Notes for Gauge Fields, Knots and Gravity

Justin Kulp

Notes taken from the book Gauge Fields, Knots and Gravity by John Baez andJavier P. Muniain. 1994 Edition. Notes updated 2018.

Contents

1 Electromagnetism and Differential Geometry 11.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Manifolds and Topolgy . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Covariant versus Contravariant . . . . . . . . . . . . . . . . . 41.3.3 Flows and the Lie Bracket . . . . . . . . . . . . . . . . . . . . 6

1.4 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Cotangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Change of Coordinates . . . . . . . . . . . . . . . . . . . . . . 131.4.4 p-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.5 The Exterior Derivative . . . . . . . . . . . . . . . . . . . . . 181.4.6 Recovering Vector Calculus . . . . . . . . . . . . . . . . . . . 19

1.5 Rewriting Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . 201.5.1 The Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.2 The Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.3 The Hodge Star Operator . . . . . . . . . . . . . . . . . . . . 291.5.4 The Second Pair of Equations . . . . . . . . . . . . . . . . . . 32

1.6 deRham Theory in Electromagnetism . . . . . . . . . . . . . . . . . . 361.6.1 Closed and Exact 1-Forms . . . . . . . . . . . . . . . . . . . . 361.6.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 421.6.3 deRham Cohomology . . . . . . . . . . . . . . . . . . . . . . . 451.6.4 Gauge Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 Gauge Fields 482.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.1.1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.2 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2 Bundles and Connections . . . . . . . . . . . . . . . . . . . . . . . . . 552.2.1 Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1 Electromagnetism and Differential Geometry

1.1 Maxwell’s Equations

Classically, the electric field ~E and magnetic field ~B are (possibly time-dependent)vector functions on “spacetime” which depend on the electric charge density ρ andcurrent density ~j, satisfying Maxwell’s equations.

Result 1.1 (Maxwell’s Equations).

1. ∇ · ~B = 0 3. ∇ · ~E = ρ (1)

2. ∇× ~E +∂ ~B

∂t= 0 4. ∇× ~B − ∂ ~E

∂t= ~j

Definition 1.2. The Lorentz transformations of spacetime are the transformationswhich leave (x, x) = (x0)2 − (x1)2 − (x2)2 − (x3)2 invariant.

1.2 Manifolds and Topolgy

Definition 1.3. A topological space is a pair (X, τ) of a set X and a set τ ∈ P(X).The elements of τ are called open sets, and are required to satisfy:

1. ∅, X ∈ τ2. If U, V ∈ τ then U ∩ V ∈ τ3. If Uα ∈ τ then

⋃α∈I Uα ∈ τ for any index set I.

An open set containing x ∈ X is a neighbourhood of x. The complement of anopen set is called closed.

Definition 1.4. f : X → Y is continuous if, given any open set U ⊆ Y , thenf−1(U) ⊆ X is open.

Note 1.5. We assume throughout that our topologies are free of pathologies. Namely,they are

1. Hausdorff : For any x, y ∈ X there exists neighbourhoods Ux and Uy, of x andy respectively, such that Ux ∩ Uy = ∅.

2. Paracompact : Every open cover has a locally finite refinement. That is, eachpoint has a neighbourhood that intersects only finitely many points in thecover.

Definition 1.6. Given a topological space X, and U ⊆ X open, a chart is a contin-uous function ϕ : U → Rn with continuous inverse, ϕ−1 : ϕ(U)→ X.

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Definition 1.7. An n-dimensional manifold is a topological space, M , with chartsϕα : Uα → Rn, where Uα are the open sets covering M , and such that the transitionfunctions, ϕα ϕ−1β defined on the overlaps, are continuous.

If the transition functions are smooth, then M is a smooth manifold. We sayf : M → R is smooth if for any α, f ϕ−1α : Rn → R is smooth.

1.3 Vector Fields

A vector field on a manifold is, intuitively, a collection of arrows on the surface. Thearrows identify directions for directional derivatives, and so we define vector fieldsalgebraically as operators.

Definition 1.8. Using xi for coordinates on Rn, the directional derivative in thedirection v = (v1, . . . , vn) of a function f : Rn → R is

vf = vµ∂µf (2)

The vector v = (v1, . . . , vn) is thus promoted to the directional derivative operator,identifying fields with operators

v = vµ∂µ (3)

Definition 1.9. Let C∞(M) denote the (commutative) algebra of smooth functionson a manifold M . A vector field on M is a map v : C∞(M)→ C∞(M) satisfying:

1. v(f + g) = v(f) + v(g)2. v(αf) = αv(f)3. v(fg) = v(f)g + fv(g)

for all f, g ∈ C∞(M). The set of all vector fields is Vect(M).

Proposition 1.10. If we define addition and multiplication on Vect(M) by1. (v + w)(f) = v(f) + w(f)2. (gv)(f) = gv(f)

for all v, w ∈ Vect(M) and f, g ∈ C∞(M), then Vect(M) is a module over C∞(M):i. f(v + w) = fv + fw

ii. (f + g)v = fv + gwiii. (fg)v = f(gv)iv. idMv = v

Proof. We show just one example: that (v + w) is a vector field. (v + w) clearlysatisfies both of the linearity properties of a vector field, so we will just show it

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satisfies the Leibniz rule:

(v + w)(fg) = v(fg) + w(fg)

= v(f)g + fv(g) + w(f)g + fw(g)

= (v + w)(f)g + f(v + w)(g)

Result 1.11. The vector fields ∂µ on Rn span Vect(Rn) as a module over C∞(Rn)and are linearly independent. Thus every v ∈ Vect(Rn) can be written

v = vµ∂µ

for some vµ ∈ C∞(Rn).

Proposition 1.12. vµ∂µ = 0 iff v ≡ 0.

Proof. Suppose v ≡ 0, then vµ∂µf = 0µ∂µf = 0 for all f ∈ C∞(Rn). Conversely, ifvµ∂µ = 0, then consider f = xi. Then 0 = vµ∂µx

i = vi for each i, so v ≡ 0.

Corollary 1.13. Every vector field v on Rn has a unique representation as a linearcombination vµ∂µ. Thus ∂µ is a basis of Vect(Rn).

1.3.1 Tangent Vectors

Given v on M and f ∈ C∞(M), we can evaluate v(f) at any p ∈M , v(f)(p), to getthe result of evaluating the v-directional derivative of f at the point p.

Definition 1.14. A tangent vector at p ∈ M is a function vp : C∞(M) → Rsatisfying:

1. vp(f + g) = vp(f) + vp(g)2. vp(αf) = αvp(f)3. vp(fg) = vp(f)g(p) + f(p)vp(g)

the set of all tangent vectors at p is the tangent space to p, TpM .

Proposition 1.15. Any vector field v defines a tangent vector vp at any p ∈M by:

vp(f) = v(f)(p) (4)

Furthermore, a vector field is completely determined by its tangent vectors on M .i.e. v = w iff vp = wp for all p ∈M .

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Proof. The fact that any v defines a tangent vector vp for any p ∈M is clear, as thedefinition of a tangent vector is simply the definition properties of a vector field withevaluation at p.

Now we show v = w iff they agree at each point, i.e. vp = wp for all p ∈M .⇒ if v = w then v(f) = w(f) for any f ∈ C∞(M); and since these are functions,

this just means v(f)(p) = w(f)(p) for any p ∈ M . Since f was arbitrary andvp(f) = wp(f) on the whole domain of vp and wp, so that vp = wp.

⇐ Suppose vp = wp for any p ∈ M . Then, as functions C∞(M) → R, vp and wpagree for all inputs f . So vp(f) = wp(f) and v(f)(p) = w(f)(p). This meansthe functions v(f) and w(f) agree for all p, so v(f) = w(f). Then since theseagree for all f , v = w.

Result 1.16. The converse of the proposition is true: Every tangent vector at p ∈Mis of the form vp for some vector field v ∈ Vect(M).

Note 1.17. Note that TpM forms a vector space over R.

Definition 1.18. A curve is a function γ : R → M that is smooth. i.e. for anyf ∈ C∞(M), then f(γ(t)) depends smoothly on t. We define the tangent vector tothe curve (at some fixed t), to be the map γ′(t) : C∞(M)→ R such that

[γ′(t)] (f) =d

dtf(γ(t)) (5)

γ′(t) sends functions in the direct that the curve γ is moving at time t. The factthat γ′(t) ∈ Tγ(t)M can be seen by noting that d/dt will ensure γ′(t) is linear ANDsatisfies the Leibniz rule at γ(t).

1.3.2 Covariant versus Contravariant

Definition 1.19. Say φ : M → N and f : N → R. We can get a function on M bycomposition f φ. We define the pullback of f from N to M by φ by

φ∗f = f φ (6)

Thus if φ : M → N , then the pullback is a map φ∗ : C∞(N)→ C∞(M).

Because functions f : N → R “go backwards” under φ : M → N , we callfunctions contravariant.

We can generalize the definition of a smoothness from functions and curves toarbitrary maps between manifolds. In particular:

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Definition 1.20. A function φ : M → N is smooth if φ∗f is an element of C∞(M)for any f ∈ C∞(N).

We recover the case for smooth functions when N = R, and smooth curves whenM = R.

Definition 1.21. A tangent vector v ∈ TpM and smooth function φ : M → Ngives a tangent vector on N , called the pushforward of the vector vp by φ. Thepushforward φ∗ is the map φ∗ : TpM → TpN given by

(φ∗vp)(f) = vφ(p)(φ∗f) (7)

Example 1.22. Let M and N be manifolds, φ : M → N , and γ(t) a curve on Mwith tangent γ′(t) ∈ TpM . Then φ γ is a curve in N with tangent vector:

(φ γ)′(t) = φ∗(γ′(t)) ∈ Tφ(p)N (8)

That is, the tangent vector to the curve in N (moved from M by φ), is the pushfor-ward of the tangent vector of the original curve in M .

To see this, take any function f ∈ C∞(N) and evaluate:

(φ γ)′(t)(f) =d

dt[f((φ γ)(t))]

=d

dt[f(φ(γ(t)))]

=d

dt[(f φ)(γ(t))]

=d

dt[φ∗f(γ(t))]

= [γ′(t)] (φ∗f)

= [φ∗(γ′(t))] (f)

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Thus far we have only defined the pushforward on individual vectors vp ∈ TpM .We can extend the definition in a well-defined way to whole vector fields.

Definition 1.23. Let M and N be manifolds, φ : M → N a diffeomorphism.Further, let v be a vector field on M . There is a well-defined way to define thepushforward of a vector field, φ∗v : C∞(N)→ C∞(N). Namely by

(φ∗v)q = φ∗(vp) when q = φ(p) (9)

Proof. q 7→ (φ∗v)q is well-defined if x = y implies (φ∗v)x = (φ∗v)y. Well, x = ymeans φ−1(x) = φ−1(y) since φ is one-to-one (it is a diffeomorphism). Thus we have(φ∗vφ−1(x)) = (φ∗vφ−1(y)) so that (φ∗v)φ(φ−1(x)) = (φ∗v)φ(φ−1(y)), or rather, (φ∗v)x =(φ∗v)y.

Now we confirm the pushforward is really a smooth vector field on N . Since wehave defined (φ∗v)φ(p) = (φ∗vp) we can see pretty easily that (φ∗v) is linear on thefunctions f ∈ C∞(N). So we will just confirm the Leibniz rule:

(φ∗v)φ(p)(fg) ≡ (φ∗vp)(fg)

= vp(φ∗(fg))

= vp((fg) φ)

= v(fg)(φ(p))

= vφ(p)(f)g(φ(p)) + f(φ(p))vφ(p)(g)

= vp(φ∗f)g(φ(p)) + f(φ(p))vp(φ

∗g)

= (φ∗v)φ(p)(f)g(φ(p)) + f(φ(p))(φ∗v)φ(p)(g)

1.3.3 Flows and the Lie Bracket

Definition 1.24. Let v be a vector field on M , and suppose γ : R → M is a curveon M satisfying:

1. γ′(t) = vγ(t) for all t ∈ R2. γ(0) = p

Then γ is the integral curve through p defined by v. A vector field v is calledintegrable if all integral curves are defined for all t.

Definition 1.25. Let φt(p) be the integral curve of v through p ∈M . For any t ∈ Rdefine φt : M →M to be the smooth map (smooth because of theorem on differentialequations) at t generated by this integral curve. The family φt on M is the flowgenerated by v.

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Proposition 1.26. Let φt be a flow on M generated by some vector field.1. φ0 is the identity map2. φt φs = φt+s

Proof. For the first part, φ0(p) is the integral curve through p at t = 0, defined tobe φ0(p) = p. For the second part, computing, we get

(φt φs)(p) = φt(φs(p))

= φt(γp(s))

= γγp(s)(t)

= γp(t+ s)

= φt+s(p)

where γp(s) denotes the integral curve through p at s.

Definition 1.27. Given two vector fields, v, w ∈ Vect(M), they define a new vectorfield [v, w] ∈ Vect(M) called the Lie Bracket by

[v, w](f) = v(w(f))− w(v(f)) (10)

Proposition 1.28. Let v, w, u ∈ Vect(M), then the Lie bracket [v, w] satisfies:1. [v, w] ∈ Vect(M)2. [w, v] = −[w, v]3. [u, αv + βw] = α[u, v] + β[u,w]4. [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0

Proof. The proofs are all straight forward algebra, using the linearity of the vectorfields. The only property that doesn’t just use linearity is in proving that Vect(M)is closed under the Lie bracket, and that just needs the Lebiniz rule:

[v, w](fg) = v(w(fg))− w(v(fg))

= v(w(f)g + fw(g))− w(v(f)g + fv(g))

= v(w(f)g) + v(fw(g))− w(v(f)g)− w(fv(g))

= v(w(f))g + w(f)v(g) + v(f)w(g) + fv(w(g))

− w(v(f))g − v(f)w(g)− w(f)v(g)− fw(v(g))

= (v(w(f))− w(v(f)))g + f(v(w(g))− w(v(g)))

= [v, w](f)g + f [v, w](g)

So the Lie bracket [· , ·] is effectively a skew-symmetric trilinear map satisfyingthe Jacobi identity. It measures the failiure of two flows to commute.

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1.4 Differential Forms

1.4.1 1-Forms

Definition 1.29. A 1-form on a manifold M is a map ω : Vect(M) → C∞(M)satisfying

1. ω(u+ v) = ω(u) + ω(v)2. ω(gv) = gω(v)

for all u, v ∈ Vect(M) and g ∈ C∞(M). The space of all 1-forms on M is Ω1(M).

Proposition 1.30. If we define addition and multiplication on Ω1(M) by1. (ω + µ)(v) = ω(v) + µ(v)2. (gω)(v) = gω(v)

then Ω1(M) is a module over C∞(M).

Definition 1.31. For and f ∈ C∞(M) we define the differential of f or exteriorderivative to be the map df : Vect(M)→ C∞(M) by

df(v) = v(f) (11)

Proposition 1.32. df is a 1-form.

Proof. We prove it by verifying the properties of a 1-form. Let f ∈ C∞(M) arbitrary,then

1. df(v + w) = (v + w)(f) = v(f) + w(f) = df(v) + dv(w)2. df(gv) = (gv)(f) = gv(f) = gdf(v)

Definition 1.33. The differential operator is the operator d : C∞(M) → Ω1(M)defined by

d(f) = df (12)

Proposition 1.34. For any f, g, h ∈ C∞(M) and α ∈ R, d satisfies1. d(f + g) = df + dg2. d(αf) = αdf3. (f + g)dh = f dh+ g dh4. d(fg) = df g + f dg

Proof. 1 through 3 are easy examples of linearity. For property 4 note:

d(fg)(v) = v(fg) = v(f)g + fv(g) = d(f)(v)g + fd(g)(v)

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Example 1.35. Differentials work exactly how we think they do. For example:

d(sinx) = cos x dx (13)

Proof. Let v ∈ Vect(R) arbitrary, then we know v = vx∂x for some component vx.Then the LHS is

d(sinx)(v) = v(sinx) = vx∂x sinx = vx cosx

and the RHS is

(cosxdx)(v) = (cos x)v(x) = (cos x)vx∂x(x) = vx cosx

Example 1.36. If f(x1, . . . , xn) is a function on Rn, then

df = ∂µfdxµ (14)

Theorem 1.37. Let xµ the usual coordinate functions on Rn and ∂µ the asso-ciated basis for Vect(Rn). The exterior derivatives dxµ form a basis for Ω1(Rn).

Proof. Any vector field v ∈ Vect(Rn) can be written v = vµ∂µ for some vµ. Nownote that dxµ(∂ν) = ∂ν(x

µ) = δµν . Now suppose ω is any 1-form on Rn, and define

ωµ = ω(∂µ)

Then to show that our set spans Ω1(Rn) we just have to show that ω is in fact a linearcombination of dxµ. In particular, we claim that it is exactly the linear combination:

ω = ωµdxµ (15)

To see this, just take any arbitrary v = vν∂ν and note that

ω(v) = ω(vν∂ν) = vνω(∂ν) = vνων

ωµdxµ(v) = ωµdx

µ(vν∂ν) = vνωµδµν = vνων

Now we show that it is a basis, i.e., that the set is linearly independent. Well,suppose ω ≡ 0. Well for any ∂ν we have 0 = ω(∂ν) = ωµdx

µ(∂ν) = ωµδµν = ων . So

ω ≡ 0 iff ων for each component ν.

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1.4.2 Cotangent Vectors

Definition 1.38. Given a manifold M and p ∈M , a cotangent vector, ωp, at p is alinear map ωp : TpM → R. The space of all cotangent vectors (at p) is T ∗pM .

Example 1.39. If ω is a 1-form on M , one way to define a cotangent vector at anyp ∈M is by defining, for any v ∈ Vect(M):

ωp(vp) = ω(v)(p) (16)

Proposition 1.40. Defining ωp : TpM → R by ωp(vp) = ω(v)(p) is well-defined.

Proof. ωp(vp) is well-defined if it depends only on the value of its input vector vp ∈TpM . That is, suppose v and u are two fields on M which agree at p, so vp = up.Then ωp(vp) = ωp(up) iff ωp(vp − up) = 0, which is true because vp = up and ωp islinear.

Proposition 1.41. A 1-form is completely determined by its associated cotangentvectors. That is, ω = µ iff ωp = µp for all p ∈M .

Proof. Let v be an arbitrary vector field. Recall that v is determined by its vp. Thuswe have as with the vector fields

ω = µ ⇐⇒ ω(v) = µ(v) for all v ∈ Vect(M)

⇐⇒ ω(v)(p) = µ(v)(p) for all v ∈ Vect(M) and p ∈M⇐⇒ ωp(vp) = µp(vp) for all v ∈ Vect(M) and p ∈M⇐⇒ ωp = µp for all p ∈M

In the same way that a tangent vector at a point can be associated with a curvethrough a point, the cotangent vector can be associated with a function at that point.

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In particular, around any point p there are level surfaces defined by a function f , andthe cotangent vectors are like little infinitessimal level surfaces there. For vp ∈ TpMwe have that df(vp) = v(f)(p) counts how many hyperplanes vp crosses in the dfstack.

Definition 1.42. Given any vector space V , the dual, V ∗, is the space of all linearfunctionals T : V → R. For any map between vector spaces, f : V → W , we havethe induced dual map f ∗ : W ∗ → V ∗, given by

(f ∗w)(v) = w(f(v)) (17)

for all w ∈ W ∗ and v ∈ V . So f ∗ : W ∗ → V ∗ is contravariant.

Proposition 1.43. Let V , W , and X be a vector spaces. Let id : V → V be theidentity map on V , and f : V → W and g : W → X maps on the vector spaces, then

1. (id)∗ : V ∗ → V ∗ is the identity on V ∗.2. (gf)∗ = f ∗g∗

Proof. Begin by noting that id : V → V is defined so that id(v) = v for all v ∈ V , sothen for any w ∈ V ∗ we have ((id∗)w)(v) = w(id(v)) = w(v). So (id∗)w = w and id∗

is the identity on V ∗. For the second part, note that gf : V → X so (gf)∗ : X∗ → V ∗.Now let x ∈ X∗ and v ∈ V arbitrary, then we see that

((gf)∗x)(v) = x((gf)(v))

= x(g[f(v)])

= (g∗x)[f(v)]

= [f ∗(g∗x)](v)

= [(f ∗g∗)(x)](v)

We have a mechanism for pushing forward individual vectors on a manifold, andthis in turn gives us a well defined pushforward of vector fields. We can naturallypullback our cotangent vectors globally as follows, and this will give us a way topullback entire 1-forms.

Definition 1.44. Let φ : M → N , φ(p) = q, and recall that the pushforward is alinear map φ∗ : TpM → TqN . The dual to the pushforward of tangent vectors is thepullback of cotangent vectors, φ∗ : T ∗qN → T ∗pM . It is given by

(φ∗ωq)(vp) = ωq(φ∗vp) (18)

for ω ∈ T ∗qN , v ∈ TpM , p ∈M all arbitrary, and q = φ(p).

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That is, we have a natural/forced way to define the pullback of cotangent vectors,from how we defined the pushforward of tangent vectors.

Theorem 1.45. There exists a unique pullback operation for 1-forms, φ∗ : Ω1(N)→Ω1(M) by

(φ∗ω)p = φ∗(ωq) (19)

Proof. For existence, note that a 1-form φ∗ω (whatever it may be) is defined by itspoints. Thus we can take any p ∈ M and v ∈ Vect(M) arbitrary, and recalling thatv is also defined by its points, we have (with the assistance of our definition of thepullback of a cotangent vector and pushforward of tangent vectors) that

LHS : (φ∗ω)p(vp) ≡ [(φ∗ω)(v)](p) = [ω(φ∗v)](q) = ωq(φ∗v)q

RHS : [φ ∗ (ωq)](vp) = ωq(φ∗vp) = ωq(φ∗v)q

For uniqueness, suppose there were two possible outputs for the pullback of acotangent vector, µ1 and µ2. Then let µ = µ1− µ2, and evaluate an arbitrary vectorv ∈ Vect(M) at any p ∈ M . Then µp(vp) = 0 by linearity, so µ = 0 and thusµ1 = µ2.

Theorem 1.46. The pullback of the differential is the differential of the pullback.

φ∗(df) = d(φ∗f) (20)

Proof. These are both 1-forms, so it suffices to show that they agree for any p ∈M , that is, that they produce the same cotangent vector at p for arbitrary p. Inequations, φ∗(df) = d(φ∗f) iff [φ∗(df)]p = [d(φ∗f)]p. Take v ∈ Vect(M) arbitrary,since it is determined uniquely by its tangent vectors, then

[φ∗(df)]p(vp) = [φ∗(df)q](vp)

= (df)q(φ∗vp)

= [(φ∗vp)f ](q)

= [vp(φ∗f)](p)

= (d(φ∗f))p(vp)

Example 1.47. Let φ : R → R by φ(t) = sin t. Let dx the usual 1-form on R(target). Then (φ∗dx) = d(φ∗x) = d(x sin t) = d(sin t) = cos t dt.

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1.4.3 Change of Coordinates

The charts of a manifold are diffeomorphisms, ϕ : U ⊆ M → Rn, which turn localdata into calculations on Rn.

Definition 1.48. Let xµ be coordinates on Rn with associated coordinate vectorfields ∂µ, then ϕ pullsback the coordinates of Rn to the manifold M by

xµ ≡ ϕ∗xµ (21)

these are the local coordinates on U ⊆ M . Any function f : U ⊆ M → Rn can bewritten as f(x1, x2, . . . , xn).

Definition 1.49. Since ϕ is a diffeomorphism, it has a continuous inverse, we definethe coordinate vector fields on U ⊆ M , associated to the local coordinates xµ onU , by

∂µ ≡ (ϕ−1)∗∂µ (22)

Corollary 1.50. Any vector field v on U ⊆M can be written as

v = vµ∂µ (23)

Definition 1.51. The coordinate 1-forms on U ⊆ M , associated to the local coor-dinates xµ on U , are

dxµ ≡ ϕ∗dxµ (24)

Proposition 1.52. From our theorem, we see that the differentials of the local coo-ordinates are the coordinate 1-forms. That is

d(xµ|U) = (dxµ)|U (25)

Thus we have that any 1-form on U can be written as

ω = ωµdxµ (26)

Definition 1.53. A passive coordinate transformation is a change of local coordi-nates on the chart. We do not ‘move’ points of space. An active coordinate trans-formation is a diffeomorphism φ : M →M .

Theorem 1.54 (Passive Coordinate Transforms).Let xµ and x′µ be two sets of coordinates on Rn with associated bases ∂µ

13

and ∂′µ the associated bases for Vect(Rn). Similarly, let dxµ and dx′µ be theassociated basis for 1-forms. Then if we let

T νµ =∂x′ν

∂xµ(27)

For the vector fields the basis elements and components transform as:

∂µ = T νµ ∂′ν and v′ν = T νµ v

µ (28)

For the 1-forms the basis elements and components transform as:

dx′ν = T νµ dxµ and ωµ = T νµ ω

′ν (29)

Proof. Let v ∈ Vect(Rn) arbitrary, then we may write, uniquely, that v = vµ∂µ =v′ν∂′ν , in the respective bases. Then we can relate the bases by some transform, callit T νµ , satisfying

∂µ = T νµ∂′ν

Acting on x′λ we have

∂µ(x′λ) = (T νµ∂′ν)(x

′λ) = T νµ δλν = T λµ

or rather

T νµ =∂x′ν

∂xµ

proving the first part about basis elements. Now for some field v we have

v = v′ν ∂′ν = vµ ∂µ

v′ν ∂′ν = vµ T νµ∂′ν

and since ∂µ forms a basis, we have that

v′ν = T νµ vµ

Now suppose that ω ∈ Ω1(Rn) arbitrary, then we may write, uniquely, thatω = ωµdx

µ = ω′νdx′ν , in the respective bases. Then we can relate the bases by some

transform, call it Sνµ, satisfying

dx′ν = Sνµdxµ

14

Acting on ∂λ we have

(dx′ν)(∂λ) = Sλµ(dxµ)(∂λ)

∂λ(x′ν) = Sλµ∂λ(x

µ)

∂x′ν

∂xλ= Sνµδ

µλ

so we see that Sνµ = T νµ from before, and we have our first result. The second resultis as before.

Note 1.55. Let φ : Rm → Rn (m = n, but we will use this to track our data). Writex1, . . . , xm for the Rm coordinates, and x′1, . . . , x′n for the Rn coordinates. Since wecan’t take derivatives of primes (non-primes) with respect to non-primes (primes) weuse the shorthand:

T λµ =∂x′ν

∂xµ≡ ∂

∂xµ(φ∗x′ν) (30)

Theorem 1.56 (Active Coordinate Transforms).If φ : Rm → Rn as above, and xµ and x′ν as above, and using our new definition forT νµ , then:

1. We can pushforward the coordinate vector fields ∂µ on Rm

φ∗(∂µ) = T νµ ∂′ν (31)

2. We can pullback the coordinate 1-forms dx′ν on Rn

φ∗(dx′ν) = T νµ dxµ (32)

Proof. 1. Consider any x′λ of Rn, then

LHS : (φ∗∂µ)(x′λ) = ∂µ(φ∗x′λ) = T λµ

RHS : T νµ (∂′ν)(x′λ) = T λµ

2. Consider any ∂λ, then

LHS : φ∗(dx′ν)(∂λ) = d(φ∗x′ν)(∂λ) = ∂λ(φ∗x′ν) = T νλ

RHS : T νµdxµ(∂λ) = T νµ δ

νλ = T λµ

15

We can generalize our coordinate transformations to non-orthonormal bases ofvector fields and 1-forms. But first we use the resulting theorem

Result 1.57. Let ∂ν be the coordinate fields associate to local coordinates xν onU ⊆ Rn. Define a set of vector fields

eµ = T νµ∂ν (33)

where T νµ is some matrix-valued function on U . Then eµ are a basis of fields on Uiff for each p ∈ U , T νµ (p) is invertible.

Theorem 1.58. If eµ is a basis for Vect(Rn) it induces a unique dual basis of1-forms, fµ, on U such that

fµ(eν) = δµν (34)

Proof. Let fµ be the set of 1-forms defined by the above equation. Then δµν =fµ(eν) = fµ(T λµ ∂λ) for some transformation T . Now we can write the 1-form fµ asfµ = Sµδ dx

δ for some S. Thus δµν = Sµδ Tλν (dxδ∂λ) = Sµδ T

λν δ

δλ = SµλT

λν so ST = I. T is

invertible by our result, so S is unique by uniqueness of inverse.

Theorem 1.59 (Generalized Passive Coordinate Transforms).Let eµ be a basis for Vect(U), where U ⊆ Rn, and fµ the associated dual basisof 1-forms. If we define e′µ by

e′µ = T νµ eν (35)

then we have thatf ′µ = (T−1)µν f

ν (36)

In terms of the components we have that if v = vµeµ = v′νe′ν is an arbitrary vector

field and ω = ωµfµ = ω′νf

′ν is an arbitrary 1-form, then the components transformas

v′µ = (T−1)µν eν (37)

andω′µ = T νµ ων (38)

The proof is identical, so we skip it. This shows that:

Vectors 7→ Covariant

Components of Vectors 7→ Contravariant

1-Forms 7→ Contravariant

Components of 1-Forms 7→ Covariant

16

Example 1.60. If an object has value 1 for length when we measure it in the basiswhere ex = 1m, then in the basis e′x = 1cm, we divide the reference scale by 100,but our length multiplies by 100. So the length vector (in terms of its components)is contravariant since it changes opposite to the shift of basis.

1.4.4 p-Forms

Definition 1.61. Let V be a vector space. The exterior algebra over V is∧V , the

algebra generated by v with the relation

v ∧ w = −w ∧ v (39)

Define∧p V to be the subspace of

∧V of linear combinations of p-fold products

v1 ∧ · · · ∧ vp, where∧1 V ≡ V and

∧0 V ≡ R.

Proposition 1.62. Let V be a vector space and dim(V ) = n. Then1.∧V =

⊕p

∧p V2. dim(

∧p V ) = C(n, p)3. dim(

∧V ) = 2n

Proof. The first thing to note is that for p > n that∧p V is vanishing, this comes

from the antisymmetry. In which case, part one comes basically as the definitionof the algebra generated by the relation. The second part comes from the fact thatthere are n choose p ways to choose the bases for

∧p V . Then we use the fact thatthe sum of C(n, p) from p = 0 to n is 2n.

Definition 1.63. We define Ω(M) to be the algebra of differential forms on M ,generated by Ω1(M) over C∞(M) with ω∧µ = −µ∧ω, with only locally finite linearcombinations allowed. We also say Ω0(M) ≡ C∞(M) and define

f ∧ ω = fω (40)

for any f ∈ C∞(M) and ω ∈ Ω(M). An element of Ωp(M) is a p-form.

Definition 1.64. Given a vector space V ,∧V is supercommutative if ω ∈

∧p V andµ ∈

∧q V , thenω ∧ µ = (−1)pqµ ∧ ω (41)

Note 1.65. Ω(M) is clearly supercommutative.

Theorem 1.66. If φ : M → N is a map, then there is a unique pullback mapφ∗ : Ω(N)→ Ω(M) satisfying:

17

1. φ∗(αω) = αφ∗ω for α ∈ R2. φ∗(ω + µ) = φ∗ω + φ∗µ3. φ∗(ω ∧ µ) = φ∗ω ∧ φ∗µ

which agrees with the usual 0 and 1-forms.

Proof. The 3 criteria are definitions, and clearly they agree with the definition of apullback on functions and on 1-forms because of properties 1 and 2. So really we justneed to prove uniqueness, and that should probably involve property 3. To see it isunique, note that property 3 shows that we can break down the pullback of a p-forminto p 1-forms, each of which are unique and well-defined, and so is the wedge of allthose p 1-forms, so it could be nothing else.

1.4.5 The Exterior Derivative

Definition 1.67. The differential, or exterior derivative, is the unique set of mapsd : Ωp(M)→ Ωp+1(M) satisfying:

1. d : Ω0(M)→ Ω1(M) by df(v) = v(f)2. d(ω + µ) = dω + dµ3. d(cω) = c dω for c ∈ R4. d(ω ∧ µ) = dω ∧ µ+ (−1)pω ∧ dµ for ω ∈ Ωp(M) and µ ∈ Ω(M)5. d(dω) = 0 for any ω ∈ Ω(M)

Proof. First we show this map d is well-defined, really that it behaves nicely on theequivalent wedge made by reversing the objects in the wedge:

d(−µ ∧ ω) = −d(µ ∧ ω) = −dµ ∧ ω + µ ∧ dω = −ω ∧ dµ+ dω ∧ µ = d(ω ∧ µ)

For uniqueness, note any 1-form is a locally finite linear combination of those of theform df . Then take any differential form fdg ∧ dh and note that

d(fdg ∧ dh) = df ∧ (dg ∧ dh) + f ∧ d(dg ∧ dh) = df ∧ dg ∧ dh

Proposition 1.68. Let I be the multi-index standing for the p-tuple (i1, . . . , ip) ofdistinct integers between 1 and n. So that

dxI = dxi1 ∧ · · · ∧ dxip (42)

Consider the p-formω = ωIdx

I (43)

then the (p+ 1)-form dω is

dω = (∂µωI)dxµ ∧ dxI (44)

18

Proof.

dω = d(ωIdxI) = dωI ∧ dxI + (−1)pωI ∧ d(dxI) = dωI ∧ dxI = (∂µωI)dx

µ ∧ dxI

Theorem 1.69. The differential of the pullback is the pullback of the differential.That is, for any ω ∈ Ωp(M) we have

φ∗(dω) = d(φ∗ω) (45)

Proof. φ∗ is real-linear, so we can consider just the case of ω = f0df1∧· · ·∧dfp. Thenwe have

φ∗(dω) = φ∗(df0 ∧ df1 ∧ · · · ∧ dfp)= φ∗df0 ∧ · · · ∧ φ∗dfp= d(φ∗f0) ∧ · · · ∧ d(φ∗fp)

= d(φ∗f0 ∧ d(φ∗f1) ∧ · · · ∧ d(φ∗fp))

= d(φ∗f0 ∧ φ∗d(f1) ∧ · · · ∧ φ∗d(fp))

= d(φ∗(f0 ∧ df1 ∧ · · · ∧ dfp))= d(φ∗w)

1.4.6 Recovering Vector Calculus

In R3 we have the operator d behaves as

0→ Ω0(R3)d0−→ Ω1(R3)

d1−→ Ω2(R3)d2−→ Ω3(R3)

d3−→ 0 (46)

We also have that, writing V = R3, that

dim(∧2

V)

= C(3, 2) = 3 (47)

dim(∧3

V)

= C(3, 3) = 1

so then we have ∧0V = C∞(R3)∧1V = R3 (48)∧2V ' R3∧3V ' C∞(R3)

19

where the choice of isomorphism chooses if we are in a “left” or “right” handedcoordinate system.

Consider a 1-form on R3. Then ω = ωxdx+ ωydy + ωzdz so

dω = (∂xωxdx+ ∂xωydy + ∂xωzdz) ∧ dx+ (∂yωxdx+ ∂yωydy + ∂yωzdz) ∧ dy (49)

+ (∂zωxdx+ ∂zωydy + ∂zωzdz) ∧ dz= (∂yωz − ∂zωy)dy ∧ dz

+ (∂zωx − ∂xωz)dz ∧ dx (50)

+ (∂xωy − ∂yωx)dx ∧ dy

Similarly, if we consider a 2-form on R3, ω = ωxydx ∧ dy + ωyzdy ∧ dz + ωzxdz ∧ dx,and we get:

dω = (∂zωxy + ∂xωyz + ∂yωzx)dx ∧ dy ∧ dz (51)

Now, we note that

C∞(R3)∇−→ R3 ∇×−−→ R3 ∇·−→ C∞(R3) (52)

so in combination with our work above we get

∇ ∼ d0 (53)

∇× ∼ d1 (54)

∇· ∼ d2 (55)

1.5 Rewriting Maxwell’s Equations

Maxwell’s first pair of equations, without any time dependence, transform to

∇ · ~B = 0→ ∇ · ~B = 0 (56)

∇× ~E +∂ ~B

∂t= 0→ ∇× ~E = 0 (57)

So if we treat ~B = (Bx, By, Bz) as a 2-form

B = Bx dy ∧ dz +By dz ∧ dx+Bz dx ∧ dy (58)

and ~E = (Ex, Ey, Ez) as a 1-form

E = Ex dx+ Ey dy + Ez dz (59)

20

Then these first two equations just become

dB = 0 (60)

dE = 0 (61)

In the dynamic case, on a general Minkowski spacetime R4 (x0, x1, x2, x3), wedefine

F = B + E ∧ dt (62)

In local coordinates

F =1

2Fµνdx

µ ∧ dxν (63)

with Fµν the matrix of components

Fµν =

0 −Ex −Ey −EzEx 0 Bz −By

Ey −Bz 0 Bx

Ez By −Bx 0

(64)

Note 1.70. Over Rd+1 (where d is the number of spatial dimensions) we can separateany form into space and time components. i.e. if ω = ωIdx

I then

dω = ∂µωIdxµ ∧ dxI (65)

= (dt ∧ ∂tω) +(∂iωI dx

i ∧ dxI)

(66)

We define dSω = ∂iωIdxi ∧ dxI to be this differential of the spacelike components.

Theorem 1.71. Maxwell’s first pair of equations are equivalent to

dF = 0 (67)

Proof. Begin by noting that

dF = dB + dE ∧ dt (68)

= (dSB + dt ∧ ∂tB) + (dSE + dt ∧ ∂tE) ∧ dt (69)

= dSB + (∂tB + dSE) ∧ dt (70)

From the first line, we see if dB = 0 and dE = 0 then dF = 0. Conversely, if dF = 0we have from the last line that dSB = 0 and ∂tB + dSE = 0; which are Maxwell’sequations!

21

Note 1.72. We see on M = R4 that this formulation is equivalent to Maxwell’s firstpair of equations. On a general manifold, M , we take dF = 0 as the definition forMaxwell’s first equation.

Proposition 1.73. For any form ω on R× S, there is a unique way to write

dω = dt ∧ ∂tω + dSω (71)

such that for any local xi on S and t = x0, that:

dSω = ∂iωI dxi ∧ dxI (72)

dt ∧ ∂tω = ∂0ωI dx0 ∧ dxI (73)

Corollary 1.74. On a general manifold, M , dF = 0 is Maxwell’s first pair ofequations. The preceding proposition shows if we can write M = R×S, then we canuniquely decompose

F = Fi0 dxi ∧ dt+

1

2Fijdx

i ∧ dxj (74)

and that we can define Ei = Fi0 and Bij = Fij so that

E = Ei dxi (75)

B =1

2Bij dx

i ∧ dxj (76)

F = B + E ∧ dt (77)

1.5.1 The Metric

Definition 1.75. In Minkowski spacetime (with c = 1) we define the metric as

v · w = −v0w0 + v1w1 + v2w2 + v3w3 (78)

Furthermore, if V is a Minkowski space, and x ∈ V arbitrary, then1. if x · x > 0 then x is spacelike.2. if x · x < 0 then x is timelike; the velocity of a particle moving slower than c is

timelike.3. if x · x = 0 then x is null or lightlike.

Definition 1.76. Let V be a vector space, a semi-Riemannian metric is a symmetricbilinear non-degenerate form, g : V × V → R. If g(v, w) = 0, then v and w areorthogonal. Given a metric g on V , we may always construct an orthonormal basiseµ such that g(eµ, eν) = ±δµν . If the number of +1 is p and the number of −1 isq, then g has signature (p, q).

22

Definition 1.77. A metric g on an arbitrary manifold M assigns to each p ∈ M ametric gp on the tangent space TpM in a smoothly varying way. By smoothly varying,we mean if v and w are smooth fields on M , then gp(vp, wp) is smooth on M .

Result 1.78. Smoothness implies the signature of gp is constant on any connectedcomponent of M .

Definition 1.79. If the signature of a metric is1. (n, 0) then it is Riemannian.2. (n− 1, 1) then it is Lorentzian.

Note 1.80. The easiest way to make a Lorentzian (3, 1) manifold, is to take M =R× S for S a 3-dimensional Riemannian manifold with metric 3g and define

g = −dt2 +3 g (79)

gµν =

−1 0 0 000 3g0

(80)

This metric is static, i.e. independent of time.

Definition 1.81. We naturally define measures of length of a path γ : [a, b] → Mfor spacelike curves ∫ b

a

√g(γ′(t), γ′(t))dt (81)

called the arclength. Similarly, for timelike curves the proper time is∫ b

a

√−g(γ′(t), γ′(t))dt (82)

Theorem 1.82. If (V, g) is a semi-Riemannian vector space, then V ' V ∗ via theisomorphism v → g(v, ·).

Proof. The map T (v) = g(v, ·) is clearly a linear map from V → R, so T (v) ∈ V ∗.Now T (v1) = T (v2) iff T (v1)(u) = T (v2)(u) for all u, which happens iff g(v1, u) =g(v2, u) iff g(v1 − v2, u) = 0. Since u was arbitrary and g is non-degenerate, thenv1 − v2 = 0, and so v1 = v2, and T is one-to-one. Next, let w ∈ V ∗ arbitrary, thenw = wνf

ν so w(eµ) = wµ. Then note that if v = vµeµ and we let wν = g(v, eν) =

23

vµgµν , and combine it with the invertibility of g (from the non-degeneracy of g), wecan solve for vµ = g−1µνw

ν . Thus, if w ∈ V ∗ arbitrary and we consider

v = vµeµ = (g−1µνwν)eµ ∈ V

then T (v) = w ∈ V ∗.

Corollary 1.83. If (M, g) is a semi-Riemannian manifold, then1. For each p ∈ M , TpM ' T ∗pM , so each tangent vector vp is associated with a

tiny stack of hyperplanes, perpendicular to the vector.2. There is an isomorphism Vect(M) ' Ω1(M), which is how we think of 1-forms

as vectors.

Note 1.84. Now that we can associate a 1-form with a vector, we can associate ap-form with the parallelepiped formed by the vectors. However, this formulation isnot perfect, because

(dx+ dy) ∧ (dy + dz) = (dy + dz) ∧ (dz − dx) (83)

but these do not form the same parallelepiped. The two parallelepipeds lie in thesame plane, have the same area, and form a basis with the same orientation (left-handed or right-handed). In fact, it can’t be taken super literally because there areelements of

∧i T ∗pM which are not wedges of i cotangent vectors.

We convert from upper and lower indices by use of the metric.

Proposition 1.85. Let eµ be a basis of vector fields on a chart. Define the com-ponents of the metric as

gµν = g(eµ, eν) (84)

24

Then non-defeneracy implies the matrix gµν is invertible. We write the inverse

gµν = (gµν)−1 (85)

This inverse transforms between raised and lowered indices:1. if v = vµeµ is a vector field, the corresponding 1-form is:

g(v, ·) = vνfν where vν = gµνv

µ (86)

2. if ω = ωµfµ is a 1-form, the corresponding vector field is:

ωνeν where ων = gµνvµ (87)

i.e. gµν lowers vµ to vν, and gµν raises ωµ to ων.

Proof. The proof proceeds by direct calculation.1. The corresponding 1-form to v = vµeµ is g(v, ·). Now take eν , then:

LHS : g(v, eν) = vµ g(eµ, eν) = vµgµν

RHS : vµfµ(eν) = vµδ

µν = vν

2. Suppose ω = ωµfµ is a 1-form, and ω = ωµeµ is the corresponding vector field,

then

ω = g(ω, ·) ⇐⇒ ωµfµ(eν) = g(ωµeµ, eν)

⇐⇒ ωµδµν = ωµgµν

⇐⇒ ων = ωµgµν

⇐⇒ ωµ = gµνων

so the corresponding vector field is ωνeν where ων = gµνωµ.

Corollary 1.86. For any objects, we have raising and lowering by the metric:1. Aα

β···γδε···ξ = gαµA

µβ···γδε···ξ

2. Aαβ···γδε···ξ = gµδAαβ···γµε···ξ3. gµλ = gµαgαλ = δµλ

The metric allows us to naturally define an inner-product on p-forms.

25

Definition 1.87. The inner product of two p-forms ω and µ on M is a bilinearfunction 〈ω, µ〉 on M . Since it is bilinear, we can define it on p-fold products of1-forms by:

〈e1 ∧ · · · ∧ ep, f 1 ∧ · · · ∧ fp〉 = det[〈ei, f j〉

](88)

where the RHS is the determinant of the p× p matrix whose entries are 〈ei, f j〉 andon 1-forms behaves like

〈ω, µ〉 = gαβωαµβ (89)

Note 1.88. Note how this inner product on 1-forms is analogous to the inner producton vectors

g(v, w) = gαβvαwβ (90)

Theorem 1.89. The inner product of p-forms, 〈·, ·〉, is non-degenerate. Further-more, the wedge products ei1 ∧ · · · ∧ eip form an orthonormal basis of p-forms with

〈ei1 ∧ · · · ∧ eip , ei1 ∧ · · · ∧ eip〉 = ε(i1) · · · ε(ip) (91)

where ε(i) = g(ei, ei) = ±1.

Proof. First let eI = ei1 ∧· · ·∧ eip and eJ = ej1 ∧· · ·∧ ejp be basis vectors. Obviouslyeach of the ik is distinct, or else eI = 0, similarly for eJ and it’s indices. Now, supposeeI and eJ agree on m indices, for some m. WLOG, since it just changes a sign, wecan take these to be the first m, so that i1 = j1, i2 = j2, . . . , and im = jm. Then wehave that

〈ei1 ∧ · · · ∧ eip , ej1 ∧ · · · ∧ ejp〉 = det

0

M......

0 · · · · · · 0

where M is the m ×m matrix whose entries are only along the diagonal and thoseentries are 〈eik , ejk〉 at diagonal k, giving out result. Non-degeneracy is apparentbecause g is invertible.

1.5.2 The Volume Form

Definition 1.90. Given a vector space, V , with bases eµ and fµ, there is aunique linear transform T : V → V such that

Teµ = fµ

which is necessarily invertible. We say eµ and fµ have the same orientation ifdet(T ) > 0, and opposite orientation if det(T ) < 0.

26

Definition 1.91. An orientation on V is a choice of equivalence classes of bases,where two bases are in the same equivalence class if they have the same orientation.

Example 1.92. Even permutations of a basis have the same orientation. Oddpermutations of a basis have the opposite orientation.

Definition 1.93. Suppose V is an n-dimensional vector space with basis eµ, thenthe element

e1 ∧ · · · ∧ en ∈∧n

V (92)

is called the volume element associated to the basis eµ. So that any ω ∈∧n V can

be written ω = ce1 ∧ · · · ∧ en for c ∈ R

Proposition 1.94. Suppose fν is a new basis of V such that fν = T µν eµ. Thenthe volume element associated to fµ is

f1 ∧ · · · ∧ fn = det(T )(e1 ∧ · · · ∧ en) (93)

Proof. The proof is straightforward algebra and noting the definition of a determi-nant in terms of sums of permutations weighted by the signs of the permutations.Any terms with duplicate entries will vanish.

f1 ∧ · · · ∧ fn = (T 11 e1 + · · ·+ T n1 en) ∧ · · · ∧ (T 1

ne1 + · · ·+ T nn en)

= det(T )(e1 ∧ · · · ∧ en)

Definition 1.95. A volume form ω on a manifold M is a nowhere-vanishing n-form.

Example 1.96. For any p ∈M , ωp ∈ T ∗pM is a volume element on T ∗pM .

Example 1.97. The standard volume form on Rn is ω = dx1 ∧ · · · ∧ dxn.

Definition 1.98. M is orientable if there exists a volume form on M . An orientationon M is a choice of equivalence class of volume forms on M , where ω ∼ ω′ if ω′ = fωfor some positive function f . We say volume forms int he chosen equivalence classare positively oriented, and the others are negatively oriented.

Proposition 1.99. Let M be an oriented manifold, then we can cover M with ori-ented charts ϕα : Uα → Rn, such that the basis dxµ of cotangent vectors on Rn,pulled back to Uα by ϕα, is positively oriented.

27

Proof. since M can be oriented, we can define a volume form ω on M to definethe “positive orientation” on M . In the chart Uα we can define a volume form bydx1 ∧ · · · ∧ dxn. For it to be positively oriented when pulled back, we should haveω = cϕ∗α(dx1∧· · ·∧dxn) for some c > 0, c ∈ R. Well, we just use whatever our usualcharts are, say φα, and pull them back. Then, ω = cφ∗α(dx1 ∧ · · · ∧ dxn), and if c > 0let ϕα = φα, else if c < 0 just interchange dx1 and dx2.

Definition 1.100. Given a diffeomorphism φ : M → N from one oriented mani-fold to another, we say φ is orientation preserving if the pullback of any standardorientation basis for the cotangent space of N is a standard orientation basis of thecotangent space of M .

Proposition 1.101. If we can cover M with charts such that the transition functionsϕa ϕ−1b are orientation preserving, then M can be made into an oriented manifold.

Proof. In each chart, just pullback the standard volume form on Rn to M . Since eachchart is connected by an orientation preserving transition function, the orientationis the same in any chart, and the manifold is oriented everywhere.

Theorem 1.102. Let M be an oriented n-dimensional manifold with metric g. Thenthere is a canonical volume form on M given by

vol =√|det gµν |dx1 ∧ · · · ∧ dxn (94)

for dxµ the local coordinates on M .

Proof. Cover M with oriented charts ϕα : Uα → Rn. In any chart set gµν = g(∂µ, ∂ν)where ∂µ is the pullback of the basis for vector fields from Rn. Well, then volas defined is a volume form on Uα, so now we must show it agrees on all of M byshowing that if ϕ′ : U ′ → Rn is an overlapping chart with ϕ : U → Rn that vol = vol′

on U ∩ U ′.Well, on the overlap: dx′ν = T νµdx

µ for the usual T νµ = ∂x′ν/∂xµ. So that we have

dx′1 ∧ · · · ∧ dx′n = (detT )(dx1 ∧ · · · ∧ dxn).

This means vol = vol′ if√∣∣det g′µν

∣∣ = (detT )−1√|det gµν |. Well, we have:

g′µν = g(∂′µ, ∂′ν)

= g((T−1)αµ∂α, (T−1)βν∂β)

= (T−1)αµ(T−1)βνgαβ

so det g′µν = (detT )2 det gαβ. We can take the square root to get our result by notingthat detT > 0 since the charts are oriented.

28

Note 1.103. The volume form is typically denoted√|det g| dnx (95)

or in the Lorentzian case √−g dnx (96)

If we don’t want to work in the standard basis dxµ for our cotangent spaces,we can generalize it simply.

Proposition 1.104. Let M be an oriented n-dimensional semi-Riemannian man-ifold. Let eµ be any oriented orthonormal basis of cotangent vectors at p ∈ M .Then we have

e1 ∧ · · · ∧ en = volp (97)

where volp is just the volume form associated to the metric evaluated at p on themanifold.

Proof. If eµ is a standard orientation orthonormal basis, it is related to the stan-dard basis dxµ by an orthogonal transformation, T , so detT = ±1. But it isorientation preserving so detT = 1. Since in transforming e1 ∧ · · · ∧ en into thestandard basis at p we collect a factor of detT we get

e1 ∧ · · · ∧ en = (detT )(dx1 ∧ · · · ∧ dxn)|p = volp

]

1.5.3 The Hodge Star Operator

The difference in the second pair of Maxwell’s equations, besides non-homogeneity,is that ~E now behaves like a 2-form, and ~B is now like a 1-form.

∇ · ~E = ρ (98)

∇× ~B − ∂ ~E

∂t= ~j (99)

Definition 1.105. Let M be an n-dimensional oriented semi-Riemannian manifold,and let the inner product of two p-forms ω and µ is 〈ω, µ〉. The Hodge star is theunique linear map ? : Ωp(M)→ Ωn−p(M) satisfying:

ω ∧ (?µ) = 〈ω, µ〉 vol (100)

for any ω, µ ∈ Ωp(M).

29

Note 1.106. To compute the Hodge Star: Suppose eµ are a basis of positivelyoriented orthonormal 1-forms on some chart. i.e. 〈eµ, eν〉 = δµν ε(µ). Then for anydistinct 1 ≤ i1, . . . , ip ≤ n:

?(ei1 ∧ · · · ∧ eip) = σeip+1 ∧ · · · ∧ ein (101)

where σ = sgn(i1, . . . , in)ε(i1) · · · ε(ip), and ip+1, . . . , in = i1, . . . , in−i1, . . . , ip.

Note 1.107. Recall that if W is a vector space, the Riesz representation theoremsays that every (continuous) linear functional f ∈ W ∗ has a unique vector v ∈ Wwhich emulates its behaviour with the help of the inner product. I.e. for any w ∈ Wthat

f(w) = 〈w, v〉 (102)

Which gives the isomorphism between W and W ∗.The Hodge star is effectively the analogous structure for wedges. If V is an n-

dimensional vector space with basis eµ, then for 0 ≤ k ≤ n the exterior power

spaces∧k V and

∧n−k V are effectively dual. Take λ ∈∧k V and θ ∈

∧n−k V ,then taking the exterior product together effectively gives a scalar since it maps toa one-dimensional vector space, i.e. λ ∧ θ ∈

∧n V , and so it is a scalar multiple ofe1 ∧ · · · ∧ en.

Now fix λ. There exists a unique linear function fλ ∈(∧n−k V

)∗such that for

any θ:λ ∧ θ = fλ(θ)(e1 ∧ · · · ∧ en) (103)

By construction of ?λ, it is the element of∧n−k V such that for any θ we have

fλ(θ) = 〈θ, ?λ〉 (104)

Result 1.108. Let V an n-dimensional vector space with basis eµ, and λ ∈∧k V

and θ ∈∧n−k V arbitrary. Then

λ ∧ θ = 〈θ, ?λ〉(e1 ∧ · · · ∧ en) (105)

Theorem 1.109. We can formalize our results about vector calculus from earlier1. ?(ω ∧ µ) emulates ω × µ.2. ?dω emulates the curl of ω, ∇× ω, for ω ∈ Ω1(M).3. ?d ? ω emulates the divergence of ω, ∇ · ω, for ω ∈ Ω1(M).

Theorem 1.110. Let M an n-dimensional semi-Riemannian oriented manifold withsignature (s, n− s). Then ?2 : Ωp(M)→ Ωp(M) by

?2 = (−1)p(n−p)+s (106)

30

Proof. ω ∧ ?ω = 〈ω, ω〉 vol and ?ω ∧ ?2ω = 〈?ω, ?ω〉 vol. Now note that the secondequation can be reordered: ?2ω ∧ ?ω = (−1)p(n−p)〈?ω, ?ω〉 vol.

Substituting for the vol in both expressions

?2ω ∧ ?ω = ω ∧ ?ω (−1)p(n−p)〈?ω, ?ω〉〈ω, ω〉

so?2 = (−1)p(n−p)〈?ω, ?ω〉/〈ω, ω〉

Now let ω be a basis p-form, ei1 ∧ · · · ∧ eip , then 〈ω, ω〉 =∏p

j=1 ε(ij). We also

know form our explicit construction of the Hodge star that ?ω = σeip+1 ∧ · · · ∧ ein so〈?ω, ?ω〉 =

∏nj=p+1 ε(ij). Combining these results

〈?ω, ?ω〉〈ω, ω, 〉

= 〈?ω, ?ω〉〈ω, ω, 〉

=n∏

j=p+1

ε(ij)

p∏j=1

ε(ij)

= (−1)s

which gives our result.

Definition 1.111. LetM an n-dimensional oriented semi-Riemannian manifold withsignature (s, n − s). Let eµ be an orthonormal basis of 1-forms on a chart. TheLevi-Civita symbol for 1 ≤ i1, . . . , in ≤ n is

εi1···in =

sgn(i1, . . . , in) for all ij distinct

0 otherwise(107)

Theorem 1.112. For any p-form

ω =1

p!ωi1,...,ipe

i1 ∧ · · · ∧ eip (108)

we have

(?ω)j1,...,jn−p =1

p!εi1···ipj1···jn−pωi1···ip (109)

Proof. For any basis vector

?(ei1 ∧ · · · ∧ eip) = εi1···inε(i1) · · · ε(ip) eip+1 ∧ · · · ∧ ein (no sums)

31

Then

(?ω)j1,··· ,jn−p = 〈ej1 ∧ · · · ∧ ejn−p , ?ω〉ε(j1) · · · ε(jn−p)

= ε(j1) · · · ε(jn−p)1

p!

∑i1,...,ip

ωi1,...,ip〈ej1 ∧ · · · ∧ ejn−p , ?(ei1 ∧ · · · ∧ eip)〉

= ε(j1) · · · ε(jn−p)1

p!

∑i1,...,ip

ωi1,...,ipεi1···ipε(i1) · · · ε(ip)〈ej1 ∧ · · · ∧ ejn−p , eip+1 ∧ · · · ∧ ein〉

= ε(j1) · · · ε(jn−p)1

p!

∑i1,...,ip

ωi1,...,ipεi1···ipε(i1) · · · ε(ip)ε(j1) · · · ε(jn−p)δj1···jn−pip+1···in

= ε(j1) · · · ε(jn−p)(−1)s

p!

∑i1,...,ip

ωi1,...,ipεi1···ip

=1

p!εi1···ipj1···jn−pωi1···ip

1.5.4 The Second Pair of Equations

Let M = R4 the Minkowski spacetime, then we can split F into

F = B + E ∧ dt (110)

which in component form gives

Fµν =

0 −Ex −Ey −EzEx 0 Bz −By

Ey −Bz 0 Bx

Ez By −Bx 0

(111)

With the help of the metric, we have a Hodge star, computing we get

(?F )µν =

0 Bx By Bz

−Bx 0 Ez −Ey−By −Ez 0 Ex−Bz Ey −Ex 0

(112)

Definition 1.113. Given a vector current density : ~j = j1∂1+j2∂2+j3∂3 and chargedensity ρ, we can form the current

~J = ρ∂0 + j1∂1 + j2∂2 + j3∂3 (113)

32

or as a 1-formJ = j − ρdt (114)

Theorem 1.114. Maxwell’s second pair of equations can be written

?d ? F = J (115)

or if we define ?S to be the Hodge star on R3 (or “space”) we have

?SdS ?S E = ρ (116)

−∂tE + ?SdS ?S B = j (117)

Proof. We have E = Ejdxj and B = 1

2εjklBjdx

k ∧ dxl. Calculating:

?SE =1

2Ejε

jkldx

k ∧ dxl

dS(?SE) =1

2εjkl(dSEj) ∧ dxk ∧ dxl

=1

2εjkl∂m(Ej)dx

m ∧ dxk ∧ dxl

?SdS ?S E =1

2εjkl∂m(Ej) ? (dxm ∧ dxk ∧ dxl)

=1

2εjkl∂mEjε

mklε(xm)ε(xk)ε(xl)

=1

2εjkl∂mEjε

mkl

=1

2∂mEj(2δ

mj )

= ∂jEj

= ∇ · ~E

33

Similarly, working on B we have

?SB =1

2εjklBj ?S (dxk ∧ dxl)

=1

2εjklBjε

klmdx

m

= Bjdxj

dS ?S B = ∂mBjdxm ∧ dxj

?SdS ?S B = ∂mBj ?S (dxm ∧ dxj)= ∂mB

jεmjkdxk

= εijk∂iBjdxk

= ∇× ~B

Now

?F = ?SE − ?SB ∧ dtd ? F = ?S∂tE ∧ dt+ dS ?S E − dS ?S B ∧ dt?d ? F = −∂tE − ?SdS ?S E ∧ dt+ ?SdS ?S B

setting ?d ? F = J we have the result.

Definition 1.115. Let M be any spacetime manifold. Then the electromagneticfield is a 2-form on M , the current J is a 1-form on M . The first Maxwell equationssay dF = 0. If M is semi-Riemannian and orientable, the second Maxwell equationssay ?d ? F = J .

Result 1.116. To introduce E and B fields we must assume M = R× S and writeF = B + E ∧ dt, and J = j − ρdt. The first pair split into

dF = 0 =⇒ dSB = 0 and ∂tB + dSE = 0 (118)

and if dim(S) = 3 and there is a static metric g = −dt2 +3 g then the second pairsplits

?d ? F = J =⇒ ?SdS ?S E = ρ and − ∂tE + ?SdS ?S B = j (119)

Definition 1.117. The vacuum Maxwell equations are dF = 0 and d?F = 0. Theseequations are preserved by F → ?F .

34

Note 1.118. In 4-dimensional Riemannian space ?2 = 1, so the eigenvalues of ? are±1. Thus we can split F into self-dual F+, and anti-self-dual F−, parts, and writeF = F+ + F−, where ?F± = ±F±, and F± = 1

2(F ± ?F ).

In 4-dimensional Lorentzian space, ?2 = −1, so the eigenvalues are ±i. Thus wecan define F± = 1

2(F ∓ i ? F ), so that F = F+ + F− and ?F = ±iF±.

Note 1.119. On Minkowski M = R4 the vacuum equations give light movingthrough empty space. ?F = ?SE − ?SB ∧ dt so F is self-dual if ?SE = iB and?SB = −iE. These conditions hold for all t if

E = Eidxi and B =

−i2εjklEkdx

k ∧ dxl (120)

Definition 1.120. A planewave is an equation of the form

E(x) = ~Eeikµxµ

(121)

where ~E = Ejdxj is a constant C-valued 1-form on R3, and k ∈ R4 − 0, is a fixed

covector called the energy-momentum

Proposition 1.121. If F is self-dual (as in a vacuum in 4D), and the electric field

is a plane wave: E = ~Eeikµxµ. Then

1. B = ~Beikµxµ

where ~B = −i ?S ~E2. 〈 ~E, 3k〉 = 0, so ~E is perpendicular to the momentum, and light is a transverse

wave.3. 3k ∧ ~E = k0 ~B, so the cross product of 3k and ~E is proportional to ~B by k0, the

frequency of the wave.4. The energy-momentum of light is light-like, i.e. kµkµ = 0.

Proof. If F is self-dual and its electric field is a plane-wave, then the previous resultsgive that the magnetic field is

B(x) = ~Beikµxµ

where ~B = i ?S ~E.The first Maxwell equation holds when

0 = dS ~B

= ~B ∧ dSeikµxµ

= ~B ∧ (eikµxµ3k)

0 = ~B ∧ 3k

35

But in terms of ~E, this means ?S ~E ∧ 3k = 0 or

〈 ~E, 3k〉 = 0

In addition, we have

0 = ∂tB + dSE

= ik0B + dS(eikµxµ

Ejdxj)

= ik0B + iklEjdxj ∧ dxl

= ik0B − i3k ∧ E

which means 3k ∧ ~E = k0B.To see that the energy-momentum of light is light-like 3k∧ ~E = k0 ~B = −ik0 ?S ~E,

implying 〈3k ∧ ~E, 3k ∧ ~E〉 = k20〈?S ~E, ?S ~E〉. Expanding the LHS

〈3k ∧ ~E, 3k ∧ ~E〉 = klk∗l′EjE

∗j′〈dxl ∧ dxj, dxl

′ ∧ dxj′〉= klk

∗l′EjE

∗j′(δ

jj′δll′ − δj′lδjl′)

= 〈3k, 3k〉〈 ~E, ~E〉 −∣∣∣〈3k, ~E〉∣∣∣2

= 〈3k, 3k〉〈 ~E, ~E〉

Expanding the RHS

〈?S ~E, ?S ~E〉 =1

4εjklε

j′k′l′EjE

∗j′〈dxk ∧ dxl, dxk

′ ∧ dxl′〉

=1

4εjklε

j′k′l′EjE

∗j′(δ

kk′δll′ − δk′lδkl′)

= 〈 ~E, ~E〉

So 〈3k, 3k〉〈 ~E, ~E〉 = k20〈 ~E, ~E〉, so that kµkµ = 0.

1.6 deRham Theory in Electromagnetism

1.6.1 Closed and Exact 1-Forms

On an n-dimensional manifold, M , the exterior derivative defines a (co)-chain com-plex, or de Rham complex :

0→ Ω0(M)d0−→ Ω1(M)

d1−→ · · · dm−2−−−→ Ωm−1(M)dm−1−−−→ Ωm(M)

dm−→ 0 (122)

36

Definition 1.122. If dn−1 : Ωn−1(M)→ Ωn(M) and dn : Ωn(M)→ Ωn+1(M), thenω ∈ Ωn(M) is called

1. exact if ω ∈ im(dn−1). We write Bn(M) = im(dn−1).2. closed if ω ∈ ker(dn). We write Zn(M) = ker(dn).

Theorem 1.123. d2 = 0

Corollary 1.124. All exact forms are closed. i.e. Bn(Z) ⊆ Zn(Z) for all n.

Definition 1.125. Let E be a 1-form and B a 2-form. A scalar potential is a 0-form(function) φ such that E = −dφ. A vector potential is a 1-form A such that B = dA.We also use A as the potential for F , F = dA.

When is a closed form exact? What are the conditions such that Z1(M) =B1(M)? Under these conditions then a form can be written as the differential ofsome potential and vanishes under a second application of the exterior derivative.

Definition 1.126. Let S be a manifold. A (smooth) path in S is a smooth mapγ : [0, T ] → S. We can integrate a 1-form over the path, E ∈ Ω1(S), in a naturalway: if γ is a path, γ′(t) is a tangent vector at γ(t), and then Eγ(t) is a cotangentvector at γ(t). Thus we define∫

γ

E =

∫ T

0

Eγ(t)(γ′(t))dt (123)

Definition 1.127. If there is a path between any two points in S, S is (path)connected. If not, a maximal connected subset is called a connected component.

In general we assume we are on a connected manifold, but if not, lots of ourtheorems apply with the caveat that you restrict yourself to specific connected com-ponents of the manifold.

Definition 1.128. Let γ0, γ1 : [0, T ] → S be two paths from p to q in S. Then γ0and γ1 are homotopic if there exists a γ : [0, 1]× [0, T ]→ S such that γ(s, ·) is a pathfrom p to q for any s ∈ [0, 1], and γ(0, t) = γ0(t) and γ(1, t) = γ1(t). The function γis called a homotopy between γ0 and γ1.

Definition 1.129. Given a manifold S, it is simply connected if any two pathsbetween any two (fixed) points are homotopic.

Theorem 1.130. The integral of a closed 1-form is the same over homotopic paths.That is if γ0 ∼ γ1 and dE = 0 ∫

γ0

E =

∫γ1

E (124)

37

Proof. Suppose γ0 and γ1 are paths from p to q which are homotopic. Then thereexists a γ : [0, 1]× [0, T ]→ S such that γ(s, 0) = p and γ(s, T ) = q for any s ∈ [0, 1]and γ(0, t) = γ0(t) and γ(1, t) = γ1(t). Define

I(s) =

∫ T

0

Eγ(s,t)(γ′(s, t)) dt

Working on a patch, we can switch to local coordinates, say eµ with dual basisfµ, so that we can expand in the patch as

Eγ(s,t) = Eµ(γ(s, t))fµ

γ′(s, t) = [γ′(s, t)]νeν

which we can combine into

Eγ(s,t)(γ′(s, t)) = Eµ(γ(s, t))[γ′(s, t)]νfµ(eν)

= Eµ(γ(s, t))[γ′(s, t)]µ

= Eµ(γ(s, t))∂tγµ(s, t)

Now we can compute

∂sIs =

∫∂s[Eµ(γ(s, t)∂tγ

µ(s, t)] dt

=

∫[∂sEµ(γ(s, t))∂tγ

µ(s, t) + Eµ(γ(s, t))∂s∂tγµ(s, t)] dt

=

∫[∂sEµ(γ(s, t))∂tγ

µ(s, t)− ∂tEµ(γ(s, t))∂sγµ(s, t)] dt

=

∫∂νEµ(γ(s, t))[∂sγ

ν∂tγµ − ∂tγν∂sγµ] dt

Noting that: dE = (∂µEν − ∂νEµ)dxµdxν we get

∂sIs =

∫(dE)µν∂sγ

µ∂tγν dt

so ∂sIs = 0 if dE = 0, and so we have our result.

Theorem 1.131. Let S be simply connected and E a closed 1-form on S, then E isexact.

38

Proof. Pick and fix any p ∈ S. For any q ∈ S define

φ(q) = −∫γ

E

where γ is a path from p to q. Since S is simply connected and E is closed, by theprevious theorem, this function is well-defined. We will show E = −dφ, so φ is ascalar potential for E, and so E is exact (E = d(−φ)).

Consider any vq ∈ TqS, then −dφ(vq) = −vq(φ). Pick a path γ : [0, 2] → S withγ(0) = p, γ(1) = q, and γ′(1) = vq. Then

E(vq) = E(γ′(1))

=d

dt

∫ t

0

E(γ′(s)) ds∣∣∣t=1

= − d

dtφ(γ(t))

∣∣∣t=1

= −vq(φ)

= −dφ(vq)

Hence, for any vq ∈ TqS we have E(vq) = −dφ(vq) so E = −dφ.

Proposition 1.132. Rn is simply connected.

Proof. Let p, q ∈ Rn arbitrary. Suppose γ0(t) and γ1(t) are paths from p to q. Thenlet γ(s, t) = γ0(t)(1− s) + γ1(t)s. Then γ(s, t) is a homotopy.

Example 1.133. R2−0 is NOT simply connected. Consider the paths on [0, 1]→S given by

γ0(t) = 〈cos(π(1− t)), sin(π(1− t))〉γ1(t) = 〈cos(π(1− t)),− sin(π(1− t))〉

39

Consider the 1-form

E =xdy − ydxx2 + y2

Let r = x2 + y2, then

E =x

rdy − y

rdx

dE = (∂xE) ∧ dx+ (∂yE) ∧ dy

=

[y2 − x2

r2dy − −2xy

r2dx

]∧ dx+

[−−2xy

r2dy − x2 − y2

r2dx

]∧ dx

= 0

so E is closed.Now we compute∫

γ0

E0 =

∫ 1

0

Eγ0(t)(γ′0(t)) dt

=

∫ 1

0

Eµ(γ0(t))∂t(γµ0 (t))) dt

=

∫ 1

0

[Ex(γ0)γ

′,x0 (t) + Ey(γ0)γ

′,y0 (t)

]dt

=

∫ 1

0

[− sin(π(1− t))π sin(π(1− t))− cos(π(1− t))π cos(π(1− t))]

= −π

Similarly, the opposite integral is π. So∫γ0E 6=

∫γ1E for our closed form, so γ0

and γ1 must not be homotopic, so R2 − 0 is not simply connected.

Definition 1.134. A path γ : [0, T ] → S is a loop if it ends where it starts. Ifγ(0) = γ(T ) = p we say the loop is based at p or p is the basepoint. The loop iscontractible if it is homotopic to a constant loop based at p, ηp(t) ∼= p for all t.

Note 1.135. On a simply connected manifold, all loops are contractible.

Theorem 1.136. Let E be a 1-form on any manifold S.1. E is closed iff

∫γE = 0 for all contractible loops, γ.

2. E is exact iff∫γE = 0 for all loops, γ.

Proof. We proceed by construction in each case.

40

1. For part 1.⇒ Suppose E is closed and let γ be any contractible loop based at p. Then

it is homotopic to ηp, so by our previous theorem∫γE =

∫ηpE = 0.

⇐ Assume∫γE = 0 for all contractible loops γ based at any p. Pick a chart

giving coordinates xµ about p ∈ S. Then consider the integral of Earound a tiny square of width/height ε in the xµ − xν plane. Then byGreen’s theorem ∫

γ

E =

∫ ε

0

∫ ε

0

(∂µEν − ∂νEµ) dxµdxν

using∫γE = 0 on the LHS and letting ε→ 0 we have

0 = ε2(∂µEν − ∂νEµ)|p = ε2(dE)µν |p

so dE = 0 if∫γE = 0 for all contractible loops.

2. For part 2.⇒ Suppose E is exact, then E = dφ for some 0-form φ. Let γ be any loop

based at p0 ∫γ

E =

∫ T

0

Eγ(t)(γ′(t)) dt

=

∫ T

0

(dφ)γ(t)(γ′(t)) dt

=

∫ T

0

[γ′(t)](φ)|γ(t) dt

=

∫ T

0

[ddsφ(γ(s))

]|s=t dt

=

∫ T

0

[φ(γ(t))]′ dt

= φ(γ(t))− φ(γ(0))

= 0

⇐ Suppose∫γE = 0 for all loops γ. Then E is at least closed. Suppose,

for contradiction, that E is NOT exact, then S is not simply connected.Then we may choose two points p, q ∈ S such that there exist paths γ1and γ2 from p to q, such that

∫γ1E 6=

∫γ2E. Now let γ = γ1 ∪ γ∗2 , i.e.

γ1 forward then γ∗2 in reverse. So γ is a loop based at p, so∫γE = 0 by

hypothesis, but∫γE =

∫γ1E −

∫γ2E 6= 0, a contradiction.

41

Example 1.137. For any manifold M , S1 ×M is NOT simply connected. Choosethe local coordinates xµ, µ = 0, 1, . . . ,m, where m = dimM , where x0 is theS1 coordinate with 0 and 2π identified. Consider the 1-form ω = (1, 0, 0, . . . , 0),then dω = 0, but if we try to construct a 0-form φ such that ω = dφ by φ(q) =φ(0) +

∫γω = φ(0) + x0, we see for any path γ fixed in M and travelling around S1

we have multiple values for φ based on if γ wraps around once, twice, . . . , etc. i.e.φ is a multi-function, not a 0-form.

1.6.2 Stokes’ Theorem

Definition 1.138. In Rn with coordinates xµ the closed half-space is

Hn = (x1, . . . , xn) : xn ≥ 0 (125)

Definition 1.139. A function on Hn is smooth if it extends to a smooth functionon the manifold

(x1, . . . , xn) : xn + ε > 0 for some ε > 0 (126)

Definition 1.140. An n-dimensional manifold with boundary is a (Hausdorff para-compact) topological space M equipped with charts of the form:

ϕα : Uα → Rn or ϕα : Uα → Hn (127)

where Uα are open sets covering M such that the transition function ϕα ϕ−1β aresmooth where defined.

Definition 1.141. If M is an n-dimensional manifold with boundary, the boundary∂M is the set of p ∈M such that some chart ϕα : Uα → Hn maps p to a point in

∂Hn = (x1, . . . , xn) : xn = 0 (128)

Definition 1.142. A function on a manifold with boundary is smooth if f ϕα issmooth as a function on Rn or Hn for any chart ϕα.

Note 1.143. The tangent space to a point on the boundary is still an n-dimensionalvector space. The coordinates x1, . . . , xn−1 are still as before, and xn = 0 causes noissues because functions are required to extend to smooth functions on −ε < xn < 0.

42

Definition 1.144. Let ω be any n-form on Rn. Let xµ a coordinate system onRn. Then we define the integral of the n-form, ω = f dx1 ∧ · · · ∧ dxn, as:∫

Rnω =

∫Rnf dx1 · · · dxn (129)

Proof. This map is well-defined. That is, if x′µ is any other coordinate systemthen: ω = f dx1 ∧ · · · ∧ dxn. We know if T is the map such that:

dx′µ = T µν dxν

where

T µν =∂x′µ

∂xν

i.e. the Jacobian. Thus we have that dx′1∧ · · · ∧ dx′n = (detT )dx1∧ · · · ∧ dxn meansf = f ′(detT ). So we have well-definedness by the Jacobian change of variablesformula:∫

Rnω =

∫Rnf dx1 · · · dxn =

∫Rnf ′ (detT )dx1 · · · dxn =

∫Rnf ′ dx′1 · · · dx′n =

∫Rnω

so the integral of ω is coordinate-independent.

Result 1.145. Let M be an n-dimensional manifold with boundary (Hausdorff andparacompact). Let ϕα an atlas of charts for M , ϕα : Uα → Rn or ϕα : Uα → Hn.Then we may always find a collection of smooth functions fα on M called partitionsof unity such that:

1. fα is zero outside Uα.2. Any point p ∈M has an open set containing it on which only finitely many of

the fα are non-zero.3. For any p ∈M ,

∑α fα(p) = 1.

Note 1.146. This allows us to basically stitch together integrals over charts byweighting our form against an fα, ensuring each point only gets counted with totalweight 1 when integrating. That is, we decompose

ω =∑α

fα ω (130)

so that fα ω vanishes outside Uα. We thus have, in the Uα coordinates,

fα ω = gα(x1, . . . , xn) dx1 ∧ · · · ∧ dxn (131)

where gα vanishes outside Uα.

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Definition 1.147. We define the integral of ω ∈ Ωn(M) as∫M

ω =∑α

∫gα(x1, . . . , xn) dx1 ∧ · · · ∧ dxn (132)

where fα ω = gα(x1, . . . , xn) dx1 ∧ · · · ∧ dxn and fα is a partition of unity.

Note 1.148. Integration is independent of the charts and partition of unity.

Proposition 1.149. Let M be an oriented manifold with boundary. Then ∂M is anoriented manifold in a natural way, with dimension n− 1.

Proof. Take an atlas of charts for M , and only consider those of the form ϕα : Uα →Hn. Define Vα = Uα ∩ ∂M , so Vα are open subsets of ∂M . Define ψα = ϕα|Vα , thenψα : Vα → Rn−1 is continuous with continuous inverse, and so ψα ψ−1β are smooth(and orientation preserving!) It is oriented by the fact that diffeomorphisms (can)preserve orientation.

Result 1.150 (Stokes’ Theorem). Let M be a compact n-manifold with boundary,and let ω be an (n− 1)-form on M . Then∫

M

dω =

∫∂M

ω (133)

Note 1.151. We can drop M compact if M vanishes outside a compact set.

Definition 1.152. Given S ⊆M , S is a k-dimensional submanifold of M , if for anyp in S there is an open set U ⊆M , containing p, and a chart ϕ : U → Rn such that

S ∩ U = ϕ−1Rk (134)

Or for a k-dimensional submanifold with boundary

S ∩ U = ϕ−1Hk (135)

Definition 1.153. If N is a manifold, and φ : N → M is a smooth map such thatφ(N) is a submanifold of M , then φ is an embedding of N in M .

Proposition 1.154. Let M be a manifold.1. Any open subset of M is a submanifold.2. Any submanifold of M is a manifold in a natural way.

Further suppose M has boundary.

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3. If S ⊆M is a k-dimensional submanifold with boundary, then S is a manifoldwith boundary in a natural way.

4. ∂S is a (k − 1)-dimensional submanifold of M .

Proof. We proceed directly.1. Let U ⊆ M open. U is obviously coverable by charts of M restricted to U .

The transition functions are smooth on U since they are smooth on M . If not,M would not be a manifold anyway, for we could just add U to any chart forM .

2. Suppose S ⊆ M is a submanifold of M of dimension k, then there are setsVα = S ∩ Uα = ϕ−1α Rk, which cover S and are open in the natural subsettopology inherited from M to S. As for charts, we choose ψα : Vα → Rk byψα = ϕα|Uα .

3. Exactly as above, but we also have ψα : Vα = S ∩Uα = ϕ−1α Hk mapping to Hk,by ψα = ϕα|Uα with appropriate projection.

4. Suppose ϕα : Uα → Rn or Hn are the charts of the manifold of M , and S isa k-dimensional submanifold with boundary, covered by the charts ψα : Vα =S ∩ Uα → Rk or Hk by ψα = ϕα|Uα . The boundary ∂S is the set of s ∈ S suchthat some ψα : Vα → Hk map p to a point in ∂Hk. So in the same way we choseVα = ϕ−1α Rk we choose the Wα = ψ−1α ∂Hk, with all appropriate projections.

1.6.3 deRham Cohomology

Definition 1.155. The pth deRham cohomology group of M is the vector space

Hp(M) =Zp(M)

Bp(M)(136)

An element ofHp(M) is an equivalence class, where two closed p-forms ω, ω′ ∈ Zp(M)are equivalent if

ω − ω′ = dµ (137)

for some (p−1)-form, µ. In this case, ω and ω′ are cohomologous and lie in the samecohomology class.

Note 1.156. Intuitively, dim(Hp(M)) is the number of “p-holes,” i.e. objects pre-venting homotopies for p-surfaces. For example a 1-hole stops a curve homotopy likein R2 − 0.

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H0(M): A 0-form is a function f . It is closed if 0 = df = ∂µfdxµ. That is, if

it is locally constant. The most general locally constant function is one that takeconstant value ci on the Mi component of a manifold M .

H0(M) ' space of locally constant functions on M (138)

dim(H0(M)) = number of connected components of M (139)

H0(M) = R iff M is connected (140)

H1(M): H1(M) = 0 if M is simply connected. More arbitrarily, if we find a set

of d closed 1-forms ω1, . . . , ωd on M such that no (non-trivial) linear combinationis exact, then H1(M) is at least d-dimensional.

We can show a closed 1-form is not exact by Stokes’ theorem. Consider S1 ⊆Man embedded circle. If ω = df for some f ∈ Ω0(M) then

∫Sω =

∫Sdf = ω∂Sf = 0

since ∂S = ∅. So if we find a circle such that∫Sω 6= 0 then ω is not exact.

Theorem 1.157. Suppose ω = dµ ∈ Bp(M) is an exact p-form on M . Then forevery compact p-dimensional manifold S and map φ : S →M∫

S

φ∗ω =

∫S

φ∗dµ =

∫S

d(φ∗µ) =

∫∂S

φ∗µ = 0 (141)

Corollary 1.158. If S ⊆M any compact orientable submanifold then∫S

ω = 0 (142)

This occurs for φ = id in the previous theorem.

Result 1.159. The converse is also true: if∫Sφ∗ω = 0 for every map φ : S → M

of a p-dimensional manifold S to M , then ω is exact.

Note 1.160. Combining these two results: Exact forms are those that integrate to0, always.

Proposition 1.161. The pullback of a closed form is closed. The pullback of anexact form is exact.

Proof. Suppose ω is closed, then d(φ∗ω) = φ∗(dω) = φ∗(0) = 0. Suppose ω is exact,ω = dA, then φ∗ω = φ∗(dA) = d(φ∗A).

Proposition 1.162. Given any map φ : M → N there is a linear map φ∗ : Hp(N)→Hp(M), by

φ∗([ω]) = [φ∗ω] (143)

for all ω ∈ Zp(M). Further, if ψ : N → S, then (ψφ)∗ = φ∗ψ∗.

46

Proof. Since φ∗ preserves closed/exact forms it preserves linear combinations, and soit preserves the linear combination: ω − ω′ = dµ by φ∗ω − φ∗ω′ = d(φ∗µ). Linearityis built in, so we see it is well-defined, linear, and composition is trivial:

(ψφ)∗[ω] = [(ψφ)∗ω] = [φ∗ψ∗ω] = φ∗ψ∗[ω]

1.6.4 Gauge Freedom

Definition 1.163. Given a potential A for B, that is B = dA, we can add anydifferential to A without changing B. A → A + df is called a gauge transform andA is said to have gauge freedom. Picking and f is called choosing the gauge.

Example 1.164. Suppose we are working on M = R × S with metric dt2 − 3g. Ifthe 1-form A satisfies

A(∂t) = 0 (144)

on R× S. In this case, we say we are in temporal gauge.

Proposition 1.165. Given any exact 2-form on M = R× S, and F = dA, we canalways choose A so that it is in temporal gauge.

Proof. A = A0dt+ AS for some A0 and AS on S. Define f on M by

f(t, p) =

∫ t

0

A0(s, p) ds

and then let A′ = A− df . Then dA′ = F , and

A′(∂t) = A0(t, p)− (df(∂t))(t, p)

= A0(t, p)− (∂tf)(t, p)

= A0(t, p)− ∂t∫ t

0

A0(s, p) ds

= 0

so A′ is in temporal gauge.

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2 Gauge Fields

2.1 Symmetry

“You probably already know what a Lie group is...”

William Fulton and Joe Harris, Representation Theory, 2004

2.1.1 Lie Groups

Definition 2.1. The general linear group GL(n,F), is the set of all n× n invertiblematrices over F. The special linear group is the subset with the property that anyelement has determinant 1. The orthogonal group, O(p, q), is the set of n×n matriceswhich preserve an inner product on Rn with signature (p, q). The C-analog to O(n)is U(n).

The Lorentz group is SO(3, 1).

Definition 2.2. The Poincare group is the group of symmetries of Minkowski space.The group of all diffeomorphisms preserving spacetime intervals.

Result 2.3. Any element of the Poincare group is a product of a translation, Lorentztransform, and possibly a parity or time reversal.

Definition 2.4. Let G be a group and suppose it is also a (smooth) manifold under· : G×G→ G and −1 : G→ G. Then G is a Lie group.

Definition 2.5. If G and H are Lie groups, and H is a subgroup and submanifoldof G, then H is a Lie subgroup of G.

Definition 2.6. A Lie group homomorphism, ρ : G→ H, is a group homomorphismfrom G to H which is also a C∞-map.

Theorem 2.7. GL(n,R) and GL(n,C) are open subsets of Mn(R) and Mn(C) re-spectively. GL(n,R) and GL(n,C) are Lie groups (and thus so are their Lie sub-groups).

Proof. We will do just one example. First note that Rn2is a manifold of dimension

n2 in its own right. Now consider f : GL(n,R) ⊆ Rn2 → R by f(M) = det(M),since f is just polynomial in the entries of M it is continuous. Now consider theopen set V = R− 0, then GL(n,R) = f−1(V ), so GL(n,R) is open and thus is asubmanifold of Rn2

with usual topology.

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Definition 2.8. Given a Lie group, G, define its identity component G0 as theconnected component containing the identity.

Proposition 2.9. G0 is a subgroup of G, and a Lie group.

Proof. Really, we just need to show G0 is closed. Consider Lh : G→ G by Lhg = hg.Then Lh is continuous since multiplication on a Lie group is smooth. Now, since G0

is connected there exists a γ : [0, 1] → G0 such that γ(0) = e and γ(1) = g. SinceLh is continuous, Lhγ is a path in G0 and Lhγ(0) = h and Lhγ(1) = hg. Now letα : [0, 1] → G0 a path in G0 such that α(0) = e and α(1) = h, then concatenationα ? Lhγ is a path in G0 from e to hg. So G0 is closed. The other group propertiesare straightforward.

Proposition 2.10. Let G be a connected Lie group and U ⊆ G any neighbourhoodof e, then G = 〈U〉.

Proof. Let H = 〈U〉. For any h ∈ H, Lh(U) ⊆ H. Since Lh is a homeomorphism,Lh(U) is an open neighbourhood of h and Lh(U) is open in H. H ⊆

⋃Lh(U) so

H is open. Let g ∈ G − H, and say x ∈ Lg(U) ∩ H, then x = gu ∈ H for someu ∈ H. By closure, xu−1 = g ∈ H, a contradiction, thus Lg(U) ∩ H = ∅. HenceG − H =

⋃Lg(U) is open, so H is closed. Since H is clopen and non-empty in a

connected space, H = G.

Example 2.11. Every element of O(3) is a rotation about an axis or a rotationabout an axis plus a reflection through some plane. The former are in the identitycomponent, that is SO(3).

First, realize T ∈ O(3) is a purely real element of U(3) and so we can diagonalizeany element T with its orthogonal eigenvectors, say as

U−1TU =

λ1 0 00 λ2 00 0 λ3

and we know that each λi = eiθi . Furthermore, we know det(T ) = ±1 and that rootsof the characteristic equation come in conjugate pairs, so we may continue WLOG

∼

eiθ 0 00 e−iθ 00 0 ±1

and say that the ±1 eigenvalue corresponds to the eigenvector ~a. Then we see if it is+1, it is just an element of O(2) about ~a. If it is −1, it is a rotation about the axis~a after reflection. The last bit is obvious by continuity of the determinant.

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Result 2.12. There is no path from I ∈ SO(3, 1) to PT in SO(3, 1). We callSO0(3, 1) the connected lorentz group. More generally O(3, 1) has 4 components:

1. The identity component: SO0(3, 1); Λ00 positive and det = +1.2. The parity component: Λ00 positive and det = −1.3. The time component: Λ00 negative and det = −1.4. The time-parity component: Λ00 negative and det = +1.

So each element is a product of a proper orthochronous transform and a discretetransform I, P, T, PT.

Definition 2.13. We say a group G acts on a vector space V is there is a mapρ : G→ GL(V ) such that

ρ(gh)v = ρ(g)ρ(h)v (145)

for any v ∈ V and g, h ∈ G. We call ρ a representation of G on V .

Definition 2.14. Let ρ : G → GL(V ) and ρ′ : G → GL(V ′) be representations ofG. ρ and ρ′ are equivalent if there is a bijective linear map T : V → V ′ such thatρ′(g) T = T ρ(g) for all g ∈ G.

Example 2.15. If G is a matrix group, G ≤ GL(n,F), then it naturally defines arepresentation, called the fundamental representation, on V = Fn.

Definition 2.16. Let ρ : G → GL(V ) and ρ′ : G → GL(V ′) be representations ofG on V and V ′. Then the direct sum representation ρ⊕ ρ′ is the rep (ρ⊕ ρ′) : G→GL(V ⊕ V ′) by

(ρ⊕ ρ′)(g)(v, v′) = (ρ(g)v, ρ′(g)v′) (146)

Definition 2.17. Let V and V ′ be vector spaces with bases ei and e′j respec-tively. Then the tensor product V ⊗V ′ is the vector space with basis ei⊕e′j. Givenv = viei ∈ V and v′ = v′jej ∈ V ′, we say

v ⊗ v′ = viv′jei ⊗ e′j (147)

Proposition 2.18. For any bilinear function f : V × V ′ → W to some vector spaceW , there exists a unique linear function F : V ⊗ V ′ → W such that

f(v, v′) = (F ϕ)(v, v′) = F (v ⊗ v′) (148)

so that the following diagram commutes

V × V ′ V ⊗ V ′

W

ϕ

fF (149)

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Proof. f is bilinear so f(v, v′) = viv′jf(ei, e′j), so we will define F by F (ei ⊗ e′j) =

f(ei, e′j). The basis ei ⊗ e′j are linearly independent, so the function is unique.

Definition 2.19. Let ρ : G → GL(V ) and ρ′ : G → GL(V ′) be representations ofG on V and V ′. Then the tensor representation ρ ⊕ ρ′ is the rep (ρ ⊗ ρ′) : G →GL(V ⊗ V ′) by

(ρ⊗ ρ′)(g)(v ⊗ v′) = ρ(g)v ⊗ ρ′(g)v′ (150)

Definition 2.20. Suppose ρ : G → GL(V ) is a rep, and suppose V ≤ V ′ a vectorsubspace such that ρ(g)v′ ∈ V ′ for all v′ ∈ V ′ and g ∈ G. Then V ′ is an invariantsubspace of V . Furthermore, the emergent representation ρ′ of G on V by

ρ′(g)v′ = ρ(g)v′ (151)

for all v′ ∈ V ′, is called a subrepresentation.

Definition 2.21. ρ is irreducible if it has no non-trivial proper invariant subspaces.w

Definition 2.22. If G is compact, every rep is equivalent to a direct sum of irreps.

Theorem 2.23 (Schur’s Lemma). If ρV : G → GL(V ) and ρW : G → GL(W )are irreps of G and ϕ : V → W is a G-module homomorphism (a homomorphism-equivalency of ρV and ρW ) then

1. ϕ is an isomorphism, or ϕ = 0.2. If V = W , then ϕ = λ I for some λ ∈ C.

Proof. This is one of my favourite theorems in representation theory.1. Let g ∈ G arbitrary, and v ∈ kerϕ arbitrary, then ϕ(ρV (g)v) = ρW (g)(ϕv) = 0,

so ρV (g)v ∈ kerϕ. So kerφ is an invariant subspace of V . Similarly, imϕ isan invariant subspace of W . Thus we have kerϕ = 0 or V and imϕ = W or0 respectively, giving the two cases.

2. If V = W then since ϕ must have an eigenvalue, λ ∈ C, by algebraic closureof C, then ϕ− λ I has a non-zero kernel which is atleast 1-dimensional. Fromproving 1 we know that the kernel of the map ϕ− λ I is an invariant subspaceof V , thus ϕ− λ I = 0 and we get ϕ = λ I.

Corollary 2.24. Every irrep of an abelian group is 1-dimensional.

Proof. If G is abelian, all ρ(g) commute, so it must be that ρ(g) = λg I for all g, andso every subspace is invariant and ρ is one-dimensional.

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Definition 2.25. For any rep ρ of G on a vector space V , there is a dual or contra-gradient representation ρ∗ : G→ GL(V ∗) by

(ρ∗(g)w)(v) = w(ρ(g−1)v) (152)

for w ∈ V ∗. I.e. we have ρ∗(g) = ρ(g−1)T .

Definition 2.26. A rep is unitary if ρ(g) is unitary for all g ∈ G. A rep is projectiveif it holds up to a phase

ρ(1) = eiθ (153)

ρ(g)ρ(h) = eiθ(g,h)ρ(gh) (154)

the phase function eiθ(g,h)is called the cocycle of g and h.

Note 2.27. Cocycles satisfy the cocycle condition

eiθ(g,h)eiθ(gh,k) = eiθ(g,hk)eiθ(h,k) (155)

We can change any projective rep to an equivalent one by multiplication by aphase, in particular, we may choose ρ(1) = I. If we cannot make θ(g, h) vanish forall g, h ∈ G, then the cocycle eiθ(g,h) is called essential.

2.1.2 Lie Algebras

Definition 2.28. A Lie algebra g is an algebra equipped with a bilinear skew-symmetric map satisfying the Jacobi identity:

1. [v, w] = −[w, v]2. [u, αv + βw] = α[u, v] + β[u,w]3. [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0

Definition 2.29. A Lie algebra homomorphism is a linear map f : g→ h such that

f([v, w]) = [f(v), f(w)] (156)

When f is a bijection, it is an isomorphism.

Note 2.30. Given a Lie group (manifold) G, we defined the commutator on Vect(G)by [v, w](f) = v(w(f))− w(v(f)). This is an infinite dimensional Lie algebra

Note 2.31. The left multiplication map for any g ∈ G is a diffeomorphism. Thuswe may consider the pushforward of vector fields that it defines

(Lg)∗ : Vect(G)→ Vect(G) (157)

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Definition 2.32. A vector field v ∈ Vect(G) is left-invariant if (Lg)∗v = v for allg ∈ G.

Proposition 2.33. Let M a manifold, v, w ∈ Vect(M), and φ a diffeomorphism ofM . Then

φ∗[v, w] = [φ∗v, φ∗w] (158)

Proof. Using N = M to denote the tangent space, the pushforward of a vector fieldon M is defined pointwise by:

(φ∗v)φ(p) = φ∗(vp)

⇐⇒ (φ∗v)φ(p)(f) = [φ∗(vp)](f)

⇐⇒ [φ∗v]f(φ(p)) = v(φ∗f)(p)

⇐⇒ (φ∗v)(f) φ = v(φ∗f)

Since a vector field is determined by its points and φ : M → M is a diffeomor-phism

(φ∗[v, w])φ(p)(f) ≡ φ∗([v, w]p)(f)

= [v, w]p(φ∗f)

= vp(w(φ∗f))− wp(v(φ∗f))

= vp([phi∗w](f) φ)− wp([φ∗v](f) φ)

= vp(φ∗([φ∗w](f)))− wp(φ∗([φ∗v](f)))

= [φ∗vp]([φ∗w](f))− [φ∗wp]([φ∗v](f))

= [φ∗v]φ(p)([φ∗w](f))− [φ∗w]φ(p)([φ∗v](f))

= [φ∗v, φ∗w]φ(p)(f)

Thus φ∗[v, w] = [φ∗v, φ∗w].

Corollary 2.34. The left-invariant vector fields form a subalgebra of Vect(G).

Proof. We see immediately that Vect(G)L is a vector subspace, since (Lg)∗ is linear.Vect(G)L is a subalgebra because it inherits the bracket and Lg : G → G is adiffeomorphism so

(Lg)∗[v, w] = [Lg∗v, Lg∗w] = [v, w]

Theorem 2.35. Given any Lie group G, the subalgebra Vect(G)L ≤ Vect(G) isisomorphic to TeG. Hence we call g = TeG the Lie algebra of G since it is the onenaturally associated with G.

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Proof. Take any ve ∈ g, that is, a tangent vector at e ∈ G, then define v ∈ Vect(G)by translating this vector from the tangent space at the identity around, that is

vg = (Lg)∗ve

We see that such a v is left-invariant, because for any h ∈ G, we have:

(Lg)∗vh = (Lg)∗(Lh)∗ve = (LgLh)∗ve = (Lgh)∗ve = vgh = vLg(h)

so that (Lg)∗v = v by the definition of a pushforward of a vector field. So eachelement of TeG generates a vector field in Vect(G)L.

Conversely, consider any v ∈ Vect(G)L and consider ve ∈ g. So each element ofVect(G)L generates an element of TeG.

Corollary 2.36. There is a natural Lie bracket on g, following from the isomorphismVect(G)L ∼= g.

Result 2.37. Let G be a Lie group, g its associated Lie algebra, then there exists asmooth map

exp : g→ G (159)

satisfying:1. exp(0) is the identity of G.2. exp(sX) exp(tX) = exp((s+ t)X) for all X ∈ g and s, t ∈ R.3. d

dt(exp(tX))|t=0 = X

Thus, by the inverse function theorem, exp maps any sufficiently small open neigh-bourhood of 0 ∈ g onto an open set containing e ∈ G, and generates the whole ofG0.

Corollary 2.38. Any element of the identity component of G is the product of ele-ments of the form exp(X) for X ∈ g.

Note 2.39. For G a matrix Lie group, [·, ·] is just the matrix commutator and expis the matrix exponential.

Theorem 2.40. For any homomorphism ρ : G→ H, the map dρ = ρ∗ : TeG→ TeHis a homomorphism between the Lie algebras.

Proof. First we note that ρ∗ really is a map TeG→ TeH since it just pushes forwardvectors veG to veH . Now let v, w ∈ Vect(G)

dρ([v, w]) = ρ∗[v, w]

= [ρ∗v, ρ∗w]

= [dρ(v), dρ(w)]

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Result 2.41. The following results sum up the Lie algebra/group correspondence:1. G determines a g uniquely2. g determines a G that is simply connected uniquely, and all other Lie groups

with g as an algebra are covered by g.

g h

G H

(dρ)e

exp exp

ρ

Definition 2.42. A Lie algebra representation of a Lie algebra g on a vector spaceV is a Lie algebra homomorphism

f : g→ gl(V ) (160)

where gl(V ) is the Lie algebra of linear operators on V .

2.2 Bundles and Connections

Fields are sections of bundles, not maps F : M → V , because those maps reallyonly exist when we work locally in a chart. Connections allows us to compare vectorfields at different parts of the manifold.

2.2.1 Bundles

Definition 2.43. A bundle is a triple (E,M, π) consisting of a manifold E, called thetotal space, a manifold M , called the base space, and a projection map π : E →M .

For each p ∈M , the fiber over p is Ep = π−1(p) = q ∈ E : π(q) = p. E is thusthe union of its fibers over M , E =

⋃p∈M Ep, and we say E is a bundle over M .

M is typically a physical space/spacetime.

Definition 2.44. The tangent bundle, TM , of a manifold M is simply

TM =⋃p∈M

TpM (161)

The projection map π : TM → M takes each vp ∈ TpM to p ∈ M . Thus the fiberover p is TpM .

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Note 2.45. If M is an n-dimensional manifold it looks locally like Rn. Since eachTpM ∼= Rn, specifying a point of TM is picking a point p ∈ M and a vector field vevaluated at p. Thus locally TM ∼ Rn × Rn.

Theorem 2.46. TM is an n2-dimensional smooth manifold.

Proof. Given a manifold M of dimension n we have smooth charts ϕα : Uα ⊆ M →Rn covering M . We define Vα to be all tangent vectors to M associated to points inthe chart Uα, so Vα = vp ∈ TM : π(vp) ∈ Uα.

Note, in this picture, elements of TM are not tangent planes, those are the fibersabove the points p and q. The actual elements of TM are vectors.

Now we show the Vα cover TM . Let vp ∈ TM arbitrary, so vp ∈ TpM for somep ∈ M and thus π(vp) = p ∈ M . Since p ∈ M and Uα cover M , π(vp) ∈ Uα forsome α, so vp ∈ Vα.

Now we define the map: ψα : Vα → Rn × Rn by

ψα(vp) = (ϕα(π(vp)), (ϕα)∗vp)

The first part of the map takes the vp to p and then to its corresponding point inthe “first” Rn, and the second part of the map pushes vp forward to a vector in Rn.

Now we equip TM with the topology generated by unions of sets O, satisfying:O ⊆ Vα and ψα(O) ⊆ Rn × Rn is open.

Then we see the ψα are smooth charts (and so TM is smooth) since ϕα are smoothand ψα is just a function of ϕα and its pushforward (ϕα)∗. The transition functionsare just compositions of ψα, which are then smooth by the ϕα argument again. π issmooth since on a patch π : Rn × Rn → Rn by

1. Use ψ−1α to take x ∈ Rn × Rn back to vp.2. Take vp to p with π.3. Take p to Rn by ϕα

i.e. π = ϕα π ψ−1α .

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Definition 2.47. Given manifolds M and F , the trivial bundle over M with standardfiber F is defined by

E = M × F π(p, f) = f (162)

for all (p, f) ∈ E. In this case, the fiber of p is just a respective copy of F , i.e.Ep = p × F .

Definition 2.48. Suppose π : E →M and π′ : E ′ →M ′ are two bundles. A bundlemorphism from the first to the second is a map ψ : E → E ′ and a map φ : M →M ′

such that ψ maps each fiber Ep into the fiber E ′φ(p)

Proposition 2.49. Given bundles π : E → M and π′ : E ′ → M ′, then ψ : E → E ′

and φ : M → M ′ defines a bundle morphism if and only if π′ ψ = φ π. Thus ψuniquely determines φ and the following diagram commutes

E E ′

M M ′

ψ

π π′

φ

(163)

Proof.

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