Fortschritte der Physik / Progress of Physics: Band 28, Heft 8/9 1980 [Reprint 2021 ed.] 9783112522943, 9783112522936

Citation preview

FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN

REPUBLIK

VON F. K A S C H L U H N , A. L Ö S C H E , R. R I T S C H L U N D R. R O M P E

H E F T 8/9 • 1 9 8 0 • B A N D 28

A K A D E M I E

- V E R L A G

EVP 2 0 , - M

ISSN 0015 - 8208

31728

•

B E R L I N

BEZÜGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an den Post zeitungsvertrieb, an eine Buchhandlung oder an den AKADEMIE-VERLAG, DDR - 1080 Berlin, Leipziger Str. 3 — 4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD nnd Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7000 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferunesstelle 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 Str. 3 — 4

Zeitschrift „Fortschritte der P h y s i k " Herauggeber: Prof. Dr. Frank Kaschluhn, P r o f . Dr. Artur Lösche, Prof. Dr. Rudolf Ritsehl, Prof. D r . Robert Rompe, ¡in A u f i n g der Physikalischen Gepellschaft der Deutschen Demokratischen Republik. " Verlag: Akademie-Verlag, D D R - 1080 Berlin, Leipziger Straße 3—4; Fernruf: 22 36221 und 2236229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. 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 Demokratisehen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 7400 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines J a h r e s bilden einen B a n d . Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis f ü r die D D R : 120,— M). Preis je H e f t 15,— M (Preis f ü r die D D R : 1 2 0 , - M). Bestellnummer dieses Heftes: 1027/28/8/9. © 1980 by Akademie-Verlag Berlin. Printed in the German Demoerati« Republie. AN (EDV) 57 618

Portschritte der Physik 28, 427-464 (1980)

Differential Geometry and Nonlinear Field Models B . M . BARBASHOV a n d V . V . NESTERENKO

Joint Institute for Nuclear Research, Dubna,

USSR

Contents 1. Introduction

428

2. The 2.1. 2.2. 2.3.

Gauss Surface Theory 429 The definition of the surface and its fundamental quadratic forms 429 The moving frame on the surface. The central theorem of the surface t h e o r y . . . . 430 The coordinate systems on the surface 432

3. The 3.1. 3.2. 3.3.

Basic Ideas of the Differential Geometry of the Riemannian Manifolds The definition of the Riemannian space The embedding of the Riemannian manifolds The Riemannian spaces of a constant curvature

435 435 436 437

4. The Differential Geometry and the Inverse Scattering Method 439 4.1. The inverse scattering method for the solution of the nonlinear evolution equations. 439 4.2. The embedding of the Riemannian manifolds and the inverse scattering method . . 440 5. Nonlinear Two-Dimensional cr-Model 5.1. The Lagrangian and equations of Motion 5.2. The nonlinear cr-model with 0(3)-symmetry 5.3. 0(4)-symmetry 5.4. The instantons in [0(3)]2 tr-model 6. The 6.1. 6.2. 6.3. 6.4. 6.5. 6.6.

444 444 445 445 447

Geometrical Approach to the Relativistic String Theory 449 The variational principle and the equation of motion 449 The geometrical approach to the string theory 450 The string theory in a Lorentz invariant gauge 451 The time-like gauge i = r 454 The relativistic string in a space-time of a constant curvature 454 The generalization of the relativistic string model in the framework of the geometrical approach. 455

7. The Solitons in the String Theory 7.1. The soliton solutions of the Liouville equation and their stability 7.2. The semiclassical quantization of the solitons

458 458 460

8. Conclusion

462

References

462

28

Zeitschrift „Fortschritte der Physik", Heft 8/9

428

B . M . BABBASHOV a n d V . V . N E S T E K E N K O

1. Introduction The survey is devoted to the use of the differential geometry methods in the theory of the two-dimensional field models. The basic idea of this approach is the following. From the geometrical point of view the field function dependent on two variables describes the two-dimensional surface embedded into some space, that is in general case a Riemannian space. The specific properties of this enveloping space are determined by the field interaction in the model under consideration. The embedding theorem of the differential geometry says that the surface embedded into enveloping Riemannian space can be described not only by its radius-vector (in the field theory this means by the field function) but also by the fundamental differential forms of the surface. The coefficients of these forms considered as a function of the coordinates on the surface have to obey the Gauss-Codazzi-Ricci equations. These coefficients can be taken as new dynamical variables and the Gauss-Codazzi-Ricci equations as the new equations of motion. I n this way we obtain the description of the initial field model in terms of the new differential equations essentially different from the Eiler-Lagrange equations in the usual approach. Furthermore, in the differential geometry the Gauss-Codazzi-Ricci equations are derived as the compatibility conditions of the system of the first-order partial differential equations which describe the motion of the moving basis on the surface. I t turns out t h a t these later equations give the constructive method for obtaining the Lax pair operators (or the LM pair operators) required for the solution of the Gauss-CodazziRicci nonlinear equations by the inverse scattering method. This fact is outstanding as the problem of the obtaining the Lax operators for a given nonlinear equation or the set of equations in the general case is not yet solved. The survey starts with the Gauss theory of the surfaces in the three-dimensional Euclidean space. The definition of the surface, its fundamental quadratic forms and the moving trihedral on the surface are considered here. In Section 3 the basic ideas of the differential geometry of the Riemannian manifolds are outlined. A special attention is paid to the theorems of the embedding of the Riemannian spaces into Euclidean ones and to the embedding of one Riemannian space into another. These embeddings are described by partial differential equations of Gauss, Codazzi and Ricci (GCR-equations) on the coefficients of the fundamental forms of t h e manifolds. In Section 4 we apply the Gauss and Weingarten formulas, describing the motion of the moving basis on the manifold, for the construction of the Lax operators for the GCRequations. Section 5 is devoted to the geometrical approach to the two-dimensional nonlinear 0,

(2.2)

where g,j = rtirtj, rj = dr/dui, j = 1, 2, then we can go from (2.1) to the definition of the surface by one of the following formulas: z — z(x,y),

y = y(x, z)

or

x = x{y,z).

(2.3)

It is the usual definition of the surface in the Monge-Euler form. Indeed, if the condition (2.2) is satisfied then from the equation 9

/ 8(x, y) V \ w2) /

/ 8(y,z) V \ 8(u\ U2) )

/ 8(z,x) V 8{u\ U2)}

'

it follows that at least one of the functional determinant in the right-hand side of (2.4) is different from zero. Using this fact we can express the parameters u 1 and u 2 as functions of either x, y (or y, z, or z, x depending on what determinant in (2.4) is different from zero) and reduce (2.1) to one of the formulas (2.3). So, when g > 0 the surface defined by the parametrical equation (2.1) does not degenerate into the curve and has no singular points. The vectors r , and r , are the tangent vectors at a given point of the surface. The plane containing these vectors is a tangent plane of the surface. If the condition (2.2) is satisfied, then the vectors r A and r 2 are by virtue of (2.3) noncollinear |[f.iXr i g ]| = V g > 0 . The unit vector perpendicular to given point and is defined by

and r

2

is- the normal vector of the surface at a

In the Gauss surface theory the central role plays the fundamental differential forms of the surface. The first fundamental form is the squared distance between the two neighbouring points on the surface 2, n) on the surface.

432

B . M . BABBASHOV a n d V . V . NESTERENKO

Equations (2.13) and (2.16) can be represented in the matrix form

where A and B are the 3 x 3 matrices with the elements defined by the fundamental tensors g¡j and b¡¡ and X¡ — r 1; X2 = r 2 , X3 = n. The set of equations (2.17) will be completely integrable, or a complete system [2], if

This matrix equation is the condition that the tensors and b^ define the surface with the metric tensor g^ and the tensor of the second fundamental form Indeed, if (2.18) is satisfied, then the integration of Eq. (2.17) gives r { and n as the functions of u1 and u2. Further integration of these relations results in the radius-vector of the surface r as the function of u1 and w2. The later step is obviously possible because the integrability conditions in this case r = r ^ are satisfied by virtue of the symmetry of rfa and £>i;- in expansions (2.13) and (2.14) with respect to the indices i and j. After these two integrations in the expressions for r(w1, w2) there appear the constants of integration. The different values of these constants correspond to the motion of the surface as a whole [2], By this conclusion, the proof of the Bonnet theorem is completed. The matrix relation (2.18) gives only three equations on gy and instead of the nine equations as it may be supposed taking into account the dimension of the matrices A and B. We have one Gauss equation

Riiki = bikbji — bubjk

(2.19)

and two Codazzi equations ¡ = 0, In these formulas

t=H,

»,7 = 1,2.

(2.20)

is the Riemann tensor of the surface curvature

Rijkl = ~2 (9il.jk + 9ik.il — 9ik.jl — 9)1.ik) + ffS{^a.ik^r.il —

ik) •

For the two-dimensional surface the tensor -R^j has only one essential component R1212. 2.3. The coordinate systems on the surface Three equations (2.19) and (2.20) do not fix completely six coefficients of the first and the second fundamental forms. In addition to these equations the tensors g,j and b tj can be subjected in general case to the two subsidiary conditions for the following reason. The Gauss and Codazzi equations are covariant, i.e. they conserve their form under arbitrary coordinate transformations with the nonvanishing Jacobian m* = u^u1, w2),

¿=1,2.

(2.22)

It is a direct consequence of the tensor form of (2.19) and (2.20). Let us assume that in the initial coordinate system ul the tensor gV) was not subjected to any conditions. Then we can introduce the new coordinates ui according to Eq. (2.22) so that in the new co-

Differential Geometry and Nonlinear Field Models

433

ordinate system the transformed tensor gkt

will satisfy two conditions. It can be always made by a corresponding choice of the functions u^u1, w2) in transformation equations (2.23). In this reasoning the metric tensor gy can be replaced obviously by the tensor of the second fundamental form If 9» = r^r, = F = 0, (2.24) then such a coordinate system u1, u2 on the surface is orthogonal. In this case the vectors rA and r 2 are orthogonal to each other. In addition to (2.24) we can put gn = E = 1. This coordinate system is called the almost geodesic system. The line element has the form ds2 = (du1)2 + G{u\ u2) (du2)2. (2.25) The example of such a coordinate system is the usual geographical coordinates 6,

=1

= 0,

£ Wri" = e,

P=1

e=±l.

The derivative formulas (2.13) and (2.16) become now

x% = ebijVx, (i

= 1, 2 , . . . ,

n+

1,

i,j,k=

(3.7) 1, 2, ..., n.

(3.8)

437

Differential Geometry and Nonlinear Field Models

The integrability conditions of these equations reduce to the Gauss and Codazzi equations Rijki

=

e{bikbj,

—

K* i,

j,

k,

bnbjk),

(3.9)

= 0,

(3.10)

= 1, 2, ...,

I

n ,

where -Rhh is the Riemannian curvature tensor in space V„ defined by (2.21). In general case this tensor has n (n — 1)/12 essential components. If the dimension of the enveloping flat space is greater than n + 1 (let it be equal to n + P> P > 1)> then at each point of the Riemannian manifold Vn one can construct p unit vectors r/^, ¡i = 1, 2, ..., n + p, tx = 1, ...,p perpendicular to each other and normal to Fn 2

2

n+p z

W

(»/A/,*) >

a= l

Codazzi equations

(3.13)

Km-jc — Ktik-.j = 27 efofrlkbpiij — Tfaijbpiik)

(3-14)

and Ricci equations V/hli*

—

v

Mk;i

+

27 ^(»VMV/* —

v

V

rNkv7«li)

+

S'^fllfia/mk

—

bp/iAini)

= 0.

(3.15)

Thus, we have the following embedding theorem. The symmetric tensor §rj;-, p symmetric tensors and p(jp — l)/2 torsion vectors —v^n), i, j = 1, ..., n, a, = 1, 2, ..., p determine the Riemannian manifold F„ with the fundamental metric tensor gri; embedded into the flat real space R (the Riemann curvature tensor of this space vanishes identically) then and, only then the Gauss-Codazzi-Ricci equations (3.13) —(3.15) are satisfied. The manifold F„ is defined in this case up to the motion as a whole in the flat space R . n + P

n + p

3.3. The Riemannian spaces of a constant curvature The natural generalization of the usual sphere in three-dimensional Euclidean space are the spaces of the constant scalar curvature R = g^Rij in the Riemannian geometry. Here R- j is the Ricci tensor R^ = gklRklThe scalar curvature of the two-dimensional t

B. M. Babbashov and V. V. Nesterenko

438

surface is connected with its total or Gauss curvature K = det ||&;,-||/det \\gij\\ in the following way B = — 2 K . F o r the sphere with radius a we get, owing to (2.10), K = 1/a 2 . I t turns out that any m-dimensional Riemannian manifold of the constant curvature can be considered as a hypersphere embedded into the flat space with dimension ra -f- 1 [¿>]The equations defining this embedding are m+1 I Z c f W = -=r,

p,=l

C

A

U

=±1.

(3.16)

The quantities z?, /i = 1 , 2 , . . . , to + 1, which are the Cartesian coordinates in flat space with dimension TO + 1, can be considered as special coordinates (the so-called Weierstrass coordinates) of the wi-dimensional Riemannian space of the constant curvature. As their number is greater b y one than it is required for the space Vm, the Weierstrass coordinates have to obey one constraint (3.16). The advantage of these coordinates is that the fundamental quadratic form of the space of constant curvature in terms of z1' is diagonal m +1

vi =

E^n2-

f=i

The normal to this space is a vector z*. L e t us have the m-dimensional Riemannian space of the constant curvature, Vm in which there is embedded other Riemannian manifold F„ with n < to. I n the space Vn we introduce the coordinates ul, i — 1 , 2 , . . . , « and in the space of a constant curvature Fm the Weierstrass coordinates z^, ¡i = 1, 2, . . . , to -f- 1 satisfying E q . (3.16). The embedding of V„ into F m means, as usual, that there is the functional dependence z1" = 2'*(m2'

2 = cos 0e 2 .

From the equation of motion (5.4) written in the basis (5.11) by means of the Eqs. (3.17) and (3.18) we get (5.13)

bn=b22. The compatibility condition of Eqs. (5.12) is given by the Gauss equation (3.19) 0

b2 — b2 =_ii^ 11. +

0 •22

11

Sine

cos 0

sin 0 cos 0

and by the Codazzi equations (3.20) ^12.1 — ^11,2 —

^12,2 — ^11,1

v

sin 0—cos 0 ' &120.1 &1J0.2 bllQ.l

(5.14) '

(5.15)

—

sin 0 cos 0

B y the substitution bn

= cot 0Ai2 ,

612 = cot 0Afl.

Eqs. (5.15) reduce to one equation for the function ¿(w1, u2) (cot2 0A.O,! = (cot2 0A,2),2.

(5.16)

Thus, in a geometric approach the nonlinear cr-model with the 0(4)-symmetry is described by the system of two nonlinear equations 0.11 - 0,22 - sin 0 cos 0 + ^ ^

(A2! - 1%) = 0

(5.17)

and Eq. (5.16). The derivative formulas (5.12) enable us to construct the X ; -operators (see Eq. (4.3)) required for the solution of the nonlinear equations (5.16) and (5.17) by the inverse scattering method [26—29]. One of the possible applications of such an approach is,

Differential Geometry and Nonlinear Field Models

.447

for example, the derivative of the infinite series of the conservation laws in the cr-model by using the corresponding conservation laws of Eq. (5.10) or Eqs. (5.16) and (5.17) [30,

31\

5.4. The instantons in [0(3)] 2 c-model The two-dimensional nonlinear tr-model with the 0(3)-symmetry has, as the Yang-Mills theory [23], the instanton solutions [22], The geometrical approach enables us to get them in the most simple way. The instantons are considered in the Euclidean space-time u 1 , w2 and we define them as such configurations of the fields that give the local minimum of the Hamiltonian in the functional space n a (u l , u 2 ). The functions n a(u 1, u 2), a = 1, 2, 3 give the mapping of the infinite Euclidean plane u 1, u 1 onto the unit two-dimensional sphere S 2 in the isotopic space n 2 — 1. If we are interested in the fields n a(u x, u 2) with the following asymptotic at infinity

where n0 a = (0, 0, —1), z = u 1 + iu 2 then all points of the plane u 1, u 2 at infinity will be indistinguishable from the physical point of view and equivalent to one point. The topology of such a plane is the topology of the sphere. Any point of this plane with the coordinates u 1 , u 2 can be uniquely associated by Eqs. (2.33) with the corresponding point of the unit sphere ¿72 with the spherical coordinates 0, 0, g22 = x'2 < 0 and g < 0. We use in the Minkowski space the following metric a 2 = a ^ = (a0)2 — a 2 . Thus, from the geometrical point of view the world surface of the relativistic string is the two-dimensional Riemannian manifold with the indefinite first fundamental form g^.The principle of least action, as applied to the functional (6.1), reads that the world surface of the string is the minimal surface [48, 49]. In the isometric coordinate system gn = x2 = -gr 22 = —x'2,

gn = xx' = 0

(6.2)

the Eiler equation for (6.1) 8 dii1

dj-9

= 0

(6.3)

is reduced to the D'Alembert equation for xu(a, r) x„ - V

= 0

(6.4)

with the boundary conditions xn'(