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.
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.
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 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.
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.
Relativistic quantum mechanics was developed to reconcile the principles of quantum mechanics with Einstein’s theory of relativity. Despite its success in describing high-energy particles, the theory continues to face unresolved conceptual and mathematical difficulties, particularly in relation to the nature of time, causality, and relativistic consistency. In recent years, the idea of extending space-time into the complex domain has emerged as a useful and potentially meaningful approach to these problems. In this work, a theoretical synthesis is presented to examine the role of complex space-time in relativistic quantum mechanics. The study discusses how complex time arises naturally in relativistic wave equations, quantum field theory, and path integral formulations. Established methods such as Wick rotation, complex coordinate transformations, and non-Hermitian but parity–time (PT) symmetric Hamiltonians are reviewed to illustrate how complex space-time contributes to mathematical stability and clearer physical interpretation (Dirac, 1928; Wick, 1954; Bender & Boettcher, 1998). The relevance of complex space-time is further explored in the context of relativistic particle dynamics and quantum cosmology, where it helps address, issues related to vacuum structure, time evolution, and causality (Feynman & Hibbs, 1965; Hartle & Hawking, 1983). The paper suggests that complex space-time should be viewed not only as a computational technique, but as a framework that may reflect deeper features of relativistic quantum phenomena.
Some aspects of the interpretation of microscopic physics in terms of quantum theory are discussed. It is first emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutation relations between ''canonically conjugated'' coordinate and momentum operators leads to a wrong version of quantum mechanics. In this connection the Feynman integral formalism is also discussed. In this formalism the measure is not well defined, and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one. In this respect, the Feynman approach consists of a mnemonic rule to generate perturbation series from an undefined zero-order term. The origin of time in the quantum framework is then analyzed in detail and illustrated by the example of atomic collisions. It is shown that the time-dependent Schr ¨ odinger equation for the closed three-body (two nuclei + electron) system has no physical meaning because in the high-impact energy limit it transforms into an equation with two independent time-like variables; time naturally appears in the stationary Schr ¨ odinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system. Finally, following the well-known Einstein-Rosen-Podolsky experiment and Bell's inequality, we reiterate that the wave function must be interpreted as an actual field of information, in a form as elementary as the usual material particles and electromagnetic fields. In fact, experimental measurements transfer this quantum information field into the classical world, which is directly discernable. In my conclusion, the relation between physical reality and its mathematical formulation is discussed. 2012 Physics Essays Publication. (DOI: 10.4006/0836-1398-25.1.27) R ´
Negative energy states are applied in Krein space quantization approach to achieve a naturally renormalized theory. For example, this theory by taking the full set of Dirac solutions, could be able to remove the propagator Green function's divergences and automatically without any normal ordering, to vanish the expected value for vacuum state energy. However, since it is a purely mathematical theory, the results are under debate and some efforts are devoted to include more physics in the concept. Whereas Krein quantization is a pure mathematical approach, complex quantum Hamiltonian dynamics is based on strong foundations of Hamilton-Jacobi (H-J) equations and therefore on classical dynamics. Based on complex quantum Hamilton-Jacobi theory, complex spacetime is a natural consequence of including quantum effects in the relativistic mechanics, and is a bridge connecting the causality in special relativity and the non-locality in quantum mechanics, i.e. extending special relativity to the complex domain leads to relativistic quantum mechanics. So that, considering both relativistic and quantum effects, the Klein-Gordon equation could be derived as a special form of the Hamilton-Jacobi equation. Characterizing the complex time involved in an entangled energy state and writing the general form of energy considering quantum potential, two sets of positive and negative energies will be realized. The new states enable us to study the spacetime in a relativistic entangled "space-time" state leading to 12 extra wave functions than the four solutions of Dirac equation for a free particle. Arguing the entanglement of particle and antiparticle leads to a contradiction with experiments. So, in order to correct the results, along with a previous investigation [1], we realize particles and antiparticles as physical entities with positive energy instead of considering antiparticles with negative energy. As an application of modified descriptions for entangled (space-time) states, the original version of EPR paradox can be discussed and the correct answer can be verified based on the strong rooted complex quantum Hamilton-Jacobi theory [2-27] and as another example we can use the negative energy states, to remove the Klein's paradox without the need of any further explanations or justifications like backwardly moving electrons. Finally, comparing the two approaches, we can point out to the existence of a connection between quantum Hamiltonian dynamics, standard quantum field theory, and Krein space quantization [28-43].