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Journal of Physics Conference Series

EPR & Klein Paradoxes in Complex Hamiltonian Dynamics and Krein Space Quantization

Farrin Payandeh

Published Jul 3, 2015• 0 citations
Publisher

Abstract

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].

DOI
10.1088/1742-6596/626/1/012053

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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].
Based on complex quantum Hamilton-Jacobi theory and its natural complex spacetime configuration, we realized new states for both positive and negative values of energy and momentum, and we discuss this complex configuration in a relativistic entangled "space-time" state and the resultant 12 extra wave functions than the four solutions of Dirac equation for a free particle. Accordingly, we calculate the quantum forces between particles and antiparticles in a relativistic entangled "space-time" state.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.
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.
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.
The Schrodinger variational approach (1926) to quantization of the natural Hamilton mechanics in 2n2n2n-dimensional phase space is revised in the modern paradigm of quantum mechanics in application to the system the Hamilton function of which is a positive-definite quadratic form of the n momenta with the coefficients depending on coordinates in the generic configuration space VnV_nVn​. The quantum Hamilonian thus obtained has a paradoxical potential term depending on choice of coordinates in VnV_nVn​, which was discovered first by B. DeWitt in 1952 in the framework of canonical quantization of the system by a particular ordering of the operators of observables of momenta and coordinates. It is shown that the Schrodinger approach in the standard paradigm of quantum mechanics determines uniquely the ordering selected by DeWitt among a continuum of other possibilties to determine the quantum Hamiltonian. Two particular classes of observables of localization in VnV_nVn​ are considered in detail. It is shown that, in general, the quantum-mechanical potential does not vanish even in the Euclidean configuration space VnV_nVn​ except the case when Cartesian coordinates are taken as the observables of localization of the system. It is noted also that, in the quasi-classical approach to the quantization considered by DeWitt in 1957, the quantum Green function (propagator) is also non-unique and depends on the choice of a line in VnV_nVn​ connecting these points. All three formalisms have the same local asymtotics of the quantum Hamiltonian if the normal Riemannian cordinates are used, at least implicitly, to localize the system in VnV_nVn​. Keywords: Hamilton function; Quantization; Quantum-Mechanical Potential; Observables of Localization; Quantum Anomaly of (non-relativistic) General Covariance.