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Modern Physics Letters A

Klein's paradox and quantum Hamiltonian dynamics in complex spacetime

Farrin Payandeh

Published May 28, 2014• 3 citations
Publisher

Abstract

In this paper, along with previous investigation in Payandeh et al., Chin. Phys. C 37, 113103 (2013), we will use the full set of free Dirac's solutions to remove the Klein's paradox. However, we choose a different approach, namely the relativistic quantum Hamilton–Jacobi equations and the resultant quantum states. We show that this full set of solutions can also be retained as a single wave function, and also possess the potential of removing Klein's paradox.

DOI
10.1142/s0217732314500953

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In this paper, along with previous investigation in Payandeh et al., Chin. Phys. C 37, 113103 (2013), we will use the full set of free Dirac's solutions to remove the Klein's paradox. However, we choose a different approach, namely the relativistic quantum Hamilton–Jacobi equations and the resultant quantum states. We show that this full set of solutions can also be retained as a single wave function, and also possess the potential of removing Klein's paradox.
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.
The Hamilton-Jacobi method is generalized, both, in classical and relativistic mechanics. The implications in quantum mechanics are considered in the case of Klein-Gordon equation. We find that the wave functions of Klein-Gordon theory can be considered as describing the motion of an ensemble of particles that move under the action of the electromagnetic field alone, without quantum potentials, hidden uninterpreted variables, or zero point fields. The number of particles is not locally conserved.
We have calculated normalized wave function solutions to the time-dependent Dirac equation on a space-time lattice grid. Using these solutions we systematically explore phenomena that are consistent with the classical relativistic Liouville phase space approach and those that do not have a direct classical counterpart. The examples reviewed are the paradoxical Zitterbewegung, the time-resolved Klein-paradox, the dynamics of spin in an inhomogeneous magnetic field, the suppression of the spatial spreading rate and new dynamical features that occur close to relativistic resonances.
The Klein paradox is one of the cornerstones in the development of quantum mechanics, and its consequences were used in various branches of physics, ranging from elementary particles to solid state. Yet its mathematical derivation is questionable in a number of steps, resulting in the wrong solution of Dirac equation. In this paper, the paradox is analysed in more detail, and its mathematical content is emphasized in order to show that in the study of extreme conditions of matter simple arguments may result in erroneous predictions.