Lesson 14 Web

Lesson 14 Groups in Special Relativity

Introduction

The last lesson treated a single world-line and the frame that travels with it. The four-velocity, the four-acceleration, the Frenet tetrad, and the 2-form that rotates that tetrad were all objects attached to one observer. Special relativity is not only about one observer. It is about the family of all inertial frames, and about the maps that send one such frame to another. Those maps form groups.

A Lorentz transformation is not merely a matrix that acts on four-vectors. It is an element of a topological group, then of a Lie group, and finally of the isometry group of Minkowski spacetime. The Poincaré group enlarges that picture by translations. Once those groups are in view, the “index gymnastics” of earlier lessons becomes the representation theory of a familiar object, and the local frames of Lesson 13 become the fibers of a principal bundle.

We begin with topological groups, after a short aside on equivalence relations and the fundamental group, so that “continuous group” means something precise. The first examples are vector spaces, general linear groups, and translations. A brief physical interlude—constitutive laws—shows why a group action on tensors is not a luxury: it is how a material or a field theory is told which tensors are allowed. Product bundles and fiber bundles, including the Möbius band, then make the same idea geometric. Principal bundles are the setting in which frames live. Spacetime itself, with its Lorentz frames, is the example we are aiming at.

More background follows, where the tangent space at a point, smooth fiber bundles, the tangent bundle, a return to differentials, and the Lie bracket are presented. Those tools turn a group of transformations into an algebra of infinitesimal generators.

Lie groups and their Lie algebras occupy the center of the lesson. After some motivation we define the Lie derivative on scalars and on tensors, then Lie groups, one-parameter subgroups, generators, and the Lie algebra of a Lie group. Only then do we name the groups that special relativity actually uses, introducing the Lorentz group and the Poincaré group. Their one-parameter subgroups recover the boosts and rotations already written in rapidity language. Isometries close the circle, where the Poincaré group is the group of maps that preserve the Minkowski interval.

The last two sections do the same constructions in Mathematica and point to further reading. By the end of the lesson a Lorentz matrix will no longer be an isolated 4×4 array. It will be a point of a Lie group, a generator will be a vector in the corresponding Lie algebra, and a local frame will be a point of a principal bundle over spacetime. That is the language in which symmetries, conservation laws, and—later—gauge fields are written.

Topological Groups

Recall from Lesson 1 that a group (G,◦) is a set with an associative product, an identity, and inverses. A subset of G that is itself a group under the same product is a subgroup.

Say that we have two groups, G and H. Also say that we have some mapping between these groups, φ:G→H. If such that φ(g◦ h)=φ(g)◦φ(h), then we call φ a group homomorphism. If the group homomorphism is also a bijection, then we call it a group isomorphism.

If we have a topological space having a group structure such that the mappings φ◦G→G and the inversion are continuous then the elements of the space that form the group are called a topological group. We call this relationship between the algebraic structure of the group and topological structure of the space a compatibility.

We are not here to study abstract algebra, or even topology. We are using these mathematical structure to get some clue as to their use in representing physical processes mathematically, thus formulating new theories. We can get some idea of how to do this if we look at what a group does.

One thing we can say is that if for every element of a group we can associate a mapping of some set, then we say the group acts on that set. Such mappings are a bijection of the set with itself.

An example is in order. If we take a square as our point set and a counterclockwise rotation by 90° as the mapping, we can form a group out of the composition of these mappings. The mapping is an example of the identity mapping.

We can see then that we can use groups as important representations of the symmetry of a system. We will also see that groups are important components of precise definitions of other structures that we have been using where our terminology might have been a little sloppy up to now.

We can make this idea more precise. We can say that a group G acts on the left on a set S if, then such that the following properties are true:

LA1: For the group identity .

LA2: For all g ∧ h ∈ G, .

Without further assumptions we can say that every map is a bijection.

Exercise 14.1: Show that each is a bijection, with inverse .

In this way, the action of a group on a set can be thought of as a homomorphism between the group G and the group of mappings that are induced on the set being acted upon.

If where , then we call the action effective.

If this condition holds for some, but not all, elements of S, then the action is said to be free.

If such that , then we call the action transitive.

For an element s∈S, all points t∈S that can be reached by is called the left-orbit , or just the orbit (if there is no confusion associated with the term), of the point s.

The set of all elements that are orbits of S is called a quotient space.

We can say that a group G acts on the right on a set S if, then such that the following properties are true:

RA1: For the group identity .

RA2: For all g ∧ h ∈ G, . Note the order of composition. This difference distinguishes the right action for the left action.

We can also have right orbits in direct analogy with the left-oriented orbits we have been discussing.

For example, we have discussed the space-time bundle, , for classical mechanics. We also noted the nature of the Galilean group in Lesson 4. We can sort of see that the action of the Galilean group relates all inertial frames within the space-time bundle by making sure that the laws of physics look the same. This also provides the degrees of freedom of a system. Motion not constrained by symmetries will have a number of degrees of freedom equal to the spatial dimensions multiplied by the number of bodies (or particles) of the system. Symmetries simplify the equation, for example, adopting a spherical symmetry allows us to say that instead of three degrees of freedom per particle, we now have only one degree of freedom per particle.

An Aside on Equivalence Relations

We can define an equivalence relation (symbol) between two other symbols, for example a R b, if R is a symmetry relation a R a, is a reflexive relation a R b ⇒ b R a, and is a transitive relation a R b ∧ b R c ⇒ a R c. We can denote an equivalence relation between a and b by a ~ b.

Exercise 14.2: Show that the equality symbol = is an equivalence relation.

Exercise 14.3: Explain, for our example of counterclockwise rotation of a square by 90° is an equivalence relation.

If we have an element of a set, a ∈ S, such that there is an equivalence relation involving b, we can construct a new set [a]={b|b∈S∧a~b}. The new set [a] is called an equivalence class.

In this case, we can think of each orbit as an equivalence class.

The Fundamental Group

Recall from Lesson 12 that we defined an arcwise connected space, and we described a way of thinking about it, “A way of thinking about this is that we can make a continuous curve between any two points in a space where the entire curve is contained in the space.” We can now expand on this idea, that curve is what we can call a path between the end points.

Take a space S and two of its points, x and y. Ask whether those points can be joined by a curve that never leaves S. Precisely: is there a continuous map

(14.1)

with f(0)=x and f(1)=y? Continuity is what forces the image to be an unbroken curve. The value f(0) is the source, or initial point; f(1) is the target, or final point.

If such a map exists for every pair of points in S, the space is path-connected, and each such map is a path.

A single path depends on one parameter s∈I. To deform one path into another we use two parameters at once, where we have a continuous map

(14.2)

For each fixed p∈I, the slice F( ⋅ ,p) is a path in S. As p runs from 0 to 1, those slices form a continuous one-parameter family of paths. That family is the raw material of a homotopy; we will see what that means next.

Hold the endpoints fixed while the family runs. Require

(14.3)

for every p∈I, not merely for p in the open interval (0,1). Then every slice F( ⋅ ,p) starts at the same point q and ends at the same point r.

The two ends of the family are themselves paths from q to r

(14.4)

As p varies, F deforms t into u without ever moving the endpoints. In that case t and u are said to be homotopic relative to the point q and r. More generally, they are homotopic relative to any specified set of points not allowed to move. A homotopy is any continuous deformation of one map into another. We add the term rel A to indicate that an additional rule is being imposed, where every point in A stays put during a deformation. For paths it is normal to write A as {0,1}, the two endpoints of the interval. We write this

(14.5)

saying that is is homotopic rel {q,r}.

Without “rel,” a homotopy could slide the endpoints around. That is a different equivalence relation, and it does not give the fundamental group. Loops based at are the special case , where the whole deformation is pinned at the base point.

The function f then represents a continuous deformation of the curve t into u that we call a path homotopy. For each fixed p, f(s,p) is a curve from q to r and is an intermediate curve between t and u, like an interpolation, and must be within the space. For each fixed s, f(s,p) is a curve from t to u and is called a variation.

It is possible to show that this homotopy is an equivalence relation.

Problem 14.1: Can you show this?

Given this equivalence relation, there is an associated equivalence class denoted [γ]. These are, reasonably enough, called homotopy classes.

We can compose paths to form new paths. This composition is well-defined on the homotopy classes.

Problem 14.2: The proof of this is beyond the scope of this discussion. Consult a book on algebraic topology and write a proof of this, explaining each step.

Such compositions do not form a group.

In Lesson 12 we also defined a loop. The end-points specified for a loop are termed its base-point. If we take the homotopy classes of groups formed by loops with a given base-point, then we can form a group called the fundamental group of the topological space relative to the base-point.

Problem 14.3: Consult your book on algebraic topology and figure out how much the fundamental group depends on the base-point.

Assume that we have some group G acting on a topological space S. This action is usually required to be continuous. The mappings are also required to be homeomorphic. Specifically, a topological group G acts on the left on a topological space if the following properties are met:

TGA 1: g induces a homeomorphism .

TGA 2: and .

TGA 3: The left action l:G×S→S is a continuous map. Specifically, is continuous in both g and x.

If these conditions are met, then G is called a topological group of mappings of S.

Every group induces a group of mappings by left action are called left translations. Every group also induces a group of mappings by right action called right translations. These actions are then free, effective, and transitive.

In an Abelian group the left and right actions coincide.

Examples of Topological Groups

Vector Spaces

Every vector space is an Abelian topological group under vector addition.

General Linear Groups

The set of linear mappings of a finite-dimensional vector space, V, under composition forms a topological group called the general linear group and is denoted, G L(V). Such a group consists of all nonsingular square matrices of nth order under matrix multiplication.

For the vector spaces we write the general linear group G L(n;R).

There are three proper subgroups:

The group of all nth order matrices whose determinant is 1 is called the special linear group and is denoted S L(n,R).

The group of all orthogonal matrices is called the orthogonal group O(n,R).

The group of all orthogonal matrices with determinant 1 is called the proper orthogonal group . The action is not free. The orbits are circles whose centers are at the origin.

Translations

Since is a vector space, it is also a group. This group acts on itself to produce all possible translations.

Exercise 14.4: What can you say about left and right actions on this group.

Exercise 14.5: What is the orbit of this group?

The Application to Physics: Constitutive Laws

Going back to classical mechanics, we can consider a material point as an infinitesimally small volume in subject to deformation by linear maps that we discussed in Lesson 10. These linear maps can be represented by nonsingular matrices that we will denote F. If there is some scalar-valued function of the state of the system we can call it a constitutive relation. If we identify such a function, say φ, we can say that φ depends on F through a constitutive equation,

(14.6)

We can apply the general linear group, G L(3;R), formed by all 3×3 matrices, G, that acts on the right of the matrix containing all possible constitutive equations of the form (14.6),

(14.7)

Such a matrix, G∈G L(3;R), constitutes a material symmetry of the constitutive equation φ if

(14.8)

or, specifically,

(14.9)

We say that the constitutive equation φ is invariant under the right action of G. The collection of such symmetries is called the material symmetry group of φ.

Exercise 14.6: Is the right action of G L(3;R) on the collection of matrices F?

We can also examine the left actions of G L(3;R) by

(14.10)

If we look at the orthogonal group, O(3,R)⊂G L(3;R), we can interpret this as the consequence of our choice of Galilean frame on φ.

We can say that

(14.11)

In other words, all constitutive equations are invariant under this action. This is called the principle of material frame indifference.

Product Bundles

We are now in a position to explain that a topological fiber bundle is a topological space having some extra structure. We have already introduced the idea of a fiber bundle, but we are now in a position to go into greater depth. We have spoken of Galilean space-time as the product set .

Exercise 14.7: Can you specify the topology of ?

We can now state that the base is not merely a space, it is a manifold in its own right. From now on we will no longer call it the base space, we will call it the base manifold.

In this way we can see that a point within a fiber is also a point within the manifold that we can call the product bundle,

(14.12)

where S is the product bundle, B is the base manifold, and F is the common fiber.

For any product bundle we have two projection maps,

(14.13)

called the first projection, and

(14.14)

called the second projection. These maps must be surjective and continuous in the product topology.

If we have an open subset of B, we can call it U. We can then write .

Exercise 14.8: Can you apply ?

Let’s say that we have an atlas in B and an atlas in F. The product of these atlases is itself an atlas of the product bundle. For the product bundle, as a manifold, this is reasonable enough—as we can always cover a manifold with an atlas.

More About Fiber Bundles

The fiber bundle is more general than a product bundle. The way that we think about this is that we want to keep the first projection and drop the second projection. What does this mean? If we think about Galilean space-time the notion of simultaneity is dependent on the observer (the base manifold—that specified by the first projection) and not on the spatial orientation (the fiber—that specified by the second projection).

A fiber manifold having a base manifold B, a typical fiber F, and a group structure G that acts on the left on F is a manifold S. The manifold S has a continuous surjective map π:C→B. This is called a bundle preservation map. With this map there is an open cover of B. The open cover has subsets we can denote . We can define homeomorphisms

(14.15)

These homeomorphisms have the property . Such are termed local trivializations.

Here and are fibers of the product . This means that for every point s in an open set the fiber is mapped onto the fiber . We can make this more precise by saying that a local trivialization consists of both the identity map of a set of open sets and a continuous family of fiber-wise homeomorphisms .

What about the group structure? What role does it play?

Assume that we have two trivializations, and . Say also that their open sets are not disjoint, . Then we gain the homeomorphisms and .

We can define the composition

(14.16)

We call this the transition map at s between the first and second trivializations.

Each transition map must be a mapping of F that is induced by some element g(s)∈G.

Let’s say that we have the same fiber bundle as before with a different open cover we will call . We will place on this cover the trivializations . This can then be called a trivialization cover. If we have the composition belongs to G then we have the same fiber bundle as before. When this happens we can say that the two trivializations are compatible.

From this we can see that the union of two trivialization covers forms a new trivialization cover.

We can denote a fiber bundle as (C,φ,B,F,G) and these are of course placeholder labels for the symbol of the fiber bundle, the symbol of the trivialization, the symbol of the base manifold, the symbol of the typical fiber, and the symbol of the group providing the structure. In situations where there is no chance of confusion we can denote it in a more compact form, (C,φ).

Every point in a base manifold has a neighborhood. The restriction of the fiber bundle to this neighborhood is itself homeomorphic to its product with a typical fiber.

Exercise 14.9: Is a fiber bundle always locally trivializable?

We can state without proof the following theorem.

Theorem 14.1: This is also called the Fundamental Existence Theorem of Fiber Bundles: Assume that we have the manifolds B and F and a topological group G that acts effectively on the left on F. Further assume an open cover of B. We also assume the continuous mapping, , of an element of G to every point in every non-disjoint intersection of the cover. Then there will exist a fiber bundle (C,φ,B,F,G) having local trivializations based on the cover and where elements of G are assigned as transition maps. Any two bundles having this property are equivalent.

Problem 14.4: Consult a book on differential geometry and prove this theorem.

There is an important application of Theorem 14.1, we can use it to associate bundles that have the same base manifold and group structure, but having a different typical fiber.

How do we do this? It is almost algorithmic:

Choose a trivialization cover for the given bundle.

Calculate the transition maps.

Define the associated bundle (C',φ',B,F',G) using Theorem 14.1.

This associated bundle will be modulo an equivalence.

Exercise 14.10: What does modulo an equivalence mean?

A bundle whose typical fiber coincides with the structure group G is said to be a principal bundle.

Thus, an associated bundle whose typical fiber coincides with the structure group G is said to be an associated principal bundle.

The Möbius Band

We begin by taking a two-dimensional long strip, twisting one end of it by 180°, and then gluing the ends together. This is called a Möbius band, it looks like this.

Figure 14.1: A Möbius band.

As we have discussed in Lesson 12 we have a product space of a 1-sphere, , and a real interval (say [0,1]). This forms a fiber bundle having its base and the typical fiber is the real interval, and we can call that I.

For any point in there will exist an open neighborhood U. We will assert that the part of the Möbius band corresponding to U will be homeomorphic to the product U×I. How do we know this? Think about it, if we cut a thin slice we get a rectangular slice no matter where we cut.

We can then cover the entire band by two such slices that overlap at their ends. Here is where we need to be careful, as those end pieces are where we need to figure out how the homeomorphisms are related to each other along the fibers. We need to have the necessary freedom to perform the twist in the band. This amounts to performing the gluing process in two different ways on the two ends.

Where do we get these degrees of freedom from? The underlying structure group is where such ideas of symmetry are located. We need to map each point of a typical fiber to its inverse with respect to the center of that fiber. From this we can see that the underlying structure group must be the permutation group of the two ends.

More About Principal Bundles

As we said above a principal bundle is one whose typical fiber and underlying group structure coincide. The action of this group on the typical fiber is assumed to be on the left.

If an action is independent of trivializations then we call it a canonical action.

This being the case, the action of the underlying structure group in a principal bundle is the canonical left action of the group on itself.

If we can show the existence of a canonical right action of a group with itself, then we can show that a principal bundle has a canonical right action from the group to the principal bundle itself’. The canonical right action is free and fiber-preserving.

Exercise 14.11: What does this mean?

It turns out that the existence of a free right action on a manifold is sufficient, on its own, to define a principal fiber bundle. This is an important definition as it is free from the transition maps. It is possible to extend this to associated principal bundles, also without resort to transition maps.

Assume the we have a topological manifold, M, with an underlying structure group, G. Let’s say that G acts freely on the right on M. This tells us that there is a continuous map where . Further we can then write and for the group identity element . The action being free tells us that for some element of the group, if , then g=e.

Were we to define the base space as a quotient, B=M/G. The first thing to do is make sure that B is a topological manifold.

Exercise 14.12: How would you do that?

We also need to show that the canonical projection is continuous. The set is called the fiber over the points b∈B.

One way of looking at this, by the definition of a quotient, is that an element of the base manifold is an equivalence class of M by the action of G. Put another way, every element of the base manifold (the quotient) represents an orbit of G. Here the projection map assigns the appropriate orbit to each element of M. The fiber over the points in the base manifold are formed by the elements of M that are part of the orbit represented by that point in the base manifold.

There is only one remaining piece left before we can fully define a principal bundle—local triviality. This tells us that for every point in the base manifold there will exist a neighborhood U of M where the fiber is isomorphic to the product U×G. Thus we will have a fiber-preserving homeomorphism where such that . This must be consistent with the group action,

(14.17)

So this completes the formal definition of a principal bundle.

Spacetime

When Einstein derived special relativity, he destroyed the bundle structure of space-time and replaced it with the four-dimensional spacetime manifold. Special relativity combines the need for both a spatial and a temporal metric into a single spacetime metric, the spacetime interval. This metric structure is not positive-definite since we can have space-like and time-like distances between events. This is much simpler to formulate than the fiber bundle structure.

More Background

If special relativity discards the fiber bundle structure of space-time why spend all of this time on it? It turns out that every manifold has a bundle structure on it where the tensor fields live. We need to explore that to really understand how tensor fields exist in manifolds.

The Tangent Space at a Point

The set of all tangent vectors at a point can be denoted and is called the tangent space to the manifold M at .

Problem 14.5: Can you prove that is a vector space?

Assuming that is a vector space, then the set of vectors forms a basis that we call the natural basis. For any tangent vector we write

(14.18)

Smooth Fiber Bundles

Say that we have a fiber bundle. Show, or assume, that the base manifold and the typical fiber are smooth manifolds. Show, or assume, that the conditions of continuity are replaced by conditions of smoothness. We call this a smooth fiber bundle.

If we have some smooth m-dimensional manifold M, then we can define canonically fiber bundles with M as their base manifolds and the general linear group as their structure group. All of these bundles are associated.

Exercise 14.13: Explain why.

This is important if the base manifold has physical relevance, since then it is possible that some or all of the associated bundles will also have such physical relevance.

The Tangent Bundle

The most basic associated bundle of this type is found by attaching to each point in M its tangent space . Such a bundle is called a tangent bundle. This tangent bundle can be denoted T M.

In order to specify a tangent vector in a tangent space we need find the m coordinates of in M that the tangent vector is attached to. Then we need the m components of the vector within the vector space . The collection of tangent spaces projects on the original manifold. A point of the tangent bundle consists of where p ∈ M and . We have the projection map where .

For the chart (U,φ) we can assign to each point . We can specify this point as the 2 m components , thus it also represents point in . This gives us a chart on . We can continue this for the entire atlas of T M. In this way we can see that T M is a 2 m-dimensional differentiable manifold. Where the set is the fiber at the point p. As each fiber is an m-dimensinal vector space, we can say that the typical fiber is .

A Return to Differentials

We have already examined differentials on a manifold in Lesson 12. Say that we have the differentiable map between manifolds f:M→N. Given an open interval of , I, we can say that the curve λ:I→M through the point can be mapped to another curve through the map where . So we say . Here we collect the curves into equivalence classes. What do we call these equivalence classes, we call them tangent vectors.

Exercise 14.14: How can we use this to map tangent vectors at to tangent vectors at ?

We can thus use this chart computation to define a new linear mapping between the tangent spaces and . We can call this mapping the differential and denote it D f:T M→T N. Another term used for this is the tangent map.

We can call this an example of a fiber-preserving map. We can build up a diagram of the facts that we know about this. We begin with the first fact, f:M→N.

Figure 14.2. f:M→N.

We then have a projection map from T M to M labeled .

Figure 14.3. Adding a a projection map from T M to M.

We then have a projection map from T N to N labeled .

Figure 14.4. Adding a a projection map from T N to N.

Then we have the differential.

Figure 11.5. Taking the differential.

This is called a commutative diagram. We can say that the differential executes a push forward of tangent vectors located at to tangent vectors located at .

The Lie Bracket

There is an operation that we need to define called the Lie bracket

(14.19)

This is pronounced “Lee” and is named after the Norwegian mathematician Sophus Lie (1842-1899).

Exercise 14.15: What is the relationship between and ?

It is important to note that any vector space having a Lie bracket is called a Lie algebra.

Lie Groups and Their Lie Algebras

Thoughts Leading to Lie Theory

A smooth vector field on a manifold determines a parametric curve from a point in the manifold whose behavior is determined by some parameter. Such a curve as we have described is called a flow. Such a flow is defined over some interval of the parameter that depends on the point in the manifold. In this way we can define a one-parameter group mapping in a neighborhood of a point on the manifold to a neighborhood that smoothly depends on the chosen parameter.

Such a flow of a vector field will, reasonably enough, transport geometric entities with it. This makes sense if you think that at every point there is a vector, the definition of a vector field. If this field flows, then those vectors will be carried with it. This is sort of like transforming the vector field to a new vector field as it evolves. So a vector at one point becomes a new vector at another point.

One can imagine this being easy to analyze. Subtract the two vectors. Divide by the change in parameter. Then find the limit of the difference as the change in parameter is allowed to go to 0. This should give us the parametric rate of change, right?

The problem with this approach is that the vectors are attached to the manifold at different points, so we cannot just subtract them unless there is some structure in the manifold that allows us to do that.

All hope is not lost, though. Since the vector field pulls vectors around as it flows, we can wait for an interval in the parameter for a new vector to arrive at the same point in the manifold even though it originated at a different point. These vectors can be subtracted and then divided by the parameter interval. Taking the limit as this interval goes to 0 is equivalent to taking the Lie derivative of the evolved vector field with respect to its original state. We can do the same sort of thing for tensors and differential forms.

The Lie Derivative and the Lie Bracket

So, what is the Lie derivative of a vector field with respect to some other vector field? Say that we have the vector field in our manifold M. One element of is the vector connecting the points p and q, we can label this vector . If we assign the parameter s to the vector field then we can say that the vector points from p to q having the new coordinates .

Figure 14.6. The vector connecting two points in our manifold.

As flows due to the passage of another parameter t, it might change. The vector maps to a new set of points, in essence the vector field is no longer described by . Say that we call it . Then the vector connects p' and q'.

Figure 14.7. The vector connecting two transformed points in our manifold.

To connect to we require another vector field to describe how geometric objects (such as ) are dragged to a new form (). We can call this new vector field having the parameter t. The vector connecting p to p' is then called . We can then write,

(14.20)

at least to a first approach.

Figure 14.7. The new vector connecting a point to its transformed point.

Similarly we can call the vector connecting q to q' .

Figure 14.7. Another new vector connecting the second point point to its transformed point.

What is the expression for ? This would be , or . The coordinates can be found by Taylor expanding this.

(14.21)

Exercise 14.16: Derive this result.

We can find by subtracting by ,

(14.22)

We are not quite done yet. This describes a tangent vector at p'. We can use this to find the tangent vector at p, by noting . We can think of this as the shifted version of and we can label it , and by linear approximation we have,

(14.23)

Exercise 14.17: Derive this result.

We can now subtract these vectors,

(14.24)

We can now divide by Δ t,

(14.25)

Of course, we can write this in terms of Lie brackets,

(14.26)

This is what we call the Lie derivative of the vector field with respect to the vector field in the neighborhood of the point p,

(14.27)

The Lie Derivative of Scalar Fields

The Lie derivative of a scalar field is just the directional derivative of the scalar field.

(14.28)

The Lie Derivative of Tensors

The Lie derivative of a mixed rank-two tensor is

(14.29)

Lie Groups

As a first approach we can say that any group that looks like it is a manifold is what we call a Lie group. This kind of captures it, but it is somewhat sloppy. Let’s see if there is a way to tighten it up a bit.

How does a group, “Look like a manifold?” To be a manifold it must first be a topological space, right? So every Lie group is also a topological group. Second, the group must be locally Euclidean. Finally, the group must have a continuous group operation that admits a differentiable structure.

Any topological group, G, will have an identity element, . The connected component containing the identity element is a normal subgroup of G.

What is a normal subgroup? If J is a subgroup of G, and we denote elements of G as g and elements of J as j, then J is normal iff for the binary operation ◦ defining G. This expression is called conjugation. Thus a normal subgroup is a subgroup that is invariant under conjugation.

In this way, we can think of connected components as copies of . In a Lie group such connected copies are diffeomorphic. There is a term sometimes used for a connected Lie group, we call it an analytic group.

This leads us to a theorem that we will state without proof.

Theorem 14. This is also called Cartan’s Theorem: Any closed subgroup of a Lie group is itself a Lie group.

Since a Lie group is also a manifold it can canonically act on itself by left translations and right translations. Such actions are smooth, free, and transitive.

Since Lie groups are smooth manifolds we get extra structures not available to a topological group. We will explore some of those in the following pages.

One-Parameter Subgroups of a Lie Group

Say that we have some differentiable curve λ:E→G. Suppose further that , the group identity element. λ(a+b)=λ(a)λ(b).

Exercise 14.18: Can you show that is a consequence of λ(a+b)=λ(a)λ(b)?

We call this a one-parameter subgroup of the Lie group G.

Let G act on a smooth manifold M to the left. We can write this left action on a point p of the manifold as for any g∈G. Further, we can combine the action with a one-parameter subgroup λ to give us a new group. This group is a one-parameter group of mappings of M. We can write for t ∈ E. The one-parameter group of mappings of M represents the flow of a vector field, it is called the fundamental vector field.

The left action l(g,p) is a smooth mapping of an element of the group G and a point of the manifold. If we fix p then . So the fixed point p maps to the points of the manifold under the action of the group element, .

What role does the one-parameter subgroup λ:E→G play? We can compose this with the left action of the Lie group, . This map is the integral curve through p of the fundamental vector field. We can see this,

(14.30)

In other words maps the parameterized curve λ in G to the integral curve through p of the fundamental vector field.

The left action of the differential mapping is a tangent vector to the curve λ at the group identity unit to the fundamental vector field at p. We can write it

(14.31)

We can say that the tangent to the curve λ at the group identity element generates the fundamental vector field using the derivative of the left action .

A truly remarkable thing is that this trick works for any tangent vector at the identity element—it generates a vector field in the same way. The identity element need not even originate from the one-parameter subgroup.

There is a bijection between vectors that are tangent to G at the identity and one-parameter subgroups.

We can make two direct connects, or identifications if you like. The first is between the manifold M and the group G. The second identification is between the group action and the canonical left translations of the group.

In this way we can rewrite (14.26)

(14.32)

Here we get the fundamental vector field at the point h ∈ G by taking the partial derivative of the map l(g,h) with respect to the first variable.

We are not quite done, if we think about it, we will see that this map is the canonical right action of G on itself, . Then we can write,

(14.33)

Such a field, defined by the derivative, is called the generator of the Lie group G.

Generators

What is the practical use of generators? It would be nice to cope with groups without having to consider operations involving infinite numbers of elements.

Since Lie groups have a continuous dependence on variable parameters over closed intervals, then the corresponding set of parameters include the limit of any converging sequence of parameters.

Aside from forming a theoretical structure, we can use generators to describe a Lie group. In this approach the generator becomes a minimal set of values that can produce each element of the group we are considering. Put another way, if we have some parameter π that controls some operation of the group. We can introduce a generator (and we will use Gothic letters for this), G, so long as it has the property that the infinitesimal parameter, , can be represented (where we use R(a) as the representation of a),

(14.34)

Since we can generate operations for larger values of π by multiplying a small value (π/n) n times. Thus we get the representation

(14.35)

Since this defines an exponential, we then have

(14.36)

We, for any representation of our Lie group, we can take the derivative of (14.30) to find the generator evaluated at , we can write

(14.37)

This tells us everything about the behavior of the representation R that can be deduced from its behavior in a neighborhood or in the parameter space around the identity operation.

There are several facts that can be proved about generators that we will not prove. If you are so inclined go ahead and prove them.

The number of independent generators of a Lie group equals the order of the Lie group.

We can develop relations between generators.

In general, generators will not commute.

The maximum number of independent generators that mutually commute is the rank of the group.

The Lie Algebra of a Lie Group

If we have a diffeomorphism between two smooth manifolds, φ:M→N. We can say also that we have the vector field over M and over N. Since φ is a diffeomorphism then the derivative map D φ:T M→T N will map the vector field onto the pushed-forward vector field that is well-defined over N.

We then say that the vector fields and are φ-related. We can then write the condition for such φ-relatedness

(14.38)

There is a special case where M=N. In this case, if a vector field is φ-related to itself, then we call it invariant under φ. We can write the condition of invariance under a diffeomorphism,

(14.39)

Exercise 14.19: Can you show that (14.33) is equivalent to (14.32)?

Every left translation of a Lie group with itself, is a diffeomorphism, . In this way a vector field defined over the Lie group is left-invariant if it is invariant under all left translations. We can then write,

(14.40)

Problem 14.7: Can you prove that a vector field is left-invariant iff it is generated by such that,

(14.41)

Problem 14.8: Can you show that the fundamental vector field induced by a one-parameter subgroup acting on the left is right-invariant?

Problem 14.9: Can you show that the one-parameter subgroups of a Lie group occur simultaneously with the integral curves through the Lie group identity element of left- or right-invariant vector fields?

Problem 14.10: Can you show that the one-parameter subgroups of a Lie group can be characterized by a tangent vector to the Lie group at the group identity.

Problem 14.11: Can you prove that the Lie bracket of two left- or right-invariant vector fields is itself left- or right-invariant?

The set of left-invariant vector fields of a Lie group, or the tangent space having a Lie bracket operation, is called the Lie algebra of the Lie group G. In this way, the Lie algebra can be thought of as representing a first approximation of the Lie group. In fact, the Lie bracket measures to first-order how non-commutative the Lie group is.

The Lorentz and Poincaré Groups

We have already examined Lorentz transformations in Lessons 11 and 13. We have stated that the Lorentz transformations form the Lorentz group. If we include spacetime displacements along with the Lorentz transformations, then we have the Poincaré group. Lorentz and Poincaré groups are both Lie groups.

If we write the infinitesimal velocity of our frame then the Lorentz transformation becomes

(14.42)

and

(14.43)

We can write this in matrix form

(14.44)

We can rewrite this in terms of generators,

(14.45)

(14.46)

where we have the Pauli spin matrix

(14.47)

We can apply this to finite velocity by using the rapidity φ instead of . Recall that we introduced this concept in Lesson 13.

(14.48)

Here we passed from the infinitesimal velocity to φ by repeated transformations.

A Lorentz transformation that does not involve rotation is called a boost. We can study successive boosts by analyzing the Lorentz group. Say we compose a rapidity φ with another rapidity φ'. This gives us the representation,

(14.49)

Exercise 14.20: Can you use the Lorentz group to derive the velocity composition formula?

One-Parameter Subgroups of the Lorentz Group

If we are looking at rotations in the plane, then we have the one-parameter subgroup for the θ parameter

(14.50)

It’s generator is

(14.51)

Exercise 14.21: Can you show that these are correct?

If the parameter is rapidity φ

(14.52)

It’s generator is

(14.53)

Exercise 14.22: Can you show that these are correct?

We can also get a one-parameter subgroup by combining a boost and a rotation

(14.54)

It’s generator is

(14.55)

We can also get a one-parameter subgroup for null rotations.

(14.56)

It’s generator is

(14.57)

Why go through all of this? There is a nice theorem.

Theorem 14.3: For a given inertial frame, every one-parameter subgroup of L can be represented by if the trace of G(L)≠0 and by if the trace of G(L)=0.

Problem 14.12: Can you prove Theorem 14.3?

Isometries

Mappings of the manifold into itself that preserve the metric are called isometries. When the mappings are represented by field vectors on the manifold, isometries give us the components of a Lie derivative (one reason they are so useful). We can write, for the vector field

(14.58)

as we might expect.

We can write the condition for an isometry as

(14.59)

or,

(14.60)

We call this the Killing equation after the German mathematician Wilhelm Killing (1847-1923). The vector field is called the Killing field or the Killing vector.

The set of isometries form the isometry group, also called the Poincaré group.

Solutions of the Killing equation form a 10-dimensional Lie algebra of the Poincaré group.

Doing this Stuff in Mathematica

We now turn to another relativistic effect, say we have a mirror that is moving. The question is what happens to Snell’s law when this motion is relativistically significant? We begin in the rest frame of the mirror, R. Here Snell’s law works as normal, where the angle of incidence and reflection are equal. The incident 4-momentum of the photon is written, in terms of the incident frequency , Planck’s constant h, and a normalized spatial vector under the condition ,