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1 Chernikov-Barbashov Algebra for 2nli~ comc1dale~
Transcript of 1 Chernikov-Barbashov Algebra for 2nli~ comc1dale~
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 ·
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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.
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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
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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 .
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(21)
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+~ 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.
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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].
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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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