On the Completeness of the Chernikov-Barbashov Algebra for

St~ings

N .S.Shavokhina Joint Institute for Nuclear Reserch, Dubna

Russia

Now in Cosmology such objects as strings are very widely used. A connection between the nonlinear scalar and electromagnetic field of the Born - lnfeld type [1] - [2] and the minimal surface, which is the world surface of strings, was established by N.A.Chernikov, B.¥.Barbashov and N.S.Shavokhina ( see, for example [3] - [6] ). The theory of the nonli­near two-dimensional scalar field was constructed by N.A.Chernikov and B.M.Barbashov in 1965-1966 [3]-[4]. Further, this theory has been known as the classical and quantum theory of the relativistic string. In the pre­sent paper it is shown that quantum algebra of the relativistic string has no c-nurnerical terms in the Minkowski space-time of an arbitrary dimension.

The action of the linear scalar fields has the form

J J 1 2 -2 2 S = k 2( Ct'x - c Ct't )dctdx, (1)

where k is a dimensional constant, c is the light velocity. Further, we suppose, that k=l,c=l.

The action of the nonlinear scalar field is defined as

S = j j J1 - a2 cp~ + a2cp';dtdx, (2)

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Chernikov-Barbaslwv Algebra for Strings 259

ft has . the dimensional constant selected so that the qua~ J Y a'!' h

where a is . . . d _ (t x ) can be cons1dered mt e them asure ofl ngth. The scalar ~el cp di <p , • n It is obvious that the

Minkowski sp~ce .- tim;i:~ ~e :b~:~~ea~:r::1

:f·surfaces. The equation action ( 2nli~ comc1dale~ . field is th; extremal condition for integral the ( 2 ). of the 1to 1tear sc a.1 •

It has the form: 2 2] - 0 (3)

- <{)tt + 'Pxx + a[-t.ptt'Px + 2t.ptx'Pt'Px - 1-Pxx'Pt -

The differential equation of minimal surfaces

y = ac.p(t, x) (4)

t x are the Cartesian coordinates of the coincides with eq.( ~ ). Here h' ,y M has the following metric form: Minkowski space - time M3· T e space 3

ds2 = -dt2 + dx2 + dy2. (5)

h f ( 5 ) is represented as The induced metric form on t e sur ace

d<12 = -dt2 + dx2 + a2(c.ptdt + t.pxdx)2 (6)

( ) th t · tangent to the minimal The square of the norm of a vector N ni' n2 a is . surface ( 4 ) at any its point is equal to

N2 = -n12 + n22 + a2(c.ptnl + <Pxn2)2 (7)

The solution of eq.( 3 ) is sought in this way : at first, th:n~:::: p::~ f ( 3 ) is solved with the Fourier method; secondl.y, the n Pt

o eq. for the admissible state vector a . a of eq.( 3 ) j s satis:fi.~df, on th(~ a)vr:;;;lng the works [3]-[4]. The linear part qua.ntum level. Sa.tis Y eq. of eq.( 3 ) is d'Alembert equation

- 'Ptt + 'Pxx = 0,

The nonlinear part of eq.( 3 ) is equal to

2 D = -t.ptt'P! + 2t.ptx'Pt'Px - 'Pxx'Pt ·

(8)

(9)

260 N.S.Shavokhina

The solution of eq.( 3 ) has the form

! 1 ;_00 dk . _ ,1 .cp(t,x) = . fiL --[a+(k)e'(wt-kx) + a-(k)e-i(wt-kx)]

v21l' -oo ~ , (10)

where 2 k2

- w + · = O, w > O, k = ±w (ll)

Fourie.r's coeffi.cients a+(k),a-(k) are being imposed on the following com­mutation relations

We represent eq. ( 10 ) as

[a-(k),a+(m)] = o(k- m)

(12)

1 Loo dk . cp(t,x) = . ~ -[a+(k)e'k(t-x) + -(k) -ik(t-x)] v 21!' 0 y'2k a e + 1 / 00 dk .

+ V2Jr Jo V2/i[a+(-k)e'k(t+x) + a-(-k)e-ik(t+x)J, (13)

Substituting ( 13 ) into the nonlinear part ( g ), we have

. /;\ 100 k2dk . . v27r D = { -[a+(k)e'k(t-x) + a-(k)e-ik(t-x)]}( )2

O V'if 'Pt + 'Px + {oo k2dk .

+{Jo y'2k [a+(-k)e'k(t+x) + a-(-k)e-lk(t+x)]}( 'Pt - Sox)2. (14)

Further, we suppose that the vector N = (w, k) is tangent to th · ·-mal surface ( 4 ) d · · e mm1 an 1sotrop1c so that the conditions ( 11 ) are t' fi d All th d•t• alid sa 1s e . ese con I 10ns v if

( Si'tW + Si'xk )2 = 0.

Using ( 11 ) , we obtain

('Pt+ 'Px)2

= 0, ('Pt - 'Px)2 = 0. (15)

At a classical level the conditions ( 15 ) have only the trivial solutions.

Chernikov-Barbashov Algebra for Strings 261

Following works [2]- [3], we suppose that the additional nonlinear con­ditions ( 15 ) are satisfied on average for admissible state vectors of our system. We ·have

< A'I : ('Pt+ 'Px) 2 : IA>==< A'I : ('Pt - 'Px) 2

: IA >= 0 (16)

where the colon denote the normal operator product. We emphasize tha.t if the conditions ( 16 ) are satisfied, the nonlinear

part ( 9 ) is satisfied for the state vectors too and the metric form ( 6 ) is equal to zero, on the average, at any point of the minimal surface ( 4 ).

Above we set forth the Chernikov - Barbashov method of quantization of the nonlinear scalar field or the relativistic string in the nonorthogonal coordinates t,x on the minimal surface ( 4 ). These coordinates are the Cartesian in the Minkowski space-time M3.

The operators ('Pt - 'Px) and ('Pt + 'Px) are equal to

'Pt+ 'Px = ~ fo 00 JJi, d JL [a+(-Jt)eiµ(t+x) - a-(-µ)e-iµ(t+x)],

We represent the operators ('Pt - 'Px )2 and ('Pt + 'Px )2 in the form

+a-(µ )a-(v )e-i(µ+11)(t-x) - a+ (µ)a-(v)ei(µ-11)(t-x) _

-a-(µ )a+(v )e-i(µ-v)(t-x)],

('Pt+ 'Px) 2 = -- y'µVdµdv[a+(-Jt)a+(-11)ei(µ+v)(t+x)+ 1100100 7r 0 0

+a-(-µ )a-(-v )e-i(µ+v)(t+x) - a+(-µ )a-( -v)ei(µ-v)(t+x) _ (18)

-a-(-Jt)a+(-ii)e-i(µ-v)(t+x)].

Using the equality

262 N.S.Shavokhina

we obtain from ( 18 )

( _ )2 _ . 2 1 Loo Loo 'Pt 'Px - . ('Pt - 'Px) : +- #6(µ - v)e-i(µ-v)(t-x)d d 7r 0 0 µ v.

( 2 1 LooLoo (19) 'Pt+ 'Px) = : ('Pt+ 'Px)

2: +- #6(µ - v)e-i(1i-v)(t+x)d d

7r 0 0 µ v.

Introducing new variables of the type

µ + v = p, -µ + v = q,

we obtain

: ('Pt - 'Px )2 := ; Loo dp [bt(p )eip(t-x) +bl (p )e-ip(t-x)J,

: ('Pt+ 'Px)2 := ; Loo dp [bt(p)eip(t+x) + b2(p)e-ip(t+x)],

where the operators b are equal to .

(20)

(21)

(22)

Chernikov-Barbaslwv Algebra for Strings 263

+~ 100 dq.jq2 - p2a+ (p; q) a- (-p2- q) ' The operators b1 and b2 satisfy the following commutation relations

[bt(p), bt(q)] = -(p - q)bt(P + q),

[bi(p), bi(q)J = +(p- q)bi(P + q),

[bt(p), bi(q)] = -(p + q)bt(P - q),

[bi(p), bt(q)J = +(p + q)bi(P - q),

(i = 1,2)

and possess the property of the Hermite conjugate

The operators b1 and b2 commute.

(23)

(24)

There is a one-to-one correspondence between the operators ( 22 ) and the operators from works [2]-[3]

b1(P) ¢> bi(p), bi(P) <==> b1(p), bt(v) ¢> bt(p), b2(P.) <==> b2(p).

Here, t he right-hand sides a.re the operators from [2]-[3].

(25)

Finally, we formulate our result as follows: it is established that the quantum algebra of the string or the nonlinear scalar field of the Born­Infeld type first obtained in [3]- [4] does not contain a c-numerical term. This conclusion does not depend on the dimension of the space - time.

References

[1] Born M.,1934,Proc. Roy. Soc. A. v. 143, A 849, p. 410

[2] Born M.,Infeld L.,1934,Proc. Roy. Soc. A.,v.144,A 852 p.425.

[3] Barbashov B.M.,Chernikov N.A.,1965,Preprints JINR P-2151, P-2311, Dubna.

264 N.S.Shavokhina

(4] Barbashov RM., Chernikov N.A.,1966,Soviet Physic (JETP). v. 23,No 5, p. 861.

[5] Chernikov N .A.,Savokhina N .S.,1981, Communication JINR P2-81-434 Minimal surfaces in nonlinear electrodinamics. Dubna.

(6] Shavokhina N.S.,1988,Preprint JINR P2-88-132 The Born-Infeld equa­tions and minimal surfaces . Dubna.

· (7] Eizenhart L.P.,1948,Riemannian geometry,M.: Gostehizdat.

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