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Home , Associated bundle, Clifford bundle, Dual bundle, Fiber bundle, Frame bundle, Metric connection

A Principal Bundles, Vector Bundles and Connections

We recall in this Appendix some of the main definitions and concepts of the theory of principal bundles and their associated vector bundles, including the theory of connections in principal and vector bundles, exterior covari- ant derivatives, etc. which we shall need in order to introduce the Clifford and -Clifford bundles and to discuss some other issues in the main text. Propositions are in general presented without proofs, which can be found, e.g.in[1,2,3,4,5,6,7,8,9].

A.1 Bundles

Definition 535 A fiber over M with Lie G will be denoted by (E,M,π,G,F). E is a topological called the total space of the bundle, π : E → M is a continuous surjective , called the canonical and F is the typical fiber. The following conditions must be satisfied: (a) π−1(x), the fiber over x, is homeomorphic to F . (b) Let {Ui,i∈ I},whereI is an index set, be a covering of M, such that: • Locally a fiber bundle E is trivial, i.e. it is diffeomorphic to a product −1 bundle, i.e. π (Ui)  Ui × F for all i ∈ I. −1 • The diffeomorphisms Φi : π (Ui) → Ui × F have the form

Φi(p)=(π(p),φi,x(p)) , (A.1) | ≡ −1 → φi π−1(x) φi,x : π (x) F is onto, (A.2)

The collection {(Ui,Φi)}, i ∈ I, are said to be a family of local trivializations for E. • The group G acts on the typical fiber. Let x ∈ Ui ∩ Uj.Then, ◦ −1 → φi,x φj,x : F F, (A.3)

must coincide with the action of an element of G for all x ∈ Ui ∩ Uj and i, j ∈ I.

W. A. Rodrigues and E. Capelas de Oliveira: Principal Bundles, Vector Bundles and Connections, Lect. Notes Phys. 722, 407–431 (2007) DOI 10.1007/978-3-540-71293-0 14 c Springer-Verlag Berlin Heidelberg 2007 408 A Principal Bundles, Vector Bundles and Connections

• We call transition functions of the bundle the continuous induced mappings (see Fig. A.1) ∩ → ◦ −1 gij : Ui Uj G,wheregij (x)=φi,x φj,x. (A.4) For consistence of the theory the transition functions must satisfy the cocycle condition gij (x)gjk(x)=gik(x). (A.5)

π−1(x)

φ i,x F

p ◦ −1 φi,x φj,x = gij (x) U j φj,x U ∩ U x i j M Ui

Fig. A.1. Transition functions on a fiber bundle

Definition 536 (P, M, π, G, F ≡ G) ≡ (P, M, π, G) is called a (PFB) if all conditions in Definition 535 are fulfilled and moreover, if there is a right action of G on elements p ∈ P , such that: (a) the mapping (defining the right action) P × G  (p, g) → pg ∈ P is continuous. (b) given g,g ∈ G and ∀p ∈ P , (pg)g = p(gg). (c) ∀x ∈ M,π−1(x) is invariant under the action of G, i.e. each element of p ∈ π−1(x) is mapped into pg ∈ π−1(x), i.e. it is mapped into an element of the same fiber. (d) G acts free and transitively on each fiber π−1(x), which means that all elements within π−1(x) are obtained by the action of all the elements of G on any given element of the fiber π−1(x). This condition is, of course necessary for the identification of the typical fiber with G.

Definition 537 Abundle(E,M,π1,G =Gl(m, F),F = V),whereF = R or C (respectively the real and complex fields), Gl(m, F) is the linear group, and V is an m-dimensional over F is called a .

Definition 538 Avectorbundle(E,M,π1,G,F) denoted E = P ×ρ F is said to be associated with a PFB bundle (P, M, π, G) by the linear representation ρ of G in F = V (a linear space of finite over an appropriate A.1 Fiber Bundles 409

field, which is called the carrier space of the representation) if its transition functions are the images under ρ of the corresponding transition functions of the PFB (P, M, π, G). This means the following: consider the following local trivializations of P and E respectively

−1 Φi : π (Ui) → Ui × G, (A.6) −1 → × Ξi : π1 (Ui) Ui V , (A.7)

Ξi(q)=(π1(q),χi(q)) = (x, χi(q)) , (A.8) −1 χ | −1 ≡ χ : π (x) → V , (A.9) i π1 (x) i,x 1 where π1 : P ×ρ V → M is the projection of the bundle associated with (P, M, π, G).Then,forallx ∈ Ui ∩ Uj, i, j ∈ I, we have ◦ −1 ◦ −1 χj,x χi,x = ρ(φj,x φi,x ) . (A.10)

In addition, the fibers π−1(x) are vector spaces isomorphic to the representa- tion space V.

Definition 539 Let (E,M,π,G,F) be a fiber bundle and U ⊂ M an . A local of the fiber bundle (E,M,π,G,F) on U is a mapping

s : U → E such that π ◦ s = IdU . (A.11)

If U = M we say that s is a global section. Remark 540 There is a relation between sections and local trivializations for principal bundles. Indeed, each local section s, (on Ui ⊂ M) for a principal −1 bundle (P, M, π, G) determines a local trivialization Φi : π (U) → U × G, of P by setting −1 Φi (x, g)=s(x)g = pg = Rgp. (A.12)

Conversely, Φi determines s since

−1 s(x)=Φi (x, e). (A.13) Proposition 541 A is trivial, if and only if, it has a global cross section. Proposition 542 A vector bundle is trivial, if and only if, its associated principal bundle is trivial

Proposition 543 Any fiber bundle (E,M,π,G,F) such that M is a para- compact and the fiber F is a vector space admits a cross section Remark 544 Then, any vector bundle associated with a trivial principal bun- dle has non zero global sections. Note however that a vector bundle may admit a non zero global section even if it is not trivial. Indeed, any Clifford bundle 410 A Principal Bundles, Vector Bundles and Connections possess a global identity section, and some spin-Clifford bundles admits also identity sections once a trivialization is given (see Chap. 6).

Definition 545 The structure group G of a fiber bundle (E,M,π,G,F) is reducible to G if the bundle admits an equivalent structure defined with a subgroup G of the structure group G. More precisely, this means that the fiber bundle admits a family of local trivializations such that the transition   functions takes values in G , i.e. gij : Ui ∩ Uj → G .

A.1.1

The TM to a differentiable n-dimensional manifold M is an % to a principal bundle called the frame bundle F (M)= ∈ x∈M FxM,whereFxM is the set of frames at x M. The structure group of F (M)isGl(n, R). Let {xi} be the coordinates associated with a local chart { ∂  } (Ui,ϕi) of the maximal of M. Then, TxM has a natural ∂xi x on Ui ⊂ M. Σ { | | } Definition 546 AframeatTxM is a set x = e1 x , ..., en x of linearly independent vectors such that   | j ∂  ei x = Fi j  , (A.14) ∂x x j j ∈ R and where the (Fi ) with entries Ai , belongs to the the real general R j ∈ R linear group in n Gl(n, ).Wewrite(Fi ) Gl(n, ). −1 A local trivialization φi : π (Ui) → Ui × Gl(n, R)ofF (M) is defined by

φi(f)=(x, Σx), π(f)=x. (A.15) j ∈ R ∈ The action of a =(ai ) Gl(n, )onaframef F (U)isgivenby (f,a) → fa,wherethenewframefa ∈ F (U) is defined by φi(fa)=(x, Σ x), π(f)=x,and Σ {  |  | } x = e1 x , ..., en x ,  | | j ei x = ej x ai . (A.16) Σ Σ j ∈ R Conversely, given frames x and x there exists a =(ai ) Gl(n, )such that (A.16) is satisfied, which means that Gl(n, R)actsonF (M) actively. Let {xi} and {x¯i}be the coordinates associated with the local charts   ∈ ∩ (Ui,ϕi)and(Ui ,ϕi) and of the maximal atlas of M.Ifx Ui Uj we have     | j ∂  ¯j ∂  ei x = Fi j  = Fi j  , ∂x x ∂x¯ x j ¯j ∈ R (Fi ), (Fi ) Gl(n, ) . (A.17) A.1 Fiber Bundles 411 , - k  j ¯j ∂x  Since Fi = Fk ∂x¯i we have that the transition functions are x   k  k ∂x  ∈ R gi (x)= i  Gl(n, ) . (A.18) ∂x¯ x Remark 547 Given U ⊂ M we shall also denote by Σ ∈ sec F (U) asection of F (U) ⊂ F (M). This means that given a local trivialization φ : π−1(U) → U × Gl(n, R), φ(Σ)=(x, Σx), π(Σ)=x. Sometimes, we also use the sloppy notation {ei}∈sec F (U) or even {ei}∈sec F (M) when the context is clear. Moreover, we recall that a section of F (U) is also called a for H(U), the module of differentiable vector fields on U.

A.1.2 Orthonormal Frame Bundle

∈ 0 Suppose that the manifold M is equipped with a metric field g sec T2 M of signature (p, q), p+q = n. Then, we can introduce orthonormal frames in each Σ { | | } TxU, . In this case we denote an orthonormal frame by x = e1 x , ..., en x and   j ∂  ei| = h  , (A.19) x i ∂xj  x g e | e |  − − ( i x , j x) x =diag(1, 1, ...1, 1, ..., 1), (A.20) j ∈ with (hi ) Op,q, the real in n dimensions. In this case we say that the frame bundle has been reduced to the orthonormal frame bundle, which will be denoted by POn(M). A section Σ ∈ secPOn(U) is called a vierbein.

Remark 548 The principal bundle of oriented orthonormal frames P e (M) SO1,3 over a Lorentzian manifold modeling spacetime and its covering bundle called spin bundle P e (M) discussed in Chap. 6 play an important role in this Spin1,3 book. Also, vector bundles associated these bundles are very important. Asso- ciated with P e (M) we have the bundle, the exterior bundle and the SO1,3 Clifford bundle. Associated with P e (M) we have several bundles, Spin1,3 in particular the spin-Clifford bundle, whose sections are the DHSF. All those bundles and their relationship are studied in Chap. 6. Remark 549 In complete analogy to the construction of orthonormal frame bundle we may define an orthonormal coframe bundle that may be denoted by POn(M). Since to each given frame Σ ∈ sec POn(M) there is a natural coframe field Σ ∈ sec POn(M), the one where the covectors are the duals of the vectors of the frame. It follows that POn(M)  POn(M).Inparticular P e (M)  P e (M). SO1,3 SO1,3 412 A Principal Bundles, Vector Bundles and Connections A.2 Product Bundles and Whitney Sum

 Given two vector bundles (E,M, π,G,V)and(E,M,π,G, V )wehavethe definitions: Definition 550 The product bundle E×E is a fiber bundle whose basis space is M × M , the typical fiber is V ⊕ V, the structural group of E × Eacts separately as G and G in each one of the components of V ⊕ V and the × projection π × πis such that E × E π→π M × M . Definition 551 Given two vector bundles over the same basis space, i.e. (E,M,π,G,V) and (E,M,π,G, V), the Whitney sum bundle E ⊕ E is the of E × E by h : M → M × M, h(p)=(p, p).

Definition 552 Given vector bundles (E,M,π,G,V) and (E,M,π,G, V) over the same basis space, the bundle E ⊗ E is the bundle  −1 ⊗ −1 obtained from E and E by assigning the tensor product of fibers πx πx for all x ∈ M. Remark 553 With the above definitions we can easily show that given three vector bundles, say, E,E,E we have

E ⊕ (E ⊗ E)=(E ⊗ E) ⊕ (E ⊗ E) . (A.21)

A.3 Connections

A.3.1 Equivalent Definitions of a in Principal Bundles

To define the concept of a connection on a PFB (P, M, π, G), we recall that since dim(M)=m,ifdim(G)=n,thendim(P )=n + m. Obviously, for all x ∈ M, π−1(x)isann-dimensional submanifold of P diffeomorphic to the structure group G and π is a , π−1(x) is a closed submanifold of P for all x ∈ M. −1 The TpP , p ∈ π (x), is an (n + m)-dimensional vector −1 space and the tangent space VpP ≡ Tp(π (x)) to the fiber over x at the −1 same point p ∈ π (x)isann-dimensional linear subspace of TpP called the 1 vertical subspace of TpP .

1 Here we may be tempted to realize that as it is possible to construct the vertical space for all p ∈ P then we can define a horizontal space as the complement of this space in respect to TpP . Unfortunately this is not so, because we need a smoothly association of a horizontal space in every point. This is possible only by means of a connection. A.3 Connections 413

Now, roughly speaking a connection on P is a rule that makes possible a correspondence between any two fibers along a curve σ : R ⊇ I → M,t → σ(t). If p0 belongs to the fiber over the point σ(t0) ∈ σ,wesaythatp0 is parallel translated along σ by means of this correspondence. Definition 554 A horizontal lift of σ is a curve σˆ : R ⊇ I → P (described by the of p). It is intuitive that such a transport takes place in P along directions speci- fied by vectors in TpP , which do not lie within the vertical space VpP .Since the tangent vectors to the paths of the basic manifold passing through a given x ∈ M span the entire tangent space TxM, the corresponding vectors Y p ∈ TpP (in whose direction parallel transport can generally take place in P ) span a n-dimensional linear subspace of TpP called the horizontal space of TpP and denoted by HpP .Now,themathematicalconcept of a connection can be presented. This is done through three equivalent definitions given below which encode rigorously the intuitive discussion given above. We have the following definitions: Definition 555 A connection on a PFB (P, M, π, G) is an assignment to each p ∈ P of a subspace HpP ⊂ TpP , called the horizontal subspace for that connection, such that HpP depends smoothly on p and the following conditions hold: (i) π∗ : HpP → TxM , x = π(p), is an . (ii) HpP depends smoothly on p. (iii) (Rg)∗HpP = HpgP, ∀g ∈ G, ∀p ∈ P .

Here we denote by π∗ the differential of the mapping π and by (Rg)∗ the differential of the mapping Rg : P → P (the right action) defined by Rg(p)=pg. Since x = π(ˆσ(t)) for any curve in P such thatσ ˆ(t) ∈ π−1(x)andˆσ(0) = p0, we conclude that π∗ maps all vertical vectors in the zero vector in TxM, 2 i.e. π∗(VpP ) = 0 and we have ,

TpP = HpP ⊕ VpP. (A.22)

Then every Yp ∈ TpP can be written as h v h ∈ v ∈ Yp = Yp + Yp , Yp HpP, Yp VpP. (A.23)

Therefore, given a vector field Y over M it is possible to lift it to a horizontal h ∈ ∈ vector field over P , i.e. π∗(Yp)=π∗(Yp )=Yx TxM for all p P with h π(p)=x. In this case, we call Yp the horizontal lift of Yx. We say moreover that Y is a horizontal vector field over P if Yh =Y .

2 We also write TP = HP ⊕ VP. 414 A Principal Bundles, Vector Bundles and Connections

Definition 556 A connection on a PFB (P, M, π, G) is a mapping Γp : TxM → TpP , such that ∀p ∈ P and x = π(p) the following conditions hold: (i) Γp is linear. ◦ (ii) π∗ Γp = IdTxM . (iii) the mapping p → Γp is differentiable. ∈ (iv) ΓRg p =(Rg)∗Γp, for all g G. We need also the concept of parallel transport. It is given by the following definition:

Definition 557 Let σ : R ⊃ I → M, t → σ(t) with x0 = σ(0) ∈ M,bea curve in M and let p0 ∈ P such that π(p0)=x0. The parallel transport of p0 along σ is given by the curve σˆ : R ⊃ I → P, t → σˆ(t) defined by d d σˆ(t)=Γ ( σ(t)) , (A.24) dt p dt with p0 =ˆσ(0) and σˆ(t)=pt, π(pt)=x.

In order to present yet a third definition of a connection we need to know more about the nature of the vertical space VpP .Forthis,letY ∈ TeG = G be an element of the G of G. The vector Y is the tangent to the curve produced by the exponential map   d  Y = (exp(tY)) . (A.25) dt t=0

Then, for every p ∈ P we can attach to each Y ∈ TeG = G a unique element v ∈ − → → Yp VpP as follows: let f :( ε, ε) P , t p exp tY be a curve on P . Observe that it is obtained by right translation and then π(p)=π(p exp tY)= x and so the curve lies in π−1(x) the fiber over x ∈ M. Next let F : P → R be a smooth function. Then we define   v d  Yp F(p)= F (p exp(tY)) . (A.26) dt t=0

By this construction we attach to each Y∈TeG = G a unique vector field over P , called the fundamental field corresponding to this element. We then have the canonical isomorphism v ←→ v ∈ ∈ Yp Y, Yp VpP, Y TeG = G , (A.27) from which we get VpP  G . (A.28) Definition 558 AconnectiononaPFB (P, M, π, G) is a 1-form field ω on P with values in the Lie algebra G = TeG such that ∀p ∈ P we have, ω v v ←→ v ∈ ∈ (i) p(Yp )=Y and Yp Y,whereYp VpP and Y TeG = G. (ii) ωp depends smoothly on p. −1 (iii)ωp[(Rg)∗Yp]=(Adg−1 ωp)(Yp),whereAdωp = g ωpg. A.3 Connections 415

i ∗ It follows that if {Ga} is a basis of G and {θ } is a basis for T P then

ω a ⊗G a i ⊗G p = ωp a = ωi (p)θp a , (A.29) where ωa are 1-forms on P . Then the horizontal spaces can be defined by

HpP =ker(ωp) , (A.30) which shows the equivalence between the definitions.

A.3.2 The Connection on the Base Manifold

Definition 559 Let U ⊂ M and

−1 s : U → π (U) ⊂ P, π ◦ s = IdU , (A.31) be a local section of the PFB (P, M, π, G). Definition 560 Let ω be a connection on P . The 1-form s∗ω (the pullback of ω under s) given by ∗ω ω ∈ ∈ (s )x(Yx)= s(x)(s∗Yx),Yx TxU, s∗Yx TpP, p = s(x) , (A.32) is called the local gauge potential. It is quite clear that s∗ω ∈ sec T ∗U ⊗G. This object differs from the gauge field used by physicists by numerical constants (with units). Conversely we have the following proposition: Proposition 561 Given ω ∈ sec T ∗U ⊗ G and a differentiable section of π−1(U) ⊂ P , U ⊂ M, there exists one and only one connection ω on π−1(U) such that s∗ω = ω.

Consider now ω ∈ T ∗U ⊗ G,ω=(Φ−1(x, e))∗ω = s∗ω,s(x)=Φ−1(x, e) , (A.33) ω ∈ T ∗U  ⊗ G,ω =(Φ−1(x, e))∗ω = s∗ω,s(x)=Φ−1(x, e). Then we can write, for each p ∈ P (π(p)=x), parameterized by the local trivializations Φ and Φ respectively as (x, g)and(x, g)withx ∈ U ∩ U ,that

−1 −1 −1  −1   ωp = g dg + g ωxg = g dg + g ωxg . (A.34) Now, if g = hg , (A.35) we immediately get from (A.34) that

 −1 −1 ω¯x = hdh + hω¯xh , (A.36) which can be called the transformation law for the gauge fields. 416 A Principal Bundles, Vector Bundles and Connections A.4 Exterior Covariant Derivatives

k k Let (P, G)= T ∗P ⊗ G, 0 ≤ k ≤ n, be the set of all k-form fields over P with values in the Lie algebra G of the gauge group G (and, of course, the 1 connection ω ∈ sec (P, G)).

k Definition 562 For each ϕ ∈ sec (P, G) we define the so-called horizon- k tal form ϕh ∈ sec (P, G) by

ϕh ϕ h h h p (X1, X2, ..., Xk)= (X1 , X2 , ..., Xk ) , (A.37) where Xi ∈ TpP , i =1, 2, .., k. ϕh ∈ Notice that p (X1, X2, ..., Xk) = 0 if one (or more) of the Xi TpP are vertical. k Definition 563 ϕ ∈ sec T ∗P ⊗ V (where V is a vector space) is said ϕ to be horizontal if p(X1, X2, ..., Xk)=0, implies that at least one of the Xi ∈ TpP , i =1, 2,...,k is vertical.

k Definition 564 ϕ ∈ sec T ∗P ⊗ V is said to be of type (ρ, V) if ∀g ∈ G we have ∗ϕ −1 ϕ Rg = ρ(g ) . (A.38) k Definition 565 Let ϕ ∈ sec T ∗P ⊗ V be horizontal. Then, ϕ is said to be tensorial of type (ρ, V).

k Definition 566 The exterior of ϕ ∈ sec (P, G) in relation to the connection ω is k+1 Dωϕ =(dϕ)h ∈ sec (P, G) , (A.39)

ωϕ ϕ h h h h where D p(X1, X2, ..., Xk, Xk+1)=d p(X1 , X2 , ..., Xk , Xk+1). Notice that k a a ∗ dϕ = dϕ ⊗Ga where ϕ ∈ sec (T P ), a =1, 2, ..., n.

i j Definition 567 The commutator of ϕ ∈ sec (P, G) and ψ ∈ sec (P, G), i+j 0 ≤ i, j ≤ n, denoted by [ϕ, ψ] ∈ sec (P, G) such that if X1, ..., Xi+j ∈ sec TP,then

[ϕ, ψ](X1, ..., Xi+j ) (A.40) 1 = (−1)σ[ϕ(X , ..., X ), ψ(X , ..., X )], i!j! ι(1) ι(i) ι(i+1) ι(i+j) σ∈Sn A.4 Exterior Covariant Derivatives 417

σ where Sn is the permutation group of n elements and (−1) = ±1 is the sign of the permutation. The brackets [ , ] in the second member of (A.40) are the Lie brackets in G. Writing

i j a a a ∗ a ∗ ϕ = ϕ ⊗Ga, ψ = ψ ⊗Ga, ϕ ∈ sec T P, ψ ∈ sec T P, (A.41) we have3 a b [ϕ, ψ]=ϕ ∧ ψ ⊗ [Ga, Gb] (A.42) c ϕa ∧ ψb ⊗G = fab( ) c, c where fab are the structure constants of the Lie algebra. With (A.42) we can prove easily the following important properties in- volving commutators:

[ϕ, ψ]=(−1)1+ij [ψ, ϕ] , (A.43)

(−1)ik[[ϕ, ψ], τ ]+(−1)ji[[ψ, τ], ϕ]+(−1)kj[[τ, ϕ], ψ]=0, (A.44) d[ϕ, ψ]=[dϕ, ψ]+(−1)i[ϕ,dψ], (A.45) i j k for ϕ ∈ sec (P, G), ψ ∈ sec (P, G), τ ∈ sec (P, G). We shall also need the following identity

[ω, ω](X1, X2)=2[ω(X1), ω(X2)] . (A.46)

The proof of (A.46) is as follows: (i) Recall that a b [ω,ω]=(ω ∧ ω ) ⊗ [Ga, Gb] . (A.47)

(ii) Let X1, X2 ∈ sec TP (i.e. X1 and X2 are vector fields on P ). Then,

a b a b [ω, ω](X1, X2)=(ω (X1)ω (X2) − ω (X2)ω (X1))[Ga, Gb] (A.48) =2[ω(X1), ω(X2)] .

1 Definition 568 The form of the connection ω ∈ sec (P, G) is 2 Ωω ∈ sec (P, G) defined by

Ωω = Dωω. (A.49)

Definition 569 The connection ω is said to be flat if Ωω =0.

3 In this Appendix in order to obtain formulas that can be easily compared with the ones appearing in standard texts we use the exterior product as defined in Remark 22. This, we hope, generates no confusion. 418 A Principal Bundles, Vector Bundles and Connections

Proposition 570

ωω ω ω ω D (X1, X2)=d (X1, X2)+[ (X1), (X2)] . (A.50)

a (A.50) can be written using (A.48) (and recalling that ω(X)=ω (X)Ga) as

1 Ωω = Dωω = dω + [ω, ω] . (A.51) 2

Proof. See [1]. "# Proposition 571 (Bianchi identities):

DΩω =0. (A.52)

Proof. (i) Let us calculate dΩω.Wehave,   1 dΩω = d dω + [ω, ω] . (A.53) 2

We now take into account that d2ω = 0 and that from the properties of the commutators given by (A.43), (A.44), (A.45) above, we have

d[ω, ω)] = [dω, ω] − [ω,dω], [dω, ω]=−[ω,dω] , [[ω, ω], ω]=0. (A.54)

By using (A.54) in (A.53) gives

dΩω =[dω, ω] . (A.55)

(ii) In (A.55) use (A.51) and the last equation in (A.54) to obtain

dΩω =[Ωω, ω] . (A.56)

(iii) Use now the definition of the exterior covariant derivative [(A.39)] h together with the fact that ω(X )=0,forallX ∈ TpP to obtain

DωΩω =0,

that proves the proposition. "# We can then write the very important formula (known as the Bianchi identity), DωΩω = dΩω +[ω, Ωω]=0. (A.57) A.4 Exterior Covariant Derivatives 419

A.4.1 Local curvature in the Base Manifold M

Let (U, Φ) be a local trivialization of π−1(U)ands the associated cross section as defined above. Then, s∗Ωω := Ωω (the pullback of Ωω) is a well defined 2-form field on U which takes values in the Lie algebra G. Let ω = s∗ω (see (A.33)). If we recall that the differential operator d commutes with the pullback, we immediately get 1 Ωω = s∗Dωω = dω + [ω,ω] . (A.58) 2 It is convenient to define the symbols

Dω := s∗Dωω , (A.59) DΩω := s∗DωΩω, (A.60) and to write DΩω =0, (A.61) DΩω = dΩω +[ω,Ωω]=0. Equation (A.61) is also known as Bianchi identity.

Remark 572 In gauge theories (Yang-Mills theories ) Ωω is (except for nu- merical factors with physical units) called a field strength in the gauge Φ.

Remark 573 When G is a matrix group, as is the case in the presentation of gauge theories by physicists, Definition 567 of the commutator [ϕ, ψ] ∈ i+j i j sec (P, G) (ϕ ∈ sec (P, G), ψ ∈ sec (P, G)) gives

[ϕ, ψ]=ϕ ∧ ψ − (−1)ij ψ ∧ ϕ , (A.62) where ϕ and ψ are considered as matrices of forms with values in R and ϕ ∧ ψ stands for the usual matrix multiplication, with entries multiplied by the exterior product. Then, when G is a matrix group, we can write (A.51) and (A.58) as

Ωω = Dωω = dω + ω ∧ ω, (A.63) Ωω := Dω = dω + ω∧ω. (A.64)

A.4.2 Transformation of the Strengths Under a Change of Gauge

Consider two local trivializations (U, Φ)and(U ,Φ)ofP such that p ∈ π−1(U ∩U )has(x, g)and(x, g)asimagesin(U ∩U )×G,wherex ∈ U ∩U . Let s, s be the associated cross sections to Φ and Φ respectively. By writing 420 A Principal Bundles, Vector Bundles and Connections s∗Ωω = Ωω, we have the following relation for the local curvature in the two different gauges such that g = hg

Ωω = hΩωh−1, ∀x ∈ U ∩ U  . (A.65)

We now give the coordinate expressions for the potential and field strengths μ ⊂ ∂ in the trivialization Φ.Let x be a local chart for U M and let ∂μ = ∂xμ and {dxμ}, μ =0, 1, 2, 3, be (dual) bases of TU and T ∗U respectively. Then,

a ⊗G a μ ⊗G ω = ω a = ωμdx a , (A.66) 1 Ωω =(Ωω)a ⊗G = Ωa dxμ ∧ dxν ⊗G . (A.67) a 2 μν a a a ⊃ → R C where ωμ, Ωμν : M U (or )andweget

a a − a a b c Ωμν = ∂μων ∂ν ωμ + fbcωμων . (A.68)

The following objects appear frequently in the presentation of gauge the- ories by physicists, 1 1 (Ωω)a = Ωa dxμ ∧ dxν = dωa + f a ωb ∧ ωc , (A.69) 2 μν 2 bc ω a G − Ωμν = Ωμν a = ∂μων ∂ν ωμ +[ωμ,ων ] , (A.70) aG ωμ = ωμ a . (A.71)

We now give the local expression of Bianchi identities. Using (A.61) and (A.69) we have

ω 1 ω DΩ := (DΩ ) dxρ ∧ dxμ ∧ dxν =0. (A.72) 2 ρμν By putting ω ω (DΩ )ρμν := DρΩμν , (A.73) we have ω ω ω DρΩμν = ∂ρΩμν +[ωρ,Ωμν ], (A.74) and ω ω ω DρΩμν + DμΩνρ + Dν Ωρμ =0. (A.75) Physicists call the operator

Dρ := ∂ρ +[ωρ, ] . (A.76)

the covariant derivative. A.4 Exterior Covariant Derivatives 421

A.4.3 Induced Connections

Let (P1,M1,π1,G1)and(P2,M2,π2,G2) be two principal bundles and let F : P1 → P2 be a bundle homomorphism, i.e. F is fiber preserving, it induces a diffeomorphism f : M1 → M2 and there exists a homomorphism λ : G1 → G2 such that for g1 ∈ G1, p1 ∈ P1 we have F F (p1g1)=Rλ(g1) (p1) . (A.77)

Proposition 574 Let F : P1 → P2 be a bundle homomorphism. Then a connection ω1 on P1 determines a unique connection on P2.  Remark 575 Let (P, M, π , Op,q)=POp,q(M) be the orthonormal frame bundle, which is as explained above reduction of the frame bundle F (M).Then, a connection on POp,q(M) determines a unique connection on F (M).This is a very important result that has been used implicitly in Sect. 4.7.8 and the solution of Exercise 310. Proposition 576 Let F (M) be the frame bundle of a paracompact manifold M.Then,F (M) can be reduced to a principal bundle with structure group Op,q, and to each reduction there corresponds a Riemannian metric field on M. → e Remark 577 If M has dimension 4, and we substitute Op,q SO1,3 then with each reduction of F (M) there corresponds a Lorentzian metric field on M.

A.4.4 Linear Connections on a Manifold M

Definition 578 A linear connection on a smooth manifold M is a connection ω ∈ sec T ∗F (M) ⊗ gl(n, R). Remark 579 Given a Riemannian (Lorentzian) manifold (M,g) a connec- tion on F (M) which is determined by a connection on the orthonormal frame bundle P (M)(P e (M)) is called a . After introducing Op,q SO1,3 the concept of covariant derivatives on vector bundles, we can show that the covariant derivative of the with respect to a metric connection is null. | → Rn Consider the mapping f p : Tx(M) (with p =(x, Σx)inagiven trivialization) which sends v ∈ Tx(M) into its components relative to the { | | } { j } { | } frame Σx = e1 x , ..., en x .Let θ x be the dual basis of ei x .Wewrite  | j  f p (v)=(θ x (v)) . (A.78) 422 A Principal Bundles, Vector Bundles and Connections

Definition 580 The canonical soldering form of M is the 1-form θ ∈ ∗ n sec T F (M) ⊗ R such that for any v ∈ sec TpF (M) such that v =π∗v we have θ θa| ( (v)): = p (v)ea a| = θ x (v)ea, (A.79)

n a ∗ where {ea} is the canonical basis of R and {θ } is a basis of T F (M),with θa ∗ a, θa| a| = π θ p (v)=θ x (v). ω ∈ ∗ ⊗ Definition 581 The torsion form of a linear connection sec T F (M) gl(n, R) is the 2-form Dθ = Θ ∈ sec 2 T ∗F (M) ⊗ Rn . As it is easy to verify, the soldering form θ and the torsion 2-form Θ are tensorial of type (ρ, Rn), where ρ(u)=u, u ∈ Gl(n, R). ω Using the same techniques employed in the calculation of D ω(X1, X2) (A.50) it can be shown that

Θ = dθ +[ω,θ], (A.80) where [ , ] is the commutator product in the Lie algebra of the affine group A(n, R)=Gl(n, R)  Rn,where means the semi-direct product. Suppose b R R that (Ea, ec) is the canonical basis of a(n, ), the Lie algebra of A(n, ). Recalling that ω ωa b (v)= b (v)Ea , (A.81) a θ(v)=θ (v)ea , (A.82) we can show without difficulties that

DωΘ =[Ω,θ]. (A.83)

A.4.5 Torsion and Curvature on M

Let {xi} be the coordinate functions associated with a local chart (U, ϕ)of ∈ j ∂ θ θa the maximal atlas of M.LetΣ sec F (U)withei = Fi ∂xj and = ea. Take π∗v = v.Then θ | | j | j k −1 ( p(v)) = f p (v)=f p (dx (v)∂j)=f p (dx (v)(Fj ) ek) k −1 j =((Fj ) dx (π∗v)) . (A.84) With this result it is quite obvious that given any w ∈ Rn, θ determines a horizontal field vw ∈ secT F (M)by

(θ(vw(p))) = w . (A.85) A.5 Covariant Derivatives on Vector Bundles 423

With these preliminaries we have the following proposition: Proposition 582 There is a bijective correspondence between sections of ∗ ∗ ⊗ r ⊗ Rnq T M Ts M and sections of T F (M) , the space of tensorial forms Rnq r of the type (ρ, ) in F (M),withρ and q being determined by Ts M. Using the above proposition and recalling that the soldering form is ten- Rn sorial of type (ρ(u), ), ρ(u)=u, weseethatitdetermineson M a vector 4 a 1 ∗ valued differential 1-form θ = ea ⊗ θ ∈ sec TM⊗ T M. Also, the torsion Θ Rn is tensorial of type (ρ(u), ), ρ(u)=u and thus define a vector valued a 2 ∗ 2-form on M, Θ = ea ⊗ Θ ∈ sec TM ⊗ T M. We can show from (A.80) that given u, w ∈ TpF (M), a a a b − a b Θ (π∗u, π∗w)=dθ (π∗u, π∗w)+ωb (π∗u)θ (π∗w) ωb (π∗w)θ (π∗u) . (A.86) On the basis manifold this equation is often written:

a Θ = Dθ = ea ⊗ (Dθ ) ⊗ a a ∧ b = ea (dθ + ωb θ ) , (A.87) wherewerecognizeDθa as the exterior covariant derivative of index forms introduced in Sect. 3.3.4. Ωω Rn2 Also, the curvature is tensorial of type (Ad, ). It then defines Ω = ⊗ b ⊗Ra ∈ 1 ⊗ 2 ∗ ea θ b sec T1 M T M which we easily find to be given by ⊗ b ⊗Ra Ω = ea θ b ⊗ b ⊗ a a ∧ c = ea θ (dωb + ωc ωb ) , (A.88) Ra ∈ 2 ∗ where the b sec T M are the curvature 2-forms introduced in Chap. 2, explicitly given by Ra a a ∧ c b = dωb + ωc ωb . (A.89) a Ra a Note that sometimes the symbol Dωb such that b := Dωb is introduced in some texts. Of course, the symbol D cannot be interpreted in this case as the exterior covariant derivative of index forms5. This is expected since 1 ω ∈ sec T ∗P ⊗ gl(n, R)isnot tensorial.

A.5 Covariant Derivatives on Vector Bundles

6 Consider a vector bundle (E,M,π1,G,V) associated with a PFB bundle (P, M, π, G) by the linear representation ρ of G in the vector space V over the field F = R or C. Also, let dimF V = m. Consider again the trivializations of P and E given by (A.7)–(A.9). Then, we have the definition:

4 θ is clearly the identity operator in the space of vector fields. 5 See Sect. A.3. 6 Also denoted E = P ×ρ V. 424 A Principal Bundles, Vector Bundles and Connections

Definition 583 The parallel transport of Ψ0 ∈ E, π1(Ψ0)=x0,alongthe curve σ : R  I → M, t → σ(t) from x0 = σ(0) ∈ M to x = σ(t) is the element Ψt ∈ E such that: (i) π1(Ψt)=x, ◦ −1 (ii) χi(Ψt)=ρ(ϕi(pt) ϕi (p0))χi(Ψ0). (iii) pt ∈ P is the parallel transport of p0 ∈ P along σ from x0 to x as defined in (A.24) above.

Definition 584 Let Y be a vector at x0 tangent to the curve σ (as defined above). The covariant derivative of Ψ ∈ sec E in the direction of Y is denoted E ∈ (DY Ψ)x0 sec E and

E ≡ E 1 0 − (DY Ψ)(x0) (DY Ψ)x0 = lim (Ψt Ψ0) , (A.90) t→0 t 0 ≡ ∈ where Ψt is the “vector” Ψt Ψ(σ(t)) of a section Ψ sec E parallel transported along σ from σ(t) to x0, the only requirement on σ being   d  σ(t) = Y. (A.91) dt t=0

In the local trivialization (Ui,Ξi)ofE (see (A.7)–(A.9)) if Ψt is the element in V representing Ψt,wehave

0 −1 χi(Ψt)=ρ(g0gt )χi|σ(t)(Ψt) . (A.92)

By choosing p0 such that g0 = e we can compute (A.90):   E d −1  (D Ψ)x0 = ρ(g (t)Ψt) Y dt   t=0     dΨt   dg(t) =  − ρ (e)  (Ψ0) . (A.93) dt t=0 dt t=0 This formula is trivially generalized for the covariant derivative in the direction of an arbitrary vector field Y ∈ sec TM. With the aid of (A.93) we can calculate, e.g. the covariant derivative of ∈ ∂ ≡ Ψ sec E in the direction of the vector field Y = ∂xμ ∂μ.Thiscovariant derivative is denoted DE Ψ. ∂μ   dg(t)  d We need now to calculate dt . In order to do that, recall that if dt is t=0   d a tangent to the curve σ in M,thens∗ dt is a tangent toσ ˆ the horizontal d ∈ ⊂ −1 lift of σ, i.e. s∗ dt HP TP. As defined before s = Φi (x, e)isthecross section associated with the trivialization Φi of P (see (A.6). Then, as g is a mapping U → G we can write ' ( d d s∗( ) (g)= (g ◦ σ) . (A.94) dt dt A.5 Covariant Derivatives on Vector Bundles 425

To simplify the notation, introduce local coordinates {xμ,g} in π−1(U)and write σ(t)=(xμ(t)) andσ ˆ(t)=(xμ(t),g(t)). Then,   d μ ∂ ∂ s∗ =˙x (t) +˙g(t) , (A.95) dt ∂xμ ∂g in the local coordinate basis of T (π−1(U)). An expression like the second member of (A.95) defines in general a vector tangent to P but, according to d its definition, s∗ dt is in fact horizontal. We must then impose that     d μ ∂ ∂ μ ∂ a ∂ s∗ =˙x (t) +˙g(t) = α + ω G g , (A.96) dt ∂xμ ∂g ∂xμ μ a ∂g for some αμ. ∂ aG ∂ We used the fact that ∂xμ + ωμ ag ∂g is a basis for HP, as can easily be verified from the condition that ω(Y h)=0,forallY ∈ HP. We immediately get that αμ =˙xμ(t) , (A.97) and

dg(t) − μ aG =˙g(t)= x˙ (t)ωμ ag, (A.98) dt  dg(t) − μ aG  = x˙ (0)ωμ a . (A.99) dt t=0 With this result we can rewrite (A.93) as     E dΨt  −  dσ  (DY Ψ)x0 =  ρ (e)ω(Y )(Ψ0),Y=  . (A.100) dt t=0 dt t=0 which generalizes trivially for the covariant derivative along a vector field Y ∈ sec TM.

E Remark 585 Many texts introduce the covariant derivative operator DY act- ing on sections of the vector bundle E as follows. Definition 586 A connection DE on M is a mapping DE :secTM × sec E → sec E, → E (X, Ψ) DXΨ. (A.101) E → such that DX :secE sec E satisfies the following properties: E E (i) DX(aΨ)=aDXΨ, E E E (ii) DX(Ψ + Φ)=DXΨ+DX Φ , E E (iii) DX(fΨ)=X(f)+fDXΨ , (A.102) E E E (iv) DX+YΨ = DXΨ+DYΨ , E E (v) DfXΨ = fDXΨ. 426 A Principal Bundles, Vector Bundles and Connections

∀ X, Y ∈ sec TM, Ψ, Φ ∈ sec E, ∀a ∈ F = R or C (the field of scalars entering the definition of the vector space V of E) ∀f ∈ C∞(M),whereC∞(M) is the set of smooth functions with values in F. Of course, all properties in (A.102) follows directly from (A.100). However, the point of view encoded in Definition 586 may be appealing to physicists. To see, first recall that E = P ×ρ V. Recall that 0 stands for the representation of G in the vector space V. ∗ ∗ Definition 587 The of E is the bundle E = P ×ρ∗ V ,where V∗ is the of V and 0∗ is the representation of G in the vector space V∗. Example 588 As examples we have the tangent bundle which is TM = n F (M) ×ρ R where ρ :Gl(n, R) → Gl(n, R) denotes the standard representa- ∗ n ∗ ∗ tion and T M = F (M) ×ρ∗ (R ) where the dual representation ρ satisfies ρ∗ (g)=ρ(g−1)t.Also,thetensorbundleoftensorsoftype(r, s),thebundle of homogeneous k-vectors and the bundle of homogeneous k-forms are: r r r n T M = TM = F (M) × r ( R ) , s s s s k k n TM = F (M) × k R , ρ k k ∗ n T M = F (M) × k R , (A.103) ρ∗ r k k where s, ρ and ρ∗ are the induced tensor product and exterior powers representations. Definition 589 The bundle E ⊗ E∗ is called the bundle of endomorphisms of E and will be denoted by End E. ∗ Definition 590 A connection DE acting on E∗ is defined by E∗ ∗ ∗ − ∗ E (DX Ξ )(Ψ)=X (Ξ (Ψ)) Ξ (DXΨ) , (A.104) for ∀Ξ∗ ∈ sec E∗, ∀Ψ ∈ sec E and ∀X ∈ sec TM. ∗ Definition 591 A connection DE⊗E acting on sections of E ⊗E∗ is defined for ∀Ξ∗ ∈ sec E∗, ∀Ψ ∈ sec E and ∀X ∈ sec TM by ∗ E⊗E ∗ ⊗ E∗ ∗ ⊗ ∗ ⊗ E DX Ξ Ψ =DX Ξ Ψ + Ξ DXΨ. (A.105) ∗ We shall abbreviate DE⊗E by DEndE. Eq(A.105) may be generalized in an obvious way in order to define a connection on arbitrary tensor products of bundles E ⊗ E ⊗ ...E.... Finally, we recall for completeness that given two bundles, say E and E and given connections DE and DE there is an obvious connection DE⊕E defined in the Whitney bundle E ⊕ E (recall Definition 551). It is given by

E⊕E ⊕  E ⊕ E  DX (Ψ Ψ )=DXΨ DX Ψ , (A.106) for ∀Ψ ∈ sec E, ∀Ψ ∈ sec E and ∀X ∈ sec TM. A.5 Covariant Derivatives on Vector Bundles 427

A.5.1 Connections on E Over a Lorentzian Manifold In what follows we suppose that (M,g) is a Lorentzian manifold (Definition 229). We recall that the manifold M in a Lorentzian structure is supposed ∗ r paracompact. Then, according to Proposition 543 the bundles E,E , Ts M and End E admit global cross sections. We then write for the covariant derivative of Ψ ∈ sec E and X ∈ sec TM, E 0E W DXΨ =DX Ψ+ (X)Ψ, (A.107) where W∈sec EndE ⊗ T ∗M will be called connection 1-form (or potential ) E 0E for DX and DX is a well defined connection on E, that we are going to determine. Consider then a open set U ⊂ M and a trivialization of E in U. Such a trivialization is said to be a choice of a gauge. { } | ∈ | −1 Let ei be the canonical basis of V.LetΨ U sec E U = π (U). Consider the trivialization Ξ : π−1(U) → U × V, Ξ(Ψ)=(π(Ψ),χ(Ψ)) = (x, χ(Ψ)). In this trivialization we write | Ψ U := (x,Ψ(x)) , (A.108) ∈ ∀ ∈ → { }∈ | Ψ(x) V, x U,withΨ : U V a smooth function. Let si sec E U , −1 | { }∈ si = χ (ei) i =1, 2, ..., m be a basis of sections of E U and eμ sec F (U),μ =0, 1, 2, 3abasisforTU.Letalso{θν }, θν ∈ sec T ∗U,bethe { } { ∗i}∈ ∗| ∗| dual basis of eμ and s sec E U , be a basis of sections of E U dual to the basis {si}. We define the connection coefficients in the chosen gauge by E Wj Deμ si = μisj . (A.109) i μ Then, if Ψ = Ψ si and X = X eμ E μ E i D Ψ = X D (Ψ si) X eμ μ i Wi j = X eμ(Ψ )+ μj Ψ si . (A.110) μWi j Now, let us concentrate on the term X μj Ψ si. It is, of course a new  μWi j | μWi j section := (x, X μj Ψ si)ofE U and X μj Ψ si is linear in both X and Ψ. WU ∈ | ⊗ ∗ This observation shows that sec(End E U T U), such that in the trivialization introduced above is given by WU Wi ⊗ ∗i ⊗ μ = μj sj s θ , (A.111) is the representative of W in the chosen gauge. ∈ ∈ | Note that if X sec TU and Ψ := (x,Ψ(x)) sec E U we have U U μWi ⊗ ∗i ω (X):=ωX = X μj sj s , U μWi i ωX(Ψ)=X μj Ψ sj . (A.112) We can then write E U DXΨ = X(Ψ)+ωX(Ψ) , (A.113) 428 A Principal Bundles, Vector Bundles and Connections

0E 0E thereby identifying DX Ψ = X(Ψ). In this case DX is called the standard flat connection. Now, we can state a very important result which has been used in Chap. 2 to write the different decompositions of Riemann-Cartan connections. Proposition 592 Let D0E and DE be arbitrary connections on E then there exists W∈¯ sec EndE ⊗ T ∗M such that for any Ψ ∈ sec E and X ∈ sec TM, E 0E W¯ DXΨ = DX Ψ+ (X)Ψ. (A.114)

A.5.2 Gauge Covariant Connections Definition 593 A connection DE on E is said to be a G-connection if for any u ∈ G and any Ψ ∈ sec E there exists a connection DE on E such that for any X ∈ sec TM E E DX (ρ(u)Ψ)=ρ(u)DXΨ. (d11) E 0E W¯ ∈ ∈ Proposition 594 If DXΨ =DX Ψ+ (X)Ψ for Ψ sec E and X sec TM, E 0E W¯  then DX Ψ =DX Ψ+ (X)Ψ with W¯ (X)=uW¯ (X)u−1 + udu−1 . (A.115) Suppose that the vector bundle E has the same structural group as the orthonormal frame bundle P e (M), which as we know is a reduction of the SO1,3 frame bundle F (M). In this case we give the definition: Definition 595 A connection DE on E is said to be a generalized G- connection if for any u ∈ G and any Ψ ∈ sec E there exists a connection E 4 D on E such that for any X ∈ sec TM, TM = P e (M) × TM R SO1,3 ρ E E DX (ρ(u)Ψ)=ρ(u)DXΨ, (A.116) where X = ρTMX ∈ sec TM.

A.5.3 Curvature Again E Definition 596 Let D be a G-connection on E. The curvature operator RE∈ sec 2 T ∗M ⊗ EndE of DE is the mapping RE:secTM ⊗ TM ⊗ E → E, (A.117) E E E − E E − E R (X, Y)Ψ = DXDYΨ DYDXΨ D[X,Y]Ψ E E E − E E − E R (X, Y)=DXDY DYDX D[X,Y], (d14) for any Ψ ∈ sec E and X, Y ∈ sec TM.

If X = ∂μ, Y = ∂ν ∈ sec TU are coordinate basis vectors associated with the coordinate functions {xμ} we have RE RE E E (∂μ,∂ν ):= μν = D∂μ ,D∂ν . (A.118) A.5 Covariant Derivatives on Vector Bundles 429

∗j In a local basis {si ⊗ s } of EndE we have under the local trivialization used above E a ⊗ ∗b Rμν = Rμνbsa s , a Wa − Wa Wa Wc −Wa Wc Rμνb= ∂μ νb ∂ν μb + μc νb νc μb . (A.119) (A.119) can also be written E W − W W W Rμν = ∂μ ν ∂ν μ +[ μ, ν ] . (A.120)

A.5.4 Exterior Covariant Derivative Again r ∗ s ∗ Definition 597 Consider Ψ⊗Ar ∈ sec E ⊗ T M and Bs ∈ T M .We define (Ψ⊗Ar) ⊗∧ Bs by

(Ψ⊗Ar) ⊗∧ Bs = Ψ⊗(Ar ∧ Bs) . (A.121) ⊗ ∈ ⊗ r ∗ ⊗ ∈ ⊗ Definition 598 Let Ψ Ar sec E T M and Π Bs sec EndE s ∗ T M. We define (Π⊗Bs) ⊗∧ (Ψ⊗Ar) by

(Π⊗Bs) ⊗∧ (Ψ⊗Ar)=Π(Ψ)⊗(Bs ∧ Ar) . (A.122)

Definition 599 Given a connection DE acting on E, the exterior covariant E derivative dD acting on sections of E ⊗ r T ∗M and the exterior covariant End E derivative dD acting on sections of EndE ⊗ s T ∗M (r, s =0, 1, 2, 3, 4) is given by: (i) if Ψ ∈ sec E then for any X ∈ sec TM

DE E d Ψ(X)=DXΨ, (A.123)

r ∗ (ii) For any Ψ⊗Ar ∈ sec E ⊗ T M

DE DE d (Ψ⊗Ar)=d Ψ⊗∧Ar + Ψ⊗dAr , (A.124) s ∗ (iii) For any Π⊗Bs ∈ sec End E ⊗ T M

DEnd E DEnd E d (Π⊗Bs)=d Π⊗∧Bs + Π⊗dBs . (A.125)

s ∗ Proposition 600 ConsiderthebundleproductE=(EndE ⊗ T M) ⊗∧ r ∗ s ∗ (E ⊗ T M).LetΠ = Π⊗Bs ∈ sec End E ⊗ T M and Ψ = Ψ⊗Ar E ∈ sec E ⊗ r T ∗M . Then the exterior covariant derivative dD acting on sections of E satisfies

DE DEndE s DE d (Π ⊗∧ Ψ)=(d Π) ⊗∧ Ψ +(− 1) Π⊗∧d Ψ. (A.126) 430 A Principal Bundles, Vector Bundles and Connections

Exercise 601 The reader can now show several interesting results, which make contact with results obtained earlier when we analyzed the connections and on principal bundles and which allowed us sometimes the use of sloppy notations in the main text: (i) Suppose that the bundle admits a flat connection D0E. We put 0E dD = d.Then,if χ ∈ sec E ⊗ r T ∗M we have

D0E d χ = dχ+W⊗∧χ. (ii) If χ ∈ sec E ⊗ r T ∗M we have

DE 2 E (d ) χ = R ⊗∧χ. (A.127) (iii) If χ ∈ sec E ⊗ r T ∗M we have

DE 3 E DE (d ) χ = R ⊗∧d χ. (A.128)

(iii) Suppose that the bundle admits a flat connection D0E. We put D0E d = d.Then,if (iv) Π ∈ sec End E ⊗ s T ∗M we have

End E dD Π = dΠ +[W, Π] . (A.129)

(v) End E dD RE =0. (A.130) (vi) E R = dW + W⊗∧W . (A.131)

End E Remark 602 Note that RE = dD W. We end here this long Appendix, hopping that the material presented be enough to permit our reader to follow the more difficult parts of the text and in particular to see the reason for our use of many eventual sloppy notations.

References

1. Choquet-Bruhat, Y., DeWitt-Morette, C. and Dillard-Bleick, M., Analysis, and Physics (revisited edition), North Holland Publ. Co., Amster- dam, 1982. 407, 418 2. Coquereaux, R., and Jadcyzk A., Riemannian , Fiber Bundles, Kaluza-Klein Theories and all That..., World Sci. Publ., Singapore, 1988. 407 3. Frankel, T., The Geometry of Physics, Cambridge University Press, Cambridge, 1997. 407 4. Kobayashi , S., and Nomizu, K., Foundations of Differential Geometry,vol.1, Interscience Publishers, New York, 1963. 407 References 431

5. Naber, G. L., , Geometry and Gauge Fields. Interactions, Appl. Math. Sci. 141, Springer-Verlag, New York, 2000. 407 6. Nicolescu, L. I., Notes on Seiberg-Witten Theory, Graduate Studies in Mathe- matics 28, Am. Math. Soc., Providence, Rohde Island, 2000. 407 7. Nash, C. and Sen, S., Topology and Geometry for Physicists, Academic Press, London, 1983. 407 8. Osborn, H., Vector Bund les, vol. I, Acad. Press, New York, 1982. 407 9. Palais, R.S., The Geometrization of Physics, Lecture Notes from a Course at the National Tsing Hua University, Hsinchu, Taiwan, 1981. 407 Acronyms and Abbreviations

ADM Arnowitt-Deser-Misner CCSP Classical Charged Spinning Particles CDS Covariant DEC for a DHSF DHE Dirac-Hestenes Equation DHS Dirac-Hestenes Spinor DHSF Dirac-Hestenes EP Equivalence Principle ELE Euler-Lagrange Equation EMFS Even Multiform Fields FECD Fake Exterior Covariant Differential GR GRT General Relativity Theory IMTγ Inertial Moving Tetrad Along γ IRF Inertial Reference Frame LIASF Left Algebraic Spinor Field LLC Local Lorentz Coordinates LLCC Local Lorentz Coordinates Chart LLF Local Lorentz Frame LLRFγ Local Lorentz Reference Frame Associated with γ MDE Maxwell-Dirac Equivalence ME Maxwell Equations MTW Misner-Thorne-Wheeler NLDHE Non-Linear Dirac-Hestenes Equation PWS Plane Wave Solution PFB Principal PLLI Principle of Local Lorentz Invariance PIRF Pseudo Inertial Reference Frame RCST Riemann-Cartan Spacetime 434 Acronyms and Abbreviations

RCWS Riemann-Cartan-Weyl Space RIASF Right Ideal Algebraic Spinor Field SRT Theory SAP Stationary Action Principle i.e. Id Est e.g. Exempli Gratia List of Symbols

V VectorSpace ...... 19 ∗ V Dual Space of V ...... 19 ⊕ DirectSum ...... 19 dim Dimension ...... 19 N SetofNaturalNumbers ...... 19 R RealField ...... 19 Tk(V) Space of k-Cotensors ...... 19 ∞ SeeDefinition3 ...... 19 k=0 T (V∗) SpaceofMulticotensors ...... 19 k k-PartOperator ...... 20 ⊗ TensorProduct ...... 20 r Ts (V) Space of r-Contravariant, s-Covariant . . . . . 21 T (V) of V ...... 21 T(V) SpaceofMultitensors ...... 21 ∧ MainInvolution ...... 21 ˜ Reversion ...... 21 − Conjugation ...... 22 g Metric Tensor of V ...... 22 g Metric Tensor of V∗ ...... 22 · Scalar Produc Induced by g ...... 22 g · ScalarProduct ...... 22 {ek} Basis of V ...... 22 k {ε } Dual Basis of {ek} ...... 22 k {e } Reciprocal Basis {ek} ...... 22 k {εk} Reciprocal Basis of {ε } ...... 23 ∗ V ExteriorAlgebra ...... 23 ∧ ExteriorProduct ...... 23 A AntisymmetrizationOperator ...... 24

i1...ik Permutation Symbol of Order k ...... 25 436 List of Symbols

˚g Fiducial Metric Tensor of V ...... 25 ∗ ˚g Fiducial Metric Tensor of V ...... 25 ∗ (V ,˚g) MetricVectorSpace ...... 25  HodgeOperator ...... 25 Hodge Operator Associated with  ...... 26  ˚ ˚g g  LeftContraction ...... 27 RightContration ...... 27  Left Contraction Associated with ˚g ...... 27 ˚g Right Contraction Associated with ˚g ...... 27 ˚g ∗ ( V ,˚g) GrassmannAlgebra ...... 27 ∗ C(V ,˚g) Clifford Algebra of (V∗,˚g) ...... 28 AB Clifford Product of A and B ...... 28 A → B A is Embedded in B and A ⊆ B ...... 31 ∗ ext-(V ) SpaceofExtensors ...... 33 ext( p V∗, q V∗) Space of (p, q) Extensors ...... 33 † AdjointOperator ...... 34 ExteriorPowerExtensionOperator ...... 35 − tr TraceOperator ...... 36 det DeterminantOperatorofa Extensor ...... 36 bif(t) Biform of t ...... 37 g Endomorphism Associated with g ...... 39 • ScalarProduct ...... 40 S[ab] Skew-Symmetric Part of Sab ...... 44 R1,3 MinkowskiVectorSpace ...... 46 ||v|| Norm of v ...... 47 ↑ TimeOrientation ...... 47 L LorentzTransformation ...... 48 O1,3 LorentzGroup ...... 48 SO1,3 ProperLorentzGroup ...... 48 e ≡L↑ SO1,3 + ProperOrthochronousLorentzGroup ...... 48 P Poincar´eTransformation ...... 49  Semi-DirectProduct ...... 49 P Poincar´eGroup ...... 49 ∧ ·  Any one of the Products , , , or Clifford Product ...... 52 ∂Y F (Y ) One of the Derivative Operators Acting on F (Y )..53 C ComplexField ...... 61 H √ QuaternionSkewField ...... 61 i= −1 ImaginaryUnity ...... 61 A Associativealgebra ...... 62 D DivisionAlgebra ...... 64 D(m) m × n MatrixAlgebra ...... 64 Rp,q Real Vector Space with Metric of Signature (p, q)..65 List of Symbols 437

p,q Rp,q Clifford Algebra of R ...... 65 R0 R p,q Even Subalgebra of p,q ...... 65 R1 R p,q Set of Odd Elements of p,q ...... 65 R0,2 QuaternionAlgebra ...... 66 R3,0 PauliAlgebra ...... 66 R1,3 SpacetimeAlgebra ...... 66 R3,1 MajoranaAlgebra ...... 66 R4,1 DiracAlgebra ...... 66 epq Idempotent of Rp,q ...... 67 Γp,q Clifford-LipschitzGroup ...... 69 0 Γp,q SpecialClifford-LipschitzGroup ...... 69 R The Set R −{0} ...... 69 Pinp,q PinorGroup ...... 70 Spinp,q SpinGroup ...... 70 e Spinp,q SpecialSpinGroup ...... 70 p,q Op,q Pseudo Orthogonal Group of R ...... 70 p,q SOp,q Special Proper Pseudo Orthogonal Group of R .. 70 e Special Orthochronous Pseudo Orthogonal SOp,q Group of SOp,q ...... 71 γ Standard Dirac γ Matrices ...... 74 μ

φΞu Representative of Dirac-HestenesSpinor ...... 80 β TakabayasiAngle ...... 82 B Boomerang ...... 84 xj(x), xμ(x) CoordinateFunctions ...... 96 TM Tangent Bundle of M ...... 99 T ∗M of M ...... 99 r Ts M Bundle of r-covariant, s-covariantTensors ...... 99 T M Tensor Bundle ...... 99 φ∗f Pullback of Function f : N → R ...... 100 φ∗ Derivative Mapping of Diffeomorphism φ ...... 100 £v LieDerivative ...... 105 {ej} Basis for Sections of TU ⊂ TM ...... 106 i ∗ ∗ {θ } Basis for Section of T U ⊂ T M ...... 106 T ∗M Cartan (Exterior) Bundle ...... 107 d ExteriorDerivative ...... 107 ivα Interior Product of Vector and Form Fields ...... 108 Hr(M) r-de Rham Group ...... 109 Hr(M) r HomologyGroup ...... 114 g Metric of TM ...... 117 (M,g) RiemannianorLorentzianManifold ...... 117 τg MetricVolumeElement ...... 117 M =(M,g,τg) OrientedMetricManifold ...... 117 ∗ g Metric of T M ...... 117 (M) Hodge Bundle ...... 118 438 List of Symbols

δ,δ Hodge Codifferential Associated with g ...... 118 g , Associated with g ...... 118 g ♦ HodgeLaplacian ...... 118 g Θ TorsionTensor ...... 119 R Curvature(Riemann)Tensor ...... 119 [eα, eβ] CommutatorofVectorFields ...... 120 D ExteriorCovariantDerivative ...... 121 ρ ωβ Connection1-forms ...... 120 Θρ Torsion2-forms ...... 120 Rρ μ Curvature2-forms ...... 120 ˚g Fiducial Metric for TM ...... 124 ∇ GeneralCovariantDerivative ...... 124 D˚ Levi-Civita Covariant Derivative of ˚g ...... 124 D Levi-Civita Covariant Derivative of g ...... 124 ↑ Time of Lorentzian Manifod ...... 125 M LorentzianSpacetime ...... 125 M MinkowskiSpacetime ...... 125 C(M,g) Clifford Bundle of (M,g) ...... 126 T M Covariant Tensor Bundle ...... 126 {ei} for Sections of TU ⊂ TM ..... 127 {θa} Orthonormal Cobasis for Sections of T ∗U ⊂ T ∗M 127 g Metric for T ∗M ...... 127 ˚g Metric for T ∗M ...... 127 ∂| Dirac Operator Associated with D˚ and C(M,˚g) . . 129 [[ α, β]] StandardDiracCommutator ...... 131 {α, β} StandardDiracAnticommutator ...... 131 ∂ Dirac Operator Associated with ∇ and C(M,g) . . 136 ∂ See (4.173) ...... 136 ∂∧ See (4.173) ...... 136 ∂∨ See (4.188) ...... 139 [[[ α, β]]] DiracCommutator ...... 139 {α, β} DiracAnticommutator ...... 139 Ricci RicciTensor ...... 142 Covariant D’Alembertian Associated  = ∂| · ∂| with D˚ and C(M,˚g) ...... 144 ∂| ∧ ∂| Ricci Operator Associated with D˚ and C(M,˚g) . . 144 R˚μ Ricci 1-forms Associated with D˚ and C(M,˚g) . . . . 146  EinsteinOperator ...... 148 G˚μ Einstein 1-forms Associated with D˚ and C(M,˚g) . 148 ∂ · ∂ See (4.244) ...... 149 ∂ ∧ ∂ See (4.244) ...... 149 T a = −T a Energy-Momentum1-forms ...... 161 ∇+ See (4.320) ...... 164 List of Symbols 439

∇− See (4.321) ...... 165 (nacs|Q) Naturally Adapted Coordinate Chart with Q ..... 175 { a} Moving Orthonormal Frame Over σ ...... 176 {a} Moving Orthonormal Coframe Over σ ...... 176 F Fermi-WalkerConnection ...... 175 P e (M) Principal Bundle of Oriented Lorentz Tetrads . . . . 233 SO1,3 P e (M) Principal Bundle of Oriented Lorentz Cotetrads . . 233 SO1,3 P e (M) Spin Frame Bundle ...... 235 Spin1,3 P e (M) Spin Coframe Bundle ...... 236 Spin1,3 S(M,g) Real (Left) ...... 237 S(M,g) Dual Real Spinor Bundle ...... 238 Sc(M,g) Complex Spinor Bundle ...... 238 Sc (M,g) Dual Complex Spinor Bundle ...... 238 l C e (M,g) Left Real Spin-Clifford Bundle ...... 238 Spin1,3 r C e (M,g) Right Real Spin-Clifford Bundle ...... 239 Spin1,3 r 1i (x) See(6.33) ...... 245 l 1Ξ (x) See(6.34) ...... 245 dv PfaffDerivative ...... 248 ∇s V SpinorCovariantDerivative ...... 251 ∂s Spin-DiracOperator ...... 255 ∇(s) ∇s C V Representative of V in (M,g) ...... 258 ∂(s) Representative of ∂s in C(M,g) ...... 258 ∂x See(7.14) ...... 272 L(x, Y, ∂x Y ) LagrangianDensity ...... 273 L(x, Y, ∂x Y ) Lagrangian ...... 273 δv VerticalVariation ...... 274 δh HorizontalVariation ...... 274 δ TotalVariation ...... 274 JX Canonical Extensor ...... 286 † LX OrbitalAngularMomentumExtensor ...... 287 † SX SpinExtensor ...... 287 1 ∗ ∗ J ( T M) 1- Bundle over T M →C(M,g) ...... 293 J 1[( T ∗M)n+2] See(8.2) ...... 294 Lm LagrangianDensityMapping ...... 294 Σ(φ) Euler-LagrangeFunctional ...... 295 tc,Sc See(8.68) ...... 306 LEH, Lg See(8.91) ...... 313 dE Exterior Covariant Differential Operator ...... 346 D FakeExteriorCovariantDifferential ...... 351 Rμν CurvatureBivector ...... 355 Ta EnergyMomentum1-vectorFields ...... 359 τL DegreeofLinearPolarization ...... 368 τC DegreeofCircularPolarization ...... 368 440 List of Symbols

Π HertzPotential ...... 381 S StrattonPotential ...... 381 (E,M,π,G,F) Fiber Bundle ...... 407 F (M) Frame Bundle ...... 410 E ⊕ E Whitney Sum of Bundles E and E ...... 412 E DY Covariant Derivative on Vector Bundle E ...... 424 E dD See (A.123) ...... 429 End E dD See (A.125) ...... 429 Index

Action Principle Cartan Bundle, 107 Stationary, 275 Cartan’s Magical Formula, 108 Adjoint Operator Change of Gauge, 419 Properties, 34 Charge Conjugation, 85 Algebra Charge Conservation, 157 Associative, 29 Chart, 96 Clifford, 27, 28 Classical Determinant, 42 Universality, 29 Classical Spinning Particles, 178, 393 Exterior, 23 Classification of , 124 Grassmann, 23 Clifford Algebra, 377 Tensor, 19 Clifford Bundle, 126, 159, 233, 257, 398, Algebraic Dirac , 78 402 Algebraic Spin Frames, 75 Clifford Fields, 235, 248 Amorphous Spinor Fields, 259 Clifford Group, 68 Angular Momentum Clifford-Lipschitz Group, 69 Extensor Cliforms, 343 Canonical, 285 Closed Forms, 109, 116 Anticommutator, 132 Coframe, 233 Antisymmetrization, 24 Cohomology Groups, 109 Associated Dirac Operators, 134 Commutator, 132, 417 Associative Algebras, 61 Commutator Product, 71 Atlas, 96 Complex Clifford Algebras, 64 Complex Spinor Bundle, 238 Connections, 412, 427 Base Manifold, 419 Conservation Laws Berezin-Marinov Model, 404 Riemann-Cartan Spacetimes Bianchi Identities, 123, 418 Lorentz Spacetimes, 293 Bilateral Ideal, 62 Conservation laws, 336 Bilinear Covariants, 83, 260 Contractions Bradyons, 173 Properties of, 27 Bundle of Algebras, 241 Coordinate Functions, 96 Bundle of Modules, 241 Cotangent Space, 98 Bundle Structure, 234 Cotensors, 19 442 Index

Covariant ‘Conservation’ Laws, 297 Exact Forms, 109 Covariant Derivative, 248, 250, 423 Extensor Covariant Dirac Fields, 254 Belinfante Covariant Dirac Spinor, 80 Energy-momentum, 284 Curvature, 161, 422 Exterior Power Extension, 35 D’Alembertian Operator, 144, 146 Spin Darboux Biform, 178 Canonical, 285 de Rham Periods, 107, 116 Free Electromagnetic Field, 289 Derivative, 100 Spin Density Diffeomorphism Dirac-Hestenes Spinor Field, 288 Induced Connections, 123 Extensor Diffeomorphism Invariance, 193, 198 Calculus, 19 Differential Geometry, 126 Extensor Field, 271 Dilation, 43, 139 Extensors, 33 Dirac Algebra, 66 Characteristic Biform, 36, 37 Dirac Anticommutator, 131 Characteristic Scalars, 36 Dirac Equation, 254, 363 Determinant, 36 Dirac Operator, 129, 136, 256 Generalization Operator, 38 Dirac-Hestenes Equation, 256, 321, 363, Exterior Covariant Derivative, 416, 429 395 Exterior Covariant Differential, 345 Dirac-Hestenes Lagrangian, 324 Exterior Power Operator, 35 Dirac-Hestenes Spinor, 80 Properties, 35 Canonical Form, 82 External Synchronization, 208 Dirac-Hestenes Spinor Field, 242 External Synchronization Processes, Dirac-Like Equation, 371 207 Direct Sum, 19 Dotted Algebraic Spinors, 86 Fake Exterior Covariant Differential, Einstein Equations, 160 343 Einstein Operator, 144, 148 Fermi Transport, 175, 178 Einstein Synchronization, 184 Fiber Bundle, 407, 410 Einstein-Hilbert Lagrangian, 332 Fiducial Frame, 76 Electromagnetic Gauge, 261 Fiducial Sections, 243 Electromagnetic Potential, 197 Field Equations, 160 Endomorphisms, 43 Field Strengths, 419 Energy-momentum Fierz Identities, 83 1-forms, 298 Flux Conservation, 158 Energy-Momentum Conservation, 307 Forms, 108 Energy-Momentum Extensor Integrations of, 110 Canonical, 279 Frame Bundle, 410 Dirac-Hestenes Field Frenet Frames, 177 Canonical, 284 Friedmann Universe, 218, 222 Electromagnetic Field Friedmann-Robertson-Walker Universe, Canonical, 282 183 Energy-Momentum Tensor, 160 Frobenius Theorem, 186 Equivalence Principle, 216 Function Euler-Lagrange Equations, 275, 294, Extensor Valued, 271 401 Functional Derivatives, 293 Index 443

Gauge Covariant Connections, 428 Lagrangian Gauge Invariance, 259 Density, 272 Gauge Metric Extensor, 40 Dirac-Hestenes, 278 General Dirac Operator, 149 Mapping, 273 General Relativity, 198 Maxwell, 277 Teleparallel Equivalent, 338 Varied, 275 General Relativity Theory, 213 Lagrangian Formalism, 269 Generalized Lichnerowicz Formula, 262 Left Spin-Clifford Bundle, 238 Generalized Maxwell Equation, 379 Levi-Civita Connection, 139, 329 Global Inner Product, 119 Lie Algebra, 71 Gravitational Theory, 327 Lie Bracket, 131, 417 Graviton, 336 Lie Covariant Derivative, 303 Lie Derivatives, 104 Linear Connections, 421 Hamilton-Jacobi Equation, 396 Local Curvature, 419 Hertz Potential, 382 Local Lorentz Coordinates, 214 Hertz Potential Equation, 380 Local Lorentz Invariance, 213, 218 Hestenes Ideal Spinors, 72 Logunov’s Objection, 199 Hodge Bundle, 118, 156 Lorentz Hodge Codifferential, 118 Diffeomorphism, 275 , 48 Ideals, 92 Non Standard Realization, 209 Idempotent, 62 Proper Orthochronous, 48, Induced Connections, 421 72, 75 Inequality Lorentz Invariance, 324 Anti-Minkowski , 48 Lorentz Tetrads, 233 Anti-Schwarz, 47 Lorentz Transformation, 378 Integral Curves, 99 Lorentzian Structure, 117 Interior Product, 108 Lorenz Gauge, 198 Internal Synchronization Processes, 207 Luxons, 173 Invariance Action Integral Majorana Algebra, 66 Diffeomorphism, 296 Majorana Spinor, 85 Rotational, 285 Manifold Involution Differential, 95 Conjugate, 22 Hausdorff, 125 Main, 21 Lorentzian, 236, 427 Reversion, 21 Smooth, 96 Maxwell Equation, 159, 363 Maxwell Equations, 156, 194, 369 Jacobi Identity, 32, 106 Maxwell like Equations, 358 Maxwell-Dirac Equivalence, 375, 383, Generalized, 272 387 Jet Bundles, 293 Minimal Lateral Ideals, 67 Jones Representation, 368 Minimal Left Ideal, 89 Minimal Right Ideal, 89 Killing Coefficients, 133 Minkowski Spacetime, 393, 402 Killing Vector Mother Spinor, 72 Timelike, 310 Multicotensors, 19 444 Index

Multiform Pullback, 100, 419 Calculus, 19 Multiform Functions Quantization of Action, 158 Continuity, 49 Quaternion Algebra, 88 Derivatives, 50, 53 Radon-Hurwitz Numbers, 67 Differentiability, 51 Real Clifford Algebras, 64 Directional Derivatives, 51 Reference Frames, 171, 175, 179, 186, Limit, 49 204 Real Variable, 49 Relativistic Spacetimes, 171 Several Multiform Variables, 50 , 61 Multiform Lagrangian, 398 Ricci Operator, 144, 147 Ricci Tensor, 142 Nature of Spinor Fields, 258 Riemann-Cartan Spacetime, 321 Nunes Connection, 161 Riemannian Structure, 117 Right Spin-Clifford Bundle, 238, 239 Operator Rotation, 175, 178 Adjoint, 34 Hodge Star, 25 Sachs Representation, 372 k-part, 20 Sagnac Effect, 187 Orthonormal Coframe Bundle, 411 Sallh¨ofer Representation, 373 Orthonormal Frame Bundle, 411 Schwarzschild Solution, 200 Seiberg-Witten Equations, 363, 388, Parallel Transport, 414, 424 389 Pauli Algebra, 66, 87, 89 Shear, 43, 139 Pauli-Lubanski Spin 1-Form, 178 Sl(2,C) , 357 Periodicity, 65 Spacetime Algebra, 66, 86 Pfaff Derivative of Form Fields, 248 Spacetime Theory, 192 Pfaff Derivatives, 322 Spacetimes, 124 Pinor Group, 68 Spin Bundles, 237 Poincar´e Diffeomorphisms, 197 Spin Frame, 243 Poincar´e Group, 48 Spin Source, 289 Polarization, 366 , 235 Position Covector, 270 Spin-Dirac Operator, 255 Poynting Vector, 367 Spinor Primitive Idempotents Reconstruction of a, 83 Algorithm for, 68 Spinor Equation, 393 Principal Bundles, 412 Spinor Fields, 237, 250, 393 Principal Fiber Bundle, 423 Spinor Group, 68 Principle of Relativity, 205 Spinor Representation, 72 Product Spinors, 75, 89 Bundles, 412 Standard Clock Postulate, 173 Clifford, 28 Standard Dirac Commutator, 131 Properties, 29 Standard Dirac Operator, 129 Commutator, 32 Stokes Parameters, 366 Left Contraction, 27 Stokes Theorem, 107, 115 Right Contraction, 27 Strain, 43, 139 Scalar, 22 Stratton Potential, 381 Multicovectors, 25 Structure Equations, 119, 143 Tensor, 20 Superfields, 402 Index 445

Superparticle, 397, 402 Undotted Algebraic Spinors, 86 Superpotentials, 306 Symmetric Automorphisms, 39 Variation Symmetry Groups, 203 Horizontal, 274 Synchronizability, 184 Total, 274 Synchronizable Reference Frame, 186 Vertical, 273 Takabayasi Angle, 82, 385 Vector Bundles, 423 Tangent Space, 98 Vector Fields, 99, 108 Tangent Vector, 98 Vector Space, 19 Tangent Vectors, 97 Dual, 19 Tensor Minkowski, 43, 45 Contravariant, 21 Volume Element, 26 Covariant, 21 Tensor Bundles, 99 Wedderburn Theorem, 64 , 106 Weitzenb¨ock Spacetime, 125 Tetrad Postulate?, 164 Weyl Spinor, 85 Torsion, 139, 422 Whitney Sum, 412