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On c=1 matrix model and 2D gravity - with emphasis on chiral formalism Wang, Daoyan 2008

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On c=1 Matrix Model and 2D Gravity With Emphasis on Chiral Formalism by Daoyan Wang  B.Sc., University of Science and Technology of China, 2004  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics)  The University Of British Columbia (Vancouver) October, 2008  ©  Daoyan Wang 2008  Abstract In this thesis, we study the relationship between the effective spacetime the ory of Liouville string theory in two spacetime dimensions and the collective field of the c=1 matrix model by finding exact solutions on both sides. The correspondence between the matrix model and the effective spacetime theory turns out to be nonlinear in their fields. By comparing the exact solutions on both side, we show the nonlinearity begins to appear at the second order in terms of the incoming tachyon field. In particular, we employ the chiral formalism in the matrix model the formalism allowing to write down solutions to equations of motion explicitly to find out exact solutions. We show the chiral formalism is much simpler than the more traditional classical field method. Also it is more powerful as it enables us to study the behavior around the singular point in the background of the matrix model. —  11  Table of Contents Abstract  ii  Table of Contents  iii  List of Figures  v  Acknowledgements  vi  Dedication  vii  1  Introduction  1  2  Background  3  3  4  2.1  2D Liouville String Theory and Its Effective Spacetime Action  2.2  c  =  1 Matrix Model  3 5  2D Gravity  8  3.1  Gravity + Dilaton  3.2  Adding a Dynamical Field  3.3  The Strong String Coupling Limit  13  3.4  Two Incoming Pulses  15  c  =  8 -  Tachyon  1 Matrix Model  11  25  4.1  Description of Fluctuation  25  4.2  Collective Field Description  27  4.3  Classical Field Perturbation Theory  28  111  Table of Contents 4.4  5  7  32  Chiral Formalism  38  5.1  The Exact Solution  38  5.2  Time Delay and S-Matrix  40  5.3  Chiral Formalism Calculation  42  5.3.1  Given Incoming Field  42  5.3.2  Given Outgoing Field  44  5.4  6  Physical Pictures and Relations  Chiral Formalism for the Original Background  46  5.4.1  The Formalism  46  5.4.2  Justification of the Formalism  48  5.4.3  Behavior around the Singular Point  50  Relationships Between MM and 2D Gravity  53  6.1  The Correspondence at the Boundaries  53  6.2  The Nonlinearity  55  6.3  Constraints on L 1 and L 2  58  6.4  The Correspondence in the Bulk  59  Conclusion  Bibliography  63 .  .  .  65  Appendices A (Modified) Kruskal Gauge  67  B Relationship between ri and  70  C Integrals involving the leg-pole kernel K  72  D Derivation: the Third Order Tachyon from MM  73  iv  List of Figures 2.1  The Pictures for the Left Hyperbola  7  3.1  Penrose Diagram for Tachyon Field  14  3.2  Two Separated Incoming Fields  16  3.3  Two Localized Pulses  17  4.1. The Penrose Diagrams for Left and Lower Hyperbolae 4.2  The Pictures for Left and Lower Hyperbolae  .  .  .  .  30 36  V  Acknowledgements I would like to thank Joanna Karczmarek. As my supervisor, she helped me so much, from ideas to the calculation details. Especially, thank her for her great patience when I lost my direction. She is the person who brought me into theoretical physics research. Thank my family for their infinite love. Special appreciation to Tao Tao for her support during my tough time.  vi  To my Tao Tao.  vii  Chapter 1  Introduction The effective spacetime theory 1 of Liouville string theory contains a single propagating degree of freedom in two spacetime dimensions, whose dynam ics is equivalent to that of the c  =  1 matrix model . The correspondence 11  between the gravity and the matrix model at the incoming and outgoing boundaries of the Penrose diagrams is already known[1]. Holographic prin ciple tells us that the behavior in the bulk of spacetime is encoded on the boundary, thus it is nature to study the correspondence in the bulk of spacetime. The study on the interior behaviors of gravity theory and the matrix model and their correspondence will be the main goal of this thesis. This thesis starts with the explanation of each side of the correspondence the gravity theory and c  1 matrix model in chapter 2. In order to study  the behavior of the gravity in the bulk of spacetime, in chapter 3 we find the exact solutions to the first three orders in tachyon field on the gravity theory side in the limit that the tachyon background is negligible. By these exact solutions, we study two gaussian incoming pulses as an example and illustrate the statement in [1] that “pulse 2 scatters off the gravitational field of pulse 1 before it ever reaches the wall”. In [1], Natsuume and Polchinski studied the correspondence between the scattering in the matrix model and the scattering in the gravity theory. In this thesis we will employ the collective field description (e.g., [15]) to ‘We will refer to it as gravity theory for short. “Since there is no other matrix model studied in this thesis, we will just call it matrix model or MM for short.  1  Chapter 1. Introduction express the matrix model and reproduce the main results in [1]. In chapter 4 we review the collective field description of the matrix model. We then use classical field method to study the behavior of the collective field behavior in the bulk of its “spacetime”. In chapter 5 we study the chiral formalism for collective field in great detail and employ it to reproduce the results in [1]. Besides, we use it to study the behavior of the collective field around the singular point in the background of the matrix model and study the interior behavior. Finally the correspondence between the matrix model and the gravity theory is studied in chapter 6.  “We will often loosely call it the interior behavior.  2  Chapter 2  Background In this chapter we will briefly collect the facts that are relevant to this thesis about the 2D Liouville string theory and the c  =  1 matrix model in  the literature[2] [3] [4J[5] [6] [7] [8] [16] [17] [18] [21] [22].  2.1  2D Liouville String Theory and Its Effective Spacetime Action  The Polyakov action for a string in d dimensional flat target spacetime is  s  (2.1)  =  The X’’s stand for the spacetime coordinate, so the flat spacetime metric, and  ‘yab  i runs from  is the world sheet metric.  ‘  1 to d.  i,  is  is a constant  with dimension of squared length. To fix conformal gauge, we write the metric 7ab  ‘yab  in terms of some fixed  as 7ab  The Liouville field  =  (2.2)  e/ab.  become dynamical. Taking into account the Fadeev  Popov determinant and the Weyl anomaly, the action eventually becomes [21] [22]  S  = —  f  du {ab (8aX0bX + 0 aUb)  —  +  (2.3) 3  Chapter 2. Background  Actually the term e 2 is added by hand.  One can find the following  relation =  25— d  b+b . 1  (2.4)  The Liouville field q is like another space dimension of the target spacetime. Therefore, we can think of this as a non-critical string in d dimensions with metric ‘yab, or a critical string in d + 1 dimensions with metric Yab [3]. If we take the second point of view, we can find an effective spacetime theory [21]. Let us consider the string-sigma model S  =  —  fd2{ 7 ( a bG + iEabB) 8aX8bX ’ + ‘R + 1 (2.5)  where ab is the Levi-Civita tensor. G,,, is the metric of the target spacetime. is an antisymmetric tensor, and  + t9B is the  +  =  field strength for B, which is also antisymmetric. I and T are scalar fields which are named dilaton and tachyon. In order to let Weyl invariance hold, some constraints among these fields need to be imposed. These constraint equations could be written in power of a’. To the first order in a’, these constraint equations can be derived from the following spacetime action eff 8  =  f  dDXe_2{  2(26— D)  2 + R + 4(V)  —  - 2(VT) 2 + T2}.  (2.6)  These fields could take the following background IV =  ; B9)  =  0;  =  lvXv; T°  =  exp(kX),  (2.7)  IVNatsuume and Polchinski took a different background for T in [1]. However, these two backgrounds are probably connected by field redefinition. Anyways, the choice in the background will not affect the study in this thesis since we will study the limit that the tachyon background goes to zero.  4  Chapter 2. Background where (k  —  )(kIL 1  —  1’)  If we take D  the 3-order antisymmetric tensor  q  =  XD, j,,  =  Q  öD  0 k  =  =  2 (in which dimension  is always zero) and let D  =  d + 1,  li,, we see that the string-sigma model (2.5) is  the Liouville theory (2.3). Therefore, we will focus on the effective spacetime action (2.6). Let us write down the effective spacetime action in the D  Seff  f  =  With c’  =  dtdçb /e 2  {  2 + R + 4(V)  —  2 + 2(VT)  =  2 case —  4V(T)}. (2.8)  1, this is the action (35) in [1]”. Here we added in a potential  V(T) which is —T in [1].  VI  There is also another important property of the D  =  2 case. If we define  a new field S in terms of the tachyon field and the dilaton field as  T  =  eS,  (2.9)  then it is easy to see the field S is massless. This coincides with the fact that there are D  —  2 oscillating degrees of freedom and one tachyon with  the mass square proportional to 2  —  D. If D > 2 its mass square is negative  and this is why we call it a tachyon. However, if we take D  2, there is no  transverse oscillator and the mass of the tachyon is zero.  2.2  c = 1 Matrix Model  There are different matrix models. The particular one which we will consider in this thesis is the one with the Lagrangian [3] [16] vr =  {  tr (8M) 2 + M2},  (2.10)  VActually our Sg is twice of the action (35) in [1]. VlThere are different selection of the potential, eg, [21]. VIIThis is the effective Lagrangian under the double-scaling limit, in which we take  N  —  o while at the same time zooming in on the quadratic maximum of the original,  non-universal potential.  5  Chapter 2. Background where M is an N x N Hermitian matrix and ‘tr’ means the trace. In absence of long strings, Liouville string theory is described by the singlet sector which depends only on the eigenvalues ) of the matrix M. Since the Lagrangian is a trace, it is invariant under the transformation:  M  UMU’, where U is an arbitrary invertible matrix. M could be diagonalized by such transformation and the action will be the function —*  only of the eigenvalues of M since the nonzero entries of the diagonalized matrix will be the eigenvalues of M. i.e., our matrix model contains only the singlet sector. The singlet wavefunct ion I’ (\) is necessarily completely symmetric under exchange of the eigenvalues. When we change the variable from the matrix  M to its eigenvalues, we need to multiply the wavefunction by the square root of the Jacobian, which is antisymmetric in )j’s. Therefore our new wavefunction as a function of )j’s is also antisymmetric in ?j’s. When acting on this new wavefunction, the Hamiltonian becomes 1  /  2  H  211  i.e., the system represents N (which goes to oc in the double-scaling limit) uncoupled fermions in the inverted harmonic oscillator potential. In the classical limit we can describe the collective motions of the fermions in terms of a time-dependent Fermi surface[3]. Fermions in the Fermi sea move freely in the inverted harmonic potential. problem in the x  —  Let us think about the  p phase space of a single fermion, where x means  position and p means the momentum and they satisfy ±  =  p; j  =  x.  First, it is quite easy to figure out the Fermi surface of the ground state is p 2  —  =  Energy  =  constant, which is a hyperbola. For perturbations  that are not too large, the Fermi sea can be described by its upper and lower surfaces p±(x, t). Figure 2.1 shows the left branch of the hyperbola” 11 2 p  —  VIII  =  E with E < 0.  We will call this left hyperbola.  6  Chapter 2. Background  p  p+  p—  ‘SSs  ,/  Figure 2.1: The Pictures for the Left Hyperbola This graph shows an example that the fluctuation is flowing downwards.  7  Chapter 3  2D Gravity The action (2.8) (set c’  =  1  IX)  is our action on the 2D gravity theory  side. In this chapter, we will study its properties by finding exact solutions which will be useful in finding the correspondence to the matrix model. In particular, we will study two incoming localized tachyon field pulses in great detail and illustrate the statement “...the actual physical picture is that pulse 2 scatters off the gravitational field of pulse 1 before it ever reaches the wall.” in[1].  3.1  Gravity + Dilaton  In order to investigate the properties of action (2.8), let us start with a simpler case and then add in more fields. This simpler case is a gravitational field coupled to dilaton S =  f  d x 2 / (R + 4(V) 2 + 4A ) 2  (3.1)  The equation of motion for g is —  +4  (aa  —  gai(V)2) —  2A 1 g 2 1  In 2D theory, the cosmological constant must be zero. Einstein tensor  —  =  0.  (3.2)  This is because  is zero in 2D. When we take the trace of  IX When we did calculations, we wanted the formalism to be as general as possible. We  will keep the explicit form of the constants ai, a2, )... rather than some specific values for them, as long as it is possible to do so.  8  Chapter 3. 2D Gravity equation (3.2), the second term is also zero. Thus ).. must be zero. The case is changed when the action changes intoX  f  S =  xe (R + 4(V) d 2 2 + 4A ) 2  (3.3)  This is because when we vary gv to find out SS/g, the e 2 will be affected. The equations of motion become[13] [14]  R, + 2VLVV =0, R + 4V 2  —  2 + 4\2 = 4(V)  (3.4)  In conformal gauge XI, the equations of motion[9] are 2(90_  —  4U8_  —  e = 0, 2 )  )=0, 2 2(9±p8±—8 P = 0, e 2 —4t98_ + 48U + +ô_p 28 + \ 88_(p—) ==0.  (35)  The exact solution to equations (3.5) is a black hole [9] [13]. In Kruskal Gauge coordinate system (primed coordinate), it is e_2p’  =  e_2’  =  m —  (x+ 2 A —  xj’)(x’  —  x).  (3.6)  In the later sections, we will use its perturbation form with respect to the backgroundXhl P0  0, = ax+ + bx  =  + c,  (3.7)  where a, b, c are arbitrary constants satisfying 4ab = —A 2 (in [1], a = 1, b = = o)X. —1, c Let us follow the method given by eq (41) in [1]. To linear [lj )2 = 4• I9,g = 2 XIFirst define x = x 0 ± x’, then take the metric: g+— = _(1/2) =0. xIIThis background is of course a solution to (3.5), XIII c could be absorbed by redefining o, so we could ignore c. However, here we will 11 X  keep it.  9  Chapter 3. 2D Gravity order in gravity and dilaton, the equations of motion are (6-i(8_  —  —  88_6  2a)2  =  2b)2 =  0, 0,  (3.8)  —b Q,  0+8_(ö—p) =0, where S  o, and f2  —  —  (a0_ + b8±) S + 4ap. Here  =  —4ab is  used. It is easy to find the solution °, 2 =Ae S  =  p  =  + B(xj + C(x),  —  —  (3.9)  Ae20 + —0±B(xj +  where A is an arbitrary constant, and B, C are arbitrary functions.  In  the Modified Kruskal Gauge coordinate system (unprimed coordinate) ax+ + bx + c, p = 5, thus the solution becomes 0, 4 p o  XIV  Ae°,  =  S  =  p  =  (3.10)  ° + Be2a + Ce 2 —Ae , 2  where A, B, C are arbitrary constants. We expect that the solution to the first order equations (3.8) is the lin earization of the exact solution, i.e., we expect that (3.9) is the first order of (3.6). In order to compare them, we express (3.6) in the Modified Kruskal Gauge coordinate system. In the Modified Kruskal Gauge coordinate sys tem, the exact solution (3.6) becomes XV 2 e  =  23 e  =  1+  (  e + Fe2a + Ge ° . 2 + FGe_2c) 2  (3.11)  When p and S are small, the linear approximation is p  =  = —  r 0 XIVF  {(  ° + Fe20 + Ge 2 2 + FGe_2c) e  },  (3.12)  Kruskal Gauge (KG) and Modified Kruskal Gauge (MKG), please refer to Ap  pendix A. XVPlease refer to Appendix A for the derivation.  10  Chapter 3. 2D Gravity which is just the solution (3.10). Relations between m, F, G and A, B, C are easy to figure out F=-2B,  m  3.2  ).  (-  —  G=-2C, c) 2 4BCe_  Adding a Dynamical Field  -  (3.13)  Tachyon  As mentioned in section 3.1, the most general solution of the Gravity + Dilaton system is a black hole, which means there is no dynamical degree of freedom. Let us add one the tachyon T. The Gravity + Dilaton + Tachyon -  system has only one dynamical degree of freedom. In principle, one could “integrate out” gravity and dilaton and leave the action with tachyon only, which is an effective action of tachyon. This is part of the motivations of my project. The tachyon field T is added to the action (3.3) in such a way[1]  s  =  f  xe {ai (R + 4(V) 2 d ) 2 2 + 4A  — (VT) 2 + 4T 2— a T3} 2 (3.14)  which implies the equations of motion  — ai0T0T 0, T 2VVT + 4T — a 2 V T =0, 2 R + 4V 2 — 4(V) —2 (VT) + 4a 1 a 2 T 1 2+ + 2VVV  =  (3.15)  —  42  —  ’T 1 a 2  =  0.  They come from varying graviton, tachyon and dilaton, respectively. We will refer to them as the graviton equation, the tachyon equation and the dilaton equation. From now on, we will follow the notation in the literature to define =  t; x 1  =  q.  11  Chapter 3. 2D Gravity Start with the background [1] P0 0, o = 2, T 0 = 1 (2b + 2 )e XVI b ’ Here b ,b 2 2 = bi(—1 + 47 + mu), where ‘y is Euler’s constant 1 = —2j/a —F’(l). Define field S in the same way as [1] e0S.  T  (3.16)  In conformal gauge coordinate, we have the equations of motion to the first three orders in the tachyon field equation as equation (42) in [i]XVII 6+0_S(1)  =  8+8—5(2)  =  —  (i)2 —  eZ+_  —  +  =  S(2), 0 T  +  (3.17) —  a2  ——T (3 S S 0 0 p 2 —a T (1) To the linear order in dilaton and graviton field and the second order in tachyon field, the dilaton and graviton equations now become [1] 2)Q  =  2+T —(D±T) , 2  ai(8_ + 2)2  =  2 (8_T)  2a88_6  2a2 + T , 2  ai(6+  —  (ó  —  =  o)  —  , 2 T  XVIII  (3 18)  =  where the definition l is the same as it in last section. With the specific  value a= 1,b= —1,  2(8_ + 6 + 0 — p 4 ) .  XVIThe procedure to obtain the background is: first we ignore the tachyon field, and in this case P0  =  0, o  =  2 is a solution. Then let it be the background and only linearize  the tachyon equation. We then get equation  the general static solution T 0  =  —88_T + 8_T  —  8T + T  0, which has  1+2 (b )e b ’ . Therefore, this background is actually not  a solution to equation (3.15). One could think this method is valid due to the higher order  equations in  ‘.  xvIIIt seems that [1] missed some terms in the 2nd order and the 3order equations. Here we add them in. XVIIIThe last equation is missed in [1].  12  Chapter 3. 2D Gravity  3.3  The Strong String Coupling Limit  The tachyon background term in equation (2.3) is to the scenario to allow for a well behaved string perturbation expansion, shielding the strongly coupled region [5][17J. Since we are working at the tree level in string theory, we can then reimpose the strong string coupling limit by setting j to be small so that the tachyon background T 0 —* 0. From the gravity theory point of view, in the T 0  —*  0 limit we are studying the effects independent of the  background. In this case equation (3.17) is modified as =  0, , 2 (S(1))  =  =  +  (3.19) —  e_SS(2).  We can think of the left hand side of these equations as fields and the right hand side as sources. Furthermore, the first order equation is just the free field equation. The second order can be treated as a self-scattering process. The third one can be treated as the process that the field is scattered by the dilaton+gravity and the second order tachyon field 8(2) which are all produced by the incoming tachyon field.  Among them, dilaton+gravity  is produced via equation (3.18) and the second order tachyon field  8(2)  is  produced via the second equation of (3.19). In order to investigate the physics process further, let us consider the case where the incoming tachyon field is given, and try to find its effects on other fields (including higher order tachyon fields). If we let 8(1) be a field coming from the boundary of Penrose diagram as shown in Figure 3.1, 8(1) will be a function of ir only. From the first equation of (3.19) we know that s’ = f(xj + g(xj. i.e., the form of 8(1) will stay the same as it at incoming boundary. Since we only care about the effects due to the incoming tachyon field, the boundary condition should be all the fields are zero at x  =  —oo. Now we are ready to find solutions. 13.  Chapter 3. 2D Gravity  t  x  Figure 3.1: Penrose Diagram for Tachyon Field It shows the incoming field is only a function of x. And the little arc on the right of the diagram shows we take the limit ) is small. First, the second order tachyon field is easy to find by integrating its equation  -  8(2)  =  —e  f  dv  e_V  . 2 (s(1)(v))  (3.20)  In order to find the third order tachyon field, we need to find ö and p first. They can be obtained from the first three equations of (3.18). The solution is =  e2(_)  =  {_  2 (S(1)(x))  dv ((v)+  +  f  dv  (s(1)!(v))2},  ) 2 (T()(x+v))  (3.21)  (3.22) 14  Chapter 3. 2D Gravity By definition p  1 + (T(1) (x+,v))2 dv ((x+v)  =  )  -  2 (T(1))  (3.23)  One can check that in the special case which we consider here (To  s’  —*  0,  includes only the incoming component), the last equation of (3.18) is  automatically satisfied. i.e., when we substitute the solution of p, 6 into this equation, we just get an identity. Now we could integrate the equation for  (the third equations of  (3.19)) to obtain its solution. Explicitly, the solution can be expressed in terms of the incoming  s’  as =  e2(H + I + J),  (3.24)  where H  =  1=  f _f  dv e_vS(v)  j  dv e2vS(1) (v)f  J= __S(1)(x_)f  dv  V 2 e_  du du  e_U  , 2 (S(1)(u))  , 2 (s(1)I(u))  [(S(1)(v))  _2f  du(S(1)1(u))  2  ].  The physical meaning of these solutions is going to be studied in the next section.  3.4  Two Incoming Pulses  As stated in last section, the third order tachyon field can be treated as a result of the scattering process where the tachyon field is scattered by the dilaton+gravity and the second order tachyon field 8(2) produced by 15  Chapter 3. 2D Gravity the tachyon.  In order to justify this, we would like to clearly separate  the tachyon field producing dilaton+graviton+S 2 with the tachyon field scattered by the dilaton+graviton+S 2 produced earlier. In order to do so, we consider the interaction of two incoming tachyon pulses denoted as i.e., we expect that the first’° pulse comes in and produces  , 1 S  2 fields, then the second pulse is scattered by these dilaton+graviton+S fields produced by the first pulse. Please refer to Figure 3.2.  Figure 3.2:  Two Separated Incoming Fields  The two incoming gaussian tachyon pulses are centered at  =  0 and x  =  T,  respectively. And T>> 1. XIX  They ‘can also be denoted as S, S. We will use either S or S’ depending on  what we are emphasizing. XX “First” and “second” means their order in time. ‘Ve will use subscript “1” and “2” to stand for them.  16  Chapter 3. 2D Gravity Before moving on to the problem of two incoming pulses, let us study a general function relation that is essential for studying the problem. Given two localized functions (for example, gaussian functions) G , G 1 2 with sepa ration D (Figure 3.3)  Figure 3.3:  Two Localized Pulses  Two Localized Pulses with separation D. when D is huge so that the overlap between G 2 is negligible, we 1 and G have 00  00  2 ) fdx (x 2 G f  00  1 Gi(xi) fdx dx 2 ) (x 2 G f  1 Gi(xi). dx (3.25)  This is because 00  X2  2 ) fdx (x 2 G f  =  p 00  fX2  p  j  1 Gi(xi) dx  (x G 2 dx ) J  —00  dxiGi(xi)+]  —00  fX2  (x G 2 dx ) J  dxiGi(xj),  —00  X(J  (3.26) where xo is a point where the values of both G 1 and G 2 are negligible. The first term of the right hand side of (3.26) is approximately zero and the second term is approximately  J  p00  pxo  p00  2 ) dx (x 2 G  J  J  1 Gi(xi) dx  —00  00 P  2 ) dx (x 2 G  —00  j  1 Gi(xi). dx  —00  (3.27)  Similarly, we have p00  J  -00  pxi  1 Gi(xi) dx  J  (x G ) 2 2 dx  0,  (3.28)  -00  17  Chapter 3. 2D Gravity and the cross term Gi(x) G (x) 2  0,  (3.29)  for any x. Now let us come back to our problem of two incoming pulses. Let us investigate  (2),  2, p and S first. From equations (3.20), (3.21), (3.22) and  (3.23) we know that they are all combinations of of (S (1 ))  2  (S(1))2  .  In the case where we are considering S (1 )  and the integration (1)  .  S  =  (1)  + 82  ,  equation  (3.29) implies that the cross terms (of pulse 1 and pulse 2) are zero and leaves the results as terms either only including Sf1) or only including 1)• (2),  We denote the quantity (among that quantity with subscript i  l, p and S  )  only including  1)  as  XXI•  Let us consider quantities with subscript 2, which are fields produced by pulse 2. Equation (3.24) implies that the interaction between pulse 1 and the fields produced by pulse 2 is (3)  =  K + L + M,  (3.30)  where K  =  _  f_  L  =  al =  du  e_U (S’)(u)) 2  -00  dv e_2vS(v)  00  du  , 2 (s1)I())  00  2+  M  f f  dv e_vS(v)  -00  _s1)(x_)fX  V  —  dv  V 2 e_  [(s1)(v))2 —  2fdu (S1)(u))2],  which all include the form  Fi(u)fdvF ( 2 v)0, or the derivative of  where F 1 is either  (3.31)  2 is either the fields pro F  duced by pulse 2 or their derivatives. The vanishing result is physically r example, S2) 0 XXIF  =  —e  f dv  e_V  . 2 (S1)(v))  18  Chapter 3. 2D Gravity reasonable since the pulse 1 can not be scattered by the fields that are produced later. Now let us consider the opposite: pulse 2 scattered by the fields produced by pulse 1.  The result is just equation (3.30) with the subscript 1 and  subscript 2 exchanged.  Furthermore, equation (3.25) drives the result a  little further to the approximation (3)  =  K + L + M,  (3.32)  with a2e2  K  —j—-—j—  L  ____f e2  M  f  X  00  dv eS1)(v)f 00  du  e_U  (s1u)  2  00  X  00  dv e_2vS1) (v)j  _:s’)f  {e2v  du (S’)’(u))  [(S(1(v))2  -  2 (S’)’(v)) 2 +e  2  ] 2 (S1)!(v))  }.  Detailed Physical Picture Equation (3.32) can also be obtained by the following more detailed physical picture. Let us still start with p, ö (produced by pulse 1). When  XXII  is large,  their solutions (3.23)(3.22) are p  —  {e20  + e2 xxIIAs discussed above, x  f  2 dv (S’)’(v))  V 2 dve_  2 ((sI’(v))  (si1(v))2) —  },  (3.33)  will be the variable of pulse 2. Thus dominate when it is large  since the separation between two pulses is large.  19  Chapter 3. 2D Gravity which happen° to have the MKG form of the black hole solution (3.12). Even we can figure out the mass of the black hole in terms of the incoming field  m  dv (S’)’(v)) , 2  =  (3.34)  --  where we have set )  =  2. (3.33) implies  dv (S’)’(v)) . 2  f  al  (335)  Since the solution is a black hole, one might think that there was only dilaton+gravity here. i.e., the tachyon field producing it and the second order tachyon field 8(2) are all negligible. Although the tachyon field pro ducing dilaton+gravity could indeed be chosen so that it is negligibleXV, we do not need this assumption when we derive p and S. The only assump tion we need is that the incoming pulse is localized so that it falls zero when  x  —*  oc. Furthermore, we are not sure about whether  8(2)  is also negligible.  Under the same “degree of approximation” as p and 5, the approximation of 8(2) 8(2)  _ex+f  which does not depend on x.  dv  e_V  , 2 (S’)(v))  (3.36)  Thus it is not negligible.  However, one  could still expect that its contribution to the final outgoing field  was  negligible. Let us see whether this is the case. Let us substitute (3.33)(3.36) into the last equation of (3.19) to get to figure out their contribution to the final outgoing  The contribution  xxIIIIt  “happens” to agree with (3.12), since we did not require p = 6 when we solved the equations. We will see this again in the following example. xxlvSince we already see that the dilaton+gravity change with x exponentially as we can set the incoming tachyon field to fall faster than this exponential tail. Eg, gaussian pulses.  20  Chapter 3. 2D Gravity  from p & ö is 2x+ —  —  2x+ —  f  p00  X  dv e_2vS(v) P00  dv e_2vS(v)]  s— j  f  00  1 4a  2  du  (s1)’(u))  (337)  2  du (S’n)  (3.38)  —00  where the ö in (3.37) is shown in (3.33) and the subscripts 1 & 2 stand for the first and the second incoming tachyon pulses, respectively. The contribution (2)  from S  is 2e2 2 ±a  f f  dv  e_vS1)(v)2f  -00  x e 2 a2  du e_uS(u)  (3.39)  du e_US)(u).  (3.40)  -00  dv eS(v)2  -00  f  -00  The sum of (3.37) and (3.39) is just equation (3.32), while the sum of (3.38) and (3.40) is e2{f  dv e_vS1)(v)f  —00  1  dv e_2vS)(v)  —  00 alJ_  2 (s(1)(u))  poo  p00  —  due_U  -00  /  du (8’(u))  ,  (3.41)  00 j_  which is of the same form as the equation (46) in [1]. Actually if we substi tute the value  a2  =  —2/ into the equation, we will get the equation (46)  in [1]. i.e., (3.41) is equation (46) in [1] with 2 a explicitly expressed. These two terms are comparable, which means that 8(2) is not negligible compared with dilaton+gravity. However, the field 8(2) is not coupled with the dilaton+gravity system. We just simply add their contributions together. Hence the physical picture is: the first pulse produces fields, which is the combination of 82) and the dilaton + gravity system. Then the second pulse is scattered off by these fields before it reaches the wall. Therefore, the picture in the conclusion section of [1]  “...  one believes that the actual physical picture is that pulse 2 21  Chapter 3. 2D Gravity scatters off the gravitational field of pulse 1 before it ever reaches the wall” should be modified to include the tachyon-tachyon interactions. Now the outgoing  (from the interaction of the two pulses) can be  calculated from three different (sets of) equations: 1) Equation (3.30) (with the subscript 1 and 2 exchanged). No approximation is made in deriving it. 2) The sum of equation (3.37) and equation (3.39). We assumed that x is large when we calculate the fields produced by the first pulse. The sum of equation (3.38) and equation (3.40).  3)  We assumed that x  is large when we calculate the fields produced by the first pulse and we assumed x  —*  oo when we derive the outgoing  ), 3 s(  which means this set  of equations is only valid at the outgoing boundary. We will refer to these three (sets of) equations as equation set 1, equation set 2 and equation set 3. Therefore, it is clear that our equation set 1 and set 2 can give more information (specifically, the interior of the Penrose diagram) than [1].  Example: Two Incoming Gaussian Pulses In this section we will perform explicitly the calculation via equation set 2 with a specific example. The purpose of this example is tO: 1) justify the approximations that we made for p, ö,  8(2)  are indeed of the same “degree  of approximation”; 2) show something beyond [1] ‘s result by the specific example. When we derived equation set 2, we only assumed that the pulses were localized. Thus we can pick an easy-calculated function as the example. Let us consider two gaussian pulses° (denote as Sf1) and S ). The 1 first is centered at  ir  =  0, and the second at x  =  T. i.e., S( )(xj 1  =  2 2 Ae_B(i +ae_1(_T) as shown in Figure 3.2. T (the separation between  the two pulses) is large as discussed above. XXV  Another minor reason is that every well-behaved function could be expanded as a combination of gaussian functions (one can think about the coherent state of Harmonic Oscillator, gaussian functions form an over complete basis). 22  Chapter 3. 2D Gravity We will use equation set 2 to perform the calculation. will first use equation (3.23) (3.22) (3.20) to calculate p, 3,  8(2)  However, we and show the  approximations we made to find equations (3.33) (3.36) are indeed of the 2 Let us consider S’(x) = Ae_B() only and get the field (2, 6, p, Sf2)) produced by s’ based on equation  same “degree of approximation”. (3.23)(3.22)(3.20). We get —  8(2)  (1+Ef[V/(1 +xj])  e 2 A2a  8/  e 2 frA2a  V2 jrrA ‘  V2  8/  2 e_2B(w)  (2  k\  v(+x)  2a e8B 2  4/  Here the Erf is the error function. To get to the last line, we used Erf(x) 1  when x >> 1. Throwing away the small (the relative magnitude  —  is equal or below the order of e()  )  terms, we get the term after the  ,  which is the same as the result from (3.36). For the p, 6, we performed the same approximation and get  p  2 A  e2  2 A —  e2x+_2.  We want to emphasize that their exact forms are different (which can also be seen from equations (3.22) and (3.23)) and only after performing the approximation procedure mentioned above, they appear to be the same as the result calculated from (3.33). we can now substitute the fields above into (3.37)(3.39), and obtain the  which is divided into 8(2) contribution (denoted as S) and p, 6  23  Chapter 3. 2D Gravity (3) contribution (denoted as S,)  (3)  2 2x-T++8 aA 2a e 2  1  (3.44)  aA2ae2_Tir  (3 45)  —  aA2e2_b(T_) _ 2 2  (_i +  e+2)  1 4a aA2e_2T+2ir  (i + Erf (1+b(_T+z))) (346)  —  8va/ —  aA2f  eI2T+2ir  (1+b(_T+x_)))  (i + Erf  8/a/ —  aA2vei6_2T+21r  (3 7)  4vai  The sum of (3.44) and (3.46) is the result based on equation set 2. On the other hand,  3 could also be calculated from equation (34) of [1] s  (which  is the same as equation set 3) and the result is —  —  aA2 ae2_T  16/  —  aA2e_2T+27r  1 4/a  (3 48)  which is the same as the sum of (3.45) and (3.47). i.e., our result includes more information than [1].  24  Chapter 4  c  =  1 Matrix Model  As stated in chapter 2, we will consider the collective motion of noninter acting fermions in the inverted harmonic oscillator. We describe the Fermi surface of a ground state as a hyperbola p 2 We will be interested in the “excited state”  —  —  =  ±2i, here ii  small fluctuation to the sur  face. In [1], Natsuume and Polchinski described the fluctuation around the incoming and outgoing boundaries. Some authors[15] employed a specific collective field description, which is valid everywhere. In this chapter, we will just collect these two descriptions and find relations between them. We will try applying classical field theory method to the collective field directly, which motivates us to change the field background of the collective field. Furthermore, the complexity of the classical field theory method motivates us to find a simpler approach, which will be introduced in the next chapter.  4.1  Description of Fluctuation  To be specific, let us consider the left branch of the hyperbola 2 p  —  =  —2i.  (4.1)  In order to write down the fluctuations around this background, [1] define the field S in the asymptotic region as p(x,t)  =  =  ±ii(q,t) —8q(q,t),  (4.2) 25  Chapter 4. c  =  1 Matrix Model  where the q is defined as x  when x  —e  =  <  0.  (4.3)  In terms of the field S, the Hamiltonian can be written as H  2+ dq{I  =  (0q)2  + e2QO(3)}.  (4.4)  L  Thus this is a massless scalar field in the “space” q and time t. On the other hand, some later works (e.g., [15]) used collective field to rewrite the field as (x,t)  =  p+(x,t) —p_(x,t)  (4.5)  and then write the small fluctuations about a fixed solution o(x, t) as  (4.6)  o=o+v8x. The following relation [15] will hold p+(x,t)+p_(x,t)  where Z(x, t) terms of  j.  f d’a(’, t).  Z  (47)  We are able to express  From (4.2) we can also write down  in terms of  Thus we should be able to find the relation between S and fixed solution be the left hyperbola (so po and large we will have  =  p+(x,t)p_(x,t)  /x2  —  j.  in .  If we let the  2t), when x is negative  (4.8) p+ +p— 2  =  1 —(e 2x  —  —  _)  =  x  -  —  ,  (4.9)  which imply that (x,t)  (x,t).  (4.10)  26  Chapter 4. c  =  1 Matrix Model  We can therefore reproduce all the results about scattering problem in [1] in terms of  .  Besides, another advantage of collective field description is that  its definition does not depend on specific background. Also, we will develop the chiral formalism of collective field to investigate the interior behavior.  Collective Field Description  4.2  Let us collect the relevant parts of collective field description[1O] [11] [12] [15]. The action for the collective field is  dtdx  =  f (  2 + (z  —  —  2/1)).  (4.11)  As mentioned above, the small fluctuation about a fixed solution ço (x, t) 0 is defined as (4.12) Then the action can be rewritten in terms of the small fluctuation  f  dtdx ((Zo+8t??) 2  +  -  +  (x  -  )(yo + 2 (4.13)  where Zo(x, t) powers of  7)  =  f dx’ Utyo(x’  ,  t). Terms in this action can be grouped in  (the field that we are interested in) as (terms linear in  vanish  by the equation of motion) S  =  ) + 2 S(o) + S(  (4.14)  where S(o) has no 7)-dependence, the other two terms are  ) 2 S(  =  f  dtdx  —  —  2(o7))2}  (4.15)  and  Sint  =  f  dtdx  —  {_o(a7))3 +  (at?]  2 (ra)n} Z) (4.16) 27  Chapter 4. c  =  1 Matrix Model  It is proposed that coordinates (r, o) exist in which S( ) takes a standard 2 form of a kinetic term for a field in a conformally fiat metric. For example, the below two cases that we will consider later. Case 1 (the left hyperbola): T  =  t,  e  (—00,0],  =  —/cosh(),  P0(J)  (4.17)  /sinh(u)  =  r  o(o)  =  —/sinh(a).  Case 2 (the upper and lower hyperbolae): T  =  a E (—oc,oc), x  ,/sinh(a).  =  p(a)  =  (4.18)  /cosh(a)  o(a)  p(a)  =  —/cosh(a)  /2jcosh(a).  For case 2, we will let the upper hyperbola stay fixed and let the fluctuation only occur at the lower hyperbola, this is a key future we will need later. The action S  f  drda{  —  S(o) for these two cases is  ((8)2  —  ) 2 (a)  (3(8)2(8)  —  +  (8)2  00  i.e., now the system is a massless scalar field  + (O))  (_)fl  j  }.  (4.19)  with interactions in the  Minkowski “spacetime” (a, r).  4.3  Classical Field Perturbation Theory  Let us investigate the scattering of  .  Since we have the action, it is quite  natural to apply the classical field perturbation theory (for example, equa 28  Chapter 4. c  =  1 Matrix Model  tion (2.5) in [19]) to calculate S-Matrix  XXVI.  In our case, we can treat the  theory as a massless scalar field with interactions. Assuming we already have the form of (1) (The incoming free field. The superscript (i) indicates the order in terms of the incoming field (‘) ), we can calculate the second and the third order as (2)  (T,  a) =  3)  (T,  a) =  f f  (P  dr’da’G(r, a; r’, a’)F( ) 2  (1)  dT’da’G(T, a; T’, a’) {F(3)  21 + cross terms in F  (T’,  a’)),  (T’,  (?]() (T’, a’)  +  a’)) 2)  (4.20)  (r’, a’))  where G(T,  f  1 a;  T’,  a’) =  2 (2n)  },  ei{_ko(T_T’)+ki(o_dl}  dk  k?  —  (ko +  (4.21)  i)2  is the retarded Green function. And F( ) and Ff 2 ) are the second and third 3 order terms in terms of  in the equation of motion. Explicitly, the equation  of motion is 82  T-  82  /::  f  ((8?)2  + (0)2\  (()2)  +  28  2  ((8)2)}  _{  +...  (4.22)  F( ) () + +3 We are going to start with the first order field as a free field coming from a  =  —00 (‘)(T,  a) =  f  —  dw__+(w)eT_.  21r\/iw  (4.23)  The problem we want to consider is Case 1. However, the fixed back  ground o = —\/sinh(a) is singular at a = 0 and this results in the formula (4.21) being not integrable. Thus we switch our problem to Case xxvI• 2  The Penrose Diagrams are  xXVIThis corresponds to the tree level Feynman Diagram. XXVIIThe motivation of this switch and the relation between case 1 and case 2 will be stated in the next section.  29  Chapter 4. c  =  1 Matrix Model  Case 1  Case 2  T  T  0•  ‘7  Figure 4.1: The Penrose Diagrams for Left and Lower Hyperbolae For the left hyperbola, the incoming field is bounced back at some place and can not go through the whole “spacetime”. For the lower hyperbola, the incoming field is able to go through the whole spacetime, but can only emerge at the outgoing boundary. We will refer to the left hyperbola as the original background and the lower hyperbola as the alternative background. To distinguish the quantities in the alternative background with those in the original background, we denote quantities in the alternative background with a tilde sign, eg,  .  For  example, the first order field is tilded as P+OO  =  J  dw. jm(w)eT_. 2ir/iw  (4.24)  After calculation via equation (4.20), the second order result is (2)  = —  When  —*  f  eh()  16/2•  (4.25)  oc, this becomes  f  —  dw 1 dw  (4.26)  ), 2 ei(w12)(T_)äjn(wi)äjn(w  8ir / 2  which gives the scattering matrix to the second order Qout(W) =  fdwidw v 2 . 2ir 2ir 2u  j  —  Wi  —  w2).  (4.27)  30  Chapter 4. c  =  When expressed in terms the mode of c_(w)  2 I dwi dw  -  = —  j  1 Matrix Model  ,  f[’\iW  it is  XXVIII  -  -  )2ir6(w 2 -—zwo+(wi)c+(w  —  —  w2),  --—-—  (4.28) which is the same as equation (14) in [1] when n  2 (and of course (gs)’  =  =  2k). For the third order, I can only figure out the case when  f  dw 2 dwidw 3  e0(T  62(2)2  a  —*  00  (w )äj (w2) in (w ), 3  (4.29)  which gives the scattering matrix to the 3rd order  dw 2 [dwidw 3  (w) 0 ci =  I  4ir 2qi-2qr2ir 2 3(2i)  (4.30)  Expressed in terms the mode of 5, it is  1 dw [dw 2 dw 3 —  J  -—-—-—  w  47r  iw(iw ) 2 ( 3  )  -  -  -  —  —  (4.31) which is the same as equation (14) in [1] when n 2 W  +  =  3. Where  +  i  W3.  Thus we can conclude that indeed we can use the collective field § to study the matrix model instead of S. The classical field theory method does work for the scattering problem. However, the work we need to do increases dramatically with the order (i).  Alternatively, we will use the  chiral formalism to study the scattering problem systematically to all the orders in chapter 5, which is much more powerful and simpler than this classical field theory method. XXVIHThe relation among and  ,  j  and  a is expressed as equation  S  will be given in section 4.4. The relation between  a  (4.57).  31  Chapter 4. c  4.4  1 Matrix Model  =  Physical Pictures and Relations  In the last section we investigated the scattering problem in terms of the collective field  .  However, the physical picture is not clear yet. Eg, the field  stands for the fluctuations of the left hyperbola, basically one can imagine the fluctuations can come from either upper part or from lower part. Can they both happen? In this section we will study the physical picture of and i. Besides, the relationships among j and i will be listed in this section.  Physical Picture of For the matrix model describing fluctuations around the left hyperbola back ground x 2 —p 2 = 2i, we parametrized x as x and had the relationships =  +  O  =  2 +p- — 2 —  +  =  2’ — —  x2  —/ cosha with a  —2+8 zTh  V’T?]  —  çO  =  — 2 + /F8  X  <  0,  (4.32) (4 33)  which gives  —2+ _x2 — 2i —  p When j  =  p =  +  (4.35)  (r + a) (left traveling), these two equations become =  when  (4.34)  +  r,(r  —  =  — 2ji, _Vx2  — 2 — 2/t9j;  (4.36) (4.37)  a) (right traveling), these two equations become =  =  — 2i +  (4.38)  — 2i.  (4.39)  _/x2  32  Chapter 4. c  1 Matrix Model  =  This means the perturbation comes in from upper half part and goes out via the lower half part of the hypobola. And it can not come from lower half part and go out via the upper part. This can also be seen by the classical equation of motion j=x, ±=p.  (4.40)  Since x < 0, p decreases with time so that the perturbation comes from the upper part and goes out via the lower part.  Relationships Between S and ‘q In section 4.1 we already figure out that when x  Since the parametrization of x for S is x  —*  —oo, (x,t)  —e and x  =  =  (x,t).  —/ coshu with  u<0for.Sowhenx—*oo,x---*—7e,wehave (4.41)  —q—a+ln/7.  Combining this with the fact that when x T  =  —* —,  (x, t)  —*  t as mentioned in equation (4.17)(4.18)), and .(x,t)  (r + u) (here  —  (t ± q), we  have (rq) More relation between S and  j  (r+u±ln).  (4.42)  can be found. Equation (4.8) and (4.9)  tell us —x +  +  )  —  €)  —x +  (4.43) (4.4)  which imply that i  + /(x6 + + /(x8  —  t), 8  (4.45) (4.46)  33  Chapter 4. c Thus for incoming field  (j  =  =  1 Matrix Model  —  — 2/(8) For outgoing field  (j  =  r,i(T  c  ;  (4.47)  .  + o)) ;  i  — -  2/(8).  (4.48)  These relationships will be used for finding the time delay and the scattering relation.  Relationships Between  ‘ij  and i  In order to remove the singularity of the left hyperbola at u nature to consider the exchange, x  ÷—*  =  0, it is quite  However, there exists another  p.  better transformation which can also achieve the same purpose and has better property  it is the canonical transformation =  —p,  5=  x.  (4.49)  Since this transformation is canonical, the dynamics of the transformed vari ables  and  5 can  also be described by the same collective field theory. This  is totally the same as the collective field theory if we replace the original background with the lower branch of the hyperbola feature is much better than just x  ÷—*  2  —  52  —2i. This  =  p. Now we have a fixed upper branch  /2  + 2 and fluctuations only occur at the lower branch around the background of the lower branch (0) = _/2 + 2. The superscript (0) de  5+  =  notes the background. Parameterize  as  sinhã For the collective  =  field of the new variables, we still employ the definitions such as = =  =  2 —  + 2+a  which imply (for simplicity, we drop the  —  subscript of  5  X  (4.50)  when it is not  confusing) x  =  5  =  _/2  + 2i.i  —  (4.51) 34  Chapter 4. c Since  =  1 Matrix Model  can be positive or negative, the above equation implies the following  two equations  5  =  x  +  x  — —  2/a ;  —  —  when  2/8 ;  —*  when  —cc.  (4.52)  cc.  (4.53)  —*  Combining these two equations with (4.38) and (4.37), we get incoming:  < 0 ;  =  outgoing: >0; Thus we conclude that when r and x r  —  —f  —cc; and ffi  cc, the field  field  —+  x (ö  =  u) ;  =  —cc, the field  (4.55)  ;  =—x(ãrr—u)  —*  is the function of (r  —cc, the field i is the function of (r  is the function of (r + u) and x  is still the function of (r  —*  —cc;  Again, this can also be seen by equation of motion 0,  —  —*  o).  —  u)  When  cc and the  —  Thus the physical picture is that the field goes from  5<  (4.54)  —.  increases with time. The picture for  i  =  =  —cc to  —5;  =  —.  =  cc.  Since  and i is shown as in Figure  4.2. Note: the relation between  and  we got here is only exact at the  incoming and outgoing boundaries. In the bulk, the relation is not linear. Please see Appendix B for higher order relationships.  Relationships Between S and We can now obtain the relationships between S and  from equations (4.41),  (4.42), (4.54) and (4.55) —  )  —  )  =  —(t  —  a)  (t + a)  =  =  —S(t — q)  S(t + q)  =  = —  (t  —  (t &  —  —  ln  + ln  ),  (4.56)  35  Chapter 4. c  =  1 Matrix Model  3 p/I  Figure 4.2:  The Pictures for Left and Lower Hyperbolae  For the left hyperbola, the fluctuation is flowing downwards. For the lower hyper bola, the fluctuation is flowing from left to right. which are for incoming and outgoing fields, respectively. Or we could also rewrite these relationships in terms of the modes of i and S f/\iW/2  o(w)  =  =  —a+(w) w)  )  ()  —iw/2 .  (4.57)  From the leg-pole transformation between S and S (equation (20) in [1])  (qj 0 S  (+)  =  fdzK (q  =  f  —  dzK (x  z) S(z)  —  z) 0 (z),  (4.58)  36  Chapter 4. c  =  1 Matrix Model  we can now write down the leg-pole transformation between = —  out() =  f_ooZ  f  K  (  dz K (z  + in  —  + in  —  and S  z) S(z),  )  t(z), 0 S  (4.59) (4.60)  where K is defined as equation (21) in [1] K(z)  —  I  J_ =  (iW/4  2r  —  \21 Z  f(_iw) F(iw)  . 2 e 8 2(2/)  (4.61)  here J 1 is the first Bessel function. We are now able to perform the transformation between the collective field  (at the incoming and outgoing boundaries) in the matrix model and  the tachyon field (at the incoming and outgoing boundaries) in the gravity theory.  37  Chapter 5  Chiral Formalism In chapter 4 we used classical field perturbation theory to figure out the scattering problem of the collective field. It turned out to be difficult to do it, especially for higher orders. In this chapter, we will investigate the possibility of using only the equation of motion. To be specific, now let us focus on the alternative background case, thus the equation of motion is  =  —;  =  —.  It is quite easy to find the exact solution since any  function with the form F(Q5 — )e , T =  — and  =  (5 + )eT)  =  0 implies the equations  —, which are equations of motion of the system. In this  chapter, we just need to find the equations which are of such a form, then relate them to the physical field  ñ  at the boundaries. Since the form implies  equations of motion, as long as the , ffl match our physical condition at some region, they will match our physical field everywhere. For the original background, the case is similar. Please refer to section 5.4.  5.1  The Exact Solution  The purpose of this section is to relate the exact solution in the bulk to boundary condition. First, a good feature is that when --  a  r—aln  —*  / +5 7 N  ±oo,  5  —ln  We have  /-5 N .  (5.1)  which are of the form F((5—)e_’, (5+)ej. On the other hand, we know that the field at incoming and outgoing boundaries are of the function of 38  Chapter 5.  Chiral Formalism  -r — 6- only. Thus we can replace r — 6- by in (_er) or — in In fact, we should replace r —6- by ln  (_eT)  for the incoming field. This is because  rather than — in  (f,e_T).  (,je_T)  for incoming field, hence small  fluctuation can affect the relative change  significantly while  + )/( + ) can remain small. Similarly, we should  the relative change replace r — 6- by — ln  (,je_T)  rather than ln  (_,eT)  for the outgoing  field. Therefore, if we can find a equation involving the incoming (or outgoing field), we can rewrite it by replacing r—6- by in  (_,eT)  (or — ln  (e_T)  and keep the function form unchanged. In this way, the equation is of the form  F((5_ff)e_T,  (+)eT) =  0, which is an exact solution. To be specific,  let us consider the equation (4.52), which implies the following relation at the incoming boundary =  p  2i —  — 6-)  =  2 + 2f (in (_er)) (5 2)  —  where we have defined the function form f(6)  Since  =  has only one argument &, the function definition is valid. Once again, let us emphasis that the definition is the definition of function form, not of the function value. Or we omit the intermediate term l  2 +2 p  (  /-e))  p+x  which is an exact solution since it is of the form  F((p_)e_T,  (+)eT)  =  0.  Since equation (5.3) represents our physical system at the incoming bound ary and it implies the equation of motion, it should represent the physical system anywhere. Similarly, (4.53) tells us ii—,-1o  1  1  1n  —e —T (5.4) 39  Chapter 5. Chiral Formalism where the function form definition is f t() 0 Therefore  fin  and  0 f  =  are of order one in terms of the incoming field  and outgoing field, respectively (or we loosely say there is one ö in them). These two equations can be used when the collective field is known at incoming boundary and outgoing boundary, respectively.  5.2  Time Delay and S-Matrix  Let us consider equation(5.3) and (5.4). Physically the motion is unique, thus equation (5.3) and (5.4) must be equivalent. They are just different representations of the same solution. We obtain  fin /ln /  N N e),}  -  =  /-ln /  0 f  -  e NN  )).  (5.5)  We can use equation(5.3) to express (i5 ) on the right hand side of (5.5) in terms of ( + ) and f. Simplifying, we get —  fin(a)  =  t (a 0 f  -  (  ln  + Ln(a)))  (5.6)  Similarly, from equation (5.4), we get fout(a)  =  fin  (a+ln  (+fout(a)))  (5.7)  Equations (5.6) and (5.7) can be thought of time delay equations if we substitute  fin(a)  =  ain(a) and f 8 2/ t(a) 0  =  _2/8aijout(a). i.e., ex  press these two equations in terms of fields at boundaries. Equations (5.6) and (5.7) can also be expressed in term of e-j. Combining equations (4.47), (4.48) and (4.41) with (4.54) and (4.55), we get =  +  fin  (a  —  ln  ),  (5.8) (5.9)  40  Chapter 5. Chiral Formalism From (5.8), (5.9) and (5.6), (5.7), we get the time delay =  (a+ln  +  (a  =  —  in  (E_(a)))  (5.10)  (+(a)))  (5.11)  From relations (5.10) and (5.11), one can find the S-Matrix for the field S [1][20]. Furthermore, by the relationships among 5,  and  j  at the bound  aries as discussed in section 4.4, we could also find the S-Matrices for  and  i. However, it is possible to find the S-Matrix for i directly from the chiral formalism. If we define =  fi/i,  -  =  (5.12)  f t 0 /i,  then equations (5.6) and (5.7) become (x + in (1 + ±(x)))  (5.13)  which is exactly the same as equation (2.5) in [20]. We then get the same result as equation (3.4) in [20]  f  00  1  dx +(x)e =  1  f  00  dx e_{(1 +(x))’  Taylor expanding the right hand side in terms of  =  I-: f-: 1  dx e_{(l  i}. (5.14)  we get  (1 zw)(+iw) iw)(z) +  (())2  +  eW  =  ,  —  n! F(2-n+iw)  ((x)),  (5.15)  41  Chapter 5. Chiral Formalism or expression in terms of modes =  F(2_ n —w)  { Hf  out(Li)}2n6 (w  —  (5.16) =  F(2-n±iw)  { Lf  c1wi()}2 (w  -  w). 1 L (5.17)  Now equation (5.17) is the scattering matrix. If we combine this with equation (4.57), we can reproduce equation (14) in [1]. In this sense, the collective field description could totally replace the description in terms of  S. An interesting fact: one can substitute (5.17) into (5.16), then get an equation which left hand side is orders in  jl?,.  and right hand side is a sum over different  The first order is a trivial identity. All the higher orders must  be zero, leaving to be some mathematical identities.  Chiral Formalism Calculation  5.3 5.3.1  Given Incoming Field  Now we are trying to find the interior behavior in the case that the incoming boundary is given. Let us write the =  as a sum (5.18)  where the superscript (i) indicates the order in terms of the incoming field or its mode 42  Chapter 5.  (  If the incoming field  Chiral Formalism  6) is given, it is equivalent to say the fin is already given. We could use (5.3) to calculate fields order by order via the calculation of (i) since we have the following relation between ) and ) —  from (4.51) (i)  —  j(i) 8 j 2  ()  (5.19)  The calculation of ) is quite straight forward. We substitute  IJ  (i)  into (5.3) and sort the equation in different orders in term of the in (note: the order of fi is one by its definition). Then solve the equation order by order. The results for the first three orders are (0) (1)  —n_  P  1 /fp(1) ( in px+p’  (2)  p(3)  —  7  1  1  ((1)2\ —  2  /  I  (2) i(0)  p--(1) p-(2)  —  I  P Jjn X+p  (5.20)  j  f çFI(0)  kjin  —  c’(O)\ ((1)’2 )  un I P 2 2(i+(°))  where the derivatives of fin means the derivatives with respect to its own argument (in (_ej). The superscript (0) on or its derivatives indicates the value of fin or its derivatives at  =  (0)  XXIX  In this way, we can investigate the field in the whole “spacetime”. As an example, let us calculate the second and the third order results and compare it with the classical field perturbation theory. Given  iin(T  —  o)  =  the second order result is  = —  163/2  f  ei12T_in(wi)äin(w dwid ) 2  co:h()’  (5.21)  which is exactly the same as the result (4.25) from classical field perturbation theory. XXIXE  f’(O)  =  (i —J’tn  (_P±eT  43  Chapter 5. Chiral Formalism The third order result is 3) =  f  39622 d 2 dwidw e0(T_) w  wç = w  g(w,) =  )  —*  f  )  (w  )in(w3)g(wo, &), (5.22)  where I denote  When  (w  2+ +w  W3  and  )()()}  cosh{  (5.23)  becomes  1 dw d 2 w  eo(T_)  62(2)2  a  1 (w  )äj, (w2)  ), 3 (w  (5.24)  which is the same as the result (4.29) from classical field perturbation theory. Furthermore, here it is quite easy to figure out the interior behavior, while it is quite hard to find by classical field perturbation theory, as the integrals involved are nontrivial.  5.3.2  Given Outgoing Field  Similarly, if the outgoing field  out(r  —  6) is given, we can use (5.4) to  calculate fields order by order via the calculation of (i) (Now the superscript indicate the order in terms of the outgoing field, or its mode  aQU).  Equation  (5.19) is still valid here. The results are (1)  —  (2)  =  /çt (O)-(1) Jout  ( p()x—pi 1  5(3)  (_(1)(2)  ((1)2\ —  +  P  2 j j, +  —p( )  p  (5.25)  (f°  +  f)  ((1))2  2(—5(°))  Again, the derivatives of f t means the derivatives with respect to its own 0 argument ln (_e_T)), and the superscript (0) on f 0 or its deriva  (—  tives indicate their values at j5  =  44  Chapter 5. Chiral Formalism In this way, we can also investigate the field in the whole “spacetime”. However, what is the relation with the formalism introduced in section 5.3.1? First we already have the scattering relation equation (5.17), i.e., combination of various orders in  is a  i.e., the outgoing boundary condition  is totally fixed by the incoming boundary condition. If we choose  &j,  as  our “unit”, then we can sort the fields calculated from outgoing boundary in orders of äjj. On the other hand, the same order field could of course calculated from the incoming boundary. If the formalism is self-consistent, the two result should be the same. For example, if we want to calculate the field which is of the second order in  We can calculate it from  the incoming boundary (denote the result as  )).  On the other hand, we  can also calculate from the outgoing boundary. Let us write the scattering relation as order in U 0  —  then take  +  —  =  (denote the result as  from ci t 0 &out  +  =  +  _(1) out  ...  (the superscript (i) indicate the  to calculate the second order result in  the subscript 1 in outl indicate the result is  t), and take 0 c  out 0  (denote the result as  =  2).  out  to calculate the first order result in  The result from incoming boundary and  outgoing boundary should match. i.e. XXX .(2) l‘ 1 in  Indeed (5.26) does hold.  =  _(2) l7 outl  +  _(1) l7 out2  (5.26)  One can check it by explicitly performing the  calculation. Even the third order results also match. i.e. +  + cross terms in  ?lout1+out2  (5.27)  where cross term in 1+out2 is obtained from the following procedure: we (2) -(1) (1) (2) let 0 ci, then only take the cross terms in ci 0 and ciQU. out = 0 OUt + Again, one can check it by explicitly performing the calculation. These two matched results confirmed the consistence of the formalism. XXX  The subscripts in and out on  are to denote i is calculated from incoming and  outgoing boundaries. Please do not mix them with the incoming and outgoing boundary fields.  45  Chapter 5. Chiral Formalism  5.4  Chiral Formalism for the Original Background  5.4.1  The Formalism  Exact Solutions The exact solutions are 2t + 2 ljn (ln  (_zE±eT))  (5.28) x  —  L l 2 out L+ (—in (_±ze_r))  (5.29)  x+p  First we think Thn and T a) and (-r + a) respec l‘ out as functions of (r tively. Then here we define the function relation l(a) = 2./9ariin(a) —  and lout(a)  =  2\/Da7out(a). So again, the 1 of order one (with one c(w)  in it). Equations (5.28) and (5.29) are exact solutions since they are of the form F((x ±  —  p)eT, (x  +  p)e_T)  =  0 which implies the equation of motion  x. As seen in the equations, one includes + while another includes p. So they are not equivalent. However, since we know the =  p;  j  =  scattering matrix for i and the relationship between  and  at incoming  and outgoing boundaries, we can still find out the scattering relation =  n —iw) —  00  (\/)nl  {!I  {1[f  i()}2ö (w  —  (5.30)  }  -  (5.31) 46  Chapter 5. Chiral Formalism Chiral Formalism Calculation  Given incoming boundary field, we have (1)  p+  —  (  XXXI  ——y, p+  (2)  p+  =  1 —j p+  (0)  x—p+  p+  —  (i)2\  I  P+)  (5.32)  I’  2  —  I  (2),, (0)  1  (3)  (  (1),,  p+t (0)  + p+ x  p+  —  p  Here the superscript (i) (except for the (0) on cates the order in  and its derivatives) mdi-  . 2 Tj  Given outgoing field, we have (1)  p_ =  1(0)  tout p  1  /  ((i))2  (1),i(0)  I p_ t 0 P- =—?oiI (0) P_ \ap (2)  (5.33)  2  —  (  (2)  1 P--  P— out (0)  x+p  2 (i)) —  —  \ (iç + i”°” out)1  2(x+p(°))2  Here the superscript (i) (except for the (0) on o 1 ut and its derivatives) mdicates the order in ‘ 1out T And  is given by p+—p  (5.34)  = O + 8o4]  (n)  cc where P± = Zn=o P±  XXXIThe superscript (0) on lr or its derivatives means the value when superscript (0) on lout or its derivatives means the value when p  =  p. The  =(O)  47  Chapter 5. Chiral Formalism The procedure for calculating to a higher order is similar to the chiral formalism for the alternative background. The only difference is that here we need to calculate both p and p  and then apply (5.34). For example,  if we want to calculate the nth order in  the procedure is: 1) calculate  p; 2) apply the scattering relation (5.31) to write the outgoing modes as m  ci_(w) (m),  m—1 .  s with n  (m)  ci (w) as what we did in section 5.3.2; 3) calculate all the m  it;  4) apply equation (5.34).  One can see there is a weakness about the chiral formalism for original background compared with the chiral formalism for the alternative back ground: here we need to borrow results from alternative background’s to figure out the scattering relation. We will talk more about this in section 5.4.3.  5.4.2  Justification of the Formalism  First, it is clear that the exact solutions follow the equation of motion i x; th  =  p.  =  Thus we only need to justify that the boundary condition at  incoming and outgoing boundaries are satisfied. Or equivalently, we need to: 1) justify that the first order  give us the correct boundary condition;  2) justify that the results from p) go to zero when x  —*  —oc for n> 2.  Justification of 1) Given the incoming field  )  =  f  dw +(w)eT_, 2ir’/iw  f  2ir/iw  _  (535)  and the outgoing field u)  =  _  ___(w)eT,  (5.36)  we expect the ffrst order field (‘) should be just the sum of equation (5.35) and equation (5.36). At the same time, by chiral formalism we can figure 48  Chapter 5. Chiral Formalism (1)  out  (1)  and p_  .  .  .  Substitute them into equation (5.34). The result is indeed  the sum of (5.35) and (5.36).  Justification of 2) First let us find out the second order. The calculation is straight forward and XXXII  easy to do. The result of  at boundaries is like  goes to zero at boundaries. Actually 0 -i a form  e(T+)  which  e(T)e2  and it is also of such  i—’  2 at boundaries, which means integration and derivative e  do not change its limit property at boundaries. Assume the (n— l)th order pp and lower orders go to zero at least a, now let us prove that p)p goes to zero at least as fast as 2 as fast as e  . 2 e Let us take pp as example. The proof for  is the same.  Let us look at the The nth order equation of (5.28). It will be of such a form  2 ()  —  2 = ((o))  rn=O  ((i)(°)  .p...pm))  (537)  rn=1 (0)  (a)  (m) (n—rn) The left hand side is 2+ ii+ + b. Here b is a combination of °. The right hand side is a combination 2 which go to zero at least as fast as e .  where ii +  of form of Lj,, (8pmljn)  ‘  .  + jrn  =  .  n. First, by the special  ai is of the form  XXXIII• x—p+  i.e., it  ) is a combination of derivatives that smaller and equal to m divided by m n e_aia. So the term in right hand side is either Oi (x  ()  —  means  of  the part of  which is from p part by equation (5.34).  means the part  which is from p_ part by equation (5.34).  XXXIIIThe superscript ((m)) means the derivative of  Here the argument is (in  (_ej).  derivatives means the value when p  m times with respect to its argument.  And again, the superscript (0) on  in 1  or its  = p . 1  49  Chapter 5. Chiral Formalism mj(O)p)pm) (which goes to zero at least as fast as e = e ) or 8 2 (here all the ii’s 2 in p3) , so p’s go to zero at least as fast as e ). So 2  every term in the rith order equation goes to zero at least as fast as e . Of 2 course . So ?7 goes 2 pp will go to zero at least as fast as e ‘-  to zero at least as fast as e °. 2  5.4.3  Behavior around the Singular Point  Now the chiral formalism is justified. We can use it to investigate the interior behavior. Specifically, we can find the behavior around the singular point (u  =  0) of the background. Let us see if we can use the formalism to find  boundary condition at the singular point. From the scattering relation (5.31), we can obtain c(w)  /iw{  -  =  -  —  2  dwi  c(w)  iriw(i—  (w)  +  l){jfdwi  () +  ()}26  ()}25(  () +  (w  —  w) + 1 E  (5.38)  ....  Now let us calculate the field which is of second order in ple. Let us denote such field as (2)  We take o_(w) .  (1)  .  -  out p_ then add the result with p_ obtained from  =  Tj  as an exam  c(w) and figure  ci (w) =  c  (2) (w).  We  also calculate p and then substitute them into equation (5.34). We get  =  a+(wl)+(w2) 2 / 3 16ii  2 [dwidw j  (e0(T)  ‘  +  e0(T))  —e smhu  20(T_  +  awo 2 e  2i+W  1 (i F + 2  _,  1  1 (i 1 F 2  —  iWO  —  e2J)]  ,  2  },  2  —  ; 2  e2)  (539)  50  Chapter 5. Chiral Formalism where the 2 (a, b, c; z) is the hypergeometric function. When u 1 F the limit of  is just outgoing field with  reasonable. When  o  —*  c_(w)  —*  —cc,  which is  =  0, we will get its behavior around the singular point.  The limit is 2 fdwidw  —  ) 2 +(wi)+(w  {  — e0(T_)  ±  [3 + 6nwo  —  e0(T)  —  2iwoe0(T_J)  0 (7 + Log(2) + 2iw  ln(a)  (_))] }  (5.40)  where ‘y is Euler’s constant and ‘(z) (2)  around the singular point is like  So the degree of singularity of + ln(a).  Another Motivation for Background Changing we have been able to use the chiral formalism to study the behavior around the singular point. However, since the field here is singular and probably the  higher orders (in  aj)  are more singular, as a result the higher orders will be  bigger than the lower orders (thus nonconvergent, thus singular), which indi cates that the perturbation method breaks down around this point. Maybe one would argue that although perturbation method breaks down, one can still (formally) write the field as a sum of different orders in  at,.,.  One can  also match the result from incoming and outgoing boundaries by the con tinuity condition and even get the scattering relation out of it. However, this expectation cannot come true since that the field is singular so that the continuity condition can not apply XXXIV On the other hand, we know that every point on the small fluctuation follows its hyperbola and should be finite at any point and the field should XXXIVOne can assume  the field is continuous at this point, namely, p. = p. To the lowest  order we should have  =  ?7 and obtain c]  =  crj. However, this disagrees with  the lowest order of equation (5.38), which indicates that the continuity condition is not applicable.  51  Chapter 5. Chiral Formalism be continuous everywhere thus not be singular at a singularity should be due to the definition  0. Therefore the  treating the fluctuation as a  function of x is not appropriate around this point. An possible explanation is that the fluctuation in the fermi sea moves the endpoint  hence  is not  well-defined because the fermi sea itself has shifted. Instead, one can try to treat the fluctuation as a function of p, which pushes us to do the coordinate exchange x  —>  p, which finally changes into the alternative background as  discussed in section 4.4.  52  Chapter 6  Relationships Between MM and 2D Gravity We are now ready to study the correspondence between matrix model and 2D gravity in the bulk. We will first show that the boundary behavior of the tachyon on the gravity side can be derived from the matrix model via the chiral formalism.  Or that’s the correspondence at the incoming and  outgoing boundaries. Inside the bulk, we will first show the correspondence is nonlinear and then prove the nonlinearity of the correspondence begin to appear at the second order in tachyon field. After that we will find some constraints and other hints about the form of the correspondence. Also, we will try finding some correspondence in the bulk.  6.1  The Correspondence at the Boundaries  In this section, we will study correspondence at the incoming and outgoing boundaries [1]. Since we already have the correspondence (4.59) (4.60) at the boundaries, here we actually just check their validity. Since we already have the evolution in chiral formalism (5.21)(5.22), we can use the transformevolve-transform back strategy°°’ in [1]. The key formulae are listed in appendix C. When deriving them, we assumed x—xj to be large and negative xxxv(J) transform from the gravity picture to the matrix model side via (4.59), (II) evolve the pulse in matrix model via the chiral formalism, and (III) transform back to the gravity picture.  53  Chapter 6. Relationships Between MM and 2D Gravity (later x will be used as x±, and x will be used to stand for the coordinate of the incoming field), which means that the formulae are only valid when the x is much smaller than the “main region” (the region where the field dominates) of the incoming tachyon field. Since the correspondences at the boundaries are linear, the nth order tachyon field is transformed from the nth order collective fie1d°°” . First, 1 let us study the outgoing tachyon field from the first order outgoing collective field. From equations (4.59),(4.60) and (C.3), we have  dK  S(x) -  too  (+ + ln  K (ln  +  -  u) S(u)  -  du e_u((x+  —  u) —2+47+ ln())S(u).  (6.1)  On the other hand, from the first equation of (3.17), we have  S(xj  =  u(bi(x+  _fdu e  Comparing these two, we get b 1  =  -  u)  2 —2P/a2; b  + (b 2 =  -  bi))S(n).  (6.2)  (—1 + 47 +ln/). In this 1 b  way, we justify that the choice of tachyon background T 0 is correct. The second order: from equation (5.21), we have 1  ?outT 1  -  ,  —  —  2 v  ((1)f 7] (7  2  L  Combining (6.3) with (C.2), we get  S(xj  f  = —  ddudv K  —  OK (in/+&  f  du  (+ + ln  e  _) OK (ln/+ _) S(u)S(v) , 2 (S(u))  (6.4)  which agrees with the gravity result (3.36). XXXVISince the transformations at the incoming and outgoing boundaries are linear, we will just loosely say “the nth order” instead of “the nth order in 54  Chapter 6. Relationships Between MM and 2D Gravity The third order: from equation (5.22), we have =  (8  ()1(o))3. —  1)  (6.5)  In the case where there are two localized incoming tachyon pulses which are separated widely as shown in Figure 3.2, we can express the incoming field as s’ + where comes in later than Sf1). Combining (6.5), (C.4), (3.25), (3.28) and (3.29), we get the interaction between these two pulses S(x)  = —  32  f  — i  didndvdw K  —  {a (ln  UK (ln+ e  /  _)  00  e  6.2  XXXVII  }  f  —  n) 8K (in  +  S)S(v)S(w),  /  00  dv  —  v).  (6.6)  2  e_V  (s1)(v))  du  , 2 (s1)’(u))  o 0 J—  dv e_2vS(v)  -00  which  +  du e_uS(u)  0 J—o —  (+ + ln  f  (6.7)  -00  agrees with the gravity result (3.41).  The Nonlinearity  In last section, we confirmed the correspondence at the boundaries; now we shall study the correspondence in the buik. Let us write the most general XXXVII  We omitted a lot when we wrote down the final equation. However, the calculation is as appendix D.  nontrivial, thus we include it  55  Chapter 6. Relationships Between MM and 2D Gravity transformation as  XXXVIII  S(x,x) +  ....  (6.8)  If we assume the correspondence between S and i is linear, then we have to assume that the correspondence between S and Therefore the correspondence between  and  is also  linear)X.  is linear, too.  from Appendix B we know that the correspondence between  However,  and i is not  linear, thus we get the conclusion that the correspondence between S and i is nonlinear. However, since the correspondence between S and  is linear at incoming  and outgoing infinity boundaries, the natural question is: what is the lowest order where the nonlinear correspondence begin to appear. The answer is the second order in tachyon field. Here is a proof.  Let us assume a linear correspondence between S and i is compati ble with the equations of motion at the second order, thus the function Li(x+, x; ö-l, &) needs to satisfy S(x, x)  =  =  XXXVIIINotation 1,2 is  basically  f —  .  Li(x+, x; , 2)(2)(&1, 2 d&d  f  dze_Z (S(z)) , 2  no preference between 7)  and  .  (6.9)  However, since there is correspondence involving second,  third... order in , subscript 1,2... will be handier. XXXIXLogically it is possible that the correspondence between S and correspondence between S and  J2)  ñ  is linear and the  is nonlinear, or vice versa. However, physically there is One could show that the correspondence between S and  is nonlinear by showing the linear correspondence (between S and  )  is not compatible  with the possible form of L ,L 1 2 (which will be discussed in section 6.3) and the relationship (B.6).  56  Chapter 6. Relationships Between MM and 2D Gravity or  f  6+8_L 2 daida ( 1 x, x; a , 1  ) a ) 2 , 1 (a ( 2)  g(x, ) (S(’)(x)) , 2 (6.10)  where g(x+,xj Since ii” in  —  f  =  ‘(u1,u2)  2  ) (82 2 f(i,a  =  and  (o2))  i ‘(o ) 2  =  2+ f dxK(u  zjS(’) (xj, this becomes  dd 1 da d 2 a  K (a 2 8 2 + in  —  )  K (a 2 2 + in  —  )(q) 1 q) S(’)(p)S(  , 2 g(xx) (s(1)(x_))  (6.11)  or  H(x+, x, 2 2 fda )0 a K =  g(x, x)ö(x  —  where H(x,x,a ) 2  =  2 (a  p)ö(x  + in —  —  )  2 + in K (a 2 8  —  q)  (6.12)  q),  )f(ai,a fdai8+8_Li(x+,x;ai,a ) 2 . Now iet  muitipiy both sides by K  (  + in  K  —  (  + in  —  q) and in  tegrate over p and q, we get  f  H d& ) 2 0o(a (x,x,a g(x, x)K  (  —  + in  2 x)ao(a K  —  where we used fdy K(z—y)K(z—y)  =  —  (  + in  —  xj,  (6.13)  (x—z). If we integrate over x, the  left hand side is zero while the right hand side is not. Hence the assumption  that the correspondence between S and i is linear is not compatible with the second order equation of motion.  57  Chapter 6. Relationships Between MM and 2D Gravity  Constraints on L 1 and L 2  6.3 On L Since  =  and  8(1)  (1)  that the x  o  ,  ?] o, 2 1 a  (6.14)  =  will stay the same as their incoming form. Thus we conclude and 52 dependence of the kernel L 1 should stay the same as it  is at the boundary. Since the incoming S and i are related by the leg-pole transformation K, the form of 1 L is limited to ) 2 Li(x,x,&i, where  1 L  satisfies  =  K  (2  —)  +ln  ,ã 1 diLi(x,x,6 ) I2  G(x,i) +  0. Since  =  (1)  —  L,  (6.15)  and (‘) satisfy  (6.14), we could guess the form of G must be G(x o). We also want to go to —oc when x+ —oc since (1) and are related at the incoming boundaries, thus we expect G(z)  —*  0 when z  —  ±oc.  On L 2 By the same method we applied in section 6.2, we can also write down  , which are 2 constraints for L  f  3 d 1 d d 2 80_L ( x, x; i, ; 3,  g(x,x)  f  f  K 2 d&  (2  u4)  K  + ln  —  (4  (x,x;i, 8+D_L , 3 ; 2 ) d d 1 & d 4  K 4 g(x+,x_)fd  (2  K  + ln  —  =  + ln  —  xj,  (6.16)  xj,  (6.17)  , 2 L  (6.18)  =  (4  + ln  —  thus  f  3 d 1 d& &  g(x, x)K  =  (2  K  + ln —  (4  + ln  + —  58  Chapter 6. Relationships Between MM and 2D Gravity where L 2 satisfies  f 6.4  (6.19)  The Correspondence in the Bulk  First let us make some conventions. We just do similar things as what we did for the chiral formalism when we argued the consistence of results calculated from the incoming boundary and the results calculated from the outgoing boundary.  Since we denoted the incoming collective field as ij and the  incoming tachyon field as Sj, and we know (refer to section 6.1) that the correspondence between  and S is linear, in this section we will call the  fields including n S’s or fj’s as nth order field. The uth tachyon field can be expressed as a combination of 8 r as shown in equation (3.20) and (3.24) which we will refer to as exact forms of tachyon field. On the other hand, we expect that a correspondence (6.8) holds. Thus the uth order can also be expressed as  in a similar way to the case of  the chiral formalism, expressing the field in terms of the incoming boundary and the outgoing boundary respectively and matching them. Let us use an example to explain this. Consider the second order tachyon field. First, it can be expressed as equation (3.20). On the other hand, it can be expressed as  (x, x) 2 S =  f  L(x, x; , 2)(2)(1, 2) 2 did (6.20)  Please pay attention to the superscripts (i) indicating the order. Let us assume that we already know the exact form of L , then we can 1  . This is because we know the 2 use the two equations above to figure out L exact form of the tachyon field which should replace the left hand side. The form of  ) 2 j(  is also known by the chiral formalism. Hence the difference 59  Chapter 6. Relationships Between MM and 2D Gravity between left hand side and the  (2)  term in the right hand side will be the  second term of the right hand side which includes 2 L L . 2 could then be solved for. Explicitly  LHS  =  =  —  —  eX+  eX+  f f  dv  e_V  2 (Sj(x,v))  (6.21)  dv e_V  The first term of the right hand side is 1 RHS =  f I  =  I j  Li(x, x; 2 did  ,  2)(2)(1,2)  2 e(a12)/ ir Li(x,x;6 2 did , 1 ) — 5 2i cosh ((o’ 2 )/2) —  2 1 (fds’(& _u)@ u )). The LHS-RHS 1  =  (6.22)  , which will give us an expression for L 2 RHS 2  4 ( L ; 2 , 3 ) x,x;&i, ö ã px  a2÷I  jdve —V Li(x + 1 )L 2 ,v;ai,o ( x + ,v;u ,o4) 3  -e 4 =—-—  f  -  dyLj(x, X;  2 e’)’ ‘cosh((i —  • ö(o 1  —  )5’(o 3 2  —  y)6’(ó  —  y).  (6.23)  This method can be generalized to any order. However, this method is not satisfying since a key point is artificial  —  at  the boundaries there is only one variable and writing it in the two-variable form is totally artificial since the fields at the boundary do not depend on the other variable, one could also guess other two-variable forms that reduce to the one-variable form at the boundaries. Any choice of the two-variable form may cause the loss of generality. And the essence of this method is to throw all the terms that can not be included in lower order to higher 60  Chapter 6. Relationships Between MM and 2D Gravity order correspondence. If we made different assumption in changing onevariable functions to two-variable functions, we would get different answers for higher order correspondence. However, this method itself can not tell which assumption is more reasonable. In another word, the method can not tell whether there is something wrong.  Another critique is that the  real correspondence (assuming it exists) may include many terms and the correspondence we found in this way is only a part of use this method, we will need higher order equations  it-.  (iii  If we insist to  both the graviton  and dilaton) on the gravity side. To solve this one-variable  vs  two-variables problem, we suggest that the  correspondence between the matrix model and the gravity system should be an equal-time correspondence the  r  —  the time t in the gravity theory and  in the matrix model should be the same. It would be plausible to  have an equal-time relationship, since both Liouville string theory and the matrix model are hamiltonian systems, and if their Hamiltonians somehow correspond to each other (which they do) then the time evolution should match as well. Under this assumption, we could perform the correspondence in the following way. First we can get the following equation from the gravity part (to the second order)  S(x,x) = S(x)  f  —  dv  e_V  . 2 (S(v))  (6.24)  On the matrix model side, to the second order we have °  - -+  --  )°  ‘°_  ir e (&+——)/2 (8&—i(a 2tcosh((+—j/2)  Equal-time condition reads x  —  =  +  ,o-  ))  2 .  (6.25)  .  fl 1 XL  e  fact we encounter this case before. For example, K(x) is a combination of tails when x <0. When x is large and negative, ex is the most important term. Recalling  the process to get the boundary behavior of the tachyon field in gravity theory from the matrix model (section 6.1), we would say it is highly possible.  61  Chapter 6. Relationships Between MM and 2D Gravity At the incoming boundary  S(x)  =  fdK  (  — -)  + in  jTh().  (6.26)  Also, the time t should be the same as r in this equation. We can substitute (6.25) into (6.26), then substitute into (6.24), and we will get the relation to this order. This expression make sense since the i actually has only one = ö effective variable under the condition x Thus this result + —  .  strongly depends on the equal-time assumption. Again, if we release the condition, we can not judge the condition for general case and this only serve as a necessary condition.  62  Chapter 7  Conclusion The goal of this thesis is to try finding the correspondence between the gravity theory and the matrix model in the bulk of the spacetime. In or der to reach the goal, we studied the interior behavior on each side of the correspondence. For dilaton gravity, it is handy to work in the Modified Kruskal Gauge. Once we add in the tachyon field, there is one dynamical degree of freedom. The exact solutions for the first three orders in tachyon field are found in the limit where the tachyon background is negligible. Based on these exact solutions, it is possible to study in detail evolution of two incoming pulses. The picture described in [1]  “...  the actual physical picture is that pulse 2  scatters off the gravitational field of pulse 1 before it ever reaches the wall” is modified as  “...  the actual physical picture is that pulse 2 scatters off  the combination of the gravitational + dilaton field and the higher order tachyon field of pulse 1 before it ever reaches the wall”. The collective field description is a good tool to study the scattering problem and the interior behavior in matrix model. All the results in [1] are easily reproducible by collective field methods. Classical field perturba tion theory can be used to study the behavior inside the Penrose diagram. However, the chiral formalism is much more convenient and powerful than classical field perturbation theory to study collective field description of ma trix model since it is much easier to calculate and enable us to study “degree of singularity” about the behavior around the singular point in the original  63  Chapter 7. Conclusion background. Finally, the chiral formalism help us to realize that the collec tive field is ill-defined around the singular point. Once the interior behaviors of the matrix model and the gravity theory are found, it is possible to study the correspondence between them. The cor respondence is nonlinear and the nonlinearity begins to appear at the second order. Some constraints and hints about the form of the correspondence are also found.  64  Bibliography [1] M. Natsuume and J. Polchinski, “Gravitational Scattering In The e 1 Matrix Model,” Nuci. Phys. B 424, 137 (1994) [arXiv:hep-th/9402156]. [2] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,” Cambridge, UK: Univ. Pr. (1998) O2 p. [3] J. Polchinski, “What is string theory?” (1994) [arXiv:hep-th/9411028]. [4] T. Mohaupt, “Introduction to string theory,” Lect. Notes Phys. 631, 173 (2003) [arXiv:hep-th/0207249]. [5] I. R. Klebanov, th/9108019].  “String theory in two dimensions,”  [arXiv:hep  [6] S. Alexandrov, “Matrix quantum mechanics and two-dimensional string theory in non-trivial backgrounds,” [arXiv:hep-th/03 11273]. [7] 5. Y. Alexandrov, V. A. Kazakov and I. K. Kostov, “Time-dependent backgrounds of 2D string theory,” Nuci. Phys. B 640, 119 (2002) [arXiv:hep-th/0205079]. [8] S. Alexandrov, “Backgrounds of 2D string theory from matrix model,” [arXiv:hep-th/0303190]. [9] C. G. Callan, S. B. Giddings, J. A. Harvey and A. Strominger, “Evanescent black holes,” Phys. Rev. D 45, 1005 (1992) [arXiv:hep th/9111056]. [10] 5. R. Das and A. Jevicki, “String Field Theory And Physical Interpre tation Of D = 1 Strings,” Mod. Phys. Lett. A 5 1639(1990). [11] S. R. Das, “The one-dimensional matrix model and string theory,” [arXiv:hep-th/92 11085]. 65  Bibliography [12] A. Jevicki, “Development in 2-d string theory,” [arXiv:hep-th/9309115]. [13] W. T. Kim, J. Lee and Y. J. Park, “Stability Analysis Of The Dila tonic Black Hole In Two-Dimensions,” Phys. Lett. B 347, 217 (1995) [arXiv:hep-th/95021 14]. [14] W. T. Kim and J. Lee, “ADM, Bondi mass, and energy conservation in two-dimensional dilaton gravities,” Tnt. J. Mod. Phys. A 11, 553 (1996) [arXiv:hep-th/9502078]. [15] M. Ernebjerg, J. L. Karczmarek and J. M. Lapan, “Collective field description of matrix cosmologies,” JHEP 0409, 065 (2004) [arXiv:hep th/0405 187]. [16] J. L. Karczmarek, “Time-dependence and holography in the c Matrix Model,” Can. J. Phys. 99, 1-7 (2008).  =  1  [17] E. J. Martinec, “Matrix models and 2D string theory,” [arXiv:hep th/04 10136]. [18] M. Li, “Some remarks on tachyon action in 2-d string theory,” Mod. Phys. Lett. A 8, 2481 (1993) [arXiv:hep-th/9212061]. [19] Gerard ‘t Hooft, “The Concept Basis of Quantum Field Theory,” http://www.phys.uu.nl/ thooft/. [20] G. W. Moore and R. Plesser, “Classical scattering in (1+1)-dimensional string theory,” Phys. Rev. D 46, 1730 (1992) [arXiv:hep-th/9203060]. [21] Y. Nakayama, “Liouville field theory: A decade after the revolution,” Tnt. J. Mod. Phys. A 19, 2771 (2004) [arXiv:hep-th/0402009]. [22] P. Di Francesco, P. H. Ginsparg and J. Zinn-Justin, “2-D Gravity and random matrices,” Phys. Rept. 254, 1 (1995) [arXiv:hep-th/9306153].  66  Appendix A  (Modified) Kruskal Gauge The last gauge There are three degree of freedom implied by diffeomophism invariance. After we fix conformal gauge, there is still one gauge degree of freedom left. Since the coordinate transformation is .9x”c 9I/3 =  --—-g.  (A.1)  Maintaining conformal gauge, the coordinate transformation is x’ and x”  =  =  z(x), and we have 1 PP1+{ln()  +inQ-)}.  (A.2)  Kruskal Gauge [13] Since  3Q1  —  p)  =  0, we have —  p  =  p(xj + q(xj,  (A.3)  where p, q are arbitrary functions. Using the last gauge freedom mentioned above, we can find a coordinate system in which p gauge is called Kruskal Gauge (KG).  =  1. This choice of the  Modified Kruskal Gauge Since we want o = ax+ + bx + c, P0 = 0 to be the background, we prefer to fix the gauge with p = o. This gauge is called Modified K’ruskal 4 —  67  Appendix A. (Modified) Kruskal Gauge Gauge (MKG). This choice of gauge is allowed since the form of the right hand side of eq. (A.3).  satisfies  Relation between KG and MKG In order to convert results from Kruskal Gauge (primed coordinate) to Mod ified Kruskal Gauge (unprimed coordinate), we need to find the coordinate transformation  the specific form of equation (A.2).  Since we expect  Fo in the new coordinate system, we need p p’ = p = 1’ = —Fo. The first equality is because of p’ = ‘, which p = is the KG. The second equality is due to the fact that dilaton is a scalar  p  =  —  —  —  —  —  field. The third equality is our requirement. The fourth is nothing but a definition. Therefore, if we can find a coordinate transformation satisfying —Io, p = 6 will hold automatically since these two equations are equivalent due to the fact that p’ = ‘ and dilaton is a scalar field. p  —  p’  =  Let us express the coordinate transformation p ii  in  —  p’  =  —o explicitly  / 6z’\ /0x’ N ã—-)+1n——)j=_(ax++bx +c). —  (A.4)  We now need to find a coordinate transformation satisfying (A.4). If we let in  =  —2ax + + d,  (A.5)  then in  =  —2bz  —  2c  —  d,  (A.6)  where d is an arbitrary constant. Equations (A.5) and (A.6) imply the coordinate transformation I )  1  Again, here  X ‘+_I  _2ax+)djç. J,  —  x’  =  2 _e_  _2c_d  + g.  (A7)  f, g are arbitrary constants. 68  Appendix A. (Modified) Kruskal Gauge  In particular, if we have the following solution in the KG =  m  ’ 2 e  (A.8)  — —  —  in MKG we will have ’ 2 e  e20e_2)’  =  26 e  =  e2{  =  2 (_e_2a  +  f —  • (_e2  =  where F  =  1+  (  __2c_d+g_xj_)  }  ° + Fe2 + Ge 2 . 2 + FGe_2c) e  (f _xje_d  and G  =  (A.9)  . Since d,f,g are 2 (g _xje  arbitrary constants, we can choose convenient values for them.  69  Appendix B  Relationship between r, and  ‘i  Without approximation, we will get the following two equations from equa tion (4.32) and equation (4.33)  —  =p  =  x2  —  2+  (B.1)  —  ,  2 \/x  2 + “O  —  (B.2)  ,  \/x2  for  < 0 and  —  2+  V’x  > 0 respectively. Expanding these up to third order in 1/x  and third order in j, we get —  P+  (/3)2 —  (/8)  —  2  +  —x +28 +  (in)  ‘  —  + (8  2 +2(aX)2  ()3  +  +  .  (out)  Equation (4.51) implies when  —2++ 2  <0;  when>0.  Now we are able to find relation between i and  .  Take the case  (B.3)  <0  for example: combining equation (in) arId (B.3) we will get the following  70  Appendix B. Relationship between  and i  relationship 28()  =  +  —  + x =x  2 + (8)2  //87 + (/8)2  —  2/a  (/8)  22  +  +2 5IL(/877) 3 2x  (B 4  + —  —  2+J)2  B5 (.)  3 2x  —  which is nonlinear. And we see that the linear result (4.54) is just the lowest order of these two equations. Actually we can do one more stepXM to the second order in order in  to find the function relation between i and  i  and second  is  2(a)=O(3)+O() — 2/ / (a) +  2 irii’(a)  —  —  (—3t ‘(a) + (a) 2  4ir,j , (a) // (a) +  2(  —  1 + /j”(a) + 2iri/’(a) ) 2  —  —  a 2 4rr /(a)r”(a)  (B.6)  where a is negative and large.  XLIBasically it is an iterative method. Given two equations: F(a) (or  b  =  a  —  G(b) and a = b+g(b) g(b)), where g(b) << a, then we can replace the bin G(b) by a g(b), then  replace b in the expression G(a  =  —  —  g(b)) by a  —  g(b)  ...  Until the order we want, then replace  the last b by a. Then expand the huge expression. It is very handy to done by computer program.  71  Appendix C  Integrals involving the leg-pole kernel K K is the kernel of leg-pole transformation (4.61).  We can get have the  following relations  f For x  f f  —  dy K(y  —  xi)K(y  —  ) = 2 x  —  X2).  x large and negative, we have  dy K(x  —  y)8K(y  —  xi)8K(y  —e’ö(xi  —  —  z2),  (C.2)  dy K(x—y)K(y—xi)—e’ (x_xl+47_2+ln) (C.3)  where ‘y is Euler’s constant. For x <<x 3 and x >>  f  X3,  we have  )8K(y—x 2 {(1_8)K(x_y)}8K(y—xi)8K(y—x ) dy 3  1 2 e {e  6(x2  —  x3)  +  e1  “(x2  —  x3)}.  (C.4)  72  Appendix D  Derivation: the Third Order Tachyon from MM Let us start with equation (6.6). First, we perform integration by parts to move the derivative 8- to the first K. Thus the interaction between pulse 1 and pulse 2 is XLII  f:ddud {(1_8x+)K(x++ln_). 8K (in  +  8K (in  —  8K (in+ _w)  }  {(i  + 8K (in  +  -  —  +  —  (u)Si(v)Si(w) 2 S  —  8+)K  8K (i  (D.1)  (+ +in-). +  -  —  Si(n)S ( v)S w). 8K(ln+_w)} 2  (D.2)  i.e., the interaction are the terms including either one subscript 2 and two subscripts 1 or one subscript 1 and two subscripts 2. We will refer to these two terms as term 211 and term 122, which should represent the scattering XLIIThe factor three in the denominator is canceled by the three in + 3S?S 2 + 3SS +  =  1 + S2) (S  =  Also, since there is no higher orders in the tachyon field, we  omit the superscript (1) in this appendix.  73  Appendix D. Derivation: the Third Order Tachyon from MM process that pulse 2 is scattered by fields produced by pulse 1 and vice versa, respectively. Let us first study term 211 (equation (D.1)). Since the two pulses are localized and widely separated, the integration is approximately  J  rxo  rxo  00 p  du 82(u)...]  dv S(v)...  —00  x0  J  dw Si(w)....  (D.3)  —00  where x 0 is again a point between pulse 1 and pulse 2 where the values of both pulses are negligible. Since now u >> w, equation (C.4) implies that equation (D.1) is approximately eX+  f  00  du e_uS(u)  00 J—  e  —  f  00  dv e  (S1)(v))  2  00 J—  f°°  e2vS(v)  dv  -00  f°°  du (S’)’(u)) , 2  (D.4)  -00  which is the outgoing third order field (6.7). Physically this represents the scattering process that pulse 2 is scattered by fields produced by pulse 1. Let us study term 122 (equation (D.2)). Since the two pulses are localized and widely separated, the integration is approximately p 00  ]  fZ0  f0  du S (u)...J 2  dv S (v)...] 2  xo  dw S(w)....  (D.5)  —00  Since now u >> w, equation (C.4) implies that equation (D.2) is approxi mately fOO  J  Xfj  pXJ  p00  du ...  J  X0  dv  ]  dw  ...  {5(v  —  w) + ...S”(v  —  w)},  (D.6)  —00  which is zero. This is because there is no overlap between the integration region of v and the integration region of w, delta functions are zero.  74  

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