Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

Подождите немного. Документ загружается.

316 13 Some Problems on Lorentz Manifolds

see that that the coeﬃcients of the metric play the role of the potential for

the ﬁeld strength.

13.2 A Two-Point Boundary Value Problem on a

Lorentz Manifold Arising in A. Poltorak’s Concept

of Reference Frame

13.2.1 Discussion of the problem

In [196, 197] A. Poltorak suggested a concept in which a reference frame in

general relativity is deﬁned as a certain smooth manifold with a connection.

In the most simple cases this is Minkowski space (see Example 13.4) with

its natural ﬂat connection but in more complicated cases some more general

manifolds and connections may also appear.

In the reference frame, the gravitational ﬁeld is described as a (1, 2)-tensor

G (see Theorem 2.36) that on any pair of vector ﬁelds X and Y takes the

value

G(X, Y )=∇

Y −

∇

where

∇ is the covariant derivative of the Levi-Civit´a connection of the

Lorentz metric while ∇ is the covariant derivative of the connection in the

reference frame. Denote by

dτ

the covariant derivative of the connection in

the reference frame along a given world line with respect to some parameter

τ. Then the geodesic m(τ) of the Levi-Civit´a connection in M (a world line in

the absence of all force ﬁelds except gravitation) is described in the reference

frame by the equation

dτ



(τ)=G

m(τ)



(τ),m



(τ)) (13.9)

where G

(X, Y ) is the value of G(X, Y )atpointm (cf. equation (12.1)).

Notice that the right-hand side of (13.9) is quadratic in the velocity m



(τ).

We refer the reader to [196, 197] for more details on Poltorak’s concept

and for a physical interpretation of the covariant derivative of a connection

in the reference frame, of the tensor G and several other objects associated

with it. The subsequent development of this idea can be found in [198].

We suggest a version of the concept where the manifold of the reference

frame is the tangent space T

at an event m ∈ M

and where a Lorentz-

orthonormal basis e

,whereα =0, 1, 2, 3, is speciﬁed (the time-like vector e

is the observer’s 4-velocity). We assume that this reference frame is valid in a

neighborhood O of the origin in T

, which is identiﬁed with a neighbor-

hood U of the event m by the exponential map of the Levi-Civit´a connection

of the Lorentz metric (the normal chart).

13.2 A two-point boundary value problem on Lorentz manifold 317

We deal with two choices of connection on the manifold T

. In the ﬁrst

one, we consider on T

its natural ﬂat connection of Minkowski space

(the main case considered by A. Poltorak). In the second case, we involve a

Riemannian connection of a certain (positive deﬁnite) Riemannian metric on

. This case is motivated by a natural development of the idea yielding

Euclidean models in quantum ﬁeld theory. Observe that the above-mentioned

Riemannian connection may not be the Levi-Civit´a connection, and that a

non-zero torsion connection compatible with the metric is also allowed. In

principle, this also allows one to consider electromagnetic interactions.

For the two reference frames mentioned above, we investigate the question

of whether it is possible to connect two events m

and m

in M

by a time-

like geodesic if they are connected in the reference frame by a geodesic of the

corresponding connection whose initial vector is time-like, i.e., lies inside the

light cone in the space T

. This question can be interpreted as follows:

does the event m

belong to the proper future of the event m

if this is true

in the reference frame? (See Remark 13.16.) The fact that this is not always

so is illustrated by the following example:

Example 13.28. For the sake of simplicity we deal with a 2-dimensional space-

time. Consider the manifold R

= {(q

) ∈ R

> 0} with the Lorentz

metric g = −dq

⊗ dq

+(q

)

4/3

⊗ dq

. This is a 2-dimensional Lorentz

submanifold in the Einstein-de Sitter space-time (see Example 13.5). The ﬂat

Minkowski connection of the metric h = −dq

⊗dq

+dq

⊗dq

plays the role

of a connection in the reference frame while g is the metric transferred into

the reference frame from the space-time by the exponential map as described

above.

Let m

=(a

) ∈ R

be an event. The proper future I

of m

with

respect to the ﬂat Minkowski connection is the interior of the light cone

at m

, i.e., the events from its future are located between the lines (a

t, b

+(a

−2/3

)t)and(a

+ t, b

− (a

−2/3

)t)wheret>0. On the other hand,

the proper future I

of m

with respect to the Levi-Civit´a connection of g

consists of the events located between the curves (a

+ t, b

√

+ t)and

+ t, b

− 3

√

+ t), t>0. One can easily see that the closure

(simply

called “the future”), except the event m

, is a proper subset of the open set

. This means that if an event m

∈ I

is “far enough” from m

and “close

enough” to the boundary of the light cone I

, it may not lie in I

, hence

it may not be connected to m

by a time-like geodesic of the Levi-Civit´a

connection of g. But by construction m

is connected to m

by a time-like

geodesic of the ﬂat Minkowski connection (its initial vector is time-like).

If m

is conjugate with m

along all geodesics joining them in a reference

frame, it may be impossible to connect m

and m

by a geodesic of another

connection (in particular, of the connection in the space-time; see general

examples of this sort in Section 12.2).

For the two reference frames described above we ﬁnd geometric conditions

that in each case guarantee a positive answer to the aforementioned question.

318 13 Some Problems on Lorentz Manifolds

The conditions take the form of a certain interrelation between the tensor G

and some geometric characteristics measuring in particular “how far” m

from m

and “how close” m

is to the boundary of the “proper future” of

and to conjugate points (if they exist) in the reference frame. For this

investigation we use the machinery developed in Chapter 12.

13.2.2 The reference frame with ﬂat connection

In this section we investigate the reference frame of the ﬁrst type mentioned

above, i.e., the manifold of the reference frame is T

with a basis e

where α =0, 1, 2, 3, such that the time-like vector e

is the 4-velocity of some

observer, and the connection in T

is the ﬂat connection of the Minkowski

space.

In this case, it is convenient to regard O as a domain in a linear space

on which there are given the Lorentz metric, the tensor G and other objects

described above. Here we can make use of the familiar facts of linear algebra.

In particular, the tangent space T

¯m

at ¯m ∈Ocan be identiﬁed with

by a translation and for any ¯m ∈Owe may consider the light cone

in T

¯m

(generated by the Lorentz metric tensor at ¯m)aslyinginT

but depending on ¯m.

Geodesics in T

with respect to the ﬂat connection are straight lines.

Thus, in this reference frame, the question under consideration takes the

following form: is it possible to connect the events m

and m

on M

by a

time-like geodesic if they are connected by the straight line a(τ) in O so that

a(0) = m

, a(T )=m

and which lies inside the light cone in T

? Here

τ is a parameter that can be, say, the proper time on M

or the natural

parameter in the reference frame, etc. In this case the fact that the straight

line a(τ) belongs to the light cone in T

is equivalent to the fact that the

vector of the derivative a



(0) =

dτ

a(τ)

|τ=0

lies inside that cone, as postulated.

Since the covariant derivative with respect to the ﬂat connection coincides

in this case with the ordinary derivative in T

, equation (1) takes the

form

dτ



(τ)=G



). (13.10)

Thus the main problem is reduced to the two-point boundary value problem

for (13.10). Since the right-hand side of (13.10) has quadratic growth in

velocity, for some pairs of points the two-point problem may have no solutions

(see Sections 12.1 and 12.2).

Recall that the tangent space T

to the Lorentz manifold M

has

the natural structure of a Minkowski space whose inner product is the metric

tensor of M

at the event m. Since the speciﬁed basis e

,whereα =0, 1, 2, 3,

is Lorentz-orthonormal, the Minkowski inner product of X = X

and

Y = Y

has the form

13.2 A two-point boundary value problem on Lorentz manifold 319

X · Y = −X

+ X

where X

= X

for i =1, 2, 3. Introduce a Euclidean inner product in T

by changing the sign of the time-like summand, i.e., by setting

(X, Y )=X

+ X

Hereafter in this section all norms and distances are determined with respect

to the latter inner product.

By a linear change of time introduce a parameter s along a(·) such that for

the line ˜a(s) obtained from a(τ ) we get ˜a(0) = m

and ˜a(1) = m

. Consider

the Banach space C

([0, 1],T

) of continuous curves in T

with the

usual supremum norm.

Lemma 13.29 There exists a suﬃciently small real number ε>0 such that,

for any curve ˜v(s) from the ball U

⊂ C

([0, 1],T

) of radius ε centered

at the origin, there exists a vector

˜v

∈ T

belonging to a bounded neigh-

borhood of the vector ˜a



(0) =

˜a(s)

|s=0

and such that

˜v

lies inside the light

cone of the space T

and the curve m



(˜v(t)+

˜v

)dt takes the value

at s =1. The vector

˜v

continuously depends on ˜v(·) and 

˜v

 <C for

any curve ˜v ∈ U

for some C>0.

Proof. By explicit integration one can easily prove that C

˜v

such that m



(˜v(t)+

˜v

)dt = m

exists and is continuous in ˜v. Then by continuity, from

thefactthatthevector˜a



(0) lies inside the light cone of the space T

it follows that for a perturbation ˜v(·) suﬃciently small with respect to the

norm, the vector

˜v

also lies inside the same light cone. Take for C the

upper bound of the set of norms of the vectors

˜v

from the above-mentioned

bounded neighborhood of

˜a(s)

|s=0

. 

C is an estimate of the Euclidean distance between m

and m

We turn back to the parametrization of the line a(·) by the parameter τ.

Consider the Banach space C

([0,T],T

Lemma 13.30 Let a real number k>0 be such that T

−1

ε>k,where

ε is as in Lemma 13.29. Then for any curve v(t) from the ball U

⊂

([0,T],T

) of radius k centered at the origin, there exists a vector

∈ T

from a bounded neighborhood of the vector a



(0) =

dτ

a(τ)

|τ=0

such that the vector C

lies inside the light cone of the space T

and the

curve m



(v(t)+C

)dt takes the value m

at τ = T . The vector C

continuous in v(·).

Proof. Changing the time along a(τ) construct the straight line ˜a(s)=a(Ts)

that meets the conditions ˜a(0) = m

and ˜a(1) = m

as in Lemma 13.29.For

any curve v(·) ∈ U

⊂ C

([0,T],T

), construct the curve ˜v(s)=Tv(Ts)

that lies in U

⊂ C

([0, 1],T

), i.e., which satisﬁes Lemma 13.29.In

320 13 Some Problems on Lorentz Manifolds

particular, for this curve, there exists a vector

˜v

such that 

˜v

 <C from

Lemma 13.29. By explicit calculation one can easily derive that



(˜v(s)+

˜v

)ds = m



(v(t)+C

)dt = m

where C

= T

−1

˜v

. 

By construction C

 <T

−1

C for any v ∈ U

For the tensor G, introduced above, deﬁne the norm G

 by the standard

formula

G

 =sup

X∈T

,X≤1

G

(X, X).

The deﬁnition immediately implies the estimate

G

(X, X)≤G

X

for any X ∈ T

. (13.11)

Theorem 13.31 Let m

and m

be connected in O by a straight line a(τ )

that lies inside the light cone of the space T

and satisﬁes the conditions

a(0) = m

and a(T )=m

.Letm

and m

belongtoaballV ⊂ T

such

that for any ˆm ∈ V the inequality G

ˆm

 <

(ε+C)

holds, where ε and C are

as in Lemma 13.29.ThenonM

there exists a time-like geodesic m

(τ) of

the Levi-Civit´a connection of the Lorentz metric such that m

(0) = m

and

(T )=m

Proof. Consider the ball U

⊂ C

([0,T],T

) of radius K = T

−1

ε −

ϕ centered at the origin, where ϕ is as in Lemma 12.6.SinceK<T

−1

ε,

the assertion of Lemma 13.30 is true for U

and the following completely

continuous operator

Bv =



(v(s)+C

)ds

(v(t)+C

,v(t)+C

)dt

is well-deﬁned on this ball. Let us show that this operator has a ﬁxed point

in U

. Recall that, for any curve v ∈ U

,itsC

-norm is not greater than

K = T

−1

ε−ϕ and that by Lemma 13.30 C

 <T

−1

C. Then the hypothesis

of the Theorem, the estimate (13.11) and Lemmas 12.6, 13.29 and 13.30 imply

that



(v(s)+C

)ds

((v(t)+C

), (v(t)+C

))



≤



(v(s)+C

)ds





(εT

−1

− ϕ)+CT

−1



< (T

−2

ε −T

−1

ϕ).

From the last inequality we obtain

13.2 A two-point boundary value problem on Lorentz manifold 321





(v(s)+C

)ds

((v(t)+C

), (v(t)+C

))dt



< (T

−1

ε −ϕ)=K.

This means that the operator B sends the ball U

into itself and so, by

Schauder’s principle, B has a ﬁxed point v

(t) in this ball. It is easy to see

that m

(τ)=m



(t)+C

)dt is a solution of the diﬀerential equation

(13.10) such that m

(0) = m

and m

(T )=m

. Notice that by construction

(τ) is a geodesic of the Levi-Civit´a connection of the Lorentz metric on

. The equality Bv

= v

and the deﬁnition of B implies that v

(0) = 0,

and hence

dτ

(τ)

|τ=0

= C

, where by Lemma 13.30 the vector C

lies in

the light cone of the space T

, i.e., its Lorentz scalar square is negative.

Since the scalar square of the derivative is constant along the geodesic of the

Levi-Civit´a connection of the Lorentz metric, this geodesic is time-like. 

Remark 13.32. Recall (see above) that C estimates the Euclidean distance

between m

and m

. By construction, ε estimates in some sense “how close”

is to the boundary of the proper future of m

in the reference frame. This

clariﬁes the meaning of the condition G

ˆm

 <

(ε+C)

13.2.3 The reference frame with Riemannian

connection

In this section we investigate the reference frame at the event m ∈ M

of the

second type described in Section 13.2.1. The manifold here is identical to the

manifold in the previous section: T

with a speciﬁed orthonormal basis,

while the connection may be not ﬂat but it is assumed to be compatible with

a (positive deﬁnite) Riemannian metric on T

(see Section 13.2.1). We

do not assume this connection to be torsionless.

Here, the question of the existence of a geodesic of the Levi-Civit´acon-

nection on M

that we are looking for is reduced to the solvability of the

two-point boundary value problem for equation (13.9) in the reference frame.

An important diﬀerence between this case and the case of a ﬂat connection

is the fact that for non-ﬂat connections conjugate points may exist. This yields

additional diﬃculties since there are examples (see Section 12.2) showing

that, for a pair of points conjugate along all geodesics joining them, the

boundary value problem for a second order diﬀerential equation may have

no solutions at all. We suppose from the very beginning that m

and m

are connected by a geodesic in the reference frame along which they are

not conjugate. Under these conditions, we ﬁnd conditions under which the

problem for (13.9) is solvable.

The case of a non-ﬂat connection requires more complicated machinery

than the previous case. In particular, we replace ordinary integral operators

(used in the previous section) by the integral operators with parallel trans-

322 13 Some Problems on Lorentz Manifolds

lation from Section 3.2. In fact the method we use here is a modiﬁcation of

that elaborated in Chapter 12.

Everywhere in this section the norms in the tangent spaces and the dis-

tances on manifolds are induced by the above-mentioned positive deﬁnite

Riemannian metric.

Recall that the operator S from Section 3.2 is well-deﬁned on complete

Riemannian manifolds. Consider the neighborhood O in T

described in

Section 13.2.1. Without loss of generality we may assume that the Rieman-

nian metric on O is a restriction of a complete Riemannian metric on T

Indeed, take a relatively compact domain O

⊂Owith smooth boundary

such that O

contains the points 0 ∈ T

, m

and m

as well as the

geodesic γ(t), where t ∈ [0, 1], joining m

and m

(if O is relatively compact

one can take O

= O). Then it is possible to change the Riemannian metric

outside O

so that it becomes complete on T

, and to use O

in place of

O. Thus, the operator S is well-deﬁned in this case.

Recall that for a constant curve v(t) ≡ X ∈ T

M we get by construction

that Sv(t) = exp X, where exp is the exponential map of the given connection

(see Section 3.2).

Let the points m

∈Obe connected in O by a geodesic γ(t)ofthe

Riemannian connection so that γ(0) = m

and γ(1) = m

. In particular, we

get m

=exp(

γ(t)

|t=0

)=S(

γ(t)

|t=0

), where exp is the exponential map

of the Riemannian connection. Let m

and m

be non-conjugate along γ(·)

and the vector

γ(t)

|t=0

lie inside the light cone of T

Hereafter we denote by U

the ball of radius k centered at the origin in

some Banach space of continuous maps.

Lemma 13.33 In the conditions and notation of Lemmas 3.48 and 3.50, the

number ε can be chosen so that for the curve ˜u(·) ∈ U

⊂ C

([0, 1],T

the vector

˜u

lies inside the light cone of the space T

and, for the curve

u ∈ U

⊂ C

([0,T],T

O), the vector C

also lies inside the light cone of

the space T

Proof. The fact that, for suﬃciently small ε>0, the vector C

˜u

belongs to

the interior of the light cone is derived from continuity considerations as in

Lemma 13.29.ForC

, this statement follows from the fact that C

= T

−1

˜v

where ˜v(s)=Tv(Ts) ∈ U

⊂ C

([0, 1],T

O)(seeabove). 

Hereafter we choose ε satisfying the hypotheses of Lemmas 3.48 and 13.33.

Let γ(t)beaC

-curve given for t ∈ [0,T]andletX(t, m) be a vector

ﬁeld on O. Denote by ΓX(t, γ(t)) the curve in T

γ(0)

O obtained by parallel

translation of vectors X(t, γ(t)) along γ(·)atthepointγ(0) with respect to

the connection of the reference frame.

Imitating the previous section, we introduce the norm G

 by means

of the norms of vectors with respect to the Riemannian metric on O,as

mentioned above. Clearly, (13.11) is valid for G

, as in the previous section.

With the help of S and Γ we construct the integral operator B : U

→

([0,T],T

) of the following form:

13.2 A two-point boundary value problem on Lorentz manifold 323

Bv =



ΓG

S(v(t)+C

)



S(v(t)+C

)



dt, (13.12)

where k and T satisfy the conditions of Lemmas 3.50 and 13.33. One can

easily see that the operator B is completely continuous.

Theorem 13.34 Let m

∈Oand let there exist a geodesic γ(τ) of the

connection in the reference frame such that γ(0) = m

, γ(T )=m

, m

and

are not conjugate along γ(·) and the vector

γ(t)

|t=0

lies inside the light

cone of the space T

.Ifm

and m

belong to the ball V ⊂Osuch that

at any m ∈ V the inequality G

 <

(ε+C)

holds, where ε and C are as in

Lemmas 3.48 and 13.33, then there exists a time-like geodesic m

(τ) of the

Levi-Civit´a connection of the Lorentz metric on M

such that m

(0) = m

and m

(T )=m

Proof. Let k := T

−1

ε − ϕ,whereϕ is as in Lemma 12.6. For this k,the

hypothesis of Lemma 3.50 is satisﬁed. Hence, on the ball

⊂ C

([0,T

],T

the operator (13.12) is well-deﬁned. Recall that, for the curve v(·) ∈ U

,its

-norm is not greater than k,thatC

 <T

−1

C and that the parallel

translation with respect to the Riemannian connection preserves the norms

of vectors. Then taking into account the deﬁnition of operator S we see that



dτ

S(v(τ )+C

)



< (T

−1

ε −ϕ)+T

−1

Hence, formula (13.11), the hypothesis of theorem and Lemma 12.6 imply

that



S(v(τ)+C

)



dτ

S(v(τ )+C

dτ

S(v(τ )+C

)





≤



S(v(τ)+C

)



((T

−1

ε −ϕ)+T

−1

(13.13)

< (T

−2

ε −T

−1

ϕ).

Since the operator Γ of parallel translation preserves the norms of the vectors,

from the last inequality we obtain:





ΓG

S(v(t)+C

)



S(v(t)+C

)





≤





S(v(t)+C

)



S(v(t)+C

)





< (T

−1

ε −ϕ)=k. (13.14)

This means that the completely continuous operator B sends the ball U

into itself. Hence, by Schauder’s principle, B has a ﬁxed point v

∗

(τ)inU

324 13 Some Problems on Lorentz Manifolds

Then from the deﬁnition of the operator S and the usual properties of the

covariant derivative it follows that m

∗

(τ)=S(v

∗

(τ)+C

∗

) is a solution of

the diﬀerential equation (13.9) (see Chapter 12). By construction the curve

∗

(τ) is a geodesic of the Levi-Civit´a connection of the Lorentz metric on

and, for it, m

∗

(0) = m

and m

∗

(T )=m

From the equality Bv

∗

= v

∗

and (13.12) it follows that v

∗

(0) = 0. Then

from the deﬁnition of the operator S it follows that

dτ

(τ)

|τ=0

= C

∗

where the vector C

∗

belongs to the light cone of the space T

Lemma 13.33. This means that the Lorentz scalar square of the vector C

∗

negative. Since C

∗

is the initial vector of the derivative of the geodesic m

∗

(τ)

of the Levi-Civit´a connection on M

and since the Lorentz scalar square of

the derivative along this geodesic is constant, the geodesic m

∗

(τ) is time-like.



The meaning of condition G

 <

(ε+C)

is analogous to that in the

previous section (see Remark 13.32), but here C estimates the Riemannian

distance between m

and m

while ε depends on the geometry of the reference

frame and in some sense estimates “how close” m

is to the boundary of

the “proper future” of m

and “how close” it is to conjugate points in the

reference frame.

13.3 A Classical Particle in a Classical Gauge Field

In this Section we introduce and investigate a special class of diﬀerential

equations on the total spaces of ﬁber bundles equipped with connections. We

interpret these equations as equations of motion for a classical particle in a

classical gauge ﬁeld.

It should be pointed out that the term ‘gauge ﬁeld’, as used in contempo-

rary physics, means a connection on a ﬁber bundle. Gauge ﬁelds are generally

used in quantum physics.

The problem of the motion of a classical particle in a classical gauge ﬁeld

was ﬁrst treated by S.K. Wong (see [231]). Further developments can be

found, e.g., in [143, 201, 213, 214, 215]. The symplectic geometry and the

Hamiltonian formalism on some special ﬁber bundles were exploited in [213,

214, 215]. Here we suggest another approach using the connection (i.e., the

gauge ﬁeld itself) on bundles diﬀerent from those of [143, 213, 214, 215].

This approach seems to be more natural; we show that in the special case

of the electromagnetic ﬁeld our equation is reduced to the ordinary Lorentz

equation (13.5).

In what follows we denote ﬁber bundles and their total spaces by the same

symbols.

13.3 A Classical Particle in a Classical Gauge Field 325

13.3.1 A brief introduction to gauge ﬁelds and some

preliminary constructions

Let M

be a Lorentz manifold with a metric g(·, ·). We retain the standard

notation M

of this chapter for a Lorentz manifold since our principal ex-

ample of such a manifold is a space-time. Nevertheless please note that in

all constructions below in this section we do not use the fact that the di-

mension of the manifold is 4 and so all our results are valid in arbitrary

ﬁnite-dimensional Lorentz manifolds.

Consider a principal bundle Π : E→M

with structure group G.TheLie

algebra of G will be denoted by g.

Let H be a connection on the bundle E. Recall (see Section 2.7)thatH is

a distribution on E, i.e., a sub-bundle of TE, that is invariant with respect

to the natural right action of G on E and is complementary to the vertical

distribution V consisting of subspaces tangent to the ﬁbers of E.Notethat

all subspaces V

⊂ T

E are canonically isomorphic to g (see Section 2.7).

We introduce a semi-Riemannian (Lorentz) metric g

on the manifold E by

determining an inner product in g, and hence in all subspaces V, by deﬁning

the inner product in the subspaces H as the inverse image of g with respect

to TΠ, and by setting the subspaces V

and H

in all T

E to be orthogonal to

each other. Following this we can treat the manifold E as an ordinary semi-

Riemannian manifold, in particular, we can consider all the usual operations

with diﬀerential forms on E.

In what follows we shall also denote by V and H the projections of T

E =

⊕ H

onto the vertical (i.e., V

) and horizontal (i.e., H

), respectively,

subspaces in T

E where b ∈Eis an arbitrary point. So, for X ∈ T

E, VX

is its vertical component and HX is its horizontal component. This notation

will be used when dealing with all types of ﬁber bundles with connections.

Consider the connection form ϕ and the curvature form Φ =Dϕ of H where

the covariant diﬀerential D on the right hand side is deﬁned by the usual

formula Dϕ(·, ·)=dϕ(H·, H·) (see Section 2.7). Recall that ϕ is a vertical

1-form (i.e., equal to zero on subspaces H)andΦ is a horizontal 2-form (i.e.,

equal to zero on subspaces V), both taking values in g. We interpret Φ as the

gauge ﬁeld strength. The gauge ﬁeld, as usual, is determined by the equations

DΦ =0, D ∗Φ = ∗J, (13.15)

where J is a horizontal 1-form on E with values in g and ∗ is the star operator.

The ﬁrst equation is Bianchi’s identity (2.39). Note that (13.15) are analogous

to Maxwell’s equations (13.4).

Let F be a (real or complex) linear space with inner product h(·, ·). Sup-

pose that the left action of G on F is given and h(·, ·) is invariant under this

action. We interpret F as the space of internal states of the particle and the

group G as

the group of its internal symmetries. In addition, we suppose that