Stochastic Lorentz forces on a point charge moving near the conducting plate Department of Physics National Dong Hwa University TAIWAN Da-Shin Lee Talk given at QFEXT07, Leipzig, Germany 17 – 21 September 2007

Stochastic Lorentz forces on a point charge moving near the conducting plate When a charged particle interacts with quantized electromagnetic fields, a nonuniform motion of the charge will result in radiation that backreacts on itself through electromagnetic self-forces as well as the stochastic noise manifested from quantum field fluctuations will drive the charge into a zig-zag motion. We wish to explore the anisotropic nature of vacuum fluctuations under the boundary by the motion of the charged particle near the conducting plate.

REFERENCE Stochastic Lorentz forces on a point charge moving near the conducting plate Jen-Tsung Hsiang, Tai-Hung Wu, Da-Shin Lee e-Print: arXiv: [hep-th]

The influence of electromagnetic fields on a nonrelativistic point charge moving near the conducting plate is studied by deriving the nonlinear, nonMarkovian Langevin equation from Feynman-Vernon influence functional within the context of the Schwinger-Keldysh formalism. This stochastic approach incorporates not only backreaction dissipation on a charge in the form of retarded Lorentz forces, but also the stochastic noise manifested from electromagnetic vacuum fluctuations. Under the dipole approximation, noise-averaged result reduces to the known ADL equation plus the corrections from the boundary, resulting from classical effects. Fluctuations on the trajectory driven by the noise are of quantum origins where the dynamics obeys the F-D relation. Velocity fluctuations of the charged oscillator are to grow linearly with time in the early stage of the evolution at the rate, smaller in the parallel motion than that of the normal case. The saturated value is then to obtain asymptotically for both orientations of the motions due to delicate balancing effects between F & D. Summary

Velocity fluctuations ( quantum effect ) It is of interest to compute velocity fluctuations of this charged oscillator under fluctuating electromagnetic fields in the presence of the boundary. Velocity fluctuations grow linearly in time at early stages, and then saturate to a constant at late times. Although they for two different orientations of the motion start off at different rates, the same saturated value is reached asymptotically.

Discussion on the saturated value of velocity fluctuations The change in velocity fluctuations, as compared with a static charge interacting with electromagnetic fields in its Minkowski vacuum state, arises from the imposition of the conducting plate as well as the motion of the charge. The relative importance between two effects will be estimated by taking an electron as an example. Fluctuations induced by the boundary : constrained by the electron’s plasma frequency as well as the width of the wave function Fluctuations induced by the motion of the charge Velocity fluctuations owing to the electron's motion are overwhelmingly dominant

The initial density matrix for the particle and fields is assumed to be factorizable by ignoring the initial correlations: The fields are assumed to be in thermal equilibrium with the density matrix given by: where is the free field Hamiltonian. Then, in the Schroedinger picture, the density matrix evolves in time as: The Lagrangian for a nonrelativistic charged particle coupled to electromagnetic fields is given by such a particle-field interaction ( the Coulomb gauge):

Here we have introduced an identity in terms of a complete set of eigenstates The reduced density matrix of the particle by tracing out the fields becomes: Then, the matrix element of the time evolution operator can be expressed by the path integral.

Reduced density matrix

We also assume that the particle is initially in a localized quantum state approximated by the position eigenstate: The nonequilibrium partition function can be defined by taking the trace of the reduced density matrix over the particle variable. The limits have be taken at this moment.

The Langevin equation is then obtained by extremizing the stochastic effective action. We ignore intrinsic quantum fluctuations of the particle by assuming that the resolution of the length scale measurement is greater than its position uncertainty.

Remarks: The influence of electromagnetic fields appears as the nonMarkovian backreaction in terms of electromagnetic self forces, and stochastic noise, driving the charge into a fluctuating motion. This is the nonlinear Langevin equation on the charge's trajectory since the dissipation kernel as well as noise correlation are the functional of the trajectory. The noise-averaged result arises from classical effects. Fluctuations on the particle’s trajectory driven by the noise entirely are of the quantum origin as seen from an explicit dependence on the noise term.

Fluctuation-Dissipation theorem Fluctuation-Dissipation theorem plays a vital role in balancing between these two effects to dynamically stabilize the nonequilibrium Brownian motion in the presence of external fluctuation forces. The tangential component of E fields and the normal component of B fields on the perfectly conducting plate surface located at the z=0 plane vanish.

The corresponding fluctuation-dissipation theorem can be derived from the first principles calculation: The F-D theorem at finite-T The F-D theorem in vacuum

Gauge invariant expression Retarded E and B fields are obtained by introducing the Lienard-Wiechert potentials together with the Coulomb potential. Stochastic E and B fields involve transverse components of the gauge potentials only because in the Coulomb gauge, the Coulomb potential is not a dynamical field, and hence it has no corresponding stochastic component.

Langevin equation under the dipole approximation Dipole approximation will be applied for this nonrelativistic motion to account for the backreaction solely from E fields. The charged particle undergoes the harmonic motion with the small amplitude at. An additional component of the external potential is applied to counteract the Coulomb attraction from its image charge. The initial conditions are specified as which can be achieved by applying an appropriate external potential to hold the particle at the starting position with zero velocity. Then the applied potential is suddenly switched off to the harmonic motion potential.

The noise-averaged equation ( classical effect ) Backreaction from the free-space contribution entails the retarded Green's function nonvanishing for the lightlike spacetime intervals. The charge follows a timelike trajectory where radiation due to the charge’s nonuniform motion can backreact on itself at the moment just when radiation is emitted. It is given by, electromagnetic self force + UV-divergence absorbed by mass renormalization =the ADL equation. Backreaction owing to the boundary has a memory effect where emitted radiation backscatters off the boundary, and in turn alters the charge's motion at a later time.

The kernel can be found from inverse Laplace transform: where the Browish contour is to enclose all singularities counterclockwisely on the complex s plane. The branch-cut arises from discontinuity of the kernel. Since the cut lies within the region of where imaginary part of the self-energy nonvanishing. The pole equation: The poles originally in the first Rienmann sheet move to the second sheet due to the interaction with environment fields as long as the poles are in the cuts. The pole on the first sheet located in the positive real s axis corresponds to the runaway solution to be discarded. ．

High frequency modes relevant to very early evolution are ignored. The resonance mode with the peak around the oscillation frequency is found to have dominant contributions to the late time behavior: Breit-Wigner shape

Velocity fluctuations ( quantum effect ) It is of interest to compute velocity fluctuations of this charged oscillator under fluctuating electromagnetic fields in the presence of the boundary. Velocity fluctuations grow linearly in time at early stages, and then saturate to a constant at late times. Although they for two different orientations of the motion start off at different rates, the same saturated value is reached asymptotically.

The function has a Breit-Wigner feature on k space peaked at about and its width being approximately of order at early times or at late times. The spectral density reveals the oscillatory behavior on k space over the change in k by. The integrand has the linear k dependence for large k, leading to quadratic UV- divergence with the weak time dependence in velocity fluctuations.

Growing regime: Backreaction dissipation can be ignored. Velocity fluctuations thus mainly result from the stochastic noise. Velocity fluctuations are found to grow linearly with time. The growing rate is related to the relaxation constant out of the dissipation kernel due to the F-D relation. Quadratic UV-divergence is found to vary slowly in time. The effect of the stochastic noise on the oscillator is much weaker, leading to a smaller growing rate on the parallel motion than the normal one since E field fluctuations parallel to the plate vanish, but its normal components become doubled, compared with that without the boundary. The relaxation constant shares the similar feature as a result of the F-D relation. The presence of the boundary apparently modifies the behavior of the charged oscillator in an anisotropic way.

Saturation regime: We investigate the behavior of velocity fluctuations at late times by incorporating backreaction dissipation. Backreaction from the contribution of the resonance is isotropic due to delicate balancing effects between fluctuations and dissipation, and thus is solely determined by the motion of the charge. The high-k modes probe UV-divergence as well as the strong boundary dependence for small z on backreaction. As expected, the enhancement in velocity fluctuations arises in the normal motion for small z resulting from large E fields induced in that direction.

Discussion on the saturated value of velocity fluctuations The change in velocity fluctuations, as compared with a static charge interacting with electromagnetic fields in its Minkowski vacuum state, arises from the imposition of the conducting plate as well as the motion of the charge. The relative importance between two effects will be estimated by taking an electron as an example. Fluctuations induced by the boundary : constrained by the electron’s plasma frequency as well as the width of the wave function Fluctuations induced by the motion of the charge Velocity fluctuations owing to the electron's motion are overwhelmingly dominant

The influence of electromagnetic fields on a nonrelativistic point charge moving near the conducting plate is studied by deriving the nonlinear, nonMarkovian Langevin equation from Feynman-Vernon influence functional within the context of the Schwinger-Keldysh formalism. This stochastic approach incorporates not only backreaction dissipation on a charge in the form of retarded Lorentz forces, but also the stochastic noise manifested from electromagnetic vacuum fluctuations. Under the dipole approximation, noise-averaged result reduces to the known ADL equation plus the corrections from the boundary, resulting from classical effects. Fluctuations on the trajectory driven by the noise are of quantum origins where the dynamics obeys the F-D relation. Velocity fluctuations of the charged oscillator are to grow linearly with time in the early stage of the evolution at the rate, smaller in the parallel motion than that of the normal case. The saturated value is then to obtain asymptotically for both orientations of the motions due to delicate balancing effects between F & D. Summary

Q & A