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Real-time propagators resummed with nontrivial boundary wavefunctions in a constant electric field

Kenji Fukushima, Shuhei Minato

TL;DR

This work develops a real-time in-in framework for quantum fields in a constant electric field by preserving explicit boundary wavefunctions that connect inequivalent in/out vacua. By reexpressing these boundary conditions as quadratic self-energy-like terms in the functional integral, the authors obtain propagators that resum infinite diagrams to capture vacuum-instability effects such as Schwinger pair production. The formalism yields proper-time representations with additional contours whose origin is clarified through the boundary structure, and it reproduces canonical results from operator methods. As a concrete check, a one-loop in-in vector current is computed, showing time-dependent growth consistent with pair production, demonstrating the practical utility of the boundary-resummed propagators for real-time strong-field QED calculations.

Abstract

We present the derivation of an alternative representation of the real-time in-in formalism under a spatially homogeneous and time independent electric field. Because the system exhibits instability associated with pair production of particles and antipartiles, the perturbation theory should be reorganized depending on the choice of the reference vacuum. We recast the boundary wavefunctions into the quadratic self-energy-like terms in the functional integration formalism. The resulting generating functional in the modified in-in formalism leads to the propagators that resum infinite diagrams necessary to capture the vacuum-instability effects. The proper-time representations of the propagators reproduce the known expressions from the canonical operator formalism, but our derivation based on the generating functional along the closed-time path clarifies the origin of the addtional proper-time contour and provides a better physical understanding. Finally, as a concrete example of the application, we compute the in-in expectation value of the vector current in a constant electric field, and find that the one-loop calculation results in the expression consistent with the physical intuition.

Real-time propagators resummed with nontrivial boundary wavefunctions in a constant electric field

TL;DR

This work develops a real-time in-in framework for quantum fields in a constant electric field by preserving explicit boundary wavefunctions that connect inequivalent in/out vacua. By reexpressing these boundary conditions as quadratic self-energy-like terms in the functional integral, the authors obtain propagators that resum infinite diagrams to capture vacuum-instability effects such as Schwinger pair production. The formalism yields proper-time representations with additional contours whose origin is clarified through the boundary structure, and it reproduces canonical results from operator methods. As a concrete check, a one-loop in-in vector current is computed, showing time-dependent growth consistent with pair production, demonstrating the practical utility of the boundary-resummed propagators for real-time strong-field QED calculations.

Abstract

We present the derivation of an alternative representation of the real-time in-in formalism under a spatially homogeneous and time independent electric field. Because the system exhibits instability associated with pair production of particles and antipartiles, the perturbation theory should be reorganized depending on the choice of the reference vacuum. We recast the boundary wavefunctions into the quadratic self-energy-like terms in the functional integration formalism. The resulting generating functional in the modified in-in formalism leads to the propagators that resum infinite diagrams necessary to capture the vacuum-instability effects. The proper-time representations of the propagators reproduce the known expressions from the canonical operator formalism, but our derivation based on the generating functional along the closed-time path clarifies the origin of the addtional proper-time contour and provides a better physical understanding. Finally, as a concrete example of the application, we compute the in-in expectation value of the vector current in a constant electric field, and find that the one-loop calculation results in the expression consistent with the physical intuition.
Paper Structure (34 sections, 137 equations, 7 figures)

This paper contains 34 sections, 137 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Real-time Propagators between Unstable Vacua
  3. Recasting the Vacuum with Boundary Wavefunction
  4. Generating Functionals
  5. Formal expressions of the generating functionals
  6. sQED:
  7. QED:
  8. Explicit forms of the boundary wavefunctions
  9. sQED:
  10. QED:
  11. Combined expressions of the generating functionals
  12. sQED:
  13. QED
  14. Propagators
  15. Formal relations of the real-time propagators
  16. ...and 19 more sections

Figures (7)

  • Figure 1: Proper-time path in the standard Schwinger propagator.
  • Figure 2: Time-integral contour in the standard in-in formalism. The in-vacuum at the infinite past evolves to the infinite future, and then returns to the infinite past.
  • Figure 3: Modified time-integral contour in the resummed in-in formalism for a constant electric field. The boundary state is transformed according to Eq. \ref{['eq:vac-transform']}, and the time contour starts from the infinite future to the infinite past.
  • Figure 4: Proper-time integral contours appearing in Eqs. \ref{['eq:Jmp_prop']} and \ref{['eq:Ja_prop']} as well as vanishing integrals in Eqs. \ref{['eq:fds=0at0']} and \ref{['eq:fds=0atipi']}.
  • Figure 5: Proper-time integral contours for the $(--)$ and $(++)$ components of the propagators. The contours along the positive and negative real axis are depicted slightly above and below the real axis just for the presentation purpose; both are simply real integrals over $(0,+\infty)$ and $(-\infty,0)$. Contours passing through (or starting from) $s=-\mathrm{i}\pi/(2eE)$ or $s=-\mathrm{i}\pi/(eE)$ are parallel to the real axis. The integrals crossing the singularities at $s=0$, $-\mathrm{i}\pi/(eE)$ are defined as a principal-value.
  • ...and 2 more figures