## The Vacuum Polarisation Contribution to the Lamb Shift Using Non-Relativistic Quantum Electrodynamics

##### Abstract

In this thesis we calculate the vacuum polarisation contribution to the well-known Lamb shift. The Lamb shift is a correction of order $\alpha^5$ to the non-relativistic energy spectrum of the hydrogen atom. We discuss the gauge symmetry of quantum electrodynamics (QED), and derive the photon propagator in both the Lorenz and Coulomb gauge. We then calculate the vacuum polarisation tensor of the photon in QED. It is calculated using dimensional regularisation and the modified minimal subtraction scheme. We show that the vacuum polarisation results in a finite correction to the photon propagator, and thus changes the two-point function of the photon field. We introduce non-relativistic QED (NRQED), an effective field theory that is popular for QED bound state calculations. Radiative and relativistic corrections are incorporated into NRQED by a matching procedure. The matching procedure enforces that the scattering amplitudes of QED coincide with the scattering amplitudes of NRQED. The vacuum polarisation of the photon is incorporated into NRQED by adding correction terms to the photon Lagrangian. The two-point function of the photon field in NRQED is matched to the two-point function of the photon field in QED at one loop. We use the corrected photon Lagrangian to calculate the vacuum polarisation contribution to the Lamb shift. This contribution is a relatively small fraction of the total Lamb shift, it shifts the level of the 2S$_\frac{1}{2}$-state downward by 27.1 MHz compared to the 2P$_\frac{1}{2}$-state. Lamb shift is an upward shift of around 1000 MHz. We would not have agreement between the theoretical result and experimental observations without the vacuum polarisation contribution. The Lamb shift lifts the degeneracy between the 2S${}_\frac{1}{2}$-state and the 2P$_\frac{1}{2}$-state.