Quantum gauge theories

This chapter, which is the last chapter in Part I, is devoted to an extensive discussion of quantum gauge theories, which is based on functional integrals and Lagrangian quantization. After introducing the notion of a Yang-Mills gauge theory, the Faddeev-Popov method (also known as the DeWitt-Faddeev-Popov procedure) is explained. Starting from this point, the BRST symmetry is formulated, and the corresponding Ward identities (called Slavnov-Taylor identities in some cases) established. More specialized subjects, such as the gauge dependence of effective action and the background field method, are dealt with in detail. In addition, Yang-Mills theory is analyzed as a primary example of general theorems concerning the renormalization of gauge theories.

Analysis of a Collision-Energy-Based Method for the Prediction of Ice Loading on Ships

Ships designed for operation in Polar waters must be approved in accordance with the International Code for Ships Operating in Polar Waters (Polar Code), adopted by the International Maritime Organization (IMO). To account for ice loading on ships, the Polar Code includes references to the International Association of Classification Societies’ (IACS) Polar Class (PC) standards. For the determination of design ice loads, the PC standards rely upon a method applying the principle of the conservation of momentum and energy in collisions. The method, which is known as the Popov Method, is fundamentally analytical, but because the ship–ice interaction process is complex and not fully understood, its practical applications, including the PC standards, rely upon multiple assumptions. In this study, to help naval architects make better-informed decisions in the design of Arctic ships, and to support progress towards goal-based design, we analyse the effect of the assumptions behind the Popov Method by comparing ice load predictions, calculated by the Method with corresponding full-scale ice load measurements. Our findings indicate that assumptions concerning the modelling of the ship–ice collision scenario, the ship–ice contact geometry and the ice conditions, among others, significantly affect how well the ice load prediction agrees with the measurements.

Gauge theories

After reviewing electrodynamics as the special case of an abelian gauge theory, this local symmetry is generalised to non-abelian gauge theories. The curvature of space-time is introduced as analogue of the non-abelian field-strength. Non-abelian gauge theories are quantised using the Fadeev–Popov method and the resulting Feynman rules are derived.

EMERGENCE OF THE HAAR MEASURE IN THE STANDARD FUNCTIONAL INTEGRAL REPRESENTATION OF THE YANG-MILLS PARTITION FUNCTION

The conventional path integral expression for the Yang–Mills transition amplitude with flat measure and gauge-fixing built in via the Faddeev-Popov method has been claimed to fall short of guaranteeing gauge invariance in the nonperturbative regime. We show, however, that it yields the gauge-invariant partition function where the projection onto gauge-invariant wave functions is explicitly performed by integrating over the compact gauge group. In a variant of maximal Abelian gauge the Haar measure arises in the conventional Yang-Mills path integral from the Faddeev-Popov determinant.

THE RELATIVISTIC LANDAU PROBLEM FOR FERMIONS

Using the Faddeev-Popov method an exact expression for the propagator of a relativistic spinning particle in a constant magnetic field is found. The conservation laws and the generators of the magnetic group are obtained in the path integral formalism. Both the relativistic and non-relativistic models are discussed in the planar case.

THE FADDEEV–POPOV METHOD FOR THE RELATIVISTIC LANDAU PROBLEM

We use the Faddeev–Popov method to calculate explicitly the path integral propagator for a relativistic spinless charged particle in the presence of a constant magnetic field. We obtain the conservation laws in the path integral approach. We also establish the equivalence between the Faddeev–Popov method and the Fock–Schwinger proper time approach. Finally, after proposing a suitable regularization prescription for the non-relativistic problem, we obtain the Landau levels directly from the path integral result.