This paper provides a mechanism-first, non-mathematical clarification of why negative energy solutions of relativistic quantum equations—most notably those arising from Dirac’s formulation—are never physically realized. Rather than modifying existing theories or introducing new physics, the work demonstrates that the exclusion of negative energy states is a necessary consequence of established physical constraints: thermodynamics (specifically the Second Law), locality, causality, and local energy conservation. By grounding physical existence in the requirements for causal interaction and irreversible state change, the paper shows that negative energy states fail at the coupling stage of any realizable interaction and are therefore structurally excluded from the physical manifold. The framework explains why standard Quantum Field Theory procedures such as normal ordering, renormalization, and the reinterpretation of antiparticles are mathematically effective while remaining physically mandatory. The result is a foundational consistency explanation that resolves the negative energy problem without equations, speculation, or modification of existing physical law.
This paper provides a mechanism-first, non-mathematical clarification of why negative energy solutions of relativistic quantum equations—most notably those arising from Dirac’s formulation—are never physically realized. Rather than modifying existing theories or introducing new physics, the work demonstrates that the exclusion of negative energy states is a necessary consequence of established physical constraints: thermodynamics (specifically the Second Law), locality, causality, and local energy conservation. By grounding physical existence in the requirements for causal interaction and irreversible state change, the paper shows that negative energy states fail at the coupling stage of any realizable interaction and are therefore structurally excluded from the physical manifold. The framework explains why standard Quantum Field Theory procedures such as normal ordering, renormalization, and the reinterpretation of antiparticles are mathematically effective while remaining physically mandatory. The result is a foundational consistency explanation that resolves the negative energy problem without equations, speculation, or modification of existing physical law.
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].
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht, which satisfies the anti-involution property Ht2=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ and its Hilbert-transformed partner ϕ=Ht[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ, under which the standard U(1) gauge symmetry emerges naturally as a global SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
Recent works in foundations of quantum (field) theory and relativistic quantum information try to better grasp the interplay between the structure of quantum correlations and the constraints imposed by causality on physical operations. Chapter 1 is dedicated to the study of the conceptual implications of quantum nonlocality, a concept that subsumes that of entanglement in a certain way. We detail the recent information-theoretic approaches to understanding the structure of nonlocal correlations, and the issues the latter raise concerning the ability of local observers to isolate a system from its environment. Chapter 2 reviews in what sense imposing Poincare invariance affects entanglement detection and quantification procedures. This invariance ultimately forces a description of all quantum systems within the framework of quantum field theory, which leads to the impossibility of localized finite-energy states and to the divergence of all entanglement measures for local observers. We provide a solution to these two problems by showing that there exists a generic equivalence between a finite spatial resolution of the measurement apparatus and the exclusion of high-energy degrees of freedom from the definition of the observed system. This equivalence allows for an epistemic interpretation of the standard quantum formalism describing nonrelativistic localized systems and their correlations, hence a clarification of the origin of the finite measures of entanglement between such systems. Chapter 3 presents a recent theoretical framework that predicts the existence of correlations with indefinite causal order. In analogy to the information-theoretic approaches to nonlocal correlations, we introduce some principles that constrain the structure of such correlations, which is a first step toward a clear understanding of their physical origin.
Abstract The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.
Abstract This paper presents an in-depth investigation into quantum effects and the three established principles of quantum mechanics, including the dynamics derived from the development of physical systems—specifically, the reversal of positive and negative energy directions, as well as the density expansion and contraction caused by interactions between energy systems. Additionally, this study introduces a novel theoretical framework that proposes a new scope for understanding quantum phenomena. A truly universal principle must satisfy both Galileo’s principle of relativity and the law of covariance, thereby enabling the identification of global hidden variables or underlying mechanisms behind physical phenomena, and allowing for the explanation and demonstration of the deductive logic and outcomes of all physical processes.Contemporary quantum experiments, evaluated through Bell’s inequality, suggest that quantum mechanics is currently regarded as complete. Furthermore, it is widely accepted that quantum entanglement exhibits uncertainty prior to measurement, with its process being superluminal and non-local. However, the fundamental reasons behind these characteristics—particularly their true physical principles, dynamic behavior, and evolutionary logic—remain inadequately addressed. In fact, current interpretations of quantum mechanics are largely superficial and do not fully incorporate Galilean covariance, indicating an inherent incompleteness.At present, the major frameworks of quantum mechanics—including quantum theory, relativity, quantum field theory, and the Schrödinger equation—are primarily confined to the description of the positive (observable) world. They generally deny the possibility of energy direction reversing into negative energy states or transitioning into a hypothetical negative world. Moreover, they reject the idea that quantum entanglement can occur within the negative world and subsequently return to the positive world, implying oscillatory motion, reciprocation, energy exchange, and superluminal properties.The prevailing interpretation—that the interaction between entangled quanta, involving energy, force, reference frames, and positively entangled particles, constitutes a holistic effect—is insufficient. Specifically, attempting to explain the instantaneous, superluminal, and uncertain nature of quantum effects solely based on dynamics restricted to the positive world domain appears fundamentally flawed. Current quantum mechanical explanations fail to identify the global hidden variables or the underlying mechanisms governing physical phenomena. Furthermore, they cannot account for why quantum entanglement results violate Bell’s inequality.According to Galileo’s law of covariance, quantum mechanics should not be satisfied with partial or limited covariance. Therefore, we propose a new quantum mechanical principle grounded in three hypotheses to address these limitations. Given that energy possesses both positive and negative directional properties, particularly under dynamic conditions where energy direction reverses, the instantaneous density ratio becomes extremely small. This leads to the physical realization of quantum superluminal phenomena. Through the principle of resonance—such as energy direction reversal and quantum entanglement—these phenomena become possible.To satisfy Galilean covariance, it can be inferred that only the reversal of the energy vector direction and transition into the negative world can fulfill such requirements. Due to the potential existence of negative entanglement in negative energy and the reciprocal exchange between the positive and negative worlds, there remains uncertainty regarding the parity of energy increase or spin quantity in returning quanta or systems. Furthermore, experimental observations from phase-locked reverse measurements—where "the jump value of PDV data consistently exceeds the rise"—suggest new physical interpretations involving imaginary numbers and calculus related to energy direction and the negative world. These findings demonstrate the pre-measurement uncertainty of quantum states, as well as the superluminal and non-local nature of quantum processes, thereby supporting the inference of a new foundational theory of quantum mechanics.Finally, the quantum teleportation experiment provides evidence of vector reversals in energy and information transmission, further confirming the existence of superluminal communication.
The axiomatic foundations of quantum mechanics, while operationally successful, obscure the underlying physical reality. This work introduces a mechanical topological ontology intended to provide a causal basis for the theory. We propose a single postulate: that a fundamental fermion is the 0-dimensional boundary of a 1-dimensional, massive, string-like object topologically anchored to a dynamic vacuum field. From this postulate, we argue that the core structures of single-particle relativistic quantum mechanics can be understood as derivable theorems. First, we construct the object’s state space by identifying two independent, binary properties—a local, vibrational/topological duality (spin) and a global, orientational duality (matter/antimatter)—whose tensor product naturally yields the 4-component Dirac spinor. Second, we show that the requirement of relativistic consistency uniquely determines the algebraic operators (gamma matrices and their Clifford algebra) and the lowest-order dynamical law, leading to the Dirac equation as a theorem of consistency. Third, we demonstrate that the non-relativistic Schrödinger and Pauli equations emerge as its correct low-energy limits, where the quantum potential V (x) is identified with a local variation in the object’s rest mass. Finally, we argue that the foundational tenets of quantum measurement—the spin-1/2 property, the probabilistic Born rule, and the correlations of entanglement—can be reframed as necessary consequences of the object’s intrinsic topology and interaction mechanics. This framework thus offers a path toward demoting the axioms of quantum theory to derivable theorems, aiming to resolve its long-standing paradoxes by providing a clear and coherent physical picture.