new contact terms. Time-symmetric orderings of operator amplitudes lead to causal propagators and usual diagram technique. Taking into account external gravitational field of quanta at Planck length, gravitational radius of the system, and dilation of their proper times in this field lead to gravitational self-regularization of loop diagrams. This makes perturbation theory series in Standard Model (SM) and quantum gravity finite and convergent. Contributions of chiral anomalies are finite and cancel out in SM.
Time-symmetric relativity theory (TSR) based on Zisman-Stueckelberg-Feynman (ZSF) interpretation includes reference frames going backward in time, maximally expanding their symmetry group to general Lorentz group O(1,3) with 4-inversion (Article 1). Similar time-symmetric (TS) extensions of relativistic quantum mechanics (TS RQM) (Article 2) and QFT (TS QFT) (this article) avoid problems of former RQM and QFT. Former QFT introduced unfounded hypotheses about physical quantum fields in entire space for each type of particle, and these hypotheses led to insoluble problems. TS QFT is limited to known external interaction fields, and describes particles using theory of many-particle systems. In second quantization method, single-particle wave functions and operators, which increase or decrease number of particles in system, form operator amplitudes of transition probabilities. These amplitudes are complex, act in their state spaces for given energy sign, and lead to positive state norm. As in quantum mechanics, they, together with conjugate amplitude, form canonical pair. Canonical formalism in relativity theory is consistent at using Hamilton-Jacobi (HJ) formalism, where 3-momentum and Hamiltonian enter into HJ equation in the same degree. Differential form of relativistic HJ equation is Klein-Gordon (KG) equation, expressing the relationship between energy and momentum, and is general equation for physical components of all operator amplitudes. For bosons, KG equation suffices; for fermions, linearization of HJ equation is required, leading to special first-order equations, such as Dirac equation. Operators of observables are normally ordered, and there are no zero-point fluctuations in ground states. Probability current is determined by 4-momentum and has the same sign; interaction currents are also proportional to probability current and change sign with 4-momentum, which corrects the currents in fermion theory. Introducing interactions into HJ equation leads to new contact terms. Time-symmetric orderings of operator amplitudes lead to causal propagators and usual diagram technique. Taking into account external gravitational field of quanta at Planck length, gravitational radius of the system, and dilation of their proper times in this field lead to gravitational self-regularization of loop diagrams. This makes perturbation theory series in Standard Model (SM) and quantum gravity finite and convergent. Contributions of chiral anomalies are finite and cancel out in SM.
Zisman-Stueckelberg-Feynman (ZSF) interpretation, by describing antiparticles as negative-energy particles evolving backward in ordinary time, makes relativistic quantum mechanics (RQM) covariant, but relativistic theory was not agreed with it. Consistent agreement with ZSF interpretation led to the time-symmetric relativity theory (TSRT) (Article 1). In TSRT, relativity principle is extended to rest frames of negative-energy particles going into lower light cone, and symmetry group is extended to the general Lorentz group O(1,3) with 4-inversion. In this article, a time-symmetric RQM (TS RQM) based on TSRT is formulated. The states of negative-energy particles in their rest frames are transformed into usual frames with reflecting time axis. In Hamilton-Jacobi formalism, translational symmetry leads to a general relativistic wave equation for components of wave function of any free particle with a quadratic Hamiltonian. Using in it relativistic relationship between energy and momentum gives Klein-Gordon equation, which yields a general average of 4-momentum and general probability current, time components of which describe currents in both directions of time axis and are negative at negative energy. Special wave equations with first time derivative (Duffin-Kemmer-Petiou for spins 0 and 1, and Dirac for spin 1/2) account for mixing of wave function components due to spin and interactions. In TSRT, 4-vector currents change sign upon 4-inversion, which improves theories of bosons and fermions, eliminating former paradoxes and errors. Time-symmetric transition amplitudes in propagator approach lead to causal propagators and usual diagram technique. Total energy of system of particles in loop diagrams creates external gravitational field, which slows down proper times relative to world time until they freeze at Planck length, gravitational radius of system. This leads to gravitational regularization of loop diagrams, making them finite, and fields of Standard Model (except for scalar field) and quantum gravity renormalizable. Quantization of fields in TSRT is described in Article 3.
Time-symmetric theory (TST) is formulated as a theoretical foundation of time-symmetric physics (TSP), classical, relativistic and quantum. TST is based on the widely used in particle physics fact that the description of positive energy antiparticles moving forward in space and time is equivalent to the description of negative energy particles moving backward in space and time (Zisman 1940, Stückelberg 1941, Feynman 1949). In TST, this fundamental fact is generalized by formulating the equivalence principle of particle physics and applying it to all physical phenomena in the rest frames of negative energy particles. As a result, the group of transformations TST also includes such inertial frames that move backward in space and time with coordinates related to the ordinary ones by 4-inversion. This led to the formulation of time-symmetric relativity theory and corresponding relativistic quantum mechanics. In the latter, the probabilities remain positive, but the bilinear forms of the wave functions are associated with probability flows, the signs of which depend on the direction of movement along the coordinate axes, including the time axis. In TST, therefore, the Klein-Gordon equation is a consistent equation for probability amplitudes, and the fermion theory contains the necessary corrections, which make it physically more consistent, unchanging its main observational consequences. Applications of TST to quantum fields will be considered in the second article.
We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schrodinger equation. We define a relativistic quantum principle asserting that a quantum 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 coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2) an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation, Ampere-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
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.
In special relativity, time and space are transformed according to the Lorentz transformation. Relativistic quantum mechanics is quantum mechanics that is consistent with special relativity. Nevertheless, relativistic quantum mechanics is only an approximation to a fully self-consistent relativistic theory of known particle interactions because it does not describe cases where the number of particles changes. In special relativity, mass can emerge from energy, which is not possible in quantum mechanics based on the conservation of probability. Dirac proposed the equation named after him to solve the difficulty encountered in the Klein-Gordon equation stemmed from the fact that it is a second-order differential equation in time.