Applications of Differential Geometry to Physics

Transcription

1 University of Cambridge Part III of the Mathematical Tripos Applications of Differential Geometry to Physics Lent 2013, Lectured by Dr Maciej Dunajski Notes by: William I. Jay Diagrams by: Nobody Yet Last updated on July 2, 2013

2 Preface I have typeset my notes from the Part III lecture course Applications of Differential Geometry to Physics given in the 2013 Lent term by Dr. Maciej Dunajski. In several cases, I have expanded upon the material written on the blackboard from outside sources and my own take on things. However, my main original contribution to the content in these notes is certainly the (undesired!) introduction of errors. In particular, I anticipate that the sections on integrable systems in Chapter 3 and on connections on principle bundles in Chapter 5 are rife with errors. If and when you locate problems, please let me know at william.jay (at) colorado (dot) edu. If you are interested in helping me complete the notes by producing diagrams, please get in contact at the above address. 1

3 Contents 1 Manifolds and Vector Fields Manifolds Vector Fields Lie Groups Geometry on Lie Groups The Lie Algebra of a Lie Group Metrics on Lie Groups General Kaluza-Klein Projection Hamiltonian Mechanics and Symplectic Geometry Poisson Manifolds Some Differences Between Riemannian and Symplectic Geometry The Natural Symplectic Structure on Cotangent Bundles Canonical Transformations Moment Maps Geodesics, Killing Vectors, and Killing Tensors Topological Charges in Field Theory Scalar Kinks The Degree of a Map Applications of Topological Degree in Physics Sigma Model Lumps Gauge Theory Hodge Duality The Yang-Mills Equations Yang-Mills Instantons Chern Forms (in the context of gauge theory on R 4 ) What s actually going on with Chern forms? Fiber Bundles and Connections Yang-Mills Instantons Revisited

4 Introduction In a course entitled Applications of Differential Geometry to Physics, a natural question is where the union of physics with the ideas of differential geometry began and how this union has developed over time. Arguably, the beginning came with Newton s explanation of the Kepler problem, in which he discovered the connection between the shape of planetary orbits and conic sections. Conic sections were first described algebraically by Apolonius of Perga, who showed that five points on a plate determine a conic section. Physically, we can understand this statement to mean that one can determine a conic section (i.e., and orbit) using five measurements. Apolonius equation is given by: ax 2 + by 2 + cxy + dx + dy + f = 0, where a, b, c, d, e, and f are six real constants. Since one of the constants must be non-zero (else we have the trivial case of nothing happening), dividing by the nonzero constant yields an equation with the advertised five degrees of freedom. In other words, we ve gone from the 6-dimensional space R 6 by identifying points according to an equivalence class. This gives rise to the notion of a real projective space, i.e., RP 5 def = R 6 /, where is the equivalence relation. However, what made Newton s differential geometry (instead of the algebraic geometry of Apolonius) was the fact the conic sections in his theory arose as solutions to a differential equation: r = GmM r/r 3. We now give three other principle examples of differential geometry in physics: 1.) The Hamiltonian reformulation of classical mechanics has deep ties to (and was indeed provided the motivation) for symplectic geometry as originally studied by Arnold. In Hamiltonian mechanics, one has a Hamiltonian function H(q, p, t), and the equations of evolution for p and q are given by: q = H p, ṗ = H q In symplectic mechanics, one considers a symplectic form ω = dp dq on the phase space manifold with coordinates p and q and studies the form-preserving maps (symplectomorphisms) on the manifold. Vector fields of the form: also arise. X H = H p q H q 2.) General relativity as developed by Einstein (c.1917) brought into physics the ideas of preexisting (c.1850) ideas Riemannian geometry, where the central object is the pair (M, g), where M is a manifold and g is a Riemannian metric. p 3

5 Applications of Differential Geometry to Physics Section ) Gauge theories like Maxwell and or Yang-Mills theories are more recent physical theories that involve the geometrical objects of connections on principle or vector bundles. A principle bundle associates a Lie group to each point on a manifold. In the case of Maxwell theory, the Lie group associated to each point is U(1), while in Yang-Mills theory the Lie group is SU(2) or SU(3). This course will cover the three points above with the geometry of Lie groups as the unifying theme. The course will prove some theorems (Arnold-Liouville, Topological degree, etc...), but the focus will be on providing lots of examples from physics (e.g., Kaluza-Klein theory, instantons, etc...). Often we will use examples instead of proof, as the emphasis of the course is on developing concrete calculations skills that will be useful in research. Assumed prior knowledge is the Part III course on General Relativity or Differential Geometry or the equivalent. 4 W.I. Jay

6 Chapter 1 Manifolds and Vector Fields 1.1 Manifolds Definition 1. A smooth n-dimensional manifold M is a set such that there exist open sets U α, α = 1, 2,... for which U α cover M open There exist injective maps Φ α : U α V α R n such that ϕ αβ = ϕ β ϕ 1 α : ϕ α (U α U β ) ϕ β (U α U β ) are smooth maps from R n to R n. The point is the following: A manifold is a topological space with the added structure of these maps. This additional structure is what makes calculus possible. Example 1. The trivial manifold with one chart, M = R n. The question, then, becomes whether other, non-trivial possibilities also exist. In fact, Donaldson showed in 1984 that there exist infinitely many exotic smooth structures (when n=4????). Example 2. The sphere S n = { v R n+1 : v = 1}. This manifold is covered by two charts: U 1 = S n {0, 0,..., 0, 1} def = S n N North U 2 = S n {0, 0,..., 0, 1} def = S n S South The chart functions into R n are given by stereographic projections: ϕ 1 (r 1,..., r n+1 ) = (x 1,..., x n ), where x k = ϕ 2 (r 1,..., r n+1 ) = ( x 1,..., x n ), where x k = r k 1 r n+1 r k 1 + r n+1 On the intersection U 1 U 2, Now 1+r n+1 1 r n+1 = 1 r2 n+1 (1 r n+1 ) 2 x k = r k 1 + r n+1 = = r r2 n (1 r n+1 ) 2 r k 1 r n+1 1 r n r n+1 = x k 1 r n r n+1. = x x2 n. Thus we see that: ϕ 12 (x 1,..., x n ) = ( x 1,..., x n ), where x k = x k x x2 n 5

7 Applications of Differential Geometry to Physics Section 1.2 Example 3. Let f 1, f 2,..., f k : R N R. A surface M f = { v R n : f 1 ( v) = f 2 ( v) = = f k ( v) = 0} is a smooth manifold of dimension dim M = N k if the rank of the N K matrix with elements a f i is maximal, where a = 1,..., N and i = 1,..., k. We see for example that S n R n+1 is a special case of this construction with f 1 = 1 r 2. Theorem (Whitney) All n-dimensional manifolds are surfaces in R N N n. for suitably large One can also show that a bound for N is given by N 2n + 1. Example 4. The real projective space RP n, and n-dimensional manifold. RP n = R n+1 /, where [x 0,..., x n ] [cx 0,..., cx n ] for c R. We define charts according to and ( x 0 ϕ α [x] = U α = {[x], x α 0} x α,... xα 1 x α We note that RP n = S n /Z 2 (this is the antipodal map). ), xα+1 x α,..., xn x α. Definition 2. Let M, M be manifolds of dimension n and ñ, respectively. A map f : M M is said to be smooth if ϕ β f ϕ 1 α : R n Rñ are smooth for all α, β. For example, let M = R and suppose f : M R. If f is smooth is a smooth function on the manifold in the sense above, we see that f is a smooth function on the manifold. 1.2 Vector Fields Let γ : R M be a smooth curve with γ(0) = p M. Let U M, and let x a (ϵ) be the coordinate functions of γ. Definition 3. The tangent vector V to γ at p is defined by: dγ(ϵ) dϵ = V p T p M. ϵ=p Definition 4. The tangent space is the vector space of all tangents to curves in M passing through the point p. As a vector space of dimension n, T p M R n. Definition 5. The disjoint union of the tangent spaces at each point on a manifold is known as the tangent bundle: T M = T p M p M Note that the tangent bundle is itself a manifold of dimension 2n. One can think of an element of the tangent bundle as a pair (p, v), where p M is a point on the base manifold and v T p M is a vector in the tangent space at p. Furthermore, there is the natural projection from the tangent bundle onto the manifold given by π : T M M where (p, v) p. Definition 6. A vector field assigns a tangent vector each point p M (or if we re interested in a locally-defined vector field, to each point in the open subset U M) 6 W.I. Jay

8 Applications of Differential Geometry to Physics Section 1.2 Let f : M R be a function. The rate of change of f along the curve γ is: df(x a (ϵ)) = f ϵ=0 dϵ ϵ=0 x a ẋa def = V a (x) x a f. The V a (x), a = 1,..., n defined above are the components the components of the vector field V. In other words, we can write any vector field V in the form: V = V a (x) x a, and we therefore see that vector fields are first-order linear differential operators. The derivatives { x 1, x 2,..., x n } evaluated at p form a basis for T p M. Definition 7. An integral curve (or flow) γ = γ(ϵ) of a vector field V is defined by γ(ϵ) = V γ(ϵ) or equivalently in components by γ a (ϵ) = V a (x(ϵ)), a = 1, 2,..., n. These are the defining equations that one solves in practice. They are a system of first-order ODE s, which admit a unique solution (provided the V a (x(ϵ)) are Lipshitz continuous in x and continuous in ϵ). Vector fields are often called the generators of flows. This is because given x a (ϵ) and its value for ϵ = 0, one can perform a series expansion in ϵ to find x a (ϵ) = x a (0) + ϵv a + O(ϵ 2 ), where V a is a vector field. Example 5. Let M = R 2, x a = (x, y), V = x x + y. The differential equations defining integral curves are ẋ = x and ẏ = 1. Given initial data x(0) = x 0, y(0) = y 0, these equations clearly have the unique solutions (x(ϵ), y(ϵ)) = (x 0 exp ϵ, y 0 + ϵ). Now y = y 0 + ϵ = ϵ = y y 0. Substituting into the equation for x, we see x(y) = x 0 exp(y y 0 ). Rearranging, we see that x exp( y) = x 0 exp( y 0 ) is constant along the curve γ = (x, y). Evidently this quantity is an invariant of the vector field. Definition 8. An invariant of a vector field V is a function f that is constant along the flow, i.e., along the integral cuves of V : f(x a (0)) = f(x a (ϵ)), ϵ V (f) = 0. Example 6. From curves to vector fields. Let M R 2 Consider the 1-parameter group of rotations on R 2 : (x(t), y(t)) = (x 0 cos t y 0 sin t, x 0 sin t + y 0 cos t), i.e., let the usual rotation matrix R(t) operate on the pair (x(t), y(t)). ( y(t) V = t y + x(t) ) t x = x t=0 y y x. An invariant of this vector field is evidentially r = x 2 + y 2. Moreover, any function of r will also be invariant. After one defines the notion of vector fields, a natural question is if one may sensibly combine vector fields. As usual, there is no natural notion of multiplication of vector fields (vector spaces don t have a built-in multiplication structure). However, since the vector fields are linear differential operators, composition is a natural operation at our disposal. 7 W.I. Jay

9 Applications of Differential Geometry to Physics Section 1.2 Definition 9. The Lie bracket of two vector fields V, W is a vector field [V, W ] defined by its action of functions: [V, W ]f = V (W (f)) W (V (f)), f. The point is that the Lie bracket produces another vector field (i.e., another first-order differential operator), since the second-order derivatives cancel due to the subtraction. The Lie bracket also satisfies two properties as a result of the above definition: Antisymmetry: [V, W ] = [W, V ] The Jacobi identity: [V [W, U]] + [U, [V, W ]] = [W, [U, V ]] = 0, for all U, V, W. Example 7. Consider the vector fields V = x x + y, W = x. Computing the bracket, we see: [V, W ]f = x 2 xf + xy f x (x x f + y f) = x f, f = [V, W ] = x = W This closure is special and does not always happen. However, there is a rich theory for the case when it does. This leads us to the notion of Lie algebras. Definition 10. A Lie algebra is a vector space g equipped with an anti-symmetric, bilinear operation [, ] : g g g called a Lie bracket which satisfies the Jacobi identity. If g is finite dimensional and {V α }, α = 1,..., dim g form a basis for g, then g is determined by the γ γ structure constants f such that [V α, V β ] = f V γ (summation implied). αβ Example 8. Up to isomorphism, there are two unique 2-dimensional Lie algebras: [V, W ] = W, f 2 12 = 1, sometimes called the affine case [V, W ] = 0, the abelian case αβ Example 9. g = gl(n, R) = {the space of all n n real matrices}. dim GL(n, R) = n 2, and the lie bracket is given by the matrix commutator. Theorem (Ado s Theorem). Every finite dimensional Lie algebra L (over a commutative field K of characteristic zero) can be viewed as a Lie algebra of square matrices with bracket given by the matrix commutator. Proof. Omitted. Example 10. Consider the (infinite-dimensional) space of vector fields V on a manifold M. Once endowed with the Lie bracket of vector fields, this space becomes a Lie algebra g = Diff(M). 8 W.I. Jay

10 Chapter 2 Lie Groups Definition 11. A Lie group is a group G with the structure of a smooth manifold such that the group operation G G G, (g 1, g 2 ) g 1 g 2 and inverse G G, g g 1 are smooth maps between manifolds. Example 11. G = GL(n, R) = {invertible n n matrices}, with dim GL(n, R) = n 2. Example 12. G = O(n) = {O GL(n) : O T O = 1}, with dim O(N) = n 2 n(n+1)/2 = n(n 1)/2, since the orthonormality of the n columns of the matrices imposes n(n + 1)/2 algebraic constraints. Definition 12. Let G be a (Lie) group and M be a manifold. A group action on a manifold is a map G M M, (g, p) g(p) such that: i. e(p) = p, where e is the identity in G and for all p M, and ii. g 2 (g 1 (p)) = (g 2 g 1 )(p), p M, g 1, g 2 G. Example 13. Let M = R 2 and let G = E(2) the 3-dimensional Euclidean group. We let g G act on (x, y) T R 2 : ( ) ( ) ( ) ( ) ( ) x cos θ sin θ x a x cos θ y sin θ + a g = + = y sin θ cos θ y b x sin θ + y cos θ + b Let G ϵ denote a one-parameter subgroup of G with parameter ϵ (set the other parameters equal to zero). Then for the three parameters in our current group we have the actions: { { { x x = x cos θ y sin θ x x = x + a x x = x G θ : G a : G b : y ỹ = x sin θ + y cos θ y ỹ = y y ỹ = y + b Associated to each of these one parameter subgroups G ϵ G is a vector field V ϵ (p) Γ(T M) (the smooth sections of the tangent bundle T M). We define the vector field V ϵ (p) according to the formula: V ϵ (p) = g ϵ(x µ ) ϵ ϵ=0 x µ, where g ϵ (p) denotes the group element g ϵ G ϵ acting on the point p M. Using this formula, we see that we have a vector field for each of the one-parameter subgroups of G = E(2): ( x V θ = θ x + ỹ ) θ ỹ = x θ=0 y y x ( x V a = a x + ỹ ) a ỹ = a=0 x ( x V b = b x + ỹ ) b ỹ = b=0 y 9

11 Applications of Differential Geometry to Physics Section 2.1 Jumping ahead slightly with our language, we will say that these vector fields form a basis for the (3-dimensional) Lie algebra of E(2). They satisfy the commutation relations: 2.1 Geometry on Lie Groups [V a, V θ ] = V b, [V b, V θ ] = V a, [V a, V b ] = 0. Definition 13. Let f : M M be a smooth map between manifolds. Let γ : I M be a curve in M and suppose that V T p M is vector field tangent to γ. The tangent map f : T p M T f(p) M is defined by: f (V ) = d dϵ f(γ(ϵ)) ϵ=0. In components with i, j, α = 1,..., dim M we have: (f V ) α = V i yα x i, where x i are local coordinates on M and y α are local coordinates on M. Remark. The tangent map is the same things as the pushforward and extends easily to the tangent bundle T M. Definition 14. Let V, W be vector fields on M such that V = γ (i.e., γ is the integral curve of V ). The Lie derivative is defined by: W (p) (γ(ϵ) (W ))(p) L V W = lim = [V, W ]. ϵ 0 ϵ The second equality is not obvious from the definition. However, we proved it in General Relativity last term, and it is Exercise 3 on Example Sheet 1. We define the action of the Lie derivative on functions according to L V (f) = V (f). As we saw previously, the tangent space T p M is a vector space. Associated to this vector space is the dual vector space Tp M, which consists of the linear functionals on T p M. Tp M is known as the contangent space. If a basis for T p M is given by { x 1, x 2,..., x n }, then the dual basis of covectors for Tp M is given by the differentials {dx 1, dx 2,..., dx n }, which are defined according to their action on the basis vectors of T p M: dx i ( x j ) = x j xi = δj. i Just as the union of the tangent spaces at each point on the manifold gave rise to the tangent bundle, the union of the dual spaces gives dual bundle, known as the cotangent bundle: Tp M = T M. p M One can take direct products of vector and covector fields to construct tensor fields on manifolds. A particularly useful type of product is the so-called wedge product, which uses direct products of covectors to construct antisymmetric covariant tensor fields on the manifold. These antisymmetric tensor fields are important for two reasons. First, they can be integrated in a natural (i.e., coordinate-independent) fashion 1. Secondly, they are critical in the field of de Rahm cohomology. 1 For an heuristic but enlightening explanation why antisymmetric covariant tensors (i.e., k-forms) are the natural objects for integration see Chapter 12 of J.M. Lee s Introduction to Smooth Manifolds (Springer). 10 W.I. Jay

12 Applications of Differential Geometry to Physics Section 2.2 Definition 15. Let dx i, dx j be covector fields. The wedge product is the antisymmetrized product defined by: dx i dx j = dx j dx i. Definition 16. A 1-form is a covector field: Ω = Ω i (x)dx i. A k-form is an antisymmetric tensor field of the form: Ω = 1 r! Ω ij...kdx i dx j dx r. Definition 17. The differential d of a 1-form is defined by dω = Ω i x j dx j dx i. Some notation: We will denote the contraction of the vector x i x i dxj dx j ( x i ) = xj x i = δ j i. If V is a vector and Ω is a covector, this notation lets us write: V Ω = V i x i Ω jdx j = V i Ω j = a function. This notation can also be used to write Cartan s formula: with the covector dx j with L V Ω = d(v Ω) + V dω. (2.1) This formula follows because of the Leibniz rule satisfied by the Lie derivative. We proved it in Part III General Relativity last term, though without using the hook notation. 2.2 The Lie Algebra of a Lie Group Definition 18. The Lie algebra g of a Lie group G is the tangent space T e G to G at the identity element e G. The bracket in g is the bracket of vector fields in G. The second portion of this definition requires some further explanation in order to be clear. Definition 19. We define the function known as left translation by L g : G G with L g (h) = gh for any g, h G. (Here we are imagining g to be arbitrary but fixed. If we think about letting g vary, we would probably talk about the group action of left translation.) We remark that left translation is actually a diffeomorphism of G. For details, see Lee 2. This function also induces a pushforward map of between the tangent spaces on G (i.e., from g = T e G to T g G). given by: (L g ) : g = T e G T g G with v g the vector(l g ) v T g G. Note that since this holds for all g G, we can think of the vector v in the Lie algebra as generating the whole vector field (L g ) v on G. In other words, given a vector v T e G, I assign a vector (L g ) v T g G to each point g G. 2 J.M. Lee, Introduction to Smooth Manifolds, p.93. The idea is that the map is smooth and has the obvious smooth inverse L g W.I. Jay

13 Applications of Differential Geometry to Physics Section 2.2 With these definitions, we have [(L g ) v, (L g ) w] = (L g ) [v, w] g, where the bracket on the left hand side is the bracket of vector fields on the manifold (think Lie derivative) and the bracket on the right hand side (with the subscript g) is the bracket associated with the Lie algebra. Although the lecture omitted the proof of this statement, the argument is that left translation is a diffeomorphism, so it doesn t matter if we change coordinates before or after computing the Lie bracket. Let {V i } be a basis for g. By considering the vector fields associated with the basis vectors V i, one finds n = dim G global, non-vanishing vector fields on G. Using the terminology of mathematicians, this statement says that Lie groups are parallelizable manifolds. In physics, we say that there is a global frame. A natural question is whether or not the converse is true, namely, if all parallelizable manifolds are also Lie groups. This is an involved question, but the answer is negative. A counterexample is the parallelizable manifold S 7, which is not a Lie group. Definition 20. Let X be a vector field on G. (L g ) X g = X gg for all g, g G. X is said to be a left-invariant vector field if NOTE: The following notation using L α for vector fields is a bit confusing. Tread carefully! Lemma Let Ω be a 1-form and let V, W be vector fields. Then: Proof. dω(v, W ) = V (Ω(W )) W (Ω(V )) Ω([V, W ]). dω(v, W ) = (dω) ab V a W b = ( a Ω b )V a W b ( b Ω a )V a W b = ( V Ω b )W b ( W Ω a )V a = V (Ω b W b ) ( V W b )Ω b W (Ω a V a ) + ( W V a )Ω a = V (W Ω) W (V Ω) ( V Ω a W V a )Ω a = V (W Ω) W (V Ω) [V, W ] Ω = V (Ω(W )) W (Ω(V )) Ω([V, W ]). Proposition 1 (Maurer-Cartan Relations). Let {L α, α = 1, 2,..., dim g} be a basis of left-invariant γ vector fields such that [L α, L β ] = fαβ L γ. Let {σ α } be the dual basis of left-invariant one forms, i.e., L α σ β = δα β. Then. Proof. We use the previous lemma to see that: d(σ α ) f βγ σβ σ γ = 0 (d(σ α ))(L β, L γ ) = L β (σ α (L γ )) L γ (σ α (L β )) σ α ([L β, L γ ]) α = σ α ([L β, L γ ]), since σ α (L γ ) and σ α (L β ) = const. λ = σ α (fβγ L λ) λ = fβγ σα (L λ ) α = f βγ 12 W.I. Jay

14 Applications of Differential Geometry to Physics Section 2.2 So we see that the βγ component of d(σ α ) is given by f α βγ d(σ α ) = 1 2 f α βγ σβ σ γ d(σ α ) f α βγ σβ σ γ = 0.. Antisymmetrizing, we see that: Remark. The Maurer-Cartan relations are basically just a statement of the component form of dσ α in the basis of left-invariant forms. From now on we will assume that G is a matrix Lie group. Definition 21. Let g G. The left-invariant Maurer-Cartan 1-form on G is the Lie algebra valued 1-form defined on vectors in T g G defined by: ρ : T g G T e G = g ρ(v) = (L g 1) v for any v T g G ρ def = g 1 dg. In other words, we saw that left translation induced the push-forward map (L g ) : T e G T g G between tangent spaces on G, which allowed us to identify each element of the Lie algebra g with a vector field on G. The Maurer-Cartan 1-form goes in the opposite direction, i.e., given a (leftinvariant) vector field v on G, ρ(v) gives you the element of the Lie algebra g that generated the vector field. The last line in the definition is perhaps a bit mysterious at first glance, since we haven t yet defined what we mean by the differential dg of a Lie group element. If we think of the (matrix) Lie group as some sort of symmetry group (like, say, SO(2)), then the group elements will depend continuously on the parameters of the symmetry (in the case of SO(2) the angle θ). When we write dg, we re really thinking of taking the differential of the group element with respect to these parameters. For example, ( ) ( ) cos θ sin θ sin θ cos θ g = SO(2) = dg = dθ. sin θ cos θ cos θ sin θ Let g 0 = const. Then (g 0 g) 1 d(g 0 g) = g 1 g 1 0 g 0dg = g 1 dg. Let g(s) be a smooth curve in G. Expand g 1 (s)g(s + ϵ) in a power series about ϵ = 0: g(s) 1 1 dg g(s + ϵ) = e + ϵg + O(ϵ 2 ). ds ϵ=0 By definition, the second term is in the tangent space at the identity, i.e., in the Lie algebra: 1 dg g T e G = g. ds ϵ=0 Thus we can express the Maurer-Cartan form in a basis expansion as ρ = g 1 dg = σ α T α, where T α γ are matrices forming a basis for the Lie algebra with the usual commutators [T α, T β ] = fαβ T γ and where σ α form a basis for left-invariant one-forms. Taking the exterior derivative of both sides of the expansion, we compute: dρ = d(g 1 ) dg + g 1 d 2 g = ( g 1 dgg 1 ) dg, since dg 1 = g 1 (dg)g 1 = ρ ρ. 13 W.I. Jay

15 Applications of Differential Geometry to Physics Section 2.2 Thus we have found the Maurer-Cartan equation: dρ + ρ ρ = 0. Let us check our result. Now dρ = d(σ α ) T α. On the other hand, ρ ρ = σ α σ β T α T β = 1 2 σα σ β [T α, T β ] = 1 2 σα σ β f γ αβ T γ Relabelling indices, we see that the Maurer-Cartan equation reconfirms Proposition 1 above. Example 14. The Heisenberg group is the set of upper triangular matrices of the form: 1 x z g = 0 1 y = 1 + xt 1 + yt 2 + zt The matrices T 1, T 2, T 3 span the Lie algebra of the Heisenberg group and satisfy the commutators: [T 1, T 2 ] = T 3 [T 1, T 3 ] = 0 [T 2, T 3 ] = 0. Thinking physically, we identify T 1 with the momentum operator, T 2 with the position operator, and T 3 with i. The Heisenberg Lie algebra is the central extension of the two-dimensional abelian Lie algebra. Further, for the g given above we have: 1 x z + xy 0 dx dz g 1 = 0 1 y and dg = 0 0 dy Therefore: 1 x z + xy 0 dx dz 0 dx dz xdy ρ = g 1 dg = 0 1 y 0 0 dy = 0 0 dy Thus for the Maurer-Cartan form we compute: ρ = g 1 dg = T 1 dx + T 2 dy + T 3 (dz xdy). By inspection, one sees that σ 1 = dx, σ 2 = dy, σ 3 = dz xdy. Recalling that d 2 = 0, we immediately observe dσ 1 = 0, dσ 2 = 0, dσ 3 = dx dy = σ 1 σ 2. The three σ i above are the left-invariant one-forms. The associated left-invariant vector fields are evidently L 1 = x, L 2 = y +x z, L 3 = z, since they satisfy the defining equation L α σ β = σ β (L α ) = δα β. Furthermore, we see that the L i satisfy the Lie algebra of the Heisenberg group: [L 1, L 2 ] = L 3 [L 1, L 3 ] = 0 [L 2, L 3 ] = 0. We can also define right-invariant vector fields from the right translation map R g (h) = hg 1. The general discussion is largely the same as for left-invariant vector fields. We remark that the bracket of two right-invariant vector fields carries an additional minus sign: [R α, R β ] = fαβ R γ. γ 14 W.I. Jay

16 Applications of Differential Geometry to Physics Section 2.3 We also record that fact that the bracket of the a left-invariant vector field with a right-invariant one vanishes: [L α, R β ] = 0. One defines the right-invariant Maurer-Cartan one form: ρ = dg g 1 = σ α T α, where σ α are the right-invariant 1-forms and the T α give (as before) a basis for g. How does the Lie derivative of invariant 1-forms behave? Using (2.1), one sees that L Lα σ β = L α dσ β 0 in general. Furthermore, we also have L Rα σ β = 0, ( α, β). (2.2) Proof. Proof or argument??? Example 15. Consider again the Heisenberg group. To find the right-invariant vector-fields and 1-forms, we begin by constructing the (right) Maurer-Cartan form ρ = dg g 1 : 0 dx dz 1 x z + xy 0 dx dz ydx ρ = 0 0 dy 0 1 y = 0 0 dy We read off that the right-invariant 1-forms are σ 1 = dx, σ 2 = dy, σ 3 = dz ydx. Recalling the defining equation R α σ β = δ β α, we see that the right-invariant vector fields are: 2.3 Metrics on Lie Groups R 1 = x + y z, R 2 = y, R 3 = z. The left-invariant metric on G is h = g αβ σ α σ β, where σ α is a left-invariant 1-form, g αβ = g βα is a constant, symmetric, non-degenerate matrix. Note that the notation σ α σ β is simply the symmetrized tensor product: A B def = 1 2 (A B + B A). Since g αβ is already symmetric, this notation is actually redundant. As we see from (2.2), the Lie derivative of the left-invariant 1-forms with respect to right-invariant vector fields vanishes. This implies that G acts isometrically on G. In other words, there exist n = dim G independent Killing vector fields R α, i.e., L Rα (h) = 0. Example 16. Let G be the Heisenberg group. Then h = δ αβ σ α σ β = dx 2 + dy 2 + (dz xdy) 2. We can read off the isometries: i. y y + ϵ, generated by the vector field y = R 2, ii. z z + ϵ, generated by the vector field z = R 3, and 15 W.I. Jay

17 Applications of Differential Geometry to Physics Section 2.4 iii. {x x + ϵ and z z + ϵy}, generated by the vector field R 1 = x + y z. Example 17. The Kaluza Klein Interpretation: motion on the space of orbits of Consider the Lagrangian given by z in G. L = 1 2 (ẋ2 + ẏ 2 + (ż xẏ) 2 ). (Note that this is just a specific case of the general form of the Lagrangian L = 1 2 g µνẋ µ ẋ ν that we learned in General Relativity that can be used to calculate quickly the Christoffel symbols of the Levi-Civita connection.) We the write the geodesic equations as Euler-Lagrange equations: L x µ = d L ds ẋ µ. Here x µ = (x, y, z). Now L ż = const = ż xẏ = c = constant along the geodesic. (We hang the words charge conservation around this fact.) The other equations of motion follow quickly and are given by: ẍ = ẏ(ż xẏ) = cẏ (2.3) ÿ = (ż xẏ)ẋ = cẋ (2.4) We observe now that d ds (ẋ2 + ẏ 2 ) = 2ẋẍ + 2ẏÿ = 2ẋ( cẏ) + 2ẏ(cẋ) = 0. Thus we see that ẋ 2 + ẏ 2 = const = A 2. This feature tells us that ẋ = A cos(s s 0 ) and ẏ = A sin(s s 0 ). In other words (x(s), y(s)) on R 2 is a family of circles. Reconnecting with physics, we ask ourselves what sort of objects move in circles. One answer is of course a charged particle in a constant magnetic field. Thus we compare our result above with the geodesic motion in a magnetic fields. Let (M, g = g ij dx i dx j ) be a Riemannian manifold and with the closed 2-form F = 1 2 F ijdx i dx j (the magnetic field ). Here closed means that df = 0. The geodesic equation is: ẍ i + Γ i jkẋj ẋ k = cf i j ẋ j, where the Γ i jk are the Christoffel symbols of the Levi-Civita connection. We now compare our result to (2.3) and (2.4). Take M = R 2, g ij = δ ij (This is a flat metric with Γ i jk = 0). F = dx dy, F ij = ϵ ij. In other words, the constant magnetic field is the preferred constant 2-form on the manifold: the volume form. Thus geodesics of the left-invariant metric h on the Heisenberg group project to trajectories of a charged particle moving in a constant magnetic field. INSERT PICTURE OF THIS PROJECTION!!!! 2.4 General Kaluza-Klein Projection Let h = g ij dx i dx j + (dz + A) 2 be the metric on the Lie group G, where g ij dx i dx j is the metric on a submanifold M and A = A i (x)dx i is a 1-form on M such that da = F. The Lagrangian is now given by: L h = L g + (ż + A i ẋ i ) W.I. Jay

18 Applications of Differential Geometry to Physics Section 2.4 Charge conservation is given by the statement ż + A i ẋ i = constant along the geodesic. Assume now that z S 1 is periodic. The time-independent Schroedinger equation reads: 2 ϕ = Eϕ, where 2 = L L2 2 + L2 3 is the Laplacian of the metric on the Lie group. In the case of the Heisenberg group, we have: ϕ xx + ϕ zz + ( y + x z ) 2 ϕ = Eϕ. Making the separation ansatz ϕ(x, y, z) = Ψ(x, y) exp(iez), where e is a constant (the electric charge...), leads to: ( x iea x ) 2 Ψ + ( y iea y ) 2 Ψ = (E e 2 )Ψ. Compare this with the well-known Landau Problem of quantum mechanics. We remark that if 0 z 2πL, then charge quantization arises: el Z. insert discussion of general Laplace Operators here!!! Example 18. The Killing Metric. As usual, ρ = g 1 dg = T α σ α. We define the following positive definite, bi-invariant metric: h = Tr(g 1 dg g 1 dg) = Tr(T α T β )σ α σ β For example, on SU(2), this metric gives the round metric on the 3-sphere. Let G be a group of 2 2 matrices. Then one can also define a metric according to h = det g 1 dg Suppose further that det g = 1, as in SL(2, R) or SU(2). Then det(dg) = det(d(g 0 gg 1 )), where g 0, g 1 are constant elements of the group, and we see that the metric is bi-invariant as well. Example 19. AdS (Anti-de Sitter space in 2+1 dimensions). Let g SL(2, R), so that g is a 2 2 matrix with det g = 1. Write g in the form: ( x g = 3 + x 2 x 1 + x 4 ) x 1 x 4 x 3 x 2. The define the metric as introduced above: h = det(dg) det g=1. Note that this is the same as the flat metric of signature (2, 2) on R 4 : ds 2 = (dx 3 ) 2 + (dx 4 ) 2 (dx 1 ) 2 (dx 2 ) 2, restricted by the constraint (x 3 ) 2 +(x 4 ) 2 (x 1 ) 2 (x 2 ) 2 = 1. We parametrize this equation according to: x 1 = y 1 /y 0 x 3 = y 2 /y 0 x 2 + x 4 = 1/y 0 x 2 x 4 = ((y 1 ) 2 (y 2 ) 2 + (y 0 ) 2 )/y 0. When one substitutes these variables into the constraint equation, one sees that the expression is identically equal to one. Using Mathematica to speed along (though hand calculations are still reasonably quick), we see that the resulting metric takes the form: h = 1 (y 0 ) 2 ( (dy 0 ) 2 + (dy 1 ) 2 (dy 2 ) 2). These coordinates are known as Poincare coordinates, and they cover half of AdS 3. We note that AdS 3 = SO(2, 2)/SO(2, 1). 17 W.I. Jay

19 Chapter 3 Hamiltonian Mechanics and Symplectic Geometry We begin with some physical motivation from classical mechanics. Let M be a 2n-dimensional phase space (i.e., a manifold). Given real-valued functions f, g on M, we can define the familiar Poisson Bracket: {f, g} = f g f g, q α α p α p α q α where (p α, q α ), α = 1,..., n are local coordinates (momentum and position) on the phase space. There exists a privileged function H : M R known as the Hamiltonian such that ṗ α = H q α q α = H p α. Solutions to these first-order ordinary differential equations are integral curves of the Hamiltonian vector field: X H = H H. p α α q α q α p α We see that X H (q α ) = H p α q α on an integral curve. We would now like to consider a more general framework. 3.1 Poisson Manifolds In the words of V.I. Arnold 1, A Poisson structure on a manifold is a Lie algebra structure on its space of smooth functions (i.e., a bilinear skew-symmetric operation of Poisson bracket on functions, satisfying the Jacobi identity) such that the operator ad a = {a, } (contraction of the Poisson bracket with any fixed function a) is an operator of differentiation by some vector field θ a. The vector field θ a is then called the hamiltonian vector field with hamiltonian function a. The mapping a θ a gives a homomorphism from the Lie algebra of functions to the Lie algebra of vector fields. A manifold with a given Poisson structure is called a Poisson manifold. We now return to the lecture. 1 Arnold, V.I., Mathematical Methods of Classical Mechanics (MMCM), 2nd Ed., Springer, p

20 Applications of Differential Geometry to Physics Section 3.1 Definition 22. Let M be a manifold and let f, g be smooth functions on M. A Poisson structure is the a map {, } : C (M) C (M) R defined in coordinates using ω ij = ω [ij] according to: {f, g} = such that the Jacobi identity holds as well: n i,j=1 ω ij (x) f x i g x j, {{f, g}, h} + {{h, f}, g} + {{g, h}, f} = 0, ( f, g, h C (M)). The Jacobi identity will constrain the possible explicit form of ω ij = {x i, x j }, where x i, x j are coordinates in a chart on M. Note that in this more general setting we have abandoned the distinction of momenta p i versus positions q i that we had in the Hamiltonian formulation of classical mechanics above. Example 20. Let M = R 3. Take ω ij = ϵ ijk x k. Then the Poisson brackets of the coordinates are given by: {x 1, x 2 } = x 3, {x 3, x 1 } = x 2, {x 2, x 3 } = x 1. We remark that any function of the form f(r), where r = (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2, will have vanishing Poisson brackets with the coordinate functions x i : {f(r), x i } = 0, ( i). As an explicit example, {r 2, x 1 } = {(x 2 ) 2, x 1 } + {(x 3 ) 2, x 1 } = 2x 2 {x 2, x 1 } + 2x 3 {x 3, x 1 } = 2x 2 x 3 + 2x 3 x 2 = 0. We can use this framework to apply the familiar Hamiltonian dynamics of classical mechanics to systems not typically amenable to such methods. Define a function: H(x 1, x 2, x 3 ) = 1 ( (x 1 ) 2 + (x2 ) 2 + (x3 ) 2 ) 2 a 1 a 2 We compute ẋ i according to ẋ i = {x i, H}. Doing this for the example at hand, we find: ẋ 1 = 1 {x 1, (x2 ) 2 + (x3 ) 2 } 2 a 2 a 3 ( ) a3 a 2 = = x 2 x 3 a 2 a 3 ( ) ẋ 2 a1 a 3 = x 1 x 3 a 1 a 3 ( ) ẋ 3 a2 a 1 = x 1 x 2 a 1 a 2 We see that these equations possess the same form as those of Euler s equations for the motion of a rigid body. We identify the coordinates x 1, x 2, x 3 with the components of the angular momentum vector of a rigid body relative to some point O and the coefficients a i with the principle moments of inertia (cf. Arnold 2 ). a 3 Note that in the previous we took the Poisson structure ω ij to have the form of the totally antisymmetric tensor ϵ ijk. We identify this as the privileged top form on the manifold, i.e., the volume form. As an alternative, one can also interpret the ϵ ijk as the structure constants of so(3). With this interpretation, the fact that the evolution equations take on the form of the Euler equations 2 Arnold in MMCM, p W.I. Jay

21 Applications of Differential Geometry to Physics Section 3.1 is less mysterious, as the Euler equations deal with three-dimensional rotations of a rigid body. Moreover, we note that this result is part of a much broader topic involving Lie groups. (We won t pursue the topic further in this course.) We call the function f(r) above with vanishing Poisson brackets a Casimir. On the topic of Casimirs Arnold 3 says: A Poisson structure, like a symplectic one, defines a homomorphism of the Lie algebra of vector fields on the manifold: the derivative of a function g along the field of the function f is equal to {f, g}. Such fields are called Hamiltonian; their flows preserve the Poisson structure. Hamiltonians to which zero fields correspond are called Casimir functions 4, and they form the centre of the Lie algebra of functions. In contrast to the nondegenerate symplectic case, the centre need not consist of only locally constant functions. We see that our example above was an explicit construction of just such a non-constant function in the centre of the Lie algebra of functions. Example 21. Building on the previous example, we consider level sets of the Casimir f(r), i.e., we look at r = const. This choice restricts the Poisson structure to a 2-sphere S 2 R 3. On the sphere we take coordinates: x 1 = sin θ cos ϕ, x 2 = sin θ sin ϕ, x 3 = cos θ. We d like to compute {θ, ϕ}. To do so, we need to recall the inverse formulas for θ(x, y, z) and ϕ(x, y, z). They are of course given by: Then {θ, ϕ} = ωij θ x i we find: ϕ x j θ(x, y, z) = arctan(z/ x 2 + y 2 ) ϕ(x, y, z) = arctan(y/x). = ϵijk θ ϕ x k. Using Mathematica to compute the mindless derivations, x i x j θ ϕ x 1 x 2 θ ϕ x 2 x 1 = z x 2 + y 2 (x 2 + y 2 + z 2 ) θ ϕ x 2 x 3 θ ϕ x 3 x 2 = x x 2 + y 2 (x 2 + y 2 + z 2 ) θ ϕ x 3 x 1 θ ϕ x 1 x 3 = y x 2 + y 2 (x 2 + y 2 + z 2 ) 1 = {θ, ϕ} = x 2 + y = 1 2 sin θ, where we ve evaluated on the unit sphere r = 1. This result gives a Poisson structure on S 2. We remark that the answer 1/ sin θ was given in the lecture (The opposite sign...i think mine is wrong!). Our Poisson structure is non-degenerate. This means that ω ab = 1 ( ) 0 1 sin θ Arnold, V.I., Dynamical Systems IV: Symplectic Geometry and its Applications, p.32 4 In other worlds, a Hamiltonian that is a Casimir corresponds to an everywhere-vanishing vector field 20 W.I. Jay

22 Applications of Differential Geometry to Physics Section 3.1 has an inverse, where the inverse is given by 1 2 (ω 1 ) ab dx a dx b = sin θdθ dϕ. This is an example of a symplectic structure induced by the Poisson structure. Definition 23. A symplectic manifold is a smooth manifold M of even dimension dim M = 2n with a closed non-degenerate 2-form ω Λ 2 (M). In other words, } ω ω {{ ω} 0. n times For example, on R 4 with coordinates x 1, x 2, x 3, x 4, we can take ω = dx 1 dx 2 + dx 3 dx 4. There are several important consequences of this structure. There is an isomorphism between T M and T M given by: v T M v ω T M. This is analogous to the canonical isomorphism of raising and lowering indices that appears in the presence of a metric on Riemannian or Lorentzian manifolds. If f : M R, then df is a 1-form on M, which is paired with the Hamiltonian vector field X f according to: X f ω = df. Note the presence of the minus sign on the right side of the equation above. Although this sign is just a convention, it is the critical sign that determines all the rest of the related sign conventions below. Letting f and g be functions, we define the Poisson structure: {f, g} X g f = ω(x g, X f ) = ω(x f, X g ) = X f (g) = {g, f}. In a coordinate basis, this Poisson structure takes the same form as we had above. Namely, {f, g} = 2n i,j=1 ij f g ω x i x j. We also remark that the Jacobi identity that we had previously imposed upon {, } is equivalent to the closure condition dω = 0 in the definition of a symplectic manifold. There is a Lie algebra homomorphism from the Lie algebra of smooth functions on M into the Lie algebra of vector fields on M: {, } [, ]. In particular, one finds that [X f, X g ] = X {f,g}. 21 W.I. Jay

23 Applications of Differential Geometry to Physics Section 3.2 Proof. Let f, g, h be functions. Then we have: [X f, X g ](h) = X f (X g (h)) X g (X f (h)) = X f ({h, g}) X g ({h, f}) = {{h, g}, f} {{h, f}, g} = {h, {f, g}}, using the Jacobi identity = X {f,g} h Since h was arbitrary, the result follows. Hamiltonian vector fields preserve the symplectic form in the sense that the Lie derivative vanishes along such fields: L Xf ω = X f dω + d(x f ω) = 0 + d( df) = d 2 f = 0, since d 2 = 0. The following theorem tells us, roughly speaking, that symplectic geometry is not very interesting locally. Theorem (Darboux s Theorem). Let (M, ω) be a 2n-dimensional symplectic manifold. There exist local coordinates x 1 = q 1,..., x n = q n, x n+1 = p 1,..., x 2n = p n around any point on M such that: n ω = dp i dq i i=1 and the Poisson bracket takes the standard form. Arnold remarks, This theorem allows us to extend to all symplectic manifolds any assertion of a local character which is invariant with respect to canonical transformations and is proven for the standard phase space (R 2n, ω 2 = d p d q). 5 Proof (Sketch) Induction on n = dim M/2. Begin by choosing a function p 1 : M R and find or construct another function q 1 : M R such that X p1 (q 1 ) = 0 (equivalently, q 1 along integral curves). This can always be done because of the existence of solutions for ODE s. Next, let M 1 = {x M : p 1 = const, q 1 = const}. This submanifold is locally symplectic with the symplectic form ω 1 = ω p1,q 1 =const. Now look for p 2 and q 2, and so on. Recall the equation for ω on S 2 : ω = sin dθ dϕ = d(cos θ) dϕ. Thus in the proof above we would take p 1 = ϕ, q 1 = cos θ. 5 Arnold, MMCM, p W.I. Jay

24 Applications of Differential Geometry to Physics Section Some Differences Between Riemannian and Symplectic Geometry The previous theorem shows that symplectic geometry has a decidedly different flavour than that of familiar Riemannian geometry. We recall that on a Riemannian (or Lorentzian) manifold one can always choose canonical coordinates at a particular point, but not in a full neighbourhood. If one adds the structure of a Levi-Civita connection to the Riemannian (or Lorentzian) manifold, one can always choose so-called normal coordinates, which - roughly speaking - bend slowly away from canonical coordinates in a sufficiently small neighbourhood (this feature makes precise the notion of a local inertial frame in relativity). Darboux s theorem tells us that we always (and without additional mathematical structure) can pick canonical coordinates on a symplectic manifold. It is also fruitful to think about the geometric source of the differences between symplectic and Riemannian geometry. The metric of Riemannian geometry allows one to define the idea of lengths on a manifold: ds 2 = g(x, X). Because of the antisymmetry built into a symplectic form, we see that ω(x, X) 0, so lengths are clearly not the fundamental object of symplectic geometry. Instead, oriented areas take center stage. If X and Y are two infinitesimal vectors, then the oriented area (with respect to the symplectic form) of the parallelogram spanned by X and Y is ω(x, Y ). By antisymmetry, swapping the order of X and Y reverses the orientation. By non-degeneracy, the area vanishes if and only if X = Y ; a single vector spans a parallelogram of vanishing area. Just as every metric gives rise to a different machine for measuring lengths, each symplectic form gives a different machine for measuring areas. The symplectic manifold (M, ω) also has additional geometric information, since a symplectic form ω is (by definition) closed: dω = 0. As we found on the example sheet, this constraint amounts to the fact that the symplectic form satisfies the familiar Jacobi identity. Geometrically, this statement tells us that the oriented surface area of any compact three-dimensional region vanishes. On a related note, Darboux s theorem above tells us that we can always pick canonical coordinates, which means that symplectic manifolds are always flat. A general manifold of course may be curved, and in this case the oriented surface area of a compact three-dimensional region might not vanish. Thus we see that a symplectic form and its accompanying flatness impose a rather rigid constraint on a manifold. Finally, we remark that the rigidity of a symplectic form gives rise to lots of symmetries, i.e., form-preserving maps. As we will show below, there exist infinitely many so-called symplectomorphisms f for which f (ω) = ω. As one recalls, a general Riemannian or Lorentzian manifold (M, g) will not necessarily have any isometries for which f (g) = g. 3.3 The Natural Symplectic Structure on Cotangent Bundles Let Q be an n-dimensional manifold. Then there exists a globally defined symplectic structure on the total space of T Q, the cotangent bundle. Let (q 1,..., q n ) be local coordinates on Q and (p 1,..., p n ) local coordinates on the cotangent space T q Q R n. As mentioned previously, we have a natural projection from the cotangent bundle onto the base manifold: π : T Q Q π(p, q) = q. 23 W.I. Jay

25 Applications of Differential Geometry to Physics Section 3.4 We also remind ourselves of some details concerning the pullback (cf. General Relativity or Differential Geometry notes). Definition 24. Let M, N be manifolds and let f : M N be a smooth function between the manifolds. Then the pullback, denoted f is a map between the cotangent spaces f : T f(p) N T p M. Let (x 1,..., x m ) = x a and (y 1,..., y n ) = y i be local coordinate charts on M and N, respectively. Using the function f, we can express the coordinates on N as functions of the coordinates on M, i.e., y i = f i (x a ). In these coordinates the pullback takes the following form: f (dy i ) = f i x a dxa Recall that we defined the pullback in a coordinate-free manner in General Relativity. We did this by looking at how sections of the cotangent bundle (i.e., covector fields) acted on vector fields. Namely, we said f (η)(x) def = η(f (X)), where η T N, X T M and f denotes the pushforward of X. Returning now to the discussion at hand, we consider the pullback of the natural projection: π : T (Q) T (T Q). Comparing the previous line with definition of the pullback, we see that π is f, Q is N, and T Q is M. Suppose p T (Q) is a 1-form on Q with components p 1,..., p n. We then make the following definitions: In local coordinates these objects take the form θ := π (p), ω := dθ. θ = i p i dq i, ω = i dp i dq i, where we see explicitly that ω is a closed 2-form on T Q. 3.4 Canonical Transformations Definition 25. Let (M, ω) be a symplectic manifold of dimension 2n. The one-parameter groups of (ω-preserving) transformations generated by Hamiltonian vector fields are known as canonical transformations. Note that if n = 1 the manifold is the usual two-dimensional phase space and the canonical transformations correspond to area preserving maps. We now describe the general local picture. Suppose f is a canonical transformation. Let (p i, q i ) f (P i, Q i ) be local charts, so that we may write P = P (p, q) and Q = Q(p, q) (Note that Q is a coordinate on N, i.e., not the manifold Q mentioned above.) Then we have: d( p d q) = d( P d Q) = d( Q d P ) = d( p dq + Q d P ) = W.I. Jay

26 Applications of Differential Geometry to Physics Section 3.6 In the last line we have a closed one-form. As we recall, a closed form is always at least locally exact (global properties depend on the topology of your space, cf. the Poincare Lemma). Thus we write: p i dq i + Q i dp i = ds = S q i dqi + S P i dp i, where S = S(q, P ) is a generating function. Given any S, we can simply read off a canonical transformation from the above equation, namely: 3.5 Moment Maps p i = S q i, Q i = S P i. Let (M, ω) be a symplectic manifold. Let G be a Lie group with action on M that preserves ω (G acts by symplectomorphisms ). Let g be the Lie algebra of G with dual g and, : g g R the standard pairing between g and its dual. As we ve seen previously, any ζ g induces a vector field ρ(ζ) on M. Consider the contraction ρ(ζ) ω = ω(ρ(ζ), ), which is a closed 1-form for all ζ, since G acts by symplectomorphism. Definition 26. The moment map is a function µ : M g defined by d µ, ζ = ρ(ζ) ω, ( ζ g). In the above equation µ, ζ is the function from M to R defined by µ, ζ (x) = µ(x), ζ. This construction can be slightly confusing. The general idea is the following. Let the Lie group G act on the symplectic manifold (M, ω). Let ζ g generate the vector field X ζ on M. Then X ζ ω = dh ζ, where H ζ is a Hamiltonian function associated with the generator ζ g. If we strip away ζ, we see that we then have a function µ : M g such that (µ(x))(ζ) = H ζ. The fact that X ζ ω is a closed one 1-form is essentially the assumption that G acts by symplectomorphism. The Wikipedia page gives a nice, physically-motivated example of the moment map. Let (M, ω) = T R 3 be the cotangent bundle of three-dimensional Euclidean space. Let G be the Euclidean group of rotations and translations on R 3, i.e., the semidirect product of SO(3) and R 3. Then the six components of the moment map are the three angular momenta and the three linear momenta. Note that both the definition and the previous example make it clear that the moment map will have a total of dim G = dim g components, one for each of the generators in the Lie algebra. 3.6 Geodesics, Killing Vectors, and Killing Tensors Let (M, g) be a Riemannian manifold with dim M = n. We may write the metric as g = g ij (x)dx i dx j. As we have seen previously, geodesics on the manifold are determined by the geodesic equation; ẍ i + Γ i jkẋj ẋ k = 0, (3.1) 25 W.I. Jay