3.5 An Illustrative Example of Fibre-Bundle and Connection

Fig. 3.16 Polar coordinates on the 2-sphere

We construct our fibre-bundle introducing two local trivializations respectively associated with the two open charts H − and H +:

φN−1 : π −1 H − → H − U (1) : φN−1 {x1, x2, x3} = zN , exp[iψN ]

(3.5.3)

φS−1 : π −1 H − → H + U (1) : φS−1 {x1, x2, x3} = zS , exp[iψS ]

To complete the construction of the bundle we still need to define a transition function between the two local trivialization namely a map:

In order to illustrate the geometrical interpretation of the transition function we are going to write it is convenient to make use of polar coordinates on the two sphere. Following the conventions of Fig. 3.16, we parameterize the points of S2 by means of the two angles θ [0, π ] and φ [0, 2π ]:

and we see that the azimuth angle φ parameterizes the points of the equator identified by the condition θ = π/2 x3 = 0.

The catch in the above equation is that each choice of the integer number n identifies a different inequivalent principal bundle. The difference is that while going around the equator the transition function can cover the structural group S1, once, twice or an integer number of times. The transition function can also go clockwise or anti-clockwise and this accounts for the sign of n. We will see that the integer n characterizing the bundle can be interpreted as the magnetic charge of a magnetic monopole. Before going into that let us first consider the special properties of the case n = 1.

The Case n = 1 and the Hopf Fibration of S3 Let us name P(n)(S2, U(1)) the principal bundles we have constructed in the previous section. Here we want to show that the total space of the bundle P(1)(S2, U(1)) is actually the 3-sphere S3.

To this effect we define S3 as the locus in R4 of the standard quadric:

It is immediately verified that xi (i = 1, 2, 3) as given in (3.5.11) satisfy (3.5.1) if Xi (i = 1, 2, 3, 4) satisfy (3.5.9). Hence this is indeed a projection map from S3 to S2 as claimed in (3.5.10). We want to show that π can be interpreted as the projection map in the principal fibre bundle P(1)(S2, U(1)). To obtain this result we need first to show that for each p S3 the fibre over p is diffeomorphic to a circle,

namely π −1(p) U(1). To this effect we begin by complexifying the coordinates of R4

so that the equation defining S3 (3.5.9) becomes the following equation in C2:

Then we name U − S3 and U + S3 the two open neighborhoods where we respectively have Z2 = 0 and Z1 = 0. The Hopf map (3.5.11) can be combined with the stereographic projection of the North pole in the neighborhood U − and with the stereographic projection of the South pole on the neighborhood U + obtaining:

From both local descriptions of the Hopf map we see that:

is the 3-sphere S3. Indeed we can rewrite them in the opposite direction as follows:

This time the transition function does not have to be invented rather it is decided a priori by the Hopf map. Indeed we immediately read it from equations (3.5.17):

which coincides with (3.5.8) for n = 1. This is the result we wanted to prove: the 3-sphere is the total space for the principal fibre P(1)(S2, U(1)).

The U(1)-Connection of the Dirac Magnetic Monopole The Dirac (see Fig. 3.17) monopole [22] of magnetic charge n corresponds to introducing a vector

P(n)(S , U(1)) we discussed above. Physically the discussion goes as follows. Let us work in ordinary three dimensional space R3 and assume that in the origin of our coordinate system there is a magnetically charged particle.

From the relation Fij = εij k Hk we conclude that the electromagnetic field strength associated with such a magnetic monopole is the following 2-form on R3 − 0:

Fig. 3.17 Paul Adrien Maurice Dirac (1902–1984). Together with Einstein, Schrödinger, Pauli and Heisenberg, Dirac is one of the founders of the new vision of the world encoded in XXth century physics. For the revolutionary originality of his contributions he ranks second to no-one, including Einstein. The invention of Dirac equation for the electron wave-function shares the same degree of intellectual audacity and scientific creativity as the invention of Einstein equation for the metric. Similarly to Einstein’s case Dirac’s mathematical discovery led to the prediction of new fully unexpected phenomena such as the existence of the positron and in general of anti-matter. Paul A.M. Dirac was born in England from a French-speaking father immigrated from Switzerland and a British mother. He studied engineering in Bristol and only in 1923 he joined Cambridge University where he developed most of his career. He was awarded the Nobel Prize in Physics in 1933 together with Erwin Schrödinger for the discovery of new productive forms of atomic theory. He spent the last fourteen years of his life in the United States of America where he was professor in Florida at Miami Coral Gables and then Tallahassee. Among the many scientific achievements of Dirac there is also the quantum theory of magnetic monopoles. His paper on the subject dates 1931. There he showed that if magnetically charged particles existed in Nature this would account for the mutual quantization of electric and magnetic charges. Dirac’s argument amounts, in contemporary mathematical language, to the discussion presented in this section about fibre-bundles. Quantization of the charge is a topological effect since the exponent in the transition function of a U(1) bundle, namely the number n, is necessarily an integer

Indeed (3.5.19) makes sense everywhere in R3 except at the origin {0, 0, 0}. Using polar coordinates rather than Cartesian ones the 2-form F assumes a very simple expression:

namely it is proportional to the volume form on the 2-sphere Vol(S2) ≡ sin θ dθ dφ. We leave it as an exercise for the reader to calculate the relation between the constant g appearing in (3.5.20) and the constant m appearing in (3.5.19). Hence the 2-form F can be integrated on any 2-sphere of arbitrary radius and the result, according to Gauss law, is proportional to the charge of the magnetic monopole. Explicitly we have:

Consider now the problem of writing a vector potential for this magnetic field. By definition, in each local trivialization (Uα , φα ) of the underlying U(1) bundle we must have:

where Aα is a 1-form on Uα . Yet we have just observed that F is proportional to the volume form on S2 which is a closed but not exact form. Indeed if the volume form were exact, namely if there existed a global section ω of T S2 such that Vol = dω than, using Stokes theorem we would arrive at the absurd conclusion that the volume of the sphere vanishes: indeed 3S2 Vol = 3∂S2 ω = 0 since the 2-sphere has no boundary. Hence there cannot be a vector potential globally defined on the 2- sphere. The best we can do is to have two different definitions of the vector potential, one in the North Pole patch and one in the South pole patch. Recalling that the vector potential is the component of a connection on a U(1) bundle we multiply it by the Lie algebra generator of U(1) and we write the ansätze for the full connection in the two patches:

on U −; A = AN = i 1 g(1 − cos θ ) dφ 2

(3.5.23)

on U +; A = AS = −i 1 g(1 + cos θ ) dφ 2

As one sees the first definition of the connection does not vanish as long as θ = 0, namely as long as we stay away from the North Pole, while the second definition does not vanish as long as θ = π , namely as long as we stay away from the South Pole. On the other hand if we calculate the exterior derivative of either AN or AS we obtain the same result, namely i times the 2-form (3.5.20):

The transition function between the two local trivializations can now be reconstructed from:

and by comparison with (3.5.8) we see that the magnetic charge g is the integer n classifying the principal U(1) bundle. The reason why it has to be an integer was already noted. On the equator of the 2-sphere the transition function realizes a map of the circle S1 into itself. Such a map winds a certain number of times but cannot wind a non-integer number of times: otherwise its image is no longer a closed circle.

In this way we have seen that the quantization of the magnetic charge is a topological condition.

3.6Riemannian and Pseudo-Riemannian Metrics: The Mathematical Definition

We have discussed at length the notion of connections on fibre bundles. From the historical review presented at the beginning of the chapter we know that connections appeared first in relation with the metric geometry introduced by Gauss and Riemann and developed by Ricci, Levi Civita and Bianchi. It is now time to come to the rigorous modern mathematical definition of metrics.

The mathematical environment in which we will work is that provided by a differentiable manifold M of dimension dim M = m. As several times emphasized in Chap. 2, we can construct many different fibre-bundles on the same base manifold M but there are two which are intrinsically defined by the differentiable structure

dle T M = M . Of these bundles we can take the tensor powers (symmetric or antisymmetric or none) which are bundles whose sections transform with the corresponding tensor powers of the transition functions of the original bundle. The sections of these tensor-powers are the tensors introduced by Ricci and Levi Civita in their absolute differential calculus, for whose differentiation Christoffel had invented his symbols (see (3.2.7) and (3.2.8)).

For brevity let us use a special notation for the space of sections of the tangent bundle, namely for the algebra of vector fields: X(M ) ≡ Γ (T M , M ). Then we have

Definition 3.6.1 A pseudo-Riemannian metric on a manifold M is a C∞(M ), symmetric, bilinear, non-degenerate form on X(M ), namely it is a map:

g(, ) : X(M ) X(M ) −→ C∞(M )

satisfying the following properties

(i)X, Y X(M ) : g(X, Y) = g(Y, X) C∞(M )

(ii)X, Y, Z X(M ), f, g C∞(M ) : g(f X + hY, Z)=f g(X, Z) + hg(Y, Z)

(iii){ X X(M ), g(X, Y) = 0} Y ≡ 0

Definition 3.6.2 A Riemannian metric on a manifold M is a pseudo-Riemannian metric which in addition satisfies the following two properties

(iv)X X(M ) : g(X, X) ≥ 0

(v)g(X, X) = 0 X ≡ 0