Advances in the Worldline Approach to Quantum Field Theory: Strong Fields, Amplitudes and Gravity

Abstract

This thesis is devoted to the first-quantized approach to quantum field theory, commonly known as the 'Worldline Formalism'. It collects most of the works completed by the author during the PhD, illustrating the versatility and efficiency of this formalism across a broad range of physical contexts. The applications discussed fall into two broad categories: perturbative and non-perturbative analyses. In particular, the thesis investigates how quantum particles interact with strong background fields, how scattering amplitudes can be efficiently computed, and how perturbative expansions of the heat kernel can be systematically performed. These studies highlight recent advances in extending the worldline approach to increasingly complex situations. Different field theories are examined from this first-quantized perspective and are organized according to the spin of the particle under consideration. The discussion begins with the seemingly simple scalar case, progresses through spin 1, both in the abelian and the non-abelian case, and concludes with spin-2 particles, both in the massless and in the massive realizations. Through this sequence of examples, the thesis aims to demonstrate the flexibility as well as the computational power of the Worldline Formalism in addressing fundamental open problems in theoretical physics.

Figures (8)

• Figure 1: Schematic illustration of the distinction between the two approaches commonly employed within the worldline framework. In the “top-down” approach, one starts from a given quantum field theory quantity and reformulates it in worldline form. In the “bottom-up” approach, one instead begins with a worldline model and extracts field theoretical information from it.
• Figure 2: Topologies of the worldline.
• Figure 3: Graphical representation of a virtual electron-positron pair becoming real due to the presence of strong electric field.
• Figure 4: Diagrammatic perturbative expansion of the Euler--Heisenberg effective Lagrangian with the inclusion of the classical Maxwell Lagrangian. Note that, in QED, the diagrams with an odd number of external photon legs vanish due to Furry's theorem.
• Figure 5: The left panel is a density plot of $\log\left(\frac{P^{(1)}}{P^{(0)}}\right)$ as a function of $\bar{m}$ and $\bar{\omega}$, taking $\epsilon=10^{-3}$ and $D=4$. The right panel shows $\frac{P^{(1)}}{P^{(0)}}$ as a function of $\bar{\omega}$, for three different values of the dimensionless mass: $\bar{m}=2$ (yellow, dotted line), $\bar{m}=1$ (blue, dashed line) and $\bar{m}=0.5$ (red, continuous line).

Theorems & Definitions (4)

• Proposition 1
• proof
• Theorem 1
• proof