Bosonic analog of the Klein paradox

Abstract

The standard Klein paradox describes how an incoming electron scatters off a supercritical electrostatic barrier that is so strong that it can generate electron-positron pairs. This fermionic system has been widely discussed in textbooks to illustrate some of the discrepancies between quantum mechanical and quantum field theoretical descriptions for the pair creation process. We compare the fermionic dynamics with that of the corresponding bosonic system. We point out that the direct counterpart of the Pauli exclusion principle (the central mechanism to resolve the fermionic Klein paradox) is stimulated emission, which leads to the resolution of the analogous bosonic paradox.In 1927 the Swedish scientist Oskar Klein [1] analyzed the quantum-mechanical solution of the Dirac equation for an incoming electron that scatters off a one-dimensional electrostatic barrier of height V . He observed a nonvanishing transmission if this height exceeds twice the rest mass energy of the electron (V > 2 mc 2 ), even for incoming energies E that are far below the potential height, E < V . This classically forbidden nonvanishing transmission has been dubbed the "Klein paradox." Two types of approaches have been used to investigate this paradox [2][3][4][5][6][7][8][9][10][11][12]. The first group analyzed the dynamics in purely quantum-mechanical terms, which cannot predict any of the pair creation that is unavoidably associated with the supercritical barrier by itself. The second approach used quantum field theory, which correctly describes the pair-creation process, but the effect of the incoming electron on the pair-creation process was not included. A complete description has to be quantum field theoretical and must include the incoming particle as well as the pair-creation process at the barrier.Numerical solutions to quantum field theory were used to study the interaction of the pair-creation process at the barrier with an incoming electron with full space-time resolution [10,11]. In agreement with earlier predictions, it was shown that the incoming electron suppresses the pair-creation process during the time it is being (completely) reflected at the barrier. The incoming electron occupies states that are therefore no longer available for the pair-creation process. As a result of this Pauli blocking, the reflected electron gets entangled with the pair-creation process such that it cannot be described by a single-particle wave function. The suppression due to Pauli blocking is in contradiction to statements that the incoming electron can "knock-out" a positron under the barrier [4] or that the incoming electron "stimulates" [7] the pair-creation process.While a wide number of articles have examined the Klein paradox in its historical fermionic context, an analogous analysis within the context of a bosonic system with spacetime resolution has found less attention [5,13]. There are several reasons for this. First, the threshold with regard to the electric field strength to break down the fermionic vacuum is easier to reach tha...

Non-perturbative approach to bosonic multi-pair creation in arbitrary external fields

“…As this approach is rather power ful, it is worthwhile to study whether a similar approach is also possible to describe the creation of boson-antiboson pairs under supercritical field con ditions. Preliminary analysis about the bosonic ana logue of the Klein paradox [31] and the unexpected exponential growth of bosonic pair production due to the presence of other bosons [32] have indicated the SU et al need to develop an ab initio theory to study the bosonic vacuum decay and the corresponding spatially and time dependent features of the dynamics.…”

Electron-positron pair creation induced by quantum-mechanical tunneling

“…(1). It is valid for the energy regime c 2 <E<V-c 2 with V>2c 2 , at which a portion of the wave function (modeling an incoming particle) can actually penetrate the potential barrier of total height V. This purely mathematical transmission, however, does not mean that an actual physical electron can pass through this barrier as its energy E is less than the barrier height V. This mathematical property of the Dirac (and also of the Klein-Gordon) equations is related to the famous Klein paradox [25][26][27][28].…”

“…In other words, this initial growth can be viewed as the result of an incoherent process due to each (subcritical) electric field pulse, where the created particles at one force location do not have sufficient time to affect the creation process (via Pauli blocking [27,28,32]) at the other location.…”

“…It can provide an alternative approach to the traditional S-matrix approach, which is based on the in-and out-states only and therefore cannot visualize the processes inside the interaction zone. In addition to visualizing the details of the pair creation dynamics, direct time-dependent quantum field theoretical solutions to the Dirac equation have also contributed to the resolution of various conceptual problems related to the negative energy states such as the Zitterbewegung [23], the relativistic localization problem [24] as well as the Klein-paradox [25][26][27][28].…”

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

“…They split the Klein-Gordon wave function into two components and for the components vector they arrived at a Schrödinger-like equation with first order in time derivative. Although the Feshbach-Villars formalism appear in some advanced quantum mechanics books [9,10,11,12,13,14,15], and they were utilized in gaining deeper insight into relativistic physics of Klein paradox pair production, [16,17,18,19,20,21,22,23], in exotic atoms [24,25,26], used in theoretical consideraions [27,28,29,31], study relativistic scattering [32,32] and optics [33] or demosntrate PT symmetry [34,35,36,37], they were hardly used as a computational tool. The equations look like ordinary coupled differential equations, but the components are coupled by the kinetic energy operator, which makes them very hard to solve.…”