Rainer Grobe Intense Laser Physics Theory Unit Illinois State University Illinois State University, Physics Colloquium April 2004 Resolution of the Klein paradox Support: NSF, Res. Corp www.phy.ilstu.edu/ILP
Rainer Grobe Intense Laser Physics Theory Unit Illinois State University Illinois State University, Physics Colloquium April 2004 Temporal ordering operatorsin space-time resolved fermionic quantum field theory Support: NSF, Res. Corp www.phy.ilstu.edu/ILP
Why the (+) sign matters … F(x,t)=< 0 | Y (+)(x,t) | F(t=0) >
Acknowledgement Strong-field sub-group of ILP: Additional help: Prof. S. Hassani Prof. C. Gerry Prof. S. Haan Prof. K. Rzazewski Prof. I. Bialynicka
Overview (destructive comments) Relationship: inefficient teaching and physical theories Isolated pieces without glue fall apart (constructive comments) Space-time resolved quantum field theory Resolution of three myths: • Zitterbewegung is unphysical • Localization beyond C is possible • Klein paradox is resolved
First quiz question: Why does everybody believe that physics is tough? (a) Limitation of human intelligence (b) Physics is taught very inefficiently (c) both of the above (d) none of the above [correct answer (b)] even though lots of historical and political evidence for (a)]
B 4
Problems in inefficient teaching Small problem: How we teach Recent “discovery” in pedagogy: “ active involvement enhances learning” • Big problem: What we teach • stress commonality and not differences (Newton& Coulomb) • breakup into mechanics, quantum and e&m inefficient ?? • articulate what is important (C. Wenning) • new topic selection: (PHY102, B. Clark)
Facts are accidental and therefore irrelevant if facts were different => world would look different, but physics would identical Same pursuit to understand quantitatively the world using mathematics Bad teaching Good teaching fact 1 fact 2 fact 5 fact 3 fact 4 Problem of disconnectedness persists into higher levels …
Lacking the glue: True gaps in our knowledge Classical mechanics F=ma (Newton) c h 0 Quantum mechanics c itY = H Y (Schrödinger) ? ? nconst. Field theory
e– e+ Quantum mechanics ? ? Field theory Example of problems that require “glue” E E=mc2 in spatial and temporal resolution • How is a particle born? • When is a particle born? • Are an e– and e+ born at the same spot? • Are the spins related? • What is the size of an electron ?
From quantum field theory to quantum mechanics Dirac equation for the electron-positron field operatorY i h t Y = - i h c a Y – q aA Y + m b c2 Y + VY field operator Y infinite set of wave functions F(x1, x2,..y1, y2, .., t) F(x1, x2,..y1, y2, .., t) = <0|| Y(+)(x1,t) Y(+)(x2,t) …Yc(+)(y1,t) Yc(+)(y2,t).. || F(t=0)> / √N! M! positive frequency part initial state vacuum state charge conjugation S.S. Schweber, “An introduction to relativistic quantum field theory”
Silvan.S. Schweber, “An introduction to relativistic quantum field theory” Foreword by H.A. Bethe (Nobel prize 1967) “It is always astonishing to see one’s children growing up, and to find that they can do things their parents can no longer understand. This book is a good example.”
e– E e+ How is matter born? This bubble chamber photograph shows an electron and a positron (anti-electron) spiraling in opposite directions. Required amplification factor: 1000 000 000 000 000 Required spatial resolution: 0.000 000 000 000 001 meter (femtometer) Required time scale: 0.000 000 000 000 000 000 000 1 second (zeptosecond)
Numerical solution to quantum field theory Dirac equation for the electron-positron field operatorY i h t Y = - i h c a Y – q aA Y + m b c2 Y + V(z)Y Y(z) = Spbp Wp (z,t) + Sn dnWn(z,t) 2 CPU-months later: F(x, y, t) =<0|| Y(+)(x,t) Yc(+)(y,t) || F(t=0)>
Time evolution of e– and e+ densities mc2 = 810–14J = 0.5 MeV E ≈ 81018 V/m
Stages in the evolution Energy arrives, no particles Birth process, shape invariant “Moving away stage” Steady state stage
energy e– e– e– Has the electron a minimum width? e n e r g y Dx Dx e--width Dx shrinks with the reduction of W Dx << lc is possible ! ? ?
An electron state can be arbitrarily narrow !!! Peter Miloni,” Quantum vaccuum” (1994), page 323: “.. even a point particle has an effective linear dimension ~ lC” P. Strange “Relativistic quantum mechanics” (1998), page 82 “.. impossible for a localized particle to be purely particle-like” Resolution to this contradiction: isolated e– : Y(x) Dx > lC=h/(mc) (≈ 310–13m) e– correlated with e+ : Y(x,y) Dx arbitrarily small
How far can e– and e+ be apart? e n e r g y ? ? ? ? ? ? ? ? ? D D Maximum e_ e+ distance D? W=∞ e– W=0.2lc e+ e+ e+ • W ∞: Dlc Maximum e–-e+ distance: Compton wavelength lC
Total electron population as a function of time N(t)=Sp<0|| (t) (t) ||0> = <Y(x,y,t)|Y(x,y,t)> • Transient: N(t) ~ t2 • Steady state: N(t) = a t • a =dE T(E)/2p • T(E)=single particle trans. coeff. • Schwinger 1959 • If V0≤2c2 a =0 • however N(t) ≠ 0 and bump N(t) = a t
Emitted electron spectrum P(E,t)=<0|| (t) (t) ||0> • Initially decreasing function • peak at V0/2 for long times • Steady state region • P(E,t) = T(E)/2p t • T(E)=single part. trans. coeff.
Escape direction of e– and e+ e- expect: e+ Electric field e+ e- e+ e- : e+: e- wrong way e- e+ wrong way
Evolution of electron and positron
Klein paradox Oskar Klein (1894-1977) Z Physik 53, 157 (1929)
height H speed V Classical mechanics predicts: if V2 < H bounces back if V2 > H rolls up
Traditional quantum mechanics if V2 < H bounces back Movie: Courtesy of Bernd Thaller http://www.kfunigraz.ac.at/imawww/thaller/
height H speed V Single-particle quantum mechanics predicts: even if V2 < H “some of the particle” rolls up and some of the particle bounces off
Wave packet evolution undersingle particleDirac equation Interpretation of mysterious transmitted portion unclear Braun, et al PRA 59, 604, (1999)
pair production from vacuum pair production with incoming electron
2004: Resolution of the Klein-paradox Transmitted portion is a hole in the positron density P. Krekora, Q. Su and R. Grobe, Phys. Rev. Lett. 92, 040406 (2004)
N. Nitta, T. Kudo, H. Minowa, Am. J. Phys. 67, 966 (1999)
innocent Legal disclaimer: ILP denies any involvement in any illegal activities on campus
Can one measure the Klein paradox ? Generally: no need to duplicate experiments with theory and vice versa Here: very fundamental questions Collisions of two ions: U92+ U92+ Z/r + Z/r (sub-critical) (sub-critical) ≈ 2Z/r (supercritical field) => pairs are produced: U92+ + U92+ U92+ + U92+ + e– + e+
Schrödinger’s Zitterbewegung Erwin Schrödinger (1887-1961) 1933 Nobel Prize
Zitterbewegung Y(t=0) = |E+> + |E–> <x(t)> = x0 + V0 t + jitter Schrödinger, Sit. K. Press. Akad. Wiss. Phys. Math K1 24, 418 (1930) Dirac, The Principles of Quantum Mechanics, 1964 J. Braun, Q.Su and RG, PRA59, 604 (1999) Is there a physical object that jitters? • unphysical interpretation • wrong use of operators (Newton-Wigner) • real, but practically unobservable • real and due to …. ? ?
|E+> |E_> ? ? ? ? ? Myth about the Zitterbewegung |Y> = |E+> + |E–> Common belief: • Zitterbew. associated with localization problem ? F(x,t)=< 0 | Y (+)(x,t) | F(t=0) > (+) removes |E–> states from field operator e– described by F(x,t)does not jitter !!
Summary Glue between quantum mechanics & field theory found first application: Birth process of an electron • e– arbitrarily narrow • maximum e– - e+ separation is lc • Klein paradox is resolved • Zitterbewegung does not exist P. Krekora, Q. Su and R. Grobe, Phys. Rev. Lett. 92, 040406 (2004) P. Krekora, Q. Su and R. Grobe, Phys. Rev. Lett. (submitted) P. Krekora, Q. Su and R. Grobe, Nature (to be submitted) www.phy.ilstu.edu/ILP
