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Lessons from the Klein paradox

E. T. Akhmedov, D. V. Diakonov, V. I. Lapushkin, D. I. Sadekov

TL;DR

This work reexamines the Klein paradox from a many-particle quantum-field-theory perspective in 1+1 dimensions, focusing on the current induced by strong step-like electric potentials. By analyzing constant, switched-on, and finite-duration field protocols, it demonstrates that the observed current in an eternal field is highly sensitive to the chosen mode basis, with a Fock-space ground state yielding zero current while the standard vacuum produces a nonzero, time-independent current in the Klein zone. When the field is switched on or applied for finite duration, the current in the asymptotic future can be expressed in terms of the same integral over Klein-zone modes, $J_{\mathrm{tot}}=\int_{m}^{V_0-m}\frac{d\omega}{2\pi}\frac{4\kappa_\omega}{(1+\kappa_\omega)^2}$, highlighting a robust connection between particle production, vacuum structure, and observable current via scattering data and Bogoliubov transformations. The results clarify how particle definitions arise in strong background fields, show that switching protocols yield a well-defined, protocol-independent current in the infinite-volume limit, and indicate that relaxation to equilibrium in finite systems requires genuinely 4D interacting dynamics.

Abstract

We re-examine the Klein paradox from a many-particle perspective in quantum field theory. Specifically, we compute the expectation value of the particle current induced by a sufficiently strong step-like electric potential in 1+1 dimensions. First, for a constant (eternal) potential, we calculate the current for different Fock space ground states corresponding to distinct mode bases. While one basis yields a zero current, another produces the standard nonzero value. We then consider a potential that is rapidly switched on, recovering the standard current in the asymptotic future. This result is generalized to potentials that interpolate between different constant values at spatial infinity. Finally, we analyze a potential acting for a finite duration and again reproduce the standard current. A physical interpretation of these results is provided.

Lessons from the Klein paradox

TL;DR

This work reexamines the Klein paradox from a many-particle quantum-field-theory perspective in 1+1 dimensions, focusing on the current induced by strong step-like electric potentials. By analyzing constant, switched-on, and finite-duration field protocols, it demonstrates that the observed current in an eternal field is highly sensitive to the chosen mode basis, with a Fock-space ground state yielding zero current while the standard vacuum produces a nonzero, time-independent current in the Klein zone. When the field is switched on or applied for finite duration, the current in the asymptotic future can be expressed in terms of the same integral over Klein-zone modes, , highlighting a robust connection between particle production, vacuum structure, and observable current via scattering data and Bogoliubov transformations. The results clarify how particle definitions arise in strong background fields, show that switching protocols yield a well-defined, protocol-independent current in the infinite-volume limit, and indicate that relaxation to equilibrium in finite systems requires genuinely 4D interacting dynamics.

Abstract

We re-examine the Klein paradox from a many-particle perspective in quantum field theory. Specifically, we compute the expectation value of the particle current induced by a sufficiently strong step-like electric potential in 1+1 dimensions. First, for a constant (eternal) potential, we calculate the current for different Fock space ground states corresponding to distinct mode bases. While one basis yields a zero current, another produces the standard nonzero value. We then consider a potential that is rapidly switched on, recovering the standard current in the asymptotic future. This result is generalized to potentials that interpolate between different constant values at spatial infinity. Finally, we analyze a potential acting for a finite duration and again reproduce the standard current. A physical interpretation of these results is provided.
Paper Structure (8 sections, 113 equations, 3 figures)

This paper contains 8 sections, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Constant electric field
  3. Switching on the electric field
  4. Current for a generic potential of the form $U(x)\theta(t)$
  5. Finite duration of the external field
  6. Conclusions and acknowledgments
  7. Free Dirac equation
  8. Single-particle solution

Figures (3)

  • Figure 1: Illustration of the in-modes.
  • Figure 2: Illustration of the CP transformation.
  • Figure 3: Contour of integration around the poles. The zigzag line indicates non-analyticity of the integrand elsewhere.