Geometric interpretation of magnetic fields and the motion of charged particles

Nuclear Physics B166 (1980) 162-188 © North-Holland Publishing Company

G E O M E T R I C I N T E R P R E T A T I O N OF M A G N E T I C F I E L D S A N D T H E M O T I O N OF C H A R G E D P A R T I C L E S B. F E L S A G E R Institute of Mathematics, Unicersity o[ Odense, Campusvej 55, DK-5230 Odense M and The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark

J.M. L E I N A A S NORDITA, Blegdamsvei 17, DK-2100 Copenhagen C), Denmark Received 2 October 1979

A geometric representation of magnetic fields in terms of plane bundles in R 3 is examined, and for two examples, the magnetic flux string and the monopole, it is shown how the field configurations can be related to the geometry of cones and spheres. Using the normal vector of the planes as a dynamical variable we treat the structure of the magnetic field in a way similar to that of an ordered medium. We especially compare it with the structure of the 0(3) non-linear g-model. Finally a geometric interpretation of electric charge is introduced by associating with charged particles a rotating, body-fixed frame which is constrained to rotate around the normal vector of the planes. The close connection with the Kaluza-Klein model is discussed.

1. Introduction

The "geometrization" of classical field theories is part of an important tradition in modern physics. The structure of gravitational fields was geometrized in Einstein's theory of general relativity. There it was shown that space-time is curved and that gravity is associated with the curvature of space-time. More recently it has also been shown that the theory of Yang-Mills fields [1] (including electromagnetic fields) can be geometrized in a similar way [2]. Such a geometrization has thrown important light on the structure of gauge theories. In this new formulation the Yang-Mills field is associated with the curvature of an internal space. The understanding of such a geometrization is generally obscured by the use of various complicated concepts from modern differential geometry, like fiber bundles, characteristic classes, etc. The purpose of this paper is to re-examine the geometrization of electromagnetism by using only concepts based on elementary euclidean geometry. We make use of a model, similar to the one outlined in ref. [3], where the geometry of the electromagnetic field is given an explicit representation in terms of geometric structures in ordinary physical space. 162

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The paper is organized as follows: In the first section (sect. 2) we examine the geometric representation of static magnetic fields. By restricting the fields to be purely magnetic we are able to work entirely inside three-dimensional, euclidean space, and the field configurations can then be visualized directly as geometric structures. The basic element of this geometric representation of the magnetic field is the construction of a family of planes in R a. With each point in space we associate a plane passing through this point. The distribution of these planes is characterized by a curvature, and this curvature can directly be related to the magnetic field strength. This geometric model is closely related to the usual formulation of electromagnetism as a gauge theory. A gauge can be interpreted as a distribution of orthonormal frames in the planes, and a gauge transformation corresponds to a rotation of these frames. We illustrate these general considerations by examining two simple examples: the magnetic flux string and the magnetic monopole. For the monopole we show in particular how thc problem of choosing a smooth gauge is directly related to the problem of constructing a coordinate system covering a sphere. In sect. 3 we examine the field equations, with the magnetic field expressed in terms of the normal vectors if(r) of the planes. Expressed in this way the description is similar to that of an ordered medium with ~(r) as the order parameter. We discuss the close relation which exists between this model, and the O(3)-non-linear ormodel, which also has a unit vector glr) as order parameter. The boundary conditions of the if-field are compared for the two models, and the close connection between the soliton configuration of the non-linear o'-model in two dimensions and the magnetic monopole in three dimensions is demonstrated. In the last part (sect. 4) we show how the geometric model of the magnetic field can be extended to also give a representation of the motion of an electrically charged particle in the field. Geometrically, the charged particle is represented as a small spinning body, where the spin vector is constrained to be perpendicular to the plane at the position of the particle. Apart from this constraint the particle is free, and the Lorentz force then appears as a gyroscopic effect which follows from the constraint acting on the direction of the spin. We discuss, as the final point, the close connection which exists between this geometric model of the charged particle and the KaluzaKlein model of electromagnetism [4].

2. The geometric model of the magnetic field We begin this section by examining in an elementary framework the geometry of plane bundles in R 3. The connection with magnetism is introduced later in this section, but we anticipate the connection by identifying already at this introductory stage some of the geometrical concepts with concepts from the gauge theory of electromagnetism. (An elementary introduction to the basic concepts of differential geometry can be found, e.g., in ref. [5].)

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2.1. P l a n e bundles in R 3

With each point r in three-dimensional euclidean space R 3 w e associate a plane, d e n o t e d by ~'(r), which passes t h r o u g h the point. Such a field of planes will be referred to as a p l a n e bundle. The orientation of the plane 7r(r) is specified by a unit vector li(r), perpendicular to the plane. To begin with we will assume the plane bundle to be s m o o t h , i.e., tl (r) is a s m o o t h vector field. Later on we will admit isolated point singularities in the a-field. In each plane ~-(r) we choose an o r t h o n o r m a l frame (~l(r), ~2(r)), such that ( ~ ( r ) , ~2(r), ti(r)) is a s m o o t h positively oriented frame field in R 3. A specific choice of local frames will be referred to as a gauge, and the transition from one choice of local frames to a n o t h e r as a g a u g e t r a n s f o r m a t i o n : e'l (r) = cos s#(r) • ~l(r) +sin ~(r) • ~2(r), (1) ~ (r) = - s i n £(r) • 'el (r) + cos ~:(r) • ,~2(r) • These transformations form the g r o u p SO(2), which is referred to as the gauge g r o u p (fig. 1). We will be interested in the curvature of the bundle. To investigate this we introduce the first-order derivatives O,~, and Oili. T h e y generate the following t h r e e - v e c t o r fields: a = e'lVe~ , l =eiVni

(i=1,2,3;a=1,2).

(2)

T h e vector field a ( r ) is the connection of the plane bundle. T h e vector fields l~, l:, which control the variation of the tl-vector, are called the fields of extrinsic curvature. F r o m the o r t h o n o r m a l i t y of the vector fields ( ~ , ~2, ~) it follows that all the derivatives of these vector fields can be expressed in terms of a and i~ : Ve'l = - a e 2 -,l t n i) i Vei2 = aeil _ 1 2 n i I V e ~ = - e . b a e b - l~n i ,

(3)

V n i = lle'l + !2e'2 ,

Fig. 1. The plane bundle in R 3. With each point r in R 3 there is associated a two-dimensional plane ~(r) with normal vector ti(r). A gauge transformation is a rotation of the basis vectors in this plane ea ~ e'.

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These equations are k n o w n as the Cartan structure equations. It follows from (2) that the connection a transforms i n h o m o g e n o u s l y u n d e r a gauge transformation (1):

a--,a'=a -V~:.

{4)

T h e vector fields of extrinsic curvature In, on the o t h e r hand, transform h o m o genously: !1 ~, l'i = cos ~!1 + s i n ~ ! 2 , t5) 12 ~ I" = - s i n ,f !1 + cos 6 12.

2.2. Vector transport in the plane bundle Having i n t r o d u c e d the structure of a plane bundle in space, we can now distinguish between two types of vectors. The first type consists of o r d i n a r y vectors in R s, e.g., !~, while the second type consists of vectors always constrained to lie in the planes, e.g., go. (For the latter type we will mostly use greek letters, e.g., ¢b.) T h e first type of vectors can be uniquely transported from one point to a n o t h e r using standard parallel displacement. This will not work for the vectors of the second type, unless the bundle is flat, i.e., the planes are parallel. H o w e v e r , since the planes are e m b e d d e d in a euclidean space it is possible to define the so-called canonical vector transport in the plane bundle. But this vector transport is path dependent. A n infinitesimal displacement of the type of a vector ~ = d~"~,, from a point r to a n e i g h b o u r i n g point r + d r can be t h o u g h t of as an operation in two steps. O n e first parallel displaces ~b as a vector in R 3 from r to r + d r and then projects it down into the plane ~-(r + dr) (fig. 2). It follows from the Cartan structure equations (3) that this operation is given by ~ " ( r ) ~ d~'"(r + d r ) = ~b" (r) - e~b&h(r)a(r) • d r .

(6)

By integration, eq. (6) can be used to generate the canonical vector transport along a curve F connecting two distant points r and r'. It follows from (6) that the length of a vector is preserved u n d e r the canonical vector transport. T h e r e f o r e the intrinsic c o m p o n e n t s d)" of the vector qb(r) and the intrinsic c o m p o n e n t s ,;b'" of the transO(r)

~ (r * dr i

.111 --':U, .......

\~

~__.

• dr~

J

Fig. 2. The two steps of an infinitesimal, canonical vector transport O(ri-* d~'ir 4-dr). The vector is first parallel transported as a vector in R3(1), and then projected down into the plane ,-r(r * dr) (2 I.

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B. Felsager, J.~4. Leinaas / Geometric interpretation of magnetic fields

ported vector do'(r') are related by an equation of the form

[a "l

l-c°s¢ -sin ¢,I[6'I

~b'2J = [ sin Ct cos Cr J[~b-~J '

(7)

where Cr represents the angular increase measured relative to the local frames at r and r'. For an infinitesimal transport we get from eq. (6): d~ct- = a • d r .

(8)

Consequently the angle ~Pr is given by = f a • dr. Ji

(9)

This shows that the connection a generates the angular increase of the vector d~ under the canonical vector transport. The canonical vector transport determines what is known as the intrinsic geometry of the plane bundle. The most important invariant of this geometry is the intrinsic curvature, which is related to the vector transport around closed curves. For a closed curve F the angular increase (9) can be converted, by Stokes' theorem, to a surface integral over an arbitrary surface S having F as its boundary,

w,.=q~ a.dr=f 3 1 3

b.ds,

b=V×a.

(10)

S

The vector field b(r) is the vector field of intrinsic curvature (or gaussian curvature). It follows from eq. (10) that b satisfies

V.b=O.

(11)

In differential geometry this is known as the Bianchi identity. The intrinsic curvature can be derived from the extrinsic curvature !,, using eqs. (2) and (3): b=Vxa

= V e ' l xVe~ = 1 1 x ! 2 .

(12)

This equation is known as Gauss' equation. It shows that b is normal to the plane spanned by I1 and 12. Observe that, starting from a plane bundle characterized by the triple (g~, e2, r~), we have now generated a secondary bundle characterized by the triple (11, 12, b). By construction b is a gauge-invariant quantity, since the angular increase around a closed curve does not depend on the choice of local frames. In fact b can be expressed directly in terms of ~: b = !1 x 12 = ( e i V n ~) x (ekVn k ) = l~eiikniVni × V n k ,

(13)

or equivalently: |

^

b, = ~_eijkn • (ai~ x ak~) .

(14)

B. Felsager, J.M. Leinaa.~ / Geometric interpretation of magnetic fields

167

H e r e the gauge invariance of b is manifest. For later use we will also express b in terms of the polar angles of tl. We have d = [sin 12 cos o2 ; sin 12 sin o2; cos .Q],

(15)

b = sin 12 V.Q x Vo2.

(16)

which gives

If we in addition choose the local frame el = [ c o s / 2 cos to ; c o s / 2 sin to ; - sin 12], (17) ~2 = [ - sin to ; cos to ; 0 ] , this fixes the connection, a = - c o s 12 Vo2.

(18)

O b s e r v e that with this choice of gauge we have a • b =0.

(19)

2.3. Connection with the magnetic field T h e connection b e t w e e n the geometric quantities introduced magnetic field quantities has already been indicated by our choice curvature b is equivalent to the magnetic field strength B and the equivalent to the magnetic vector potential A. To be m o r e precise A =,~a,

B =,~b,

above and the of notation: the gauge field a is we should write (20)

where A is an unspecified constant of the correct dimension. With such an identification it follows from eq. (4) that a gauge transformation of the usual type,

A~A'=A-VO,

(21)

is r e p r e s e n t e d by a rotation in the plane ~r(r) with ~:(r)= (1/,~)O(r) as the rotation angle. F r o m eq. (10) we derive the usual relationship between B and A : B =V×A.

(22)

V. B = 0,

(23)

Finally the magnetic field equation,

is geometrically interpreted as the Bianchi identity (11). T h e o t h e r field equation Vx B

=/~,j,

(24)

where j is an external electric current, is still to be interpreted as a dynamical equation.

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B. Felsager, J.M. Leinaas / Geometric interpretation o/magnetic [ields

This identification immediately raises the question if an arbitrary magnetic field can be g e n e r a t e d as the curvature of a plane bundle. We will show that this is indeed the case at least locally. Suppose, therefore, we have a given magnetic field B satisfying the field equation (23). We will use the formulas (15)-(19) to reconstruct a plane bundle c o r r e s p o n d i n g to this field. First, it follows from eq. (23) that B can locally be g e n e r a t e d from a vector potential A. This vector potential is not uniquely d e t e r m i n e d and we will fix the gauge, so that it satisfies the gauge condition [~f. eq. (19)] A • B--A

• (VxA)=0.

(25)

We therefore m a k e a gauge transformation A ~ A - ~ r 0 where O(r) is defined in the following way: we choose a surface S which is not tangential to B. T h e vector potential A is then integrated along the magnetic flux line which passes through r, from the point ro in S to the point r (fig. 3); r

0(r)

= I,

A • dr.

i26)

V0. B =A . B.

(27)

o

It follows that

C o n s e q u e n t l y the t r a n s f o r m e d vector potential A ' = A - V O satisfies the gauge condition (25). W e note that eq. (25) is precisely the condition that A is a normal to a family of surfaces in R 3. If we introduce a function w(r) which is constant over each of these surfaces, it follows that A is p r o p o r t i o n a l to Vw : A = tz(r)V~o.

t

-

(28)

/

Fig. 3. The gauge transformation A --, A - V # which transforms A to be orthogonal to B. The function O(r) is determined as the line integral of A along the magnetic flux line F.

t3. Felsager, J.M. Leinaas / Geometric interpretation of magnetic fields

169

By rescaling the function w we can locally obtain 1# (r)/A I < 1 and therefore write eq. (28) as

1

- A a

=

- c o s .O VoJ,

(29)

with cos .(2 = -tz/A. We consequently get

1 - B = sin 1"2 V.O × VoJ. A

(30)

C o m p a r i n g with eq. (16) we see that in this way we have constructed a plane bundle generating B if (,(2, oJ) are interpreted as the polar angles of the associated normal vector field tJ(r). Thus, we can associate with any magnetic field B (at least locally) a plane bundle, with the field strength being p r o p o r t i o n a l to the field of intrinsic curvature of the bundle. To study this " g e o m e t r i z a t i o n " of the magnetic field s o m e w h a t closer, we will consider the s m o o t h map from R 3 into the unit sphere, r--, ~(r), generated by the field of normal vectors rJ(r). This is k n o w n as the spherical m a p associated with the bundle (see fig. 4). The spherical m a p gives a geometric m e a n i n g to the total magnetic flux t h r o u g h a surface S in R 3. To see this we make use of eq. (16) to obtain the following expression for the magnetic flux: q~--

B.

=A

(

sin

=A I

- 8r ~ x~

0,)

dAldA2

[SAtaa z

sin .(2 d.O do).

8A2~ 3 dAIdA 2

(31)

1($1

H e r e n(S) d e n o t e s the surface on the sphere which S maps into u n d e r the m a p p i n g r ~ t~(r). This expression shows that the magnetic flux t h r o u g h S is equal to h times the area of n(S). We will refer to this as the area theorem.

\

,

Fig. 4. The spherical map r --, ~(r). According to the area theorem the magnetic flux through a surface S in R 3 is proportional to the area of the corresl:x)ndingsurface n(S) on the sphere.

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B. Felsager, J.M. Leinaas / Geometric interpretation of magnetic fields

The area theorem is particularly useful when we study the geometric interpretation of m o n o p o l e singularities in the magnetic field. A magnetic monopole is characterized by the fact that there is a rton-vanishing total magnetic flux 4) through any closed surface S surrounding the monopole. It then follows from the area theorem that the area of the corresponding surface element n(S) is non-vanishing too. However, since n(S) is a closed surface, this area can only have values which are integer multiples of 4~r. This is due to the fact that n(S) must wrap an integer number of times around the unit sphere. Consequently the total flux of the monopole, which is identical to the m o n o p o l e charge g, is quantized; =g

=

4~rmA,

m integer.

(32)

H e r e m denotes the winding number, which indicates the number of times n(S) wraps around the sphere. Note that the charge quantization is not a quantum mechanical effect. It follows simply as a consequence of the geometrical interpretation of the magnetic field. We also see that 4~-A can be interpreted as the unit of magnetic charge, but at this classical level, there is nothing to fix the size of A. When the closed surface n(S) is characterized by a non-vanishing winding number m, this means that there is a singularity in the plane bundle inside S. This singularity, which represents the magnetic monopole, appears as a topological singularity in the field of normal vectors ~i(r). Such singularities are well-known from other branches of physics, where a field of unit vectors ri(r) appears as the dynamical variable of the physical system [6]. The most simple example corresponds to a radially directed si-field r

~i(r) = r

(33)

This field is characterized by the winding n u m b e r m = 1 and therefore represents the magnetic field of a m o n o p o l e of unit strength. We will in a later example study the plane bundle associated with this field in more detail. One should note that for monopoles characterized by m > 1, the field ri(r) becomes more complicated, and is no longer simply a radially directed field. Observe that the magnetic field B does not uniquely fix the field of unit vectors ti(r). It only determines the intrinsic geometry of the corresponding plane bundle, but not the particular embedding of this bundle in R 3. The area theorem shows that the magnetic flux through some surface element S is unchanged under a transformation t~(r) -+ ti'(r),

(34)

if this transformation preserves the area of n(S). Consequently the magnetic field B is invariant under transformations (34) of the sphere, which preserve the area of n(S) for any surface S in R 3.

B. Felsager, J.M. Leinaas / Geometric interpretation of magnetic fields

171

A particular example of such an area-preserving transformation is given by an r-independent rotation ~ , t~(r) ~ t~'(r) = (~t~)(r).

(35)

This transformation obviously only rotates the surface element n(S) on the sphere, but does not change its area. The geometric model of the magnetic field outlined above is closely related to the gauge description of electromagnetism originally introduced by Weyl [7]. Let us conclude this section by briefly discussing how this connection is obtained. Weyl's theory applies to the quantum description of a charged particle in the electromagnetic field. The state of this particle is described by a wave function ~(r), which in the spinless case is simply a complex-valued function. Thus, for fixed r one can identify ¢(r) as a vector in a (local) complex plane. Let us write this vector in the following way: ~(r) = ~(r)~(r) = ~ l ( r ) ~ ( r ) + ~2(r)i~(r).

(36)

where ~(r) is the real unit vector of the complex plane, ~l(r) the real component of ~(r) and tO2(r) the imaginary component. We will now make an identification of the complex plane associated with the point r with the plane rr(r) through this point. In terms of the unit vectors (~, ,~2) this identification is obtained by the relations ~(r) = ~(r),

e2(r) = i ~ ( r ) .

(37)

As shown by eq. (36) this identification implies that the wave function is represented by a vector field constrained to the planes zr(r). The complex phase of the wave function is represented by the angle between the vector O(r) and the unit vector ~ (r). Consequently a rotation of the unit vectors in the plane z'(r) [eq. (1)] corresponds to a gauge transformation of the second kind of the wave function ~9(r) ~ ~p'(r) = e-i~l'l~(r) .

(38)

The path dependence of the vector transport in the plane bundle corresponds to the basic assumption in Weyl's theory of non-integrability of the local phase of the wave function. This path dependence forces one to introduce the covariant gradient Dtb = (V - ia)~P

(39)

as a consequence of the fact that the ordinary gradient V does not have the correct transformation properties of an observable under gauge transformations. In Weyl's theory the connection with the standard description of a charged particle is then obtained by identifying this covariant gradient with the operator obtained from minimal coupling of the charged particle to the magnetic field: D

= V -

iqA h

o

(4O)

172

13. Felsager, J.M. Leinaas / Geometric interpretation of magnetic ]ields

We note that this identification determines the constant A, which we so far have left unspecified, h A =-. q

(41)

With A specified by eq. (41), the quantization rule for the magnetic charge, eq. (32), takes the form 4zqg = ~ m,

m integer.

(42)

This is almost identical to the quantization rule originally obtained by Dirac [8], the only difference being that Dirac's quantization rule allows m to take half-integer values. This difference can be understood as a consequence of the fact that the abstract bundle which represents the magnetic field for m = ±~, ± ~. . . . . cannot be e m b e d d e d as a plane bundle in R 3. The geometric model described above is in fact a specific representation of the fiber-bundle description of Weyl's theory. Such a formulation of the gauge theory of electromagnetism can be found for example in ref. [9]. However, x0e would like to stress the point that whereas the fiber-bundle formulation is only concerned with the intrinsic geometry of the bundle, the representation in terms of a plane bundle also introduces an extrinsic geometry. This makes it possible to represent some of the abstract structures from the fiber-bundle formulation in a more direct way, as we will see in two examples below. Let us finally make a c o m m e n t about the possibility of generalizing the geometric model to include also the electric field E. One can show that it is possible to incorporate electric fields in the model by making the planes rr(r) time dependent. But, as long as the planes are confined to three-dimensional space, R 3, only fields which obey the constraint E 'B =0

(43)

can be represented in this way. However, it has been shown by Gliozzi [3] that if the planes are instead embedded, as space-like planes, in four-dimensional Minkowski space, then it is possible to represent (at least locally) the full electromagnetic field F,~. We will, nevertheless, in the following restrict the discussion to the purely magnetostatic case. This is because we are mainly interested in using the planebundle description as a simple model of a gauge theory.

2.4. Two examples: the flux string and the magnetic monopole In these examples we will consider a particular simple class of plane bundles, where the planes are tangential to a family of disjoint surfaces in R 3. Such a plane bundle is said to be integrable and the surfaces are called characteristic surfaces. For

B. Felsager, J.M. Leinaas ,/Geometric interpretation o]"magnetic" fields

173

an integrable plane bundle the g e o m e t r y of the bundle is intimately related to the g e o m e t r y of the characteristic surfaces. Thus the flux density of b t h r o u g h a characteristic surface is precisely the gaussian curvature of this characteristic surface. F u r t h e r m o r e the canonical vector transport along a curve in a characteristic surface reduces to the geodesic vector transport on the surface. In the first example we consider a flux string (or rather a flux tube) lying along the z-axis. The flux string, which is assumed to be axially symmetric, can be considered being p r o d u c e d by an external circular, current Ca solenoid). In cylindrical coordinates (p, ~, z) the magnetic field is given by B ( p ) = 0 for p > po.

B = B(p)l~,

(44)

The c o r r e s p o n d i n g plane bundle is not uniquely fixed by the magnetic field, but it seems natural to consider an axially symmetric bundle. In terms of the polar angles of ~(r) this is expressed by the conditions O =.O(p),

oJ = ¢ .

(45)

F r o m eq. (16) it follows that the associated magnetic field is expressed by B = A sin .(2 d-~-~Vp x V ¢ op _

A d {cos.O)/~. p dp

(46)

We will assume that the magnetic flux density is positive for all p. This means that cos .(2 is a decreasing function of p. For p = 0 one h a s / 2 = 0, since the planes on the z-axis, for s y m m e t r y reasons, must be horizontal. The function cos .O(p) therefore has the f o r m shown in fig. 5. Outside the radius of the flux tube it assumes a constant value cos .(2o. Such an axisymmetric bundle is integrable and the form of the characteristic surfaces is shown in fig. 6. Outside the radius p0 they have the form of circular cones, characterized by a half-angle ,1~ - .(20. This angle is closely related to the flux of the

c..

COS

COS

9:

~t" o

•

-]

I

-

.

.

.

.

• I

.

Fig. 5. The polar angle .O(p) of the vector si(p, ¢,) shown as a function of p. The function cos .O(pl is monotonically decreasing and reaches a constant value cos ,(2ooutside the flux tube.

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B. Felsager, J.M. Leinaas

Geometric interpretation of magnetic fields

B{r)

It ....

- ~ ./'~-'~:

i~

a -.<4 ~.

/

.....

•

_

_

._

' ...........

'.

-...._-

--

....

~° Fig. 6. T h e c h a r a c t e r i s t i c surfaces a s s o c i a t e d with a m a g n e t i c flux tube. O u t s i d e the t u b e they form circular cones. T h e n o r m a l vectors ~(r) on o n e of the surfaces are shown.

string. T h e spherical m a p r ~ ti(r) m a p s the whole x, y - p l a n e into the region defined by [2 <~/'2o. T h e area of this region is 21r(1 - cos 12o) and due to the area t h e o r e m (31) the total flux in the string is therefore = 2~rh (1 - cos -0o).

(47)

(We note that this gives a m a x i m u m value for the flux, 4) ~< 4~lh I. But this is only due to the fact that we have constrained the plane bundle to be axially symmetric.) A characteristic feature of the cones is the fact that the gaussian curvature vanishes at all points (corresponding to the fact that B = 0 for p > Po). T h e r e is nevertheless a global curvature associated with the cone. This can most easily be seen if one considers a closed geodesic on the cone as shown in fig. 7. Such a geodesic will intersect itself with an angle )¢ which is d e t e r m i n e d by the opening angle of the cone X = 2zr(l - cos -O0).

(48)

,/A,\

Fig. 7. The g l o b a l c u r v a t u r e of a cone. G e o d e s i c t r a n s p o r t of a v e c t o r a r o u n d the cone r o t a t e s the vector by an angle x. W h e n ~ < 180 ° the global c u r v a t u r e also leads to a self i n t e r s e c t i o n of geodesics with X as i n t e r s e c t i o n angle.

B. Felsager, J.M. Leinaas / Geometric interpretation of magnetic fields

175

If one transports a tangent vector around the closed geodesic, it will return to the starting point as a vector which is rotated by the angle X. Comparing with eqs. (9) and (10) we see that, apart from the factor A, this angle is identical to the flux carried by the string. In the quantum description of a charged particle moving in the region outside the flux string, the angle X gets a physical significance. It corresponds to a phase difference in the quantum mechanical phase associated with charged particles moving on each side of the string. This is known as the A h a r o n o v - B o h m effect [10]. It follows from what we have shown above that this effect is related to the presence of a global intrinsic curvature of the plane bundle outside the string. In the other example we consider a spherically symmetric magnetic monopole. As we have seen it is represented by a plane bundle having a point singularity characterized by a certain winding n u m b e r m. Let us examine the simplest case, m = I. It can be produced by a spherically symmetric bundle, where tl is radially directed outwards, i.e., the polar angles of ~i(r) are given by .(2 = 0,

w = ~.

(49)

From eq. (16) it follows that the associated magnetic field is expressed by A B = A sin 0 V0 x ~r¢ = ~ r ,

(50)

r

corresponding to a m o n o p o l e of strength g = 4rrA located at the origin. Also in this case the bundle is integrable, but the characteristic surfaces are now concentric spheres. If we therefore consider a fixed radial distance r, the geometry of the plane bundle reduces to the geometry of the corresponding sphere. We also note that a choice of gauge, which means a choice of a frame field (~l(r), ~2(r)), corresponds to a coordinatization of the sphere. It is well-known that such a coordinatization cannot be carried out over the entire sphere without introducing singularities. The corresponding singularities in the description of the magnetic m o n o p o l e are known as Dirac strings. Since the Dirac strings in the present formulation appear as coordinate singularities, this stresses the fact that they should not be interpreted as physical singularities. They merely reflect defects in the choice of coordinates. A standard choice of coordinates on the sphere is shown in fig. 8a. It corresponds to ~1 pointing in the direction of the longitudes and g2 in the direction of the latitudes: ~ = [cos ~ cos 0, sin q~ cos 0, - s i n 0], ~2 = [ - sin ~, cos ~,, 0].

(51 )

The corresponding vector potential is obtained from eq. (18): A = -A cos 0 V ~ .

(52)

This is known as the Schwinger gauge for the monopole field. It has a Dirac

! 76

B. Felsager, J.M. Le#,aas / Geometric interpretation of magnetic fields N

:'~i

:

,L

,

,

,'

N

;

, ",. )(,.

-S

,,.

.....

..._x__....i A

Fig. 8. Two choices of gauge for the monolx)le of unit charge. The gauge is represented as a coordinatization of the characteristic surfaces, which are spheres surrounding the monopole. The standard choice of coordinates corresponds to the Schwinger gauge ta). A n o t h e r set of coordinates obtained by stereographic projection of the cartesian coordinates in the plane, corresponds to the Dirac gauge (b).

singularity along both the positive and the negative z-axis, corresponding to the singularities in the coordinate system for 0 = 0 and 0 = rr. A n o t h e r gauge, known as the Dirac gauge is obtained if one rotates the unit vectors ~1 and ~2 by the angle + ~. This produces the vector potential A'= A-AVe

= -A(1 +cos O)Vq:.

(53)

The Dirac potential has only one Dirac string, which in eq. (53) is located along the negative z-axis. One can also, in this case, relate the choice of gauge to a coordinate system on the sphere. This is shown in fig. 8b, where the coordinate lines are constructed by stereographic projection of the cartesian coordinate lines in the x, y-plane. The resulting coordinate system on the sphere is apparently singular at the "north pole". The problem of fixing the gauge of the monopole field has been examined in great detail by Wu and Yang [2]. Using the more abstract fiber-bundle formalism, they have shown how one can avoid Dirac strings. This is done by operating with two different choices of gauge, each of which only covers a part of the sphere. Together they cover the whole sphere and in the intersection region they are joined together by a gauge transformation. In the present formalism the necessity of using two gauges corresponds simply to the fact that we need two coordinate systems to cover a sphere.

3. Relations with the 0 ( 3 ) non-linear ~r-model We have in sect. 2 discussed the geometric interpretation of the magnetic field in terms of plane bundles in R 3. Since B can be expressed in terms of the normal vector of the planes, fi, it is possible to reformulate the description of the magnetic field using the unit vector tJ rather than B as dynamical variable. The magnetostatic energy density then has the form ~b = ½{(OM • 0~) 2 - (OM • Oi~)(~M • ~jtJ)}.

(54)

B. Felsager, J.M. Leinaas / Geometric inte(pretation of magnetic fields

177

(For simplicity we will use the convention A = 1 in this section, which means that we have B = b). F o r m u l a t e d in this way the description shows a close similarity with that of an ordered m e d i u m , with n then representing the order parameter of the system. W e will, in this section, c o m p a r e the above description of the magnetic field with that of a n o t h e r o r d e r e d medium, namely the well-known O(3~ non-linear or-model. T h e dynamical variable in this model is also a field of unit vectors ti(r), but the energy density has the m o r e simple form, ~,~ = ~,'~ • ;~itl = ~V'n i • V n ' .

(55)

As a physical interpretation of this model one can consider ti(r) as describing the local spin in a ferromagnet. The non-linear ~r-model has also been much used as a simple model of a gauge theory. As shown in ref. [11 ], one can in fact give the model a ~'gauge theoretical" formulation by introducing a plane rrlr) normal to ti(r) in the same way as described above for the magnetic field.

3. 1. The field equations To have the closest possible similarity between the non-linear o--model and the geometrical model of the magnetic field described above, we will consider static spin configurations in three euclidean dimensions. It is f u r t h e r m o r e convenient to introduce the following c o m p l e x vector field of extrinsic curvature, I = I1 +il~ ,

(56)

which transforms as a gauge vector under gauge transformations, ! ~ I' = e-i~l.

(57)

By use of eq. (12) one can then express the magnetostatic energy density as a gauge scalar constructed from l, Nt, = ~b 2 = -~(l* x l) "~. Also, the of L and products equation

(58)

e n e r g y density of the non-linear or-model can be expressed purely in terms it in fact c o r r e s p o n d s to the other of the two quadratic gauge-invariant of ! and !*. The expression, which is o b t a i n e d by use of the structure (3) is ~,, = ~l* • l.

(59)

(Note that, in general, the e n e r g y density of an o r d e r e d m e d i u m with ~i(r) as dynamical variable cannot be expressed purely in terms of l.) H o w e v e r , there is a characteristic difference between the two expressions for the energy density, given by eqs. (54) and (55). W h e r e a s Nb d e p e n d s only on the intrinsic g e o m e t r y of the plane bundle, g,r d e p e n d s explicitly on the extrinsic geometry. In fact, the close c o r r e s p o n d e n c e between the two models b e c o m e s most a p p a r e n t

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when the magnetostatic model is expressed in terms of the intrinsic curvature b, while the non-linear or- model is expressed in terms of the extrinsic curvature l. To study this more explicitly we consider the field equations satisfied by !. There are two such equations, corresponding to the Bianchi identity (11) and the dynamical equation (24) satisfied by b. Applying the structure equations (3) to the identity V x (~n ~) = 0 ,

(60)

we obtain the following equation satisfied by i, Dxi

=Vxi-ia

xl =0,

(61)

with D denoting the covariant derivative. This equation is known as the CodazziMainardi equation. It corresponds to the Bianchi identity (11) satisfied by b, and the equation in fact implies this identity. The dynamical equation, which follows from extremizing the static energy (55) with the subsidiary condition h 2 = 1, can be written as V . (Vn i) = ~tn ~,

(62)

with ~z as a lagrangian multiplier. This gives, by use of the structure equations, the second equation satisfied by 1:

D. I=V.

l-ia

•

i=0.

(63)

The similarity between the field equations (61) and (63) and the equations satisfied by the magnetic field becomes most apparent when these are expressed in terms of the field tensor f~i Eiikbk rather than b. Assuming a source-free magnetic field, we can relate the two sets of equations in the way shown in the following table: =

Magnetostatic case

Non-linear g - m o d e l

Geometric equations eiikOifik = 0

eqkDil i = 0

Dynamical equations O~[ii = 0

Dili = 0

However, even if there is a close similarity between the two sets of equations, one also notes the characteristic difference. The field equations of the non-linear ~r-model involve the covariant derivative, whereas the magnetostatic field equations involve only the ordinary derivative. This difference is a consequence of the fact that I transforms as a gauge vector, but b as a gauge scalar. Since the gauge potential a (r) then appears explicitly in the field equations of the non-linear o'-model, a third equation which relates a to i has to be added. This is Gauss' equation (12), which can

B. Felsager, J.M. Leinaas / Geometric interpretation of magnetic fields

179

be written as 1

V × a =~-~ !* x !.

(64)

The appearance of the covariant derivative in the field equations of the non-linear ~r-model is in fact the reason why this set of equations has non-trivial solutions. This is different from the magnetostatic case, where the sourceless field equations have only the trivial solution b = 0 (assuming the usual boundary condition b ~ 0 at infinity). We note that for non-abelian gauge theories the field strength, which generalizes b, is no longer a gauge scalar. The corresponding field equations, the Yang-Mills equations, also have non-trivial solutions. In this respect the non-linear o--model is more similar to non-abelian gauge theories, even if the gauge group of the non-linear o--model is the abelian group SO(2) = U(1) just as in the electromagnetic case. 3.2. B o u n d a r y conditions

We will now make a reduction to two dimensions. This reduction we can consider as being due to translational invariance of the plane bundle along the z-axis, ~d --

az

= O.

(65)

For the magnetic field this means that it is directed along the z-axis, and therefore gets the simple form

bx =by = 0 ,

bz = d

•

(Oxd×ayd).

(66)

In two dimensions the boundary conditions of the non-linear ~r-model, implied by the requirement of finite energy, are known to give rise to the existence of soliton configurations. In terms of the spherical map r o d ( r ) these configurations are characterized by a non-trivial winding number. We will briefly review this point and then show that the boundary conditions of the geometric model of the magnetic field can also be given a geometric interpretation. This time however the spherical map is not characterized by a winding number. (Implicitly we assume that external currents can be present to produce non-trivial field configurations.) The requirement of finite energy (per unit length in the z-direction) implies that the energy density tends to zero more rapidly than 1 / p 2 as p ~

p2~

O --~ o O

,0.

(67)

For the non-linear o--model, with the energy density given by eq. (55), this gives the following boundary conditions for the n-field, a,~

p~cC

~ O,

pap~

p~oC

, O.

(68)

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B. Felsager,J.M. Leinaas / Geometric interpretation of magnetic fields

If we in particular consider the field ti(p, ~) along a circle of radius p in the x, y - p l a n e we conclude from the angular part of the b o u n d a r y conditions that the unit vectors a r o u n d the circle b e c o m e parallel when 19-+ pc. This means that the spherical m a p r ~ ti(r) maps the circle into a closed curve on the sphere, which shrinks to a point as p ~ pp. T h e radial part of the b o u n d a r y condition shows that the curve tends to a fixed point on the sphere (fig. 9). As a c o n s e q u e n c e of this, the whole x, y - p l a n e R 2 is m a p p e d into a closed surface in S 2. This surface can therefore be characterized by a winding n u m b e r m, in the same way as we have previously discussed for the d-field associated with a magnetic m o n o p o l e . T h e picture is s o m e w h a t different when we consider the b o u n d a r y conditions of the magnetostatic model. T h e e n e r g y density, which then can be written as gb = ~

1

(a~

x

ao~ ) 2 ,

(69)

gives a w e a k e r b o u n d a r y condition for the n-field,

a,~

x

aotl

0~30

,0.

(70)

This condition does not imply that the d - v e c t o r s a r o u n d the circle of radius p b e c o m e parallel in the limit p ~ oo, but only that the vectors c3~ and apti in this limit b e c o m e parallel. C o n s e q u e n t l y a circle of radius p in the x, y - p l a n e is m a p p e d into a closed curve which does not have to shrink to a point as p ~ oo. H o w e v e r , since aoti b e c o m e s parallel to 0¢~ it follows that 0p~ in this limit points along the tangent of the closed curve. T h e r e f o r e for large p the point ~(p, q~) on the sphere moves along the curve, as p is increased. As a c o n s e q u e n c e of this the closed curve on the sphere tends towards a fixed curve rather than a fixed point as p-~ o0 (fig. 10). Thus, in the magnetostatic case the plane R 2 is in general not m a p p e d into a closed surface in S 2. As a c o n s e q u e n c e of this there is no magnetic flux quantization. W e have already seen an example of this in the discussion of the A h a r o n o v - B o h m effect. T h e magnetostatic e n e r g y of the flux string has finite value per unit length for any value of the total flux. This follows from the fact that the magnetic field vanishes

,,-ic~)

n {c~

r -- fi(r)

c Fig. 9. Boundary conditions of the non-linear o--model. A circle C is mapped into a closed curve n(C) on the sphere. The image n(C) contracts to a point n(C~) when the radius of the circle tends to infinity.

B. Felsager, J.3/l. l.einaas / Geometric interpretation o[ magnetic fields

181

f ;

-

-

t,

Fig. I0. Boundary conditions in the magnctostatic case. The circle C' is now mapped into a closed curve niC) which tends to a fixed curve n(C:~) when the radius of the circle tends to infinity.

outside the radius po. But the vectors t~(r) are in general not parallel for p >p(). They are normal vectors to cones of fixed opening angles, and the region outside the flux string is therefore m a p p e d into a fixed circle on the sphere (corresponding to a polar a n g l e / 2 =/2o). Since the spin configurations of the non-linear c - m o d e l are characterized by a winding n u m b e r associated with the spherical map r-* 6(r), they are in this respect similar to the magnetostatic field configurations defined on closed surfaces enclosing a magnetic monopole. We will end this section by making a c o m m e n t on how the representation of the m o n o p o l e field in terms of a plane bundle makes it possible to give an explicit representation of this correspondence. We will then consider a compactification of the x, y-plane, defined by a stereographic projection R2-~S 2. This projection, which maps the point (p, w) in the x, y-plane into the point (0, ~) on the sphere, is given by p = c o t ½0,

~ =~.

(71)

If the vector t~(0, ¢) defined in R 2 is associated with the transformed point (0, ~b), this defines a vector field tJ (0, 4)) on the sphere. Due to the asymptotic condition this field is everywhere continuous, even at the "'north pole" 0 = 0, where the vector field assumes the asymptotic value of tJ(p, ¢) for p ~ o e (fig. 11). Let us assume the sphere S 2 to be imbedded in R 3 and to surround a magnetic monopole. We can then also use the stereographic projection the other way around to project the m o n o p o l e configuration of t~(r) from the sphere into the x, ),-plane. This transformed field automatically satisfies the asymptotic conditions of the non-linear o--model. We will consider the simplest case, where we have a monopole of unit strength. The vector field on the sphere is simply given by t~(r) = v/r, with the planes of the plane bundle then being tangential to the sphere. Stereographic projection of this into the x, y-plane gives the following transformed field:

( x , , ~(x,y)= l+p 2 1+02'1 -

-

,

.

172)

182

B. Felsager, J.M. Leinaas I Geometric interpretation of magnetic fields

'

x

. . . . .. . . . . .

f

Fig. 11. Compactification of the plane. By a stereographic projection the plane bundle in the x, y-plane [represented by its normal vector field rJlp, ¢ ~] is transferred to a plane bundle on S2 [represented by d(0. 4,)]. One can now readily verify that this field configuration in fact solves the field equation of the non-linear or-model. It describes a soliton of winding number m = 1 [12]. The stereographic projection thus gives a direct correspondence between the monopole of unit strength in R 3 and the soliton of winding number m = 1 in R 2.

4. Geometric interpretation of the motion of a charged particle In this section we will examine the geometric description of the magnetic field from a somewhat different point of view. We will focus our interest on the motion of a charged particle rather than on the field itself. Previously we have briefly discussed how the quantum description of a charged particle, when expressed as an abelian gauge theory, gives support to a geometric interpretation of the magnetic field. We will now show that the motion of a charged particle even at the classical level, can be described in a way which fits into the scheme given by the plane bundle description. The charged particle is g~ometrically represented as a small spinning body, where the direction of the spin is constrained to be normal to the plane 7r(r). This constraint gives rise to the Lorentz force acting on the particle. Also here we limit the discussion to the case of a purely magnetic field. To include electric fields the model has to be generalized to give a fully relativistic description of the spinning body.

4.1. Electric charge interpreted as spin

The motion of a small spinning body can be described using the c.m. coordinate r and a body-fixed cartesian frame as dynamical variables. For simplicity we will consider the body to be a small sphere, but the actual shape is of no importance since we will examine the motion of this body only in the point-particle limit. The particle moves freely in three dimensions, except for the presence of a constraint on the orientation of the spin of the particle. Thus the rotation of the particle about its center of mass is assumed, at any point along its trajectory, to be confined to the plane 7r(r) at the position of the particle. This means that the body-fixed frame can be chosen such that one of the unit vectors coincides with the normal vector tJ(r) of the

B. Felsager,J.M.Leinaas/ Geometricinterpretationofmagneticfields

183

plane. The frame is then determined by a single unit vector th lying in the plane and therefore has the form (th, tJ × ih, ~) (fig. 12). The unit vector th is related by a rotation to the fixed frame (~, ~2) in the plane It(r): th = cos a'el +sin a'e2 •

(73)

The rotation angle )¢ together with the c.m. coordinate r determines the motion of the rotating sphere. We note that the angle X is a gauge-dependent quantity. Using the structure equations (3) we derive the following expression for the time derivative of the unit vector ~h : dt -

(t] x t h ) - [ v • (m'Vn')]t]

(74)

where v is the velocity of the particle and D x / d t denotes the covariant derivative of X, D__Xg= dx dt dt

-a

• v.

(75)

From this equation we get the angular velocity of the body-fixed frame: dth ee=~x--= dt

Dr ~+[v.(m'Vni)]~xth. dt

(76)

The lagrangian which determines the equation of motion of the spinning particle is assumed to be the free-particle lagrangian. In the non-relativistic case it is simply the kinetic energy L = lrnv 2 + 1 1 ~ 2 , (77) where I is the moment of inertia of the sphere about its center-of-mass. Inserting eq. (76) the full expression for the lagrangian becomes , ~L~-~t 1 , [ D g '}~ 2 + ½ i [ v . (miVni)]2 . L = ~1 m v 2 -t-

(78)

. X " / ~^ ~ ~

Fig. 12. The motion of a spinning particle. The rotation is confined to the plane ~r(r) of the plane bundle at the position of the particle.

184

B. Felsager, J.M. Leinaas / Geometric interpretation of magnetic fields

This shows that the angular variable X couples minimally to the vector potential a. Let us write the moment of inertia as I = m a 2. The parameter a is the radius of gyration, which measures the size of the small sphere. The last term of the lagrangian can now be estimated as follows: ~I[v _

"

( m i ~ , n , ) ] 2 ~:_. 2mr; t 2~[ a t•r n iF, vrn ,,~2 )] .

(79)

This shows that if the size of the sphere is small compared with the variations in the -field, (80)

]a(rniVn~)[<< 1 ,

then we can neglect the last term. (This corresponds to neglecting the component of to perpendicular to ~). In the point-particle limit, the lagrangian thus reduces to /-. = gray

~I

.

(81)

The lagrangian (81) does not have the same form as the usual one describing a charged particle in a magnetic field. It is nevertheless closely related as one can see more clearly by considering the corresponding hamiltonian. We introduce the canonical momenta

DX p = rnv-1

~

D~ a,

(82)

p~ = l d t

In terms of these, the hamiltonian is expressed by 1 H =p

• v + p , , • f¢ - L

=~m

~ (p+p~a)-

1 +-~p-~.

(83)

Since X is a cyclic coordinate, the conjugate m o m e n t u m p~ is a constant of motion. If this constant is related to an electric charge q by q = p,,/a,

(84)

then the first part of the hamiltonian (83) is identical to that of a charged particle moving in the magnetic field generated by the vector potential A = Aa. The last term, which can be interpreted as a spin energy, is a constant and therefore does not affect the equation of motion of the c.m. coordinate r. The motion of the spinning sphere described above is therefore, in fact, identical to the motion of a charged particle in the magnetic field. We note that the momentum p~ gives the magnitude of the spin 5 of the small sphere relative to its centre of mass $ = I~ =p~.

(85)

Therefore, the electric charge of the moving particle is geometrically represented as the intrinsic spin $ of the rotating sphere. The Lorentz force acting on the particle in

B. Felsager, J.M. Leinaas / Geometric interpretation o[ magnetic fields

185

this geometric representation comes from the constraint on the direction of S. It thus has the character of a gyroscopic effect. We can illustrate the above discussion by a simple example, which describes the motion of a~electrically charged particle in a spherically symmetric monopole field. The representation of the charged particle as a particle with spin can in this case be realized (at least in principle) as a purely mechanical construction. (For more details about this particular construction we refer to ref. [13].) We have previously seen that a monopole of unit strength is geometrically represented by a plane bundle with radially directed normal vectors t~(r) = r/r. This means that when we introduce a small, rotating sphere to represent the charged particle in the m o n o p o l e field, then the spin S of the sphere is constrained to point in the radial direction, S = Sr/r. Such a constraint can be produced by the following mechanical construction: we introduce a thin, rigid rod which is fixed at the origin but which otherwise can freely be oriented in any direction in space. This rod, which we consider to be massless, is assumed to pass through the center of mass of the small sphere. The sphere can, without friction, rotate around and move along the thin rod. Thus the effect of the rod is to constrain the spin of the sphere to point in the radial direction, but apart from that there is no force acting on the sphere. According to the previous discussion the small sphere will then, when given an initial velocity, follow the trajectory of a charged particle in the monopole field. This motion is illustrated in fig. 13. As described, e.g., in ref. [ 13], the particle moves along a geodesic on the surface of a circular cone, such that the total angular m o m e n t u m J of the particle points along the symmetry axis.

0

Fig. 13. Simulation of the motion of a charged particle in a monopole field. The spinning sphere moves along and rotates around a thin massless rod t which has one fixed point O. The trajectory t of the particle follows the surface of a circular cone which lies symmetrically around the direction of the total angular momentum & L denotes the angular m o m e n t u m associated with the c.m. motion and S the spin relative to this point.

186

B. Felsager, J.M. Leinaas / Geometric interpretation of magnetic fields

4.2. Connection with the K a l u z a - K l e i n model

As we have seen above, the configuration space of the spinning body is a four-dimensional space which is described in a natural way by the coordinates (r, , ) . We will now introduce a simple metric in this space. The distance ds between two neighbouring points is then defined by the equation ds 2 = dr 2 + a 2 d~o2 ,

(86)

where d~0 is the infinitesimal rotation vector of the body-fixed frame of the particle, and a is the radius of gyration mentioned before. As follows from eq. (76), this vector is related to the infinitesimal displacement (dr, dg) in configuration space by d~o = (Dx)t] + [ m i ( V n ~) • dr]tJ x ~

(87)

where D X is the gauge-invariant increment in X: DX = d x - a

• dr.

(88)

In the point-particle limit the metric thus reduces to ds 2 = dr 2 + a Z ( D x ) 2 •

(89)

With the metric given by eqs. (86) and (87), the lagrangian (78) takes the simple form 1

[ds'~ 2

L =~m~)

.

(90)

It is simply the "free-particle" lagrangian in the four-dimensional configuration space. Consequently the particle moves with constant speed along a geodesic in this space. The motion of the position variable r of the particle is then obtained by projecting this geodesic into the three-dimensional subspace R 3. When formulated in this way, the above model is intimately related to the Kaluza-Klein model of electromagnetism [4]. In their model, which is a relativistic one, a fifth angular variable is added ad hoc to the four coordinates of space-time. A metric is then introduced in this space, and this has a form analogous to the one given in eq. (89). As a consequence of this the path of a free particle, which is a geodetic curve in the five-dimensional space, reproduces the trajectory of a charged particle when projected down into four-dimensional physical space. This is completely analogous to what we have shown above. Note, however, that in our non-relativistic model the angular variable X is given a direct physical meaning and the lagrangian is not introduced ad hoc but appears simply as the kinetic energy of the particle. Recently the Kaluza-Klein model has been re-examined in terms of concepts from fiber-bundle theory [14]. We will, as a final point, show that such a description in fact comes out in a natural way in our formalism. Let us then for a m o m e n t neglect the constraint on the orientation of the spinning particle in space. The configuration

B. Felsager. J.M. Leinaas / Geometric interpretation of magnetic fields

187

space consequently has the form R3×SO(3), where R 3 corresponds to the c.m. coordinate r and SO(3) to the rotation group which determines the orientation of the body-fixed frame. This configuration space carries a natural metric, which is exactly of the form given by eq. (86). As long as the constraint on the rotation of the particle is neglected, the rotation angle d~ is independent of dr, and the geodesic curve in the configuration space R3x SO(3) therefore simply describes a body which moves with constant speed and constant angular velocity along a straight line. The configuration space described above can naturally be considered as a bundle space. We then consider a copy of the rotation group SO(3) to be associated with each point r in R 3. However, this bundle has a trivial structure since there is a unique correspondence between rotations at different points. This correspondence is simply determined by ordinary parallel transport of frames from one point to another. When we re-introduce the constraint on the orientation of the body-fixed frame, the configuration space still has the form of a bundle space. But now there is with each point in R 3 associated a two-dimensional rotation group SO(2), corresponding to rotations around d(r). The structure of this bundle is no longer trivial, since there is no unique way to transport a frame in ~-(r) from one point to another. Similarly the metric in this space, which is now given by eq. (89), is a non-trivial metric, since it mixes the coordinates dr along the horizontal direction of the bundle and d,~ along the vertical direction. Thus, the configuration space of the constrained, spinning body, has a natural bundle structure and the non-trivial metric in this space arise naturally from the higher-dimensional bundle with trivial metric. Note the close connection which exists between this reduction from higher dimensions and the one described at the beginning of this paper, where a non-trivial vector transport was introduced in the plane bundle by projection of the trivial parallel transport in three-dimensional space.

5. Conclusions We have in this article examined the geometrization of electromagnetism from several different points of view, and shown how this leads to a unified picture which includes both the geometrical structure of the field itself and the geometrical interpretation of electric charge. The discussion has been limited to the magnetostatic case in order to m a k e the geometrical structure as simple as possible, but, as pointed out, a relativistic generalization including electric fields is clearly possible. However, such a generalization involves some complications, for example in the description of a relativistic spinning particle.

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/3. Felsager, J.M. Leinaas / Geometrw interpretation of magnetic fields

A s we have p o i n t e d out, the g e o m e t r i c m o d e l of m a g n e t i c fields can also be v i e w e d as a s i m p l e r e p r e s e n t a t i o n of the g e o m e t r i c s t r u c t u r e s k n o w n from the a b s t r a c t f i b e r - b u n d l e d e s c r i p t i o n of Y a n g - M i l l s fields. H o w e v e r , a m o r e d i r e c t g e n e r a l i z a t i o n of the m o d e l to n o n - a b e l i a n t h e o r i e s is possible. This is d o n e by a s s u m i n g the p l a n e s 7r(r) to be of h i g h e r d i m e n s i o n t h a n two. F o r a real p l a n e of d i m e n s i o n n, the g a u g e g r o u p , c o r r e s p o n d i n g to r o t a t i o n s in rr(r), w o u l d then be S O ( n ) . But, w h e r e a s in the e l e c t r o m a g n e t i c case the p l a n e b u n d l e can be i m b e d d e d in o r d i n a r y physical space, this is no l o n g e r the case for n o n - a b e l i a n t h e o r i e s w h e r e the (hyper) p l a n e s m u s t be i m b e d d e d in s o m e h i g h e r - d i m e n s i o n a l flat space. A p a r t f r o m this d i t t e r e n c e the a n a l o g y is close. O n e can, for e x a m p l e , r e f o r m u l a t e the g a u g e t h e o r y as that of an o r d e r e d m e d i u m with the n o r m a l v e c t o r (or a set of n o r m a l vectors) as the o r d e r p a r a m e t e r . F u r t h e r m o r e , o n e can d e s c r i b e the m o t i o n of a n o n - a b e l i a n c h a r g e d p a r t i c l e by a s s o c i a t i n g with the p a r t i c l e a r o t a t i n g o r t h o n o r m a l f r a m e in the h i g h e r d i m e n s i o n a l space. W h e n n of the unit v e c t o r s of this f r a m e are c o n s t r a i n e d to lie in the h y p e r p l a n e at the p o s i t i o n of the p a r t i c l e , this i n t r o d u c e s a force which is a g e n e r a l i z a t i o n of the L o r e n t z force in the e l e c t r o m a g n e t i c case.

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