2. — Gauge potentials and fields.

We shall now recall the data of a classical theory as understood by physicists and then reinterpret them in geometrical form.

We begin by fixing a compact Lie group G, typically SU{2) or SU(3), but not excluding yet the abelian group Z7(l). We consider its Lie algebra L(G), which for 8U(n) consists of skew-hermitian n x n matrices of trace zero. A gauge potential is then a set of functions A^x) taking values in L(G), where x = ... x4) is a point of Minkowski or Euclidean space and H — 1, ..., 4 is a spatial index. Associated with this potential we also con­sider the operator

 

 

where dll = d/dx». This operator acts on a vector function (fi(x),..., /„(«)) whenever we are given an m-dimensional representation of G: for example when G = 8U(n) we can take m = n using the standard representation.

Computing the commutator of and V, we get the gauge field l\v, given by

 

 

where the commutator [J.^, Av] is taken in the Lie algebra of G. The im­portant point to note is that for non-abelian G this commutator does not vanish and so J* is a non-linear function of A. For G — 11(1) however this term drops out and we get the usual linear relation between the field and vector potential that characterizes Maxwell theory.The usual non-uniqueness of the potential has its counterpart in the general case in the form of gauge transformations. By definition a gauge transformation is a function g{x) taking values in G and transforming the potential An by the formula which corresponds to sending V^ into fir1^ (here we consider G as a group of matrices so that d^g is simply the differentiated matrix). The gauge field F„v then transforms by

 

 

It is important to observe that the A„ transform inhomogeneously whereas the Ff,v transform homogeneously. In other words F„v is a vectorial (or ten- sorial) object whereas A„ is an affine object (with no preferred zero).

Geometrically or mechanically we can interpret this data as follows. Imagine a structured particle, that is a particle which has a location at a point x of R* and an internal structure, or set of states, labelled by elements g of G. We then consider the total space P of all states of such a particle. In general we conceive of the internal spaces Gx and Gv for x # y as not being identified and so we draw the picture of P as a collection of « fibres »

 

 

In the absence of any external field however we consider that all Gx can be identified to each other so that in addition to the vertical lines or fibres we can also draw horizontal lines (called sections) making the usual Car­tesian type of grid

 

Now we imagine an external field imposed which has the effect of distorting the relative alignment of the fibres so that no coherent identification is possible between the Gx at different points. However we assume that Gx and Gv can still be identified if we choose a definite path in jR4 from x to y. In more physical terms we imagine the particle moving from x to y and car­rying its internal space with it. In Minkowski space such a motion would take place along the world line of the particle. This identification of fibres along paths is called «parallel transport». If we now imagine two different path joining x to y then there is no reason for the two different parallel transports to agree and they are assumed to differ by multiplication with a group element, which could be viewed as a generalized «phase shift». This phase shift is interpreted as produced by the external field. In geome­trical terms it is viewed as the total «curvature» or distortion of the fibre bundle over the region enclosed by the two paths.

If we now infinitesimalize this picture in Newtonian style we get the infinitesimal parallel transport at a point i in a given direction. This will be an infinitesimal shift A of the fibre Gx into the nearby fibre, and is called a connection. The infinitesimal curvature F depends on two directions at x and takes values in the Lie algebra of Gx, i.e. it is an infinitesimal « phase shift». As usual the infinitesimal picture, that is the connection, can be integrated up to give the global picture of parallel transport along curves: the two points of view are mathematically equivalent.

If we now compare this picture with the situation where we had no field and all fibres Gx were coherently identified we can view parallel transport as a change of phase in a fixed copy of G, and the connection as an element A„{x) of the Lie algebra depending on the point x and the /i-th direction. Thus we recover the gauge potential of the physicists' language. Similarly the curvature F becomes the gauge field F„r(x), taking values in the fixed Lie algebra of G. Thus the curvature F can be thought of as the distortion produced by an external field, or it can be identified with the field when we think of a field of force as measured by its local effects. This identification of fields with geometrical distortion is of course at the heart of Einstein's theory of gravitation. The difference here is that the distortion is not taking place in the geometry of space-time but in the geometry of some fictitious state-space of internal structure super-imposed on space-time. This diffe­rence makes the relevant geometry less obvious and historically, both in physics and in mathematics, the geometry of fibre-bundles came later than the geometry of space. It is significant however that both mathematicians and physicists, each for their own reasons, were led to study these objects which in fact turn up naturally in a great variety of contexts.

Despite its later historical appearance the geometry of fibre bundles ofthe type we have been describing is much simpler technically than the Biemann-Einstein geometry of space. This is because the relevant group of our theory was taken as a finite-dimensional group G whereas in Bieman- nian geometry we have to deal with the group of all coordinate transforma­tions. To clarify this point we shall now return to our fibre bundles and re-examine their relation to gauge theory.

In order to describe our geometrical connection in algebraic terms we compare our parallel transport with the situation in the absence of a field. In this case we used a coherent identification of all the fibres Qx. Now it is important to emphasize that this coherence represents the absence of a field but the particular choice of coherent identification is at our disposal. A particular choice is called picking a gauge and a change from one choice to another is a gauge transformation. Pictorially we imagine two different sets of horizontals for our fibre bundle and the change from one to the other is described by a function g(x), taking values in G. Ko particular choice is regarded as being preferred (despite the appearance of the picture!). Once a gauge has been picked the connection and curvature can be written down in coordinate form. The group of gauge transformations plays a role analogous to that of coordinate transformations in Biemannian geometry. Since it is basically a simpler group the geometry of fibre-bundles is an easier theory: it is, in a definite sense, «less non-linear ».

It should be emphasized that a connection is a definite geometric object and is more primitive than the curvature. As a consequence one should consider the gauge potential as more primitive than the gauge field. This is borne out physically even in electro-magnetism by an experiment which shows that the field may be identically zero but physical effects are still detected due to the fact that parallel transport need not be trivial if the region of space is not simply-connected. The vanishing of curvature only gives information about parallel transport round very small closed paths. In physical terminology parallel transport in general is described by talking about non-integrable phase factors. K on-integrability locally refers to a non-vanishing- field, whereas large scale non-integrability is topological in character (going round a wire for example) and may arise even for zero fields (outside the wire). Classically potentials were introduced as a mathematical

device to simplify the field equations and the ambiguity (or gauge freedom) in choice of potential was taken as an indication that the potential had no genuine physical meaning. The geometrical point of view shows that this is too narrow an interpretation. The connection is a geometric object and so the potential should be regarded as physical. The unphysical thing is the choice of gauge in which one chooses to describe the potential, cor­responding to the fact that our geometrical fibre bundle where the con­nection sits has no natural horizontal sections. These remarks about the predominant role of the potential will acquire more substance when we discuss the field equations in the general non-abelian case.

So far we have talked only about fibre bundles in which the fibre was the group G. These are called principal fibre bundles by differential geom­eters. However for applications one is usually interested in associated bundles in which the fibre is a vector space C" corresponding to a representa­tion of G. Again the typical case is to take G = U(n). The geometric picture is essentially similar in that we consider a space (the vector bundle) E, fibred over 124 so that the fibre Fx is thought of as a vector space depending smoothly on x. Parallel transport from x to y is regarded as a unitary trans­formation from Ex to Ey. Thus parallel transport in the principal bundle gives rise to parallel transport in the vector bundle and the same applies to connection and curvature. In particular a section of the vector bundle, namely a function f(x) defined on R* and taking values in the variable vector space JEX is thought of in terms of its graph. A connection enables us to shift this graph infinites imalJy in a given direction of R1. This shift is precisely the covariant derivative V^/. This is a geometric notion independent of any choice of gauge. Once we choose a gauge we can describe f algebraically by a m-vector (f-i{x), ..., fn(x)) of ordinary functions and the covariant deri­vative is then given explicitly by formula (2.1). The curvature Fuv defined as the commutator V»] is again seen to be geometric in character, appearing now as an (algebraic) operator on sections of the vector bundle.