Fortschritte der Physik / Progress of Physics: Band 29, Heft 10 [Reprint 2022 ed.] 9783112656020

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HEFT 10 • 1981 • BAND 29

A K A D E M I E

31728

- V E R L A G

EVP 10,- M

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B E R L I N

ISSN 0015-8208

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Zeitschrift „Fortschritte der Physik 4 1 Herausgeber: Prof. D r . Frank Kagchluhn, Prof. Dr. Arthur Lösche, Prof. D r . Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1086 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2236229; Telex-Nr.: 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: D r . Lutz 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 Druckhaus „Maxim Corki", D D R - 7400 Altenburg, Carl-von-Ossietzky-Strafle 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines Jahres bilden einen B a n d . Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 15,— M (Preis f ü r die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/29/10. © 1981 by Akademie-Verlag Berlin. Printed in the German Democratic Republlc. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 29, 4 4 1 - 4 6 2 (1981)

Extended Objects in Field Theory with Non-Abelian Group Symmetry H . MATSUMOTO a n d H . UMEZAWA

Department of Physics, University of Alberta, Edmonton, Alberta, Canada T60 2J1 and M . UMEZAWA

Centre de Recherches Nucléaires, Université Louis Pasteur, 67037 Strasbourg, France Abstract This paper presents a general formalism for the description of topological objects created in quantum field systems with non-Abelian symmetries. This formalism utilizes relations of differential form and therefore is suitable for the description of the local properties of the topological structure. The presentation of the general considerations is followed by several examples. E v e n when there is no gauge field, the analysis requires use of a gauge transformation matrix. This coincides with the usual gauge transformation matrix when a gauge field is introduced.

1. Introduction

In our recent publications [1, 2] an attempt has been made to formulate a theory for quantum field systems in which certain extended objects are self-consistently created and interact with quanta. These extended objects are created by the condensation of certain bosons, which was treated mathematically by the boson transformation. In ref. [2], it was shown that a specific choice of the solution of a classical Euler equation uniquely determines the Heisenberg field for the system in the state with extended objects (i.e. the Heisenberg field in a soliton sector). This Heisenberg field contains the full quantum corrections and describes not only the behavior of extended objects but also the interactions between extended objects and quanta. It was also proven that [2], when a theory without extended objects is renormalized, the creation of extended objects does not upset the renormalizability. This proof takes into account all possible quantum corrections. A particularly interesting kind of extended objects are those with certain topological singularities (i.e. topological objects). It has been shown \3, 4] that the creation of topological objects by a boson condensation is possible when and only when the boson energy is gapless. Here, energy means that the energy, say co(fc), vanishes at k — 0. This explains the commonly known fact that most of the topological objects which have been observed occur in certain ordered states. Indeed, these orders are maintained by certain Goldstone bosons whose energies are gapless. These topological objects carry certain topological constants. When some of these topological constants assume certain discrete 1

Zeitschrift „Fcirtsehritte der Physik", Bd: 29, Heft 10

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H . MATSUMOTO a n d H . UMEZAWA

values (i.e. the topological quantum number), the topological objects appear to be stable. In ref. [5], we presented a systematic way of treating these topological objects when a system has an Abelian symmetry. A merit of this method lies in the fact that it describes the topological structure by local rather than global properties of fields. This provides more detailed information about the local field configurations. There have been many excellent works [6—10] on the topological classification of extended objects. Most of these works were based on the use of the global properties of the topological structure. In practice one can identify the local structure of topological objects as well, while the global boundary conditions do not specify the details of the local structure. Therefore, it is desirable to describe the topological nature in terms of the spacetime positions of extended objects. To look for such a local description in the case of non-Abelian symmetries is the main purpose of this paper. This local description depends heavily on the use of the concept of "gauge". The primary significance of this concept is directly associated, not with gauge field theory, but with the local description of topological objects. Once the concept of gauge is established in the realm of topological problems, one can define the gauge transformation in which the transformation parameter is the gauge function. This concept of gauge coincides with the usual gauge when one considers gauge theories. The considerations in this paper are an extension of a formalism for the study of topological objects in systems with Abelian symmetry. We therefore present in the next section a very brief sketch of this formalism for Abelian symmetry. In section 3, the formalism is extended to cover non-Abelian symmetries. In the case of Abelian symmetry the local properties of the topological structure are described by the quantity [8^, 8y] U(x) in which U(x) is the phase factor. In the case of a non-Abeiian symmetry (say G) we replace the phase factor by the gauge transformation matrix, through which the gauge function is introduced, even though there is no gauge field. I t is pointed out that the whole domain of the topological singularities associated with the gauge transformation matrix is not necessarily observable. The observable part of the topological singularity appears only through a projected part of the gauge transformation matrix. The direction of the projection is determined by the stability subgroup. This is a new situation which does not appear in the case of Abelian symmetry. The machinery needed for a local description of the topological structure is presented. Part of this machinery is a set of equations (c.f. (3.41) und (3.44)) which specify the local properties of the topological structure. The topological quantum number originates from the requirement of singlevaluedness of the order parameter. The statement that topological objects can be created by a boson condensation when and only when the boson energy is gapless has been proven only in the case of Abelian symmetry. The proof is now extended to cover the cases of non-Abelian symmetries. The general formalism in section 3 is applied to several examples in section 4. In section 5 the above formalism is applied to a gauge theory. This application clarifies the relation between the common methods based on the use of gauge fields and the method presented in section 3.

2. Topological Objects in Field Theories with Abelian Symmetry Consider a complex Heisenberg field yi, the equation for which is invariant under the phase transformation ip -» eisip. The complex field ip consists of two real fields (N = 2). We assume that it is possible to realize ip as a matrix acting on the vectors in the Fock space of certain free fields (say q>°); (a| y>(x) \b) = {a\ f(x; q?°) [6). Here |a) and |6) stand for the vectors in the Fock space. In the following the above relations among the matrix elements (i.e. the weak relations) will be simply written as ip(x) = ip(x;

«

(2.2)

though there can be free fields other than I t was shown in ref. [11] that, when the phase symmetry is spontaneously broken, the dynamical map (2.2) takes the form f(x) = :exp{iX0}F(x-,8x0):,

(2.3)

where is suitably normalized and 8%° stands for x° carrying derivatives. The phase transformation y> ei6y> is induced by - » x° + ® (the dynamical rearrangement of phase symmetry). To create an extended object, we make the replacement / - > / + / ,

"

(2-4)

where / is a c-nurdber function satisfying D{d) / = 0 .

'

.

(2.5)

Defining = U(x) : exp { i f } F[x; 8tf + 8f]:

(2.6)

with U(x) = exp \if{x)},

(2.7)

we can prove [12,1, S\ that ft and ip satisfy the same Heisenberg equation. As was shown in ref. [J],-the boson-transformed Heisenberg operator ipf describes a system in which the %°-quanta and certain extended objects exist and interact among themselves. The classical behavior of the extended objects is described by the c-number \vs(x) — v°], where v° = . (In this sense vc is a topological constant). Therefore G\,{x) is an assembly of two dimensional ¿-functions and describes (d — 2)-dimensional objects in eZ-dimensional ja^}-space. The existence of the topological constant leads to the conclusion that the domain 9> does not have any closed boundary (i.e. 3> consists of endless lines). For example, in three-dimensional space, the singularities are endless lines, planes (continuous distributions of endless lines), etc. In general, the

Extended Objects in Field Theory

445

domain 3) of the topological singularities is a (continuous or isolated) assembly of (d — 2)-dimensional objects which are expressed by the coordinate y^(e and also those conditions which are obtained from the singlevaluedness of e(x). The topological properties of singularities can be determined by the mapping of all possible closed surfaces C onto the sphere Se. These mappings will be written as C Ce. Let dc denote the dimension of C. Then, we need dc parameters in order to specific G. Since the dimension of Se is N — 1, through each mapping of C on Se some parameters or some combinations of them should remain constant when dc is larger than N — 1. Therefore we need to consider only those G whose dimension is not larger than N — 1. We therefore assume dc ^ N — 1 in the following consideration. Let us begin our analysis with an (N — l)-dimensional closed surface G in (x^}-space, assuming N d (d is the dimension of the {«„(-space). When C does not cross any singularities in @)e, G is mapped onto the closed surface Se. When G intercepts a certain singularity in 3>e, the corresponding Ge forms some region in 8 e . The surface area occupied by Ge is obtained by the oriented surface area integral as Qvc = f a , c.

(3.32)

where co is an exterior differential form on an oriented manifold \19\ given by w =

A

(ff - 1)!

^

A

"'

A dei

™

(3"33)

and co = f co. se

(3.34)

In (2.33), (ex ..., eN) are the coordinates in the representation space J f and £ ( i s the totally anti-symmetric tensor of rank N. Since Se is the (N — l)-dimensional unit sphere, it can be parametrized in terms of the polar angles {61 ... 0jv-i) : e1 — (sin

(sin 62) ••• (sin 6N~2) cos

e2 = (sin 0j) ... eN = cos 0!.

(sin dN_2) sin 0jV-i (3.35)

450

H . MATSTJMOTO a n d H . UMEZAWA

Then co is given by CO = DB(sin 0i)N~2 (sin 62)N~3 • • • (sin 6N-Z) d8l ••• ddN^,

(3.36)

where the ± sign is specified in accordance with the orientation of the surface Ce. The total surface area Q is given by Q = (27i){fi)N-ilr(N2).

(3.37)

The surface integral (3.32) is rewritten in terms of the integration of the coordinate system as follows: where a

(x)

1

-

.

c l x U

(3.38)

dxp A dx^ ••• A dx^N_it

&vc = J g^-n^ix) c

eiUx))

= (_l)UM)lff + i)/« (sin B^x))»-* ... sin 6N-2(x)

v

,

^JlJL

.

(3.39b)

B y use of the Stokes' theorem, we have [20] Qvc = f dllg/11...„N_1{x) dx„ A eZz^ A ••• A dx^ sc =

FT/

dx «> A

A

•

(3.40a)

(3-40b)

Sc

Here the function Glll.../iN(x) is defined by ) = X W - - i *

(3-41)

A

and Sc is the region enclosed by C. The symbol £ means the summation over those terms A

which are given by anti-symmetrizing the indices. Since r'a a a

1

- (N -

f 1)! r

%,(*) -

eiK{»))

e(xMl...

x,j

^ +

f

o

,

a w

and ^

-

^

L

^

8{ek(x) ... eiN(x))) J (3.42) (3.43)

which is obtained from e2(x) = 1, the support of @l1.../tN(x) is the domain of the topological singularities (i.e. 3>e). Therefore the integration (3.40b) shows that vc remains constant throughout a continuous deformation of Sc as long as the cross section between C and 2fe is not changed. In this sense, vc is a topological constant. The relation (3.40b) shows that (x) is an assembly of N-dimensional d-furictions. Then it specifies (d — .^-dimensional objects in ¿-dimensional {a^J-space. Denoting the position of the object by y^a 1 ( . . . , ad^N) which depends on (d — iV)-parameters, we find G^.-.^ix) to be

= W~NV.

/** -

'•'.'.6ii){x -y{ai (3.44)

Extended Objects in Field Theory

451

The relation (3.41) is rewritten in terms of the dual tensor

as with =

f da, - dad_N ^

"'

d^(x - y(0l - aA_N)).

(3.47)

(In the case of Abelian symmetry, Eqs. (3.39), (3.46) and (3.47) become (2.22), (2.21) and (2.20) respectively.) I n this way, the topological singularities, whose topological constant is the (2V — l)-dimensional closed surface integral vc and whose dimension is (d — N) can be identified locally in space-time. I n particular when a singularity is isolated in the sense that it can be separated from other singularities by a closed surface C, Ge forms a closed surface on Se because of the singlevaluedness of e(x). I n this case, vc in (3.32) takes an integer value (topological quant u m number), since Ge must envelope Se an integral number of times. When an assembly of singularities forms an object which is enclosed by G, an integer value of vc is attached to this assembly. The weight Qvc (vc integer) in (3.47) determines the topological quant u m number associated with the object whose positions of singularities are specified by i fd-jv). This integer is the same as the index called the Brower degree [21], An object with a non-vanishing topological quantum number is stable or semi-stable, since it cannot disappear unless it meets another object with the opposite quantum number. The above consideration shows that the origin of the topological quantum number is based on the singlevaluedness condition for e(x) together with the condition that the closed surface G does not cross any part of the domain of the singularities. Note t h a t (3.46) gives = 0, (3.48) which is an extension of (2.23). When the condition (3.48) is applied to the expression (3.47), it indicates t h a t the object is a (d — jV)-dimensional object without any closed boundaries. This is consistent with the fact that vc remains as a topological constant unless the closed surface G cuts the object. Note that the point (i.e. N = d) is considered to be an object without a closed boundary. We have so far considered only the (N — l)-dimensional closed surface G. Let us now turn to the question asking how the lower-dimensional closed surfaces C in {^J-space (i.e. dc < N — 1) specify the topological singularities in ¿¡¿e. When the surface C is mapped onto 8 e , the result is a ¿¡.-dimensional surface Ge. Consider a situation in which G cannot be reduced to a point without crossing certain singularities in Then, for any deformation of G, the dimension of Ge cannot be changed as far as the deformation of C does not cross any singularity in 3>t. Therefore, throughout the course of this deformation, Ge can always be mapped on to a ¿¿-dimensional sphere S e '. Note that S e ' is not required to be on Se. Since the Ce cannot be reduced to a point on Se, there is a certain singular region in the