1.2 Gauge theories

The prototype of all gauge theories is electro- magnetism. From the geometrical point of view the electro­magnetic potential aM (/x = 1,..., 4) defines a connection for a U( 1) bundle over Minkowski space M. The field is the corresponding curvature

 

Maxwell's equations in vacuo take the form

 

where/ is now viewed as a 2-form, d is the exterior derivative and d* is its formal adjoint (relative to the Minkowski metric).

Non-abelian gauge theories are obtained by replacing (7(1) with a compact non-abelian Lie group G, e.g. SU(n). A potential is then a connection A over Minkowski space, with components AM in the Lie algebra of G, and the field is the curvature F with components

 

The most straightforward generalization of Maxwell's equations are the Yang-Mills equations

dF = 0, d*F = 0.

Gauge theories possess an infinite-dimensional symmetry group given by functions g: M-» G and all physical, or geometric, properties are gauge invariant.

 

where the norm and volume are those of Minkowski space. For Yang-Mills theory the Lagrangian is

 

To specify a physical theory the usual procedure is to define a Lagrangian or action L. This is a functional of the various fields obtained by integrating over M a Lagrangian density. For example, for a scalar field theory where the only field is a scalar function <p, the simplest Lagrangian is

where the norm here also uses an invariant metric on G.

These Feynman integrals are not very well defined mathemati­cally but they can, when used skilfully, be a useful heuristic

Having fixed a Lagrangian L(<p) the 'partition function' of the theory (by analogy with statistical mechanics) is the Feynman functional integral

 

 

More generally, for any functional W{<p), the unnormalized expectation value of the 'observable' W is defined by the integral

 

 

tool. In particular, perturbation expansions can be computed explicitly.

The Feynman integral provides a relativistically invariant approach. This is its main purpose. In a non-relativistic treat­ment a quantum field theory is described by a time-evolution operator e"H in a certain Hilbert space X. The infinitesimal generator H is the Hamiltonian of the theory. There are formal rules which, starting from the Lagrangian formulation via the Feynman integral, produce the Hilbert space "X and the Hamiltonian H. The fundamental relation between the two approaches rests on the formula

 

 

where (p0, <pT are scalar fields on R3 (space) and the Feynman integral is taken over all fields <p(x, t) which interpolate between (p0 = (p(x, 0) and <pT = <p{x, T) for 0< / < T. In par­ticular

 

where, in the Feynman integral, <p is a function on R3 x S'T where SV is the circle of length T.

 

Witten's version of the Jones theory is defined by a suitable choice of Lagrangian in 2 +1 dimensions and this will be described in Chapter 7. Until then we shall be following the non-relativistic Hamiltonian approach, which is mathemati­cally more rigorous.

In gauge theory, classical fields of force are described in terms of curvature. However, gauge theories have global features which can be non-trivial even when all curvatures vanish. This is fundamental for the relations with quantum field theory which are our basic interest. The prototype of this is the Bohm-Aharonov effect in the quantum theory of the electron. This concerns a solenoid with an interior magneticflux but with no external magnetic field. A beam of electrons travelling past the solenoid produces interference patterns indicating a phase-shift. This physical effect takes place even though the electrons travel in a force-free region.

 

Electron

Beam

 

Mathematically the wave-function of the electron in the external region is a section of a flat line-bundle, with non- trivial holonomy round the solenoid.

In non-abelian gauge theories wave-functions are sections of vector bundles and the holonomy lies in a non-abelian group. This is the starting point for the relation between topology and quantum field theory that is embodied in the Jones-Witten theory.