Classical and Quantum Fields in Brane Worlds

Transcription

1 Classical and Quantum Fields in Brane Worlds Alan Knapman A Thesis Submitted to the University of Newcastle Upon Tyne for the Degree of Doctor of Philosophy School of Mathematics & Statistics Newcastle Upon Tyne, England December 2004

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3 Acknowledgments I would like to acknowledge my supervisor, David Toms, my internal examiner and head of the Newcastle group Ian Moss, and my external examiner Stuart Dowker. Thanks also to Misao Sasaki, for collaboration on our latest research along with Wade Naylor and Nino Flachi and for successfully arranging funding for a visit to Kyoto University. i

4 During the completion of this work, I have jointly published and collaborated in the preparation of the following: A. Knapman and D. J. Toms, Stress-energy tensor for a quantised bulk scalar field in the Randall-Sundrum brane model, Physical Review D 69 (2004) ; A. Knapman and D. J. Toms, Non-constant ground states for symmetrybreaking fields in brane world models, Physics Letters B 594 (2004) 213; A. Flachi, A. Knapman, W. Naylor and M. Sasaki, Zeta functions in brane world cosmology, arxiv:hep-th/ , to be published in Physical Review D. These articles form the basis of most of this thesis. ii

5 Abstract We discuss various aspects of the behaviour of classical and quantum fields in brane world models, with applications to particle physics and cosmology. We review the concept of extra dimensions and brane worlds and present the necessary background in terms of the theory of quantised fields in curved manifolds with boundaries. We first consider higher-dimensional spontaneous symmetry breaking. In a situation where the boundary conditions rule out a constant field configuration except for the zero solution, we obtain an approximate analytical solution for the ground state of a bulk scalar field with a double-well potential in the Randall-Sundrum brane world background. The stability of the zero solution is determined by the brane separation. We find our approximation near the critical separation at which the zero solution becomes unstable to small perturbations. Next, we calculate the vacuum expectation value of the stress-energy tensor for a quantised bulk scalar field in the Randall-Sundrum model, and discuss the consequences of its local behaviour for the self-consistency of the model. We find that, in general, the stress-energy tensor diverges in the vicinity of the branes. We conclude that the stress-energy tensor is sufficiently complicated that it has implications for the effective potential, or radion stabilisation, methods that have so far been used. Finally, we directly evaluate the zeta function and one-loop effective action for a bulk scalar field of arbitrary mass and general curvature coupling in the background of the bulk inflaton model of brane world inflationary cosmology. Our calculation avoids possible subtleties associated with less direct methods such as conformal transformation and dimensional reduction. iii

6 Contents 1 Introduction 1 2 Background The idea of extra dimensions Brane world models The hierarchy problem The Randall-Sundrum model Observational limits on extra dimensions Quantum fields in brane world models Quantum field theory in curved spacetime Dimensional reduction Symmetry breaking in brane worlds Introduction Spontaneous symmetry breaking The Higgs mechanism Symmetry breaking in brane world models Symmetry breaking fields with nonconstant ground states A twisted field in the Randall-Sundrum model Stability of solutions Approximate analytical solution Conclusion Self-consistency of brane world models Introduction Massless, conformally coupled field General mass and curvature coupling Surface divergence Self-consistency Conclusion iv

7 5 Brane cosmology Introduction The hot big bang model Inflation Brane world cosmological models The bulk inflaton model Bulk inflaton effective action Some comments on conformal transformation and dimensional reduction Direct solution of the field equations Zeta function evaluation General method Explicit evaluation of the zeta function Discussion and Conclusion Conclusion 82 A The scalar Green function in a slice of AdS D+1 84 B The uniform asymptotic expansion of Legendre functions 88 v

8 Chapter 1 Introduction The idea of extra dimensions in its current meaning has a history spanning close to one hundred years. It at least dates back to the proposal of Nordström [1] in 1914 for a special-relativistic description of gravitation united with electromagnetism, in which the fundamental spacetime was five-dimensional. The well-known Kaluza-Klein model [2, 3] was essentially a general-relativistic successor to Nordström s idea, which united Einstein gravity with electromagnetism, again in a five-dimensional setting. The Kaluza-Klein picture was also noteworthy for providing a geometrical interpretation of gauge invariance. The model was later extended to more than five dimensions to incorporate the nonabelian gauge interactions of the weak and strong nuclear forces. In the 1980s, much work focused on eleven-dimensional supergravity, due partly to Witten s observation [4] that a minimum of eleven dimensions was necessary for a higher-dimensional model to incorporate the standard model of particle physics, as well as Nahm s earlier observation [5] that a maximum of eleven dimensions can be supported by supergravity, if one restricts the field content to spins not exceeding two. Presently, motivation for extra dimensions mainly stems from work on superstring theory and M-theory, which also require ten or eleven dimensions as currently formulated. Extra dimensions have conventionally been thought of as compactified down to the scale of quantum gravity and have therefore been totally inaccessible to any reasonably conceivable measurement. However, the past six years have seen the concept of the brane world begin to take off, though it had been suggested much earlier [6, 7, 8]. In models based on this idea, the scale of compactification can essentially be anything one would like to suit a particular model, since they are hidden from everyday observation by the construction that the ordinary matter fields are confined to a membrane in a higher-dimensional spacetime. The brane world idea has generated much interest, because for the first time it opens up the possibility that quantum gravity phenomenology may be observed directly in particle accelerators and high energy cosmic rays, and indirectly via 1

9 signatures imprinted in the cosmic microwave background radiation. Deviations from Newton s law may also show up in precision measurements of the gravitational force. The subject of this thesis is the behaviour of classical and quantum fields in various brane world models. The original brane models assumed the higher dimensional spacetime away from the brane representing the universe to be devoid of matter. However, there are many reasons one would want to extend these models to include fields in this extra spacetime. One problem has been the issue of how to stabilise the sizes of the extra dimensions in these models, and the inclusion of extra fields has often been considered as an appropriate stabilising mechanism, either classically or quantummechanically via the Casimir force. An issue then is whether the quantum effects of these fields renders the geometry of the spacetime self-consistent in the presence of these effects. Another idea has been that the standard model fields are not in fact confined to a brane, but merely localised around the brane. This is a more universal approach to the brane world concept, attractive from the point of view that gravity is not artificially separated off from other matter and forces, and phenomenologically interesting. Further, a fast-developing field has been brane cosmology. In some models, higher-dimensional scalar fields in play a central role in the cosmological evolution of the brane universe. In Chapter 2, we begin with a brief chronological overview of the development of the idea of extra dimensions, starting with the early work at the beginning of the last century and finishing with the present ideas on brane worlds. This will, hopefully provide the reader with some context for the work presented in this thesis. With regard to brane worlds, we briefly describe the hierarchy problem, one of the main motivations for the proposals of such models, and outline the Randall- Sundrum model, which is probably the most satisfactory brane world solution to that problem that has been constructed. It is certainly the most well-known and widely studied. The chapters of this thesis are mainly based upon this model. We finish the chapter by giving the essential methods of quantum field theory in curved spacetime used in the thesis, including the technique of dimensional reduction. In Chapter 3, we describe spontaneous symmetry breaking in brane worlds. We give an account of spontaneous symmetry breaking, which plays a central role in standard particle physics, before discussing the effect in brane world models. In particular, we study the case in which the boundary conditions on a higher-dimensional symmetry-breaking field prevent constant non-zero ground states from occurring. This could have interesting consequences for high energy particle physics phenomenology. 2

10 In Chapter 4, we discuss the self-consistency of brane world models in which bulk fields are present. We start by describing various approaches to the problem of radius stabilisation, which is a typical feature of Kaluza-Klein models, and then relate this to the problem of self-consistency under the inclusion of quantum effects. This is the requirement that the brane world metric be a solution of the quantum-corrected Einstein equations. In Chapter 5, we describe brane cosmology. After giving a short review of the main developments in this area, we focus on the particular example of the bulk inflaton model, in which a higher-dimensional scalar field drives inflation on the brane. We calculate the zeta function and one-loop effective action of this inflaton field. We sum up the thesis in Chapter 6, discussing our main results and conclusions. 3

11 Chapter 2 Background 2.1 The idea of extra dimensions The world of everyday experience is three-dimensional. However, phenomena that in three-dimensional space may seem unrelated or require complex explanations can sometimes be shown to be describable in a simpler or more unified manner in higher numbers of dimensions. Intrinsic to any such description is how it can be reconciled with the observed three-dimensionality of space. By 1907, for example, Minkowski [9] had realised that both electromagnetism and special relativity are best understood if time and space are considered together as part of a single four-dimensional spacetime continuum. Along with the familiar three spacetime coordinates x 1, x 2, x 3, there is a fourth x 0 = ct, where c is the velocity of light and t is time. Physics had always appeared three-dimensional since, firstly, the fourth coordinate does not mark distance, and secondly, the large size of the parameter c ensures that the effects of mixing space and time coordinates appear only at speeds approaching that of light. These are the well-known effects of length contraction and time dilation predicted by special relativity. The concept of spacetime later proved essential to Einstein in developing the general theory of relativity. Later, in 1914, Nordström [1], a contemporary of Einstein, proposed a model of gravitation that was special-relativistic, in which gravity was described by a scalar field coupled to the trace of the stress-energy tensor. Though this approach was doomed to fail, he took the analogy with electromagnetism and from that proposed to unite electromagnetism and gravity into a single force. He achieved this treating the four-dimensional spacetime as a projection of a five dimensional continuum. Considering the fundamental force as being five-dimensional electromagnetism, in other words an abelian 5-vector field obeying the higher dimensional Maxwell s equations, he identified the fifth component as his scalar gravity. He was able to recover ordinary four-dimensional electromagnetism plus scalar grav- 4

12 ity by requiring that all dynamical variables be independent of the new coordinate. Nordström considered the introduction of the fifth coordinate as merely a mathematical trick and his idea soon faded into obscurity with the advent of general relativity a year later. However, the idea was rediscovered independently by Kaluza [2] in 1921, who formulated a five dimensional model from a general relativistic starting point. His approach was the reverse of Nordström s: rather than starting with Maxwell s equations in five dimensions, deriving gravity from them, he started with five-dimensional general relativity. Again, by artificially barring the fifth coordinate from appearing in the equations, which he termed the cylinder condition, he showed that this contains the four-dimensional Maxwell s equations plus four-dimensional general relativity. The model also predicted the existence of a massless scalar field, left uninterpreted by Kaluza at the time. The idea in its basic form was completed in 1926 by Klein [3], who introduced an idea from quantum mechanics into the picture. He showed that the cylinder condition naturally arises if the extra dimension has a circular topology. This circular topology gives rise to periodicity in the extra dimension, which allows any field quantity to be expanded in terms of an infinite series of modes, now known as Kaluza-Klein modes. According to quantum mechanics, the wavelengths, and therefore energies, of these modes are determined by the radius of the circle the smaller the radius, the higher the energies of the modes. Effective independence of four-dimensional physics from the extra dimension was achieved by assuming the energies of all modes above the ground state are too high to have ever been observed, meaning that the extra dimension is compactified to a very small scale. Klein s contribution makes it much more intuitive why the idea of introducing an extra dimension in such a way gives rise to electromagnetism. Invariance under general coordinate transformations, implicit in general relativity, combined with the periodic nature of the extra dimension, is in fact nothing but a U(1) gauge invariance, the symmetry group of electromagnetism. The Kaluza-Klein concept therefore suggests a possible geometrical interpretation of gauge invariance. Despite these initial successes, the Kaluza-Klein model offered no explanation as to why the extra dimension should differ so markedly in form from the other four. It was also not understood how to interpret the predicted additional scalar field. Meanwhile, quantum theory was developing at a rapid rate. During the 1930s and 40s, quantum field theory became the main preoccupation of cuttingedge research, as was particle physics after the particle explosion in the 1950s. A relatively few people continued to hope that a more sophisticated version of the Kaluza-Klein idea would underlie these developments, including Einstein who was looking for an explanation of quantum mechanics (e.g. [10]). Like Nordström s original idea, that of Kaluza and Klein became submerged by the tide of change. 5

13 However, these other developments brought with them the emergence of nonabelian gauge theory (which describe the then newly-discovered weak and strong nuclear forces). It was only after these discoveries that Kaluza-Klein theory was generalised to more than five dimensions. DeWitt [11] was the first to suggest in 1964 that a model based on a higher-dimensional extension of Kaluza- Klein theory could lead to the unification of the Yang-Mills interactions and gravitation. This idea was first correctly elaborated by Kerner [12] in 1968, who gave the higher-dimensional metric and Cho and Freund [13] in 1975, who gave the first complete derivation of four dimensional gravity plus the Yang- Mills fields and a scalar field from a compactified higher-dimensional general relativitistic model. Interest in higher-dimensional field theory arose again with the birth of supergravity (the theory of quantum fields that respect local supersymmetry, which turn out to include a massless spin 2 field identified as the graviton) in the late 1970s. In particular, much work in the early 1980s focussed on eleven-dimensional supergravity. This interest was based in part upon Witten s observation [4] in 1981 that a minimum of eleven dimensions is necessary for a model based on Kaluza-Klein theory to admit the gauge group SU(3) SU(2) U(1) of the standard model of particle physics, which together with Nahm s earlier observation [5] that a maximum of eleven dimensions can be supported by a supergravity model (consistent with a single graviton and particle spins not greater than two), seemed to single out eleven dimensions as interesting. These ideas were plagued with problems of their own, which we will not detail here. It suffices to say that research has moved on from these problems via the growth in interest in superstring theory in the 1980s. Superstrings turn out to be formulated most naturally in ten dimensions [14]. Furthermore, the five different ten-dimensional anomaly-free string theories that exist, together with eleven-dimensional supergravity, have since been shown to be aspects of a single underlying theory, as yet unknown but named M-theory, which is also higher-dimensional [15]. This brief overview illustrates that the idea of extra dimensions in modern physics has a history spanning close to a century. Current ideas and approaches are based, or at least inspired, by developments in string or M-theory. String theory has traditionally suffered from the problem that it is phenomenologically totally removed from the energy scale of any conceivable experiment. The extra dimensions have conventionally been compactified down to the order of the Planck length, the scale of quantum gravity, of the order of m, corresponding to the truly enormous energy M Pl GeV. However, the discovery [16] in 1995 of objects in the theory known as branes has changed this situation. 6

14 2.2 Brane world models The past six years have seen a renewed interest in extra dimensions due to so-called brane world models. String theory contains objects called p-branes, where p refers to the number of spatial dimensions of the brane (thus a 3-brane is four-dimensional), which are dynamical in their own right and exist in the higher-dimensional spacetime of the theory. These branes have the remarkable property that certain fields can be confined to the branes while gravity can propagate away from the branes, in the bulk space between them. This has led to the suggestion that the entire universe we see could be just such a brane. The possibility that our universe could be confined to a higher-dimensional defect had actually already been raised in the early 1980s in a field theoretic context [6, 7] and in pre-brane string theory in 1990 [8]. Inspired by the idea, Arkani-Hamed, Dimopoulos and Dvali (ADD) [17] proposed a model in 1998 in which the extra dimensions could be compactified on the millimetre scale. Following that work, Randall and Sundrum [18, 19] in 1999 proposed two five dimensional models. In the second of these, the extra dimension is not compactified at all The hierarchy problem The ADD and Randall-Sundrum brane world models were originally motivated as new frameworks in which to solve the hierarchy problem, though they have since taken on a life of their own. The fundamental scale of the standard model is the electroweak scale M EW 10 3 GeV. On the other hand, the fundamental scale of gravity is the Planck scale mentioned above. The question of why nature seems to contain two so very different scales is what constitutes the hierarchy problem, in one of its forms. Put another way, it asks, Why is gravity so weak compared to the other forces? Experimentally, the electroweak interactions have been investigated up to energies approaching M EW. Gravitational forces, on the other hand, have only just begun to be measured to distances less than the millimetre scale, some 32 orders of magnitude above the Planck scale. Our interpretation of M Pl as a fundamental scale rests on the assumption that gravity goes unmodified over this huge range of distances. This is not as absurd as it might sound it is known that gravity acts according to standard theory between at least the millimetre scale and the galactic scale. In the ADD scenario, the gravitational force becomes unified with the other interactions at M EW, which is taken to be the only fundamental scale in nature. The Planck scale is not fundamental: it s size, and the corresponding weakness of gravity, is a consequence of the large size of the extra dimensions. Suppose that there are n extra dimensions compactified to a radius R, giving 7

15 a (4 + n)-dimensional universe. The Planck scale M Pl(4+n) in (4 + n) dimensions is taken to be M EW. The corresponding gravitational constant, analogous to Newton s G, is M n 2 Pl(4+n). Two masses m 1, m 2 placed within a distance r R of each other will be gravitationally aware of the extra dimensions and will feel a gravitational potential given by Gauss law in (4 + n) dimensions: V (r) m 1m 2 M n+2 Pl(4+n) 1 rn+1. (2.1) If, alternatively, the masses are separated by a distance r R, they are aware only of the usual four dimensions, giving the familiar 1/r potential: V (r) m 1m 2 1 M n+2 Pl(4+n) Rn r. (2.2) The effective four-dimensional Planck scale M Pl that is measured in the r R régime is M 2 Pl Mn+2 Pl(4+n) Rn. (2.3) Putting M Pl(4+n) M EW implies that for one extra dimension (n = 1), R m, which is clearly discounted by experience. For n = 2, however, their radii must be mm, which is near the limit of current experiments. This case is therefore particularly exciting, as deviations from the usual laws of gravity could be found in the very near future at the laboratory scale. For n = 3, R 1 nm. Also, whatever the size of the extra dimensions, this model would place quantum gravity phenomenology within reach of the next round of collider experiments. ADD hide the extra dimensions from sight by confining the standard model fields to a 3-brane that exists in this spacetime. Gravity, on the other hand, freely propagates in all (4 + n) dimensions, which is why it becomes modified as explained. The other forces are stronger than gravity simply because they are concentrated onto the brane. This confinement is put in by hand in the original model The Randall-Sundrum model While the ADD brane world does eliminate the hierarchy between the electroweak and Planck scales, it introduces a new hierarchy between the electroweak scale and the compactification scale 1/R M EW. It is therefore a geometrical reformulation of the problem rather than a resolution. 1 1 However, Aranda et al. [20] have pointed out that for n = 7, the scale 1/R M EW. The problem in this case therefore disappears. The fact that this happens in the same number of dimensions as in string/m-theory is an interesting coincidence. 8

16 The first Randall-Sundrum model [18] is an alternative brane world model that was proposed in order to avoid exactly this problem. In this scenario, there is just one extra spatial dimension and the fundamental scale of the theory is the Planck scale. The essence of the idea is that the metric is non-factorisable: the four-dimensional metric is multiplied by a warp factor, an exponential function of the fifth dimension: dŝ 2 = ĝˆµˆν dˆxˆµ dˆxˆν = e 2σ(y) η µν dx µ dx ν dy 2, σ(y) = k y, (2.4) where k M Pl is the warp scale and x µ are the usual 4D coordinates. Thus, the bulk is a slice of 5D anti-de Sitter spacetime. We use a caret to donate a higher dimensional quantity. The extra coordinate y is periodic and lies in the range πr < y < πr. Points (x µ, y) and (x µ, y) are identified, giving the compact space the orbifold topology of S 1 /Z 2. Two 3-branes of fixed tensions sit at the orbifold fixed points y = 0 and y = πr. The y = πr brane is called the visible brane and represents the universe as we know it. Conversely, the other brane is referred to as the hidden brane. Again, gravity is the only field present in the bulk spacetime. The ordinary standard model fields are confined to the visible brane. The metrics induced on each brane are simply given by g vis µν (xµ ) ĝ µν (x µ, y = πr), g hid µν (xµ ) ĝ µν (x µ, y = 0). (2.5) The classical action S is S = S gravity + S vis + S hid, (2.6) where S gravity = πr d 4 x dy ) ĝ (2M 3 ˆR Λ, πr S vis = d 4 x g vis (L vis V vis ), S hid = d 4 x g hid (L hid V hid ). (2.7) Here, ĝ = det ĝˆµˆν, etc., M is the five-dimensional mass scale, ˆR is the Ricci scalar constructed from ĝˆµˆν, Λ is a bulk cosmological constant and V vis, V hid are constant vacuum energies on the branes (brane tensions), which act as gravitational sources even in the absence of particle excitations. The other details of the 3-brane Lagrangians contained in L vis, L hid are not considered. 9

17 Given the orbifold geometry of the extra dimension, the metric (2.4) is a solution of the five-dimensional Einstein equations for the above action if V hid = V vis = 24kM 3 and Λ = 24k 2 M 3, (2.8) showing that the branes have opposite tensions, and relating both tensions and the cosmological term to the five-dimensional scales k and M. The interesting physical implications of this model can be extracted using an effective four-dimensional description. The first step is to consider the massless gravitational fluctuations about the metric (2.4). In doing this, the compactification radius r becomes the vacuum expectation value of a modulus field T(x). T(x) has zero potential and consequently the radius is not determined by the dynamics of the model. This is the radius stabilisation problem, to which we return in Chapter 4. It is a typical problem of Kaluza-Klein models. For now, we assume r is stabilised by some mechanism. The perturbed four-dimensional metric ḡ µν can be expanded in terms of the background η µν plus fluctuations h µν : so that ḡ µν = η µν + h µν, (2.9) ĝ e 8k y ḡ, (2.10) where ḡ = det ḡ µν and is used to mean contains the term. Substituting (2.9) and (2.10) into the original action (2.6) and concentrating on the curvature term, from which we can derive the scale of gravitational interactions, we get πr S eff d 4 x dy 2M 3 e 2k y ḡ R, (2.11) πr where R is the four-dimensional scalar curvature derived from ḡ µν. Alternatively, S eff d 4 x 2MPl 2 ḡ R, (2.12) where M 2 Pl πr = M 3 dy e 2k y = M3 k πr [ 1 e 2πkr ]. (2.13) This is an important result of the Randall-Sundrum model. It gives the four dimensional effective gravitational scale M Pl in terms of the fundamental five dimensional scale M and shows that M Pl changes only slightly with r for kr 1. In other words, the exponential factor in the metric has very little effect 10

18 on determining the Planck scale. The model takes k M Pl and so all three quantities are of the same order of magnitude for all r. In the case of fields confined to the visible brane, however, the exponential factor has a dramatic effect. Consider the action of a fundamental Higgs field Φ H confined to the visible brane: S vis d 4 x g vis {g µν vis D µφ H D νφ H λ 24 ( Φ H 2 a 2 0 )2 }, (2.14) where a 0 is the symmetry-breaking scale by which masses in the fundamental higher-dimensional theory are set. D µ is the gauge-covariant derivative. Eq. (2.5) tells us that gµν vis = e 2πkr ḡ µν, and therefore S vis d 4 x ḡ e 4πkr {ḡ µν e 2πkr D µ Φ H D νφ H λ 24 ( Φ H 2 a 2 0) 2 }. (2.15) After normalisation of the fields through the replacement Φ H e πkr Φ H, to get the action into the correct canonical form, we obtain S eff d 4 x ḡ {ḡ µν D µ Φ H D νφ H λ 24 ( Φ H 2 e 2πkr a 2 0 )2 }. (2.16) In the effective description, therefore, physical masses are set by the symmetry breaking scale a e πkr a 0. (2.17) In general, any mass parameter on the visible 3-brane will rescale in the same way. On the other hand, on the hidden brane, gµν hid = ḡ µν, so that the rescaling factor is unity. This mechanism produces electroweak physical mass scales from fundamental scales of GeV if e πkr This requires kr 12, meaning that no significant hierarchy exists between any of the fundamental parameters a 0, k, M and 1/r. In this way, the Randall-Sundrum model provides the basis for a possible solution to the hierarchy problem without directly reintroducing it in another form. An alternative to compactification The second Randall-Sundrum model [19] is a modification of the first, in which the size of the extra dimension is now extended to infinity, thus removing the need for compactification entirely. On the other hand, this version of the model does not solve the hierarchy problem. Randall and Sundrum reverse the labels for the visible and hidden branes in their previous model. There is still the same non-factorisable warped metric in the bulk, but we now live on the brane where gravity is strongest. In the first 11

19 model, living on the weak-gravity brane accounted for the observed hierarchy between gravity and the electroweak forces; in the present model, living on the strong-gravity brane turns out to ensure that gravity is bound to the visible brane and appears four-dimensional. More precisely, when the hidden brane is sent to infinity and removed from the picture, the curved background supports a massless bound state of the higherdimensional graviton, which is localized in the extra dimension around the visible brane. This bound state is identified as the four-dimensional graviton. In addition, there exists a tower of Kaluza-Klein modes: a continuous spectrum with no gap between the tower and the bound state. However, fourdimensional physics is extremely well approximated since the continuum modes give only a small correction. Indeed, the potential behaves as V (x) = G N m 1 m 2 x ( ), (2.18) x 2 k 2 where G N is Newton s gravitational constant. x is the distance between the masses m 1 and m 2. The leading term due to the bound state mode is the usual potential (giving the inverse-square force law). The size of k, of the order of the the fundamental Planck scale, ensures that the correction term, generated by the Kaluza-Klein tower, is extremely suppressed. The observed Planck scale is related to the fundamental scale by the limit of Eq. (2.13) as r : M 2 Pl = M3 k. (2.19) The second Randall-Sundrum model is also noteworthy in that it involves no negative-tension branes. For this reason, it is the form of the model that has become the basis for the construction of many brane-world cosmological scenarios. 2.3 Observational limits on extra dimensions Brane world models are intriguing because they offer the only possibility yet conceived that string theory could make direct contact with experiment in the foreseeable future. A significant part of the current widespread interest in string theory is almost certainly attributable to the brane idea and the chance it offers of observing exotic phenomena in the new particle accelerators both planned and in the process of being built. Brane worlds have quickly become a playground in which to build models of particle physics and cosmology. The resulting scenarios provide some interesting background spacetimes for the study of quantum field theory, as is indeed the focus of this thesis. 12

20 However, an important fact must be borne in mind: there is as yet no evidence whatsoever for the existence of extra dimensions. At present, they belong entirely to the realm of speculation. Their current non-detection can be used to provide various observational constraints on the sizes of extra dimensions. The brief review of observational constraints on extra dimensions that we give here is not intended to be exhaustive. The purpose is merely to indicate the kinds of limits that have been derived. Precision tests of the gravitational inverse-square law provide the most direct test of higher dimensions with sizes greater than 10 4 m. Before the year 2000, the shortest-distance test of this law was only at the order of several millimetres, due to the difficulty in measuring the gravitational force between two bodies at small scales. In 2000, Hoyle et al. [21] provided a step decrease in distance down to 218 microns using a piece of apparatus known as a torsion pendulum, in which the gravitational attraction between two plates is arranged so as to produce a twisting effect. The group has recently improved their experiment and tested the inversesquare law at separations down to 138 microns, finding no deviation from standard behaviour at the 95% confidence level [22]. This, the shortest distance direct measurement of gravity at the time of writing, essentially sets an upper limit on the size of any large extra dimensions in a flat bulk (as in the ADD model). Specifically, they set a constraint on a toroidally compactified single extra dimension that the radius R 0.16 mm, and on two equal-sized large extra dimensions of R 0.13 mm. Observations of astrophysical phenomena also provide constraints on large extra dimensions. Hanhart et al. [23] have derived constraints on ADD-type extra dimensions from the nearby supernova SN1987a observed in High energy nucleon-nucleon interactions in the supernova would have produced Kaluza-Klein gravitons, causing a rate of energy loss into the bulk. If these losses were too high, then the neutrino pulse from the supernova would have differed from what was measured. They find that for consistency with the observed SN1987a neutrino signal, R mm for two extra dimensions and R mm for three extra dimensions. A related analysis has been performed by Hannestad and Rafelt [24], who point out that the large fluxes of Kaluza-Klein gravitons produced in supernovae mentioned above would produce a cosmic background of these particles with energies and masses up to about 100 MeV. Radiative decays would then give rise to a diffuse cosmic gamma-ray background with energies up to that value, which is well in excess of the observations if more than 0.5 to 1% of the supernova energy is emitted into the bulk graviton channel. They derive limits of R mm for two extra dimensions and R mm for three. These limits are very conservative because they used the present-day supernova rate as representative for the entire cosmic evolution. They suggest that a more realistic 13

21 assumption could lead to a photon flux of up to a factor of 10 to 100 times larger than their estimate. Big Bang nucleosynthesis provides a detailed and accurate understanding of the observed light element abundances. Without going into details, production of bulk gravitons could unacceptably affect these predictions. On the basis of such considerations, Hall and Smith [25] have derived, for the case of two extra dimensions, a conservative estimate of R mm, again for an ADDtype scenario (ignoring the possible existence of additional branes). Hannestad has derived results of the same order of magnitude [26]. 2.4 Quantum fields in brane world models We finish this chapter with an overview of quantum field theory in curved spacetime together with some of the methods that we will use in this thesis. Section is based upon Refs. [27, 28, 29]. We discuss only scalar fields, as we do not consider fields of higher spin in this thesis Quantum field theory in curved spacetime Canonical quantisation on a flat background Let us first briefly summarise the canonical quantisation of a real scalar field in n-dimensional flat (Minkowski) spacetime. A (non-interacting) real scalar field φ of mass m in such a background satisfies the Klein-Gordon field equation ( + m 2 )φ = 0 (2.20) where η µν µ ν and η µν is the Minkowskian metric tensor. This field equation is derived from the Lagrangian L = 1 2 ( η µν µ φ ν φ m 2 φ 2), (2.21) by demanding that the variation of the action S[φ] = L (x)d n x (2.22) with respect to the field vanish, i.e. that δ φ S = 0. In flat space, the field may be expanded in terms of Fourier modes as φ(x) = ] [a k u k (t,x) + a k u k (t,x), (2.23) k 14

22 with u k = e ik.x (2π) n 1 2ω k, (2.24) where ω k = k 2 + m 2. In order to quantise the field, it is promoted to the status of an operator. Together with its momentum conjugate π L / ( t φ) = t φ, it is required to obey the equal-time commutation relations [φ(t,x), φ(t,x )] = 0, [π(t,x), π(t,x )] = 0, [φ(t,x), π(t,x )] = iδ n 1 (x x ). (2.25) Correspondingly, the Fourier coefficients are promoted to the creation and annihilation operators a k and a k respectively, which obey [a k, a k ] = 0, [a k, a k ] = 0, [a k, a k ] = δ kk. (2.26) In the Heisenberg picture, the quantum states of the field span a Hilbert space. A useful basis for this space is the Fock representation, in which the basis vectors can be constructed from the vacuum state 0, which is defined by a k 0 = 0, k, (2.27) by successively operating on this state with a. In zero-temperature quantum field theory, which is what this thesis is concerned with, one is interested in the vacuum expectation values (VEVs) of observables, such as the stress-energy tensor. VEVs of products of fields are given by Green functions G(x, x ), or propagators, of the operator that appears in the field equations and action. All of the content of the zero-temperature theory is contained in these Green functions. In general, the Green function is defined as the solution to the field equation with a point source. In our case, ( + m 2 )G(x, x ) = δ(x x ), (2.28) where δ(x x ) is the Dirac delta function. A particular Green function is selected according to the boundary conditions. The Green function has a Fourier integral representation with poles on the real axis corresponding to the mass of the field, which is where the contour of integration must lie. There is then a choice of contour that must be made according to how to avoid the poles. 15

23 On the other hand, in a space with Euclidean signature, the poles lie on the imaginary axis, and this issue is avoided. The Green function can then simply be defined as the VEV of the time-ordered product of fields: G(x, x ) = 0 T(φ(x)φ(x )) 0. (2.29) Canonical quantisation on a curved background We now proceed to illustrate the procedure of quantisation of a real scalar field in curved spacetime by proceeding in formal analogy with the Minkowski space case. In place of Eq. (2.21), we begin with the Lagrangian density L = 1 2 g ( g µν µ φ ν φ m 2 φ 2 + ξrφ 2), (2.30) Here, we have introduced the coupling to the gravitational field represented by the term ξrφ 2, where ξ is a numerical constant and R is the Ricci curvature scalar. It is the only local scalar coupling of this sort with the right dimensions. If ξ = 0, then the field is said to be minimally coupled. The generalised Klein-Gordon field equation that follows from the above Lagrangian is ( + m 2 + ξr)φ = 0, (2.31) where now g µν µ ν = 1 g µ [ g g µν ν ]. (2.32) An interesting special case is the conformally invariant case, in which m = 0 and ξ = ξ c, the conformal coupling value, given by ξ c = 1 ( ) n 2. (2.33) 4 n 1 This refers to invariance of the classical action under the conformal transformation in which case, g µν (x) ḡ µν (x) = Ω 2 (x)g µν (x), φ(x) φ(x) = Ω (2 n)/2 (x)φ(x), (2.34) ( + ξ c R)φ = 0 and ( + ξ c R) φ = 0. (2.35) Massless, conformally coupled fields are usually much simpler to analyse, though it should be noted that their behaviour is often not typical. Also, quantum effects 16

24 break the invariance via the introduction of conformal anomalies, the details of which we do not give here. Proceeding in a covariant manner, the field may be expanded as in Eq. (2.23) φ(x) = ] [a i u i (x) + a i u i (x), (2.36) i where we have replaced the vector k with the schematic index i that represents whatever quantities are necessary to label the modes. The explicit forms of the modes u i depend upon the background spacetime and any boundaries present [27]. The field is then quantised by imposing the commutation relations [a i, a i ] = δ ii, etc., (2.37) in analogy with Eq. (2.26). The vacuum state can be constructed as in Eq. (2.27). In flat spacetime, this is the state with zero particle number. However, in curved space, there is an inherent ambiguity in the vacuum [30]. This is related to the lack of any natural or privileged coordinate system (corresponding to the Cartesian coordinates in flat spacetime), and the consequent lack of any natural set of modes u i. Therefore, the vacuum may or may not contain particles. In a real or apparent gravitational field, particles may be created from the vacuum, even in the absence of sources. The effective action and zeta function An object of great interest in quantum field theory is the vacuum-to-vacuum transition amplitude. It turns out that the response of the vacuum state of a quantum field to a driving force encodes all of the information about the field (at zero temperature). In the path integral approach to quantum field theory, the vacuum-to-vacuum amplitude in the presence of a source J, which is adiabatically switched on and off, is defined as 0 0 J Z[J] N { Dφ exp S[φ] + } dσ J(x)φ(x), (2.38) where we work in Riemannian space for the purposes of this subsection. dσ represents the invariant volume element and N is a normalisation constant. Z[J] is called the generating functional, or the partition function in analogy with thermal field theory. For an interacting field, functional derivatives of Z[J] generate the n-point Green functions via 1 δ n Z[J] Z δj(x 1 ) δj(x n ) = T(φ(x 1 ) φ(x n )). (2.39) J=0 17

25 In particular, the first derivative gives the mean field. Before setting the source J to zero, this is a function of J. Let us define the classical field φ c by φ c = φ(x) J = 1 Z δz[j] δj(x). (2.40) We can formulate an effective classical description of this field, where the action contains corrections due to quantum effects. This effective action Γ is defined by Γ[φ c ] = ln Z[J] dσj(x)φ c (x), (2.41) Essentially we treat the classical field as the field. The field equations for the classical field follow from extremising the effective action in the limit J 0. The effective action is particularly useful in analysing how quantum effects change the classical behaviour of a system. The effective action is most often treated perturbatively 2, as a loop expansion, equivalent to an expansion in powers of : Γ[φ c ] = S[φ c ] + Γ (1) [φ c ] + O( 2 ) + (2.42) It can be shown that, in Riemannian space, the one-loop effective action Γ (1) can be represented formally by Γ (1) = 1 2 ln det(µ 2 ), (2.43) where is the operator appearing in the classical action. We have introduced the renormalisation scale µ with the correct dimensions (mass or inverse length) in order that the argument of the logarithm be dimensionless. The advantage here of working in Riemannian space is that the operator is then positive definite. This allows us to define the functional determinant in terms of a generalised zeta function. The generalised zeta function is defined in terms of the eigenvalues λ n of by λ s n. (2.44) ζ(s) = n This sum is only convergent for Res > d/2. However, the determinant is generally related to the zeta function at some non-convergent value of s. This is a typical feature of quantum theory, that its useful quantities are formally divergent and require regularisation. This is achieved in this instance by using complex analysis to analytically continue the above sum to a regular function around a pole at s = 0. 2 S[φ c ] is known as the classical action in this context, though (somewhat confusingly) it the same functional as the action S[φ] describing the quantum field. 18

26 After such a continuation, the one-loop effective action can be redefined as Γ (1) = 1 2 ζ (0) 1 ζ(0) lnµ. (2.45) 2 The stress-energy tensor An important physical quantity is the stress-energy-momentum tensor, or stress tensor for short, which gives the local distribution of energy-momentum due to a (classical or quantum) field in a region of spacetime. In some sense, it describes the physical structure of the field. The stress tensor is also significant as the source of the Einstein field equations. It therefore plays an important part in any semiclassical model, in which a quantum field propagates on a non-quantum gravitational background. In particular, it allows the study of the self-consistency of such a model to the backreaction of the quantum field on the background geometry. This will be the subject of Chapter 4. The stress tensor is defined as the variation of the action with respect to the metric: T µν (x) = 2 δs g δg µν (x). (2.46) In quantum field theory, T µν is an operator, and we are concerned with its VEV T µν. Though we do not employ this in this thesis, this is actually given by the variation of the effective action in an analogous manner. For a scalar field with action S = 1 d n x g ( g µν µ φ ν φ m 2 φ 2 ξrφ 2), (2.47) 2 Eq. (2.46) leads to T µν = µ φ ν φ 1 2 g µνg ρσ ρ φ σ φ 1 2 g µνm 2 φ 2 + ξg µν φ 2 + ξ [ g µν φ 2 µ ν φ 2], (2.48) where G µν R µν 1 2 Rg µν is the Einstein tensor. For a quantum field, the VEV of the stress tensor is generically infinite and must be renormalised. The method we employ in this thesis is to work in a Riemannian spacetime and make use of the Green function (2.29). Then the stress tensor can be written in the form T µν (x) = lim x x O x,x G(x, x ), (2.49) where O x,x is a differential operator. We illustrate this explicitly in Chapter 4. This allows the infinite terms, which are due to taking the coincidence limit above, to be isolated. 19

27 2.4.2 Dimensional reduction Dimensional reduction, or Kaluza-Klein decomposition, is a key technique developed in Kaluza-Klein theory for providing an effective four-dimensional description of a particular model. It is therefore crucial for constructing any realistic higher-dimensional model. A simple scalar field example is sufficient to illustrate the idea. Consider a real scalar field in a five-dimensional spacetime, in which one spatial dimension, labelled y, is compactified to a circle. The metric is then ds 2 = ĝˆµˆν dˆxˆµ dˆxˆν = η µν dx µ dx ν dy 2, (2.50) where the x-coordinates are unbounded but πr < y πr. The field φ has the higher dimensional action S = 1 πr d 4 x dy ĝˆµˆν ˆµ φ ˆν φ = 1 πr d 4 x dy ( φ φ φ y 2 2 πr 2 φ). (2.51) πr The reduction is accomplished expanding the field into harmonics, called Kaluza- Klein modes, φ(x, y) = ψ n (x)f n (y). (2.52) n Substituting this into the action, we obtain S = 1 πr d 4 x dy [ ( (ψ n ψ n )f n f n ψ n ψ n fn y 2 2 f )] n. (2.53) n n πr Let us orthonormalise the modes f n (y) according to πr πr We also define a set of eigenvalues m n by dy f n f n = δ nn. (2.54) 2 yf n = m 2 nf n. (2.55) Then, on integration over the extra dimension, the action reduces to S = 1 ( ) d 4 x ψ n + m 2 2 n ψn, (2.56) n Identifying m n as the mass of the mode ψ n gives the important result: the massless five-dimensional scalar field appears to a four-dimensional observer as an infinite tower of massive scalar fields. The masses m n are determined from Eq. (2.55) together with the appropriate boundary conditions on the field in the extra dimension in this case, periodic 20

28 conditions and the requirement that the field be continuous and regular. This introduces the dependence on the size of the extra dimension. Clearly, if the radius r is small, the modes f n will have short wavelengths and the masses m n will be large. By ensuring that the radius is small enough to generate masses that are too large to be observable other than that of the lowest mode is the conventional method of hiding the extra dimensions from the four-dimensional world. The brane world scenario provides an alternative mechanism. So far, we have considered the field classically. There are two approaches one could take to quantisation in a Kaluza-Klein background. One could either quantise the single field in the higher dimensional spacetime, or one could perform the reduction as described, and quantise the effective four-dimensional infinite set of fields. Though we take the former approach in this thesis, it is worth mentioning that the issue of dimensional reduction is not trivial. It turns out that one cannot renormalise each mode separately and then sum over all of the modes. This mode sum, which should be fully renormalised, still diverges. One must first do the infinite mode sum and then carry out the renormalisation 3. In other words, some care must be taken if the lower-dimensional Kaluza-Klein tower is quantised [31, 32, 33, 34]. We merely mention this point in order to highlight the danger of interpreting a single Kaluza-Klein mode as a fully independent quantum field in its own right. Bulk scalar field in the Randall-Sundrum model We now apply dimensional reduction to the two-brane Randall-Sundrum model. Consider a bulk scalar field φ of mass m and coupling ξ to the higherdimensional scalar curvature ˆR. Such a field has the action S = 1 πr d D x dy [ĝˆµˆν ĝ ˆµ φ ˆν φ m 2 φ 2 ξ 2 ˆRφ ] 2. (2.57) πr As for most of this thesis, we work in (D+1) spacetime dimensions for generality (we have two (D 1)-branes and one extra dimension). Integrating by parts with respect to y gives S = 1 πr d D x dy [ ( e (2 D)σ η µν µ φ ν φ + φ y e Dσ y φ ) e Dσ m 2 φ 2], 2 πr (2.58) where m 2 = m 2 + ξ ˆR. Making the decomposition into Kaluza-Klein modes as in Eq. (2.52), we obtain S = 1 πr d D x dy [ e (2 D)σ η µν µ ψ n ν ψ n f n f n 2 n n πr (2.59) ( ] +ψ n ψ n f n y e Dσ y f n ) e Dσ m 2 ψ n ψ n f n f n. 3 However, one can indeed regularise each mode, as long as the regularisation is not removed until the sum is performed, as in Ref. [35], for example. 21