Fortschritte der Physik / Progress of Physics: Band 28, Heft 11 1980 [Reprint 2021 ed.] 9783112522981, 9783112522974

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN. A. LÖSCHE. R. RITSCHL UND R. ROMPE

H E F T 11 • 1980 • B A N D 28

A K A D E M I E - V E R L A G EVP 1 0 , - M 31728

•

B E R L I N ISSN 0015 - 8208

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an den Postzeitungsvertrieb, an eine Buchhandlung oder an den AKADEMIE-VERLAG, DDR - 1080 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7000 Stuttgart I, Wilhehnstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR - 7010 Leipzig, Postfach 160, oder an den AKADEMIE-VERLAG, DDR - 1080 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Heraasgeber: Prof. D r . F r u k Kasohluhn, Prof. Dr. A r t u r Lösohe, Prof. Dr. Rudolf Ritsohl, Prof. Dr. Robert Rompe, i m A u f t r a g der Physikalisehen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1080 Berlin, Leipziger Strafle 3 - 4 ; Fernruf: 22 36221 und 2 2 3 6 2 2 9 ; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: D r . Lötz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R • 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Dniokhaus „Maxim Corki", D D R - 7400 Altenburg, Carl-von-Ossietzky-StraBe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik'* erscheint monatlich. Die 12 Hefte eines Jahres bilden einen B a n d . Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis f ü r die D D R : 1 2 0 , - M). Preis je H e f t 13,— I I (Preis f ü r die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/28/11. © 1980 by Akademie-Verlag Berlin. Printed in t h e German Democratio Republio. A N (EDV) 57618

Fortschritte der Physik 28, 5 7 9 - 6 3 1 (1980)

Topological Charges in Gauge Theories* J. M . LEINAAS

NORDITA, Copenhagen, Denmark** Abstract Topological and geometric aspects of gauge theories are examined. The geometry of the fiberbundle formulation of gauge theories is discussed and compared with the formalism of general relativity. The basic role played by the parallel displacement operator of this geometry is examined. With this operator a gauge independent characterization of various topological singularities and non-singular soliton configurations is carried out.

Contents 1. Introduction

580

2. The geometric meaning of gauge invariance

581

2.1. The internal spaces 2.2. Parallel transport. The gauge group 2.3. Gauge invariance

581 582 584

3. The 3.1. 3.2. 3.3.

electromagnetic case The gauge description of electromagnetism Topological singularities Magnetic monopoles

587 587 591 594

4. Elements from homotopy theory

597

4.1. Homotopic curves 4.2. Higher homotopy groups

597 599

5. Generalized monopole singularities 5.1. Point singularities in three dimensions 5.2. Point singularities in two dimensions 5.3. Generalization to higher dimensions 6. Non-singular configurations characterized by topological quantum numbers 6.1. 6.2. 6.3. 6.4.

The Nielsen-Olesen vortex line Generalized flux strings T h e ' t Hooft-Polyakov monopole The SU(2) instanton

601 601 605 611 612 613 618 621 625

7. Concluding remarks

630

8. References

631

* Based on lectures given at the University of Oslo in the spring semester of 1977, * * DK-2100 Copenhagen 0 , Blegdamsvej 17 39

Zeitschrift „Fortschritte der Physik", Heft 11

Fortschritte der Physik 28, 5 7 9 - 6 3 1 (1980)

Topological Charges in Gauge Theories* J. M . LEINAAS

NORDITA, Copenhagen, Denmark** Abstract Topological and geometric aspects of gauge theories are examined. The geometry of the fiberbundle formulation of gauge theories is discussed and compared with the formalism of general relativity. The basic role played by the parallel displacement operator of this geometry is examined. With this operator a gauge independent characterization of various topological singularities and non-singular soliton configurations is carried out.

Contents 1. Introduction

580

2. The geometric meaning of gauge invariance

581

2.1. The internal spaces 2.2. Parallel transport. The gauge group 2.3. Gauge invariance

581 582 584

3. The 3.1. 3.2. 3.3.

electromagnetic case The gauge description of electromagnetism Topological singularities Magnetic monopoles

587 587 591 594

4. Elements from homotopy theory

597

4.1. Homotopic curves 4.2. Higher homotopy groups

597 599

5. Generalized monopole singularities 5.1. Point singularities in three dimensions 5.2. Point singularities in two dimensions 5.3. Generalization to higher dimensions 6. Non-singular configurations characterized by topological quantum numbers 6.1. 6.2. 6.3. 6.4.

The Nielsen-Olesen vortex line Generalized flux strings T h e ' t Hooft-Polyakov monopole The SU(2) instanton

601 601 605 611 612 613 618 621 625

7. Concluding remarks

630

8. References

631

* Based on lectures given at the University of Oslo in the spring semester of 1977, * * DK-2100 Copenhagen 0 , Blegdamsvej 17 39

Zeitschrift „Fortschritte der Physik", Heft 11

580

J . M . LEINAAS

1. Introduction Gauge theories have in recent years attracted much attention. This is in particular due to the success of the unified gauge theory of electromagnetism and weak interactions [i] and of the theory of strong interactions which has been developed within this framework. Much of the interest is also due to the fundamental character of the gauge formalism. It can thus be given a geometric interpretation, which is very similar to the geometric description found in the general theory of relativity. As a matter of fact gauge theories were originally developed as an extension of the general theory of relativity, to provide a unification of gravitation and electromagnetism. Although the first form of this theory, presented by H. W E Y L in 1 9 1 9 [2], was not so successful, a later work of W E Y L ( 1 9 2 9 ) [3] gave the gauge description of electromagnetism in a form which we refer to as the C(l)-gauge theory to-day. Within this formalism P . A. M . D I R A C discussed the interesting possibility of including magnetic monopoles in the theory [4]. These appear as point singularities of the gauge field. A peculiar feature of the formalism, as pointed out by Dirac, is that the strength of the monopoles is related to the strength of the electric charges. This relation is known as the Dirac charge quantization condition. A generalization of Weyl's theory was suggested by 0 . K L E I N [5] in 1 9 3 8 and later introduced in a similar form by C . N . Y A N G and R . L . MILLS in 1 9 5 4 [6\. In their work the gauge transformations appear as local rotations in an internal isospin space. Thus, the transformations there are not pure phase factors, but are elements of the group SU(2). As opposed to the Abelian Í7(l) theory this is known as a non-Abelian gauge theory. The U( 1) and SU{2) gauge theories are in fact only two examples from the general framework of gauge theories. In recent years much work has been done, parallel with the development of specific models, to understand this general formalism and to give it a simple and precise formulation. The use of concepts from fiber-bundle theory was introduced by E. L U B K I N in 1963, in a study of the geometric meaning of gauge theories [ 7 ] . Later T. T. W U and C . N . Y A N G [ 5 ] and others have examined the fiber-bundle formulation of gauge theories in more detail. It has in particular been shown that this formulation is useful for the description of magnetic monopoles [9] and other topological singularities of the gauge fields. We will in these notes discuss some of the simple geometric concepts which lie behind the present understanding of gauge theories. This mean that we will mainly be interested in kinematical aspects of the theories and therefore not offer much attention to dynamical problems. The main object is to examine the generalizations of the monopole singularities which appear in gauge theories. These are either singularities of the gauge field or they are non-singular soliton-like configurations. We will in both cases show how a topological classification of the field configurations can be given in terms of a parallel displacement operator which is associated with closed curves in space-time. There are, as one can see in the literature [10], also other (equivalent) ways to carry out such a topological classification, however the approach we will use here has the advantage of being explicitly gauge independent. The organization of these notes is as follows. We first (sect. 2.) discuss the geometric formulation of gauge theories in terms of fiber-bundles and show part of its close relation with the general theory of relativity. A simple notation is used, where the wave function of the charged (test-) particle is, at each point in space, an abstract vector in an internal space [11]. Gauge transformations correspond simply to a change of basis in this space. We then introduce the parallel displacement operator for vectors of the internal space, and discuss in particular the fundamental role played by the operators associated with closed curves.

581

Topological Charges

In sect. 3. we examine the description of electromagnetism within this formalism and in particular study the nature of the magnetic monopole. This discussion involves some elements from homotopy theory. We therefore use sect. 4. to review some of the basic concepts from this mathematical formalism. In sect. 5. this is used to classify topological singularities within generalized non-Abelian gauge theories. In the last section we examine the non-singular configurations which can be characterized by topological quantum numbers. As opposed to the point singularities, some dynamics is involved in this case. Due to the general form of the Lagrangian, boundary conditions are imposed on the gauge fields, which are necessary for the topologically non-trivial configurations to appear. The discussion is illustrated by the N I E L S E N - O L E S E N vortex string [12], the 'T H O O F T - P O L Y A K O V monopole [13, 14] and the SU(2) instanton [15]. 2. The Geometric Meaning of Gauge Invariance 2.1. The internal spaces We take the classical configuration space of the physical system, which we will denote by X, as a basis for the description. When discussing the general formalism we do not have to specify this space; it may either be the configuration space of a single particle or of a more complicated physical system. In the traditional quantum description the state of the system is associated with a vector V" of a Hilbert space J f , which is the tensor product of an external space JÍ?ext and an internal space in t, J f = Jf? ext (x) Jifint (2.1) ¿4?ext is the space of quadratically integrable complex-valued functions defined on X, while J f int is a finite-dimensional vector space describing the internal degrees of freedom of the system. The state of the system can thus be written as • / » = n(x) Xic

(2.2)

where {%k\ is a set of basis vectors in .yf int and \pk(x) are complex-valued functions. In the description outlined above, vectors of the internal space are defined without reference to any particular point x í X. This means that the internal states of the system when located at different points in X, can be compared in an unambiguous way. The transition to the gauge formalism, in its most general form, is obtained if we assume this no longer to be true. An internal vector should be associated with a particular point i f ! There is consequently not one internal space, but one internal space for each point. We will denote this space by h(x). These spaces will certainly all have identical structure, but the point is that it is in general not possible to compare vectors belonging to different internal spaces in an unambiguous way. Within the gauge formalism the full Hilbert space will then not be the tensor product of an external and an internal space. But the state of the system can still be associated with a vector 1" which can be written in a form similar to that of eq. (2.2) W(x) =

Wk(x) tk{x)

'

(2.3)

where {y.k(x)\ now is a set of basis vectors in h(x). From the discussion above one can already see a clear resemblance between the gauge formalism and the formalism which is known from general theory of relativity (GTR). In the special theory of relativity space-time is flat, and consequently there is a distant 'parallelism of vectors. This means that vectors are well-defined without reference to any 39*

581

Topological Charges

In sect. 3. we examine the description of electromagnetism within this formalism and in particular study the nature of the magnetic monopole. This discussion involves some elements from homotopy theory. We therefore use sect. 4. to review some of the basic concepts from this mathematical formalism. In sect. 5. this is used to classify topological singularities within generalized non-Abelian gauge theories. In the last section we examine the non-singular configurations which can be characterized by topological quantum numbers. As opposed to the point singularities, some dynamics is involved in this case. Due to the general form of the Lagrangian, boundary conditions are imposed on the gauge fields, which are necessary for the topologically non-trivial configurations to appear. The discussion is illustrated by the N I E L S E N - O L E S E N vortex string [12], the 'T H O O F T - P O L Y A K O V monopole [13, 14] and the SU(2) instanton [15]. 2. The Geometric Meaning of Gauge Invariance 2.1. The internal spaces We take the classical configuration space of the physical system, which we will denote by X, as a basis for the description. When discussing the general formalism we do not have to specify this space; it may either be the configuration space of a single particle or of a more complicated physical system. In the traditional quantum description the state of the system is associated with a vector V" of a Hilbert space J f , which is the tensor product of an external space JÍ?ext and an internal space in t, J f = Jf? ext (x) Jifint (2.1) ¿4?ext is the space of quadratically integrable complex-valued functions defined on X, while J f int is a finite-dimensional vector space describing the internal degrees of freedom of the system. The state of the system can thus be written as • / » = n(x) Xic

(2.2)

where {%k\ is a set of basis vectors in .yf int and \pk(x) are complex-valued functions. In the description outlined above, vectors of the internal space are defined without reference to any particular point x í X. This means that the internal states of the system when located at different points in X, can be compared in an unambiguous way. The transition to the gauge formalism, in its most general form, is obtained if we assume this no longer to be true. An internal vector should be associated with a particular point i f ! There is consequently not one internal space, but one internal space for each point. We will denote this space by h(x). These spaces will certainly all have identical structure, but the point is that it is in general not possible to compare vectors belonging to different internal spaces in an unambiguous way. Within the gauge formalism the full Hilbert space will then not be the tensor product of an external and an internal space. But the state of the system can still be associated with a vector 1" which can be written in a form similar to that of eq. (2.2) W(x) =

Wk(x) tk{x)

'

(2.3)

where {y.k(x)\ now is a set of basis vectors in h(x). From the discussion above one can already see a clear resemblance between the gauge formalism and the formalism which is known from general theory of relativity (GTR). In the special theory of relativity space-time is flat, and consequently there is a distant 'parallelism of vectors. This means that vectors are well-defined without reference to any 39*

582

J . M.

LEINAAS

particular point in space-time, in the same way as for the internal vectors when the Hilbert space is the tensor product of an external and an internal space, as discussed above. However in GTR space-time is curved, and vectors at different points cannot be compared in an unambiguous way. One has to associate with each point x of space-time a vector space V(x), namely the tangent space at that point. Vectors at different points therefore belong to different vector spaces. This is similar to the gauge formalism, where the internal space h(x) corresponds to the tangent space 'V(x). There is however a basic difference between the formalism of gauge theories and that of GTR. The vector spaces "V(x) are not internal spaces. Their geometry is uniquely determined by the geometry of space-time. This is not the case in the gauge formalism. There the spaces h(x) form a structure which is not determined by the geometry of the underlying space X. Let us give some of the mathematical notations which are used for such a structure [i6]. The space X is called the base space, and the vector spaces h(x) associated with each point in X are called fibers. These space together form a continuous structure, as we shall discuss below, which is called a fiber bundle. A function W(x), describing a particular state, is a cross-section of the fiber bundle. Such a structure, where the meaning of the notations given above is more clearly seen, is schematically shown in fig. 2.1.

yVM X X

^hlxl Fig. 2.1 Schematic illustrations of a fiber bundle. The base space X and the fibers h(x) are here both one-dimensional.

2.2. Parallel transport. The gauge group In GTR the curvature of space-time appears as a second order effect in the separation dx beteeen close-lying points. Thus, to first order in dx the space is to be considered as flat, and there is consequently a local parallelism of vectors. This can be expressed in terms of a parallel displacement operator, P(x x + dx), which transports vectors from -fix) to + dx). Locally parallel transport of tangent vectors is therefore well defined. However, parallel transport between two distant points x and y is not an unambiguous operation. I t can be defined by integration of the operator P(x -> x dx), but the operator P(C, x^-y) which is found in this way will depend on the path G followed between x and y. Parallel transport around a closed curve will in general define an operator different from the identity. This is illustrated in fig. 2.2 by parallel transport of a vector on a curved surface, there represented by a sphere. When we turn to the formalism of gauge theories something similar is there assumed about the internal vectors. Although one cannot compare vectors at distant points directly, one assumes that this can locally be done. We will use the same notation as above and define P(x -> x - f dx) as the operator which transports an internal vector from h(x) to h(x + dx). This operator can be thought of as "gluing" the neighbouring spaces together by associating the vectors of these spaces in a unique way. As a conse-

Topological Charges

583

quence the internal spaces h(x) will together form a continuous structure, and the geometry of this structure is the subject of much of the discussion to follow. As pointed out above, parallel transport between two distant points is a path-dependent operation. We will now in particular consider parallel transport around closed paths (loops) L which starts and ends at a given point x. The corresponding operators P(L, x ->x) act linearly within the internal space h(x). (We will later also use the more simple rw

notation P(L) for these operators, when it is not important to specify the start and end point of the loops.) The set of operators P(L, x x), for all loops L starting and ending at x, forms a group. To show this we introduce the following notations: L0 is the trivial loop which includes only the point x itself, L'1 is the loop L traversed in the opposite direction and L2LX is the combined loop composed by Ll and L2. The parallel displacement operators then satisfy the following relations P{L0, x -> x) = I P{L2LU

x->x)=

P(L2, x-+x)

P(L-\ x->x)=

P{Lu x-+X)

P(L, x

(2.4)

x)'

1

where I is the identity operator in h(x). The group structure of the operators P{L, x -> x) readily follows from these equations. This group, which will be denoted by 0, is the gauge group of the theory. (The group G, as defined above, is also referred to as the holoTwmy group. It is often regarded as being only a subgroup of the full gauge group. This means that the full gauge proup is defined not only by referring to a specific parallel transport, but to all possible parallel transports allowed by the theory). In the definition of G given above we explicitly refer to a particular point x, and to be more precise we should therefore have denoted the gauge group by G(x). There is however a simple relation between the gauge groups at different points. Let x and y be two points in X and Lx a loop starting and ending at x (fig. 2.3). We can then connect x and y by a continuous curve G and define a corresponding loop Ly at y by the relation Lu = CLXC~*

(2.5)

The parallel displacement operators of these curves are related by the equation P(Ly, y->y)

= P(C, x-*y)

P{LX, x->x)

P(C, x

G(y) defined by eq. (2.6) depends on the path C between x and y, in a way similar to the correspondence h(x) -> h(y) between vectors of the internal spaces. The groups G(x) in fact also form a fiber bundle, which is closely related to the fiber bundle formed

x

Y

a) b1 Fig. 2.3 The connection between the loop Lx having z as the start and end point and the corresponding curve Z„ having y as the start and end point.

by the spaces h(x). Note, however, that the isomorphism between G(x) and G(y), as discussed above, depends on the fact that the two points x and y can be connected by a continuous curve. This we will, in the following, assume to be true for any pair of points in X, which means that X is a connected space. In GTR the scalar product of two four-vectors u and v, u • v = ghVu*vv

(2.7)

has a direct physical meaning even if that is not the case for the abstract vectors u and v themselves. As a consequence parallel transport of vectors cannot change the vectors in an arbitrary way. I t has to leave invariant the scalar product of eq. (2.7). This means that P(L, x —> y) is a Lorentz transformation in the tangent space In gauge theories we have a similar situation. The scalar product of internal vectors has a direct physical meaning. Therefore the scalar product has to be left invariant by the parallel displacement operator. If we assume the scalar product to have the standard form < T | 0 ) = fk*4>k (2.8) then P(L, x x) has to be a unitary operator acting in h(x). For an internal space of dimension n this means that the gauge group G is a subgroup of the unitary group in n dimensions G cz U(n)

(2.9)

2.3. Gauge invariance When the state of the system is described by the complex-valued wave function yk{x) instead of the abstract function W(x), then we refer to a particular choice of basis vectors y-k(x) 6 h(x), as shown by eq. (2.3). This choice of basis vectors gives a coordinatization of the fiber-bundle {h(x)). In the general case, when there is no distant parallelism of vectors, the choice of basis vectors at different points cannot be correlated in a unique way. A particular choice of basis vectors at one point x determines (by parallel transport) the basis at another point y only to within a gauge transformation. In that sense two coordinatizations related by a gauge transformation of the form X*(*) ^ 7Jx) = g(x) X*(®), g(x) e G(x) (2.10) are equivalent. Since ipk{x) a r e the coordinates of W(x) relative to a basis {/¿(a;)} the corresponding transformation of y>ic{x) is given by fk(x) -> fk{x)

= (g{x)" 1 ) w

x).

(2.11)

Topological Charges

585

Gauge invariance of the theory imposes certain constraints on the observables acting on the wave functions y>k{x). In the present formulation this requirement represents the more general requirement that physical measurable quantities should be defined without reference to any particular coordinatization of the theory. If we for example consider an internal observable 0 acting in h(x), this invariance determines the transformation properties of the components 0 kl of the observable under the gauge transformation (2.10), Okl -> 0kl = {g(x)~%

Omng(x)„i

(2.12)

We note that also the state function Hr{x) and the operator P(x, x + dx) are defined without reference to any particular basis and therefore are gauge-independent quantities. When we express the parallel displacement operator P(x -> x + dx) in terms of a given basis {%k{x)\ it has the general form P(x

x + dx) %k(x) = {dkl + iyA'kfl(x) dx*} fj(x + dx)

(2.13)

where A'kft is referred to as the gauge potential and y is a coupling parameter (identical to e/hc in the electromagnetic case). The above expression is similar to the one wellknown from the general theory of relativity, which describes an infinitesimal parallel transport of a basis vector e„ in the tangent space "Vix), P(x -> x + dx) eM(x) = {6% — r*M(x) dx*} ev{x + dx)

(2.14)

r ^ J x ) , which is called the affine connection, corresponds to the gauge potential A"k/i(x). This potential is therefore also called the connection of the fiber bundle. When we compare eqs. (2.13) and (2.14) we also note the difference between the two cases, which we have discussed earlier. Whereas r v m only has indices referring to the external coordinates, A1klt has indices referring bdth to the internal and external coordinates. The gauge independence of P(x -> x + dx) determines the transformation properties of the gauge potential under a gauge transformation. From eq. (2.13) one readily finds this to be = {g~i)km Amnflgnl + ± (g^)lm ^ .

(2.15)

In the electromagnetic case, where the gauge transformation is a pure phase factor g(x) = exp {i(e/ftc) 0(x)} this equation reduces to the well-known form A ^ A ^ A . - d . e

(2.16)

We see that the gauge potential does not represent a gauge independent operator in h(x), since it does not have the correct transformation properties of an observable. It is a coordinate-dependent quantity. Therefore, in the present formulation, to fix the gauge by choosing a particular potential is equivalent to choosing a set of basis vectors {Xk{%)} in the internal spaces h(x). Also the differentiation operator 8^ has transformation properties different from that of eq. (2.12), and therefore does not represent an observable. However, tha gauge-covariant differentiation, which is defined by Dki, = dkl dp - iyA*lft

(2.17)

has the correct transformation properties and therefore represents a physical observable. A characteristic feature of this operator is the non-commutativity of the differentiation

586

J . M. LEINAAS

in different directions. Written in matrix form we have Dv] = -vyF„

(2.18)

where F ^ x ) is the gauge field, defined by = d„A*„ - dvAki, - vy{AkmrA»„ - A"mvAmlh]

(2.19)

We note the quadratic .4-term in the expression for F ^ . This is characteristic for nonAbelian gauge theories. It gives rise to the „self coupling" term of the Lagrangian, which in turn leads to the non-linearity of the field equations. In GTR the quantity which corresponds to F ^ is the Riemann curvature tensor, =

0„R»„ -

+

-

R^R^).

(2.20)

We can therefore regard the gauge field as describing the curvature of the fiber bundle {h(x)}. This point can be more directly demonstrated in terms of parallel transport of vectors. Parallel transport of a vector v around an infinitesimal loop 8L in space-time changes the vector by [17] 1 dve = — — R ^ v d a ^ (2.21)

¥

where 8a>" is the surface element of the loop. A corresponding calculation of parallel transport for an internal vector IP" gives ^hyyiV-Fi^'.

[(2.22)

This means that the parallel displacement operator P{8L, x -> x) of the infinitesimal loop is given by P(dL,

x

x) = I + - 1 yP^M"

•

(2.23)

Eq. ( 2 . 2 3 ) shows that is an operator which generates gauge transformations in h(x). It can therefore also be expressed in terms of the generators Ta(x) of the gauge group F„v(x)=F\v{x)Ta{x). (2.24) The components F"lxv can be regarded as forming an a-component vector which transforms under the regular representation of the gauge group. These components can be used to give an alternative description of the gauge field instead of the matrix components Fki,,v. The connection between these two sets of fields is given by the matrix components of eq. ( 2 . 2 4 ) . For example, in the case where the gauge group is SU(2) and F^, (for fixed ¡i, v) are 2 x 2 matrices, this relation has the form F%r = sklaF%,.

(2.25)

Also for the gauge potential A^ a similar set of fields can be introduced by the equation A'llt = A%TM. In terms of these fields eq.

(2.19)

(2.26)

gets the form

F\, = dpA', - d,A% + yc\cA\A%

(2.27)

Topological Charges

587

where cabc are the structure constants of the gauge group. However, one should note that the matrices A^, as opposed to F^, do not represent gauge-independent operators in the Lie algebra of G(x). From the discussion above we see that the gauge potential Ak!tt and the gauge field Fkl/À, are kinematical quantities in their origin. They describe the geometry of the fiber bundle {h(x)}. Nevertheless, when we consider the equation of motion of the particle field W(x), they will have an influence on the dynamics of the system. This is because the gauge potential appears in the kinetic part of the Hamiltonian, in the form of the gauge-covariant differentiation Dp. Such a coupling of the particle field to the gauge field generalizes the minimal coupling of a charged particle to the electromagnetic field. We will return to this interpretation of the operator D^ in the discussion of the gauge theory of electromagnetism below. We let this conclude the review of the general description of the gauge formalism. In this review the geometric understanding of gauge theories has been stressed. The gauge field Fu„ has been related to a curved structure of internal spaces, h(x), and gauge transformations have been described as a change of basis vectors in these spaces. Gauge invariance of the theory then simply means that thé choice of basis is irrelevant for the description of the physically observable quantities. One should keep this simple geometric interpretation in mind when, in the following, we will study the topological classification of field configurations in terms of the gauge-independent operators P(L, x —> x).

3. The Electromagnetic Case 3.1. The gauge description of electromagnetism The simplest gauge theory is the gauge theory of electromagnetism, which was introduced by W E Y L [3] and further discussed b y D i R A C [4], I t gives a description of the interaction of a charged particle with an external electromagnetic field. We will briefly discuss how the general formalism, outlined in the previous section, applies to this case, and then turn to the description of topological singularities in the electromagnetic field. iXtx)

V(x)

For the sake of simplicity we consider a spinless particle, which is also without other internal degrees of freedom. The internal space h(x) is consequently a one-dimensional, complex vector space. This space we can picture as a complex plane associated with the point x, and the basis vector y_(x) is the real unit vector of this plane (fig. 3.1). The abstract function W(x), which describes the state of the charged particle, can be written as x).

3. The Electromagnetic Case 3.1. The gauge description of electromagnetism The simplest gauge theory is the gauge theory of electromagnetism, which was introduced by W E Y L [3] and further discussed b y D i R A C [4], I t gives a description of the interaction of a charged particle with an external electromagnetic field. We will briefly discuss how the general formalism, outlined in the previous section, applies to this case, and then turn to the description of topological singularities in the electromagnetic field. iXtx)

V(x)

For the sake of simplicity we consider a spinless particle, which is also without other internal degrees of freedom. The internal space h(x) is consequently a one-dimensional, complex vector space. This space we can picture as a complex plane associated with the point x, and the basis vector y_(x) is the real unit vector of this plane (fig. 3.1). The abstract function W(x), which describes the state of the charged particle, can be written as (x) depends partly on the state of the function and partly on the choice of basis vectors %{x). A change of basis obtained by a rotation in the local complex plane h(x) leads to a gauge transformation of the wave function of the well-known form y>{x)

- >

e x p

( i 6 { x ) )

f { x ) .

( 3 . 2 )

The gauge group G is consequently the group of phase factors, U{1). We would here like to stress that this picture of the wave function, namely that which is partly a coordinate-dependent quantity, is not necessarily restricted to the case where we consider a charged particle in the electromagnetic field. We will later discuss an example (sect. 5.2) where no gauge field is present, but where it nevertheless is useful to assume that the state of the system is primarily defined by an abstract function * l { x ) , rather than by the complex-valued wjive function y>(x). The characteristic feature of the gauge description of electromagnetism is the assumption that the orientation of the real (and imaginary) axis of the complex planes h(x) and h(y) associated with two distant points x and y cannot be correlated in a unique way. This means that the relative phase of the wave function y>(x) between two distant points does not have a definite physical meaning. One refers to this as the non-integrability of the local phase of rp(x). It is regarded as a generalization of the fact that the global phase of ip(x) in the quantum formalism has no direct physical significance. The fact that the non-integrability of the phase of tp(x) can be related to the presence of an electromagnetic field can be seen by considering the gauge-covariant differentiation which in the present case has the form r

D„

=

d„

-

i y A , , .

( 3 . 3 )

This can be recognized as the operator which defines a minimal coupling of the charged particle to the

Fig. 3.2 Parallel transport in a £7(1) gauge theory. The internal spaces are represented as complex planes with the basis vector x as the real unit vector. Parallel transport oi a vector V from a point x along two dif ferent paths (1 and 2) to another point y will in general define two different vectors < p y and in Hy).

electromagnetic field, if we identify A^ as the electromagnetic four-potential and y as being essentially the electric charge of the particle, y = el he. When this operator is substituted for dM in the Lagrangian of a free particle this gives the corresponding Lagrangian of a charged particle in an external electromagnetic field. It follows from this that the gauge formalism gives a geometric interpretation of the minimal coupling. The charged particle is in a generalized sense moving as a free particle. Deviations from the motion with no field present can be ascribed to the presence of a „curvature", much in the same way as described in the general theory of relativity. However, the curvature

Topological Charges

589

described by the electromagnetic field is not the curvature of space-time itself, but is of a more abstract nature, as we have discussed in the previous section. Since in the electromagnetic case we are dealing with an Abelian gauge group, the expression for an infinitesimal parallel displacement (eq. (2.13)) can be directly integrated to give the following expression for vector transport between two distant points x and y. (3.4) P{C, x-+y) yjx) = exp jiy J AM dx" . y.iy) • This operator is identical to the one which plays an important role in Yang's so-called integral formalism for gauge fields £]. We will here in a similar way ascribe a fundamental role to this parallel displacement operator, but be somewhat more restrictive and consider mainly operators associated with closed curves. The reason for this is that the phase factor defined in eq. (3.4) is not gauge independent, as a consequence of the fact that the operator acts between two different internal spaces h(x) and h(y). However, for a closed curve L the phase factor is gauge independent, and by use of Stoke's theorem it can be expressed directly in terms of the electromagnetic field P(L, x -> x) = exp jiy

A^ da;'') j

= exp { j y j F ^ d c j

(3.5)

where S denotes a surface with L as boundary. Note that the above equations (3.4) and (3.5) are dependent on the fact that we are dealing with an Abelian gauge group, since the operators associated with infinitesimal displacements along the curve should commute in order to give results expressed in terms of exponential functions. However, for non-Abelian one often introduces a formal operation, denoted by T, which is assumed to give the correct ordering of this sequence of operators. The parallel displacement operator can then be expressed in a way similar to that given by eq. (3.4) P(C, x->y)

tk{x) = T exp {iyA„ dx«} /,(?/).

(3.6)

The field F/Jv is uniquely determined by the operators P{L). This can be seen by specifying L to be an infinitesimal loop, as shown by eq. (2.23). On the other hand eq. (3.5) shows that in the Abelian case the field Fp_, also determines the operators P{L). However, eq. (3.5) is valid only if L can be shrunk to a point without passing through any singularity. This will usually be the case, but we will later also discuss some cases where the phase factors P(L) are not uniquely determined by the gauge field F ß v . In these cases a global gauge effect is present which is not determined by the local effect described by F!„. This means that the operators P(L) in this formalism play a more fundamental role than the gauge field F ^ itself. For non-Abelian gauge groups this is even more clear. One can for example show that when G ~ SU{2) there exist potentials A^ which are not gauge equivalent, i. e. cannot be connected by a gauge transformation, but nevertheless give rise to the same gauge field F ß v [10]. This implies that the gauge field does not determine the operators P(L) in a unique way. The operators P{L) associated with closed curves in fact also determine the operators P(C) associated with open curves — up to a gauge transformation. To show this let us assume that with the curves in X there are associated two sets of parallel displacement operators, P(G, x -> y) and P(C, x -> y), such that these operators for all closed curves

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J . M . LEINAAS

are identical up to an arbitrary gauge transformation x) = r(x)~1 P(L, x->x) r(x),

P(L, x

(3.7)

We will show that the two operators P(C) and P(C) associated with an arbitrary curve C must then also be identical up to a gauge transformation. We first assume two curves, C and C', both to connect the same pair of points, x and y. Consequently the composite curve OC"-1 is a closed one and eq. (3.7) can therefore be applied to give P(G, x ^ y ) P(C', x

y)-i = r(y)~i P(C, x

y) P(C', x

y)~i r(y)

(3.8)

which further gives

r(y) P(G, x ^ y ) = P(C', x -> y)'1 r(y) P{C', x->y).

P(C, x ->

(3.9)

We note that the two sides of this equation refer to different curves between x and y. This means that they define an operator which depends only on the end points of the curve, and which we will denote by A{x, y), P(C,x^y)-ir(y)P(C,x^y)=A(x,y). (3.10) I t follows from this definition that A(x, y) which acts in the internal space at the point x (or more precisely between the two internal spaces h(x) and h(x) associated with the two sets of parallel displacement operators P{C) and P(G), is a gauge transformation in the sense that it preserves the scalar product in the internal space.

We next introduce two new curves: A which connects x with an arbitrary point x0, and B which connects xg with y (fig. 3.3). Making use of eq. (3.7) for the closed curve B ^ C A ' 1 , we derive the following expression for P(C, x y), P(G,x^y) = P(B, x0

y) P(B, x0 -> 2/)"i P(G, x ^ y ) P(A, x 1

= P(B, x0 -> y) r(xo)" P(B, x0

x0)-* P(A, x -> x0)

P(G, x -> y) P(A, x

x0)~i r(x0) P(A, x

= A(y, x0)-i P(C, x -> y) A(x, x0).

x0) (3.11)

This expression is independent of x0 and can thus be written as P(C, x->y)

= A{y)-i P(C, x-+y) A(x)

(3.12)

where A(x) = A (x, x0) for an arbitrary x0. The equation shows that the two sets of operators P(G) and P(G) are identical up to a gauge transformation for arbitrary open curves C.

591

Topological Charges

I t follows from this that the operators P(C) for arbitrary curves C are determined (up to a gauge transformation) by the more restricted class of operators P(L) associated with closed curves L. Note that this is true not only in the Abelian case but also in the more general case where the gauge group is non-Abelian. 3.2. Topological singularities We will now use the parallel displacement operators P(L, x -> x) described above to examine the gauge description of electromagnetism from a topological point of view. The basic element of this discussion is the mapping from loops in Minkowski space into the gauge group 0 ~ U{V) defined by these operators. We will in particular study the magnetic monopole singularity and show how it can be described topologically in terms of this mapping. Let us, then, consider a one-parameter set of loops in Minkowski space (fig. 3.4). We denote these curves by L(r, X), where r is the curve parameter, i. e. one traverses the Llxl

,•

U(1)

L0(v)

Fig. 3.4 Parallel transport defines a mapping from a surface L(i, A) in Minkowski space into a continuous curve g(M in the group space of 17(1). £ 0 ( r ) is the trivial curve at the base point and H r ) the perimeter of the surface.

loop for fixed A when r interpolates between 0 and 1, and X is a deformation parameter which deformes the loop. We assume that all loops have a fixed start and end point x, and that the loops change continuously with X. In addition we assume X = 0 to correspond to the trivial loop L0(r), which consists only of the point x itself, and X = 1 to correspond to a given loop L(r),

L(r, X) : L(t, 0) = L0{r), L{t, 1) = L{t). We note that since Minkowski space is simply connected, any loop L(r) can be reached in this way from the trivial loop L0(t). The set of loops L(r, X) we will consider as defining a two-parameter surface in Minkowski space. This is a surface with the loop L(r) as boundary. Parallel transport around the loops defines a mapping from this surface into a curve in the gauge group 0 Z7(l). Formally we write this as L(r, X) A- g(X) (3.13) The group element g(X) is then identical to the phase factor defined by parallel transport around the loop L(r, X), g{X) = P{L(r, X), x

x)

This curve in U( 1) is characterized by the end points «7(0) = 1 g(l)=P(L(r),x-+x).

(3.14) (3.15)

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J . M. LEINAAS

Let us next consider another two-parameter surface L(T, X) which also has the curve L(T) as boundary (fig. 3.5 a). Since Minkowski space is without holes (like the holes of a Swiss cheese) it is possible to deform this surface continuously into L(T, X) without changing the boundary. Let us then consider a specific distortion which interpolates continuously between L(T, X) sCnd L(r, X). To each intermediate surface there will correspond a curve in £7(1) with fixed end points gr(0) and gr(l). If we assume the electromagnetic field to be singularity free everywhere, then the operators P(L, x —> x) will change

aI

b)

Fig. 3.5 Surfaces LU, A) corresponding to a given boundary Z(r). a) shows two possible surfaces, L(r, A) and L(r, A), corresponding to a finite loop L(R). b) shows a case where LIT) 1st the trivial curve. L{T, A) is then a closed surface.

continuously under continuous variations of the loop L. Under this assumption the curve g{X) will therefore change continuously for continuous distortions of L(r,).). Consequently the two curves g(X) and g(X) corresponding to L(r, X) and L (r, X) are homotopic in the sense that they can be continuously distorted into each other while keeping the end points g(0) and g( 1) fixed. It follows from this that when the electromagnetic field is singularity free, then all surfaces L(r, X) with a given boundary L(r) are mapped into curves in £7(1) which are homotopic. One can therefore consider the loop L(T) as defining a homotopy class of curves in £7(1), with fixed end points given by eq. (3.15), corresponding to all surfaces with L(r) as boundary. We will next assume L(r) to be identical to the trivial curve L0(r) at the point x. In this case the surfaces L(r, A) with L(T) as boundary are closed surfaces (fig. 3.5b). Consequently the curves g(/.) in £7(1) corresponding to these surfaces are closed curves, characterized by 0(0) = g(l) = 1. (3.16) As a particular surface of this type we can pick out the trivial one, which consists only of the point x itself. To this surface, the trivial curve g(X) = 1 obviously corresponds. Since all closed curves g(X) corresponding to closed surfaces L(r, X) are homotopic, it follows that all these curves are homotopic to the trivial one. We therefore conclude that when F ^ is everywhere a singularity free field, then the closed curves in £7(1) which are defined by the mapping L(r,X)-^g(X) (3.17) from closed surfaces L(r, X) in Minkowski space, are all homotopic to the trivial curve, i. e. they are contractible to a point. Topologically the group space of £7(1) is equivalent to a circle, (S11. This means that there exist infinitely many classes of homotopic curves in £7(1). Such a class is characterized by an integer n, the winding number, which tells how many times the closed curves of this class wind around the circle. As we have discussed above, the closed curves in

593

Topological Charges

Z7(l) which are defined by the mapping L(t, A) and let L(r, A],A2) denote

Fig. 3.6 Expansion of a closed surface £(r, As) from the trivial surface Lt(r, A) for = 0 to a surface i ( r , A,) characterized by a winding number n =t= 0 for — 1. The surface has to pass through a singularity 5 for some value of where the winding number changes discontinuously from n = 0 to n 4= 0.

a set of closed surfaces which changes continuously with A2. This set we assume to interpolate between the trivial surface at x, L0(t, A,), for A2 = 0 and the surface L(r, A/) for ^2 = 1, L(r,

;.], A 2 ):

L(t,

Au

0) =

L0(T,

AJ,

L(T,

A1; 1) =

L(r,

A,).

The surfaces L(r, Aj, A2) are mapped into a set of closed curves in £7(1), £(t,^2)-^>2 = 0 corresponds to the trivial surface, the corresponding curve is the trivial one g(Au 0) = 1

(3.19)

which obviously is characterized by winding number n — 0. The curve corresponding to >2 - 1, g{X„ 1) = g(X,) (3.20) is on the other hand assumed to be characterized by n 4= 0. As a consequence of this the winding number n has for some value of A2 to change discontinuously from n = 0 to n =fc- 0 as is increased. This demonstrates the presence of a singularity in the field F ^ somewhere inside the surface L(r, A,) (fig. 3.6). The singularity of F ^ has to be a line singularity in Minkowski space. The reason for this is that the singularity has to be enclosed by the surface L(t, A^, since the set of surfaces L(t, Ax, A2) has to pass through the singularity for any continuous interpolation between the trivial surface and the surface L(r, A,). A point singularity, for example, would not be enclosed by a two-dimensional surface in Minkowski space. The possibility, that the singularity be of dimension higher than one, can also be excluded since t he surface L(r, A,,A2) in that case would in general not pass through the singularity for a definite value of A2, but rather in an interval a < A2 < b. We know, on the other hand,

594

J . M. LEINAAS

that n has to change discontinuously for a definite value of A2, since it can take only integer values. In the following we will study this line singularity of FßV in more detail and show that it in fact describes the world line of a magnetic monopole. 3.3. Magnetic monopoles Let us now make a simplification by assuming the time coordinate of Minkowski space to have a definite value and examine the intersection of the line singularity described above with this three-dimensional subspace. The singularity then appears as a point singularity, and this point can now be enclosed by a two-parameter surface L(r, A). We recall that L(T, A) also denote a one-parameter set of loops which covers this surface when A is increased from 0 to 1. Parallel transport around the loops defines a closed curve in U(\), which we write as g(A) =exp{vp{A)).

(3.21)

We consider first an infinitesimal distortion of the loops, L(r, A) -> L(r, A + dA). This distortion can be performed by means of infinitesimal loops in the r, A-plane (fig. 3.7). x

Fig. 3.7 The distortion of a loop, £(r, a) —* L(T, A + x) that it has been of interest. Physically we can regard this field as describing a test particle, which is used to examine the electromagnetic field. The influence of this particle back on the field appears only through the dynamical equation d,F"v = — 4 n f

(3.29)

where is the current density. The magnetic monopole, on the other hand, appears as a source of the electromagnetic field independent of the dynamical equation (3.29). I t is a topological singularity of the electromagnetic field and is not associated with an independent particle field. Let us end this section by studying the world line of the monopole in more detail to see how the conservation of magnetic charge appears as a consequence of topological properties of the singularity, We have to re-introduce the time coordinate. In the four dimensions of Minkowski space which we then consider, the two-parameter, closed surface L(r, X) will no longer enclose a definite three-dimensional volume. If Q denotes the volume enclosed by L(r, X) for fixed time coordinate, then another three-dimensional region bounded by L(T, X) can be obtained from Q by shifting the interior points of Q arbitrarily 'n the time direction, while keeping its boundary fixed. This is illustrated in fig. 3.8. If we assume a monopole singularity to be present within the volume Q, then « =f= 0 for the closed curve g(X) corresponding to L(r, X). But this mean that, if L(r, X) is contracted to a point by passing through the region Q' instead of through Q, it has to pass through a singularity also there. Consequently the monopole singularity has to pass 40

Zeitschrift „Fortschritte der Physik", Heft 11

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J . M . LEINAAS

through any three-dimensional region which has L(r, I) as the boundary. The monopole therefore has to define a continuous curve in Minkowski space without end points. Since the total magnetic charge inside a volume Q is determined by the winding number associated with its boundary L{r, X), it has to be equal for two volumes Q and Q' having

surfaces, m denotes the world line of a monopole, which intersects C in xm and fi' in xm'.

the same boundary. However, this does not mean that the number of monopole singularities has to be equal for the two volumes. In fig. 3.9a a situation is illustrated where a monopole line intersects the volume Q' in two points, but does not intersect the volume Q. The winding number associated with the boundary L(r, X) in this case is zero. Therefore the two monopole singularities of Q' should be assigned charges, or winding numbers,

Fig. 3.9 Monopole singularity appearing with magnetic charge of alternating sign. I n a) it is shown how the sign is determined by the relative direction of the monopole line and the direction of orientation of the three-dimensional volume Q'. I n b) the time direction gives the orientation of the volumes. The sign of the monopole charge is then determined by the orientation of the monopole line in the time direction.

of opposite sign. This means that the winding number of a monopole singularity is not a number which is characterized by the world line of the monopole alone. The sign of the winding number also depends on how the line „intersects" the volume we refer to. This can be expressed more precisely by assigning to both the line singularity and to the volume a direction of orientation. The monopole singularity will then appear with positive charge when it intersects the volume in the positive direction and with negative charge when it intersects the volume in the negative direction (fig. 3.9a).

Topological Charges

597

If we consider only volumes characterized by a fixed time coordinate, then the positive time direction can be chosen as „direction" of the three-dimensional volumes (fig. 3.9b). I n this case the monopole has a positive magnetic charge when the line singularity is pointing forward in time and a negative charge when it is pointing backwards in time. The picture of the monopole we have reached in this way is just the standard picture of particles and anti-particles. The world line of the monopole is a continuous line which can run forwards and backwards in time. When it runs forward in time it appears as a positively charged monopole, and when it runs backward in time it appears as a negatively charged anti-monopole. We would like to stress t h a t this picture appears here as a natural consequence of the topological properties of the monopole singularity. I t does not involve the dynamical equations of the system. This is also true for the conservation of total magnetic charge. I t is not a consequence of a particular symmetry of the equations of motion. Instead, it follows from continuity of the monopole lines, which in turn is a consequence of the topological properties of the monopole. For this reason magnetic charge is often referred to as a topologically conserved quantum number, or just as a topological charge. 4. Elements From Homotopy Theory 4.1. Homotopic curves I n the discussion of the gauge description of electromagnetism we have introduced some concepts from homotopy theory. We will now give a more systematic survey of these concepts \20\ and show that they allow us to formulate the description of monopole singularities in a more compact way. This is useful for the study of generalized monopole singularities in gauge groups different from ?7(1). We start with a review of the definition of homotopic curves. If X is a topological space, then it is possible to define continuous curves in X. A formal definition of a continuous closed curve can be given in the following way: I t is a continuous mapping, L(r): I X, where I is the unit interval on the real axis, I = [0, 1] cz R, which satisfies the condition L(0) = L( 1) = x. The point x is referred to as the base point of the curve. Two curves L and IJ are homotopic if there exists a continuous mapping, L(r, r'): I x l -> X, such that L(r, 0) = L{r) and L(r, 1) = L'(T). The mapping L{r, T'), which defines a continuous interpolation between L(r) and L'(T) is called a homotopy. This obviously defines an equivalence relation between curves, which divides them into classes of homotopic curves. Such a class is called a homotopy class. The number of homo.topy classes in X describes one of its topological properties. If X is a connected space, a single homotopy class means t h a t X is simply connected, two classes means t h a t it is doubly connected, etc. We can in fact distinguish between two types of homotopy classes. The one described above, where we do not fix the base point under the interpolation between L and L', defines unrestricted homotopy classes. On the other hand, one can define restricted homotopy classes, by constraining the homotopy L{r, r') to a fixed base point. As we will discuss below these two types of classes do not necessarily have to be identical. Let us now consider loops with a fixed base point x. I t is then possible to define a composition rule for such curves. The composite curve L2LX corresponding to two curves Ll and L2 is defined by 1 2r)

LJL^t)

= •

L2( 2 T - 1 ) 40*

1 - ^ r ^ l .

(4.1)

Topological Charges

597

If we consider only volumes characterized by a fixed time coordinate, then the positive time direction can be chosen as „direction" of the three-dimensional volumes (fig. 3.9b). I n this case the monopole has a positive magnetic charge when the line singularity is pointing forward in time and a negative charge when it is pointing backwards in time. The picture of the monopole we have reached in this way is just the standard picture of particles and anti-particles. The world line of the monopole is a continuous line which can run forwards and backwards in time. When it runs forward in time it appears as a positively charged monopole, and when it runs backward in time it appears as a negatively charged anti-monopole. We would like to stress t h a t this picture appears here as a natural consequence of the topological properties of the monopole singularity. I t does not involve the dynamical equations of the system. This is also true for the conservation of total magnetic charge. I t is not a consequence of a particular symmetry of the equations of motion. Instead, it follows from continuity of the monopole lines, which in turn is a consequence of the topological properties of the monopole. For this reason magnetic charge is often referred to as a topologically conserved quantum number, or just as a topological charge. 4. Elements From Homotopy Theory 4.1. Homotopic curves I n the discussion of the gauge description of electromagnetism we have introduced some concepts from homotopy theory. We will now give a more systematic survey of these concepts \20\ and show that they allow us to formulate the description of monopole singularities in a more compact way. This is useful for the study of generalized monopole singularities in gauge groups different from ?7(1). We start with a review of the definition of homotopic curves. If X is a topological space, then it is possible to define continuous curves in X. A formal definition of a continuous closed curve can be given in the following way: I t is a continuous mapping, L(r): I X, where I is the unit interval on the real axis, I = [0, 1] cz R, which satisfies the condition L(0) = L( 1) = x. The point x is referred to as the base point of the curve. Two curves L and IJ are homotopic if there exists a continuous mapping, L(r, r'): I x l -> X, such that L(r, 0) = L{r) and L(r, 1) = L'(T). The mapping L{r, T'), which defines a continuous interpolation between L(r) and L'(T) is called a homotopy. This obviously defines an equivalence relation between curves, which divides them into classes of homotopic curves. Such a class is called a homotopy class. The number of homo.topy classes in X describes one of its topological properties. If X is a connected space, a single homotopy class means t h a t X is simply connected, two classes means t h a t it is doubly connected, etc. We can in fact distinguish between two types of homotopy classes. The one described above, where we do not fix the base point under the interpolation between L and L', defines unrestricted homotopy classes. On the other hand, one can define restricted homotopy classes, by constraining the homotopy L{r, r') to a fixed base point. As we will discuss below these two types of classes do not necessarily have to be identical. Let us now consider loops with a fixed base point x. I t is then possible to define a composition rule for such curves. The composite curve L2LX corresponding to two curves Ll and L2 is defined by 1 2r)

LJL^t)

= •

L2( 2 T - 1 ) 40*

1 - ^ r ^ l .

(4.1)

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J . M. LEINAAS

This composition rule leaves invariant the class structure of the curves. Thus, if L l and Li are homotopic and L2 and L2' are homotopic, then also the composite curves L i L i and will be- As a consequence the composition rule defines a multiplication law for (restricted) homotopy classes. One can readily verify t h a t this multiplication lawgives a group structure to these classes. For example, the class containing the trivial curve L0 acts as a unit element. And if we, with a loop L , associate another one, L~ x, which is the loop L traversed „backwards",

x) = L( 1 - r)

(4.2)

then the class to which L 1 belongs acts as an inverse to the class containing L. The group which is formed in this way is called the fundamental group of x, or the first homotopy group, and is denoted by I J i ( X , x). When X is a connected space, there is a close connection between the groups 77, ( X , x) and 77j ( X , y) associated with two different base points x and y. This can be demonstrated in a similar way to what we previously have done for the gauge groups, by introducing a continuous curve between the two points, C: x ->• y (see fig. 2.3). The curve C defines a mapping from loops L x with x as base point to loops Ly with y as base point, by tJie equation ,Ly = CLxG-K (4.3) (The composition of curves which we then refer to is an obvious generalization of the one given in eq. (4.1).) This mapping preserves the homotopy class structure, and therefore defines a mapping between the groups I J ^ X , x) and 77, (X, y). Eq. (4.3) in fact defines an isomorphism between the two groups. As an abstract group the fundamental group is therefore independent of the base point. We thus denote this group simply by nx{X). However we note t h a t the mapping I I i ( X , x) n x ( X , y) is not necessarily uniquely defined by the prescription given above, since we refer to a particular path G between x and y. If G and C denote two different curves joining x with y, then these curves will, by eq. (4.3), define two different loops Ly and Ly at y corresponding to one and the same loop L x at x. The relation between Ly and L y is given by

L y = (OC-1) Ly{CG-^)-K

(4.4)

This defines an automorphism of 77, (X, y) onto itself. If the group is non-Abelian this mapping can be different from the identity. In that case there is an ambiguity present in the correspondence between closed curves in and elements of the fundamental group 77, ( X ) . We note that this ambiguity is related to the difference between restricted and unrestricted homotopy classes which we have discussed above. If the automorphism defined by (4.4) is non-trivial, this means that Ly and L y belong to different restricted homotopy classes. On the other hand they must obviously belong to the same unrestricted class, since they can be continuously connected by moving the base point around the loop CC~\ In figs. 4.1 and 4.2 the discussion we have given of homotopic curves is illustrated by two simple examples. In the first case X is a region in the two-dimensional plane which is bounded by two concentric circles, X = {r; E1 < |r| < R2). I n fig. 4.1a two curves from different homotopy classes are shown. L, is contractible to a point and can be characterized by winding number n = 0. The curve L2 winds once around the smallest circle (n = 1) and can therefore not be continuously distorted into L l . Fig. 4.1b shows a curve of winding number n = 2. I t cannot be deformed into any of the two curves in fig. 4.1a. In this case, which topologically is equivalent to the circle S 1 , the homotopy classes are characterized by integers n — 0, ± 1 , ± 2 , ... The fundamental group is

599

Topological Charges

the additive group of integers, (n, n') —.> n + n'. This group is isomorphic to the cyclic group of infinite order, Z. In the other example, which is shown in fig. 4.2, X is the two-dimensional disk r| < R, with diametrically opposite points at the boundary identified. There are now two different classes of homotopic curves, as shown in fig. 4.2 a. These classes can be characterized by integers n = 0 and n = 1. In fig. 4.2 b it is shown that a

a)

b)

Fig. 4.1 Closed curves in the region between two concentric circles, R, < r < Rt. a) shows one curve with winding number n = 0(£,) and one with n = 1 (L,), b) shows a curve with winding number n = 2 ( L ) .

A

A

al

b)

Fig. 4.2 Closed curves in a two-dimensional disk where diametrically opposite points are identified, a) shows curves belonging to the two different homotopy classes n = 0(Z,) and n = l(Lj). In b) it is shown how a curve characterized by n = 1, when traversed twice, can be contracted to a point. I , I I and I I I are three stages in this process

curve characterized by n = 1, when traversed twice defines a curve which is contractible to a point. There is consequently no class corresponding to n = 2. The fundamental group in this case is the cyclic group of order 2, 4.2. Higher homotopy groups As we have discussed above, the fundamental group of a topological space X, ITi(X), can be used to characterize one of the topological properties of X. But there are other topological properties which cannot be expressed in terms of this group, but rather in terms of the higher homotopy groups of X. To give a simple example, let us assume X to be a part of three-dimensional space with regions in the interior being excluded from X . (X has holes like a Swiss cheese.) Even with these holes present, if X is simply connected, all closed curves in X are contractible to a point, i. e. II^X) is the trivial group. On the other hand we realize that there must be closed surfaces in X which are not contractible to a point. This motivates the introduction of the concept of homotopic surfaces in analogy with the concept of homotopic curves.

J. M. Leinaas

600

Formally we can define a closed surface in X as a continuous mapping from the unit square in R2 into X, S(RLT T2) : / X I — X , with the perimeter of the square mapped into a single point X. Two surfaces S and 3' are homotopic when there exists a continuous mapping 8(t,, t 2 , t'), t ' € [0, 1], connecting S to S':

S{ri, t 2 , 0) = S f a , r 2 ) S(Tu t 2 , 1) = S'{r1;

(4.5)

r2).

In the same way as for the set of closed curves, the homotopy relation divides the set of closed surfaces into homotopy classes. When we restrict the surfaces S(rx, t 2 ) b y assuming a fixed base point x, then a composition rule for closed surfaces can be defined by 2r a , r 2 )

0 ^ Ti ^

1 (4.6)

T2) =

St{ 2r,

l,r2)

-

^

1.

And the "inverse" surface can be defined by S-Vi.*») = S { 1 - t „ T s ) .

(4.7)

I n a similar way as for homotopic curves, these two operations give a group structure to the homotopy classes of closed surfaces. This group is the second homotopy group which is denoted b y II2(X, x). We now realize that this construction can be carried out to define the n'th homotopy group ITn(X, x) for any integer n I. This is done in strict analogy to the construction described above, b y considering continuous mappings % , r ,

r.Jii'cR'-»-!

(4.8)

with the boundary of I" mapped into a single point x. The mappiiig aS'(t], r 2 , ..., r m ) defines a w-parameter closed (hyper) surface, and the homotopy classes of such surfaces form a group which is the n'th homotopy group IIn(X, x). We note that I " with the boundary points identified is topologically equivalent to the w-dimensional hyper sphere S". As an alternative, one can therefore consider ITn(X, x) as being defined b y the homotopy classes of continuous mappings S n ~> X, where one fixed point of the sphere is mapped into the base point x. If we, in particular, asssume X to be the group manifold of a topological group G, then it is also possible to define the zero'th order homotopy group IJ0(G). The elements of this group are the classes of points in G which can be continuously connected, and the multiplication of these classes is defined by the group multiplication of G. From this definition we realize that IT0(G) is non-trivial only when G has disconnected parts. I t is identical to the factor group G/G0, where G0 is the subgroup of G consisting of the elements which can be continuously connected with the identity. L e t us also mention that it is possible to show that the higher homotopy groups I J ( X ) , n ^ 2 are Abelian groups [20], This is not necessarily true for the fundamental group n,(X). However, when X is identical to the group manifold of a Lie group then also the fundamental group is Abelian [16].

Topological Charges

601

5. Generalized Monopole Singularities 5.1. Point singularities in three dimensions] We will now return to the discussion of the magnetic monopole and show how it can be reformulated in terms of the homotopy groups described above. To be more general we will not restrict the gauge group G to be identical to U{\), but only exemplify the discussion by the electromagnetic case. We will also show two cases with G ^ U( 1), which admits monopole-like singularities. The basic element of the discussion is, as before, the mapping L{r, X)

g{X)

(5.1)

from loops in Minkowski space into the gauge group, where the group elements g{k) are defined by parallel transport around the loops g(k)=P{L(

T,X),X^X).

(5.2)

We will as before assume the loops L(x, A) to form a closed surface, i.e. such that a = 0 and 1 — 1 both correspond to the trivial curve at the base point of the curves. The group elements g(X) consequently define a closed curve in G. (We note that L{t, A) as defined above in fact satisfies the more formal definition of a closed surface given in the previous section, since all points at the unit square in the r, A — plane correspond to the base point x.) As long as all singularities in the gauge field are avoided the corespondence (5.1) is continuous. We will in the following let X denote the regular part of space-time, i. e. Minkowski space with all singular points of the gauge field excluded. If we therefore restrict the loops L(r, X) to lie within X, then the mapping (5.1) is continuous and consequently homotopy preserving. This means that homotopic, closed surfaces in X are mapped into homotopic, closed curves in G. Since the correspondence (5.1) is homotopy preserving it defines a mapping between homotopy classes. If we restrict the loops L(r, A) to have common base point, the homotopy classes can be identified as group elements of the corresponding homotopy groups, and the mapping therefore has the form n2{X,

x) - > II^G, I)

(5.3)

with I as the identity of G. This mapping is in fact a group homorphism. This readily follows from the composition rules for closed surfaces and curves, eqs. (4.1) and (4.6). The mapping (5.3) is what we in sect. 3. implicitly have used to characterize topologic a l ^ the monopole singularity of the electromagnetic field. To see this we once more specify the gauge group to be G ~ U(l). Since C(l) is topologically equivalent to a circle S1, the fundamental group of Z7(l) is 77^(1)) ~ Z

(5.4)

where Z is the cyclic group of infinite order. If we assume an isolated monopole singularity to be present, then the second homotopy group of the regular part of Minkowski space is also isomorphic to Z,

n2(X) ~

Z.

As a consequence of this the mapping (5.3) is a homomorphism of the form Z^Z.

(5.5) (5.6)

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J. M . LEINAAS

If we represent Z as the additive group of integers, then the possible mappings of this form can be written as m -»• nm, n, m integers. (5.7) The integer m is then the winding number which specifies the number of times the closed surfaces of n2(X) wrap around the singularity and nm is the winding number of the corresponding closed curves in f7(l). The integer n which characterizes the homomorphism is what we in the previous discussion have related to the magnetic charge of the monopole. We then (implicitly) assumed the closed surface L(T, X) to be characterized by TO = 1 and n was therefore identical to the winding number of the corresponding closed curve g(X) in TJ{\). The above discussion can readily be generalized to arbitrary gauge groups 0. Also in the general case we use the homorphism (5.3) to give a topological classification of singularities in the gauge field. The presence of such a singularity implies that the homorphism is a non-trivial one. We can therefore specify two conditions for singularities of this type to exist. The first one is that IJ2{X) be a non-trivial group. This can always be obtained by assuming the singularity to be a line singularity in Minkowski space, or, equivalently, a point singularity in three-dimensional space R3. The other condition is that IIi{0) is a non-trivial group. This means that the group manifold of G must be multiply connected in order to allow this type of topological singularity. The homomorphism (5.3) gives a topologically invariant characterization of the singularity in the gauge field. By invariant we then mean that the mapping is insensitive to continuous changes in the field configuration and also to continuous deformations of the line singularity. This is rather obvious, since the curve g(l) in 0 will change continuously under these transformations as long as the loops L(z, X) avoid the singularity. Tha mapping between homotopy classes is therefore unaffected. On the other hand the homomorphism II2(X, x) ni(G, I) refers explicitly to a particular base point x of the loops. If this mapping should give a truly invariant characterization of the singularity, then is should also be independent of the choice of this point. Let us in particular look at the question of whether a displacement of the base point x around a closed curve A can change the homomorphism. This displacement we assume to be performed in a way similar to what we previously have discussed (see eq. (4.4)) by associating with each loop L a new loop L defined by L=ALA~1.

(5.8)

This gives the following transformation of the group elements g(X) associated with the loops L(r, X), g{X)^g{X)=gAgWgA~1 (5.9) where gA is the group element defined by parallel transport around the loop A gA=P(A,x^x).

(5.10)

We note that if gA can be continuously connected with the identity of G then the two curves g(X) and g(X) in G are homotopic. Thus, only when G has disconnected parts the transformation (5.9) can map the curve g(X) into a different homotopy class, thereby changing the homomorphism (5.3). But if G is disconnected this means that X is multiply connected, since an element in G which cannot be continuously connected with the identity is defined by parallel transport around a loop which is not contractible to a point in X. However, as long as we restrict the singularities to be line singularities in Minkowski space, corresponding to singular points in three dimensions, then X is simply connected and no ambiguity should therefore be present.

603

Topological Charges

We will now discuss two examples where the gauge group G is non-Abelian and admits monopole-like singularities, i.e. topological point singularités in R3. In the first example we assume the internal space h(x) to be a three-dimensional real vector space and the gauge group to be the rotation group SO(3). The group space of SO(3) is doubly connected and therefore the fundamental group is isomorphic to the cyclic group of order 2 77,(50(3)) ~ Z 2 .

(5.11)

If we therefore assume / / 2 ( Z ) ~ Z by the presence of a monopole-like singularity in the gauge field, then the homomorphism (5.3) is of the form. Z^Z2.

(5.12)

There exists only one non-trivial homomorphism of this type, and it is defined by iO m even m -> { ^ 1 m odd.

5.13)

If a "monopole" singularity in this case is enclosed by a surface L(r, A) characterized by m = 1 then the corresponding curve g(X) in 0(3). Both types of curves, corresponding to n2 = 0 or n2 = 1 correspond to closed curves in S0(3), since the two elements I and —I in SU{2) both are mapped into the identity I of S0(3) by the homomorphism. The curves characterized by n2 = 1 we then recognize as the closed curves in SO(3) which are not contractible to a point. F r o m this relation it follows that two singularities with SU(2) monopole charge n2 = 1 can merge to form a pure magnetic monopole (n2 = 0). On the other hand a pure £7(1) monopole can split into two singularities with n2 = 1 (fig. 5.2). I n fact, the basic monopole charge in this U(2) gauge theory has the form (1/2, 1). This element of 772 (£7(2)) generates the full fundamental group, and the two types of singularities we have considered correspond to the sum of a n even or odd number of elements of this type, 2n • ( 1 , l j = (», 0) (5.15) (2n + l ) . ( I , 1 ) = ( » + j ,

l).

Topological Charges

605

Since the fundamental group is generated by only one element, it is also in this case isomorphic to the cyclic group of infinite order, 771(?7(2)) ~ Z. The mapping II2(X,x) - > IJ^G, I ) therefore also here has the form Z Z. However, if we express this mapping in terms of the Z7(l)-winding number nu it now has the form m - > 2w1m

(5.16)

where «i can take either integer or half-integer values. 5.2. Point singularities in two dimensions We have so far restricted the discussion of the generalized monopole singularities to point singularities in three-dimensional space R3 or line singularities in four-dimensional Minkowski space. However, the general approach we have used can readily be extended to include singularities of other dimensions. We will now look at such generalizations, first by considering singular strings in R3. In two dimensions these strings will appear as singular points. The presence of a string singularity in three-dimensional space has the effect of making the regular part X of R3 multiply connected. This follows from the fact that loops which encircle the string singularity cannot be contracted to a point without passing through the string. Consequently TJ^(X) is a non-trivial group. We will now show that parallel transport around loops in X defines a mapping nt(x,

x) ^ n0(O)

(5.17)

which can be used to give a topological classification of string singularities in R3 in a way similar to that we previously used in mapping II2(X, x)

^(G,

I)

(5.18)

to characterize the point singularities. To see this let us first- consider a loop L which does not encircle the singular string. Since this loop can be continuously contracted to a point in X this means that the corresponding element in G, g = P(L, x x), can be continuously connected with the identity I of G. A loop L' which encircles the singularity, on the other hand, cannot be continuously contracted to a point in X, it may therefore not be possible to connect the corresponding group element g' = P(L', x -> x) continuously with the identity I. This leads us to consider gauge groups G which have disconnected parts. Let us denote by Gi, i = 0, 1, 2 ..., the connected parts of G, with G0 containing the identity I. One can define a multiplication rule for these subsets of G, simply by GiGj = {gigi,

gi

€ Glt g, €

(5.19)

where g^j is the ordinary multiplication of group elements in G. The multiplication rule (5.19) gives a group structure to the set {£?,•}, with Ga acting as the identity. This is the group that we previously have denoted by II0(G). I t is identical to the factor group

G/G,.

P(L), mapped into Homotopic curves in the regular part X of are, by the relation L elements in the gauge group which can be continuously connected. They therefore belong to the same subset Gi. As a consequence of this the correspondence between loops in X and elements in G defines a mapping between homotopy classes of the form we already have mentioned, /7j(X, x) II0(G). (5.20)

606

J. M. Leikaas

This mapping preserves the composition rule of the homotopy classes and its therefore a group homomorphism. The homomorphism gives a topological characterization of the string singularity which is present. To illustrate this let us consider a simple example where the gauge transformations are pure phase factors. As we have seen, the gauge group should have disconnected parts in order to admit (topological) string singularities. Therefore the gauge group cannot be the full group of phase factors, U{\). It can however be a discrete subgroup of U( 1). Such a subgroup will be isomorphic to Z.N (the cyclic group of order N) for some integer N, and it is generated by the phase factor g1 = exp (2jii/N). The subsets On of 0 in this case consist only of one element each, gn, defined by (5.21)

exp

Consequently we have P(L) = 1 for any loop which is contractible to a point in X. In particular this is true for infinitesimal loops, and we must therefore have a vanishing gauge field, = 0, at all regular points of X (see eq. (2.23)). For a loop L which winds once around the string we have (5.22)

P{L) = exp

Therefore general P(L) = gn, with n as the winding number of the loop L around the string. , . 7T . exp ( i j )

4

Fig. 5.3 The magnetic flux string (S) as a topological singularity. The mapping from loops in three-dimensional space into the gauge group G ~ ZN is shown for the case N = 8.

If we interpret the gauge field F ^ as being the electromagnetic field (fig. 5.3), then the operator P(L) can be related to the magnetic flux through the loop L (see eq. (3.25)), P(L)=ex

(5-23)

The singular string can therefore be interpreted as a magnetic flux string with vanishing cross-section, carrying a flux ^ = - ^ ( m o d 2 ^ ) .

(6.24)

As opposed to the case of the magnetic monopole the total flux of the singular flux string is not quantized. This follows from eq. (5.24), since N can be any integer. One should also note another characteristic difference between the magnetic flux string and the monopole singularity. Whereas the presence of a magnetic monopole is reflected

607

Topological Charges

in the surrounding field, this is not the case for the flux string, since F ^ = 0 everywhere outside the string. This means that such a string would not affect the motion of a classical, charged particle (except for the highly singular situation where the particle directly collides with the string). However in the quantum description of the charged particle this is different. For any continuous and single valued gauge, the vector potential A satisfies the condition ')A-ds= m (5.25)

f

where L is a loop encircling the string once (for fixed time coordinate). As a consequence of this A cannot vanish identically outside the string. The magnetic flux of the string thus affects the motion of the particle, since the vector potential appears explicitly in the kinetic part of the Hamiltonian, in the gauge-covariant differential operator D

V —i

he

A.

(5.26)

The magnetic flux string described above can in fact be regarded as an idealized version of the solenoid referred to by Aharonov and Bohm in their discussion of the special significance of the vector potential A in quantum mechanics \21\ They considered a thought experiment, where a beam of electrons can pass on both sides of a solenoid in such a way that the electrons are prevented from entering the region where the magnetic field strength is different from zero. The magnetic flux of the solenoid nevertheless affects the interference pattern on a screen which is placed behind the solenoid. This can be shown by calculating the phase difference for the two parts of the beam passing on each side of the solenoid. The Aharonov-Bohm effect shows that, in the quantum description, the magnetic flux of the flux string will affect the motion of a charged particle in a non-local way (fig. 5.4).

•5? Fig. 5.4 A schematic representation of theAharonov-Bohm experiment. An electron beam (H) can pass through two slits in a screen Sx. In the "shadow" of the screen, between the two slits a solenoid (M) is placed. The magnetic flux of the solenoid will affect the interference pattern (I) on a second screen S 2 placed behind the solenoid.

This effect we can relate to our previous discussion (sect. 3.1.) of the more fundamental role played by the operators P{L) than by the gauge field itself. The gauge field F/lv(x) describes a local gauge effect at the point x. However, a non-local gauge effect, as given by the singular flux string, is not so easily described in terms of the gauge field, since the field in fact vanishes identically outside the string. On the other hand the operators P(L) are non-trivial for loops encircling the singularity and they are closely related to the non-local effect of the flux string. We have, in sect. 2., discussed the close relation which exists between the fiber-bundle formulation of gauge theories and the geometric description of space-time in the general

608

J . M . LEINAAS

theory of relativity. In the present case the "internal" geometry associated with the flux string can in fact be represented more directly as a two-dimensional geometric structure [22]. As we have discussed above the flux string is characterized by the fact that parallel transport around the string changes the internal vectors, but transport of the vectors around loops which do not encircle the string leaves the vectors unchanged. This is similar to the case for transport of tangent vectors on a circular cone. The cone is locally flat at all regular points and therefore vector transport around loops which do not encircle the singular apex leaves the vectors unchanged. On the other hand the cone is globally curved and therefore transport of vectors around the cone will change

Fig. 5.5 Vector transport on a circular cone. Cutting the cone along a line (1) one can unfold it into a plane. In the plane the vector transport reduces to the trivial parallel transport. I t is shown how transport of a vector v around a loop (L2) encircling the singular apex (S) changes the vector, while transport around a loop (/.i) which does not encircle the apex leaves the vector unchanged.

the vectors. This is illustrated in fig. 5.5, where the vector transport on the cone is obtained by unfolding it into a plane. In the plane the vector transport reduces to the trivial parallel transport of vectors. The geometric structure of the cone is perhaps not of much relevance to realistic spacetime models. However, it has been shown that such a structure is of relevance to the quantum description of identical particles [11]. We will make a brief review of this description and show how a non-local gauge effect appears, in a way similar to the magnetic flux string. We assume two identical particles to be moving in two space dimensions. Due to the indistinguishability of the two particles the configuration space of the system is not simply the Cartesian product of the single-particle spaces. If we introduce the particle coordinates a?, and x2 the indistinguishability implies that the configurations ( x u x2) and ( x 2 , Xx) are physically identical. In terms of the center-of-mass coordinate x and the relative coordinate z , x

=

—

(®i

+

x2)

(5.27) Z

=

X\

—

x

2

this means that the true configuration space is obtained from the Cartesian product of the single-particle spaces by making an explicit identification of the pair of points ( x , z) and ( x , —z). Since this identification does not affect the c.m. coordinate x , we will in the following suppress it. In the »-plane the true configuration space is obtained by identifying points lying symmetrically about the origin.

Topological Charges

609

We can now make an explicit construction of the configuration space in the following way: We cut the s-plane into two halves along a line through the origin, and associate the physical configurations with only one of the half-planes (fig. 5.6). In this way we exclude most of the double representation of the physical configurations. However, there

Fig. 5.6 Construction of the configuration space of t w o identical particles in t w o dimensions. I n the plane of the relative coordinate z, double representation of t h e configurations can be avoided b y excluding half the plane. The remaining half-plane i s glued together along t h e line I separating t h e t w o halfplanes, thereby forming a circular cone.

is still an identification of pairs of points along the line which separates the two halfplanes. When these pairs are "glued" together one has in fact formed a circular cone, characterized by a half-angle of 30°. Thus, the identification of points in the product space, which represents the fact that the particles are identical, makes a configuration space which is geometrically similar to a cone. It has a global curvature although it is flat at all regular points. The singularity which appears, the apex of the cone, corresponds to configurations where the two particles occupy the same point in space, xt = x2. A closed curve, on the curve which encircles the singularity, represents the interchange of the position of the particles

Fig. 5.7 Interchange of t w o identical particles in t h e two-dimensional plane. I n the configuration space of the relative coordinate which has t h e form of a circular cone, the interchange is represented b y a loop which encircles t h e singular a p e x (« = 0).

(fig. 5.7). Parallel transport of a tangent vector v around such a loop changes the vector into —v as a consequence of the fact that the half-angle of the cone is 30°. At the classical level the global property of the configuration space has no physical significance, since the motion of the particles is determined only by the local geometry. However, at the quantum level the presence of a singularity may affect the motion of the particles in a way similar to that described for the singular flux string. The ana-

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J. M. LEINAAS

logy between these two cases is in fact close, although the physical interpretation of the gauge effect which is present is quite different, as we shall see below. A quantum description of the two-particle system can be introduced by applying the general formalism described in sect. 2. We assume that with each point (x, z) in tho configuration space there is associated an internal space h(x, z). This is a one-dimensional complex vector space when the particles are without internal degrees of freedom. We further assume no interaction between the particles, and the gauge field consequently vanishes, F ^ = 0, at all regular points of the configuration space. Therefore the phase factors P(L) associated with loops which do not encircle the singularity z = 0 have to be equal to 1. However, this is not necessarily the case for the phase factors P(L) associated with loops which do encircle the singularity. The phase factor P(L) = 7] associated with a loop winding once around the singularity is in fact undetermined by the correspondence to the classical description. It describes a purely quantum mechanical effect, similar to the non-local gauge effect of the flux string. The property which the phase factor adds to the system is in fact what determines the statistics of the particles. Thus, r¡ = + 1 corresponds to the boson case and r¡ = — 1 corresponds to the fermion case. This can be shown by re-introducing the excluded half of the «-plane. When the wave functions are extended to the full s-plane one can readily show that »7 = 4-1 gives rise to symmetric wave functions and r¡ = —1 to antisymmetric functions [Ü]. However, in two dimensions the value of r¡ is not restricted to these two possibilities, r¡ can be any number with r¡\ = 1. (This is related to the fact that there is no flux quantization of the singular flux string.) Therefore, in this approach the possibility appears that the particles moving in two dimensions can obey intermediate statistics, which lie between the boson and fermion cases.

/

/

/

Fig. 5.8 Distortion of a loop L , which interchanges two identical particles, into the loop L ~ ' . In the figure this is done by rotating the loop about a symmetry axis.

The unusual statistics, which are characterized by rj 4= ± 1 appear in the formalism as a purely two-dimensional effect. This can be seen by extending the description to three dimensions. Also in three dimensions the configuration space of two identical particles has geometrical properties similar to that of a cone. Thus the identification z ~ —z makes the space globally curved, although it is flat at all regular points, and z = 0 still represents a singularity of the space. However, there is a characteristic difference between the two-and three-dimensional cases. In two dimensions the regular part of the configuration space is infinitely connected. This follows from the fact that two loops winding a different number of times around the cone cannot be continuously distorted into each other without passing through the singularity. In three dimensions this is different. If L denotes a loop encircling (once) the singularity 3 = 0, then the loop L1, which is L traversed in the opposite direction, is homotopic to L. This is illustrated in fig. 5.8, where L is transformed into L~l by rotation of the loop in the third dimension. It follows from this that the loop LP, which is L traversed twice, is contractible to a point without passing through the singularity. As a consequence of this there are only two homotopy classes of loops in the regular part of the configuration space. The first one consists of loops which do not encircle the singularity and the second one of loops which do encircle the singularity.

Topological Charges

611

The phase factor P(L) = r¡ which is associated with the loops of the second type now satisfies the condition rf = P(L3) = 1 (5.28) since L 2 is contractible to a point without passing through the singularity. Thus the unusual statistics (r¡ =j= ± 1 ) which were conceivable in two dimensions disappear when the dimension is increased to three. We are left with the boson and fermion cases as the only ones possible, and the description can therefore be reformulated in the standard way, with bosons described by symmetric and fermions by antisymmetric wave functions. One should however note the additional feature of the formalism outlined above, that the phase factor ry which determines the statistics of the particles is related to the presence of a non-local gauge effect similar to the effect of the singular flux string. This interpretation of the phase factor can in fact have a direct physical significance, as discussed in ref. [23]. There a case is studied, where the non-local gauge effect described by r¡ mixes in a non-trivial way with the local effect of the gauge field. 5.3. Generalization to higher dimensions After the above description of singularities in the configuration space of two identical particles, we will now return to the discussion of topological singularities in ordinary space R". The singular flux string we can regard as being a point singularity in two dimensions. Similary the magnetic monopole is a point singularity in three-dimensional space R3. We will now show how the discussion of these two cases can be generalized to include topological point singularities in R" for arbitrary n. The starting point of our approach is to introduce a set of surfaces in Rn which encloses the singularity. For a point singularity in n dimensions these are (n — l)-parameter surfaces We assume that the surfaces are formed by loops in the way we have previously described. This means that they can be written as L(t, A2, ..., /t„_2), with r as the loop parameter and A; as deformation parameters of the loop. We recall that all parameter values r = 0, 1 and — 0, 1 correspond to one and the same point x in R", namely the base point of the loops. In close analogy with our discussion of the two- and three-dimensional cases we next consider parallel displacement of internal vectors around the loops. This operation defines a mapping form the loops in R" into the gauge group 0 , L{r, Alt A 2 ,..., A„_2)

g{A!, A 2 ,...,

(5.29)

where the group element g is identical to the parallel displacement operator g(Ku An,

=P[L(r,Au

...,

x ^ x).

(5.30)

The group elements g(Alt A2, . . . , A„_2) form an (n — 2) — parameter, closed surface in 6?. As long as the loops are restricted to the regular part X of Rn the mapping (5.29) is homotopy preserving. I t therefore defines a correspondence between homotopy groups which is of the form nn_x(X,x)^nn-%{G,I).

(5.31)

This is the general form of the homomorphism between homotopy groups which we have used to give a topological characterization of the point singularities. We note that with one isolated point singularity present in R" the regular part of the space, X, is topologically equivalent to an (n — l)-dimensional hyper sphere X ~ S"'1. 41 Zeitschrift „Fortschritte der Physik", Heft 11

(5.32)

612

J . M . LEINAAS

The homotopy group order

II^{X)

is therefore isomorphic to the cyclic group of infinite

n n _,{X) ~ ZT^OS»"1) - Z .

(5.33)

This means that the homomorphism (5.31) has the form (5.34) 2 IIn_2{G, I). We can now reach a necessary condition, for a gauge group G to allow topological point singularities in n dimensions. The image of Z under the homomorphism (5.34) is in general a group isomorphic to ~Z.N for some integer N. A necessary condition for the homomorphism (5.34) to be non-trivial is therefore that 77„_2(6r, I) includes a non-trivial subgroup isomorphic to Z¡¡(N > 1). We can consequently examine the question of whether a particular gauge group allows topological singularities of the type described above, simply by inspection of the homotopy groups ZZB_2(Gi). For some "standard" groups such as SU(N) and SO(N) one can find these homotopy groups listed, e.g. in ref. [24], Let us end the discussion of the topological singularities by examining the question of whether one can find a gauge group G which admits point singularités in four dimensions. This would be a natural extension from the point singularity in two dimensions described by the flux string and in three dimensions described by the magnetic monopole. For n = 4 the homomorphism (5.31) has the form

n,{X,x)^n2{G,I). (5.35) The question is therefore to find a group G where II%(G) includes a subgroup isomorphic to Zjy for some integer N. However, in this case we need no table to answer the question. There is a general theorem which states that n2(G) is trivial for any Lie group G [24]. This consequently rules out the possibility of having topological point singularities of the type considered above in four-dimensional space.

6. Non-Singular Configurations Characterized by Topological Quantum Numbers In our discussion of the topological singularities in gauge fields we have paid no attention to the dynamics of the fields and therefore not considered how such singularities can appear in the complete dynamical theory. In this respect the analysis of these gaugefield singularities is similar to the analysis of topological defects in so-called ordered media, such as nematic liquid crystal or superfluid 3 He, where also dynamical considerations only play a secondary role [25]. When we now turn to the study of nonsingular gauge field configurations characterized by topological quantum numbers this is changed. The existence of such structures depends on dynamical conditions, since they arise from spontaneous symmetry breaking at infinity, which in turn is a consequence of the requirement of finite energy of the configurations. However, also in our discussion of these non-singular structures, the topological classification of the configurations will be the main point of interest. This classification will be made in terms of the parallel displacement operators P{L) in a way which is a direct generalization of the previous discussion of the topological singularities. We will illustrate the discussion by considering some of the standard examples of non-singular structures characterized by topological quantum numbers. First we will study the non-singular flux string, also referred to as the Nielsen-Olesen vortex line [12],

612

J . M . LEINAAS

The homotopy group order

II^{X)

is therefore isomorphic to the cyclic group of infinite

n n _,{X) ~ ZT^OS»"1) - Z .

(5.33)

This means that the homomorphism (5.31) has the form (5.34) 2 IIn_2{G, I). We can now reach a necessary condition, for a gauge group G to allow topological point singularities in n dimensions. The image of Z under the homomorphism (5.34) is in general a group isomorphic to ~Z.N for some integer N. A necessary condition for the homomorphism (5.34) to be non-trivial is therefore that 77„_2(6r, I) includes a non-trivial subgroup isomorphic to Z¡¡(N > 1). We can consequently examine the question of whether a particular gauge group allows topological singularities of the type described above, simply by inspection of the homotopy groups ZZB_2(Gi). For some "standard" groups such as SU(N) and SO(N) one can find these homotopy groups listed, e.g. in ref. [24], Let us end the discussion of the topological singularities by examining the question of whether one can find a gauge group G which admits point singularités in four dimensions. This would be a natural extension from the point singularity in two dimensions described by the flux string and in three dimensions described by the magnetic monopole. For n = 4 the homomorphism (5.31) has the form

n,{X,x)^n2{G,I). (5.35) The question is therefore to find a group G where II%(G) includes a subgroup isomorphic to Zjy for some integer N. However, in this case we need no table to answer the question. There is a general theorem which states that n2(G) is trivial for any Lie group G [24]. This consequently rules out the possibility of having topological point singularities of the type considered above in four-dimensional space.

6. Non-Singular Configurations Characterized by Topological Quantum Numbers In our discussion of the topological singularities in gauge fields we have paid no attention to the dynamics of the fields and therefore not considered how such singularities can appear in the complete dynamical theory. In this respect the analysis of these gaugefield singularities is similar to the analysis of topological defects in so-called ordered media, such as nematic liquid crystal or superfluid 3 He, where also dynamical considerations only play a secondary role [25]. When we now turn to the study of nonsingular gauge field configurations characterized by topological quantum numbers this is changed. The existence of such structures depends on dynamical conditions, since they arise from spontaneous symmetry breaking at infinity, which in turn is a consequence of the requirement of finite energy of the configurations. However, also in our discussion of these non-singular structures, the topological classification of the configurations will be the main point of interest. This classification will be made in terms of the parallel displacement operators P{L) in a way which is a direct generalization of the previous discussion of the topological singularities. We will illustrate the discussion by considering some of the standard examples of non-singular structures characterized by topological quantum numbers. First we will study the non-singular flux string, also referred to as the Nielsen-Olesen vortex line [12],

613

Topological Charges

6.1. The Nielsen-Oleson vortex line In the model discussed by Nielsen and Olesen the particle field W(x) (also called the Higgs field) is a one-component field, describing spinless charged particles, and the gauge field F^(x) is identical to the electromagnetic field. The Lagrangian of the system has the following gauge-invariant form & = -

V '

+ J W y ) * P M ) + « M2 - b

(6.1)

where a and b are two real-valued constants, both being assumed to be positive. The corresponding equations of motion are D^Dy

— 2 a y + 4b

= 0

(6.2)

and 8'F^

= 2niy{V*DtiV

-

(D„y>)* f )

(6.3)

with y = e/hc, where e is the charge of the Higgs particles. The Lagrangian (6.1) is essentially identical to the free energy of a superconductor in the Ginsburg-Landau theory [26]. It is known that this theory gives rise to solutions which describe magnetic flux strings which are trapped within the so-called type II superconductors [27]. These flux strings (or vortex lines of the electric current) are the topologically non-trivial structures which we want to examine. Before we show (in an approximate way) that the equations of motion (6.2) and (6.3) actually have a solution of this type, we will examine the boundary conditions and show how topological arguments indicate that such a solution in fact may exist. We then consider the Hamiltonian density corresponding to the Lagrangian (6.1), Jf =

(£2 + B*) + I {1-DoVl2 + \Df\2} + j - a M 2 + b M* + j j J •

(6.4)

Since b is assumed to be a positive constant Jf is bounded from below, and we have in eq. (6.4) added a constant 1/4 • a2/b in order to make the minimum value of J f equal to zero. The ground state, or "vacuum" of the system is characterized by all the three parts of the Hamiltonian density, indicated in eq.(6.4), being zero. This gives F„= 0 DplF = 0 1 1

(6.5)

2 fi

One notes that the vacuum state is characterized by a non-vanishing Higgs field W(x). As a consequence it is not normalizable in the usual sense, J |V|2 dsx < oo, thus indicating that the theory is not a true one-particle theory. The non-vanishing value of \W\2 in fact corresponds to a "condensate" of paired electrons in the superconductor. The vacuum value of the Higgs field, ac) breaks the gauge invariance, in the sense that it defines a "preferred" direction in the internal spaces h(x). Thus, if we consider small deviations from the vacuum state, 1 ' = */"vac + dW, the corresponding energy will depend on the relative phase between and d'/". 41*

614

J . M . LEINAAS

We will now investigate the possibility of having a stringlike field configuration, which satisfies the condition of finite energy pr. unit length. For simplicity we assume the structure to be stationary and axially symmetric, as well as translationally invariant along the symmetry axis (the z-axis). This means that the configuration is essentially a two-dimensional structure, which we can describe in terms of the polar coordinates (r, 00 •0 rDW

1 =

^ 0

• ( k t - s ) ^

(6.6) 0.

Let us in particular consider the conditions satisfied by the Higgs field W. We first note that if the condition D W = 0 is exactly fulfilled in some region, then the vectors there are parallel, i.e. W(y) = P(G,x^y) W(x) (6.7) for any curve G in the region, connecting x and y. Thus the boundary condition rl) W —> 0 implies that the vectors Wir,