[Paper Review] Pair creation by time-dependent electric fields: Analytic solutions

Iwo Białynicki‐Birula, Łukasz Rudnicki+1 more|arXiv (Cornell University)|Aug 12, 2011

Quantum Information and Cryptography3 citations

TL;DR

This paper presents exact analytic solutions for pair creation in QED vacuum under adiabatically switched-on time-dependent electric fields, using the Dirac-Heisenberg-Wigner (DHW) function formalism. It shows that the pair density is an analytic function of field strength, with the Schwinger formula's essential singularity arising not from the field itself but from the unphysical assumption of infinite-time or sudden switching, which is removed by adiabatic switching. - meta_description: Exact analytic solutions for pair creation in time-dependent electric fields show the Schwinger singularity vanishes under adiabatic switching, resolving a long-standing paradox in QED. - objective: - To resolve the paradox of the Schwinger formula's essential singularity in the context of physically realistic, adiabatically switched electric fields. - To develop an exact analytical framework for time-evolving pair production in QED vacuum under time-varying fields. - To demonstrate that the pair density remains analytic in field strength when switching is adiabatic, contrasting with the non-analytic behavior in the standard Schwinger formula. - To establish the Dirac-Heisenberg-Wigner (DHW) function as a viable tool for describing non-perturbative pair production in time-dependent external fields. - method: - Uses the Dirac-Heisenberg-Wigner (DHW) function to describe the time evolution of the vacuum state under a time-dependent electric field. - Applies an extension of the spinorial decomposition method to solve the time-dependent Dirac equation for adiabatically switched fields. - Considers two specific time profiles: exponential switching (constant rate) and hyperbolic secant squared switching (decreasing rate). - Starts from the free vacuum DHW function as the initial state, ensuring physical consistency via unitary time evolution. - Derives the pair density in phase space from the DHW function components using trace formulas involving Dirac matrices. - Demonstrates that the total pair production rate is analytic in the field strength for finite switching time scales, with convergence radius shrinking only as the switching time scale $b \to 0$. - research_questions: - Does the essential singularity in the Schwinger formula persist when the electric field is switched on adiabatically rather than instantaneously? - Can exact analytic solutions be derived for pair production in time-dependent electric fields using the DHW function formalism? - How does the analytic structure of the pair density depend on the switching time scale $b$? - Is the standard perturbative expansion equivalent to the exact solution in the adiabatic limit? - What is the role of the initial vacuum state in ensuring physical consistency of the DHW function evolution? - key_findings: - The pair density is an analytic function of the electric field strength when the field is adiabatically switched on, with a convergent power series expansion for any finite switching time scale $b$. - The Schwinger formula's essential singularity is shown to be an artifact of the infinite-time or sudden-switching limit, not a fundamental property of pair production. - The singularity appears as a pole in the switching time parameter $b$, not in the field strength, when $b \to 0$. - For the exponential switching case, the exact solution's power series matches the result of straightforward perturbation theory. - The DHW function formalism correctly captures the non-perturbative pair production process and ensures physical consistency through unitary time evolution from the free vacuum. - The pair density in phase space is expressed as $ n(\bm{r},\bm{p},t) = 1 + \frac{mf_3 + \bm{p}\cdot\bm{g}_1 + E_p f_0}{2E_p} $, with the charge density given by $ f_0 $, confirming consistency with charge conservation.

ABSTRACT

Exact analytical solutions are presented for the time evolution of the density of pairs produced in the QED vacuum by a uniform electric field that is adiabatically switched on starting at minus infinity. Pair production is described by the Dirac-Heisenberg-Wigner function introduced before [Phys. Rev. D 44, 1825 (1991)]. The explicit solution is obtained by an extension of the method of the spinorial decomposition to deal with a time-varying electric field. The main result of this work is that the pair density is an analytic function of the field strength; it can be expanded into a convergent power series. Therefore, the essential singularity present in the Schwinger formula is to be attributed to the infinitely long duration of the process of pair creation by a time-independent field.

Motivation & Objective

• To resolve the paradox of the Schwinger formula's essential singularity in the context of physically realistic, adiabatically switched electric fields.
• To develop an exact analytical framework for time-evolving pair production in QED vacuum under time-varying fields.
• To demonstrate that the pair density remains analytic in field strength when switching is adiabatic, contrasting with the non-analytic behavior in the standard Schwinger formula.
• To establish the Dirac-Heisenberg-Wigner (DHW) function as a viable tool for describing non-perturbative pair production in time-dependent external fields.

Proposed method

• Uses the Dirac-Heisenberg-Wigner (DHW) function to describe the time evolution of the vacuum state under a time-dependent electric field.
• Applies an extension of the spinorial decomposition method to solve the time-dependent Dirac equation for adiabatically switched fields.
• Considers two specific time profiles: exponential switching (constant rate) and hyperbolic secant squared switching (decreasing rate).
• Starts from the free vacuum DHW function as the initial state, ensuring physical consistency via unitary time evolution.
• Derives the pair density in phase space from the DHW function components using trace formulas involving Dirac matrices.
• Demonstrates that the total pair production rate is analytic in the field strength for finite switching time scales, with convergence radius shrinking only as the switching time scale $b \to 0$.

Experimental results

Research questions

• RQ1 Does the essential singularity in the Schwinger formula persist when the electric field is switched on adiabatically rather than instantaneously?
• RQ2 Can exact analytic solutions be derived for pair production in time-dependent electric fields using the DHW function formalism?
• RQ3 How does the analytic structure of the pair density depend on the switching time scale $b$?
• RQ4 Is the standard perturbative expansion equivalent to the exact solution in the adiabatic limit?
• RQ5 What is the role of the initial vacuum state in ensuring physical consistency of the DHW function evolution?

Key findings

• The pair density is an analytic function of the electric field strength when the field is adiabatically switched on, with a convergent power series expansion for any finite switching time scale $b$.
• The Schwinger formula's essential singularity is shown to be an artifact of the infinite-time or sudden-switching limit, not a fundamental property of pair production.
• The singularity appears as a pole in the switching time parameter $b$, not in the field strength, when $b \to 0$.
• For the exponential switching case, the exact solution's power series matches the result of straightforward perturbation theory.
• The DHW function formalism correctly captures the non-perturbative pair production process and ensures physical consistency through unitary time evolution from the free vacuum.
• The pair density in phase space is expressed as $ n(\bm{r},\bm{p},t) = 1 + \frac{mf_3 + \bm{p}\cdot\bm{g}_1 + E_p f_0}{2E_p} $, with the charge density given by $ f_0 $, confirming consistency with charge conservation.