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A COLLECTION OF THEORETICAL PROBLEMS IN HIGH ENERGY PHYSICS By THOMAS MCCARTY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1990 ACKNOWLEDGEMENTS I wish to thank Professor Thomas Curtright, my dissertation advisor, for many enlightening discussions, and Dr. Ghassan Ghandour for many helpful and illuminating discussions. I would like to thank Professor Pierre Ramond for being my surrogate advisor. I thank the University of Florida particle theory group for useful discussions during my stay here. I also thank the theory group at the University of Miami, for their hospitality during my stay there. In particular, I express my gratitude to Gary Kleppe, Dr. Jun Liu, Dr. Raju Viswanathan and Dr. Ezer Melzer for many helpful discussions. I express my gratitude to Mr. Marco Monti for his computer expertise, which was invaluable during the writing of this dissertation. I wish to thank my family, especially my brother Sam and his wife Paula, for their support and encouragement. I also thank all those friends (whom I do not list, for fear of accidentally omitting someone) who made my time in graduate school an enjoyable experience. Finally, I thank my wife Julie for her support and love, which made all this work possible. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ............... ABSTRACT . . ... INTRODUCTION ................... Quantum Backlund Transformations and Conformal Algebras Angular Realization of SU(1,1) . . Radial Realization of SU(1,1) . . Backlund Transformation ............... Extension of the Radial SU(1,1) . . . ii . iv ., 1 .8 .9 . 10 . 10 . 10 . 16 BACKLUND FUNCTIONAL APPROACH TO LIOUVILLE THEORY Invariants of the Biicklund Transformation . . LIOUVILLE PERTURBATION THEORY . . CLASSICAL STABILITY OF CLOSED RIGID STRINGS . Rigid String in a Curved Ambient Space . . deSitter Static Solution and its Stability . . Ambient Space Fluctuations to the Hoop Solution . IS THE SUPERMEMBRANE UNSTABLE? . . Classical Solutions of the Canyon Potential Model . Quantum Instability of the Canyon Hamiltonian .. .. CONCLUSIONS . . . . 19 . 21 . 25 . 35 . 36 . 39 . 42 . 44 . 46 .49 S52 APPENDIX A: COMPUTATION OF THE RADIAL EIGENFUNCTIONS 55 APPENDIX B: LIOUVILLE THEORY DETAILS ... .58 APPENDIX C: GEOMETRY OF IMMERSIONS ... .73 REFERENCES ...... ... . .82 BIOGRAPHICAL SKETCH ........ ........... .85 111 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A COLLECTION OF THEORETICAL PROBLEMS IN HIGH ENERGY PHYSICS By THOMAS MCCARTY August 1990 Chairman: Pierre Ramond Major Department: Physics A Baicklund transformation is used to relate two different potential models, which are both realizations of the algebra SU(1, 1). The Backlund transfor mation is used to extend the "angular" realization of the Virasoro algebra to that of the "radial" realization of the Virasoro algebra. This extension is pos sible, only if the Backlund transformation is a similarity transformation. This example shows how the Bicklund transformation projects out the unphysical part of the free theories spectrum to coincide with the positive spectrum of the interacting interacting model. An exact solution to Liouville theory is found by using a functional Bicklund transformation. This solution of Liouville theory has the nice property that it is much more straight forward to compute correlation functions in pertur bation theory than by using the previous solutions. This functional solution is compared to the other known solutions by computing the expectation value of exp gac to order g6. The correlation functions are found to be exactly the same at the order of g6 in the coupling constant. However, there is a term of iv order g4 that did not appear in the calculation of Braaten, Curtright, Ghan dour, and Thorn [1] that is present in our computation. This term is of the right form to renormalize the mass term that appears in the theory. The classical stability of a closed rigid string immersed in a curved ambient spacetime is examined in the third chapter of this dissertation. A zero action solution of the closed rigid string is obtained when the string is immersed into a twodimensional deSitter spacetime. This exact solution is given a radial perturbation and found to be unstable. Next, the hoop solution of Curtright et al [2], is perturbed by adding ambient spacetime fluctuations via Riemann normal coordinates and by perturbing its radial coordinate. It is observed that the spacetimes of negative curvature are closer to stabilizing the hoop solution of the rigid string than the spaces of zero and positive curvature. A renormalization group analysis for a twodimensional field theory is used to point out that conclusions on the spectrum of the supermembrane computed using finite dimensional quantum mechanics in the large N limit are not nec essarily valid. This argument arises from analogy with a two dimensional field theory whose potential has the same functional form as that of the supermem brane. It is shown that the Hamiltonian of this field theory is unbounded from below and is unstable. Hence, the continuous spectrum determined by the truncation of the su permembrane to a finite number of degrees of freedom, i.e. supermembrane quantum mechanics, is not realistic when the number of degrees of freedom is allowed to become infinite(i.e. in the limit that the theory becomes a field theory instead of a quantum mechanical model). This suggests that a phase transition can occur in the limit that the number of degrees of freedom becomes infinite. Therefore, any argument of the instability of the supermembrane based on the spectrum computed using supermembrane quantum mechanics is invalid. INTRODUCTION The last decade of particle physics has undergone a major revolution in the way that particle theorists view the unification of forces in nature and the physics of the Planck scale (1019 GeV). This revolution has occurred because until recently there was not a consistent realistic theory for quantum grav ity: either the theory was not quantum mechanically consistent, or the low energy particle multiple was inconsistent with the standard model of particle physics(see ref. 3 for a complete review). To be able to unify all the forces in nature, requires a proper description of the quantum geometry of nature. The theory of general relativity, which pro vides the classical description of geometry at distances large compared to the Planck scale, is found to be unrenormalizable [4] when it is quantized. This un renormalizability means that quantum corrections to the EinsteinHilbert ac tion have additional ultraviolet structure which were was not originally present in the action: new terms and coupling constants must be added to the action that were not originally present in Einstein's action to subtract off these ultravi olet divergences. But such terms have other physical effects besides ultraviolet subtraction. Furthermore, such terms proliferate uncontrollably as higher en ergies are probed so the model completely breaks down at very high energy scales. Supergravity theories are unrenormalizable and are also found to have an particle spectrum inconsistent with what is observed experimentally. The revolution that occurred in physics in the last decade was to view dual resonance models, or strings, as theories of just strong nuclear interactions but of all the interactions in nature. This revision of thinking has occurred because 2 the supersymmetric versions of string theories were shown to be free of anoma lies [5] for certain choices of the internal symmetry group, therefore strings reappeared in theoretical physics as the most viable models of unifying all the forces in nature, including gravity. The reason why strings are thought to be a theory of gravity is because the closed string spectrum naturally contains a massless spintwo particle [6] that is identified with the graviton of general relativity. Strings have been shown to possess excellent ultraviolet behavior which was the stumbling block of the quantized version of general relativity. There are only a few different types of string theories [7]. The simplest model is the bosonic string [8], which is consistent only when the spacetime dimension is twentysix [9]. The bosonic string is parametrized by XP(a), which are the spacetime coordinates, where p = 0, 1, .,25 labeling the one time and twentyfive space coordinates of the string, and a is a parameter labeling the individual points along the string. There are also supersymmetric strings, where the supersymmetry is either manifest on the worldsheet or on the spacetime [5]. Both versions of these supersymmetric strings have a critical dimension of ten spacetime dimensions. The string theory that is the best candidate for nature is the heterotic string [10] which combines features of both the bosonic string and worldsheet supersymmetry of one of the superstring models. There are two types of boundary conditions for the bosonic string. These are open boundary conditions, corresponding to a string with two endpoints and closed boundary conditions, where XP(0) = XP(2xr) corresponding to a closed string loop. The heterotic string and most of the other supersymmetric string theories have only closed string boundary conditions. 3 We have previously pointed out that strings are only consistent in twenty six spacetime dimensions for the bosonic string and ten spacetime dimensions for the superstrings and the heterotic string(that is strings without any addi tional degrees of freedom). This leads to one of the most important questions in particle physics today, which is, why are only four of the spacetime dimen sions presently observed? That is, what does a higher dimensional theory have to do with what is observed at present particle accelerator energies, and what happened to the extra dimensions? If we consider the case of the heterotic string, then the fact that the other six spatial dimensions have curled up has some extremely important consequences in explaining some aspects of the lower energy phenomenology [11]. For instance, the topological properties of the six dimensional space are related to the number of generations, which is the first time a theory has attempted to predict such an important feature of the stan dard model. There still is not a fundamental understanding of the dynamical reasons which cause the heterotic string to evolve into the four dimensional universe that is observed today. This problem has led to considerable effort to construct a string field theory [12] in which it is hoped that nonperturbative phenomena could lead to a proper understanding of why only four of the di mensions of the heterotic string become macroscopic, and the other six spatial dimensions remain on the order of the Planck size scale, 1032 centimeters. There are many other important questions to ask of this theory, such as how do the masses, coupling constants, families of particles, and the Higgs field arise from this higher dimensional theory? The other types of questions to ask of this theory are do black holes exist quantum mechanically, or, put another way, are singularities allowed? If so, how do they affect the evolution of the 4 early universe, the evaporation of microscopic black holes [13], and why is the cosmological constant zero [14]? One of the topics this dissertation addresses is the classical stability of a two dimensional version of a string theory that has properties similar to those of QCD. This theory is called the rigid string, and it is a theory which contains quartic derivative terms in its action. This theory can be viewed as a type of compactification from a tendimensional superstring in which the string's worldsheet becomes curved in the process of compactifying. If strings are so successful, what about other extended objects? For ex ample, membranes which instead of sweeping out a worldsheet, like strings do, sweep out a world volume. That is to say, a membrane is just one higher dimensional analogue of the string. At this time there is no definite answer on the quantum consistency of membranes. There are still many outstanding problems of membranes that have not yet been successfully answered, such as the determination of the particle spectrum of the theory. Another unanswered question, is whether bosonic membranes possess a critical dimension like string theories. These are still outstanding problems because membranes appear to be interacting theories. It has been shown that a membrane with spacetime supersymmetry [15] could propagate in elevendimensional supergravity. This result suggests that the ground state of the supermembrane is made up of the states of the elevendimensional supergravity supermultiplet. So in some sense the critical dimension for the supermembrane is known because of the constraint of supersymmetry but whether this theory is consistent quantum mechanically is unknown. Since membranes are an interacting theory, many approximate calculations [16] have been performed to try to gain some understanding of these theories. 5 The work of de Wit, Luscher and Nicolai [17], based on a quantum mechanical calculation of the mass spectrum of supermembranes, showed that the mass spectrum of this theory is continuous and that there is no mass gap. The final chapter of this dissertation examines the validity of their quantum mechanics model. This is done by examining the renormalization group equations of a twodimensional model with a potential similar to that of the potential of the supermembrane quantum mechanics. The main difference between strings and membranes is that strings possess an extra symmetry that membranes do not appear to have. This symmetry, known as Weyl symmetry, enables one to locally gauge away all of the worldsheet metric dependence. The fact that membranes do not have this symmetry will most likely rule them out as a possible candidate for a theory of nature. To properly understand strings requires one to understand twodimensional quantum gravity. This is because string perturbation theory is perturbative with respect to fluctuations of the string worldsheet topology. The quantum corrections come from the number of handles on the string worldsheet. A model of twodimensional gravity known as Liouville theory was constructed some time ago in the seminal paper by Polyakov [18]. Liouville theory is one of the first examples of a conformal field theory. Recently a major breakthrough was made in describing nonperturbative two dimensional quantum gravity [19]. These models describing nonperturbative two dimensional gravity are known as multicritical matrix models. These mod els were used to sum the entire perturbation expansion of handles on the sphere. It has been recently shown [20] that Liouville theory is intimately related to these multicritical matrix models as well as topological field theories. It ap pears that there is still much to learn from Liouville theory. 6 This leads into the first and second chapters of this dissertation where we use Bicklund transformations to solve an interacting theory in terms of a free theory. Chapter one examines how this transformation works for simple potential models and chapter two uses a Backlund transformation to relate Liouville theory to a two dimensional free field theory. This dissertation is organized as follows: Following the introduction, the first chapter examines the use of Backlund transformations to relate two simple particle models, which are both realizations of the algebra SU(1, 1). Next, the extension of the radial representation of SU(1, 1) to that of the Virasoro algebra is made using the Backlund transformation and the Virasoro extension of the angular realization of SU(1, 1). The second subject of this dissertation concerns finding a consistent quan tum Baicklund transformation for Liouville theory. We develop a perturbation expansion using the coupling constant of Liouville theory as a small parame ter. We compute expectation values for the operator expgaq. We conclude that the functional Backlund transformation solution of Liouville theory yields expressions that are almost equivalent to the exact operator results of Braaten, Curtright, Ghandhour, and Thorn [1], that is the correlation function calcu lations agree exactly for the term of order g6. However, a term appears that is of order g4 that was not present in BCGT's computation. This term in our functional of solution of Liouville theory has just the right form such that it can be consistently absorbed by a mass renormalization. In the third chapter, we construct a zero action solution and study the stability of a closed rigid string immersed in a two dimensional deSitter space time. Next, we perturb the hoop solution of Curtright et al by using Riemann normal coordinates in the ambient space. This is done to try to understand 7 how the sign of the local ambient curvature affects the classical stability of the string. It is found that a space of negative curvature is closer to stabilizing the rigid string. The fourth and final chapter concerns the construction of static soliton solutions to the cross canyon potential model of a twodimensional field the ory. Next, a renormalization group calculation is performed to show that the Hamiltonian of the crossed canyon potential is unbounded from below. Which in turn suggests that the model undergoes a phase transition QUANTUM BACKLUND TRANSFORMATIONS AND CONFORMAL ALGEBRAS Backlund transformations provide maps between various fields, for example S+ 4, and therefore relate functionals of one field to those of another, Q[4] = q[(]. If Q is a conserved charge then the Backlund transformation is just a re expression of the symmetry of the 4 system in terms of those of the 4 system and vice versa. This has been well established for classical field theories and a more detailed explanation can be found in [21]. We will investigate some of the properties that quantum Bicklund trans formations possess. We do this in the context of simple potential models for a single point particle. We consider how the Backlund transformation maps the angular realization of SU(1, 1), and its extension to the Virasoro algebra, ap propriate for a particle moving freely on a circle onto the radial representation of SU(1, 1), and determine its extension to the Virasoro algebra, appropriate for a particle moving radially in an r2 + r2 potential. The angular and radial realizations of SU(1, 1), while quite different in their appearance, are directly related via Backlund transformation, which interchanges the the two realiza tions. This is not very surprising since it is well known that all single particle models with identical spectra, have canonical transformations relating one sys tem to to the other. That is, there are unitary transformations which relates the the one basis to the other, where we have assumed that the energy spectra of the theories in question are not degenerate. For the system that we will study in this work, there is a simple canonical transformation relating the 0realization to that of the rrealization. This canonical transformation is then just the Backlund transformation. Indeed, 8 9 there are many parallels with the Liouville field theory, which will be the subject of chapter two. The major parallel is that one can view the Biicklund transformation as mapping of the Hamiltonian of an interacting theory for a particle moving in a rdependent potential to that of a free Hamiltonian with the particle moving on the circle. This can be made more precise because there exists a Biicklund transformation which maps the interacting two dimensional Liouville field theory onto a noninteracting two dimensional pseudoscalar field theory. There will be a thorough explanation of this in the next chapter of this dissertation. Angular Realization of SU(1,1) The angular realization of SU(1, 1) and its extension to the Virasoro alge bra for a free quantum particle moving on a circle is given by In = exp(inO)(i&g n/2 + + irn). (1) This realization contains two parameters, 3 and y. The In's realize the centerless algebra [In, Imr = (n m)ln+m, (2) for any choice of the parameters # and . If / and 7 are real parameters, the In's satisfy the hermiticity conditions S= n, (3) when acting on any arbitrary functions of 0. Note that 10 is the Hamiltonian for this angular realization of SU(1, 1). 10 The quadratic Casimir invariant for the SU(1, 1) subalgebra generated by 141 and 10 is given by C[O] = 1+14_1 lo(l0 + 1) = 2 + 1/4. (4) Note that the Casimir operator is independent of 3. Also, if 7 is real, as re quired by the hermiticity properties in (3), then C[O] > 1/4. The eigenvalues of 10 p are 0, 1, 2, 0oo which is the Principal series of SU(1, 1)s irre ducible representation. It will be shown that / is the groundstate energy of the radial Hamiltonian. Radial Realization of SU(1,1) Now we consider the secondorder realization of SU(1, 1) [22] in terms of the radial variable r given by the following: LO = (r)2 a/r2 + r2/16, (5) L = _()2 a/r2 r2/16 F 1/2(r + 1/2), (6) where L0 is the Hamiltonian of the radial variable system. The eigenvalues of L0 form the Discrete series of SU(1, 1)s irreducible representation. The eigenvalues for L0 0 are 0, 1, 2, oo. Where / is groundstate energy of the radial Hamiltonian. This realization of SU(1, 1) contains just one parameter a. Now the Casimir operator for the radial realization takes the form C[r] = L+1L_1 Lo(Lo + 1) = (3 + 4a)/16. (7) Comparing C[r] to C[O] we note that C[r] < 1/4 when a < 1/4. 11 Blcklund Transformation The generator of the transformation between the 0 and r theories is given by r2 i(A + 1) F(r, 0) = 30 i/2A ln(r) + cot(0/2) + 2 ln(sin(0/2)). (8) This form of the generating function comes from the well known classical results for the simple harmonic oscillator and from [23]. This generating function can be written as a similarity transformation in the following manner, S1LOS = 10, (9) where the S is the similarity transformation. The function S can be expressed in terms of the rand 0eigenfunctions as S= dqrN(Lo)l)Lo(77o10, (10) where dq = drdO and N(H) is an energy dependent function which causes (10) to be a similarity transformation rather than an unitary transformation. The functions appearing in ieta)Lo and (11to are the eigenfunctions which satisfy the following equations (Lo En) r)Lo =0, (11) (Io En))10to =0, where En are the energy eigenvalues of the operators LO and l0. Now (9) can be expressed as J dNN(H)LoI1)Lo0 (I1 = / dN(H)h7)Lo(l 1olo. (11) It is found that the Hamiltonians of the two different systems are related by LO exp(iF(r, 8)) = exp(iF(r, 9))10. (12) Then the result is (a,.F)2 i(r)F a/r2 + r2/16 = 9eF + 3. (13) In fact, (10) was used to determine F up to possible linear terms in 0. From this we can solve for A in terms of a, and obtain (2 A)A = 4a, which yields A= 1 + 1 4. (14) For the interacting rtheory, note that a > 1/4 corresponds to an attractive radial potential which is strong enough that a particle would "fall into the origin" [24]. This simple physical constraint therefore requires that a < 1/4 which was previously pointed out to be equivalent to C[r] < 1/4 Therefore, for a < 1/4 there is a possible conflict between the physics of the r and the 8 theories. The hermiticity properties of (3) imposed on the results of (4) exclude the allowed physical range for the Casimir operator of the rtheory. However, there is a way around this dilemma, because the transformation between the two different theories can be a similarity transformation instead of a unitary transformation. This will be commented on in greater detail when we discuss the extension of the radial theory to the Virasoro algebra. Now using the result (11) in (10) yields Lo exp iF(r, 8) = lo exp iF(r, 0) = 1/16 csc 2(0/2)[r 2i(A+l) sin 0] exp iF(r, 0). (15) However, (11) does not determine the parameter /. This can be solved for by relating r and 9 wavefunctions to each other. The Biicklund transforma tion can be thought of as a "nonlinear" transformation whose kernel is just the transformation exp iF(r, 0). Then with this in mind, we can relate the two different eigenfunctions to each other via 2ar PE(r) = tE J dOexpiF(r, O)E(O), (16) 0 where r7E is an energydependent normalization, which is needed to avoid the problems mentioned about the inconsistency of the Casimir operators. This transformation can be inverted. Assuming that P and y are real; we obtain 00 OEE(9) = r77E drexp(iF(r, ))%E(r). (17) 0 It is a straightforward exercise to obtain the radial eigenfunctions and the eigenfunctions of the angular Hamiltonian. For the radial eigenfunctions we use the method of series and the boundary conditions that I(r) + rx/2 as r + 0, and that T(r) + exp(r2/8) as r + oo. Then by demanding orthogonality of the eigenfunctions we can determine the proper measure for this vector space and normalize the eigenfunctions. Similarly it is simple to find the 0b(0) by asking that they be periodic functions of 0. To solve for /P we use (14) and that the groundstate eigenfunctions be g0(r) = 2(A+1)/2(r1/2(A + 1)/2))rA/2 exp(r2/8), (18) and On(0) = 1/(2r) exp(inO), n Z+. (19) Inserting these expressions and changing variables via z = exp(i9) yields S= (A + 1)/4 + N,N E Z+. (20) For simplicity we choose N = 0. This comes about by asking that there are no branch cuts about z = 0 in (14), when we perform the contour integral on z, and that the contour integral does not enclose the branch cut at z = 1. This calculation appears in Appendix A. In general the ''n(r) can be simply computed from n(r) = iE(2i)A+)/2rA/2/(27) f dz exp(r(1 + z) )z(n+), (21) 8(1 z) where again z = exp(iO). Therefore, to obtain the excited states of the radial eigenfunctions, one merely uses the residue theorem. We point out that if the value of n in appearing (17) was allowed to be negative, then (18) would be zero for the contour that we have chosen. So this is a natural restriction of the spectrum of states appearing in the 0realization of SU(1, 1) in order that it maps onto the radial realization. Now we consider the effects that the other SU(1, 1) generators have when acting on the Biicklund transformation. Equating these generators acting on the Baicklund transformation, we obtain L exp iF(r, 0) = l1 exp iF(r, 0), (22) which is valid if and only if y = i(A 1)/4 = +iv 4a. (23) Note, that the subscripts on the LHS of (19) become F on the RHS. This is necessary to be consistent with the SU(1, 1) commutators and (10). Thus we have expressed /, 7, and A in terms of a, and only one free parameter appears in the expressions for the conformal charges and the generating function. Also, note that / = 1/2 i. (24) Collecting together the results for the various realizations for the Casimir operator, we have C[O] = C[r] = 72 + 1/4 = (13 + 1), (25) or this can be expressed as C[9] = C[r] = (4a + 3)/16 = (3 A)(1 + A)/16. (26) Now returning to the problem previously mentioned about the incompati bility of the Casimir operators in the r and Orealizations, the Bicklund trans formation exp(iF(r, 8)) completely sidesteps the issue by choosing a purely imaginary value for , which is completely inconsistent with the hermiticity conditions stated in (3). However, the transformation preserves all of the commutation relations and other operator identities. This is only possible if the transformation is not unitary when acting on arbitrary states, but only a 16 similarity transformation. The r ++ 0 mapping differs from a unitary trans formation because of the presence of L0dependent factors. We have previ ously pointed out such ambiguities in exp iF. Although the final form of the 9dependent Virasoro generators does not obey the general hermiticity condi tions given in (3), there is not a problem with unitarity in the model since the space of 0dependent functions has been restricted. In particular, only the n > 0 are allowed for the single valued functions in (17), or else the condition on the spectrum resulting from (18), and that was explicitly shown in (19) to fail. The situation that occurs here is very similar to that which occurs in two dimensional conformal field theory when the "Coulomb gas" is mapped onto the the conformal series with central charge less than one [25]. Extension of the Radial SU(1,1) Next, consider the action of the full Virasoro algebra on the generating functional. This is fairly easy to do for potential models, at least formally, because the full algebra can be realized in terms of rational functions of the SU(1, 1) generators [26]. Also, by constructing L2 using difference equations, it can be seen by induction that the result [26] is (25) below. Thus for n > 0 Ln = (L + ) + ) (L1), (27) L(Lo + P + n) Ln = (Lon)r(Lo + 1 ) ( (28) Due to the fact that the Casimir operator has the form C = (/? + 1), we can interchange /3 (1 /) in these expressions. This will be very useful 17 in the following manipulations, along with the fact that the different rand 0 dependent generators of the SU(1, 1) realizations commute among themselves. Acting with Ln as given in (25) on the transformational functional yields r(Lo + 3) Ln exp iF = (Lo + n) (L + ) (L1)n exp iF. (29) r(LO + 0 + n) Now we use (19) to rewrite this as r(L/ + 1 ) Ln exp iF = (LO + n3) (Lo + (11)n exp iF. (30) r(Lo + P + n) Then we use the fact that the different generators of the realizations com mute and pull the ln's through the various functions of L0 to obtain S r(L0 + 3) Ln exp iF = (11)n(Lo + pn) exp iF. (31) r(LO + P + n) Again we replace LO with l0 acting on the generating functional to yield )r(lo + 10 + n) Ln exp iF = (1n)n(l0 + np) exp iF. (32) r(lo + P + n) Making use of the fact that (l_1)mf(lo) = f(10 m)1m1, (29) becomes F(10 n + /) Ln exp iF = (lo n + npo) (_l)n exp iF. (33) r(lo + P) That is, again for n > 0, Ln exp iF = ln exp iF, (34) n(o r(/00 + )) ( (35) Ln = (10 n + n+) (1_1)n. (35) r(lo + p) Tracing through the same steps yields Ln exp iF = In exp iF, (36) In = (l + n ng) + n (1). (37) r(io + + 1 () Now it is a simple exercise to show that the expression for In and ln reduce to (2) upon substitution of the explicit forms for 1o and l into (33) and (34) respectively. The results of (32) and (34) are a nice feature of the r <+ 0 transformation. That is, the Backlund transformation naturally provides an extension of the secondorder radial realization of SU(1, 1) to include the full Virasoro algebra. This follows immediately from the usual extension of the angular realization. BACKLUND FUNCTIONAL APPROACH TO LIOUVILLE THEORY This chapter considers the problem of using Backlund transformations to construct a functional solution to Liouville theory. As we pointed out in the previous chapter, Backlund transformations provide a reexpression of the sym metries of one dynamical system to those of a very differentappearing dynam ical system. This comes about because the interacting system is an integrable theory. The Baicklund transformation relating Liouville field theory to that of a free pseudoscalar field theory in two dimensions is given by Oab (2m/g) exp(g$) sinh(gi) = Qr = II 4a90 + (2m/g) exp(gO) cosh(gO) = 9r = rT. (38) The integrability conditions for this pair of equations give rise to the Liou ville and free field wave equations (ar2 9a2) + (4m2)/g exp 2gO = 0, (ar2 9a2), = 0. (39) The above equations (36) are just canonical transformations. A generating functional at a fixed r is given by r p (40) F[o, 0] = da[0o' 2m/g2exp gosinhgo] (40) JO 20 where 8' = a7, Canonical moment are determined from F[?Pq] in the stan dard manner through taking functional derivatives at a fixed r phone a 6F 6F n = (41) Then it is simple to evaluate these derivatives and obtain the original pair of equations appearing (36). For the classical field theory, this is essentially all that needs to be done. It is now possible to solve for exp go, and then II and thus obtain expressions for functionals of the Liouville field 0 in terms of those of the free field 4, G[II, ] = g[7r, ], at any fixed r. For the quantum theory, this is just the starting point. One way to quantize the theory is to use operator methods to convert the classical expressions for g[7r, '] into well defined free field operator expressions in such a way that the locality and conformal transformation properties that are expected to hold for the original G[II, 0] as equal time expressions involving canonical interacting fields are maintained. This is a very complicated proce dure, but it was carried out [1]. The physical properties of the expressions obtained using this procedure are not very transparent. Another equivalent way to proceed is to use the Schrodinger equaltime formalism [23]. This is in some sense a natural approach to the problem con sidering the functional derivative relations in (38). To carry out this approach requires further knowledge about the quantum interpretation of the classi cal generating functional for the Biicklund transformation, and the quantum corrections to it. The first step towards understanding this method is to expo nentiate the classical generating functional, up to factors of h. Now, the exponential of F[0, 4] acts on tensor products of two different functional spaces: one for 0, the other for 4. Then a natural way to view 21 exp iF[zC, 4] is as a transformation from one space to the other. The next question to ask is, what are the invariants of this functional transformation? Invariants of the Backlund Transformation Now we compute the invariants of the functional Backlund transformation; these will turn out to be the components of the stressenergy tensor. The stress energy tensors for the Liouville theory and the free theory are given respectively by the following 62 2 2 000 = 1/2( + 2 24 + (4m2/g2)exp 2g), (42) 01o =(i6+ia6 ),(4 62 12 000= 1/2( 2 + 2ia 6), b 64 (43) 001=(i tH +0"). Now we show that the components of the stress energy tensor yield the same expression when acting on the Backlund functional. First we point out that the Bicklund transformation can be integrated by parts to obtain 27r da( ' 2m/g2 exp gQ sinh g) = 2 dur(?b' + 2m/g2 exp g sinh gi). (44) We assume that the functional derivatives are taken at the same point, then o00 exp(iF) = 1/2[2im exp(gO) sinh(gV) + 0'2 + 4,2 2(" (45) 4m/gt' exp(g4) sinh(gO) + 4m2/g2 cosh2(2go)], where we have used S2 exp(iF) = 2im exp(gq) sinh(go) (0' 2m/g exp(g)) sinh(g))2, (46) and 6 exp(iF) = i(O' 2m/g exp(g)) sinh(go)) (47) to arrive at (43), and the fact that cosh2 gi = 1 + sinh2 go. Similarly we functionally act on exp iF with respect to 800 and obtain 800 exp(iF) = 1/2[2im exp(g) sinh(go) + ,'2 + /,2 + 4m2/g2 exp 2g cosh2 gi 24" 4m' exp g 0 sinh go], (48) so the two expressions are equal, i.e., 000 exp iF = 800 exp iF. (49) Also, it is simple to see that 001 exp iF = 801 exp iF where we have in cluded the the anomaly term in the stress energy tensors. So the equivalence of (43) and (45) means that the energy density is an invariant of the transforma tion functional and so are the other components of the stressenergy tensors. As in the case of the simple potential models in the previous chapter, the invariance of the energy densities allows the construction of energy eigenfunc tionals for the Liouville theory in terms of those for the free field theory, which are just essentially Schr6dinger wave functional transforms. Then this is ex pressed as E[] = JD exp(iF[4, )D'WE[]. (50) 23 Using equation (45) acting on (46) and functionally integrating by parts, yields Eo00oE[] = Eq[O] if and only if o00o[4'] = ET [i0],.where the integration is a functional integral over all field configurations at a fixed time, and is not a path integral. Again, equation (47) can be thought of as a "nonlinear" Fourier transformation. Now we need to solve for the free wavefunctional for the free field theory. First, we must solve equations (36) at a fixed time and with periodic boundary conditions. This yields the mode expansions for 0(a) and O(a), which are respectively given by O(a) = q + i/lV nl(an exp(ine) + bn exp(ina)), nO (a) = Q + i/V' E n1(An exp(ino) + Bn exp(ino)). (51) n0O where, for convenience, the nonzero mode part of the Liouville field will be de noted as 4 = i/V EnO nl(an exp(ina) + bn exp(ina)) and the nonzero mode part of the free field as i = i/ V EnO n1(An exp(ina)+Bn exp(ina)). Where the expressions an,bn,An and Bn are Fourier coefficients and not to be confused with creation and annihilation operators. Now, solving for '[], using 900 with the surface term discarded 62 2 1/2( 2+] = 0. (52) This has a solution of the form %(E = 0) = exp 1/2 0 (0') a(). (53) )o V aa 24 To verify that this is a solution merely take two functional derivatives of (50) where we assume the functional derivatives are at different points, so that the the infinite zero point energy can be explicitly seen, 62 Sp) (a) I(E = 0) = Io, l(a)laplC(p)W(E = 0) 1al(a p)W(E = 0), (54) where aO,.6(oa) = 1/(2r) EnO Inl exp in(ap). When a = p this expression diverges quadratically. This is just the usual zeropoint energy associated with each nonzero mode, and this must be removed to have a well defined 000. Then to show that (51) is a solution requires that 12r da = j0 da'. (55) This can be done by first showing that da (9 = 2 = d2a 1 al,, (56) and then integrating the first expression in (91) by parts using periodic bound ary conditions. Now the expression obtained from functionally differentiating T twice becomes in the limit p + a 1/2 dao, 2 2(E = 0) = dar'i'. (57) 0 6 ) Now, inserting (54) into (50), since this expression is zero, (51) is a solution to (50). Now the wavefunctional at nonzero energy but zero momentum can be achieved by multiplying (51) by exp(ipQ). Then this wavefunctional has energy 27r daoo'p = p2/4rp, (58) where we have used the fact that 1/27r(6 + ). Then the energy of the functional field Ep = p2/47r = g2k2/47r where p = gk and g is the coupling constant of the Liouville action. LIOUVILLE PERTURBATION THEORY Now we express the functional Bficklund transformation in a more con venient form, so that the evaluation of (47) in a perturbation expansion in the coupling constant g is well defined when the coupling constant is a small parameter. With this in mind, it is very useful to make the following identifi cation exp go sinh go =2expgq sinhgQ + exp g(q + Q)(exp g(i + 4) 1) (59) exp g(q Q)(exp g(O 0) 1). Using (94) in (86) with the wavefunctional at nonzero energy yields I(D) = DQ exp i(gkQ 4rm/g2 exp gq sinh gQ) x S"0 .2r (60) I D exp / do[i(q 1/218lj4] + Wi] where we have factorized the measure with respect to the zero mode and the nonzero modes. Now (57) can be expanded in terms of the function W1 where W1 is given by Wi = mi/g2[expg(q + Q)F+(pi) expg(q Q)F.(pl)], (61) where the functions F+ and F_ are defined by 027 F+(pi) = dpl(n7expg((PI) + (iPl)) 1) f2i (62) F+(PI) = 1 dpl( 7expg((pj) (pl)) 1). Note that we have inserted in the above expressions a constant r7. This constant will be chosen such that 7 = exp g2(1/(27r). The importance of this will be expanded upon shortly. Expanding the exponential in (58) in terms of W1 yields ,(0) = dQ exp i(gkQ 47rm/g2 exp gq sinh gQ x / 2l (63) Sdi 1/n!Wl exp 1/2 / d The following identity [27] is useful in relating the zero mode part of (60) to the modified Bessel functions Kv(x): dt exp[ix sinhit(k in)] = 2 expr/2(im + k)K,(x),n E Z, v = n + ik, 0O (64) where x = 47rm/g2 exp gq for our purposes. Note that for the lowest order of perturbation theory n = 0, in (61), yields (0)(k) = 2/gexp(7rk/2)Kik(x), (65) which is the zero mode solution found by D'Hoker and Jackiw [28]. With the above established, it is now a simple matter to expand I in terms of W1, use (62), complete the square of the various nonzero mode expressions involving , and then compute the Gaussian integrals over the ) nonzero modes. As an example of this last statement, consider the following expression: d\101 2i'b + 2/3g$S(a p), (66) where S(p) = ]n# exp inp, that is, the zero mode of the delta function is removed. Changing variables in (63) to lb = i + g(p), (67) then when (64) is inserted back into (63) (and note that we have dropped the index on the derivative but it is with respect to a) yields 1 11i + $ I4 #292 2(a p)l1'S(ao p) 2ifg 6(a p) = I"! (68) 5la,1 2i,'V + 203g~(a p). We point out that the expression #2Sa(_p)lll_p) = P2/rT Z 1 1/n = /321/7r is just the usual "normalordering" divergence that appears in opera tor expressions. The normal ordering divergence coefficient /, will be equal to one and hence this is why we have chosen r7 appropriately to cancel out this divergence. This choice of q will cancel all of the terms g2(1/(27r) that arise from the integration of the k nonzero modes to every order in perturbation theory. We will see later on in the calculation that there will be no "normal ordering terms coming from the integration over the 0 nonzero modes. The perturbation expansion for D is given by the following expression: S(k) = (0)(k) + 4(1)(k) + (2)(k) + 3(k)..., (69) and the various functions that appear in these terms are defined by An = 2g12nmnexp(kcr/2), 28 G = D exp 1/2 I daal I: (70) where the various expressions in (66) are defined by i(0)(k)= 2g1 exp(k7r/2)Kik(x)G, (71) 41(k) = 2AG expgqg[Kl+ik(x)F+(Pl) (72) Kl+ik(x)F(p)], 2) .(2)(k)) = A2/2G exp 2gq[K2+ikF+(pl)F+(p2) + l2+ikF(pI )F(P2) + Kik(F+(P1)F(p2) + F+(P2)F(p))]. (73) .(3)(k) = A3/6G exp 3gq[K3k+ikF+(pl)F+(P2)F+(p3)+ Kl+ik(F+(Pl)F+(P2)F(P3)+ F+(pl)F+()F(P2) +P2F+(P3)F(pl))+ K_ 1+ik( F+(pl )F 2)F(P3)+ F+(p2)F(Pl)F() 3) + F+(P3)F(pI)F(p2)+ K3+ikF(p )F(p2)F(p3))]. (74) The complex conjugate of 4*(k')) is easily obtained from (66) by changing k * k' and the term it'k changes sign which will cause the projection operators p which are defined in the appendix on Liouville theory to change to p:. The explicit form of (69),(70). and (71) appear in the app,,,ndli; on Liouville theory after integration over the v rlonzrr modes hs brn p(xrforned. Note that we have kept terms to cubic ordr in exp gq and that the subscript n 29 denotes the different shifted arguments of the Gaussian integrals, which is an overall infinite multiplicative factor which will be divided out of the correlation functions. Now we make use of the above expressions of the wavefunctional and its complex conjugate to compute the expectation value of exp ago(O) to order g6 in the coupling constant. We do this to make contact with the Liouville quantum mechanics calculation of Braaten ,Curtright, Ghandour, and Thorn [1]. Again, we complete the square of the massive modes, where the massive modes are the 4 fields, and then integrate over all possible field configurations at a given time, which yields (,P(k')lexp agO(0)l(k)) = ({(0)(k') exp ag (O) (0)(k))+ (<(1)(kc'lexp ag O(0)lD(1)(k))+ ((1)(k')exp agO(0)$(O)(k))+ ((0) (k'lexp agO(O) l(X)(k))+ (p(2)(k')[exp ag(0) (O)( k))+ (0(0)(k')jexp ag(O) I(2)(k))+ (75) ,D(3) (k') exp agc(0) 1(O)(k))+ ((O)(k' exp ,agq(0)1(3)(k))+ (4(1)(k'lexp ag(O)) (2)(k))+ (D(2)(k')lexp agO(0)JO(1)(k))+ terms of order exp g(a + 4)q. Where the brackets contain the integration over both the zero mode and the nonzero modes of the field 0. This functional form is chosen for the expectation value for two reasons, the first reason being is that the vertex operators of string theory are of this form and the second reason being, is that monomials in the 30 field 0(0) can be calculated by differentiating the correlation function with respect to a, n times and then setting a = 0 to obtain the desired power of the expectation value of 0. We point out that the terms: (('1)(k'lexp ago(0)l}(1)), ((1l)(k')lexp agc(0) 4(0)(k)), (D(O)(ck'exp agO(0)I(1l)(k)), (,(1)(k'Jexp ago(0) 1(2)(k)), and (D(2)(k')[exp ago(0) [(l)) appearing in (72) are zero. It is shown in the appendix by considering generic nonzero mode matrix elements of each of the above matrix elements are zero. Then it is explained that the same mechanism that caused the particular ma trix nonzero mode matrix to vanish, also cause the rest of the nonzero mode matrix elements to be zero. These terms vanish because there is always an an gular variable pn, which is not suitably correlated with the rest of the angular variables. When the exponentials are expanded and integrated term by term, these uncorrelated pn integrate to zero. Then the parts of the angular integrals that are nonzero when integrated always occur in pairs with a relative minus sign and cancel against each other. This lack of correlation occurs between the angular variables occurs because the in the wavefunctionals are integrated independently of each other. Then it appears that the only matrix elements that will be nonzero are of the form ((O0)(k')lexp agQ(0)1.(n)(k)) and its com plex conjugate. We have explicitly shown this to order exp g(a + 3)q in our calculation. 61 Listing the nonzero matrix elements appearing in (72): (O(0)(k') exp agtb(0) lO)(k)) =Mo f dq exp gaqKik'(z)Kik(x) ((0O)(k')lexp agq(O)')(2)(k)) =Mo/2 dq exp g(a + 2)qKik,(x)[K2+ik(x)M+2(1, 2) + iK2+ik(x)M2(1, 2) + Kik(x)(M+(1, 2) + M+(1, 2))] (1(2)(k')exp ag(0)I(0)(k)) =Mo/2 /dqexpg(a + 2)qKik(x)[K2_ik M_2(1, 2) + Ki2ik,(x)M+2(1, 2) + Kik,(x)(M+(1, 2) + M+(1, 2))] (,(O)(k'lexp ag(O)l)(3)(k)) = Mo/6 dqexp g(a + 3)qKik'(x)[ K3+ik(x)M3(1, 2, 3) + Kl+ik(x)M++(1,2,3) + 1+ik(x)M+ (1,2,3) + I3+ikM3(1,2,3)] ((O)(fk'lexpago(0) )(k)) = Mo/6 dqexpg(a + 3)qKik(x)[ K3+ik,(x)M3(1,2,3) + Klik_(x)M+(1,2,3) + Kik,(x)M+(1,2,3) + C3ik'M3(1, 2, 3)], (76) where the limits of the q integral are from oo to oo and the Mi that appear in (73) are defined by MO =4g2 exp(k + k')r/2( Dp exp 1/2 dojalf)2x (J D exp 1/2 do~I^9k) Mn =Momng12n 2 2 Mi2(1,2) =(2r)2[(9 )2(2 + 2(g)3C3] (77) 2 2 M+:F(1, 2) =(2)2 [( 2 + a2 )C31 M3 =2g6(3 M++(1,2,3) =M+(1,2,3) = 6g6g3. The form and the evaluation of these ) nonzero modes can be found in detail in the appendix on Liouville theory. We can simplify the higher order modified Bessel functions appearing in (73) by making use of the identity [27] 1 Tv+ = 2vxlT, (78) where Tv = exp ivirKv(x). This enables us to express all of the Kn+ik in terms of Kik(x),Kl+ik(x) and K_+ik(x). The same procedure can be carried out for Knik'. We give one example for IK3+ik(x), 2 1K3+ik(x) =Kl+ik(x) + 2(2 + ik)m1 exp gqKik(x) 2 (79) + 4(2 + ik)(1 + ik)(m1)2 exp 2gqK1+ik(x). 4w Now we integrate over the zero mode q and use the identity [29] j dzz K(az)K,(bz) =2(2+A)A1(r(c)1r((c + P + V)r (c + V V) (C + + V) ((cIv) x r( )r( ), 2 2 (80) where c = 1 A. To make use of (77) we change variables, exp gq = z With this change of variable, the limits of (73) change to to that of (77) and for our purposes a = 47rm/g2 Now we substitute (73) into (72) and make use of the reduction formula of (75) and as an example we list the following matrix element, 2 (4,()(k')JexpagOB(O)) (2)(k)) =Mo( g)3Zo[C3/(2a(a + 1) (2 a2)(a4 + 2a2(k12 + k2) + (k2 k2)2] + ((2 + 4C3)(a + 1/a(k'2 k2) 2k2)], (81) where the function Zo =1/(2g3r(a))(2 )(1/2(a + i(k + k'))r(1/2(a i(k + k'))x 2rm r(1/2(a + i(k k'))r(1/2(a i(k k')), (82) Z0 = (4(O)(k')lexp crg(0))D(O)(k)). We have listed (79) since every matrix element has this factor in common and it is just the matrix element of the ground state. Now adding up all of the matrix elements we obtain the final result of the correlation function which is 2 (M(k')lexp agf(O))[(k)) =MOZo[l + 2(9)2(2(a (k2 + k'2) 22r + ( )(3(a4 + 4/3a3 + 8/3a 2a(2 a)(k2 + k'2) (k'2 k2)2 + terms of order g8]. (83) The terms of order g6 correspond exactly with the computation of Braaten, Curtright, Ghandour, and Thorn who computed this correlation function using 34 Liouville quantum mechanics perturbed by the nonzero modes. The terms of order g4 can be handled in the following manner. The term k'2 + k2 can be absorbed into the normalization of the wavefunctional and the a dependent term gives a finite renormalization to the mass term that appears in prefactor of (49). This term then changes to 2 2 2Irm 2r(m + Am) where Am = g4/48. Note that any other power of a or terms with k or k' dependence multiply ing the term of order g4 could not have renormalized the mass term or been swept up in the wavefunctional normalization. The next step in the computation is to verify that D(k)) obeys the Virasoro algebra L=o(k)) =0, (84) where f2 r L = da exp ino[0o o 01] (85) 000 and O01 are defined in (42). Verifying that (84) is satisfied has not been shown yet but the calculation is in progress. This calculation is crucial to understand how this mass renormalization of order g4 is consistent with conformal invariance. CLASSICAL STABILITY OF CLOSED RIGID STRINGS Polyakov [30] and others [31] have pointed out the possibility of adding an extrinsic curvature term to the action of a relativistic string. These actions with an extrinsic curvature term can be viewed as the string's worldsheet being embedded into a higher dimensional spacetime. These rigid strings appear to arise quite naturally from GreenSchwarz actions in which the fermions have been functionally integrated out of the action [32]; so this model is in some sense a compactification which preserves many features of the original string theory. This structure is what led Polyakov to consider the rigid string as a model for QCD. This paper investigates the effects that an extrinsic curvature term coming from a curved ambient space have on the classical stability of a rigid string. The rigid string action has the novel feature that classical string configurations can be investigated whose Euler number is different from zero, depending on the ambient space's curvature. Braaten and Zachos [33] have shown that the static solution of the rigid string, that is, the hoop solution found by Curtright et al [2], is unstable against small radial perturbations. Perturbations about the other coordinates were found to be stable. This hoop solution, which assumed a flat ambient space was shown not to have a stable classical ground state; that is, the vac uum energy could be made arbitrarily negative. This implied that the radius of the hoop could be made arbitrarily large. Hence Braaten and Zachos con cluded that the rigid string should be regarded as only an effective theory and that other terms would have to be added to the action in order to guarantee the existence of a classical ground state. The present work looks for nonflat spaces 35 36 that can stabilize the rigid string against small fluctuations, in the same man ner that antideSitter Space was used to stabilize an extended supergravity theory with gauged SO(N) internal symmetries [34]. That is this supergravity theory had a Hamiltonian that was unbounded from below, until an antide Sitter background was added to the theory. The addition of the antideSitter background stabilized the supergravity theory with gauged SO(N) internal symmetries. Rigid String in a Curved Ambient Space The derivation of the rigidity term is briefly reviewed [35] in the presence of a curved ambient metric. A complete derivation of the extrinsic curvature term and its generalization appears in the appendix on the geometry of immersions. The induced metric on the string worldsheet is assumed to be the usual one of string theory, except that an ambient metric is used to contract the spacetime indices gab = X'XG (86) where Gy, is the ambient space metric. We will briefly comment on the validity of this equation at the end of this section. When one considers a single normal to the string worldsheet, one obtains the following relations G ,XP?,V = 0 and GpvIr y = 1. (87) Note that only the simplest case of just one normal is being considered here, though it is simple enough to generalize to more than one normal. Capv is the Christoffel connection of the ambient space, Da is the covari ant derivative on the string worldsheet, and b is the worldsheet Christoffel ab 37 connection. Two worldsheet covariant derivatives acting on an ambient space vector field yield DaDbX = aXP rc X, (88) where rab = g(cG X X + CaXax). (89) Now the second fundamental form can be expressed in terms of (85) and the ambient space metric as Kab = GuvDaDbXXrl' + CavXaX r"l. (90) The second fundamental form can be covariantly differentiated and anti symmetrized on the last two indices; then, using Ricci's identity, the projection of the Riemann tensor onto the string's worldsheet, and the condition for the integrability of the second fundamental form one obtains Rabcd = Kab Kd IacKbd + R 6 X ,aX XdX (91) A,8yb a c7 b di DcKab DbKac = RA XaXXX (92) where the superscript A denotes ambient space Riemann tensor in order to avoid confusing it with the worldsheet Riemann tensor. The world sheet Ricci curvature can be related to the GaussBonnet theo rem to obtain the following expression: 2 2n h= d2z/[(gabKab)2 gab cdacK J (93) +R A oQ' x37 ab vX ,6 cd, +Ra X,.a bY .,,c.,, J' where n is the number of handles, h is the number of boundaries, and since only closed strings are being considered, h will be set to zero. Since (93) is a topological invariant, one of the terms can be removed from (93). Then any two terms from (93) can be added to the minimal surface area term of the usual bosonic string, yielding a highly nontrivial interacting theory. In this work the term gabg cdKacdKbd is removed. Then the action for the rigid string in a curved ambient space becomes I = T dz g [1 R((gabKab)2 A X X tXX6 d9ab ] cd, (94) where R2 = S/T, T is the string tension, and S is the rigidity parameter. Note that this action is written as a minimal surface action plus the rigidity terms. This is because the induced metric which is the variation to the above equation, treating the worldsheet metric as an independent variable, yields equations that are no longer just algebraic equations but are nonlinear partial differential equations of the worldsheet metric, the ambient metric and the tangent vectors. This is true also for the case when the ambient space is flat. The equation for the induced metric is computed in the appendix A. So we are assuming without proof that the induced metric is gab = GpXX and it is quite possible that there could be additional terms with higher order derivatives appearing in the induced metric. This is still an unsolved problem. The equations of motion of the rigid string in a curved ambient space are given succinctly by the following expression (the full expression for them appears in the appendix of the geometry of immersions): 6I/6X 8m6I/6X,m + Dm8nl/6X'n = 0. (95) deSitter Static Solution and its Stability The classical motions of a closed string are considered in a two dimen sional deSitter space. de Sitter space is the maximally symmetric solution to Einstein's equation with scalar curvature R = 2a2. The three dimensional hyperboloid is coordinatized by r7ijYiyj = a2 where 77ij = (, +, +) embedded into R3. Now upon making the following transformations, the ambient met ric can be diagonalized YO = asinh(XO/a) Y1 = acosh(X0/a)cos X1 and Y2 = acosh(X0/a)sinX1. The range of X0 and X1 is respectively given by is oo < X0 < oo and 0 < X1 < 27r. Then the d.S. metric is d2s = (dX0)2 + a2 cosh2(XO/a)(dX')2. (96) Next the lab gauge is chosen where r = X0 and a = X1. Then the induced metric on the worldsheet is given by 1 0 (97) ab=( 0 a2 cosh2(/a) (97) The second fundamental form for this induced metric can be written as Kab = (9CGa pC tXXX + C.,pXX b). (98) This can be simplified to 'tab = 7T(CUtvr + Cpvr)X{aXv + 7OrCO V)X: Ka b = 777 3 b I ) b~y (99) 40 Then from (13) and the symmetry on the first two indices of the ambient space connection, it is simple to see that Kab = 0. Thus the static action for the rigid string reduces to S = aT d2z cosh(r/a) 1 2Ra2] (100) This deSitter geometry can only be a solution of the rigid string if the coupling constants in the theory are chosen such that 1 = 2R2a2, that is, if there is zero action. Note that in the lab gauge the second fundamental form had to vanish and the ambient space had to have constant positive curvature; otherwise the ambient curvature term could not have cancelled against the minimal surface term in (90). Now fluctuations about this deSitter background are considered. This is done by letting the radial variable a  a+ y, where y(r, o) and then expanding the action about its lowest energy state of T = 0. This is because all of the terms in the action grow as exp(r/a). Also transverse coordinates zP are added, and these directions are assumed to be flat in the ambient space. Then the perturbed worldsheet metric becomes (/1 + y2 + izy' yy' + iz' z gab Y' + Pz (a + y)2 cosh2(r/(a y)) (101) where i = ,rz and z' = Oaz. Now the fluctuations to the ambient curvature, the second fundamental form, and the square root of the determinant, are expanded out and the quadratic parts of the fluctuations are kept. Then the resulting action is I = Sa d2z[R2 2a2 + y/a(R2 2a2) 2y2/a4 +1/2(R' + 2/a2)(y'y'/a2 +2 + z z,/a2 (102) (y"/a2 )2 (z"/a2 )( ,))(/a2 Now the above action is Fourier transformed by x(r, c) = 1/27 f dw exp i(wr + no)Xn(o), n (103) Xn(w) = 1/2r J dr exp(iWr) f du exp(ino)x, so that the quadratic part of the action becomes I, = Sa f dw (lynI2w + n2/a2)2 n +1/2(R,2 + 2/a2)(w2 n2/2) 2/a4] (104) +lz12[(2 2/,2)2 + 1/2(R 2 + 2/a2)(w2 n2/2)]. Now the conditions for stability can be determined by requiring that the eigenfrequencies be real. Solving for the eigenfrequency of the radial and trans verse fluctuations, the following expressions are obtained: W2 = n2/a2 1/4(R02 + 2/a2) /2/a4 + 1/4(R2 + 2/a2)2, (105) 2zl = n n2/, 2 2/a2 1/2(Ro2 + 2a2). (106) The radial fluctuation (n = 0) eigenfrequencies are complex for all values of the parameters in the theory. The transverse (n = 0) eigenfrequencies have one zero mode and one complex mode that is also unstable. These instabilities just reflect the fact that the extrinsic curvature terms come into the action 42 with the wrong sign. The deSitter space acts as a source of negative energy density for the action. Ambient Space Fluctuations to the Hoop Solution In this section, we consider perturbations of the hoop solution of Curtright et al [2] about its Minkowski background by using Riemann normal coordinates. The ambient space metric in Riemann normal coordinates is given by G, = rlv, 1/3Rf X X 1/6DfR XXXXK +.... (107) This hoop solution also uses the lab gauge for the time, X0 = 7, and X1 = a. Then the unit radius vector of the hoop is given by i = (cos aC + sin ca) The worldsheet metric is given by the following expression when it is perturbed by the ambient curvature fluctuations and a radial perturbation, where a * a + y in just the same manner as it was done in the deSitter case except that a = 1: (Goo + Grry2 Grry' ab = Grr y' Grr((1+ y) + ') (108) The only nonflat fluctuations are assumed for G00, where G00 is given by Goo = 1 1/3R0101(1 y)2 + .... (109) The radial part of the ambient metric is assumed constant, Grr = 1. Only radial fluctuations are considered for the hoop solution, since the perturbations around the other coordinates were found to be stable by Braaten et al [33]. The quadratic part of this action is Iq = SA d2z[(y Ay")2 + M + Ny2 + Ly'y], (110) where the coefficients A, M, N, and L are respectively given by A = 1 1/3R, M = 1/2(1 RO2) R(5/2 RO2)+ 1/2R2, (111) N = 1 + R(4 5/3RO2) + R2(1/9RO2 + 37/6) + order (R3), (112) L = (1/2RO2 3/2A2 + 6A(1 R) + 1/2R2). (113) The quadratic part of the action is Fourier transformed, and the expression for w2 is given by =1/4(1 R2 R(5 1/3RO2) + 2 ) 1/2 /[1 RU2 R(5 1/3RO2) + R2]2 + 4(1 + R(4 5/3Ro2) + order (R2) (114) For w2 to be real requires that T/S = R 2 < 1 and that the scalar curva ture R should be negative. The dominant terms in the square root are 65/4 1/2RO2 + R(27/2 12/5R22) + o(R2). (115) To make wo real requires that the terms inside the square root must be zero. This cancellation forces R < 0. Since this is a weak background field, R can not be large without violating the assumption of a weak gravitational field. Therefore R can not be made large enough to cause the expression under the radical to vanish. Hence the rigid string perturbed about a Minkowski ambient space is still unstable. IS THE SUPERMEMBRANE UNSTABLE? It has been argued recently that the mass spectrum of the quantized super membrane is continuous from zero, and that smooth membranes are not stable against collapsing into fibrous configurations [17]. Thus, quantum supermem branes are to be regarded as flawed extensions of quantum superstrings. There are two logically independent features for the argument for the col lapse of the supermembrane into stringlike configurations. The first feature requires the implicit assumption that curvature terms do not appear in the effective action of the supermembranes. The second feature requires the as sumption that the mass spectrum of a quantum field theory may be obtained by first truncating the theory to a finite number of degrees of freedom, calcu lating the spectrum from the truncated theory, and then letting the number of degrees of freedom become infinite. Now, consider the first assumption, that the effective action does not con tain curvature terms. This assumption makes plausible the heuristic argument [17] that there is an energy barrier to prevent the collapse of any smooth mem brane configuration into a network of filaments with infinite extrinsic curvature, but zero area. This heuristic argument is probably correct with the given initial assumption. However, without proof that there are no such curvature terms appearing in the effective action, the argument is logically incomplete and does not establish that such a collapse can occur. Rather, it only focuses attention on the possibility that curvature terms could exist in the membrane effective action. 45 Such curvature terms have been considered [36]. In marked contrast to the assumption in de Wit et al [17], however, Curtright [36] has argued that curvature terms could indeed exist and maintain the local supersymmetry on the worldvolume swept out by the evolving membrane. The possible curvature terms were explicitly written down and shown to be locally supersymmetric to the lowest nontrivial order. Nevertheless, since no one has performed the requisite calculation to see whether such curvature terms actually arise in the membrane effective action, this issue is still open. Therefore, this first feature of the argument against quantum supermembranes is obviously not conclusive. The second feature of the argument against supermembranes is more quan titative. The supermembrane is first truncated to a finite number of degrees of freedom, essentially a SU(N) invariant quantum mechanics model [37]. Then it follows from an analysis of the resulting supersymmetric SU(N) model, that the spectrum is continuous; that is, the spectrum is essentially the same as it is for the classical limit of the theory. Since it holds for any finite N, it is claimed that the spectrum remains continuous in the limit of an infinite number of degrees of freedom, i.e., N + oo. However, it is well known that phase transitions [38] may occur to obviate such naive conclusions about the properties of SU(oo) quantum mechanical models. In the remainder of this chapter, we will not discuss any further the issue of curvature terms appearing in the effective action of membranes. Rather, we will explore the issue of finite versus infinite degrees of freedom in the quantum field theory arguments. We will examine some features of the finite versus infinite situation for a simple field theory model whose dynamical properties are very similar to those of the membrane theory. In the context of this model, we will argue that the quantum properties in the case of an infinite number 46 of degrees of freedom are indeed quite different from those of the truncated version with a finite number of degrees of freedom. We present evidence that phase transitions in such models are to be expected and lead to a quantum system whose properties are very remote from those of the classical limit. In this work we do not determine the properties of the quantum phase of supermembranes. This is still beyond reach. Nevertheless, we believe that our arguments indicate the possible directions for further study and suggest that supermembranes are still interesting and viable models. This chapter is organized in the following manner: In section one, we write down the quantum field theory model similar to the quantum mechanical supermembrane model and construct classical soliton solutions. In section two, we present two arguments to show that the field theory is unbounded from below when N + oo. Classical Solutions of the Canyon Potential Model Classically, the model we consider is given by the Lagrangian density L = 1/2[(X)2 + (aY)2] g2X2Y2 (116) and whose Hamiltonian density is H = 1/2[(9X)2 + (OY)2] + g2X2y2. (117) At this point we make contact with the membrane and the supermembrane Hamiltonian densities when written in the light cone gauge. Their respective Hamiltonians are given by 2m = = =[ Jd2a(Pa + V(X), (118) 47 where V(X) = 1/2(eabaaXVPbXp)2 and 27sm = Mm = da(pAaA + 1/4fABEfCEDXa Xb X (119) b ABa (119) i/2fABCXax A 7a . In (119) the indices A, B, are the SU(N) labels and the fABC are the SU(N) structure constants. Note the similar structure of the potential for all three models listed above and that M2 and M2m are the mass squared operators with the center of mass momentum excluded for the membrane and supermembrane respectively. Now we return to the canyon potential field theory model and examine the classical motions of this system. The equations of motion are given by OX = 2g2XY2, (120) DY = 2g2yX2, where we have chosen the metric to be goo = 1 = gaa We point out the following relation from the equations of motion: if one multiplies through the X equation of motion by X and the Y equation of motion by Y, then one obtains the identity that XOX = YOY. The physical meaning of this will be explained shortly. The above Lagrangian density can be recast into polar coordinates and this yields = 1/2[(OR)2 + R2(9)2] g2R4/8[1 cos 40]. (121) Note that for fixed R the similarity of the above equation to the Sine Gordon equation [39], where it is well known that soliton solutions exist. It is possible to construct a static soliton solution from (120) assuming that R is fixed. A solution to the 0 equation of motion can be obtained, but 48 it does not satisfy the R equation of motion. Both of the equations have the same functional form, but their arguments differ by a factor of V/2. Which is unfortunate because the equations described by the SineGordon solution exhibited the behaviour we desired. The desired behaviour, would be when one of the fields, say X(a = oo) = 1 and Y(a = oo) = 0 and then this system evolves to X(a = oo) = 0 and Y(a = oo) = 1. Physically, all this means is that the solitons can switch from one trough of a canyon to another trough of a canyon perpendicular to the one it started out in. This type of behaviour would show classically the soliton would not be restricted to moving along the same trough, which the analogy with the supermembrane would mean that, the supermembrane would not be constrained into evolving into long stringlike configurations. We were not able to construct a classical solution with this behaviour. We were able to construct one exact solution to (116) and it is a zero action solution. This solution found by setting = 0 is given by X = cosh(exp(aa)) + b), (122) Y = exp(aa). where a = gV/2 and b is arbitrary constant to be determined. We must choose the parameter b to be real for the solution to be stable. If the range of a is chosen to be 0 < a < oo, then the most physical choice of the parameter b, is to choose b = 1. Then the behaviour of the solution is as follows, X(0) = 1 and Y(0) = 1, then the solitons evolve to the configuration, X(oo) = 1.543 and Y(oo) = 0. The main thing to notice is that the X field does not go to zero, so that the soliton does not switch into a trough perpendicular to the trough it started out in. 49 The second solution that we discuss is when X = Y to obtain a solution for (114) we must add a mass term, then the solution is: 91(au) X = = (au) sn(au), (123) (4V o (au) where 0l(au) and 0o(au) are the Jacobi theta functions [40]. The argument of (123) is u = 2Ka/pi, where K is the quarter period of the expression ,12 9 K = d (124) J0 V/1 k2 sin2(8) and k is the modulus. The mass term that must be added for the above function to be a solution is m2 = (k2+ 1)a2 and the coupling constant g2 = k2a2. The function sn(au) behaves exactly like the usual sine function, that is it has a period of 4K and oscillates between 1. This solution corresponds to the particles being trapped between the maxima of the canyon potential. Quantum Instability of the Canyon Hamiltonian To define the quantum version of the model, it must be renormalized. Since this is a twodimensional spacetime, the model is superrenormalizable. Alternatively, for the quantum theory to be well defined, the Hamiltonian of the system must be free of ultraviolet divergences. This can be achieved by normal ordering the operators at some mass, say ip. 7I could be normal ordered at a different mass scale. Thus we have 'H = 1/2N,[(DX)2 + (0Y)2 + 2g2X2Y2], (125) In addition to removing the zero point energies of the fields, X and Y, which corresponds to normal ordering the kinetic terms, it is also necessary 50 to insure ultraviolet finiteness to normal order the interaction term X2y2. The theory given by 7i is now well defined, at least to any given order in perturbation theory. It is certainly a simple infinite degree of freedom version of the quantum mechanics [41] defined by H = 1/2[(X)2 + (Y)2 + 2g2X2y2]. (126) where X = OrX. The preceding facts are presumably well known. However, it is less well known that the above theory has an energy spectrum somewhat remote from that suggested by the finite degree of freedom model. In fact, the energy density is unbounded from below [42]. This may be shown by choosing an appropriate trial state and computing (H). For example, choose the state to be "coherent" so that (X) # 0 is constant over a large region, but (Y) = 0. In addition, choose the state to be "superfluidic" [43] for the Y field but not for the X field, so that (X)2 = (X2) but that (y2) < 0. The latter is possible because the normal ordered operator Np(Y2) is not positive definite. Now take the limit when (X)  oo. Then in that limit, (7) + oo. To go into a little more detail, let 10, p) be the vacuum annihilated by a Y field lowering operator of mass p. Then take an expectation value of N,(Y2) in a different vacuum, say 10, v). Then we obtain the well known result (0, INp(Y2)10, v) = (47r)1 ln(2/u2)(0, v0I, i), (127) since N,(Y2) = N_(Y2) + (47)1 ln(2/v2). (128) 51 Hence if we choose v2 > 12, then we can arrange for (7{)  oo as previously described. CONCLUSIONS The first two sections of this dissertation used Bicklund transformations to study the interacting theories in terms of free theories. For the simple potential models in chapter one we saw that the transformation had to be a similarity transformation and not a unitary transformation. This feature was consistent only because the Baicklund transformation naturally restricted the spectrum of the free theory, which normally would cause a lack of unitarity. However the Biicklund transformation simply projected out the negative energy angu lar solutions preserving the unitarity of the radial realization of the Virasoro algebra. In chapter two, we find that the functional Biicklund transformation is a solution to Liouville theory when we simply expand the nonzero modes and integrate term by term. We find that normal ordering divergences appear in the same manner that they do in the usual operator formulation. These divergences were cured by multiplying the nonzero mode interaction term by a factor of exp and this "normal ordered" the functional Backlund transformation to all orders of perturbation theory. This is because the only "normal ordering" divergences arose from the integration over the ) nonzero modes and not from the 0 nonzero modes. We next computed the correlation function exp gao and showed that the order g6 part was exactly the same as that of the correlation function computed by BCGT using Liouville quantum mechanics perturbed by the nonzero modes. Our calculation had a g4 contribution with a linear dependence on a, we showed that this could be consistently absorbed into a finite renormalization of the mass parameter appearing in the theory. To obtain 53 deeper insight into what this renormalization of the mass parameter means, we have to see if the wavefunctional we have found obeys the Virasoro algebra and compute the central charge of the Virasoro algebra. Once this calculation is carried out and properly understood there are many interesting questions to ask of this exact solution to Liouville theory. First, how does the Bicklund transformation project out the nonconformal parts of the free field functional. How is this solution related to the nonperturbative two dimensional quantum gravity results at genus one? The third chapter of this dissertation, examined closed rigid strings im mersed in a space of positive constant curvature and a Minkowski space with small ambient space curvature fluctuations were found to be unstable. When the twodimensional deSitter solution was perturbed with flat transverse co ordinates, these perturbations were found to unstable in marked contrast to results of of the hoop calculation by Braaten and Zachos. This means that a space of constant curvature only further destabilizes the rigid string. This has a simple physical explanation in this theory because the constant curvature acts like a source of negative energy density; thus, it drives the theory away from being stable. The ambient space curvature fluctuations about the hoop solution sup ports the previous comments, and suggests that an ambient space of negative curvature could classically stabilize the rigid string. This is reasonable if one considers the rigid string in the path integral formulation, and observes that the extrinsic curvature terms have an overall positive sign in the action. Integrating over the quadratic parts of the integral yields an integral that is divergent. We suggest that for a certain range of the rigidity and the string tension, a space of constant curvature might permit the rigid string to be classically stable. 54 A topic for future work would be to try to construct a solution to the rigid string in three dimensional antideSitter space. However, solving (A29) appears to be a highly nontrivial calculation. If a classical solution could be found it would be interesting to first see if it could be stable classically, and then see if it were stable with respect to quantum corrections. This is because higher derivative theories are notorious for being difficult to quantize. It would be interesting to compute the first loop corrections in a curved am bient space whose metric is Euclidean, to study how the ambient space curva ture affects the asymptotically free behavior of rigid string found by Polyakov. The last topic of this work concerns, a two dimensional field theory whose potential is quite similar to that of the supermembrane quantum mechanical model. We showed that by a renormalization group calculation that the Hamil tonian was unbounded from below, and that a system with a finite number of degrees of freedom can act rather differently when the number of degrees is allowed to become infinite, that is, the system can undergo a phase transition. Therefore the two different phases of the theory can have radically different behavior. So we make the point that the conclusion of De Wit et al is not conclusive, and that supermembranes are not ruled out as being quantum me chanically consistent theories. APPENDIX A: COMPUTATION OF THE RADIAL EIGENFUNCTIONS Now we show that there are no branch cuts for the Backlund transforma tion appearing in chapter one of this dissertation. By making the following transformation z = exp(i9), the expression for the radial wavefunction be comes ,0(r) = iNz(2i)(A+1)/2 dz/z(1 z)(A+1)/2rA/2 exp (r2(1+ z)/8(l z)). (Al) There is a simple pole at z = 0 and a branch cut at z = 1; however if the contour is distorted so that only the pole at z = 0 contributes to the integral. That is, if the contour only passes around the pole at z = 0 and avoids the point z = 1 the expression for the radial eigenfunctions are well defined. Then the radial wavefunction is given by b0(r) = 27r(2i)(A+)/2NzrA/2 exp (r2/8). (A2) Now acting with Lo on the first expression, the integral representation of the radial wavefunction is given by LOo(r) = iNz [ dzz[Eo(l + z)/(l z) r2z/4(z )2] Jc (A3) (1 z)(A+1)/2rA/2 exp(r2(1 + z)/8(1 z)) where N' is equal to N(2i)(A+1)/2 Now the above expression for L0 is replaced by E0 + 10. In (12) we used the expression of LO acting on exp iF*(r, 0) instead of acting on exp iF(r, 0). 55 56 With this in mind LO is replaced by EO + l0 in the integral representation. Where E0 = (A + 1)/4 and 10 = izdz. Then equation (A3) becomes LoV'o(r) = iN' dzzl(izz + E0)(1 z)(A+)/2 A/2 exp r2(1+ z)/8(1 z), (A4) which integrated by parts, it is simple to see that the first term when acting on exp iF(r, z) has no poles or branch cuts in the contour that we have chosen, and that the second term acting on exp iF(r, z) has a simple pole at the origin and has the proper energy eigenvalue of E0. The next topic is to determine the normalization of both the radial wave functions and the proportionality constant of the integral representation Nz. To determine the normalizations of the radial wavefunctions one must first find the proper measure. This is found by constructing o0(r) and 01(r) and demanding that they are orthogonal to each other when they are integrated over from 0 to oo. The expressions for the groundstate and the first excited state two wavefunctions are given by Co(r) =norA/2 exp r2/8 01(r) =nl(E0 r2/8)rA/2 exp r2/8. (A5) 02(r) =n2[E0(2E0 + 2) (2E0 + 1/2)r2 + r4/16]rA/2 exp r2/8. These were found from solving the differential equation for the radial vari able to construct the ground state eigenfunction and using L_1 to generate the next two higher energy wavefunctions. When this is carried out the proper measure is 1. The normalization for 0 and i1 are given respectively by no =2A/2(r((A + 1)/2)1/2 (A6) n1 =2(A+1)/2(F((A + 1)/2)(Eo + 1))1/2(A6) Now using equation (A2) and the above value for n0 is given by we can deter mine the coefficient appearing in (A4) Nz = 1/(27)(i)(A+1)/22(3A+2)/4(r(A + 1)/2)1/2. (A7) Now the nth level radial wavefunction can be computed from the following expression On(r) = iNz dzz(n+l)(1 z)(A+1)/2r/2 exp r2(1 + z)/8(1 z). (A8) This expression can be easily computed by doing a contour integral, and we obtain the following result: On(r) = 2rN'r/2z[(1 z)(A+)/2exp r2(1 + z)/8(1 z)]=0/n! (A9) Evaluating this expression gives the exact same form of the first three wavefunctions as given in (A5) . APPENDIX B: LIOUVILLE THEORY DETAILS In this appendix we will go through some of the details needed to calculate the Liouville wavefunctional to order exp 3gq and how to compute the nonzero mode matrix elements for both the 4 and < fields. We start with the expressions (71), (72), (73) and (74) and then integrate out the 4 nonzero modes. The generic integration will involve expressions of the form Pn = 1/21014 + i$'" + g(61 62 6 n) (B1) now we change variables to n = +( 8 l 91a^+g11[S1 2* * n]. (B2) Then (Bl) becomes Pn =,n lal9n g(p1 p n) (B3) g2(i 6n ,)11(i Sn). ( Where p+ =(1 + i; ) P+ =(1  ), S=(1 (B4) 4,n =4(pn) Sn =8(o pn). For concreteness we choose the following example of the linear terms in 4 are of the form 41 22. Then (B3) becomes P2 = \Q\4 9(pS1 + P62) (B5) 92(S~ \2) 61(S1 S2). Where it is understood that the operators p+ and p only act on the various delta functions and not upon the fields 0(ar). Before calculating (B5), we list some helpful identities that will be used to evaluate the rest of the nonzero mode matrix elements: 00 6n = 1/(27r) E (exp +im(cr Pn) + exp in(a Pn), m=l 00 P6n = 1/7r E expFi(a Pn), m=l 00 [oln = 1/(27r) E 1/m(exp i(a pm) + exp i(a rhon). (B6) m=l Using the results of (B6) in (B5) and integrating over a yields j daP2 = 0 da(laO g(p1 + Pi2)) oo (B7) + g2 1/n(cos n(pl P2) + 92/7r1. n=1 Now we use the above results to carry out the integration for the various nonzero modes matrix modes of the field i occurring in (68), (69), (70) and (71) with the result: ( = NoKik()exp 1/2 2 d ld8, (B8) l() = Nlexp 1/2 jf daO[i0O I dpl[Kl+k(x)(exp g(/ da(p+1(a)) 1) Kl+ik(x)((exp g(J p6Si(a)) 1)], (B9) 2(2) =1/2N2exp1/2 f27 d4a9\ 4 J dpldp2[ 00 (expg 2/x 1/n cosn(pl p2) exp g J da(p+ 1(c) + p+(a2q(')) n=l exp g J d(p+l {(a)) exp g J du(p+p2(a)) + 1)K2+ik+ (exp g2/7r 1/n cosn(pl P2) expg J da(pS1j(c') + P2'(a)) n=l expg J dc(p81(o)) expg J da(p_2(a)) + l)_2+ik + (exp g 2/7r 1/n cos n(pl P2) exp g J da(p+r ((ao) + P 2'(7)) n=l + (exp g2/r 1 1/n cos n(pl p2) exp g f do(p+8b2 t() + p 51()) n=l exp g J do(p+1 (oa)) exp g f do(p+624(o)) expg J da(p S (u)) exp g J d(p 2((o)) + 1)Kik(x)], (B10) #(3) = 1/6N3exp 1/2 2f dojai/ /dP dP2dp3[ 00 00 + (exp g2/ir 1/n cos n(p p2)exp g2/r 1/n cos n(pi P2) n=l n=l 00 exp g2/ I 1 (/n cos n(p2 P3) n=l expg J da(p+Sij(u) + p+A2 () + p+364(U) 00 exp g2/7 1/n cosn(pl P2) expg dco(p+iS4 (cr) + p+ 2q(o)) n= 1 00 expg2 /, 7 1/n cosn(p1 P2) expg do7(p4il (or) +p+A3q$(a)) n=l1 61 00 / exp g2/ 1/n cos n(P2 P3) exp g dc (p+ 2(cr) + p+ 3(o)) n=l + exp g(J dp+61(oa) l)KJ3+ik 00 00 (exp g2/r 1/n cos n(pl P2)exp g2/ir 13 1/n cos n(pl P2) n=l n=l 00 exp g2/7r E 1/n cos n(p2 P3) n=l exp g J d7(p S1 ) + P62 (() + P 3()) 00 exp g2/r y 1/n cos n(pl P2) exp g do(p16(a) + pA2(a) n=l 00 exp g2/r 1/n cos n(pi P2) exp g dor(p^i5(cr) + p83(o) n=l 00 exp g2/7r l/n cos n(P2 P3) exp g do(p2q(o) + p3A(o) n=1 + exp g J do(p1(1 (a)) + exp g J do(p 6 (cr)) + exp g do(p63b(a)) 1)K3+ik(x) 00 00 + (exp g2/7r /ncosn(pl p2)exp g2/7r 1 /ncosn(pl P2) n=1 n=1 00 exp g2/r E 1/n cos n(p2 P3) n=l exp g J da(p+61qi(o) + P+62 ()) + PS63(o)) oo exp g2/r 1/n cos n(pl p2) expg I dc(p+1 (Sa) + p+3a(~)) n=l 00 exp g2/rr 1/n cos n(pl p3) exp g J da(p+ S1(o)) + pS3(o)) exp g2/7r 1/n cosn(P2 P3) expg da(p+ 2 (O) P 3 n=1 + exp g d(p+l q(a)) + exp g J da(p+$2^(a)) + exp g J du(p_3(o)) 1 + perm(p3 P2) + perm(pl P3))Kl+ik 00 00 + (exp g2/ir E 1/n cos n(pl p2)exp g2 /x 1/n cos n(pl P3) n=l n=l 00 exp g2/7r E 1/n cos n(p2 P3) n=1 exp g J da(p i1(o) + p 2 b(o) + p+63(o)) oo exp g2/7 1/n cos(n(pl p2) exp g d(p51( ) + p2 a)) n=l exp g2/r 1/n cosn(pl P3) exp g do(pi5^5(a) + p4+3 S(3)) n=1 exp g2/ir E 1/n cos n(P2 P3) exp g f d(pS2(cr) + p+A6q()) n=l + exp g do(p61 ()) + exp g(j da(pP22(u)) + expg Jdcri(p+63^o() + 1 + perm(pl + P3) + perm(p2 + P3))Kl+ik]. (B11) Where the prefactors appearing in the various wavefunctionals were defined by Nn = An exp ngq and the An were defined in chapter two. Now we carry out the evaluation of the nonzero mode matrix elements oc curring in (72). The first matrix element we consider is (I(0)(k')jexp agq(0)l4(O)(k)) and it is of the form f27r f D exp 0 do(q jq ga4o), (B12) Where each wavefunctional contributes a factor of exp 1/2 f r" dao^O al to the evaluation of (72). A change of variables is made to complete the square by shifting (  4 S= $ ga/21a1'o, (B13) now this is inserted into the function appearing in the exponential of (B12) and the result is: ilal g2a2/46aIDo1 = ~jlal gab06. (B14) The left hand side of (B14) is put into (B12) and evaluating the a integral yields, g2 2/4/ do ollo = g2 2(1/4. (B15) Again the usual "normal ordering" divergence shows up that appears in the operator formulation of exp gaf(0). This infinite term will be divided out of the correlation function. Using the results of (B15) and (B14) (B12) becomes SDe exp + dojoa5expg a2(1/(47r). (B16) From (B16) there will be an infinite term from integrating this Gaussian integral over all the infinite number of nonzero modes. This term will be canceled by the normalization of the wavefunctional. These terms will show up in every matrix element that we compute and we will not bother with them from here on. This procedure of shifting the field q will be repeated over and over again when evaluating the nonzero mode integration of (72). Therefore, we will simply write down the original expression appearing in the various matrix elements and what it becomes after shifting variables to complete the square. 64 The next matrix elements we consider are of order exp g(a + 1)q. These are the terms are given by (4(1)(k') exp ag((0) \(0)(k)) and (.(0)(k') exp ago(O)j'(1)(k)) and their nonzero mode matrix elements are of the form I dp1(exp ,r doa[j(](exp g(pi6a8) 1). (B17) So we make the following change of variables + = g/2111(p S + aSo). (B18) Then (B17) becomes = d exp[/4 dpa(p21 + ao0)1K1(pSi + &eo)] 1). (B19) 0 0 There is an overall factor of f Dq exp fd 19810 multiplying (B19). Now to compute the expression appearing in the exponential of (B18), we use the identities in (B6) and the fact that fS2 da exp i(n m)a = 6n,m to obtain 02 7r co00 Sda[(pi6 + ao)ll1 (p + a o)] = 4/7r 1/nexp inpl + ga2L/ . n=1 (B20) Now the expression appearing in (B20) is inserted into (B18) and then expanded in a power series in g2, the result is I= dpl([1 + b 1/mexpinp + ..+ m=1 00 00 bn/mn! .. 1/(ml mn) exp fi(ml + + mn)pl] 1), m=l mn=l (B21) 65 and b = g2/(27r). Only the first term is nonzero because all integrals have either all positive or negative frequencies in the exponential, therefore they integrate to zero. All of the matrix elements of order exp(a + 1)gq are of this form and they are zero. This is simple to understand because the term appearing in the exponent for these nonzero matrix elements is either a positive or negative projection operator and only the positive frequencies or the negative frequencies are left after the a integration has been done. Where the frequencies are the positive integers. The next order in the calculation of the matrix elements of (72) involves terms of the type exp g(a+l)q. The terms to evaluate are (D(l)(k') exp agO(0)O)4()(k)) , ((O)(k')lexp ago(0))fQ(2)(k)) and and the previous term's complex conjugate. Considering the first term in the above list, its nonzero mode structure is of the form, I1, = JJ dp/dp2 Deexp J ddaPO9l[exp dog(pb 1 + aSo)4(a)) 1] [expJ dg(pTS2 + o) ()) 1]. (B22) Where it is to be understood that the integrals over the variables a, Pn are from 0 to 27r. The case that could possibly be nonzero is when the above product involves a p+ and a p_. This follows from what we observed in the evaluation of the matrix elements of order expg(a + 1)q strings of only p+ or p project out all the opposite frequency states and then there power series expansion is observed to trivially integrate term by term to zero. Considering a term with a p+ and p and examining only the exponential term with both p+ and p present because terms with a single p are equal to one after being evaluated, II+ = 41al g(p+Al + PS2 + O0)q. (B23) We change variables as before and obtain II+ = i101 g2/4(p+b1 + p 2 + ao)9\1(P+S +P62 + a6o) (B24) Now the II+,_ is integrated over a and the result is dall, = b 1/n(2 expin(p 2) + (exp inp + exp in2). (B25) n=1 We have suppressed the "normal ordering" term coming from the function exp gaf(0). (B25) is inserted back into the exponential expanded and inte grated over pl and P2. The integrated expansion of (B25) will contain only positive frequencies of pl and negative frequencies involving P2 and therefore they all integrate to zero except for the leading term in the expansion. When (B22) has been integrated it is zero because only the 1 survive the angular integration and they cancel against each other. Similarly, the term with p+ and p_ switched only flips the sign on the frequency multiplying the variables pl and P2 and it also vanishes. Next we evaluate terms of the type (4(O)(k')(expag (O)[^(2)(k)). There will be two different types of terms will occur that have not been encountered before. These terms are of the form 12 = J dpdpd2 J D exp J da(4,9\j g(p5i + P62 + Ao0)) x exp b 2/n cos n(p P2), n=l 67 I,q= = J dpdP2 f exp Jda(19i\i g(pil +PT2 + Ao)) oo (B26) x exp b E 2/n cos n(pi P2). n=1 As usual we complete the square on both terms which yields: 2 = f dpidP2expa2, (B27) I = / dpdp2 expb2, where the functions a2 and bi2 are defined respectively by J2w a2 = 2/4 do[(pSil + PA?2 + a0o)l&lj(pSl + P:2 + a6o)] (B28) + g2/ E n1 cos n(p2 P1), n=1 and 2r b2 g=92/4 da[(Pi1 + PFT2 + Ao60)(I11(pt + PFS2 + 60)] (B29) g2/7 n1 cos n(p2 Pl). n=1 Now evaluating the a integrals in appearing in (B28)and (B29), then (B28) becomes 27r M2 =g2/4 da[(pilt + PA2 b+ o60)IO1(Pi + P62 + a6)] o oo (B30) M2 =b 1/n(2(exp b E a/n(exp minp + exp inP2). n=l n=1 Note that only positive or negative frequencies appear in the exponential terms multiplied by a. Similarly, computing (B29) yields M2 =g2/4 do[(p1 + PT62 + abo)IPal '1( + P62 + (30)] oo 0 (B31) M2 =b E o/n(exp inpi + exp finP2). n=1 Finally we have the expressions for .2 and I ready to be integrated over P1 and P2, 2 = dpldp2 exp b 1/n(cos n(pi P2) + a[exp inpl + exp inp2]), n=1 (B32) 00 I+ = dpIdp2 exp b 1/n(sin n(pl P2) + a[exp inpl + exp FinP2]). n=1 (B33) To facilitate the evaluation of these integrals, it is simpler to expand out all the exponentials independently. With this in mind the expansion of (B32) becomes 2 = dpidP2[ 00 00 (1 + b /nexpin(p P2) + b2/2 Y l/nmexpi(n + m)(p + p2) + ) x n=1 n,m=l 00 00 (1 + l1/n exp in(p1 P2) + b2/2 l1/nm exp i(n + m)(pl P2) + ) x n=l n,m=l 00 00 (1 + ab 1/n exp finpl + 2b2/2 1/nmexp i(n + m)pl +..)x n=l n,m=l (1 + ab 1/n exp finp2 + +a2b2/2 1/nm exp i(n + m)p2 + ). n=l n,m=l (B34) 69 Now we perform the angular integration and list only the nonzero parts of (B34) to order g6 which is of order b3, the result is oo oo 12 = (2r)2(1 + b2 E 1/nm6n,m + b2 1/nml(61,n+m). (B35) n,m=l n,m,l=l These sums are evaluated by using the Riemann Zeta functions, which are defined by E'1 1/(nP) = (p and Em=l 1/(nm[n + m]) = 2(3. Now (B35) can be expressed as I2 = (27r)2(1 + b2 + 2b3C3 + terms of order b4. (B36) Notice that there is no a dependence in this expression because the frequen cies were either all positive or negative in the exponentials that were multiplied by a and that the sums are over the positive integers. So all of these integrate to zero. Now we quote the results for Il: using the same techniques that were used to evaluate 442, I=F = (27r)2(1 b22 + a2b3(3 + terms of order b4 (B37) Now making contact with chapter two where we have defined the various nonzero mode matrix elements, Md =0, M2 =(I2(1,2) I(1) 1(2) (27)2), M2 =(27r)2((g2/(27r))22 + 2(g2/(27r))3(3), (B38) M:F =(I:(1, 2) 1(1) I:F(2) (2r)2), M =(2x)2 ((g2/(2))2(2 + a2 ~/(27))3(3). 70 Now we move on to evaluate nonzero mode matrix elements of order exp g(a+ 3)q. The terms that we consider first are of the type (0(1)(k')\exp agO(0O)l1(2)(k)) and its complex conjugate. We show that one of the particular matrix element is zero that the same mechanism will apply to the other matrix elements of this form and that they are also zero. Now we consider the following expression M_1,2(1,2,3) = dpldp2dp3 J D exp da(^1015 a6~) [exp J dagpS3 1][exp J drg(p+ 1 + P+6S2) oo (B39) x exp b 2/n cos n(pi P2) n=l exp dagp+165 exp f dagp+862 + 1]. So we multiply out all of these terms, complete the square and perform the a integral and the result is 00 M1,2(1,2,3) = /dpiddp[dps[expb 1/n(2expin(p1 P3) + 2expin(p2 P3))x n=l exp ab E 1/n[exp inP1 + exp inp2 + exp inp3] n=l 00 x exp b 2/n cos n(pI P2) n=l exp b 1/n(a[exp inp1 + exp inp2] x n=l exp b > 2/n cos n(pI P2). n=1 (B40) Note that the other six terms had been previously evaluated in showing that the matrix element (4(1)(k')Jexp agO(0) I(1)(k)) vanished. We expand the ex ponentials appearing in (B40) and make the observation that all the exponen tials in the first term of (B40) have negative frequencies multiplying p3. All of 71 these expressions will integrate to zero and the only term that will contribute to the first term in (B40) is the cosine term and similarly the a dependent terms in the second exponential have positive frequencies and will integrate to zero. The only term that contributes to the second exponential is the cosine term again but these two exponential terms have a relative minus sign between them and cancel each other out. The other terms of this order are also of this form, so that there is always an uncorrelated pn which will not contribute once the pn integrals have been performed. The parts of the exponentials that are nonzero always come in with a relative minus sign between them and cancel. There fore matrix elements of the form (I(1)(k')lexpcagO()Jl(1)(k)) are identically zero. This lack of correlation occurs because the two different wavefunctionals Sintegrals were evaluated independently of each other. Now we consider terms of the form (4(0)(k'lexp agO(0),,(3)(k)) now that we have established how the various nonzero mode matrix elements arise and are evaluated, we will now just write the expression and quote the results: M3 = ddpldp2dp3[exp b 1/n(a exp inp + exp inp2 + exp inp3) n=l exp b : 2/n(cos n(pl P2) + cos n(p1 P3) + cos n(p2 P3)) n=1 (27r)(I2(1, 3) + 2(2,3) + I12(1,2)) + (27r)2(+ (1) + (2) + +(3)) (27,3] M3 =2g6(3, (B41) 00 M++ (1,2, 3)= /dpldp2dp3[exp ab 1/n(expinpl + exp inP2 + exp inp3) n=1 x expb b 2/n(cosn(p1 rho2) + sinn(p1 P3) + sinn(p2 P3)) n=1 (27r)(I+ (1, 3) + +(2,3) + 1++(1,2)) + (27r)2(I+(1) + I+(2) + 1(3) (27r)3 + perms(pl p3) + perm(p2 P3)] M++(1, 2, 3) = 6g6'3, (B42) and 00oo M+(1, 2,3) = dpldp2dp3[exp ab E 1/n(exp inpl + exp inP2 + exp inp3) n=1 x exp b 2/n(cosn(pl rho2) + sinn(pl P3) + sin n(2 P3)) n=l (27r)(I_+(1, 3) + +(2, 3) + (1, 2)) + (27r)2(I_(1) + (2) + +(3) (27r)3 (B43) + perms(pl + p3) + perm(p2 + P3)] M_+(1,2,3)= 6g 63. Where we have used the expressions for the various Is which have been defined earlier in the appendix. APPENDIX C: GEOMETRY OF IMMERSIONS In general one can consider a map from a n dimensional manifold into an N dimensional manifold 0 : Mn  MN. If 0 is C' and locally a one to one mapping then 01 : 0(8) + Mn for any open neighborhood 0 of a point in Mn. Then 0 is an immersion. However if 4 is a one to one mapping for all of the manifold Mn then 0 is an embedding. Immersions allow for the case of intersections while embeddings do not. For the case that we are considering Mn is the immersed or the embedded submanifold and MN is the ambient or the embedding manifold. N n is the codimension of Mn in MN. The coordinates of Mn are given by ya ,a = 1, , n and the coordinates on MN are given by X/, P = 1, , N. We assume that the ambient space metric on MN is G1v so that the invariant line element on MN is given by dS2 = GpvddX dXv. This causes an induced metric on Mn. Then the immersed manifold's invariant line element is given by ds2 = GydXudX'" = GXXbdyadyy (Cl) Where X = 5 and where the manifold MN is restricted to the image points of Mn. Then the expression for the induced metric gab(y) is given by d2s = gabdyadyb (C2) gab = G Xal X (C3) This expression is bilinear in the tangent vectors, where it is understood that XP(y) is restricted to the submanifold by the map I : fn" _, N. The 73 74 XP/ is a set of N scalar functions on M" and Xa are vectors on the submanifold M". Since the codimension of the submanifold Mn in the ambient space MN is N n, then there will be N n vectors which are orthogonal to the XI, normalized to unity, are given by the following expressions: GpXfi9Y = 0 Guvz = 67 j (C4) i,j = 1,, N n. The functions Guv and 71i are tensor functions in the ambient space but are scalars functions when restricted to the submanifold Mn. Therefore G~, and 77 are invariant under the reparametrizations of Mn. Now the GaussWeingarten and the GaussCodazziMainardi equations will be derived. Next make the identification that Xa = DaX" where Da is the covariant derivative of the submanifold Mn. Then covariantly differentiate the induced metric gab using Da to obtain the following Dcgab = agbc abgcd cbd (C5) rabc = 1/2(aagbc + abac Ocgab). Where tabc is the affine connection on Mn, gab = G yXX, and rabc is given by rabc = 1/2(Oa(GT X 'X) + Ob(G7aXjcX,) aOc(GaXpXX)) 5c b c ax a (cb) rabc = 1/2((Gyp,a + Gtp Ga,)XaXX + 2G Xf) (C6) Fabc = CaXyaXX X, + GapX Xc, where we have used the fact that DaGpv = X a Gpv (C7) OaxA 75 and Ca#7 is the affine connection on the ambient space and Xa = abX, X,ab bX that is, the comma just denotes ordinary differentiation. Now covariantly differentiating gab yields Dagbc = DaGvXbX"c + GLv[(DaXb)X + X (DaXv]. (C8) Now forming the combination Dbgac + Dcgba Dagbc = 0 and using (C7) yields the following Gv,Xa DbX + CaX'X X" = 0 (C9) and this can be rewritten as GvX, (DbX c + C XXfc) = 0. (C10) The expression inside the parenthesis is a vector that is orthogonal to the subspace M", therefore a vector perpendicular to the subspace M" without altering (C10). This new term that we add to (C10) is denoted by the following expression K]b. Then K'b is defined by DaX + X X" C = K' = K (C11) Kib = Gpv7zi(DaX1 + Ct XX ,ab) where the second line of the above equation has been projected out. It is simple to see that Kab = Kba by observing the symmetry in F'b from (A6) and since Capy is symmetric in a and P. These expressions are known as the Gauss formulas and Kab is called the second fundamental form. Then there are N n such of these symmetric second rank tensors on the submanifold .1In when the codimension is N n. 76 Now covariantly differentiating the other two equations that define the subspace Mn in (C4) to obtain the following Da(GvX u) = 0 (C12) GvDaXT~Y = G/tv,17a (C~,v + Cvpj)X aXbi7i . Now (C10) is used to plug into the left hand side of (C12) which yields Gy i({Kab XaX Cp) = GOX ,a (C/P, + C,,^P)X, X,aXb ia ,b a / X/1 L ,#bz Kab = G,,vX ,la CvzXV"aXri . (C13) Where we have used the identity G/tv,7 = Cyyv, + Cvy,7 in the previous expres sion. Now covariantly differentiating the expression Gpvii5fr' = bij yields the following result G,aX e j + Gtv(r,7 e, + 7) = 0 (C14) Again using the same identity that was used in (C13) the above expression can be rewritten as (Cp,, + CVpX7ar7ij + Gpv(77aj1 + 7 ja = 0. (C15) Which yields two equations of the same form which is given by 77(G,,,a + CypX, ) = 0 (C16) but whose normals are pointing in different directions. This equation can be rewritten in the following form Gvrl u"(?lia + C/ XA 9fl) = 0. (C17) 77 Now a vector orthogonal to Gyvqri can be added to (C17) without changing the result of (C16). This new term will be denoted by AX'b and then (C16) changes to 77a + C X ",1' = AX. (C18) Now (C18) is inserted into (C13) to express Ab in terms of the second fundamental form Kiab. Then the result is AC = gbKab. Then (C18) becomes o = gbKiabXI C(,XCC. (C19) Now we want to find the conditions for integrability of DaDbXP, to do this we make use of the Ricci identity [Da, Db]X, = Xc gef Rfabc. (C20) Where Rfabc is the Riemann tensor on the submanifold Mn which is con structed out of the gab We use the result previously derived in (C11) and express DbXc in terms of the second fundamental form and the ambient space connection multiplied by the appropriate tangent vectors. Then we covariantly differentiate with with respect to Da and obtain the following result D,,DbDXP = Da(K c?" CapX X, ) D,,DbDcs = (DaKIc), + ab + [(DaX )Xc + X,(DaXc)]CJ + XO'XDaCp. (C21) Now ,we m~ik use of (C19) along with (C11) to obtain the following ex KD a(bef1) X" + C 7 DaDbD ( = (Da. Ke'(ef Kia fX, + Cf ) +(a + KP X)Cp XX c 0C p (C22) XXXbCC + X X. b cC XOXOX"CP CI X"XAXP ,a ,C ,b 0 Ap a CM Apy Now antisymmetrizing on the a and b indices on the submanifold yields [Da, Db]Xp = (DaKib Dbiac)7ii + X gef (IacKJe KibcK e)6ij (C23) +a/37 a,b ,C* Where RA is the Riemann tensor in the ambient space MN, that is formed from Guv. Where S= GpA(Ca,/ C ,a + CC CjC p). (C24) where the same equation was used on the right hand side of (C20) but is constructed out of the subspace connections cab. Now we equate (C20) with (C23) which yields (gac l hcj abb+jgefXu lli X gefRfcba = (bKc Kh i ab+igefX + (Kc;a K,;b) (C25) +RAf Xca XX c37 ,c,b ,a" Where the semicolon denotes covariant differentiation on any tensor it acts on. Now the tangent and orthogonal components of (C23) can be projected out respectively by multiplying by GiXd to yield Rdcba =(KbKic K cd'Ib)bij + Xcfl ,b ,a (C26) and by multiplying by 77' to obtain (Kca Kc,) Rap X"flX (C27) bc;a  I ac;b )ij  Y c ,b 1Q7. 79 For the ambient spacetimes that we are considering it is simple to see that the second fundamental forms are integrable, that is (C27) is zero. Since deSitter space is a space of constant curvature, its Riemann tensor can be written as A RIva'3 = a0(GuaGv, GpGva) (C28) When (C28) is inserted into (C27) it is zero because of the first expression in (C4). Next, we show explicitly what the equations of motion are for the rigid string immersed in an ambient gravitational field. Where we have assumed that gb = GX ,X X. We vary the action which is given by I = T d2zfg [l R2((gabKab)2 RA XaxflX X X6gabgcd)] (C28) ay6 ,ay ,e ,b 9. with respect to 6XI to obtain the following equation. 6l/SXA 8m6I/6Xm + an9n6I/6Xn =0. S= (g)1/22C/, (R2 GK r bv ab ab R X ab cd (g)12 abgab (GTA b GI (gcdXc[2C(GaX, bXd)+ CL+XC,XX Xd + Gap,AXabX + C ,AX, Xd + C A) cb Id Id 0 ,f ,d a G pvK"Xb (g ,c[2CsA(GpXabX + C#XaX XXd)+ Gap,AX bX) + Capy,AXX XYX ] + C A XOX) + 4GpvKPbc"C (g)1/2gab cdA vXaX 4RA p X vX  gs)'/2s~bs~ g g k l lx~a' ,b ,d .vaca ,ca.,c,b",d' OA  am [((g)12 oGa)(XoGuXX XG,,X X) + 1 mG(X GyXXv X GX 0X 1)) [(g)(R2 Gr'P K.vbg ab RA QX X" VX 0a p. iB,a ,c ,b d 2GKv ab abgcd 1(GaX bX + Cy XX Xd)X 2GpyvK babgab gd g1(GapXbXd + Ca3yXaXX ,dX)Xc P _a abgl + Afl Xacd) 4G,, Kal i ab 9abb X1 + RA x'3XYXVXO ab cd9 [Xciab + X,"Kab]g 1G vG, (c XbX, + dm a)(GX, C X,aX,d)  G gab ab amCpX" + 6mCQX,)(GO. + G+OKb) RA 9 ab cdcmcnxpXv + .rnx 7 ,u Y ' 'v'f YrnA' v l l/ t Y^ Y a A ,c ,bX,d + c "A ,a ,d A ,a ,C ,d A d ,a ,c, R X11 X11 aXfXX 6cmbXX, + +gdm WXX+gX A ..XaX, X gd c Gax, bmXgab A(cmX +a a )] +,m9,(g)1/2G,,bbmg g ab abtl cdGpX Xzlt) g gcdG cpXIX iK . + a ((br 9 GAXX + (b gcd G#'^x )K) (C29) The last calculation of this appendix is the variation of the action with respect to the subspace metric gab, to show the highly nontrivial equation that must be solved to compute the induced metric in terms of the tangent vectors XPa, the ambient space metric Gy, and the other derivatives of these fields. 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Phys. 164 (1985) 189; M. Claudson and M.B. Halpern, Nucl. Phys. B 250 (1985) 689. 42) T.L. Curtright, (1976) unpublished. 43) D.J. Amit, Y.Y. Goldschmidt, and G. Grinstein, J. Phys. A 13 (1980) 585. BIOGRAPHICAL SKETCH Thomas McCarty was born at Ramey Air Force Base, Puerto Rico, on January 6, 1957. After graduating from South Dade High School, he attended the University of Florida and attained an undergraduate degree in nuclear engineering in 1980. He started in the Physics Department at the University of Florida in August of 1981. He was attracted to the Physics Department at the University of Florida following the formation there of a particle theory.group in 1980. He started to do research under the tutelage of Assistant Professor Thomas L. Curtright in 1984. His research, as that of his advisor, has been wide and varied. His current work is on understanding higher spin massless fields and their conformal properties in 2 and 3 spacetime dimensions. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierr amond Professor of Physics, Chairman I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard D. Field Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. es N. Fry sociate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 1A* Richard Woodard Assistant Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ulrich H. Kurzweg Professor of Aerospace Engineering, Mechanics, and Engineering Science This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1990 a~tA_ _< Dean, Graduate school UNIVERSITY OF FLORIDA 11111111 1262 085 811 11111111311 0 111130 3 1262 08553 8303 