Abstract
Electric fields can spontaneously decay via the Schwinger effect, the nucleation of a charged particle-anti particle pair separated by a critical distance d. What happens if the available distance is smaller than d? Previous work on this question has produced contradictory results. Here, we study the quantum evolution of electric fields when the field points in a compact direction with circumference L < d using the massive Schwinger model, quantum electrodynamics in one space dimension with massive charged fermions. We uncover a new and previously unknown set of instantons that result in novel physics that disagrees with all previous estimates. In parameter regimes where the field value can be well-defined in the quantum theory, generic initial fields E are in fact stable and do not decay, while initial values that are quantized in half-integer units of the charge E = (k/2)g with k ∈ ℤ oscillate in time from +(k/2)g to −(k/2)g, with exponentially small probability of ever taking any other value. We verify our results with four distinct techniques: numerically by measuring the decay directly in Lorentzian time on the lattice, numerically using the spectrum of the Hamiltonian, numerically and semi-analytically using the bosonized description of the Schwinger model, and analytically via our instanton estimate.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].
A.R. Brown, Schwinger pair production at nonzero temperatures or in compact directions, Phys. Rev. D 98 (2018) 036008 [arXiv:1512.05716] [INSPIRE].
L. Medina and M.C. Ogilvie, Schwinger pair production at finite temperature, Phys. Rev. D 95 (2017) 056006 [arXiv:1511.09459] [INSPIRE].
M. Korwar and A.M. Thalapillil, Finite temperature Schwinger pair production in coexistent electric and magnetic fields, Phys. Rev. D 98 (2018) 076016 [arXiv:1808.01295] [INSPIRE].
P. Draper, Virtual and thermal Schwinger processes, Phys. Rev. D 98 (2018) 125014 [arXiv:1809.10768] [INSPIRE].
Y. Qiu and L. Sorbo, Schwinger effect in compact space: a real time calculation, Phys. Rev. D 102 (2020) 045010 [arXiv:2005.10121] [INSPIRE].
F. Hebenstreit, J. Berges and D. Gelfand, Real-time dynamics of string breaking, Phys. Rev. Lett. 111 (2013) 201601 [arXiv:1307.4619] [INSPIRE].
B. Buyens, J. Haegeman, F. Hebenstreit, F. Verstraete and K. Van Acoleyen, Real-time simulation of the Schwinger effect with matrix product states, Phys. Rev. D 96 (2017) 114501 [arXiv:1612.00739] [INSPIRE].
C. Nagele, J.E. Cejudo, T. Byrnes and M. Kleban, Flux unwinding in the lattice Schwinger model, Phys. Rev. D 99 (2019) 094501 [arXiv:1811.03096] [INSPIRE].
I.K. Affleck, O. Alvarez and N.S. Manton, Pair production at strong coupling in weak external fields, Nucl. Phys. B 197 (1982) 509 [INSPIRE].
S.R. Coleman, More about the massive Schwinger model, Annals Phys. 101 (1976) 239 [INSPIRE].
J.B. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975) 395 [INSPIRE].
A. Carroll, J.B. Kogut, D.K. Sinclair and L. Susskind, Lattice gauge theory calculations in (1 + 1)-dimensions and the approach to the continuum limit, Phys. Rev. D 13 (1976) 2270 [Erratum ibid. 14 (1976) 1729] [INSPIRE].
L. Susskind, Lattice fermions, Phys. Rev. D 16 (1977) 3031 [INSPIRE].
T. Banks, L. Susskind and J.B. Kogut, Strong coupling calculations of lattice gauge theories: (1 + 1)-dimensional exercises, Phys. Rev. D 13 (1976) 1043 [INSPIRE].
P. Sriganesh, R. Bursill and C.J. Hamer, A new finite lattice study of the massive Schwinger model, Phys. Rev. D 62 (2000) 034508 [hep-lat/9911021] [INSPIRE].
T. Byrnes, P. Sriganesh, R.J. Bursill and C.J. Hamer, Density matrix renormalization group approach to the massive Schwinger model, Nucl. Phys. B Proc. Suppl. 109 (2002) 202 [hep-lat/0201007] [INSPIRE].
S.R. Coleman, The quantum sine-Gordon equation as the massive Thirring model, Phys. Rev. D 11 (1975) 2088 [INSPIRE].
S.R. Coleman, R. Jackiw and L. Susskind, Charge shielding and quark confinement in the massive Schwinger model, Annals Phys. 93 (1975) 267 [INSPIRE].
S. Mandelstam, Soliton operators for the quantized sine-Gordon equation, Phys. Rev. D 11 (1975) 3026 [INSPIRE].
C.M. Naon, Abelian and non-Abelian bosonization in the path integral framework, Phys. Rev. D 31 (1985) 2035 [INSPIRE].
L. Landau and E. Lifshitz, Chapter III — Schrödinger’s equation, in Quantum mechanics, third edition, Pergamon, Oxford, U.K. (1977), p. 50.
S.R. Coleman, The uses of instantons, Subnucl. Ser. 15 (1979) 805 [INSPIRE].
A. Garg, Tunnel splittings for one-dimensional potential wells revisited, Amer. J. Phys. 68 (2000) 430.
G.V. Dunne and C. Schubert, Worldline instantons and pair production in inhomogeneous fields, Phys. Rev. D 72 (2005) 105004 [hep-th/0507174] [INSPIRE].
G.V. Dunne, Q.-H. Wang, H. Gies and C. Schubert, Worldline instantons. II. The fluctuation prefactor, Phys. Rev. D 73 (2006) 065028 [hep-th/0602176] [INSPIRE].
J.D. Brown and C. Teitelboim, Neutralization of the cosmological constant by membrane creation, Nucl. Phys. B 297 (1988) 787 [INSPIRE].
J. Garriga, Nucleation rates in flat and curved space, Phys. Rev. D 49 (1994) 6327 [hep-ph/9308280] [INSPIRE].
C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].
E. D’Hoker and D.G. Gagne, Worldline path integrals for fermions with scalar, pseudoscalar and vector couplings, Nucl. Phys. B 467 (1996) 272 [hep-th/9508131] [INSPIRE].
E. D’Hoker and D.G. Gagne, Worldline path integrals for fermions with general couplings, Nucl. Phys. B 467 (1996) 297 [hep-th/9512080] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
C. Beasley, Localization for Wilson loops in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013) 1 [arXiv:0911.2687] [INSPIRE].
O. Gould and A. Rajantie, Thermal Schwinger pair production at arbitrary coupling, Phys. Rev. D 96 (2017) 076002 [arXiv:1704.04801] [INSPIRE].
C. Schubert, Lectures on the worldline formalism, in School on spinning particles in quantum field theory: worldline formalism, higher spins and conformal geometry, https://indico.cern.ch/event/206621/, Morelia, Michoacán, Mexico (2012), p. 19.
R. Bousso and J. Polchinski, Quantization of four form fluxes and dynamical neutralization of the cosmological constant, JHEP 06 (2000) 006 [hep-th/0004134] [INSPIRE].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].
G. D’Amico, R. Gobbetti, M. Kleban and M. Schillo, Unwinding inflation, JCAP 03 (2013) 004 [arXiv:1211.4589] [INSPIRE].
H.B. Nielsen and M. Ninomiya, No go theorem for regularizing chiral fermions, Phys. Lett. B 105 (1981) 219 [INSPIRE].
P. Freitas, A nonlocal Sturm-Liouville eigenvalue problem, Proc. Roy. Soc. Edinburgh A 124 (1994) 169.
F.A. Davidson and N. Dodds, Spectral properties of non-local differential operators, Appl. Anal. 85 (2006) 717.
S. Vandoren and P. van Nieuwenhuizen, Lectures on instantons, arXiv:0802.1862 [INSPIRE].
I.M. Gelfand and A.M. Yaglom, Integration in functional spaces and it applications in quantum physics, J. Math. Phys. 1 (1960) 48 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2107.04561
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Hu, XY., Kleban, M. & Yu, C. Electric field decay without pair production: lattice, bosonization and novel worldline instantons. J. High Energ. Phys. 2022, 197 (2022). https://doi.org/10.1007/JHEP03(2022)197
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2022)197