IC/86/26U
\ ••'•'•: \
\ y INTERIMATIONAL CENTRE FOR THEORETICAL PHYSICS
STOCHASTIC QUANTIZATION AND RANDOM SURFACE APPROACH TO POLYAKOV STRING THEORY
I.G. Koh
and
R.B. Zhang
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/B6/264 I. INTRODUCTION
i String theories seem to be ;in Httractive framework lor describing
all the forces of nature. A viable quantum theory of gravity may be achieved International Atomic Energy Agency by finite superstring theories. Much work i3 now in progress for multi- and United Nations Educational Scientific and Cultural Organization loop computations to understand the quantum behaviour of boaonic and super- INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS string theories. One expects that nsn-perturbative aspects of string theories
will play a crucial r3le in our understanding. To these, string field
theory and random surface approach to string theory will provide insights. STOCHASTIC QUANTIZATION AND RANDOM SURFACE APPROACH TO POLYAKOV STRING THEORY * The latter can be put into numerical computations by Monte Carlo methods. 2 Gauge theories have been successfully studied by Monte Carlo techniques.
I.G. Koh ** and R.B. Zhang These procedures inspire a new quantization technique through Langevin International Centre for Theoretical Physics, Trieste, Italy. 3 equations.
In this paper, we study stochastic quantization and random surface ABSTRACT
Polyakov string theories are quantized by stochastic methods. Langevin approach to the Polyekov string theory. equations for string coordinate and two-dimensional metric are invariant under general coordinate transformation and Wcyi scaling. In these methods, In Section II, Langevin equations for string coordinate <^(x,f) and conforms1 anomaly contributed by string coordinate and scattering amplitudes two dimensional metric 3ii(<-,») are written in general coordinate and of tt-tachyons in the tree level are computed. As a first step of analytic
computation in the random surface approach of the Polyakov stringy we evaluate t/eyl scaling invariant, way and solved perturbatlvely using heat kernel the tachyonic mass and the scattering amplitude for right triangles. methods. Conformal anomaly due to cj> is computed in stochastic quanti-
zation formulation. Tachyonic scattering amplitudes are reproduced in HIRAMARE- TRIESTE August 1986 tree diagram.
For the purpose of Implementing a Polyakov string into the numerical Monte
Carlo procedure, a random surface approach Is examined in Section III.
* Submitted for publication. ** Permanent address: Physics Department, Korea Advanced Institute of Science and Technology! P.O. Box 150 Cheongryang, Seoul. Korea. -2- II. STOCHASTIC QUANTIZATION However it i» extremely difficult to perform analytic computation. A first step towards analytic calculation of tachyontc mass and scattering Stochastic quantization is formally equivalent to canonical or path- amplitudes is taken. Finally we discuss methods of implementing the integral quantizations. Green's functions are evaluated over asymptotic random surface approach in numerical computations. distributions of configurations evolving along the fictitious time
governed by the Langevin equation. Gauge theories have been treated successfully
by using this method with and without gauge fixing , and further numerical
computations have been performed.
The non-neceS3ity of gauge-fixing and the possibility of implementing
stochastic quantization into numerical vork initiates its application
to string theories. We propose stochastic quantization procedure for the
Polyakov' string theory, which preserves both general coordinate and Weyl
scaling invariances. We briefly comment on stochastic quantization of
string field theory in light-cone.
A. GENERAL COORDINATE AND WEYL SCALE TRANSFORMATION INVARIANT LANQEVIN
EQUATION
Polyakov string action for ^ and of given topology is
where *! is Regge slope carrying negative two mass disiensionst we Bet
/- 2.TW = 1 ), and 1* is two dimensional coordinate [%t T ).
The standard procedure for stochaBtic quantization gives the Langevin
equation for • vlth fictitious tlae t as
-3- -4- Alternatively one nay set up auch a I*ngevin equation for gag without 12) splitting as eq. (3). The exponent of noise average in eq. (6) contains
To obtain Langevln equation for 8«. , we split 3«*. into t>Independent ^,. and is not Gaussian. See Ref. [6] for discussion of non-Gaussian background J and t-dependent hut as noise in stochastic quantization of Einstein gravity.
The noises L and ],. are scalar and second rank tensor, and
Its equation Is proposed as eqs. (2)-(6) have proper general coordinate transformation.
Under the following transformations
(Wa have not yet rigorously proved that eqs. (2) and (4) give a correct t , quantum theory of eq. (1).) Here "I* and •?„» are stochastic noisas of i and K«4ireaPeotlvely- For simplicity, we require that <^ $«*.,*«£ and ") satisfy the boundary condition of closed string. (7)
Ensemble averages of these noises are defined by
eqs. (2)-(6) are covarlant. In fact, this transformation is nothing hut
the local Weyl symmetry in the context of stochastic quantization.
It is difficult to solve coupled eqs. (2) and (4) analytically. However,
(6) these can be treated perturbatively order by order in W*p • To the zeroth
order in \\*\ , eq. (2) can be integrated exactly for t-independent
background Jji(See also eqs. (13) and (22)). For the case of constant Stochastic enoe«ble average (6) should not contain the metric J^A curvature metric for genus ? 2, the solution of eq. (2) is in the exponent, since V. is the noise driving Langevin equation fo (8) metric J«* . This situation is circumvented by the decomposition 'V
given In Bq..<3).
-5- -6- The heat kernel in eq. (5) can be constructed through Poincar•' series where b and c are antighost and. ghost fields. As ghost and
antighost In fact decouple from the string coordinates, we only need
(9) the Langevin equation for <^tl, i --t)
Here Qi denotes two-dimensional Laplacian with, the constant curvature metric R
In complex coordinate J^T-ti* . the action of "V - I \ with aj - JLC •= 1
+ 4, on t is The explicit form of ^ (.i, *') cTTT e in eq. (9) is well-known:
(10)
where
The propagator for 4* is worked out as
(11)
by using eqs. {5), (8) and (9).
Actually at the tree level ( i, e, Riemann surface with zero genus),
it la much easier to work ' a la Ref. [A ] with the gauge fixed action 3,
-7- -8-
r B. CON FORMAL ANOI1ALY FROM
The conformal anomaly due to quantization of q> is the non-vanishing trace where ";" denotes covsxiant differentiation. of momentum energy tensor
>. (12)
To evaluate thia, we note that heat kernel in eq. (8) for neighbouring points 1 and \ is given for arbitrary curvad background as
Here (•(? t1) ls the geodesic length defined between ^ and \'
]P and {\ are defined by
(14)
l'j*.) ~ Z (IS)
~ In two dimension, the trace in eq. (12) does not vanish only if
O^^t^) ie singular. We use dimensional regularijation
jj ^ 1+J. • arid obtain
-9- -10- C. TACHYONIC SCATTERING AMPLITUDE
As another sample computation, we compute N-tachyon scattering
amplitude to the zeroth order of *\^« in perturbation expansion. The (23) Thus eq. (23) is vertex operator la given as (24) V U> •=. %, t . <18) with g, the renormalized coupling constant, which absorbs the divergence* Here dot "," denotes the scalar product in 26 dimensions, and t is the emerging from the coincident limit of t« 11 ,• -I.' I*", fictitious time. By averaging over stochastic noise, the scattering As usual, SL(2,R) Invat/riance of eq. (24) is fixed by amplitude of N-tachyon ia given by
•"•*••
T -f From the discussion given at the end of Section II.A, we can see that eq. (25) is exact at the tree level.
(19) By noting
(20)
the scattering amplitude is reduced to
(21) Since the kernel for flat background Is
(22)
the exponent in eq. (21) becomes
-II- -12- D. STOCHASTIC QUANTIZATION OF STRING FIELD THEORIES IN THE LIGHT CONE
In thin section we will discuss the stochastic quantization of string field theory with gauge fixed as in Ref. t.4] . For simplicity, t | Il*>t-J ). (28) we will work in the light-cone, where all the nonphysical degrees of freedom Formally we can analyze eq. (28) perturbatively In orders of g by expanding have been eliminated. (For the treatment of eovariant string field theory, see Ref. t9j .) The action of open string field theory takes the form ,U;tt)i ^(Svf*:*))^-. (29)
The zeroth order in g of eq. (28) is
f (30) ( V whose solution is Here and are th* »el-l-known three and four-string interaction vertex operators respectively, whose explicit forms can be found in Ref. tlO) • Number operator M In eq. (26) acts on H , with the boundary condition the Hilbert space of the first quantized string. (32)
The first order solution of eq. (28) is Introducing a noise |n(x;t)> with Gaussian, distribution, we have
(27) (33)
where I is the identity operator in H • wltn tnlfl noise, the Langevin
equation for | $ ") reads
Here Q ^ and act on the i-th space in direct product of Hilbert
- TT1J" J'xv i 5(«,,t spaces.
-13- -14- The free propagator can be formally calculated as
diverges in the limit t —9 -t c*t for k C 2TT • leading to breakdown
of stochastic quantization. This is not surprising.because tachyon cannot i-X'J. I (3"0 or more explicitly, be quantized consistently. All the higher spin equations in eq. (37) lead
to sensible asymptotic limit5. The well-known Indefinite problem of higher
9 spin field action in the covariant formulation does not occur. This
(35) is one of the advantage of the light-cone gauge. Here | j)f} ) is the eigenstate of the string coordinate X(«") ln the
In principle one can calculate the scattering amplitude in this light-cone gauge, and t.=. x1 t' •= X'^ ^' '^' agrees with the
formulation. However, it is technically very difficult. result obtained in Ref.
However, we should point out the problem existing in Eq,. {3M. In terms of components 0.c,+ i«> t k'jU;*) Oi'Vl'
eq. (30) represents an infinite set of Langevin equations for different spins
&4>(t; ( * -t ](*;*) , (37)
(Component expansion of eq, (28) will also include couplings between different spins.) The operator - (a+3-TT) in the first equation of eq.
(37) is not positive definite. The correlation function of Wt;t$ , Fourier
transform of 4cn;y,
-15- -16-
T III. RANDOM SURFACE APPROACH TO POLYAKOV STRING tot convenience, we embed triangulated two dimensional surface into three dimensional Euclidean apace. For a particular triangle with vertices *, To Implement the Polyateov string theory further Into numerical
and paraiaetrizatlon is given as computations, one triangulates two dimensional surface. One may speculate that the fundamental structure of string world sheet nay be triangulated (41) randon surface. Similarly string coordinates inside the triangle is given as
To keep symmetries of theory as much as possible within the random surface formulation, we follow the Regge calculus idea for two dimensional where q> is represented by lengths of sides where |tc,'| is length of link between vertices i and j. Defining of the triangle (see eq. (43)), and is constant Inside the triangle. (44) The affine connection is non-iero on the links, and the curvature is concentrated on the vertices given as deficit angles. The string coordinate 12 we can rewrite action (1) on the simplicial surface as at ^t.,*^ i» now represented by ^ defined on the vertices.
5 = (45) One recalls the Euler theorem which gives relationship between numbers of vertices(N ), links(H ), and trianglea(N ) In terms of g holes as
where ^<.*\V Is the area of the triangle with vertices i, j and k. For tf. - rJ, -i iv\ = id-};. (39) a surface consisting of equilateral triangles, eq. (45) Is simplified A link is shared by two triangles, giving
(40)
-17- -18- *, SYMMETRIES OF SIHPUCIM, ACTION Tt ~?j J (46)
We can extend the random surfaces approach is & string theory in a curved We now discusa symmetries of action (45)> One can choose different coordinates a' and 8' for • and g inside a particular triangle, or target space with the following action
express the coordinates a and B of a triangle in terns of coordinates
4' and ft1 of another triangle. In both cases, «(' ar\d n' , are a where At t i\\ is the metric of the target space. Eq. (U5) is modified general coordinate transformation of < and &. Under this transformation,
to Eqs. Ct3) and {hh) are covariant and action (1*5) is invariant 11. By general
coordinate transformation, one can alvays go to a conformal gauge.
We next turn to discuss whether action (ii5) has the Weyl scale transformation
invariance. By definition, the Weyl-scale transformed metric ^
be of the same form as ^t.
where fli is a given constant for each triangle. This rescales three
3ldes of the triangle by the same factor* £ * . Since each link ia
shared by two triangles, scaling in one particular triangle would
necessitate the same rescaling for neighboring triangles. Thus one has
only a global scale transformation invariance .
-19- -20- C. APPROXIMATE TREATMENT OF TACHVONIC MASS AND AMPLITUDES IN THE
3. QUANTIZATION OF RANDOM SURFACE MODEL RANDOM SURFACE APPROACH
In path-integral quantization of action (1), one suns over all possible An important problem in the random surface approach is whether surfaces and . Similar summations are performed for quantization of eq. eq. (46) gives correct continuum limit where the number of triangles
(45). Measure for a is aimply IT^^Y ' but determination of measure is very large. For example, do theories given by eq.(4b) and eq. ittM of requires care. (, (1) provide the same phase transitions? These require deeper analyses
Surfaces are classified by the number of holes 3. The surface of which we will not undertake here. Howeve^symmetries (general coordinate genus g is approximated by N triangles. { Note that N and N are transformation and local Weyl scaling) of action (1) are larger than
fixed by eqs. (39) and (40).) Different surfaces are obtained by linking those (general coordinate transformation and global scaling) of action
the vertices in all possible, but distinct ways. Link lengths are allowed (45), it ia not clear at all that actions (1) and (45) have the same
to vary with constraint of triangle inequality. limit. As a preliminary step, we compute the tree amplitude of tachyon
ls scattering and the ground state mass in the random surface formulation of Measure of Jt. proposed Polyakov string in path-integral approach. (50)
*>«• surfaces, given by ±,i elinks We take our random surface as shown in Fig. 1 which is collection of different linkage of vertices right triangles with lengths a and b. Since we have not been able to sum
For a. given surfaes of genus g and particular l"lnkft«a of vertices, dg^ over other two dimensional surfaces analytically yet, our computations in at one triangle is Riven as vedfce product of link length dl,, of three thiB section should be taken as a first step towards calculations of sides. Sharing of a, link by tvo triangles i3 taken care by identifying sides tachyonic mass and scattering amplitudes in the random surface approach. of adjacent triangles by Dirac delta function.
For the right triangles in Fig. 1, the metric (U3) J.a diagonal
and eq. (44) is simplified as (52) - X. *<•*).}'H'.j' ^
-21- -22- Here ^J. • denotes the field on site (L,j) in Fig. 1. [Accidently eq.
(52) happens to agree with the result in Ref, I17j}where the lattice ^lu -) 'J - approach is used.) The matrices * and B for close string are here g is the coupling constant. Thesis integrated out formally to give * J / "<• -\ •• .oi o o (53)
A- 4- 0 o a o where ^li«itlt »'» \ > ls ths ProPa8ator obtained from the action (52). This propagator ia hard to obtain. However, for the extreme case ^ i
o -1 vhere a«b, one can expand ft. . ^(. •, as « 0 -I I
(57) 0 O 0 -| (54) t©t J- -1 e 0 4 o O where ^ e 0 a (58) I = j\ (.to-.) i-l^-w 0 o -I t -1 O I-1 0 o o 6
0 0
Tachyonic mass can be evaluated in the sane method as in the continuum 1-2 i-1 it. • • • theory with path integral approach which ia described in Appendix A. M- I »-• I Vt ... tf'-O 't I
These computations should be taken with qualification made earlier. Eigenvalues Va~JJ an are replaced in the random surface approach by those of A and B matrices, which are
lv . ... fT ' respectively. Computations identical to those ID the Appendix give the contribution of a particular triangle of Fig. 1 to tachyonic mass as D-l-S* - rr-l 4tZ ', TI
However^one should sun over all different lengths of triangles in eq. (45), and these have not been calculated yet.*
The tachyonic scattering amplitude in path integral quantization is
-23- -24- 0. IMPLEMENTATION OF KANDOM SURFACE INTO NUMERICAL COMPUTATIONS ACKNQWLEDGMENTS
The Polyakov string can be simulated by Monte Carlo numerical methods The authors wish to acknowledge useful discussions with E. Gosii, J.A. Magpantay and C. Rebbl. They would like to thank Professor Abdus Salam, like lattice gauge theories. One can ask non-perturbatlve questions like the International Atomic Energy Agency and UNESCO for hospitality at the whether 1) negative mass squared state in tree level spectra of bosonic International Centre for Theoretical Physics, Trieste. theory persists even in the full quantum theory, or 2) whether there are phase transitions in string theories.'
Starting with some initial triangulated surfaces with ^> at vertices,
link lengths and string coordinates are updated by Metropolis algorithms.
In the process of iteration, configurations equivalent up to global scale transformations nay occur several times, since our action is not gauge fixed. Because stochastic process is guaranteed to generate configurations, which have equal action, with the same probability, gauge inequivalent 'configurations will be duplicated with the same number of times. Thus even though the global scale transformation group
is non-compact, gauge fixing is not necessary. It is not clear at present whether eq. (25) has the correct continuum limit. Numerical
simulations of action (25) are planned.
Note also that when the curved target space action (lib) is considered, no new unification arises, even though the Integration is Impossible to do analytically. 18 Recently there is a nuaber of works in these directions.
-25- -26- APPENDIX In this appendix we compute the tachyonic mass for the closed bosonic - Z string in path-integral approach.
T The transition amplitude between the configurations ^t«? at t=-0 and
ttt is given by
(A6)
Comparison of (AS) with (A6) results in extraction of ground state mass with det(-3 ) evaluated on the strip (o,T]»(»,l]- Tne power D-2 results = - 2TT . (A7) 12 from the standard argurent for the string coordinates and ghosts. In the case fi The determinant det(-? ) can be evaluated as product of the eigenvalues of -9 , which is given by l-\,<-, ••• n'-«| !•.- (A3) that ie 04 (A4) 1*1 one obtains V '2 'hsJJ> } * The amplitude (A2) can also be computed in Haalltonian approach aa -27- -28- REFERENCES 15. M. Rocek and R. M. Williams, Phys. Lett. B104, 31 (1981) and 1. J, H. Schwarz (ed.), Superstrings; The first 1^ years or Superstrlng Z, Phya. CS_1, 371 (1984); Theory (World Scientific, Singapore, 1985) and references therein. H. W. Hamber and R. W. Williams, Nucl. Phys. B2£8, 392 (1984). 2. C. Rebbi (ed.), Lattice Gauge Theories and Honte Carlo Simulations 16. J. B. Hartle, J. Hath. Phys. 26, 804 (1985); (World Scientific, Singapore, 1983) and references therein. H. Lehto, H. B. Nielsen and H. Ninomiya, Nucl. Phy3. B272, 228 (1986). 3. G. Parlsi and V. S. Wu, Sci. Sinica 24, 259 (1981). 17. R. Giles and C. B. Thorn, Phys. Rev. D16, 366 (1977). 4. D. Zwanziger, Nucl. Phya. B192, 259 (1981); 18. A. Billoire, D. J. Gross and E. Marinari, Phys. Lett. B139, 75 (1981); I. Baulieu and D. Zwanziger, Nucl. Phys. B193, 163 (1982). J. Jurkiewi.cz and A. Krzywicki, Phys. Lett. S148, 148 (19B4) 5. A. M. Polyakov, Phys. Lett. 1Q3B, 207 (1981). D. J. Gross, Phys. Lett. B13S, 185 (1984); 6. H. Rumph, Unlversltat li/ien Preprint LJWThPh-1985-15 (1985). J. Ambjiirn, B. Durhuus and J. Frohlich, Nucl. Phys. B257 FS14 , 433 7. J. Fay, J. reine und angewendte Hath. 293, 103 (1977); (1985) ; see also E. D'Hoker and D. H. Phong, Nucl Phys. B269, 205 (19861. J. Jurkiewlcs, A. Krzywicki and B. Petersson, LPTHE Orsay 85|36 (1985); 6. B. S. De Witt, In General Relativity, eds. 6y S. W. Hawking and W. A. Billoire and F. David, FSU-5CRI-S6-44 (1966); W. Israel (Cambridge University Press, 1979) A. Duncan, and H. Meyer-Ortmanns, HPI-PAE/PTh 78/85 (1985). 9. I. Bengtsson and H. Hilffel, CERN Preprint CERN-TH. 4390-86 (1986). 10. M. Kaku and K. Kikkawa, Phy3. Rev. D10, 1110 (1974); E, Cremmer and J. L. Gervais, Nucl. Phys. B90, 410 (1975). 11. T, Hegge, Nuovo Cimento _19, 558(1961); R. Frledberg and T. D. Lee, Nucl. Phys. B242, 145 (1984). 12. M. Bander and C. Itzykson, Nucl. Phys. B257 FS14 , 531 (1985). 13. F. David, Nucl. Phys. B257 FS14 , 543 (1985). 14. A. Jevicki and M. Ninomiya, Phys. Lett. B150, 115 (1985). -29- -30- T FIGURE CAPTION Pis-! Two-dimensional surface made of right triangles with sides a and b. N+1 77777777/777/7/777 //////7777/7717////7 7///////7/77////////M 7777777///////////// -31-