Preface

Summary

In the past four decades, gauge theories have become dominant in high energy physics. All known fundamental interactions – strong, weak, electromagnetic, and gravitational, are described by gauge-invariant Lagrangians (actions). In the framework of classical physics, gauge invariance does not cause any significant theoretical problems. In the case of electrodynamics, which has always served as the benchmark for all other field theories, gauge arbitrariness can be eliminated by adding a suitable supplementary (gauge) condition on the vector potential, e.g. the Lorenz condition, to the equations of motion. In gravity, the De Donder–Fock gauge is often used. But initial studies on quantization of electromagnetic fields showed that gauge theories required a special approach. Canonical quantization demands the existence of the Hamiltonian formalism, whose construction turned out to be not an easy task. It appeared that in electrodynamics the equations that relate generalized velocities and canonical momenta cannot be solved for the former, i.e. the velocities cannot be expressed as functions of the generalized coordinates and momenta (a consequence of the Lagrangian being singular (or degenerate)). As a result, conditions on canonical variables (constraints) occur. It was required, first, to formulate the theory of dynamical systems with constraints and, second, to find a consistent procedure for their quantization.

For electrodynamics these problems had already been solved by W. Heisenberg and W. Pauli in 1930. In a further development of modern quantum electrodynamics (QED), it appeared to be possible to avoid the essential problem of quantization in the presence of constraints.