Regauging - Classical Gauge Theory - An Example: Scalar O(n) Gauge Theory

An Example: Scalar O(n) Gauge Theory

The remainder of this section requires some familiarity with classical or quantum field theory, and the use of Lagrangians.

Definitions in this section: gauge group, gauge field, interaction Lagrangian, gauge boson.

The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between fields which were originally non-interacting.

Consider a set of n non-interacting scalar fields, with equal masses m. This system is described by an action which is the sum of the (usual) action for each scalar field

The Lagrangian (density) can be compactly written as

by introducing a vector of fields

The term is Einstein notation for the partial derivative of in each of the four dimensions. It is now transparent that the Lagrangian is invariant under the transformation

whenever G is a constant matrix belonging to the n-by-n orthogonal group O(n). This is seen to preserve the Lagrangian since the derivative of will transform identically to and both quantities appear inside dot products in the Lagrangian (orthogonal transformations preserve the dot product).

This characterizes the global symmetry of this particular Lagrangian, and the symmetry group is often called the gauge group; the mathematical term is structure group, especially in the theory of G-structures. Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the currents

where the Ta matrices are generators of the SO(n) group. There is one conserved current for every generator.

Now, demanding that this Lagrangian should have local O(n)-invariance requires that the G matrices (which were earlier constant) should be allowed to become functions of the space-time coordinates x.

Unfortunately, the G matrices do not "pass through" the derivatives, when G = G(x),

The failure of the derivative to commute with "G" introduces an additional term (in keeping with the product rule) which spoils the invariance of the Lagrangian. In order to rectify this we define a new derivative operator such that the derivative of will again transform identically with

This new "derivative" is called a covariant derivative and takes the form

Where g is called the coupling constant; a quantity defining the strength of an interaction. After a simple calculation we can see that the gauge field A(x) must transform as follows

The gauge field is an element of the Lie algebra, and can therefore be expanded as

There are therefore as many gauge fields as there are generators of the Lie algebra.

Finally, we now have a locally gauge invariant Lagrangian

Pauli calls gauge transformation of the first type to the one applied to fields as, while the compensating transformation in is said to be a gauge transformation of the second type.

The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian

This term introduces interactions between the n scalar fields just as a consequence of the demand for local gauge invariance. However, to make this interaction physical and not completely arbitrary, the mediator A(x) needs to propagate in space. That is dealt with in the next section by adding yet another term, to the Lagrangian. In the quantized version of the obtained classical field theory, the quanta of the gauge field A(x) are called gauge bosons. The interpretation of the interaction Lagrangian in quantum field theory is of scalar bosons interacting by the exchange of these gauge bosons.