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The relativistic time-dependent Aharonov-Bohm effect and the topology of the electromagnetic vacuum. Athan Petridis Zachary Kertzman Drake University. The Aharonov-Bohm Effect.
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The relativistic time-dependent Aharonov-Bohm effect and the topology of the electromagnetic vacuum Athan Petridis Zachary Kertzman Drake University
The Aharonov-Bohm Effect • Phase shift due to interaction with the vector potential A even in a region where the magnetic field B = 0 (magnetic case). • Such a field is produced by an infinitely-long solenoid (azimuthal A). Interference point Solenoid Source
The A-B Effect and Topology • The A-B effect arises because the group space of the gauge group U(1) is not simply connected (the gauge function χis multi-valued): R2 with a hole. The vacuum has topological structure. • The fundamental homotopy group, π1, of U(1) is isomorphic to Z (group of integers). • This is not so for SU(2 or 3):π1(SU(2 or 3)) = 1: No A-B effect.
In the Standard Model the E/M subgroup is irregularly embedded in the gauge group SU(2)I x U(1)Y. An E/M gauge transformation by an angle γrotates the state vector of charge Q by • The point in group space is with α/β = tanθW = irrational so that π1 = 1 The A-B effect may not persist at high energies.
Current Skepticism • Possible phenomena that may mimic the A-B effect: • Stray B dipole field due to finite solenoid (Tonomura, 1986 used toroidal magnets). • Induced Coulomb charges in the solenoid (Batelaan, 2007: metal reaction times 10-14 to 10-13 seconds. Small magnets needed). • Necessity: understand dipole vs solenoid contributions and time-dependence.
Time-dependence with the Dirac Equation • Relativistic quantum equation for spin-1/2 fermions, which are described by a 4-dimentional spinor Ψ. • Including an external scalar potential, V:
Initial Conditions (2 dimensions) • The initial spinor is (N = normalization factor, m = 1, c = ħ = 1): • The probability density ρ = Ψ†Ψ at t = 0 is Gaussian with standard deviation σ0. • As σ0→∞,Ψ becomes a positive energy plane wave, which for p0=0 is a spin +1/2 eigenstate.
The Numerical Algorithm • The staggered leap-frog method is applied on a spatial grid of bin-size Δx = Δy = d and with time step Δt: • The spatial derivatives are computed symmetrically. • Reflecting boundary conditions are applied on a very large grid (running stops before reflections occur if necessary). • It works well on a PC using dynamic memory allocation.
Stability of the Algorithm • The norm is used as stability measure • The stability region: (d = spatial grid bin, Δt = time step) • Obtained via a standard stability analysis usingplane waves (for the large component) probability 1 time
Solenoid and dipole fields • Minimal substitution: p → p – e A • Infinite solenoid vector potential (r >R): • Dipole (residual) vector potential (r >R): • Cylindrical electric potential V=const. (r <R) • Initial spinor: p0=1.134, σ0= 5, R=4
Pulsed beam experiment Solenoid R = 4 σ0 = 4 p0 = 1.134 Initial probability density
Dipole field, t=000000 B0 = 0.5
Dipole field, t=600000 The diffraction pattern is asymmetric
Solenoid field, t=000000 A0 = 0.5
