Differential geometry and lie groups for physicists
The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity 1 and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data.
Lie groups and Lie algebras
The one-parameter groups are the first instance of Lie theory. The compact case arises through Euler's formula in the complex plane. Other one-parameter groups occur in the split-complex number plane as the unit hyperbola. Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere.
Its Lie algebra is the subspace of quaternion vectors. Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra. Standard treatments of Lie theory often begin with the classical groups. The initial application that Lie had in mind was to the theory of differential equations.
On the model of Galois theory and polynomial equations , the driving conception was of a theory capable of unifying, by the study of symmetry , the whole area of ordinary differential equations. According to historian Thomas W. In his work on transformation groups, Sophus Lie proved three theorems relating the groups and algebras that bear his name. The first theorem exhibited the basis of an algebra through infinitesimal transformations.
Differential Geometry and Lie Groups for Physicists
Lie theory is frequently built upon a study of the classical linear algebraic groups. Special branches include Weyl groups , Coxeter groups , and buildings. The classical subject has been extended to Groups of Lie type. We already convinced ourselves that this phenomenon is indeed realin the case of the transport of a vector around a particular spherical triangle The fact that the resulting vector had the samelength as the initial one so that the net change consisted only in the rotation is a particularfeature of the RLC connection actually its metricity.
In general, only the linearity of theoperator of the parallel transport along the closed path is guaranteed.
It turns out that an immensely important piece of information about the local dependenceof the parallel transport on the path is stored in the further tensor field characterizingthe connection, the curvature tensor. In order to motivate its formal definition, let us firstcompute what the operator of the parallel transport along a particular infinitesimal loop lookslike, namely the loop we already encountered in Chapter 4, when studying the geometricalmeaning of the commutator of vector fields. Wesawin problem 4.
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Let us investigate this useful fact from a slightly more generalperspective. Solution: on F M , each derivation is given by a vector field see Section 2. According to 4. Then it is enough see 4. It follows from the antisymmetry in the last pairof indices that the second contraction differs from the third one only in a sign and it turnsout that the first one vanishes for the RLC connection, so that it is usually ignored. In thecase of a Riemannian manifold a further contraction is possible a tool for raising the indexon the tensor of type 02 is available and one can define a further tensor field being alreadyof type 00 , a function.
A highly effectiveway of computing the curvature tensor consists in using the machinery of differential formsto be discussed in the next section. We will also mention some of its further properties there. Show that if the scalar curvature of the firstmanifold vanishes and this is not the case for the second one then the two manifolds cannotbe isometric to each other. Infer from this that the sphere S 2 is not locally isometric tothe Euclidean plane we have already proved the same result before, referring to differentKilling algebras, see 4. If we take a vector in x and perform the parallel transport along the loop,in general we arrive according to the result of For a linear connectionthis is not necessarily the whole group GL n, R , but rather it may be only a subgroup.
Let us have a look, first, at how information about the connection may be encodedinto appropriate 1-forms. A more detailed discussion of thesetopics is in order, however, since it may help the reader to develop some intuition for thework with both forms and, moreover, it paves the way for the theory of general connectionsand gauge fields, to be developed in Chapter Curvature: the role of the curvature operator R U, V within the context of translationalong a loop Appendix A.
According to Section 6. We see from the result of This matrix is non-singular and infinitesimally close to the identity matrix. This point of view reveals with no computation at all, as an example, that forthe metric connection the matrix of the connection 1-forms with respect to an orthonormalframe field has to be pseudo- antisymmetric. Since the connection determines the torsion andcurvature tensors, it determines the forms and T as well.
Consequently, there should exist. This is the way in which the Cartan structure equations areobtained. Let us have a look at the modifications of the Cartan structureequations in this particular case. That is why the solution of Cartan structure equationsshould not be a creative procedure either. Step 1 amounts to the diagonalization of the matrixof the metric tensor; in real life situations, however, one often obtains the required framefield without the formal procedure of diagonalization.
Check that i the following relations are valid it is useful to compare them with their coordinate counterpartsdisplayed in Hint: following the lines of At any given point, imagine two mutuallyperpendicular circles of appropriate radii such that they match optimally the surface in theneighborhood of the point. Let their radii be r 1 , r 2. Verify that this algorithm is consistent with theresults we obtained for the sphere and the torus.
Check that the Cartan structureequations for the RLC connection lead in these two cases to i the common resulti. The curvature andtorsion tensors enter important identities Bianchi and Ricci , which are most easily derived,and even formulated, in the language of forms. They enterthe formulas for the covariant derivative of spinor fields, cf. The formula obtained above may sometimes befound in the literature on spinors in general relativity under the noble name of the tetrad postulate. Hint: the particular case of For their relative position vector,.
This fact is so evident we knew it in advance and no computation was needed for it that the reader might beastonished as to why this trivial stuff should enter Section Imagine they start their sailssimultaneously, being say m from one another the second one eastwards from thefirst one. Both of them move uniformly along a straight line again with the same speed to the south, each one along their meridian.
Their trajectories thus also represent affinelyparametrized geodesics, but this time with geodesics on the sphere S 2. It turns out that the phenomenon already occurs at the local level andit is a manifestation of the behavior of nearby geodesics. The aim is then to learn what effect the small variation of the initial conditions has on the future course of thegeodesic.
Put another way, what is the variation of the rest of the geodesic for a given variation of its initial conditions? This object is still not the most interesting one since we can control it bymeans of the choice of its value at the time zero. Here we mayconsider meridians as a one-parameter class of geodesics RLC connection. This results in the torsion mostly remaining hidden in the shadowof its much more popular sibling, the curvature. In this section we will learn in which geometrical situation the nonvanishing torsion manifests its presence. In this waywe obtain the points Q 2 and R 2.
Check this statement bya computation. Hint: for example, in coordinates: according to Components of the transported vector v are As scientists recently discovered under microscopes, I expect this spectacular astronomical phenomenon was already prettywell known to Mayan civilization. Mayan astronomers compiled precise tables of positions for the Moon, Venus, Curvatureand Torsion and were able to predict with astonishing accuracy torsion eclipses caused by the curvature; their predictionnamely stated that it always happens. In what way do these constructions actually differ?
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In Chapter 4 we managed without any connection, here we definitely need it. In particular,there we moved along integral curves of the vector fields involved, whereas here we movealong geodesics. There we needed the fields U, V also in a neighborhood of the point P,whereas here we make do with the vectors u,vat the point P alone.
In both cases unclosed parallelograms arose; then due to non-vanishing [U, V ], now dueto non-vanishing T U, V. There is also an equivalent way of expressing the effect of torsion. Contemplate vectorsu,vat the point P. Extend them to vector fields U, V in a small neighborhood of the pointP as follows: if Q is a point in the neighborhood, we construct a geodesic from P to Q andparallel transport the vectors u,vto Q along the geodesic recall that a parametrization of thegeodesic does not matter.
All the transported vectors then constitute the vector fields U, V. The effect of torsion thus happens to coincide with the effect of minus thecommutator of these vector fields. Recall also that Section Let us illustrate non-vanishingtorsion with the example of a simple connection where the effect of the torsion may beeasily grasped visually. If it is as big as the surface ofthe Earth, it may easily happen we actually do not recognize it is a sphere it took sometime for mankind, too and we believe we walk on a Euclidean plane.
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Then it is natural toperform the parallel transport of vectors as follows. First, we measure the length of the vector to be transported and arrange the length to bethe same after the transport. Then the only issue which remains is its direction. In order to fixthe direction we use a compass and measure the azimuth of the initial vector i. We then prescribe the sameazimuth to the transported vector. This is not an accident. However,here the latter is very clear visually.
Hint: the distance between meridians gets shorter when we start to move in a directiontoward the poles. This comes into being when in a domain on a manifoldthere is a covariantly constant frame field e a alternatively it is known as a parallel framefield , i. If it happens to be holonomic i.