Search

Abstract In the first part of this series, we employed the second-order formalism and the “symbol” map to construct a particle path-integral representation of the electron propagator in a background electromagnetic field, suitable for open fermion-line calculations. Its main advantages are the avoidance of long products of Dirac matrices, and its ability to unify whole sets of Feynman diagrams related by permutation of photon legs along the fermion lines. We obtained a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the “N-photon kernel,” where this kernel appears also in “subleading” terms involving only N − 1 of the N photons. In this sequel, we focus on the application of the formalism to the calculation of on-shell amplitudes and cross sections. Universal formulas are obtained for the fully polarised matrix elements of the fermion propagator dressed with an arbitrary number of photons, as well as for the corresponding spin-averaged cross sections. A major simplification of the on-shell case is that the subleading terms drop out, but we also pinpoint other, less obvious simplifications. We use integration by parts to achieve manifest transversality of these amplitudes at the integrand level and exploit this property using the spinor helicity technique. We give a simple proof of the vanishing of the matrix element for “all +” photon helicities in the massless case, and find a novel relation between the scalar and spinor spin-averaged cross sections in the massive case. Testing the formalism on the standard linear Compton scattering process, we find that it reproduces the known results with remarkable efficiency. Further applications and generalisations are pointed out.

We investigate whether and how the quantum Zeno effect, i.e., the inhibition of quantum evolution by frequent measurements, can be employed to isolate a quantum dot from its surrounding electron reservoir. In contrast to the often studied case of tunneling between discrete levels, we consider the tunneling of an electron from a continuum reservoir to a discrete level in the dot. Realizing the quantum Zeno effect in this scenario can be much harder because the measurements should be repeated before the wave packet of the hole left behind in the reservoir moves away from the vicinity of the dot. Thus the required repetition rate could be lowered by having a flat band (with a slow group velocity) in resonance with the dot or a sufficiently small Fermi velocity or a strong external magnetic field. We also consider the anti-Zeno effect, i.e., how measurements can accelerate or enable quantum evolution.

Abstract In the first-quantised worldline approach to quantum field theory, a long-standing problem has been to extend this formalism to amplitudes involving open fermion lines while maintaining the efficiency of the well-tested closed-loop case. In the present series of papers, we develop a suitable formalism for the case of quantum electrodynamics in vacuum (part one and two) and in a constant external electromagnetic field (part three), based on second-order fermions and the symbol map. We derive this formalism from standard field theory, but also give an alternative derivation intrinsic to the worldline theory. In this first part, we use it to obtain a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the “N -photon kernel,” where off-shell this kernel appears also in “subleading” terms involving only N − 1 of the N photons. Although the parameter integrals generated by the master formula are equivalent to the usual Feynman diagrams, they are quite different since the use of the inverse symbol map avoids the appearance of long products of Dirac matrices. As a test we use the N = 2 case for a recalculation of the one-loop fermion self energy, in D dimensions and arbitrary covariant gauge, reproducing the known result. We find that significant simplification can be achieved in this calculation by choosing an unusual momentum-dependent gauge parameter.

Extending work by Gies and Karbstein on the Euler–Heisenberg Lagrangian, it has recently been shown that the one-loop propagator of a charged scalar particle in a constant electromagnetic field has a one-particle reducible contribution in addition to the well-studied irreducible one. Here we further generalize this result to the spinor case, and find the same relation between the reducible term, the tree-level propagator and the one-loop Euler–Heisenberg Lagrangian as in the scalar case. Our demonstration uses a novel worldline path integral representation of the photon-dressed spinor propagator in a constant electromagnetic field background.