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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 2.1 2D Liouville String Theory and Its Effective Spacetime Action 3 2.2 c = 1 Matrix Model 5 3 2D Gravity 8 3.1 Gravity + Dilaton 8 3.2 Adding a Dynamical Field - Tachyon 11 3.3 The Strong String Coupling Limit 13 3.4 Two Incoming Pulses 15 4 c = 1 Matrix Model 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 Physical Pictures and Relations 32 5 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 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 6 Relationships Between MM and 2D Gravity 53 6.1 The Correspondence at the Boundaries 53 6.2 The Nonlinearity 55 6.3 Constraints on L1 and L2 58 6.4 The Correspondence in the Bulk 59 7 Conclusion 63 Bibliography . . . 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 . . . . 30 4.2 The Pictures for Left and Lower Hyperbolae 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 theory1 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 model11. The correspondence 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 space- time. 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 i runs from 1 to d. i, is the flat spacetime metric, and ‘yab is the world sheet metric. ‘ is a constant with dimension of squared length. To fix conformal gauge, we write the metric ‘yab in terms of some fixed 7ab as 7ab = e/ab. (2.2) The Liouville field become dynamical. Taking into account the Fadeev Popov determinant and the Weyl anomaly, the action eventually becomes [21] [22] S = —fdu{ab (8aX0bX +0aUb) — + (2.3) 3 Chapter 2. Background Actually the term e2 is added by hand. One can find the following relation _______ = 25— d b + b1. (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{(7abG + iEabB) 8aX8bX1’+ ‘R + (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 8eff = f dDXe_2{ 2(26— D) + R + 4(V)2 — - 2(VT) + 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 — 1)(kIL — 1’) If we take D = 2 (in which dimension the 3-order antisymmetric tensor is always zero) and let D = d + 1, q = XD, j,, = Q öD k0 = 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 = 2 case Seff = f dtdçb /e2{ + R + 4(V)2 — 2(VT) + — 4V(T)}. (2.8) With c’ = 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. Let us think about the problem in the x — 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 p2 — = 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 p2 — = E with E < 0. VIII We will call this left hyperbola. 6 Chapter 2. Background p p+ ‘SSs p— ,/ 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 = fd2x/ (R + 4(V)2+ 4A2) (3.1) The equation of motion for g is — +4 (aa — gai(V)2) — 2Ag11 = 0. (3.2) In 2D theory, the cosmological constant must be zero. This is because Einstein tensor — 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 S = fd2xe (R + 4(V)2 + 4A2) (3.3) This is because when we vary gv to find out SS/g, the e2 will be affected. The equations of motion become[13] [14] R, + 2VLVV =0, R + 4V2 — 4(V)2+ 4\2 = (3.4) In conformal gauge XI, the equations of motion[9] are 2(90_ — 4U8_ —)2e = 0, 2(9±p8±—8)=0, (35) —4t98_ + 48U + 28+ô_p +\2eP = 0, 88_(p—) ==0. 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 — A2(x+ — xj’)(x’ — x). (3.6) In the later sections, we will use its perturbation form with respect to the backgroundXhl P0 = 0, (3.7) = ax+ + bx + c, where a, b, c are arbitrary constants satisfying 4ab = —A2 (in [1], a = 1, b = —1, c = o)X. Let us follow the method given by eq (41) in [1]. To linear X11 [lj )2 = 4• XIFirst define x = x0 ± x’, then take the metric: g+— = _(1/2)2I9,g = =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 keep it. 9 Chapter 3. 2D Gravity order in gravity and dilaton, the equations of motion are (6-i- — 2a)2 = 0, (8_ — 2b)2 0, (3.8) 88_6 = —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 =Ae2°, S = — + B(xj + C(x), (3.9) p = — 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) XIV p 0, 4o ax+ + bx + c, p = 5, thus the solution becomes = Ae°, S = p = —Ae2°+ Be2a + Ce2, (3.10) 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 e2 = e23 = 1 + ( + FGe_2c) e2° + Fe2a + Ge2. (3.11) When p and S are small, the linear approximation is p = = — {( + FGe_2c) e2° + Fe20 + Ge2}, (3.12) XIVF0r 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, G=-2C, m ). (- — 4BCe_2c) (3.13) 3.2 Adding a Dynamical Field - 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 = fd2xe {ai (R + 4(V)2+ 4A2) — (VT)2 + 4T2 —a2T3} (3.14) which implies the equations of motion + 2VVV — ai0T0T = 0, V2T — 2VVT + 4T — a2T =0, (3.15) R + 4V2 — 4(V)2+ 42 —a1(VT)2+4a1T2—a21’T = 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; x1 = q. 11 Chapter 3. 2D Gravity Start with the background [1] P0 0, o = 2, T0 = (2b1+b2)e’ XVI Here b1 = —2j/a2,b2 = bi(—1 + 47 + mu), where ‘y is Euler’s constant —F’(l). Define field S in the same way as [1] T e0S. (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) = — eZ+_ (i)2 — T0S(2), (3.17) = + + — a2 (3 (1) ——T0S —a2pT0S To the linear order in dilaton and graviton field and the second order in tachyon field, the dilaton and graviton equations now become [1] XVIII ai(6+ — 2)Q = —(D±T)2+ T2, ai(8_ + 2)2 = (8_T)2 — T2, (3 18) 2a88_6 = 2a2 + T2, (ó — o) = where the definition l is the same as it in last section. With the specific value a= 1,b= —1, 2(8_ —0+)64p. 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 —88_T + 8_T — 8T + T 0, which has the general static solution T0 = (b1+b2)e’. 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 T0 —* 0. From the gravity theory point of view, in the T0 —* 0 limit we are studying the effects independent of the background. In this case equation (3.17) is modified as = 0, = (S(1))2, (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 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 (s(1)(v))2. (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(_) {_ (S(1)(x))2+ f dv (s(1)!(v))2}, (3.21) = dv ((v)+ (T()(x+v))2 (3.22) t x 14 Chapter 3. 2D Gravity By definition 1 p = dv ((x+v) + (T(1) (x+,v))2 ) - (T(1))2 (3.23) One can check that in the special case which we consider here (To —* 0, s’ 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 = f dv e_vS(v) j du e_U (S(1)(u))2, 1= _f dv e2vS(1) (v)f du (s(1)I(u))2, 2 J= __S(1)(x_)f dv e_2V [(S(1)(v)) _2f du(S(1)1(u)) ]. 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+S2with the tachyon field scattered by the dilaton+graviton+S2produced earlier. In order to do so, we consider the interaction of two incoming tachyon pulses denoted as S1, i.e., we expect that the first’° pulse comes in and produces dilaton+graviton+S2fields, then the second pulse is scattered by these 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) G1, G2 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 G1 and G2 is negligible, we have 00 00 00 fdx2G2(x)f dx1 Gi(xi) fdx2G2(x)f dx1 Gi(xi). (3.25) This is because 00 X2 fdx2G2(x)f dx1 Gi(xi) = p fX2 p 00 fX2 j dx2G()J dxiGi(xi)+] dx2G()J dxiGi(xj), —00 —00 X(J —00 (3.26) where xo is a point where the values of both G1 and G2 are negligible. The first term of the right hand side of (3.26) is approximately zero and the second term is approximately p00 pxo p00 P00 J dx2 G2(x)J dx1 Gi(xi) J dx2 G2(x)j dx1 Gi(xi). —00 —00 —00 (3.27) Similarly, we have p00 pxi J dx1 Gi(xi) J dx2 G2(x) 0, (3.28) -00 -00 17 Chapter 3. 2D Gravity and the cross term Gi(x) G2(x) 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 (S(1))2 and the integration (1 2 . (1 (1) (1)of (S )) . In the case where we are considering S ) = S + 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)• We denote the quantity (among (2), l, p and S ) only including 1) as that quantity with subscript i 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 = dv e_vS(v) f du e_U (S’)(u))2 -00 -00 L = _f_ dv e_2vS(v) f du (s1)I())2,al 00 00 2+ — V M = _s1)(x_)fX dv e_2V [(s1)(v))2 — 2fdu (S1)(u))2], which all include the form Fi(u)fdvF2(v 0, (3.31) where F1 is either or the derivative of F2 is either the fields pro duced by pulse 2 or their derivatives. The vanishing result is physically XXIF0rexample, S2) = —ef dv e_V (S1)(v))2. 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 X 00 2 K —j—-—j— f dv eS1)(v)f du e_U (s1u)00 00 e2 X 00 2 L ____f dv e_2vS1) (v)j du (S’)’(u)) M _:s’)f {e2v [(S(1(v))2 - (S1)!(v))2] +e2 (S’)’(v))2}. 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 f dv (S’)’(v))2 + e2 dve_2V ((sI’(v))2— (si1(v))2) }, (3.33) xxIIAs discussed above, x 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 f _ dv (S’)’(v))2. (335) al 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 dv e_V (S’)(v))2, (3.36) which does not depend on x. 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+ X 00 2 — — f dv e_2vS(v) f du (s1)’(u)) (337) 2x+ p00 P00 2 — s— j dv e_2vS(v)] du (S’n) (3.38)4a1 —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 ±a2e2 f dv e_vS1)(v)2f du e_uS(u) (3.39) -00 -00 a22ex f dv eS(v)2 f du e_US)(u). (3.40) -00 -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 due_U (s(1)(u))2 —00 -00 1 p00 poo — — dv e_2vS)(v) / du (8’(u)) , (3.41) alJ_00 j_00 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 a2 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. 3) The sum of equation (3.38) and equation (3.40). 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 s(3), 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 S1). The first is centered at ir = 0, and the second at x = T. i.e., S(1)(xj = Ae_B(i2+ae_1(_T)2as 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. However, we will first use equation (3.23) (3.22) (3.20) to calculate p, 3, 8(2) and show the approximations we made to find equations (3.33) (3.36) are indeed of the same “degree of approximation”. Let us consider S’(x) = Ae_B()2 only and get the field (2, 6, p, Sf2)) produced by s’ based on equation (3.23)(3.22)(3.20). We get 8(2) — A2a2e (1+Ef[V/(1 +xj]) 8/ frA2a2e (2 e_2B(w) V2 8/ k\ v(+x) ‘ 2jrrA ae8B V2 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 A2 e2 — A2 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,) A2 2 2x-T++- 1(3) a a2e 8 (3.44) aA2ae2_Tir (3 45) — aA2e2_b(T_)_ (_i + e+2) 4a1 aA2e_2T+2ir (i + Erf (1+b(_T+z))) — (346) 8va/ — aA2f eI2T+2ir (i + Erf (1+b(_T+x_))) 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, s3 could also be calculated from equation (34) of [1] (which is the same as equation set 3) and the result is — aA2ae2_T — aA2e_2T+27r (3 48) — 16/ 4/a1 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 p2 — = ±2i, here ii We will be interested in the “excited state” — 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 p2 — = —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 = —e when x < 0. (4.3) In terms of the field S, the Hamiltonian can be written as H = L dq{I2 + (0q)2 + e2QO(3)}. (4.4) 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 o=o+v8x. (4.6) The following relation [15] will hold p+(x,t)+p_(x,t) Z (47) where Z(x, t) f d’a(’, t). We are able to express p+(x,t)p_(x,t) in terms of j. From (4.2) we can also write down in terms of . Thus we should be able to find the relation between S and j. If we let the fixed solution be the left hyperbola (so po = /x2 — 2t), when x is negative and large we will have (4.8) p+ +p— 1 ____ — _____ ____ = —(e — _) = — - , (4.9) 2 2x x 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. 4.2 Collective Field Description Let us collect the relevant parts of collective field description[1O] [11] [12] [15]. The action for the collective field is = f dtdx ( — + (z2 — 2/1)). (4.11) As mentioned above, the small fluctuation about a fixed solution ço0(x, t) is defined as (4.12) Then the action can be rewritten in terms of the small fluctuation f dtdx ((Zo+8t??)2- + + (x -2)(yo + (4.13) where Zo(x, t) = f dx’Utyo(x’ , t). Terms in this action can be grouped in powers of 7) (the field that we are interested in) as (terms linear in vanish by the equation of motion) S = S(o) + S(2) + (4.14) where S(o) has no 7)-dependence, the other two terms are S(2) = f dtdx — — 2(o7))2} (4.15) and Sint = f dtdx {_o(a7))3 + (at?] — Z)2 (ra)n} (4.16) 27 Chapter 4. c = 1 Matrix Model It is proposed that coordinates (r, o) exist in which S(2) takes a standard 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], (4.17) = —/cosh(), P0(J) = /sinh(u) r o(o) = —/sinh(a). Case 2 (the upper and lower hyperbolae): T = a E (—oc,oc), x = ,/sinh(a). (4.18) p(a) = /cosh(a) p(a) = —/cosh(a) o(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 — S(o) for these two cases is f drda{ ((8)2 — (a)2) — (3(8)2(8) + (O)) + (8)2 00 (_)fl }. (4.19) i.e., now the system is a massless scalar field j 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) = f dr’da’G(r, a; r’, a’)F(2) (P (T’, a’)), 3) (T, a) = f dT’da’G(T, a; T’, a’) {F(3) (1) (T’, a’)) (4.20) + cross terms in F21 (?]() (T’, a’) + 2) (r’, a’)) }, where 1 ei{_ko(T_T’)+ki(o_dl} G(T, a; T’, a’) = (2n) f dk k? — (ko + i)2 (4.21) is the retarded Green function. And F(2) and Ff3) are the second and third order terms in terms of in the equation of motion. Explicitly, the equation of motion is 82 82 /:: f ((8?)2 + (0)2\ 28T- 2 _{ (()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_. (4.23)— 21r\/iw 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 2xxvI• 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• 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 d = J w. jm(w)eT_. (4.24)2ir/iw After calculation via equation (4.20), the second order result is (2) = — f eh() 16/2• (4.25) When —* oc, this becomes — fdw1 2 ei(w12)(T_)äjn(wi)äjn(w), (4.26)8ir / which gives the scattering matrix to the second order fdwidw2v. Qout(W) = j — Wi — w2). (4.27)2ir 2ir 2u ‘7 30 Chapter 4. c = 1 Matrix Model When expressed in terms the mode of , it is XXVIII - I dwi dw2 f[’\iW - - c_(w) = — j --—-— -—zwo+(wi)c+(w2)2ir6(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 —* 00 f dwidw2w362(2)2 e0(Ta (w)äj (w2) in(w3), (4.29) which gives the scattering matrix to the 3rd order [dwidw23 4ir ci0(w) = I 2qi-2qr2ir 3(2i)2 (4.30) Expressed in terms the mode of 5, it is [dw1 dw2 dw3 w 47r - - - — J -—-—-— ) 3(2)iw(iw — — (4.31) which is the same as equation (14) in [1] when n = 3. Where i + W2 + 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 , j and S will be given in section 4.4. The relation between a and a is expressed as equation (4.57). 31 Chapter 4. c = 1 Matrix Model 4.4 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 x2 — p2 = 2i, we parametrized x as x = —/ cosha with a < 0, and had the relationships = 2 = O + = x2 — 2 +8zTh (4.32) + +p- — 2’ — — V’T?] _ (4 33)2 — çO — — 2 + /F8 X which gives — 2 + + (4.34) p _x2 — 2i — + (4.35) When j = (r + a) (left traveling), these two equations become = — 2ji, (4.36) p = _Vx2 — 2 — 2/t9j; (4.37) when = r,(r — a) (right traveling), these two equations become = — 2i + (4.38) = _/x2 — 2i. (4.39) 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 —* —oo, (x,t) (x,t). Since the parametrization of x for S is x = —e and x = —/ coshu with u<0for.Sowhenx—*oo,x---*—7e,wehave —q—a+ln/7. (4.41) Combining this with the fact that when x —* —, (x, t) —* (r + u) (here T = t as mentioned in equation (4.17)(4.18)), and .(x,t) — (t ± q), we have (rq) (r+u±ln). (4.42) More relation between S and j can be found. Equation (4.8) and (4.9) tell us —x + + ) —x + (4.43) — €) (4.4) which imply that i + /(x6 +8t), (4.45) + /(x8 — (4.46) 33 Chapter 4. c = 1 Matrix Model Thus for incoming field (j = — — 2/(8) ; c . (4.47) For outgoing field (j = r,i(T + 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 = 0, it is quite nature to consider the exchange, x ÷—* p. However, there exists another 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 2 — 52 = —2i. This feature is much better than just x ÷—* p. Now we have a fixed upper branch 5+ = /2 + 2 and fluctuations only occur at the lower branch around the background of the lower branch (0) = _/2 + 2. The superscript (0) de notes the background. Parameterize as = sinhã For the collective field of the new variables, we still employ the definitions such as = = + 2+a = 2 — X (4.50) which imply (for simplicity, we drop the — subscript of 5 when it is not confusing) x = 5 = _/2 + 2i.i — (4.51) 34 Chapter 4. c = 1 Matrix Model Since can be positive or negative, the above equation implies the following two equations x + — 2/a ; when —* —cc. (4.52) 5 = x — — — 2/8 ; when —* cc. (4.53) Combining these two equations with (4.38) and (4.37), we get incoming: < 0 ; = x (ö = u) ; = —. (4.54) outgoing: >0; =—x(ãrr—u) ; (4.55) Thus we conclude that when r —* —cc, the field is the function of (r — u) and x —f —cc; and ffi —+ —cc, the field i is the function of (r — o). When r — cc, the field is the function of (r + u) and x —* —cc; —* cc and the field is still the function of (r — Thus the physical picture is that the field goes from = —cc to = cc. Again, this can also be seen by equation of motion = —5; = —. Since 5 < 0, increases with time. The picture for i 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) = —S(t — q) = — (t — + ln — ) (t + a) = S(t + q) = (t — & — ln), (4.56) 35 Chapter 4. c = 1 Matrix Model 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) ) —iw/2 = w) () . (4.57) From the leg-pole transformation between S and S (equation (20) in [1]) (qj = fdzK (q — z) S(z) S0 (+) = f dzK (x — z)0(z), (4.58) p/I3 36 Chapter 4. c = 1 Matrix Model we can now write down the leg-pole transformation between and S = — f_ooZ K ( + in — z) S(z), (4.59) out() = f dz K (z — + in) S0t(z), (4.60) where K is defined as equation (21) in [1] K(z) — I (iW/4 f(_iw) J_ 2r \21 F(iw) = — Z 2(2/)8e. (4.61) here J1 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 — )eT,(5 + )eT) = 0 implies the equations = — and = —, 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 —* ±oo, 5 We have -- / +5 7N /-5 Na r—aln —ln . (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 (f,e_T). In fact, we should replace r —6- by ln (_eT) rather than — in (,je_T) for the incoming field. This is because for incoming field, hence small fluctuation can affect the relative change significantly while the relative change + )/( + ) can remain small. Similarly, we should 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 — = 2i — — 6-) = 2 + 2f (in (_er)) (5 2)p 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 2 +2 l ( /-e)) p 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 1o 1 1 1 —Tii—,- n —e (5.4) 39 Chapter 5. Chiral Formalism where the function form definition isf0t() = Therefore fin and f0 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 / / N N / / NNfin ln - e),} = f0 -ln - e )). (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) = f0t (a - 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) = 2/8ain(a) andf0t(a) = _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 (E_(a))) (5.10) = (a — in (+(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, j and 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, - =f0t/i, (5.12) 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] 00 1 00f dx +(x)e = 1 f dx e_{(1 +(x))’ — i}. (5.14) Taylor expanding the right hand side in terms of , we get = 1 I-: dx e_{(l iw)(z) + (1 zw)(+iw) (())2 + = f-: 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 -L1w). (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 and right hand side is a sum over different orders in jl?,. The first order is a trivial identity. All the higher orders must be zero, leaving to be some mathematical identities. 5.3 Chiral Formalism Calculation 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. Chiral Formalism If the incoming field ( — 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) — () 2j8(i) (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) ((1)2\(2) ( in — / I (5.20)px+p’ 2 j 7 (2) i(0) f çFI(0) — c’(O)\ ((1)’2 (3) — 1 1 --(1) -(2) P Jjn kjin un I P )p — p p 2 I X+p 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 fdwidei12T_in(w )äin(w)co:h()’ (5.21) which is exactly the same as the result (4.25) from classical field perturbation theory. XXIXE f’(O) = (i (_P±eT —J’tn 43 Chapter 5. Chiral Formalism The third order result is 3) = fdwidw2w39622e0(T_) (w ) (w )in(w3)g(wo, &), (5.22) where I denote wç = w + w2 + W3 and g(w,) = cosh{ )()()} (5.23) When —* ) becomes fdw1262(2)2 eo(T_) a (w1)äj, (w2) (w3), (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) — /çt (O)-(1) ((1)2\ (2) = ( Jout — P j , (5.25)p()x—pi 2 j 5(3) 1 (_(1)(2) + + (f° + f) ((1))2 p —p( ) 2(—5(°)) Again, the derivatives of f0t means the derivatives with respect to its own argument (— ln (_e_T)), and the superscript (0) on f0 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., is a combination of various orders in 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 + + + ... (the superscript (i) indicate the — — _(1) order in then take = out to calculate the second order result in 0U (denote the result as the subscript 1 in outl indicate the result is from ci0t =c0t), and take 0out = out to calculate the first order result in &out (denote the result as 2). The result from incoming boundary and outgoing boundary should match. i.e. XXX .(2) _(2) _(1) ‘1lin = 7loutl +7lout2 (5.26) Indeed (5.26) does hold. 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 - -(1) (2) (1) (2)let 0out = 0OUt + ci, then only take the cross terms in ci0 and ciQU. 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 + 2ljn (ln (_zE±eT)) (5.28) x — 2LL+lout (—in (_±ze_r)) (5.29) x+p First we think Thn and ‘Tlout as functions of (r — a) and (-r + a) respec 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 ± = p; j = x. As seen in the equations, one includes + while another includes p. So they are not equivalent. However, since we know the scattering matrix for i and the relationship between and at incoming and outgoing boundaries, we can still find out the scattering relation = — n —iw) { ! I i()}2ö (w — (5.30) 00 (\/)nl {1[f } - (5.31) 46 Chapter 5. Chiral Formalism Chiral Formalism Calculation Given incoming boundary field, we have XXXI (1) —p+ ——y,p+ (2) 1 __________ p+ = —j p+ (3) 1 p+ — ______ p+ Here the superscript (i) (except for the (0) on and its derivatives) mdi- cates the order in Tj2. Given outgoing field, we have 1 P-- Here the superscript (i) (except for the (0) on 1out and its derivatives) mdi- cates the order in ‘T1out And is given by cc (n)where P± = Zn=o P± p+—p = O + 8o4] (5.34) XXXIThe superscript (0) on lr or its derivatives means the value when = p. The superscript (0) on lout or its derivatives means the value when p =(O) ( (1),, (0) ( (i)2\p+t P+) I (0) — 2 I’x—p+ I (2),, (0) + p+ x — p (5.32) 1(0)(1) tout p_ = p / (1),i(0) (2) 1 I p_ t0 P- =—?oiI (0) P_ \ap ((i))2 — 2 (2) P— out — (0) x+p (5.33) ( (i))2 (iç + i”°” \ — out)1 2(x+p(°))2 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 (m) ci_(w) m—1 ci (w) as what we did in section 5.3.2; 3) calculate all the (m), . s with n 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_, (535)_ 2ir’/iw and the outgoing field u) = f ___(w)eT, (5.36)_ 2ir/iw 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) (1) . . out 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 easy to do. The result of XXXII at boundaries is like e(T)e2 which goes to zero at boundaries. Actually 0-i i—’ and it is also of such a form e(T+)e2 at boundaries, which means integration and derivative do not change its limit property at boundaries. Assume the (n— l)th orderpp and lower orders go to zero at least as fast as e2a, now let us prove that p)p goes to zero at least as fast as e2. 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— ((o))2 = ((i)(°) .p...pm)) (537) rn=O rn=1 (0) (a) . . . (m) (n—rn)The left hand side is 2+ ii+ + b. Here b is a combination of which go to zero at least as fast as e2°. The right hand side is a combination of where ii + + jrn = n. First, by the special form of Lj,, (8pmljn) ‘ is of the form ai XXXIII• i.e., it x—p+ ) is a combination of derivatives that smaller and equal to m divided by m n (x — e_aia. So the term in right hand side is either Oi () means the part of which is from p part by equation (5.34). means the part of which is from p_ part by equation (5.34). XXXIIIThe superscript ((m)) means the derivative of m times with respect to its argument. Here the argument is (in (_ej). And again, the superscript (0) on 1in or its derivatives means the value when p = p1. 49 Chapter 5. Chiral Formalism (which goes to zero at least as fast as e = e2) or8mj(O)p)pm) (here all the ii’s 2 in p3) , so p’s go to zero at least as fast as e2). So every term in the rith order equation goes to zero at least as fast as e2. Of course ‘- pp will go to zero at least as fast as e2. So ?7 goes to zero at least as fast as e2°. 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) = - c(w) - /iw{ 2 dwi ()}26 (w - — iriw(i— l){jfdwi ()}25( — E1w) + (w) + () + () + .... (5.38) Now let us calculate the field which is of second order in Tj as an exam ple. Let us denote such field as We take o_(w) = c(w) and figure (2) . (1) . (2) out p_ then add the result with p_ obtained from ci (w) = c (w). We also calculate p and then substitute them into equation (5.34). We get = [dwidw2 a+(wl)+(w2) (e0(T) + e0(T)) —e j 16ii3/2 ‘ smhu + 20(T_ e2awo 2F1 (i 1 — , 2 — ; e2)2i+W 2 2 + 2F1 (i _, 1 — iWO e2J)] }, (539) 50 Chapter 5. Chiral Formalism where the 2F1(a, b, c; z) is the hypergeometric function. When u —* —cc, the limit of is just outgoing field with c_(w) = which is reasonable. When o —* 0, we will get its behavior around the singular point. The limit is fdwidw2+(wi)+(w2){ ± e0(T) — 2iwoe0(T_J) ln(a) — — e0(T_) [3 + 6nwo — 2iw0 (7 + Log(2) + (_))] } (5.40) where ‘y is Euler’s constant and ‘(z) So the degree of singularity of (2) around the singular point is like + 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 0. Therefore the singularity should be due to the definition 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 transform- evolve-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, 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 S(x) - too dK (+ + 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 = _fdu e u(bi(x+ - u) + (b2 - bi))S(n). (6.2) Comparing these two, we get b1 = —2P/a2; b2 =b1(—1 + 47 +ln/). In this way, we justify that the choice of tachyon background T0 is correct. The second order: from equation (5.21), we have 1 2 ____ ((1)f 1?outT , — — - 7] (7 v2 L Combining (6.3) with (C.2), we get S(xj = — f ddudv K (+ + ln — OK (in/+& _) OK (ln/+ _) S(u)S(v) f du e (S(u))2, (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) ()1(o))3. (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 didndvdw K (+ + ln — — i {a (ln + — n) 8K (in + — v). UK (ln+ _) } S)S(v)S(w), (6.6) 00 00 2 e / du e_uS(u) / dv e_V (s1)(v))J—o0 J—0o — e f dv e_2vS(v) f du (s1)’(u))2, (6.7) -00 -00 which XXXVII agrees with the gravity result (3.41). 6.2 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 nontrivial, thus we include it as appendix D. 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 is also linear)X. Therefore the correspondence between and is linear, too. However, from Appendix B we know that the correspondence between 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) = fd&d2Li(x+,x;, 2)(2)(&1, J2) = — f dze_Z (S(z))2, (6.9) XXXVIIINotation 1,2 is basically . 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 ñ is linear and the correspondence between S and is nonlinear, or vice versa. However, physically there is no preference between and . One could show that the correspondence between S and 7) is nonlinear by showing the linear correspondence (between S and ) is not compatible with the possible form of L1,L2 (which will be discussed in section 6.3) and the relationship (B.6). 56 Chapter 6. Relationships Between MM and 2D Gravity or fdaida26+8_L1(x,x; a1,2)((a2) g(x,) (S(’)(x))2,(6.10) where g(x+,xj = 2Since ii” ‘(u1,u2) = f(i,a2)(82 (o2)) and i ‘(o2) = f dxK(u2+ in — zjS(’) (xj, this becomes fda12d 82K (a2 + in — )2K (a2 + in — q) S(’)(p)S(1) q) g(xx) (s(1)(x_))2, (6.11) or fda2H(x+, x,a2)0K(a2 + in — )82K (a2 + in — q) = g(x, x)ö(x — p)ö(x — q), (6.12) where H(x,x,a2)= fdai8+8_Li(x+,x;ai,a)f(ai. Now iet muitipiy both sides by K ( + in — K ( + in — q) and in tegrate over p and q, we get fd&2H(x,x,a)0o( — x)ao(a2— g(x, x)K ( + in — K ( + in — xj, (6.13) where we used fdy K(z—y)K(z—y) = (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 6.3 Constraints on L1 and L2 On L Since = o , a12?] = o, (6.14) 8(1) and (1) will stay the same as their incoming form. Thus we conclude that the x and 52 dependence of the kernel L1 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 L1 is limited to Li(x,x,&i,2)= K (2 +ln — ) G(x,i) + L, (6.15) where L1 satisfies I diLi(x,x,61,ã2)= 0. Since (1) 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 L2 By the same method we applied in section 6.2, we can also write down constraints for L2, which are fd1d380_L2(x,x; i, ; 3, u4) = g(x,x) f d&2K (2 + ln — K (4 + ln — xj, (6.16) fd1d8+D_L(x,x;i,;3,4= g(x+,x_)fd4K(2 + ln — K (4 + ln — xj, (6.17) thus f d&13 = g(x, x)K (2 + ln — K (4 + ln — + L2, (6.18) 58 Chapter 6. Relationships Between MM and 2D Gravity where L2 satisfies f (6.19) 6.4 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 8r 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 S2(x,x) = fdid2L(x,x;, 2)(2)(1, 2) (6.20) Please pay attention to the superscripts (i) indicating the order. Let us assume that we already know the exact form of L1, then we can use the two equations above to figure out L2. This is because we know the exact form of the tachyon field which should replace the left hand side. The form of j(2) 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 L2. L2 could then be solved for. Explicitly LHS = — eX+ f dv e_V (Sj(x,v))2 (6.21) = — eX+ f dv e_V The first term of the right hand side is RHS1 = fdid2Li(x,x; , 2)(2)(1,2) I ir e(a12)/2 = IdidLi(x,x;61,5)—j 2i cosh ((o’ —2)/2) (fds’(&2 _u)@1u)). (6.22) The LHS-RHS1= RHS2,which will give us an expression for L2 2(x,x;&i,;ö3,ã4) px a2÷I —V + + - =—--4-e jdve Li(x ,v;ai,o2)L1(x ,v;u3o4) e’)’2 — f dyLj(x, X; ‘cosh((i — • ö(o1 —3)5’(o2 — 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 one- variable 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 it-. If we insist to use this method, we will need higher order equations (iii 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 time t in the gravity theory and the r 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 (S(v))2. (6.24) On the matrix model side, to the second order we have (&+——)/2 - -+ -- ir e - 2 ° )° ‘°_ 2tcosh((+—j/2) (8&—i(a ,o- )) . (6.25) Equal-time condition reads x — = +. XL1fl fact we encounter this case before. For example, K(x) is a combination of tails e 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 effective variable under the condition x — = ö +. 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, “String theory in two dimensions,” [arXiv:hep th/9108019]. [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 = 1 Matrix Model,” Can. J. Phys. 99, 1-7 (2008). [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 = 1. This choice of the gauge is called Kruskal Gauge (KG). Modified Kruskal Gauge Since we want o = ax+ + bx + c, P0 = 0 to be the background, we prefer to fix the gauge with p = — 4o. This gauge is called Modified K’ruskal 67 Appendix A. (Modified) Kruskal Gauge Gauge (MKG). This choice of gauge is allowed since the form of satisfies the right hand side of eq. (A.3). 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 p = — Fo in the new coordinate system, we need p — p’ = p — = p — = — 1’ = —Fo. The first equality is because of p’ = ‘, which is the KG. The second equality is due to the fact that dilaton is a scalar field. The third equality is our requirement. The fourth is nothing but a definition. Therefore, if we can find a coordinate transformation satisfying p — p’ = —Io, p = 6 will hold automatically since these two equations are equivalent due to the fact that p’ = ‘ and dilaton is a scalar field. Let us express the coordinate transformation p — p’ = —o explicitly ii / 6z’\ /0x’ N —in 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 ‘+_I _2ax+)djç.) X — J, (A7)1 x’ = _e_2 _2c_d + g. Again, here f, g are arbitrary constants. 68 Appendix A. (Modified) Kruskal Gauge In particular, if we have the following solution in the KG = e2’ m — — — (A.8) in MKG we will have e2’ = e26 = e20e_2)’ = e2{ 2 (_e_2a + f — • (_e2 __2c_d+g_xj_) } = 1 + ( + FGe_2c) e2° + Fe2 + Ge2. (A.9) where F = (f _xje_d and G = (g _xje2. Since d,f,g are 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) \/x2 — 2 + “O , (B.2) \/x2 — 2 + V’x for < 0 and > 0 respectively. Expanding these up to third order in 1/x and third order in j, we get — — P + (/3)2 — (/8) — 2 + ‘ (in) —x +28 + + (8 + ()3 + 2 +2(aX)2 . (out) Equation (4.51) implies —2++ 2 when <0; (B.3) when>0. Now we are able to find relation between i and . Take the case <0 for example: combining equation (in) arId (B.3) we will get the following 70 Appendix B. Relationship between and i relationship 28() = — + 2 + (8)2 + //87 + (/8)2 + 22 + 5IL(/877)2 B 4 2x3 ( x =x — 2/a — + (/8) — 2+J)2 B5 — 2x3 (.) 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 i and second order in to find the function relation between i and is 2(a)=O(3)+O() / , // (—3t ‘(a) + (a)2— — 2/ (a) — 4ir,j (a) (a) + a2 + irii’(a)2 — 2( — 1 + /j”(a) + 2iri/’(a)) — 4rr /(a)r”(a) (B.6) where a is negative and large. XLIBasically it is an iterative method. Given two equations: F(a) = G(b) and a = b+g(b) (or b = a — 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 dy K(y — xi)K(y — x2) = — X2). For x — x large and negative, we have f dy K(x — y)8K(y — xi)8K(y — —e’ö(xi — z2), (C.2) f dy K(x—y)K(y—xi)—e’ (x_xl+47_2+ln) (C.3) where ‘y is Euler’s constant. For x <<x3 and x >> X3, we have f dy {(1_8)K(x_y)}8K(y—xi)8K(y—x2)8 (y x e21 {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) }S2(u)Si(v)Si(w) (D.1) + {(i — 8+)K (+ +in-). 8K (in + - — 8K (i + - — 8K(ln+_w)} Si(n)S2(v)w . (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 = (S1 + S2) = + 3S?S2 + 3SS + 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 p00 rxo rxo J du 82(u)...] dv S(v)... J dw Si(w).... (D.3)x0 —00 —00 where x0 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 00 00 2 eX+ f du e_uS(u) f dv e (S1)(v)) J—00 J—00 — e f°° dv e2vS(v) f°° du (S’)’(u))2, (D.4) -00 -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 f0 fZ0] duS2(u)...J dvS2(v)...] dw S(w).... (D.5)xo —00 Since now u >> w, equation (C.4) implies that equation (D.2) is approxi mately fOO p00 pXJ J du ... J dv ] dw ... {5(v — w) + ...S”(v — w)}, (D.6)Xfj X0 —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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