Two Quantum Effects in the Theory of Gravitation

Home , Gravitational anomaly, Roman Jackiw

Two Quantum Effects in the Theory of Gravitation by Sean Patrick Robinson S.B., Physics, Massachusetts Institute of Technology (1999) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 © Sean Patrick Robinson, MMV. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. MASSCHUSETSSE OF TECHNOLOGY

JUN 0 7 2005

Au th or ...... -...... LIBRARIES Department of Physics May 19, 2005

Certified by -/ 'I C ertifi ed by...... -...... Frank Wilczek Herman Feshbach Professor of Physics Thesis Supervisor ~~~~~ ~~~~/ A ccepted by ...... Professr Thom/Greytak Associate Department Head fo ducation

Two Quantum Effects in the Theory of Gravitation by Sean Patrick Robinson

Submitted to the Department of Physics on May 19, 2005, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics

Abstract We will discuss two methods by which the formalism of quantum field theory can be included in calculating the physical effects of gravitation. In the first of these, the consequences of treating general relativity as an effective quantum field theory will be examined. The primary result will be the calculation of the first-order quantum gravity corrections to the /3 functions of arbitrary Yang-Mills theories. These corrections will effect the high-energy phenomenology of such theories, including the details of coupling constant unification. Following this, we will address the question of how to form effective quantum field theories in classical gravitational backgrounds. We follow the prescription that effective theories should provide a description of experimentally accessible degrees of freedom with all other degrees of freedom integrated out of the theory. We will show that this prescription appears to fail for a scalar field in a black hole background because of an anomaly generated in general covariance at the black hole horizon. This anomaly is repaired and the effective field theory is saved, however, by the inevitable presence of Hawking radiation in the quantum theory.

Thesis Supervisor: Frank Wilczek Title: Herman Feshbach Professor of Physics

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__I__ Acknowledgments

The following body of work has benefited from the input of many individuals. Obvious among these is my thesis advisor, Frank Wilczek. I would like to thank the Nobel Foundation for making this an interesting year to write a thesis. I also need to recognize the other members of my thesis committee, Eddie Farhi and Roman Jackiw. Others who made significant contributions to the development of this work, but are not specifically cited within, include Brett Altschul, Ted Baltz, Serkan Cabi, Qudsia Ejaz, Ian Ellwood, Michael Forbes, Brian Fore, Vishesh Khemani, Joydip Kundu, Vivek Mohta, Brain Patt, Dru Renner, Jessie Shelton, and Ari Turner. I especially thank Michael Forbes for reading an early draft of this thesis. Finally, I would like to acknowledge the exceptional support and motivation provided by my wife, daughter, and parents, the importance of which cannot be overstated. In celebration of the completion of this thesis, I compose the following cautionary limerick 1 :

When working with quantum gravity, There's not much that is easy to see. The math's so opaque that it's easy to make an occasionalerror, or three.

This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DE-FC02-94ER40818.

'Incidentally, Speak in limerick? Well, maybe I did. But if I did, it was only to kid. Speaking in limerick is sort of a gimmick behind which real intentions are hid.

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_··^I_________ Contents

1 Overture 13 1.1 Quantum General Relativity and Yang-MillsTheory. . 14 1.2 Black Holes and Effective Field Theory ...... 17

2 Gravitational Corrections to Yang-Mills /3 Functions 23 2.1 Introduction ...... 23 2.1.1 One-loop Divergences ...... 26 2.1.2 Asymptotic Safety ...... 27 2.2 Technical Preliminaries ...... 28 2.2.1 Background Field Theory ...... 28 2.2.2 Definition of Newton's Constant ...... 29 2.3 Setup ...... 31 2.4 Expanding the Action ...... 32 2.4.1 Expanding the Non-Polynomial Terms . . ... 32 2.4.2 Expanding the Einstein-Hilbert Action ...... 33 2.4.2.1 Curvature with Background Derivatives . .... 33 2.4.2.2 Some Useful Definitions and Identities ...... 34 2.4.2.3 Expansion of Curvature ...... 34 2.4.3 Expanding the Yang-Mills Action ...... 36 2.5 Gauge-Fixing ...... 36 2.6 Combining the Pieces ...... 38 2.7 Compiling the Superfield ...... 41 2.8 Renormalization ...... 43 2.8.1 Computation of Functional Determinants ...... 46 2.8.2 Extracting the Function ...... 49 2.9 Enlarging the Matter Sector and the Gauge Group ...... 50 2.10 Coupling Constant Unification ...... 53 2.11 Phenomenology ...... 54 2.12 Commentary ...... 57

3 Black Hole Effective Field Theory 59 3.1 Introduction ...... 59 ...... 3.1.1 Hawking Radiation ...... 59 3.1.2 Anomalies and Anomaly Driven Currents ...... 60 3.1.3 Hawking Radiation and the Conformal Anomaly 61

7 3.1.4 Effective Field Theory Framework . 62 3.2 Spacetime Prel iminaries ...... 63 3.2.1 Sphericaal Static Metrics ...... 63 3.2.1.1 Einstein's Equation ...... 65 3.2.1.2 Horizon Structure ...... 67 3.2.2 Kruskal Extension ...... 69 3.2.2.1 The Quantum Vacua ...... 74 3.2.2.2 Euclidean Section ...... 75 3.2.3 Wave Equation ...... 75 3.2.3.1 Spherical Harmonics ...... 77 3.2.3.2 Radial Wave Equation ...... 79 3.2.3.3 Near-Horizon Action ...... 80 3.3 Thermal Radiation ...... 82 3.3.1 Hyperci ubic Blackbody Cavity ...... 83 3.3.2 Flux Versus Energy Density ...... 86 3.3.3 Sphericaal Blackbody Cavity ...... 88 3.3.3.1 Radial Mode Density ...... 90 3.3.3.2 Spectral Densities ...... 92 3.4 Calculation ...... 95 3.5 Commentary ...... 98 3.6 Blackbody Spe ctrum from an Enhanced Symmetry? ...... 99

4 Finale 101 4.1 Summary ...... 101 ...... 4.2 Open Possibilities ...... 101 4.3 Conclusion...... 102

8 List of Figures

2-1 Feynman diagrams for two typical processes contributing to the renormalization of a Yang-Mills coupling at one-loop ...... 25 2-2 The schematic Feynman diagram represented by the functional trace -TI[Mh]. A momentum p circulates in a virtual graviton loop coupled to external gluons of momentum k ...... 46 2-3 The schematic Feynman diagram represented by the functional trace -Tr[N]. A momentum p circulates in a virtual gluon loop coupled to external gluons of momentum k ...... 47 2-4 The schematic Feynman diagram represented by the functional trace 1Tr[O+O_]. A momentum p circulates in a virtual gluon-graviton loop coupled to external gluons of momentum k ...... 48 2-5 In Figure 2-5(a), the three Yang-Mills couplings of an MSSM-like the- - 2 ory evolve as straight lines in a plot of r ' =_ 47r/g versus log10 (E) when gravitation is ignored. The initial values at Mzo - 100 GeV are set so that the lines approximately intersect at 1016 GeV. When gravity is included at one-loop, the three lines curve towards weaker coupling at high energy, but remain unified near 1016 GeV. In Figure 2-5(b), g is plotted for the same theory. All three couplings rapidly go to zero near Mp, rendering the theory approximately free above this scale ...... 55

3-1 Part of the causal diagram of a black hole spacetime, with inset detail of a region near the horizon ...... 63 3-2 Sketches of three integrated mass functions and their associated h(r). In 3-2(a) the matter distribution is relatively smooth and vanishes at the origin, as in a normal star. In 3-2(b) the matter has a density singularity at the origin, but is otherwise well behaved. In 3-2(c) a potentially difficult-to-analyze situation is sketched ...... 70 3-3 In 3-3(a) typical profiles for the functions h(r) and f(r) are sketched for an asymptotically flat black hole spacetime. The horizon occurs where the functions vanish at r = rh. In 3-3(b) the corresponding profile of r is sketched along with the line r = r. Note that r diverges logarithmically at rh and approaches r at large r ...... 71

9 3-4 A sketch of a typical effective radial scattering potential. The potential for any metric qualitatively similar to the one sketched in Figure 3-3 will be qualitatively similar to the one sketched here for > 0 and d > 3. The potential falls off exponentially for negative r and as is typically dominated by the centrifugal term at large r,. r, which falls off as 1/r 2 ...... 81 3-5 The thermal integral I(a) defined in Equation (3.129) is plotted for the cases = +1 and ~ = 0. All three lines seem to converge towards e1 -a for a >> 1...... 94

10 List of Tables

3.1 A few physically interesting metrics that obey p = -P ...... 67 3.2 The angular state degeneracies for total angular quantum number 1, as determined by Equation (3.64), in a few chosen dimensions ...... 79

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_ · Chapter 1 Overture

In this thesis we shall describe two logically independent lines of research that represent small steps away from ordinary quantum field theory in flat, nondynamical spacetimes and towards quantum gravity, the as-yet-undiscovered fundamental theory of quantum spacetime dynamics. The first strategy we investigate is that of perturbatively quantizing the small field fluctuations of general relativity, the first theory of spacetime dynamics historically. This approach is famously limited in power and much-maligned, but we will show by a specific example that useful physical predictions can nevertheless be obtained in this formalism. The second strategy is to not attempt to give quantum dynamics to spacetime at all, but to instead only use quantum theory where it has already proven so successful: in the nongravitational aspects of matter. In this approach, spacetime is described in a nonquantum way, either as a nondynamical, curved background or using the classical dynamics of general relativity. This formalism is usually called semiclassical gravity. Like quantized general relativity, semiclassical gravity is rather restricted in scope and cannot be considered as more than a limited, but useful, model for quantum gravity. We will use semiclassical gravity to describe the behavior of a quantum field theory in the region outside of a black hole. The second strategy is often considered more respectable than the first, perhaps because it never attempts to be more than a model, and thus its points of failure are both more understandable and educational. We believe, on the other hand, that both formalisms are useful as model theories for quantum gravity as long as they are applied within their respective regimes of validity. By studying the conditions under which a model theory begins to fail, we can learn which aspects of the true theory the model theory lacks. Since semiclassical gravity and quantized general relativity have different regimes of validity and different failure modes, they are complimentary tools in the investigation of the properties of quantum gravity. They also have a broad overlap region of validity, which is the domain of ordinary Minkowski space quantum field theory. This provides for these models an anchor to known physics, which is often claimed to be well understood. In Section 1.1 we will describe the effects of including quantized general relativity in the calculation the function of a non-Abelian gauge theory. The calculation of this quantity in the absence of gravitation [1, 2] is considered to be one of the

13 most important calculations of Minkowski space quantum field theory [3]. The full calculation and discussion appears in Chapter 2. In Section 1.2 we will describe our attempts to import the highly successful concepts of Minkowski space effective field theory into the semiclassical description of quantum fields in a black hole spacetime. This is discussed more fully in Chapter 3.

1.1 Quantum General Relativity and Yang-Mills Theory

In Chapter 2, we will calculate to one-loop order in perturbation theory the f function of the Yang-Mills coupling constant in an arbitrary non-Abelian gauge theory coupled to quantum gravity. The core calculation of this chapter is based upon the work of [4]. Here, quantum gravity is modeled by its low-energy effective field theory, which is just quantized general relativity. This effective field theory should be an accurate description for the quantum dynamics of spacetime at energy scales below the theory's cutoff scale, the Planck mass, given in four dimensions as Mp -GN 1/ 2 1.1 X 1019 GeV, where GN is Newton's constant of universal gravitation. Before the appreciation for the proper role of effective field theories in physics be- came widespread, common lore held that general relativity and quantum mechanics are incompatible in terms of describing the physical phenomenon of gravitation. This was primarily because quantum general relativity was finally proven to be perturbatively nonrenormalizable [5, 6, 7, 8, 9] shortly after the time when renormalizability had become understood as an essential ingredient in quantum field theories of fundamental interactions. About ten years later, the discovery of quantum theories that appear to describe gravitation in terms of excited strings [10, 11, 12, 13, 14, for introductions], rather than local fields, helped to cement the common lore. Over time, however, appreciation has grown for the fact that even the best quantum field theories of reality (that is, the standard model) are, at best, effective theories containing infinite numbers of nonrenormalizable interactions. The ideas of Wilso- nian effective field theory [15, 16, 17] have taken deeper root in the intuition and made possible the use of nonrenormalizable phenomenological models such as chiral perturbation theory [18, 19, 20]. Finally, about 20 years after the proofs of the nonrenormalizability of general relativity, Donoghue [21] made compelling arguments in favor of taking seriously calculations made with quantized general relativity and treating the results of these calculations as genuine low-energy predictions of quantum gravity. It is in this spirit that we perform our calculation of the Yang-Mills function. The effective value of a renormalized coupling constant g in a quantum field theory depends on the energy scale E at which it is probed via a universal function of the theory known as the Callan-Symanzik /3 function [22, 23]:

E ag- = E). (1.1)

14 The remarkable discovery [1, 2] for four-dimensional non-Abelian Yang-Mills theories was that these theories obey

g3l 11 4 bo 3= (- )2 C2(G)-- nfC(r )I (4)23YM (1.2) for a gauge group G with nf fermions in representation r. This /3 is negative as long as nf is not too large. Equation (1.2) integrates to give a running coupling of

1 1 ___ /1 b In (1.3) gy (E)2 gy (M) 2 (47r)2 M2) (1.3) which demonstrates that the negative value of the /3function implies asymptotic freedom: gYM(E) -- 0 as E - oo, as long as b0 is positive. The only known asymptotically free theories in four spacetimes dimensions are the non-Abelian gauge theories. Thus a universe with laws of physics governed by non-Abelian gauge theories - as our universe approximately appears to be - becomes simpler and simpler as it is probed at more fundamental scales, as long as the matter content is simple enough. We now want to augment this classic calculation with quantum general relativity. The calculations will be done using the methods of background field theory, which we will sketch briefly in Section 2.2.1. We will let the spacetime background and dimension be arbitrary for as long calculationally feasible. This will require adopting a definition for Newton's constant in d dimensions. We choose a definition, described in Section 2.2.2, which preserves the interpretation of the nonrelativistic gravitational force law as describing the areal density of diverging, but conserved, gravitational flux lines. Then, in Sections 2.3 through 2.7, we perform the detailed expansion of the coupled Einstein-Yang-Mills action in terms of quadratic fluctuations about nontrivial gauge field and spacetime backgrounds. In particular, in Section 2.5 we gauge-fix the theory using the Faddeev-Popov [24] procedure and calculate the ghost and gauge- fixing Lagrangians. The gauge chosen to fix general covariance is reminiscent of the R~ gauge [25], except that the original RC gauge was for a gauge field in a scalar background and the current case is that of a gravitational field in a vector background. In Section 2.8, we finally come to the central result of Chapter 2 by evaluating the background effective action and extracting the ,Bfunction. In Section 2.9, the result is generalized to arbitrary gauge groups and matter content. We find that to one-loop accuracy, the /,3function is equal to the value calculated in the absence of gravity - such as that given in Equation (1.2) - plus a new term lA]grav that is independent of the gauge and matter content. In four spacetime dimensions, this term is given by

3 E 2 A/3grav(gy,E) =-gYM -2 (1.4) 7r Note that this term is always negative. It will dominate the running of the coupling when the energy is close to the Planck scale and the coupling constant is perturbatively small. Thus, it appears that the inclusion of quantum gravity effects renders all non-Abelian gauge theories asymptotically free. The integrated running coupling

15 coming from the combination of Equations (1.2) and (1.4) is 2 2 2 2 } 1gy _ 1 r3E -M 2 bo fEdk r3E -k g~ (E)2gy (M) 2 ex~p r Me2 + 2~(47r)2,M k e M2 (1.5)

The logarithmic running of Equation (1.3) becomes modulated by a exponential in E2 . This has little effect at low energies, where the exponential is approximately equal to one. As E approaches Mp, however, the exponential turns on very quickly in comparison to the logarithm, and the coupling gets driven rapidly to zero. This phenomenology comes with the caveat that the interesting physics is occurring very close to the cutoff scale of the theory. However, taken at face value, this result seems to indicate that Yang-Mills theories become approximately free at the Planck scale. In Section 2.10, we explore the implications of Equation (1.5) for coupling constant unification. That is, we consider a Yang-Mills theory with a simple gauge group that is spontaneously broken at some high energy scale such that the theory at low energies appears to be a Yang-Mills theory of some product gauge group with several independent coupling constants, each with its own function. Without the context of the unified theory, the low-energy values of these couplings could be taken to have arbitrary independent values. However, since all the couplings secretly derive from a unified theory at high energy with only a single coupling, the low-energy values must conspire with the 3 functions in such a way that all the couplings evolve to the same unified value at the breaking scale. The experimentally measured values of the SU(3) x SU(2) x U(1) couplings of the standard model with minimally supersymmetric matter content are consistent with such a unification in the real world with a breaking scale of MGUT - 1016 GeV [26]. No matter how many couplings are in the low-energy theory, only two of them may be chosen independently. The rest are then fixed by the condition of unification. If the field content of the low-energy theory is changed such that the functions change without a corresponding change in the values of the low-energy couplings, the unification will generically be spoiled. If the addition of gravitation spoiled unification in this way, it would indicate that the observed unification of standard model couplings is a spurious coincidence. Fortunately, as we show in Section 2.10, this is not the case for four dimensional gauge theories. Although the 3 functions are changed in a non-trivial way given by Equation (1.4), we find that theories which exhibit exact coupling constant unification in the absence of gravity continue to do so with the same values of the low-energy couplings when Equation (1.4) is taken into account. The values of the unified coupling and the breaking scale are slightly altered. For a standard-model-like situation where the measurement scale M and the putative breaking scale M0 obey a hierarchy of the form M < M0 < Mp, we find that the new breaking scale Mu is given by

FinallyMinuection2.11Mesome brief remarksegardingthe(1.6)we Finly ° S 27r rMPa] ( Finally, in Section 2.11, we make some brief remarks regarding the phenomenology

__·_ and possible experimental signatures of the calculated gravitational correction to the running of coupling constants.

1.2 Black Holes and Effective Field Theory

The core result of Chapter 3 is based primarily on the work of [27]. In the context of semiclassical gravity, we attempt to formulate an effective field theory for a scalar field that lives in a black hole background. Our prescription for constructing this theory ultimately results in a breakdown at the quantum level of the underlying gauge symmetry of gravitation, general covariance. Demanding that general covariance holds in the effective theory, as it does in the fundamental theory, forces each partial wave of the scalar field to be in a state with a net energy-momentum flux (I given by

487r' (1.7) where r, is the surface gravity of the black hole event horizon. If each partial wave mode is occupied with a blackbody frequency spectrum, then Equation (1.7) implies a temperature of TH = 2- (1.8) which is exactly the Hawking temperature of the black hole. The result (1.8) for the temperature of a black hole was originally found by Hawk- ing [28, 29] and subsequently rederived by many other methods. Hawking radiation is now understood as a kinematic effect resulting from the lack of an unambiguous global definition for a particle number basis of Fock space when spacetime is not globally flat. Our construction can be thought of as arising from the presumption that the physics observed by a given experimenter should be describable in terms of the effective degrees of freedom accessible to that experimenter. In the case of ordinary Minkowski space quantum field theory, one can apply this presumption to an experimenter with limited energy available to probe highly excited states. In that case, the effective physics observed by the experimenter is described by a theory in which states above the high-energy cutoff have been integrated out, resulting in the standard story of Wilsonian effective field theory [15, 16, 17]. The parameters and degrees of freedom of the low-energy theory may be different from those that appear in the fundamental theory. We wish to consider an experimenter who lives outside of a static, spherically symmetric black hole. Such spacetimes have a global Killing vector (spacetime symmetry generator) that appears locally like a time translation, but it is only timelike in the region outside the black hole event horizon. Thus, the conserved quantity associated with this symmetry can not be used as an energy outside of this region. Since the observer cannot see beyond the event horizon of the black hole, however, this Killing vector should be a perfectly reasonable choice with which to define the energy of quantum states in an effective theory that only describes observable physics. Unfor-

17 tunately, the "vacuum state"' obtained with this definition is exactly the one considered by Boulware [30]. The Boulware vacuum has a divergent energy-momentum tensor due to a pile up at the horizon of would-be outgoing modes (the UP modes in the language of [31]), which take arbitrarily long amounts of coordinate time to escape the near-horizon region. Our approach differs from most previous work on Hawking radiation in that we recognize the divergent energy of the horizon-skimming modes as an indicator that the experimenter who observes these modes will not be able to probe them with finite energy. Thus, the proper description of the observed physics is an effective theory with these modes integrated out. In other words, we choose to take the lessons of effective field theory seriously. The effective theory thus formed no longer has observable divergences, but it now suffers from an even worse problem; it contains an anomaly in general covariance. As shown in [32], a two dimensional scalar field theory will violate general covariance at the quantum level if the number of right-moving and left-moving modes are not identical - that is, if the theory is chiral. The breakdown of general covariance means that the energy-momentum tensor Tb of the scalar field is not conserved. In the case of a single chiral scalar field, the anomaly takes the form

VaT: = 9 dcddadaFr-c (1.9) 9o7rVf -g where the a7c are the Christoffel symbols of the background spacetime. We show in Section 3.2.3 that in the near-horizon limit, each partial wave behaves like an independent two dimensional free massless scalar field. In our case, we have eliminated the horizon-skimming part of each partial wave of the scalar field. So, this effective theory is chiral and each partial wave exhibits an anomaly given by Equation (1.9), but only in the near-horizon region. However, the fundamental theory contains all the modes, so it has no anomaly. Some new physics must be introduced into the chiral theory to carry out the job of anomaly cancellation that was formerly performed by the degrees of freedom which were integrated out in the process of forming the effective theory2 . Indeed, we find that demanding general covariance to hold in the

'The word "vacuum" is used here in the Fock space sense, meaning the state in which all momentum modes have zero occupation number. It does not mean that the state has minimal energy; the energy of a state can not be unambiguously defined in a curved spacetime. 2 The ability to form a gauge invariant effective theory for a fundamental theory which cancels anomalies between modes of very different energy seems to run contrary to decoupling theorems which state that the only effect the high-energy modes can have in the low-energy theory is in the renormalized value of low-energy coupling constants [33]. If the effective theory is at an energy scale where some, but not all, of the modes involved in anomaly cancellation have been integrated out, then decoupling should guarantee that the high-energy modes cannot cancel the remaining anomalies in the effective theory. In theories like the electroweak standard model, which have potentially anomalous chiral gauge couplings to fermions that gain a wide spectrum of masses via Yukawa couplings to a Higgs field, the problem has been partially solved [34, 35] by the discovery of a Wess-Zumino term in the low-energy effective action, but work continues on these models [36, 37, for example]. We believe the present problem, in which gravitational anomaly cancellation occurs between ingoing states of finite energy and outgoing states of divergent energy, may be another

18 effective theory places constraints of the energy-momentum tensor of the scalar field in the form of a boundary condition that must be obeyed by each partial wave at the black hole horizon. The boundary condition can then be used to solve the covariant conservation equation for the energy-momentum tensor over all of spacetime. The result is that the energy-momentum tensor must describe a flux of the form given in Equation (1.7) in each partial wave. Equations (1.7) and (1.8) are derived primarily in Section 3.4. The calculation there is relatively brief and painless. However, a great deal of formalism needs to be built-up to support those calculations. This comprises the bulk of Chapter 3. In developing this formalism, we find a number of noteworthy intermediate results, which we summarize below. We are also led to some interesting observations that are not directly relevant to the core calculation of Section 3.4 and mostly appear as footnotes in the main text. These are also summarized below. In Section 3.2.1, we study the properties of the general static, spherically symmetric spacetime in d spacetime dimensions. We compute the components of the Ricci tensor and scalar curvatures, as well as the Christoffel symbols, in a natural coordinate system. This allows us, in Section 3.2.1.1, to construct and solve the d-dimensional Einstein's equations for the most general background matter distributions allowed by the symmetries. The d-dimensional versions of a few simple, well-known four- dimensional spacetimes are listed in Table 3.1. In Section 3.2.1.2 we examine the conditions for the existence of a horizon such that the spacetime describes a black hole. We distinguish between event horizons and Killing horizons, but argue that given some rather general physical conditions, Einstein's equations imply that if an event horizon exists at some constant-radius surface of the spacetime, then a Killing horizon must also exist at the same location. We define and compute the surface gravity of the horizon. Although much of the core analysis of Chapter 3 will not depend in any way on the spacetime in question being a solution to Einstein's equations, we will find it necessary at several points to invoke a coincidence requirement for Killing horizons and event horizons. Thus, it is reassuring that this requirement is well-motivated by real physics. We also find in this section that the same physical conditions that lead to the coincidence requirement also imply that the radial pressure of the background matter seen by a static observer at the black hole horizon must be equal to the negative of the observed energy density. In Section 3.2.2, we construct the analog for this spacetime of the Kruskal extension [38] of the four dimensional Schwarzschild black hole. Unlike the Schwarzschild case, the resulting metric for the general case is not obviously non-singular at the horizon, but we prove by construction that it is. In the process, we show that the choice of the Kruskal coordinates is quite constrained. As in the original Kruskal extension, the time translation symmetry of the original coordinates now manifests as a boost symmetry. Further, we observe that translations in the Kruskal U and V coordinates become spacetime symmetries at the past and future event horizons, respectively. The existence of the Kruskal extension requires the coincidence of Killing and event horizons discussed in Section 3.2.1.2. As a bonus, we construct the Painlev-Gullstrand example of the same generic phenomenon in field theory.

19 [39, 40] coordinates and tortoise coordinates for this spacetime. In Section 3.2.2.1, we use our new Kruskal extension to reproduce the famous arguments originally made by Unruh [41] for the four dimensional Schwarzschild black hole, thus proving that Hawking radiation can be understood for this spacetime according to standard arguments. To further push this point, in Section 3.2.2.2 we follow [42] in constructing the near-horizon coordinates for the Euclideanized version of the black hole and show that the Euclideanized horizon exhibits conical singularity which is resolved by setting the period of Euclidean time to be 27r/n. This agrees with Equation (1.8) by standard arguments that associate the period of Euclidean time with inverse temperature. The existence of the near-horizon coordinates also requires the coincidence of Killing and event horizons. In Section 3.2.3, we consider the partial wave decomposition of an arbitrary scalar field theory in this d-dimensional black hole spacetime and solve the scalar wave equation by separation of variables. In Section 3.2.3.1, we derive the (d- 2)-spherical harmonics, which are an alternating product of Legendre P and Q functions. Demanding regular solutions of the wave equation quantizes the arguments of the Legendre functions and therefore sets the spectrum of angular momentum quantum numbers. We also find the degeneracy of angular momentum states with total angular momentum quantum number 1 in d dimensions to be:

Dd(l) = (21+ d - 3)(1+ d-4)! (1.10) 1!(d - 3)! In Section 3.2.3.2, we tackle the radial equation, which is not solvable in general. In the flat space limit, it is solved by Bessel functions; we define the (d- 2)-spherical Bessel functions. In the near-horizon limit, the radial equation in tortoise coordinates becomes the 1 + 1 dimensional massless free wave equation for each partial wave. Finally, in Section 3.2.3.3, we apply the partial wave decomposition to the action of a scalar field with arbitrary self-interactions. We find that whether or not an interaction is important in the far-field limit is controlled by whether it would be power-counting renormalizable in a quantum field theory. We also find that near the horizon, the vanishing of the metric functions implies the vanishing of all terms in the action except for the tortoise d'Alembertian term. Thus, the near-horizon action is that of an infinite collection of free, massless, 1 + 1 dimensional scalar fields in 1 + 1 dimensional Minkowski space. In Section 3.3, we study the spectrum of blackbody radiation. In Section 3.3.1, we derive the blackbody spectrum and energy density in d dimensions for a multi- component massive scalar field with either Bose-Einstein or Fermi-Dirac statistics. In Section 3.3.2, we find that the relationship between the energy density p and blackbody flux 1 for p massless fields of statistics in d spacetime dimensions at temperature T is given by Vol (]3d-2) (1.11) Vol(§ - 2) ' where I' is the unit n-ball and Sn is the unit n-sphere. The d-dimensional Stefan-

.____ _· Boltzmann law is given by

2 (1-)/2 (d)(d- 1)! Td (1.12) I)= p 1 -- _T (1.12) Pd)2 2d-2(d- 2)F (-2)7rd/2

In Section 3.3.3, we engage in the unexpectedly challenging calculation of the blackbody spectrum of angular momentum modes. No closed form expression is ultimately found, but various limits indicate that each partial wave behaves nearly like a 1 + 1 dimensional blackbody, including the correct statistics. We also remark on the differ- ence between canonical momenta of angular variables and actual angular momenta, which is understood using the Cartan subalgebra of SO(d- 1). This forms a pleasing connection between group theory and mechanics. Finally, in Section 3.6, we speculate on the existence of an enhanced spacetime symmetry giving rise to a thermal spectrum whose temperature is fixed by the anomaly cancellation mechanism of our core calculation. We also make a speculative remark about the possible anyon-like behavior of Euclidean black holes.

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_I___ __ Chapter 2 Gravitational Corrections to Yang-Mills Functions

In this chapter, we will calculate the one-loop contributions of virtual gravitons to the running of Yang-Mills coupling constants using background field methods in Minkowski space. We find that this renders all Yang-Mills couplings asymptotically free, independent of any additional matter content. We also find that that the addition of gravity to a theory which previously displayed coupling constant unification at a high energy MGUT does not upset this unification. Rather, the unification energy is shifted up by approximately 23-MGUT/MP, which corresponds to a slightly weaker coupling. For realistic grand unified theories, this shift corresponds to about 1010 GeV.

2.1 Introduction

Despite its problems with perturbative renormalizability [5], naively quantized general relativity can be taken as a low-energy effective theory for the true theory of quantum gravity, just as the nonrenormalizable chiral Lagrangian of mesons is a low- energy effective theory of the strong interactions. In this sense, quantum general relativity cannot be taken as a fundamental theory and its predictions should not be trusted above the built-in scale of the theory, MP- GN1/ 2 _ 1019 GeV, just as chiral perturbation theory should not be believed above its scale, f ~ 130 MeV. An interesting open possibility is that general relativity is nonperturbatively renormalizable. This will be briefly discussed in Section 2.1.2. In the case of chiral perturbation theory, we know that a new theory, namely QCD, takes over as the appropriate description of the world above the cutoff scale. Moreover, because this new theory is asymptotically free [1, 2], it is well-defined even at arbitrarily high energy scales, and thus it can be taken as a fundamental theory. Through its inverse effect, infrared slavery, asymptotic freedom also helps to provide an explanation as to why the appropriate description at low energy does not look anything like the fundamental theory, through the mechanisms of confinement [43] and chiral symmetry breaking [44, 45].

23 While the existence - and precise form - of an ultraviolet completion for the the chiral Lagrangian may be necessary for understanding its role as a description of strong interactions, knowledge of the ultraviolet completion is usually not necessary if one wants to use chiral perturbation theory simply to calculate the low-energy behavior of mesons (where "low-energy" here means below f). However, these predicted behaviors should still be taken as genuine low-energy predictions of QCD. Likewise, as compellingly argued by Donoghue [21], the predictions of quantum general relativity at scales below Mp should be taken as unambiguous predictions of whatever the true theory of quantum gravity may be. Armed with this attitude, several authors have used quantum general relativity to calculate the one-loop corrections to the nonrelativistic Newtonian potential, mostly in the Born approximation [21, 46], but some by other methods [47, 48]. By reading the numerator of the results of such calculations, one can interpret an "effective Newton's constant," altered from its bare value by short-range quantum effects. Of course, the short range in question is very close to the scale at which the theory should be cutoff, and even then the effects are weak. So, unlike the running of the QED and QCD couplings, the predicted form of gravitational running is not expected to be experimentally falsifiable anytime soon. On the other hand, allowing for the virtual production of any new species of particle will change the rate at which couplings run. That is, the form of the Callan- Symanzik 3 function [22, 23] - the logarithmic derivative of a coupling constant with respect to the renormalization scale - should be altered with each new field added to the theory. The addition of gravitons to the Standard Model should be no exception. Again, it is not anticipated that this effect is of a directly measurable magnitude for laboratory experiments, but it may disturb certain high-energy predictions of the theory. For example, the experimentally measured values of the standard model coupling constants seem to conspire together with the theoretically calculated coefficients of the Yang-Mills 3 functions (augmented with minimal supersymmetry) to give a unification near MGUT - 10-3 MP [26]. This unification is highly sensitive to the input parameters, as it is equivalent to getting three lines to meet at a point, up to experimental uncertainties. Even a small perturbation of the function coefficients could push unification out of the experimentally measured range. If virtual gravitons were to upset unification in this way, it would be somewhat disturbing, as the unification of standard model gauge couplings is a necessary prediction of all realistic grand unified theories. With this in mind, we set about calculating the the scale-dependence of a non- supersymmetric pure Yang-Mills theory coupled to quantum general relativity in 3+1 dimensions, to one-loop order. We will use effective action background field methods, since they are quite natural for gravity and are known to be useful for gauge fields. In principle, an arbitrary background spacetime could be used, in which case the renormalization of Newton's constant and the cosmological constant could be studied, too. Since renormalization depends only on ultraviolet physics, however, and all spacetimes are locally Minkowski, we will restrict our attention here to the case of Minkowski spacetime and, thus, zero cosmological constant. However, in the interest of generality, calculations will be carried out in an arbitrary background with non-zero

24 9YMI

Figure 2-1: Feynman diagrams for two typical processes contributing to the renormalization of a Yang-Mills coupling at one-loop. Curly lines represent gluons. Double lines represent gravitons. The three-gluon vertex * is proportional to 9 YM, while the gluon-graviton vertex is proportional to E/Mp. cosmological constant for as long as is feasible. The form of the gravitational contribution to the function can be guessed without calculation, since all the new one-loop Feynman diagrams of interest are essentially a three-gluon vertex with two legs connected by a graviton (See Figure 2-1). Since 9 the gluon vertex has strength YM and gravitons couple to energy-momentum, one expects' the inclusion of gravitons to add a term to the 3 function like

3 2 A/ gra (gyM, E) = aogyME /MP (2.2) at energy scale E. Sections 2.3 through 2.8 of this thesis consist of the calculation of the unknown coefficient a0 . Once the calculational method is presented for the case of a single gauge field, it can be extended to the case of multiple gauge fields with interacting matter almost by inspection. This allows for discussion of realistic theories like the standard model. It also allows for examination of any theory that exhibits high-energy coupling constant unification. Section 2.9 discusses the implications of gravitational corrections in such theories. One hindrance of applying the results to interesting supersymmetric theories, such as the minimally supersymmetric standard model, lies in the fact that we are using only bosonic gravity in our analysis. Such an application only makes

'The guess of Equation (2.2) is for the case when gym is dimensionless, which demands d = 4. Generically, each term in the ,3 function would be multiplied by an additional E d - 4 to account for the units of gyM. So, in the absence of gravity, the / function takes the form

- 3(gYME) = bogM Vol2(S (Sd ))d-4Ed_4, (2.1) where b is a number determined by the theory and the unitless, d-dependent numerical factor has been arbitrarily chosen for convenience.

25 sense in a model where supersymmetry is treated as breaking above the Planck scale in the gravitational sector while remaining unbroken in the rest of the theory.

2.1.1 One-loop Divergences Finding a proper quantum treatment of gravitation is a long standing problem of theoretical physics. The attempts to overcome the many technical difficulties of a relativistic local quantum field theory of gravity are legion. None have been completely successful, but the effort spent on this problem has not been wasted, as several techniques originally developed to cope with gravity have ultimately proven essential elsewhere in field theory (for example, the Faddeev-Popov method [24]). In [5], 't Hooft and Veltman applied the techniques they had previously used to prove the renormalizability of spontaneously broken gauge theories [25] to the problem of a scalar field coupled to quantum general relativity. They showed that the one-loop divergences in this theory can only be cancelled by counterterms that don't appear in the original action. In other words, the theory is nonrenormalizable. To cancel the divergences to all orders in renormalized perturbation theory would require an infinite number of counterterms whose coefficients would require an infinite number of experiments to determine, thus spoiling the scientific predictability of the theory. To some degree, part of the nonrenormalizability of the theory can be attributed to the matter to which it is coupled. This is because pure general relativity, without matter, is actually renormalizable at the one-loop level; new counter terms only arise at the two-loop level. One might hope, therefore, that some special combination of matter fields and general relativity might be proven renormalizable, in effect choosing a specific theory or family of theories as unique in this status. This is unfortunately not the case. Dirac fields [7], Maxwell fields [6], and Yang-Mills fields [8, 9] all yield one-loop divergences like the scalar field. So, the quantum theory we plan to compute with has unavoidable one-loop divergences. Since we wish to calculate the Yang-Mills 3 function to one-loop order, we should start with a one-loop renormalized action for general relativity. In four dimensions this can be written as the "curvature-squared" action:

2 SG= JdxV1 t{ 1 6R + aR + a2R,,vR'}X (2.3) where A is a cosmological constant, g is the determinant of the metric, R is the Ricci scalar curvature, and Rab is the Ricci tensor curvature. The allowed "Riemann- squared" term has here been eliminated by rewriting it as a sum of the other two curvature-squared terms and an unwritten topological density. The case of a, = a2 = 0 is the standard Einstein-Hilbert action for general relativity. The coefficients aei have units of action, and can thus be combined with GN and the speed of light c to 3 - 35 form a length V /GNj/C - £pV/-/h, where £p _ 1.6 x 10 m is the Planck length. The primary physical effect of the curvature-squared terms is in Yukawa corrections to the non-relativistic Newton's law with characteristic lengths given by epv/i [49], where we have returned to h = 1 units. A conservative modern limit on the length

I_·_I scale of such forces is approximately 1 mm [50], which is much larger than ep.The limit ai < 1064is thus extremely weak. Since the curvature-squared corrections to the quantum action have such a small effect on observable physics, we will ignore them throughout the rest of this thesis. Assuming ai < 1, these terms will only become important at energies very close to and above Mp.

2.1.2 Asymptotic Safety An oft overlooked open possibility is that despite all the negativity of Section 2.1.1, quantum general relativity may still be a perfectly well defined and predictive theory. This is because all that the nonrenormalizability proofs [5, 6, 7, 8, 9] really show is that perturbation theory in terms of small fluctuations around a free field theory fails for quantum general relativity. The theory may still be sensible nonperturbatively if it has the property that Weinberg has dubbed "asymptotic safety" [51, 52]. This is the case where the renormalization group flow of the theory exhibits an ultraviolet fixed point with only a finite number ultraviolet attractive directions. This condition ensures that there exists a finite-dimensional critical subspace in the infinite-dimensional space of all allowed coupling constants such that renormalization group trajectories along this subspace are confined to it. The theory on the critical surface is then parameterized by a finite number of couplings and has a well defined continuum limit given by the ultraviolet fixed point. Theories for which the ultraviolet fixed point corresponds to free field theory, such as in Yang-Mills theory, are called asymptotically free. These have the advantage that they can be calculated with standard perturbation theory and are thus perturbatively renormalizable. Generic asymptotically safe theories lack perturbatively renormalizability, but are no less suited as nonperturbative fundamental theories because of this fact. An example nontrivial asymptotically safe theory is that of a scalar field in five dimensions obeying the symmetry - - [51]. This theory has no allowed renormalizable interactions beyond a mass term, but it does exhibit an interacting Wilson-Fisher ultraviolet fixed point. This fixed point has only two attractive directions, and is thus asymptotically safe. Another theory with a nontrivial ultraviolet fixed point is that of fermions interacting via a four-fermion term in less than four dimensions [53]. There is now evidence that four dimensional quantum general relativity may be asymptotically safe with two attractive directions given approximately by Newton's constant and the cosmologicalconstant [54]. Most of the study of this fixed point has been carried out with the exact renormalization group equations for the so-called effective average action [48] of Reuter, but the fixed point has also been found with other methods, such as the proper time renormalization group [55]. These flow equations are infinite-dimensional nonlinearly coupled first-order partial differential equations. In practice, the equations must be approximated by truncating to some finite- dimensional subspace. The observed fixed point seems to be stable against changes of the truncation and acceptably insensitive to the choice of regulator and gauge. It also appears to persist for realistic matter content [56] and spacetime dimensions ranging

27 from 2 + E up to perhaps six [57]. In this chapter, we will be treating quantum general relativity coupled to matter as a perturbatively nonrenormalizable effective field theory with a cutoff. The arguments of this section show that perhaps this theory is in fact valid to arbitrarily high scales due to asymptotic safety. If it is also true that only Newton's constant and the cosmologicalconstant are essential couplings at the fixed point, then the approximation of dropping the curvature-squared terms made in Section 2.1.1 is justified even at high energy scales. Also, if the theory is asymptotically safe, then many of the caveats that would need to be stated regarding calculating and interpreting the results of the effective theory near its cutoff can be relaxed.

2.2 Technical Preliminaries 2.2.1 Background Field Theory The background field method is especially well suited to the calculation we are going to attempt. The application of the method to the calculation of one-loop Yang-Mills / functions without gravity is textbook fare. Indeed, our use of the method will follow very closely to [58, Section 16.6]. A background expansion is always necessary at some level for perturbative gravitational calculations, since these involve metric excitations which are small fluctuations about Minkowski spacetime, or some other spacetime, and perturbing about the singular state with vanishing metric would prove a poor approach. The background method is thus a convenient choice, since it accommodates this expansion naturally. While ultimately equivalent to calculations that could be done with Feynman diagrams, the background method arranges the calculation differently. Whereas Feyn- man diagrams compute results process-by-process, the background method computes them species-by-species. This is again convenient for us, since we want to examine the effect of adding one new particle species (the graviton) to an established calculation. One further advantage of this method is that a one-loop calculation corresponds to evaluating a simple Gaussian functional determinant. If we were to attempt calculating the function to higher-loop accuracy, this method would lose much of its advantage over Feynman diagram techniques. The recipe for the background field method as we will be using it (to extract 3 functions) is as follows:

1. Write down the classical action. Identify the operators whose coefficients are the renormalizable parameters of interest.

2. Expand each field that contributes to the operators of interest as a quantum fluctuation about a classical background. Leave all other fields as they are, effectively choosing zero background for them.

3. Identify the gauge freedom of the quantum fields and gauge-fix them. This will introduce Fedeev-Popov ghosts [24].

------_ _ _ _ 4. Ignore all terms that are higher than quadratic order in quantum fields. If de- sired, use the classical equations of motion for the background fields to eliminate some terms. The action is now a background-dependent Gaussian functional of quantum fields.

5. Using the generating functional, functionally integrate out all of the quantum fields into functional determinates. What remains should be a gauge-invariant functional of the classical background fields. This is the exponential of the one-loop effective action.

6. Evaluate the functional determinates as the exponential of a polynomial series in the background fields. As usual, interpret the divergent integrals encountered with a convenient scheme, such as minimal subtraction, that introduces dependence on a mass scale E. The only terms in the series that need to be retained are those that correspond to the original operators of interest.

7. Interpret the mass-scale dependent coefficients of the effective action as the running couplings in the limit that the mass-scale is differentially close to the renormalization scale.

8. Solve for the 3 functions using dg /3dE/E.

9. Integrate the 3 functions to find the running couplings.

2.2.2 Definition of Newton's Constant We will use a set of units for Newton's gravitational constant in d spacetime dimensions which reflects the physical interpretation of the nonrelativistic gravitational force law as describing the density of diverging, conserved field lines, commensurate with its origin in a Gauss law. These units are slightly different from certain other conventions [14, for example]. Start with the simplest generally covariant actions for a dynamical spacetime metric, the Einstein-Hilbert action:

S dd= Jd xv/R. (2.4)

The overall coefficient is defined as - 2 so that perturbative metric excitations have a canonically normalized kinetic energy operator when a factor of X is absorbed into the field definition. When coupled to matter, this action yields the Einstein equation:

xK2 Rab - 9gabR2 = 2~2 Tab, (2.5a) or equivalently

Rab = 2 (Tab -d 2T) (2.5b)

29 The energy-momentum tensor Tab is defined as

Tab = 2 6 matter (2.6) Jgab

The factor of 2 in the numerator ensures that tensor so defined - that is, by variation with respect to the metric, of which there is usually one power in a kinetic matter Lagrangian- actually represents the canonically normalized energy-momentum that is derived as the Noether current of translations-that is, by variations with respect to matter fields, of which there are usually two powers in a kinetic matter Lagrangian.

Consider an approximately Newtonian spacetime as seen by a nonrelativistic observer with timelike velocity ua. Contracting Equation (2.5b) with uaUband identi- fying the Newtonian potential o as the field whose gradient gives the acceleration of free-falling particles, we get X2(d-3) Vp=_2(d-2 2) p '(2.7) (2.7) where V2 is the Euclidean Laplacian and p = -To is the physical energy density seen by the observer. We have used Roo = gijRoioj = -V 2 W, where i and j run over spatial indices, UaUa= -1, and Ta - -p, since pressures are negligible in the Newtonian limit. Equation (2.6) gives a force law between two point masses of

2 m1m2 X (d- 3) IF(r) = Vol (Sd-2)rd - 2 2(d - 2) (2.8)

Of course, we have Vol(Sd - 2) = 2 r(d-1 )/ 2 FP((d-1)/2)'7

Since the physical essence of the 1/rd - 2 law is that there is a total flux generated at some small distance which is conserved as it spreads out over the area of a sphere at a larger distance r, we take the force law to represent the ratio of the areas of the 2 spheres at these different distances. Thus, the factor 2(d-2)(d-3) should be thought of as 2 2 the area of a sphere of some special unit radius, 2(d-23) Vol (Sd- )ed- . We then define GN = -2 Md as the basic unit for counting square areas. Thus,

= 2d - 32 Vol (d-2)GN. (2.9) So, this choice for the d and GN dependence of X gives a nonrelativistic Newton's Vol (Sd- 2 ) law in d dimensions of F(r) = mlm 2 vol(S2 r, ) where Vol () is the volume of an $n of radius r.

_·_ _·____ 2.3 Setup

With the exception of the overall normalization of the gravitational field, we follow the definitions and conventions of [59] for gravitational quantities as closely as possible. Boldface lowercase roman indices are gauge group indices, while normally faced lowercase roman indices are spacetime indices. The dynamics for a non-Abelian gauge field coupled to gravity in d spacetime dimensions is given by the sum of the Einstein-Hilbert and Yang-Mills Lagrangians:

L=L +Lm = 2.x/--g[R~~~~~~~~YM - X~--2A] ---- 12 gg acgbdFa ab (2.10)2cd, 3C ~~4gy~m where X 2 is defined in Section 2.2.2, A is a cosmological constant, g = det gab, R is the Ricci scalar, gYM is the gauge coupling,

Fa -/AaY~b-A~4~- -/\bAA bJ4~+ + fbcfA AaAbC~4 (2.11) is the field strength, Aaais the gauge field, fabc are the structure constants of the non-Abelian gauge group G, and /a is the derivative operator obeying Aagbc = 0, i.e. the covariant derivative. Since Fabis antisymmetric under a -+ b, the Christoffel connections arising from the derivatives in Equation (2.11) will cancel against each other 2. Thus, the covariant derivatives here could be safely replaced with partial derivatives or any other torsion-free derivatives. The Lagrangian (2.10) is non-polynomial in gab, and the configuration gab = 0 is unachievable, so we expand gab about an arbitrary classical background gab with quantum fluctuations hab: gab = gab + hab. (2.12) Indices are now raised and lowered with the background metric. We need to re-express L in terms of hab and gab, up to quadratic order in hab. Higher order terms in hab will only contribute to higher-loop processes. Once this is done, hab will look like a tensor quantum field that lives in a classical curved spacetime. We also need to expand Aa a: around a classical background configuration aa:

Aa- = a + Aa , (2.13) where aa obeys the classical equation of motion

DaFaab = 0. (2.14)

Fabis the appropriately named function of classical fields only, and Da = Va - iatr a. Here t a is a gauge group generator for the representation r, and V is the torsion-free derivative operator obeying Vagbc = 0. (2.15)

2We assume that gravity is torsion-free.

31 Of course, aaa is in the adjoint representation, so that in Equation (2.14),

[tciab = -ifabc. (2.16)

Performing the expansions (2.12) and (2.13) on the Lagrangian (2.10) is an extended calculation. It is presented in detail for the case of an arbitrary background spacetime in Section 2.4.

2.4 Expanding the Action

In this section, we show the tedious details involved in a particular method for expanding both the Einstein-Hilbert action and the Yang-Mills action in terms of perturbations about arbitrary backgrounds. The results for the gravitational portion follow closely to results presented in [5], although the calculational method differs significantly.

2.4.1 Expanding the Non-Polynomial Terms

Given the metric gab, we take the expansion as in Equation (2.12), with 9ab being an arbitrary background metric. We will now expand the inverse metric, gab using the math fact that given a matrix M = 11+ A, where is the identity, then M- 1 = ' ° 0 [-A] . This gives gab= gab_ hab+ hhCb + 0 (h 3). (2.17)

We will also need the expansion of x/I--g:

/-g = f7=exp { 1Trln[5b + ha]}.

We now Taylor series the logarithm, evaluate the trace, and Taylor series the exponential: \/= = -g[1 + lh+ 1(h2 - 2habhab)] + 0 (h3) (2.18) where h h. To sum up, we define

g 9 a gab + ab (2.19)

--g _/i-[1 + D], (2.20)

where Iab = -hab + hahcb + o (h 3), (2.21) D = h+ 1(h2-2habhab)+ 0 (h3). (2.22)

D and Iab are infinite order polynomials (that is, nonpolynomials) in hab. In fact, they are the source of all the non-polynomial graviton interactions. In practice we will truncate these polynomials at second order or less in h, but by leaving them as

___ __ D and Iab for now, we will be able to both keep the calculation more organized and keep track of the influence of the nonpolynomial terms.

2.4.2 Expanding the Einstein-Hilbert Action

In this section, we expand the Einstein-Hilbert action in terms of a quantum perturbation about an arbitrary background to second order in the quantum fields using the method of background derivatives.

2.4.2.1 Curvature with Background Derivatives

We make the definitions, as in Wald [59]:

[Aa, Ab]Vc - RabcdVd, (2.23a) Rab Racbc, (2.23b) R - Rabga , (2.23c) where va is an arbitrary vector and Aa is the derivative operator commensurate with the metric gab, that is Aagbc = 0. This derivative is unique (see, for example, THEOREM 3.1.1 of [59]) and is usually expressed in terms of the partial derivative &a as

Aavb =b Ob b + rvcb C (2.24a) AaVb = OaVb- rabvc, (2.24b) /AaVbc = aVbc- rabVdc - acVbd, (2.24c) and so forth for arbitrary tensors v, where rc 1 gcd (.5 ab = g (agbd + Obgad - '9dgab). (2.25)

Inserting this form of Aa into Equation (2.23a), we get the standard result

Rabc = brac-d d Darbc+d racreb-rbcreae d e d (2.26)

However, we can also use the slightly non-standard expression

Aavb = VaVb+ C vc, (2.27) which relates Aa to some other derivative Va. Manipulations similar to those that led to Equations (2.25) and (2.26) now lead to

Cab = g (Vagbd + Vbgad- Vdgab) (2.28) and RabcRabcd =Racd Rab d + +VbCaVbC - VaCbcCcC + CacCedb-CbecCea,C b(2.29) (2.29)

33 where [Va, Vb]vc Rabc.Vd. (2.30)

If there is an invertible symmetric tensor gab, perhaps numerically close gab, for which the derivative satisfying Vagbc = 0 is a known quantity 3, then the above constructions can become very helpful. We formally define a tensor hab such that

gab = gab+ hab. (2.31)

We then have Cab = 9 (Vahbd + Vbhad - Vdhab) (2.32)

Of course, we want to choose the arbitrary symmetric tensors gab and hab defined in Equation (2.31) to be the background metric and fluctuation, respectively, as defined in Equation (2.12).

2.4.2.2 Some Useful Definitions and Identities

We define Hab - (Vahb + Vbhc - Vchab). (2.33) Then c _abgHl-l c _ VahacVh - 2ab),VhV C (h =gabh) (2.34) where Cc is the harmonic gauge factor. Contracting Equation (2.33) over an upper and lower index gives

H = Vah. (2.35)

Another useful relation is

_j__gabrdcrrd =abcl h v 2aC bd-2VcVagbd)hcd - Vc(2habVahbc- ab). (2.36)

2.4.2.3 Expansion of Curvature

Combining Equations (2.19), (2.32), and (2.33) gives

Cab = Hab + Idb (2.37)

'The proof that the derivative operator which annihilates the metric exists and is unique can be extended easily to the derivative operator that annihilates any given symmetric, invertible tensor. That is, given any symmetric, invertible tensor Vab, there is a unique derivative operator Va such that VaVbc = 0. Thus, we can unambiguously take the Va in the above equations to be the operator that satisfies Vagbc = 0.

· __·· Equation (2.29) then becomes

Rabc = Rabcd + V bH d- V abd 3 +Vb(IdHe) - Va(Id Hbg) + Ha H -db- H Hda+ O (h ) (2.38)

Applying Equation (2.23b) to Equation (2.38) gives

Rab = Rab + VcHab-VaHCb 3 +Vc(ICHab) - Va(IeCHceb)+ HabHcee- HecbHa + 0 (h ), (2.39) where Rab is the appropriately named function of gab. Using Equations (2.23c), (2.19), (2.20), and (2.39) we find /gR = \/-(l + D)(ga + I )Rab = {R + [DR+ IabRab+ Vc(gabHacb-gcdHed) =FfnFT-,n~abD c [ab74-.lc b c ] + [DIabRab + DgabVcHa +IabVCHb DVbHccb- Ia VaHcb H(HabHce-gbHcea)+ Vc(gabJcH-gacIbe)] + 0 (h3) }.

By using Equations (2.21), (2.22), (2.34), (2.35), and (2.36) and pulling the total divergencesto the outside, this becomes

ac /--gR = 7 {R - Gabhab- hab [( 4 Gab + gabR)gcd -_ (4G + gaCR)gbd gacgbdV 2 + gabgcdV 2 + 2gacVdvb _ 2gcdvavb] hcd} + total divergence+ 0 (h3 ), (2.40) where we've made the standard definition for the Einstein tensor:

Gab Rab- 1Rg29abR. (2.41)

General spacetimes allow for a cosmological constant. We include this in the gravitational Lagrangian by adding to Equation (2.40) the term

-2v/--gA = -- {-2A-gabAhab -4hab[(4gabA - 2gabA)gcd- (4gacA- 2gacA)gbd]hcd} + 0 (h3). (2.42)

The final line of Equation (2.42) has been arranged in a form that makes obvious how it should be added to Equation (2.40). The final result is

KX2£G = /--g [R-2A] = /g { [R - 2A]- [Gab+ gabA]hab

lhab [1 (4[Gab + gabA] + gab[R - 2A]) gcd _ (4[Gac + gacA] + gac[R - 2A]) gbd -gacgbdV 2 + gabgcdV2 + 2gacVdvb - 2gcdvavb] hcd} + total divergence+ 0 (h3). (2.43)

35 2.4.3 Expanding the Yang-Mills Action The gauge field part of Equation (2.10) can be expanded as follows. First note that applying the variation of Equation (2.13) to the field strength given in Equation (2.11) results in ab Fab+ DaAa - DbAa + fabcAbAc (2.44) where F is the appropriately named function of classical fields only and Da = Va -iaat a. Using only the symmetry of the metric factors and applying Equation (2.44), we find

ac bd, ~a c a bdA-a)- ggbb g = g [FabFcd+ 2Fab(DcA - - DdA ) c +(DaA - D Aa)(D Ad - DdA-) __+ _2fabFbAAO ~~~~~abcja +Ab (A 3 A (2.45)

However, using the antisymmetry of the field strengths, applying Equations (2.19) through (2.22), and inserting Equation (2.45) gives

vgg/-gpgac bdFa~abgcd 'a _~~~~~a a ab a ab b aa fabc~ a = { [FabFa + 4FabDaAab+ 2DaAb(DaAab DbAaa) +2 abcFa AbAcb aad b 1 aaa\1 [2(F aadaadhcd1a2ab2 _ Fa Faabh)b - 4(hbd- hg )Fab(D Ad - DdA)] + [2(Fab(FaF aadhchd ~-ab FaFaabhcdhcd)ab cd --] (FabFaadhhd _IF4abF b2h -1 FabFaab(h 2 2hCdhcd) + FabFcadhachbd]} + 0 (fields 3 ) (2.46)

This can be simplified by completing total derivatives and using

2 J£vmo _ 1 [ a a 1 Lacd a\ YMO c 1 FacdFa) (2.47) Tab =- 2 gab- 2 ac b 49abF lcd, _/::g~Sgab gYM where LYMOis the appropriately named function of classical fields only. Equation (2.46) becomes -gacgbdca a 4gyM~v~ = -' ab"cd 2 {[aFaab]=~~~~~~~~~~~~ -4[DaFaAab] - 2[AaD Aab _ AabDaDbAaafaFbA -faFabA A]b] 2 ab aeb aa a aa)_ gabra rcad~ - 2gym [Ta hab] - 4 [FaCb(DcAaa- DaA ) ~1abFaDcAad] hab +hab+hab [gM[2 _ Tacgbd -mgym Iabcd g 9'ef'aFaef F (gab y gcd - 2ag2gacgb) d + F+a~bcd} aeF abd] hcd} + total divergence+ 0 (fields3). (2.48)

Before writing the fully expanded action, the contributions to the action due to gauge- fixing need to be considered.

2.5 Gauge-Fixing

We regard 9ab as fixed with respect to diffeomorphisms and aa as fixed with respect to the gauge group G. We attribute the variations of gab and 4aa to transformations of hab

36 and Aa, respectively. Thus, we take the induced infinitesimal gauge transformations due to diffeomorphisms to be

Jhab = VaTlb + Vb??a + VanJ/hcb + VbT7Chca + 7lcVchab, (2.49a) JAa = AaVa?7 + cVcAa, (2.49b) Jaa = ac Va' + rcVcaa. (2.49c)

Acting with G, we get

DaOa a fabcAbac JAa = a + (2.50a) Jhab = 0. ~~a,(2.50b)

We need to fix the gauges on hab and Aa . We take the background-covariant gauge-fixing conditions 2 Ca(h,A) -Ca(h) 2 FaabA= O, (2.51)

Ga(A) - DaAaa = 0, (2.52) where Ca h)= _ h~a a). C (h)-Vbh -Vh (h ha). (2.53) By using the Faddeev-Popov [24] method and choosing Feynman-'t Hooft gauge factors, these each add a term to the Lagrangian

AIgf:h =- 2K2 /CaC 4 ___ - ~~~aab ca+ 2 X Fa bba c V/ (CaCa 249x2 Fa abF 2b_FbA a F bc) (2.54) 2 aC abba 2:X2 _ 1 gyma~ A29f:A- 2 ,/-gGcG (2.55) 2gYM respectively. Equation (2.51) is similar to an R~ gauge [25], which is here engineered to cancel unpleasant graviton-gluon cross-terms that will appear later. We will even- tually find that a convenient choice of gauge is ~ = 1, whereas ~ = 0 reproduces the traditional harmonic gauge. Equation (2.54) can be expanded using

-C Ca = h[- abgcdV2 + gcdvavb + gabvcvd - 2gacvbvd]hh 2 a 4 42 + total divergence. (2.56)

Likewise, Equation (2.55) becomes 1a a1ra Aab -2G G 2=- - 2DaAaaDbAab 1AaaDa'bAab + -- Dzxta.~bA-t total divergence. (2.57)

37 Of course, these gauge-fixing terms appear along with the Faddeev-Popov ghost Lagrangians through the application of Equations (2.49) to Equation (2.51) and Equa- tions (2.50) to Equation (2.52). The form of the ghost Lagrangian in a non-Abelian gauge theory is a standard result, and it is only slightly altered in a curved background spacetime. We get

A/LFP:A =_/-3L-ga6G (A + 6A)ab Serb _ &aj- [6abD2 _ fabc (DaAc + ADa)] ab -f(OD ~ (2.8(2.58)

Ghosts also arise from the gauge-fixing of general covariance. Since both a and A' transform as vectors, so does6(Fa~~~~~~~' ab"the combination 'aa o FaaAab. That is, (F Aab) = bAaCVa + 7 Vb(FaAac ) (2.59)

The ghost Lagrangian is then given by

ALFP:h --/= a Ca(h + h, A + A) b

-4-z, [ gabV - [Va, Vb] + habv + Cbva + VbCa - h[V, Vb]

+2g9adHCVC-g_ (FAVa + Vb(FaCAac))] ?b. (2.60)

To second order in quantum fields, we have:

ALFP:h = A/gq [-gab V2 + [Va, Vb] + 0 (h) + O (A)] 77b, (2.61) ALFP:A = /'a [_-abD2 + 0 (A)] ab. (2.62)

2.6 Combining the Pieces

The classical fields gab and aa are governed by the classical version of Equation (2.10),

1 a L = GO + LYMO= -22/A][R-A 4 \/FaabFaab (2.63) 4gy2 M By varying this action with respect to these fields, we get the Euler-Lagrange equations that govern their dynamics:

9C2 6L£wo _ 9 2 + gabA - 2 Tab, (2.64) Gab = Vf/W 3ga b -2 and DaFaab = 0. (2.65) We will enforce Equation (2.65) on a, but we will not enforce Equation (2.64). In this way we can study the behavior of an arbitrary gauge field in any spacetime back-

38 ground, but we ignore the warping of spacetime by the background gauge field. That is, we're making a "test field" approximation. In contrast, if we were to enforce both Equations (2.64) and (2.65) and then restrict attention to, for example, Minkowski space, we would be forced to consider only gauge field configurations with vanishing stress-energy. The most satisfying approach would be to force ab and aa to be arbitrary simultaneous solutions of Equations (2.64) and (2.65), but the necessary calculational techniques required for analysis subsequent to this point have proven elusive.

By combining Equations (2.43) and (2.48), applying the equation of motion (2.65) for the background field, and dropping total divergences, we can write

= 0 + iO(h)+ JO(h2)+ -CO(A2) + LO(Ah)+ ·· , (2.66) where

1 1~---.1 Fa aab -g[R - 2A] 4g Y=FabF (2.67a) X2 ~~4gym 1 2( (h) £O~~h)= = \=-g [Gab[~ab + gababA - AXTab12Tb] hab, (2.67b)

1( -ghab [ X 2 L 1 ab cd acgbd) c2 Faac abd £O(h2) =- 4JX24 ghab 2 9 9 g-g2+2 F F

+ 2(Gabgab A - 2Tab)gCd - 4(G C + gacA -2T Tc)gbd

-g acgbdv 2 + gabgcdv2 + 2gacVdVb 2gcdvaVb] hcd, (2.67c)

a Aafcabclcab b , CO(A2) - [AD=~~~~~a2a2 a a b a 2g/2[ +AD2 Aa A + A a DbDaADD b ++a.~ AfabcFcabA .b (2.67d) 9YM

L0(hA) =-- g/-F: [-gabDcAad + 2gad(DcAab- DbAac)] hb. (2.67e) 'lOh)2g2M YM

Note that CO(A) was eliminated by the equation of motion (2.65).

Equation (2.67e) can be brought into a more symmetric form. By integrating by parts on the first and third terms of Equation (2.67e), enforcing Equation (2.65), and dropping total divergences, we get

-Mhab [9bFaadDd + DaFabC] A /CFaabA (2.68) gYM gYM Repeating the procedure on the first term of Equation (2.68) gives the similar form, 1 a[cbaad _ aac 1 aaba LhA = -2--/gA a [--g F d - LIFJ Lab- 2 -gCaF Ab. (2.69) gYM gYM

39 We now symmetrize Equations (2.68) and (2.69) to get

£hA = 2--V {hab [D F + g F Dd] AC 2gYm

+Aa [-DaFacb - gcbFaadDd]hab + 2CaFaabAa } . (2.70)

Equation (2.54) will give contributions to CO(h2), O(hA), and LO(A2), whereas Equation (2.55) will only contribute to £O(A2). Adding the gauge fixing contributions to Equation (2.67c) using Equation (2.56) we get

1 O(h2) + ALo(h2)0(h 2) I2 ~-ghb [(gacgbd 1 ab cd)aV22 + aa ___ - )V _ gX2 4) - 2gd[V ,Vc] + 9 X2FaacFabd 4X 2 ---gh ab[(gacg d M - 2T ab)gcd - 4(Gac+ gacA- 2T ac)gbd hd, (2.71) +2(Gab+ gabA 2 2 where we've used the fact that

hab (gabVcvd - gcdVavb) hcd = Va (hVbhab - habVbh) - VhVdhcd + VahabVbh = Va (hVbhab - habVbh) (2.72) is a total divergence, which we've dropped. Adding the gauge fixing contributions to Equation (2.67d) using Equation (2.57) we get

+Af[gf:h 4 O(A2)+ A/gf:A + .. O(A2)

- 1 /A~ [4-gabjab[D2]- 6ab[Va,Vb] + 2 fabcFcab+ aaFbbc] AA2b,2 2gym gYM 1 (2.73) where we've used Equation (2.16) and

a. [Da, Db] = [Va, Vb] - iFabtr (2.74)

Finally we can add the gauge fixing contributions to Equation (2.70).

CO(hA)L(hA)+ A'gf:h""''O(Ah) - 2 {hab [DaFabc + gbcFaadDd] A + A [DaFacb _ gcbFaadD] hab = ~~V--2gy-----[D F { c bd A + A - d aa --YM2 C F +2(1 - )CaFaabAa}. (2.75)

Bringing together Equations (2.67a), (2.67b), (2.71), (2.75), (2.73), (2.58), and (2.60) we get £ I: L + h + £h + hA+ A2+ L,-2+ £O,2+ . .., (2.76)

_ __ where

Lo = V/-g[R- 2A]--- aab b, (2.77a) X2 ~4gyXm

= - 2 Gab + gabA - C2Tab] hab, (2.77b) - -- -- Ia lab- 2 Lh =h= 4X12 x/L-ha-----b [(gaCgbd _l gab9cd)(_V2 _ C2 o) + 2F --acFabd2gbd[Va, Vc] 9 YM ac + 2 (Gab + gabA - 2Tab)gcd - 4(G + gaCA--2Tac)gbd] hcd (2.77c)

hA - 2 v--'g{hab [DaFabc+ gbcFaadD] Ac

+Aa [-DaFacb _ gcbFaadDd] hab + 2(1 - )CaFaabA} (2.77d)

LA2 = i \/=Aa [gabab[D 2 ] _ 6ab[Va vb]

b + 2fabcFcab + 2 X2 FaacFbbc] A , (2.77e) YM 2 La2 = g_/Z-a[_6abD ] ab, (2.77f) 2 /,2 = g-7a [-gabV + [Va, Vb]] qb*7 (2.77g)

These expressions can be evaluated for Minkowski space by taking g = -1, V , and Rab = R = A = 0. The ellipses in Equation (2.76) indicate terms of higher than quadratic order in quantum fields. Equation (2.77g) will only contribute only to the renormalization of X (or simply to an infinite constant in the effective action, in the Minkowski space limit), so it will be ignored from here in. Physically, the terms in Equations (2.77) that are proportional to background field strengths represent magnetic-moment-type interactions of the dynamical fields with the background fields.

2.7 Compiling the Superfield

The term LhA deserves special attention. It is equivalent to

[L-"' = 6 6 A hb [L6 hab] A6 = 1 h, + [6 hab A . gab 6ac I aac9 L gb J 2 { 9ab [jA j h 6a L gab9 J } (2.78) If Equation (2.77d) were in this more symmetric form, we could rewrite the quadratic terms in L by using the superoperator V4o, 1 , where (g,a) is a classical

41 superfield. That is, we wish to write

h2 + LhA + A2 £p2 = (It X5L (I), (2.79)

where I- h, A) is the superfield of quantum fluctuations. Equation (2.77d) is the sum of two parts. One of these has differential operators acting only on hab while the other part has differential operators acting only on A . If we now choose ~ = 1, then these two parts are conjugate to each other. This is exactly what we want for the "off-diagonal" terms in Ve. With this choice of gauge, then, we have

42~vtv hab~=r (Vabcd Vabbf [ A' d(2.80)' V A ' where

2 Vabcd= h~~~~~~~~~4X2 LkVTabcd(72O)- -- + 2g2,--2YM(X (FaacF oF abd + FaadFabc I (gbd[Va,Vc] + gbc[va, Vd] + gad[vb, Vc] + gac[vb, Vd]) 2 a (G C + gacA - _2TaC)gbd2 - (G(Gbc + gAA - - 2-2Tbc )gad (Gad + gadA - '2Tad)gbc_ (Gbd+ g bdA- T )gac

+(Gab + gabA- -X2Tab)cd + (G + dA- 2T )ga ], (2.81a)

Vabab= 21 [-gabab[D2] _ jab[va, vb] + 2 fabcFcab + ,2 FaacFbbc] (281b) 2g2YM M Vabac = 1 [gbCFaadD+ gacFabdDd + DaFabc+ DbFaac], (2.81c)

V = + [gcbFaadDd+ gcaFabdDd+ DaFacb + DbFaca]. (2.81d) 4gym

We have defined the tensor

Tabcd 1 (gacgbd + gadgbc _ gabgcd) (2.82)

We will later need the inverse of T in d dimensions, which can be determined from

Tabef (Tefcd + 2 d-gefg ) .abcd,= (2.83)

where ifabcd is the identity on the d(d + 1) dimensional space of symmetric 2-tensors4. 2~~~~~~~~~~~~ 4 The projector onto the space of traceless tensors in four dimensions is PT = (I + T), while the projector onto projectorthe spacespaceonto of "pure"pure trace" tensors is Pss =-](lI - T).T). InIn aa spacetime of dimension d,

_ _ I·__·I _I__ _ Finally, we can rewrite the source term (2.77b) as

£h= L4= [ ·-2 ( {Tl/Aa)_-- habV -I. (2.84)

Thus we have L £Lo + L + L2 + L2. (2.85)

2.8 Renormalization

Beginning from this section, we specialize all calculations to Minkowski space. We define the generating functional Z as

Z - DgDAeif dsC[g,A] = J DgDAeiS[gsA]

- JDhDAeis[haA] = eiw[9,a]g (2.86) where, in the last line, we have defined the effective action W as a functional of the classical fields. To evaluate W, we first gauge fix and then manipulate terms as in Sections 2.3 through 2.7:

eiW[g,a] = DhDADr )ea{ i S[ g h a A]+A S gf :h+As gf:A+/ SF P:h+As FP:A }

f JDhDADaei{So[,a]+Sh+Sh2+shA+sA2+s2 } = JDIDacei{S[9ai+s++s*2+sc2 }

We can eliminate the S. term by writing

+4,2gti)-~-g~2~ 3 ~ + ~~ = [()J + v J] vl'-e [(I +-- ~~~-v- 1J] - 4.()J(~V()-1JtV-l1J. (2.87) Then we shift - = I+ V 1iJ, which has no effect on D (that is, D( = DV4). Renaming 4Y to 4, this leaves

eiW[,a]D i{So[9a]- 4fddXJtv1J+Sp 2+S% }

In this expression, both eS ° and the -term are constants and the Gaussian the T appearing in these projector equations is given by Tabcd = abcd - 2gabgcd. The T appearing in Equation (2.81a), however, is given properly by Equation (2.82) in any spacetime dimension. The identification of these objects for d = 4 is a coincidence.

43 integrals over and a can be evaluated as functional determinants:

ezW[9a] ~ exp iSo - ddxJtVl det[V]-2 det[V ]+1

= exp {iSo - JddxJV lJj- ½Trln[V,] +Tr ln[VQ], (2.88) where Equation (2.77f) tells us that V, = -D 2 . Thus,

iW[g, a] , iSo[g, a] - JddxJtV J - Tr ln[V,] + Tr ln[VQ]. (2.89)

Note that each term in Equation (2.76) is invariant under both diffeomorphisms and a formal gauge transformation of the background gauge field where Aa is treated as adjoint matter. Since each term of Equation (2.76) is a gauge invariant functional of the classical field, so must be each term of W. Thus each term must be proportional to f ddx/ FaabFaab,at least to first order.

We begin evaluating Equation (2.89) by turning our attention to the V, term. First we extract out an overall normalization constant from the superoperator in the following way:

Vabcd Vabbf V1 = Vaecd oabef g ab 0 2 [ghcd + Mhcd Ohbf( 2 0 0 11) 292M~2gyM t _ oaecd{ _ 3abgef + M bef + Nabef (2.90) where 2 1 Mabcd= - abcd(_2+)-'Lo=~~~~~X (-b~ + j~d-4 abgf) (-a)-l[FaecF[FaecFafl

2X2 (2ab + 1jd-gabg f) (02)-i [Tecgfd 4Tefg - 4T 9g|, (2.91a) Nabab = 9YM (_2 [Faa Fbbc], (2.91c) ~YMYM2 1 2 2 MbAb= - gabsab (1 + (-0 )- D ) + 2fabc(_a )-Fcab (2.91b) a a b- +a2= X2 (T abgfcFaedD - 2DeFaf]l) ef cab NMbab= b = (--O2)-labjab( 1 [Faa+ (--2)-1cF bbc]_)-F -' 2fabc(-T _ , ef - TCdf], (2.91e)(2.91a)(2.91c) g2 M oabac 2-3C2 2 2'--l1fLaedl" eIf a d-4 gab \29d + - Ige- - ,dtD- -{ DeF ar ] (Tabef -2-'ge(91) Oac" =(--a2)-l[-g fcFadDd - DeF acf]la°bi (2.91e)

__ ___· __ Observe that each term in Mh and N is quadratic in Fb. So,

-21Trln[V~] = - 1Trn[N] - !T In [ Moh + '"'- [ O +NMAln[]-1 2T O_ MA +N 1 [ M,2+O+O_ MhO++O+(MA+N) 2 OMh+(MA+ N)O_ (MA+ N) 2 + 0_0+ + ... (2.92)

We can drop the Tr ln[N] term, as it simply represents an infinite constant. Working to the order of the fields that appear in the the classical action, we can also drop the Mh,,MAN, N2 , MAO, and NO terms. The ellipses in Equation (2.92) indicate that all the excluded terms are also of higher order in the classical fields. Thus we have,

- 1 o0 + ] 1( °- ]] -2T1in[V] 2Tr( O_ MA+ N - ( 0 M2 + +) - 1Tr[Mh- 1O+O_]- 1Tr[MA - 1 MA+ N - 1O-O+] -Tt[Mh - O+O_1- Tr[MA- MA + N], (2.93) where the last line follows from trace cyclicity.

The MA and MA terms from the second trace, along with the S0 and V terms in Equation (2.89), are exactly what would be found for a Yang-Mills theory evaluated to the one-loop level in the absence of gravitation. Evaluating these terms for d = 4 and inserting the form of So therefore gives

iW[g, a] - Tr[Mh- O+O_]- 1Tr[N]- ddxtV

(2.94) 4 ---2 + (47r)22 n ] FabM, where bo = C2(G), E is the background energy scale, and M is a renormalization scale at which gyM(E)IE=M = 9YM is imposed. Ignoring gravity, this would lead to

~(gYM) -(47r)2 yM We now return to evaluating the remaining terms in the effective action. We'll start with the J-term. Expanding VDas in Equation (2.90), we get

JddxJtV1J - 4ddxJt[[ 4 (O_- MA+N +f Mh-O+O_ MhO++O+(MA+N) O_Mh+(MA + N)O_ (MA+ N) 2++00+ +"'. [ -4C2T-1(_02)-1 0 J. (2.95) t~~ _292Y (_,02)-1J

45 p--+

Figure 2-2: The schematic Feynman diagram represented by the functional trace -Tr[Mh]. A momentum p circulates in a virtual graviton loop coupled to external gluons of momentum k.

Since J = (Tab o) , we can see that all terms in Equation (2.95) have four or ( t more powers of Fab. Since we are only working to the order of fields appearing in Lo, none of these terms make any contribution to the effective action. The final terms, - 1Tr[Mh- O+O_]- Tr[N], are each evaluated in turn in Section 2.8.1, using an ultraviolet cutoff Auv in d dimensions.

2.8.1 Computation of Functional Determinants

The term - Tr[Mh] in Equation (2.94) represents quantum contributions to the gauge field from integrating out a single graviton loop, as in Figure 2-2. It can be evaluated as follows:

-Tr[Mh]

4 2 J-ddX]abcd?2{ 2- (-O2) (tb labcdLco 9 1 ( a ef+ + d-2-gabf) _ (Q g))-l[FaecFafd] } -2 (Tabef+ --gabgef)(- ld-(ab\ 2)-l[Tecgfd- 1Tefg~cd- Tcdge]} 2 d-2 i~~~~ _ -1 X (ld(d+1)-4 + 2~ - (d- 4)(d+ 1))/ ddx {a(-a )- [FaFa ]} whee w4gw M 2 2 tf (2.96) where we've used the fact that

1 g9abT = - (d - 4)Faab = (d - 4)£o. (2.97)

9YM

46 P 4

k -*+ k-*

2 gYM

Figure 2-3: The schematic Feynman diagram represented by the functional trace -1Tr[N]. A momentum p circulates in a virtual gluon loop coupled to external gluons of momentum k.

We now evaluate the integral as

ddx{(_2)-1[Fab Faab ] = ddk FaFaab ddp 1 I I(27r)d ab b J(27r)dp2 f ddk b aab pEdd d- - (2ir)I fa aab dP E1- J (27r)d Jtab PEj(27r)d p2

d ab ' d 2 =(-7r)(2-r)d F b (27r)da~d2 / 2) irEppd-dPrEP -2i (Ad-2 _ Ed-2 ) J ddFad . ab.P Faab (2.98)

So, finally

i [ !(d(d+ 1)- 4 - (d- 4)(d + 1 d- U -2 Ed-2 - r[Mh = t4 92 (47x)d/2r(d/2) -. YM d-2 ddXFaFaab. (2.99)

The term - Tr[N] in Equation (2.94) is a gauge-fixing contribution from Equation (2.51). It is an integration over a gluon loop, as shown in Figure 2-3. We evaluate

47 p-4

9 gYM YM

Figure 2-4: The schematic Feynman diagram represented by the functional trace 'Tr[O+O_]. A momentum p circulates in a virtual gluon-graviton loop coupled to external gluons of momentum k.

this, using Equation (2.98), as

-2T[N] 292Jd x ab {(-2) -1 [Faa Fbbc] } 2 2 1 -49 2 Ad- - Ed- ' (2.100) = -i g(4r)d/2r(d/2 ) d-2 JXFa aab

The trace Tr[O+O_] represents a process where the external gauge field emits and reabsorbs a virtual graviton, as in Figure 2-4. This becomes

=1 ddX 12o+oI | 2 IdX+ OabOca°-cab} }

= 2 dx (_2) -1 [gfcFaedDd+DeFafc (ab + d-4 abg ) 2 ( [gI' { + L] (z + g g j gYM ) X (_092)-1[-gicFhagg- DhFc]l[abhi

X2 /dkdp{- 2g- f()d { a[geFacf(k)(-p- k)f + kCFade(k)]

( d + d-9ab gcd) ( )2 [gbeFa(-k)p 9 + kaF(eab-k)]}

In passing to momentum space, we dropped the gauge field part of the covariant derivatives, as these will only produce terms which are of high powers in the classical field. To the order of fields in which we are working, the classical equation of motion

___II for aaa, Equation (2.14), becomes kaFaab(k) = 0. Applying this, we get 2 I ddk ddp{ 1 (27r)d(27r)d Faa(k)Fcd(-k)p2(p+ k)2

x [(d+1d2 2 ) gacpbpd+k2gacgbd]}.

The k2 term produces a higher derivative term in the effective action whose properties we are not interested in. Ignoring it, we get

- 2 4 - (d - 4)d(d2)) Ad- 2 _ Ed- 2 111[0+0_1 __ -i1 ddXFaFaab. 5 4 g2 (4)d/2r(d/2) d-2 (2.101)

2.8.2 Extracting the 3 Function Combining Equations (2.99), (2.100), and (2.101) of Section 2.8.1 together gives Ad-2 Ed-2 - Tr[Mh- OO] - Tr[N] -i 2 (47r)d/2(/2) UV IIddXFa aab 2 4 ~gY~m(4rd2Pd d- 2 J ab' (2.102) where C(d) = 2 d(d + 1) - 4 - d)-dd)----(d -4)(d - 1)(d + 2) (2.103) is a function that is positive-valued for all integers 1 < d < 8 and negative-valued for d = 1 and integers d > 8. Note that C(4) = 6, which is the maximum for real positive d. The ultraviolet divergence in Equation (2.102) is regulated by counter terms- which we've not been writing - whose values are determined at energy scale E by a renormalization condition at scale M. Subtracting divergences by including these terms gives:

2 F _X C(d) Md- 2 - Ed-2] 1 Ad a raab - Th[Mh-O0+O-]- T4[N] I 2 ~-_I.1~4 [g92 (47w)d/2r(d/2) d-2 ]jJ - -ab' (2.104) Evaluating this for d = 4 gives

1 922 3 2 2 2 bo W[g,a]- -4 Jd4xg. + g (4-) (E - M) + (n -- Fab (2.105) gYM (4)2 M 2 Thus, when E is differentially close to M - and only then - we define the one-loop running coupling constant at scale E by

2 2 1 _ 1 X 3 2 _ bo E + ~ M! (E 2 2 ) + in (2.106) gyM(E) 2 gyM(M) 2 gym (M) 2 (47r) 2 - (47r) 2 M 2 ' 49 Under these conditions we find

2 ( d ( -( 4 )2(2EdE) + bo2 2dE (2.107) gym g--y- (-72 ( 47r) E and thus,

dgym = g d( /gd(1/gbo 2 /3(gymE) ~ - ~gyP3d(1/m) _3 Eb 3 (2.108) dlnE(Y =2Y dE (4 X)2YM (4 7r)22YM

Using the definition X2 = 167r/Mp, we find that the unknown coefficient of Equation (2.2) is now determined to be a =-3 0.95. 7r The form of the running coupling can now be found by integrating the 3 function. Equation (2.107) can be integrated using the integrating factor exp { x3 E 2 This yields

I1 2 2 1o Ed gyM(E)2 (M)> exp {3C (E2_ M2)} +2 4bo 2 dk 2 } gym(E) gy ()2 47 ( 17r) J keXP S (4r)2kLI (2.109) Notice that for E > M, 9YM (E) will always be less than the value it would have had in the absence of gravitation. Indeed, as E -- o, gyM(E) -- 0 independent of the values of b0 and gYM(M). That is to say, the addition of gravity to a pure Yang-Mills theory renders its coupling asymptotically free, even if it were not so before adding gravity. In fact, near Me the coupling turns off very rapidly in comparison to the usual logarithmic running. Remember, however, that any discussion of the theory at or above Me is dubious at best. It is interesting that quantum gravity perturbations can cause gauge couplings to run even in theories that exhibited exact conformal invariance, and thus vanishing / function, before gravity was added. Two notable examples in four dimensions are pure U(1) electromagnetism and = 4 super-Yang-Mills [60, 61, 62]. For these theories, the exponential integral in Equation (2.109) has zero coefficient, so we are left with

a(E) gyM(E) a(0) exp{ -3 () 2E2}. (2.110)

The couplings in such theories run down from their infrared values as Gaussians of width /76Mp, corresponding to a 10% reduction in a at about 0.1Mp.

2.9 Enlarging the Matter Sector and the Gauge Group

We will now examine what happens to the preceding analysis if we include scalars, spinors, or additional gauge fields. Since we are only interested in the renormalization of gauge couplings, we do not expand our new scalar and spinor fields as quantum

__ _._ ....---____ perturbations about a classical background. Therefore, every term in the Lagrangian containing these fields is of quadratic order or more in quantum fields. Thus, adding these fields doesn't change any of the terms already calculated in Equation (2.76). Each new field simply supplies one new term which is quadratic in the matter field and zeroth order in all the other fields. These terms have no effect on the functional integration over hab and Aa as described in Section 2.8. That is, adding matter to our previous discussion will not change the terms already present in Equation (2.108), but will simply add to them. Indeed, these terms will contribute in exactly the same way as they did in the absence of gravity. Namely, they add to the logarithmic running. So, adding matter to our previous discussion only changes the coefficient b0 in the function and has no effect on the gravitational term. Moreover, the matter contribution is exactly what it was in the absence of gravity. The addition of new gauge fields is slightly more complicated. Each new field contributes a term to LO and corresponding terms to Lh (via Tab) and Lh2. Likewise, the th gauge field brings in a LCA term and an LhA. cross term. Of course, gauge- fixing each new symmetry brings in a new gauge-fixing term and a corresponding ghost field. We also need to augment the gauge condition of Equation (2.51) with an additional term for each new gauge field. This adds additional couplings of the gravitational ghosts to the new gauge fields, but these terms will drop out to second order in the quantum fields. Each LA2 and LhA, receives a similar contribution from the cross terms generated by gauge-fixing. Gauge-fixing in this way also generates new cross terms between gauge fields. With so many graviton-gluon and gluon-gluon cross terms, we need to enlarge the definition of the superfield to include the graviton and each of the new gauge fields. The net effect of all this is that to second order in the fields, we get

N L ;: L + + 2 + EL,, +.., (2.111) i=1 where N is the number of gauge symmetries,

N Lo = -1 4g 2 (i)abF'(i)a aab (2.112a) 1Tab Vab

0 A%a L4=f=, O=b (2.112b) 0 A(2a

habtV h abcd Vbbf .. abbbf d e b bV2= l~ab f (2.112c)

A(5)e V a bcd Faber Ab LN~ Al)f 2~ ia

a_(iaLa2 _~b2 D a~~i ' (2.112d)

51 and

1 r N 2 Vabcd _ [abcd(_a2 - 0) + 2 (FaacFabd(i ) +FaadFabc) h L T:K2 (o°) + - '(i) (i) ) 4X2 ~~~~~~i=lgi

+ 2(Tacgbd + Tadgbc- Tabgcd + Tbc gad + Tbdgac Tcdgab)] (2.113a)

Vbab 1 [-gab6ab[D2] + 2f(a) c() + f() c () (2.113b) va[__ iCF)dbdD()d +D +DbF ] (2.113c) 2gi2

vabab = a Fbcb (2.113e)

Equations (2.112) and (2.113) are written for a Minkowski space background. The energy-momentum tensor, Tab, now refers to the total energy-momentum tensor of the background fields,

N -abg2d a.ab[r(i4dcr(i)) -1 r() +bac (b (i) +1 aF(nacd1(i)c Jab a ) (2.114) i i The covariant derivatives appearing in the above equations include a term for each gauge symmetry. However, the ghosts and gauge fields that they act on are all in adjoint representations of their respective symmetries and singlet representations of every other symmetry. Thus, these derivatives don't produce any unexpected cross terms.

We now need to evaluate the effective action. The a term can again be shifted away, as in Equation (2.87), producing a constant shift in the effective action. We again find, as in Equation (2.95), that this shift makes no contribution to lowest order in the field strengths. After extracting an overall normalization from the remaining quadratic terms and performing functional integration, we need to expand in powers of the fields. We again find that all of the terms of V, and most of the terms in V~are of higher order in the classical fields than appear in the classical Lagrangian. This includes every term involving a power of Vij. The result is that every surviving term of the operator whose trace we are evaluating is of exactly the same form as those we evaluated in Section 2.8. One copy is produced for each gauge symmetry, and each is multiplied by its own coupling constant. The exception to this is the graviton contribution, which is instead equal to a sum over multiple copies of the form presented in Section 2.8. Thus we only need to evaluate N copies of the integrals already shown in Section 2.8.1. Once this is done and terms are recombined to give the N running gauge couplings, we find that each one get renormalized separately and in exactly same way as in our analysis of a single gauge field.

II Combining these observations about adding matter and extending the gauge symmetry, we can now see how the analysis applies to a theory with arbitrary gauge and matter content. The d function for each gauge coupling is equal to what it would be in the absence of gravitation plus the new term, which is identical for each coupling. The only possible exception to this would be if the original theory is supersymmetric. In this case one might include gravitinos, which we have not included in this calculation, since including only gravitons breaks supersymmetry. We noted at the end of Section 2.8 that the addition of gravity to a pure Yang- Mills theory rendered the coupling asymptotically free. This, in itself, is not so impressive, since all pure Yang-Mills theories are already asymptotically free. We can now, however, make a much stronger statement about the gauge theory coupled to matter. Since the running is of the exact same form in the presence of matter as without it only the value of b is changed - we can now see that the addition of gravity renders all gauge couplings asymptotically free even in the presence of arbitrary numbers and types of matter fields. Of course, the meaning and utility of asymptotic freedom is somewhat ambiguous here, since none of these calculations should be extrapolated too near to or beyond Mp.

2.10 Coupling Constant Unification

Now consider a theory in the absence of gravity with N gauge symmetries and matter content such that at some high energy all couplings take on the same value. Defining the symbol yi = 1/g2 for the ith gauge coupling, each one runs as

E2 yi(E) = y7+ In - . (2.115)

The condition for unification is that there exists an energy Eo such that for any pair {i,j} we have

2 (4-7r) y~ _ y~ Eo _-(Ir)(.o ° =in-I . (2.116) 2 !b;i-/20 = n' When gravity is added to the theory, the form of the running follows Equation (2.109):

f~x2tE2_M2)}+2 boE dk 2 y\5(E)= {3 (E2-M2)+-y-exp- 2 4 3 (E - k2)} . (2.117) Y 0i ~__2 IT4-, 1 ( 7r)JM k~ ex (47r)2

The condition for unification at some energy Eu now becomes that for any pair {i, j}, we have

2 byep {3 ?2 (E2 _ M2)} =J exp{3( 2 (E 2-k2)}. 2 b-i9(4~~~~~~~)2~~~ exp 7r2kex 3(-(7 ) (2.118) But, since we assumed the theory exhibited unification before gravity was added, the ratio (Ayo/Abo)ij in Equation (2.118) has already been determined to be a constant

53 for all {i, j}. Equation (2.118) is therefore independent of the values i and j. Thus, any value Eu that solves the equation E fEu dkdk in Eo -- dk exp 3(4 )2 (M - k2)} (2.119) is an intersection point for all N gauge couplings. Note that since the exponential function is always positive, the integral in Equation (2.119) is a monotonically increasing function of Eu. Thus, Equation (2.119) has exactly one solution and the gauge couplings intersect at exactly one point. If E0 < Mp, then it should be true that Eu E0 . Assuming this is the case, then Equation (2.119) can be approximated by

InEoMn ;f:odkEo dkEexp 1 3 (M2 - k2) + Ex -o 3E 2 (M2 2 2 lnM ] k L (4r)2 Eo 0 ()2(rexp F0). (2.120)

Thus,

Eu Eo + Eo / J [1-exp{ 3 '(2 (M k2)}] exp {3 2 (M2E (2.121) Now explicitly using Mp > Eo > M, we get

Eu P_ Eo [1 + (M (2.122)

Examining Equation (2.122) for the numerical values E0 = MGUT ~ 1016 GeV and m Mp ; 10 GeV reveals that the new intersection point is shifted from E0 by less than a part per million. The one-loop flows of a - 1 - 4r/g 2 for a theory with running quantitatively similar to the minimally supersymmetric standard model with gravity are shown in Figure 2-5(a). The dramatic switching-off of the Yang-Mills interaction near Mp is made more apparent when the couplings themselves are plotted, rather than cr-1 , as in Figure 2-5(b).

2.11 Phenomenology

In this section, we will briefly comment on a few possible experimental implications of the preceding analysis. First, however, we will discuss the physical content of the result. As apparent from Figure 2-5(b), the Yang-Mills couplings turn off so quickly near Mp that free field theory (zeroth-order perturbation theory) should become an excel- lent description of the gluon dynamics. To within the accuracy of this calculation, high-energy gluons do not couple to anything but gravity. For a theory with only Yang-Mills couplings (that is, no Yukawa couplings, masses, or scalar self-couplings), this implies that all physics near Mp is well described by free fields coupled to grav-

______1__ ____1____11 ru7 -

fUO 60-

. 40-

I '~"'. "'-. ~.... 30 X~~~~~~~~~~ ~~...... 20

------'"~. -'- . -- 10

0 2 4 6 8 1 1 1 4 ' 168 20 (a) loglO(E/GeV)

1.4

1.2

- 0.8

0.6 ...... ------0.4

0.2

2 4 6 8 10 12 14 16 18 20 (b) log1 0(E/GeV)

Figure 2-5: In Figure 2-5(a), the three Yang-Mills couplings of an MSSM-like theory - 2 evolve as straight lines in a plot of a _ 47r/g versus log10 (E) when gravitation is ignored (dashed lines). The initial values at Mzo 100 GeV are set so that the lines approximately intersect at 1016 GeV. When gravity is included at one-loop (solid lines), the three lines curve towards weaker coupling at high energy, but remain unified near 1016 GeV. In Figure 2-5(b), g is plotted for the same theory. All three couplings rapidly go to zero near Mp, rendering the theory approximately free above this scale.

55 ity. On the other hand, gravitation itself may be extremely complicated at this scale. Nevertheless, it is somewhat reassuring that all of the most complicated and potentially divergent high-energy dynamics is sequestered in the gravitational sector. It is difficult to say how seriously this discussion should be taken, however, since all of the most interesting statements are being made near or above the theory's cutoff. These statements can only be strictly meaningful if the renormalization group for the quantum effective action flows to a nonperturbative ultraviolet fixed point, as discussed in Section 2.1.2. We saw in Equation (2.122) that the unification scale of a grand unified theory (GUT), as predicted from low-energy data, is slightly raised by the inclusion of gravity. One might hope that such a raising of the GUT scale is significant enough to effect predictions of GUT phenomenology, such as the proton lifetime or other "undesirable" consequences of GUT theories. We believe, however, that all effects are likely to be in the parts per million range or smaller, which is an accuracy that GUT scale experiments are not likely to reach for some time. We should consider the case where the measured values of low-energy couplings have experimental uncertainties. In this case, the hypothesis of coupling constant unification can be tested against some accepted goodness-of-fit parameter, such as x2. Accordingly, the best-fit values of the unification scale and coupling will have associated fitting uncertainties and potential cross-correlations. When gravity is taken into account in the 3 function, we expect the best-fit unification values to change according to the discussion of Section 2.10. An important open question whose answer is not obvious at this point is how the fitted uncertainties and 2 will be affected by the inclusion of gravity. Can gravitation make non-unifying models (such as the non-supersymmetric standard model) unify to within experimental error, or can it push generic almost-unified models (such as the minimally supersymmetric standard model) out of a statistically acceptable range? The first of these possibilities would suggest that coupling constant unification is not evidence for low-energy supersymmetry. The second possibility would either rule out many supersymmetric GUTs (if the coupling constants are no longer unified) or make the conditions for unification so strict that if unification persisted with the addition of gravity, the case for low-energy supersymmetry would be significantly strengthened. There has been some speculation in the literature, mostly inspired by the possibility of building large extra dimensions in string theory, that the fundamental scale of quantum gravity may in the multi-TeV range rather than at 1019GeV. It has been noted that among the predictions of such scenarios is the creation of microscopic black holes in high-energy colliders and cosmic rays experiments. The present calculation suggests that such experiments should also observe a reduced strength of standard model interactions, presumably disturbing the properties of extensive air showers and hadronic jets in some way. Working out the details of these experimental signatures, however, is beyond the scope of this thesis. A rather different probe of the TeV-scale gravity scenarios is suggested by Equa- tion (2.110). There, we saw that the fine structure constant of a theory with vanishing ,3 function in the absence of gravity runs down from its zero-energy value as a Gaus- sian of width V/76Mp. Quantum electrodynamics is such a theory at energies below

_ ...... · the threshold for electron-positron creation, approximately 1 MeV. Without gravity, the fine structure constant should be a genuine constant at these scales. Logarithmic running should only set in well above threshold, where the electron mass can be ignored. With gravity included, the fine structure constant should run as a Gaussian at low energies, with possible effects on atomic spectra and electromagnetic nuclear decays. However, even with a speculatively low Planck scale of 1 TeV, the shift in the fine structure constant at threshold would be a part in 1012, which does not appear to be within experimental reach.

2.12 Commentary

We have found that for a very generic class of four dimensional gauge theories, the addition of gravity adds a term to the one-loop 3 function of the form

3 E 2 A3grav(gyM, E) = -gYM; -2 (2.123)

The new term renders all Yang-Mills couplings asymptotically free. In fact, the couplings turn off from the usual logarithmic running quite quickly near the Planck scale. This correction to the running might be observable if the world is governed by certain unorthodox theories of particle physics. For a theory whose gauge couplings exhibited unification before gravity was added, the unique unification point is maintained, but shifted to a slightly higher energy (Eu/E 0 , 1 + 3 E/Mp) and weaker coupling. The following are some important caveats about the preceding calculation: * We have only used an effective theory for gravitation, and are trying to interpret it somewhat close to its cutoff. This is the strongest critique of the calculation. If quantized general relativity has an ultraviolet fixed point in its renormalization group flow as discussed in Section 2.1.2, however, the results given here may, in fact, be reliable to higher energies. * Quantized general relativity should properly include terms of order R2 at one- loop, as discussed in Section 2.1.1. These terms usually have little effect on observable physics, but they may begin to become important near Mp. Also, the coefficients of these operators should be allowed to run, as should X and A. The running of these coefficients could become important in about the same range as where the gravitational contribution to the Yang-Mills function become important. Again, however, an ultraviolet fixed point may mitigate these problems. * Restricting the background spacetime configuration raises the problem of back reaction. The "test field" approximation may not be valid at very high energies. That is, if the background gauge field carries a very large energy density, it should be allowed to warp the background spacetime. The caveats just given are issues of principle. There are also several warnings to be made of a less fundamental nature with regard to model building:

57 * We live in an approximately Robertson-Walker spacetime, not Minkowski, which is well modeled as having a small positive cosmological constant. We do not expect large scale structure to have dramatic effects on the ultraviolet physics discussed here, however, so this is a minor point.

* We have not considered extra dimensions as they are often implemented in realistic models of particle physics, whether they be compactified, orbifolded, deconstructed, warped, or otherwise of popular interest. We have only considered the case where all matter fields, gauge fields, and gravity propagate in the same bulk space and none of the fields are confined to lower-dimensional defects.

* We have made an inadequate treatment of supersymmetry in the gravitational sector by not including gravitinos. If we imagine that below Mp supersymmetry is broken only in the gravitational sector, while remaining valid in the gauge and matter sector, then the analysis given here is intact. However, this restriction may be too unnatural for serious application to some models.

- K Chapter 3 Black Hole Effective Field Theory

In this chapter, we will show that in order to avoid a breakdown of general covariance at the quantum level, the total flux in each outgoing partial wave of a quantum field in a black hole background must be equal to that of a 1 + 1 dimensional blackbody at the Hawking temperature. Specifically, we attempt to formulate an effective quantum field theory only in the region outside of a black hole and discover that this theory is anomalous unless the stated radiation flux is included. In Section 3.2, we will turn our attention to the physics of d-dimensional black hole spacetimes in classical general relativity and the behavior of scalar wave equations in such spacetimes. Then, in Section 3.3, we will derive some useful but nonstandard results regarding the properties of blackbody radiation in spherical coordinates and in d spacetime dimensions. Finally, in Section 3.4, we will attempt to formulate an effective field theory of scalar field modes that live only outside a black hole event horizon.

3.1 Introduction

Hawking radiation from black holes is one of the most striking effects that is known, or at least widely agreed, to arise from the combination of quantum mechanics and general relativity. On the other hand, potential sources of conflict between the central principle of general relativity, general covariance, and quantum theory may exist in certain situations in the form of gravitational anomalies. Both the anomaly and Hawking radiation result from, in a certain sense, ambiguities of the quantization process in curved spacetimes and both vanish in the absence of spacetime curvature.

3.1.1 Hawking Radiation Hawking radiation originates upon quantization of matter in a background spacetime that contains an event horizon - for example, a black hole. One finds that the occupation number spectrum of quantum field modes in the vacuum state is that of a blackbody at a fixed temperature given by the surface gravity of the horizon. The literature contains several derivations of Hawking radiation, each with strengths and

59 weaknesses. Hawking's original derivation [28, 29] is very direct and physical, but it relies on hypothetical properties of modes that undergo extreme blue shifts, and specifically assumes that their interactions with matter can be ignored. Derivations based on Euclidean quantum gravity are quick and elegant, but the formalism lacks a secure microscopic foundation [42]. Derivations based on string theory have a logically consistent foundation, but they only apply to special solutions in unrealistic world-models, and they do not explain the simplicity and generality of the results inferred from the other methods [63, 64]. In all these approaches, the Hawking radiation appears as a rather special and isolated phenomenon. Here we discuss another approach, which ties its existence to the cancellation of gravitational anomalies.

3.1.2 Anomalies and Anomaly Driven Currents An anomaly in a quantum field theory is a conflict between a symmetry of the classical action and the procedure of quantization (see [65] for a review). Anomalies in global symmetries can signal new and interesting physics, as in the original application to neutral pion decay r° - -y7 [66, 67] and in 't Hooft's resolution of the U(1) problem of QCD [68, 69]. Anomalies in gauge symmetries, however, represent a theoretical inconsistency,leading to difficulties with the probability interpretation of quantum mechanics due to a loss of positivity'. Cancellation of gauge anomalies gives powerful constraints on the charge spectrum of the standard model, which were important historically [70, 71]. A gravitational anomaly [32] is a gauge anomaly in general covariance, taking the form of non-conservation of the energy-momentum tensor. A gravitational anomaly can only occur in theories with chiral matter coupled to gravity in spacetimes of dimension 4k + 2, for integer k. The chiral matter could be either a chiral fermion or a 2k-form with an (anti-)self-dual field strength. An important case is the self-dual scalar field in 1 + 1 dimensions. This is a scalar field constrained to obey

0a4 = EabO('0. (3.1)

That is, it has only right-moving modes and is thus chiral. In this simplest case, which will be crucial for us, the anomaly then reads [32, 65, 72]

1 c ~VaT= 96 l +/ da r c (3.2)

The energy-momentum tensor of the chiral scalar is therefore not conserved in curved spacetimes [32]. There are several cases in physics where anomalies have been connected to the existence of current flows. Pair creation in an electric field has been related to a chiral anomaly [73]. The existence of exotic charges on solitons, with or without the

1'Athird kind of anomaly is a "conformal anomaly", which is a quantum contribution to the trace of the energy-momentum tensor Taa . This is the source of scale-dependent renormalization effects in otherwise scale-free theories.

__ existence of zero modes, has been related to anomalous charge flows that arise in building up the soliton adiabatically [74, 75]. Especially closely related to our problem is the connection between anomaly cancellation and the existence of chiral edge states in the quantum Hall effect [76, for example]. The effective action modeling the electromagnetic field in this state involves the Chern-Simons Lagrangian density confined to the sample area:

S oc d3 xeA(x)abCFab(x)Ac(x), (3.3) where the EA function is 1 within the sample region A and 0 outside. The gauge transformation Aa - Aa + O~adoes not leave this action invariant, because after integration by parts a term appears from the derivative acting on eA. The extra term is confined to the boundary and is proportional to a there. This variation takes the same form as the anomaly of a 1 + 1 dimensional massless, charged, chiral fermion field on the boundary. Since the theory we are modeling is gauge invariant, an adequate effective theory must cancel the boundary gauge variations. This motivates one to expect the existence of massless chiral edge states, whose anomaly cancels the boundary gauge variation. Such states do in fact arise, as can be proved from more microscopic considerations. We will demonstrate a similar phenomenon for gravitational anomaly cancellation, with Hawking radiation playing a role analogous to the edge current.

3.1.3 Hawking Radiation and the Conformal Anomaly

Many years ago Christiansen and Fulling [77] showed that it is sometimes possible to use an anomaly in conformal symmetry to derive important constraints on the energy- momentum tensors of quantum fields in a black hole background 2. This anomaly appears as a contribution to the trace Ta of the energy momentum tensor in a theory where it vanishes classically. By requiring finiteness of the energy-momentum tensor of massless fields as seen by a freely falling observer at the horizon of a 1 + 1 dimensional Schwarzschild background metric and imposing the anomalous trace equation everywhere, one finds an outgoing flux given by GN2 f2;GNm mr, Ta whra(r), i whereh m is the black hole mass, which is in quantitative agreement with Hawking's result. This is a beautiful observation, but it is quite special, and might be regarded an isolated curiosity. Specifically, the limitation to massless fields is quite essential to the analysis, as is the limitation to 1 + 1 dimensions. Indeed, only the absence of back-scattering for massless particles in 1 + 1 dimensions allows one to relate flux at the horizon - which is the simple, universal aspect of Hawking radiation - to an integral over the exterior. Also, as a conceptual matter, the central role ascribed to conformal symmetry seems rather artificial in this context.

2 For a recent application, see [78]

61 3.1.4 Effective Field Theory Framework

Our goal is to formulate an effective theory for the behavior of fields in the region outside the horizon. The relevant dynamics of the interior (that is, the part that effects the exterior) is assumed to be captured by an account of the horizon, regarded as a dynamical system. At the classical level, there is a very useful effective membrane theory of the horizon, which can be derived in a fairly straightforward way from the classical action [31, 79].

A delicate issue arises, however, when one moves to the quantum theory. To identify the ground state of a quantum field (say, for definiteness, a free field), one normally associates positive energy with occupation of modes of positive frequency. But in defining positive frequency, one must refer to a specific definition of time. In the exterior region, where the effective theory is formulated, there is a natural definition of time, for which translation t - t + to leaves the metric invariant. This time coordinate becomes mathematically ill-defined at the horizon, and the "ground state" associated with its use (the Boulware state [30]) is physically problematic, since in it a freely-falling observer would, upon passing through the horizon, feel a singular flux of energy-momentum. The singular contribution arises from modes that propagate nearly along the horizon at high frequency. In the Boulware state, these modes have non-trivial occupation. The Unruh vacuum [41], which is nonsingular, is defined instead by associating positive energy to these modes, so they are unoccupied. Mathematically, it is implemented by associating positive energy with occupation of modes that are positive frequency with respect to translation of the Kruskal coordinate U, which will be discussed in detail in Section 3.2.2.

Our proposal arises from elevating this state-choice to the level of theory-choice. That is, we suppose that the quantum field theory just below the membrane, to which we should join, does not contain the offending modes: in effect, that they can be integrated out.

There is an apparent difficulty with this, however. Having excluded propagation along one light-like direction, the effective near-horizon quantum field theory becomes chiral. But chiral theories contain gravitational anomalies, as discussed above. In our context the original underlying theory is generally covariant, so failure of the effective theory to reflect this symmetry is a glaring deficiency. Analogy to the quantum Hall effect, as in Section 3.1.2, suggests that one might relieve the problem by introducing a compensating real energy-momentum flux whose divergence cancels the anomaly at the horizon. We will show that the energy-momentum associated with Hawking radiation originating at the horizon does the job. One can extend the discussion to construct an effective theory for the interior as well as the exterior bulk, separated by a chiral bilayer membrane near the horizon. The primary features of our framework are sketched in Figure 3-1. In this context, the horizon acts as a sort of hot plate, radiating both in to and out of the black hole, similar to pair-creation in a constant electric field.

_· _ Figure 3-1: Part of the causal diagram of a black hole spacetime, with inset detail of a region near the horizon. Dashed arrows indicate unoccupied modes, while solid arrows indicate occupied modes. The white region is the infinitesimal slab near the horizon where outgoing modes are eliminated.

3.2 Spacetime Preliminaries

We will be studying the behavior of scalar fields in static, spherically symmetric background space times. It will be useful to have certain facts on hand about these spacetimes, where they come from, and properties of their wave equations.

3.2.1 Spherical Static Metrics

We wish to construct a metric ds2 = gabdxadxb for the general static, spherically symmetric spacetime in d dimensions in a suitably intuitive coordinate system. We choose coordinates (t, r, Q) such that Q are the d- 2 angular coordinates 9 i on the unit sphere S d- 2 with metric d 2 - sijdOidOj. Spherical symmetry allows setting gri and gti to zero and to declaring the remaining t-r components of the metric to be angle-independent. The static condition allows setting tr to zero and makes the metric components t-independent. We can further choose r to be the coordinate that measures the areas of spheres in the sense that a spatial surface of constant coordinate r and t has an area3 Vol(Sd-2 )rd- 2. The general metric is now given by ds2 =-f(r)dt 2 + h(r)-ldr2 + r2dQ2 (3.4)

3 The word "area" is used here only because d-2 is a boundary of a spatial region. When appropriate, we refer to the volume of such a spatial boundary as an "area" or "surface area" in order to distinguish it from the volume of the spatial region enclosed by the boundary.

63 where f(r) and h(r) are arbitrary functions of the coordinate r. The angular part of the metric is diagonal and given by d-2 2 j IH sin . (3.5) i=l j=i+l

Thus, 1 is the azimuthal angle ranging over [0, 27r] and the rest are polar angles ranging over [0, 7r]4. The determinant of the metric (3.4) is given by

g =-f r2(d-2) det Q. (3.6) h Note that the metric may not be invertible at points where f or h vanish, unless they vanish simultaneously. This condition will be explored further in Section 3.2.1.2. The Christoffel symbols are f' rttr = rtrt (3.7a) = f'h (3.7b) tit = rr (3.7c) rr = -rhsij, (3.7d) P ir. 1e (3.7e) jr- r~]rj r jk = same as unit Sd-2, (3.7f) with all other components vanishing. The primes in Equation (3.7) indicate differen- tiation with respect to r. The Ricci tensor is given by

Rab= acrab- aarcb+ rcdabr - adIcb. (3.8)

All the off-diagonal components of Rab vanish. The rest are given by

1 f f'h' 24!! 1(f ) 2 fd] ' (3.9a) 2fh 2 f) f -2 2 f 1 92f f'h' 1 Rrr 2 h'd - 2 (3.9b) 2 f 2fh 2 fj ) h r J' Rij = sij[(d-3)(1-h)- rhrln '/]. (3.9c)

4 Note that this ordering of angles is reversed from some conventions.

------For completeness, we list the Ricci scalar R =-h ff[ f'frIn 2(d- 2) ln(d- 2)(d- 3) f f h- + r2 4,lnV- + (l -h) (3.10) and Einstein tensor 1 Gab-= Rab- -gabR, (3.11) 2 which has components

3 Gtt = ~dd-2f -2 [rd (1 - h)] (3.12a)

Grr =_d 2 r In- '- (3.12b) r h fh' Gij =sijr2h [- - Orin rn + -a nj d4G] . (3.12c) 2f rd 4 h r d -2 fh J

Since Rab and gab are both diagonal, Equation (3.11) implies that Gab is, too.

3.2.1.1 Einstein's Equation While not a strict requirement for the core analysis of Section 3.4, it will be interesting to consider spacetimes that are solutions of the d-dimensional Einstein equations for some given background matter source energy-momentum tensor Tab:

5(C2 Gab = 2 Tab, (3.13a) 2 or equivalently,

Rab= 2 (Tab d- 2 9abTc)* (3.13b)

Recall that X2 was defined in Section 2.2.2 as 2 = 2- Vol(Sd-2 )GN. Since Gab is diagonal, Equation (3.13a) implies that Tb is, too. Equations (3.13a) and (3.12c) together imply that Tij is equal to sij times a scalar function of r only. Thus, we can write Tab as Ttt= p(r)f(r), (3.14a) Tr = P(r)/h(r), (3.14b) Tij = S(r)r2sij. (3.14c)

The physical energy density of background matter measured by a static observer in these coordinates is given by p. Likewise, P gives the radial pressure measured by this observer. For many matter models, the tangential pressure S is not independent of p and P. For example, a perfect, static, isotropic fluid obeys S = P, but this fact will be neither relevant here nor true in general. The trace of the energy-momentum

65 tensor is T = -p+ P+ (d-2)S. (3.15) The tt-component of Equation (3.13a) can be integrated immediately, using Equa- tions (3.12a) and (3.14a), to give

h=1- C X2 [ d 2p(x)dx, (3.16) rd-3 (d- 2)rd-3 jx p()dx

where C is constant of integration parameterizing physically different solutions with the same given matter distribution. A non-zero C is equivalent to placing an additional point mass at the origin. The explicit appearance of C could be absorbed into the lower limit of the integration. This expression can be simplified by introducing the "integrated mass" function

m(r) Vol(S d- 2 ) xd- 2p(x)dx, (3.17) /0 where we allow for a possible point-mass singularity in p at the origin. This function has the intuitive - but, in general, incorrect5 - interpretation as the total mass inside a sphere of radius r. Equation (3.16) then simplifies to 2 _ GNm(r) h(r)= 1- (d N3) r(). (3.18)

Given the solution for h(r), the rr-component of Equation (3.13a) can also be integrated, using Equations (3.12b) and (3.14b), to give

___ X[p(X)+PWx] f(r) = Ah(r) exp -{ I (x) dx (3.19)

The integration constant A allows an arbitrary rescaling of t. It can be set to any value without loss of generality, or otherwise absorbed into the lower limit of integration. Generally, it is chosen such that f = 1 at some special value or r, typically r = cc. Equation (3.19) illustrates that if the matter source obeys p = -P, as it does for vacuum, pure cosmological constant, or electrostatic field, we get the simple result f(r) = h(r), everywhere. A few simple metrics of physical interest that obey p = -P are listed in Table 3.1. The exact forms of p(r) and P(r) will depend on the dynamics or equation of state of the matter source in question, which we will not be concerned with here. It is often useful for the intuition to considerthe case of f(r) = h(r), although the general case will be studied throughout. However, as will be discussed in Section 3.2.1.2 - and as is already apparent from Equation (3.19) - Einstein's equations imply

5In general relativity, there is no global notion of "total energy" or "total mass" within a region. However, in asymptoticly flat spacetimes an unambiguous notion of the the total energy in the spacetime, the ADM mass [80], does exist. Also in such spacetimes, a Newtonian limit can be taken and the expression (3.17) does indeed agree with the Newtonian mass.

__ Name f (r) = h(r) P= -P Schwarzschild 1-(d-3) r 0

2 Reissner-Nordstrom 1- (d-3)2GN rd-3m + (d-3) 2 GNVol(Sd-2) r2 (d-3) 2 V ( r ((d-2) de Sitter 1 - 2A 2 2 de Sitter 1 (d-2)(d-1) r 22 Schwarzschild-de Sitter 1 - (d- d-2)(d- 2A

Table 3.1: A few physically interesting metrics that obey p = -P. that if a surface of constant r exists such that one of the functions f or h vanishes there, then the other must vanish there, too. We will always use this condition when needed in further sections. Interest in the special case f(r) = h(r) would be well motivated, however, by the study of matter distributions that could possibly give rise to such solutions in classical general relativity. As will be shown in Section 3.2.1.2, such spacetimes saturate the null energy condition and include several physically interesting cases (again, see Table 3.1).

3.2.1.2 Horizon Structure Two types of horizons will be encountered when studying black holes: event horizons and Killing horizons. With the coordinates chosen as they are for the static metric (3.4), the condition for an event horizon (a null hypersurface that separates spatial regions where timelike trajectories may escape to timelike infinity from regions where they may not) to exist for some constant r hypersurface is that the one-form normal to the hypersurface has vanishing norm: gabOrbr = h(r) = 0. A Killing horizon for the global Killing vector = 9t (an infinite redshift surface) will exist when the norm of vanishes: g9b7/ar/b= -f(r) = 0. We will consider the conditions under which these horizons might exist. Since the integrand in Equation (3.19) is real, the exponential is manifestly non- negative. Thus, the sign of f(r) and h(r) must always match. In particular, the zeros of f(r) and h(r) coincide. The only possible exception to this might occur when the integral in Equation (3.19) has a positive pole. Assuming the background matter is nonsingular near a given event horizon, Equation (3.19) reads

XJ2 rh[p(rh)+P(rh)] f(r) - (r - rh) + (d-2)h'(rh) (3.20) for r very close to rh, where rh is defined by h(rh) = 0. As long as this exponent is non-negative, f(rh) will vanish and the zeros of f(r) and h(r) will match. The exponent can only be negative if h(r) approaches 0 with a slope of sign opposite to that of p + P, and even then the exponent may still be positive. As will be discussed below, most spacetimes that describe stars or black holes will be asymptotically flat and obey the null energy condition. This means that near the outermost event horizon both h and p + P are non-negative. Thus, it is the case that for sufficiently well behaved matter sources, at least the outermost event horizon

67 will indeed obey f(rh) = 0. Since this is the only event horizon that can be observed from infinity, it is sufficient and unambiguous for physics outside of rh to refer to this hypersurface as the event horizon6 . The event horizons of static, spherically symmetric spacetimes are therefore also Killing horizons. We can then simply talk about "horizons" without confusion. The fact that the event horizon is also a Killing horizon means, among other things, that in the static coordinates (3.4), a patch of spacetime near the horizon looks locally like Minkowski spacetime in Rindler coordinates. If this were not the case, analysis of the behavior of wave modes near the horizon would be much more difficult (see Section 3.2.3). Arguably, the universality of black hole thermodynamics depends, at least at the kinematic level (Hawking radiation), on the ubiquitousness of Killing horizons. From the form of the metric (3.4), it can be seen that f(r) must remain positive for t to remain a timelike coordinate. If f(r) does change sign, as it does at a Killing horizon, then the global isometry t - t + to can no longer be referred to as a "time translation," and this spacetime cannot be considered truly static. Rather, such a spacetime is merely stationary. This loss of staticness allows time-irreversible processes to occur. As such, spacetimes of this sort may contain spatial regions that can be entered but cannot be escaped along timelike trajectories. Also, at some level, irreversibility may ascribe an entropy to such spacetimes [82]. The quantity often called the surface gravity of a Killing horizon, 3, is defined by /aVaqb = -lb evaluated at the horizon, with tl normalized to 1 at a conventional point, typically infinity. For the metric (3.4) this gives = f' . The surface gravity rh is so named because in a static, asymptotically flat spacetime it typically measures the magnitude of the acceleration of a static particle at the horizon, redshifted to an observer at infinity. However, in our spacetime, this acceleration is actually given by7

K; fL=l (3.21) rh

If we define the corresponding quantity for the event horizon, namely -- h' then we find /c A. (3.22)~~~~~~= We will find that the relevant quantity appearing in the equations of black hole thermodynamics is in fact nc,not a. One of the classical energy conditions often imposed on the choice of energy-

6 Nevertheless, there do exist spacetimes for which some of the zeros of f(r) and h(r) do not coincide. See, for example, the GHS black hole 81] and similar dilaton black holes. These may occur because the matter sources become singular - in which case the hypersurface is a singularity, not a horizon - or because a zero occurs inside of another horizon where h(r) has changed sign. As just discussed, such "inner horizons" are always hidden from observers at infinity by horizons obeying f(rh) = h(rh) = O. 7It may be worth noting that the acceleration as redshifted to infinity of a static point at a constant coordinate r that is not a horizon is given by f'v/h7, which limits to Equation (3.21) as r - rh.

____·II _Y momentum tensor is the so-called null energy condition (NEC) (Tabnanb > 0 for any null vector na), which demands p+ P > 0. This condition appears to be necessary for the classical stability of many systems, including perfect fluids [83]. From Equation (3.19), we see that saturating this bound yields f(r) = h(r). From the same equation, we see that the NEC also demands that for h(r) positive everywhere, as in a normal star, the integrand is always positive. Thus, the exponential is greater than 1 and f(r) > h(r). If regions of both positive and negative h(r) exist, the situation is more complicated and If(r) may be less than h(r) (but always the same sign). There is no physical significance to an intersection f = h other than f = h = 0, since such intersections can be moved around arbitrarily by rescaling t. The null radial component of Equation (3.13), Gabnanb = 2Tabnan b, reads ~~~~2 d- 2 h(I h ) = 2(p + ). (3.23) r f h Interestingly, examining Equation (3.23) near the horizon using f(rh) = h(rh) = 0 we find simply (p + P)l = 0. (3.24) rh This shows that Einstein's equations force any form of matter to saturate the NEC at a horizon in a static, spherically symmetric spacetime. Another physically reasonable condition to impose on the background matter is the weak energy condition (WEC) (TabUaub > 0 for any timelike vector Ua), which reads p > 0 here. Applying the WEC, Equation (3.17) tells us that m(r) is a positive, monotonically increasing function of r. Even with this restriction, however, h(r) may have an arbitrary number of zeros, as shown in Figure 3-2. Thus, the WEC says little about the horizon structure of these spacetimes.

3.2.2 Kruskal Extension In order to properly understand the global structure of the spacetime under consideration and the nature of its horizon, we want to define a Kruskal-like extension of this metric in analogy to the known extension of the d = 4 Schwarzschild metric [38]. Primarily, we wish to find coordinates in which the the metric is manifestly nonsingular at the horizon. But we would also like the new coordinates, in analogy to the d = 4 Schwarzschildcase, to show that the global time translation isometry of the old coordinates becomes a boost symmetry in the new coordinates. Further, we would like the new coordinates to maintain the stationarity of the metric and to exhibit an explicit on-horizon translation symmetry, which should intuitively exist given the close similarity of this spacetime to d = 4 Schwarzschild space8.

8We can also construct a Painlev6-Gullstrand [39, 40] coordinate system for this spacetime. By defining r = t + f dr S1-, the metric (3.4) becomes

ds2 =-f(r)dr 2 + 2f(1 -h) drdr + dr2 + r2dQ2. (3.25) h

69 m,h'

r r

(a) (b)

(c)

Figure 3-2: Sketches of three integrated mass functions and their associated h(r). Solid lines indicate the integrated mass function m(r), while dotted lines indicate the metric function h(r). Vertical dashed lines indicate the position of the outermost event horizon, if it exists. In 3-2(a) the matter distribution is relatively smooth and vanishes at the origin, as in a normal star. In 3-2(b) the matter has a density singularity at the origin, but is otherwise well behaved. In 3-2(c) a potentially difficult-to-analyze situation is sketched.

70 fh r, I I I I I I

rh r rh : r I : I I :. I I I I

(a) (b)

Figure 3-3: In 3-3(a) typical profiles for the functions h(r) (solid line) and f(r) (dotted line) are sketched for an asymptotically flat black hole spacetime. The horizon occurs where the functions vanish at r = rh. In 3-3(b) the corresponding profile of r is sketched along with the dotted line r = r. Note that r diverges logarithmically at rh and approaches r at large r.

The strategy will be to first transform from (t, r) to a set of coordinates (u, v) that parameterize ingoing and outgoing null geodesics. Like t and r, these coordinates will be poorly behaved at the horizon, but it will be possible to construct a rescaled set of null coordinates (U(u), V(v)) that can be extended past the horizon. We we then attempt to impose the condition that a translation in t appears as a Lorentz boost in the Minkowski-like coordinates associated with the null coordinates (U, V). This will greatly restrict the possible form of the new coordinates, leaving only three arbitrary real constants to be fixed in the coordinate transformation. We will then check whether the metric is regular at the horizon in these coordinates. The regularity condition will fix one of the free constants, while the other two are simply global rescalings of U and V that can be chosen by convenience. We will then show that translations in U or V are on-horizon isometries. The first step is to introduce the analog of the tortoise coordinate r defined by

cr / 1 (3.26) 0-~ = /fh '

The generic form of r is sketched in Figure 3-3. With this coordinate the metric becomes ds2 = (r)(-dt 2 + dr2) + r2dQ2 (3.27) where r is now thought of as an implicit function of r. These are the locally Minkowski coordinates of a free-falling observer. Radial null geodesics (dQ2 = ds2 =

In these coordinates the spacetime appears stationary, but not static. It also has the nice feature of having spatially flat equal time slices. Finally, they demonstrate that the event horizon at f = h = 0 is a non-singular surface. The case of h(r) = f(r) = 1 2GNm is the original Painlev6-Gullstrand case, which has recently been applied widely [79, 84, 85, 86].

71 0) in this spacetime obey 2 r =fh'~(at0r~)1f~~h(3.28) which can be combined with Equation (3.26) to show that t ± r, is a constant along such paths, with the plus sign for ingoing geodesics and the minus sign for outgoing geodesics. We thus define the ingoing and outgoing null coordinates u = t+r, (3.29a) v = t-r,, (3.29b) which are constants along the respectively named trajectories.

Obviously, Equations (3.26) and (3.28) exhibit a singular behavior at a horizon where either f(r) or h(r) vanishes. This is the coordinate singularity we hope to eliminate with a coordinate transformation. The singularity is made much worse if f(r) and h(r) do not have the same sign everywhere. We thus impose the physical choice of spacetimes discussed in Section 3.2.1. That is, we demand that the zeros of f (r) and h(r) coincide.

As outlined above, we wish to construct new null coordinates U(u) and V(v) such that the Minkowski-like coordinates (T, R) defined by

U = T+R, (3.30a) V = T-R (3.30b) obey

T -yT -y3(to)R, (3.31a) R - yR- y3(to)T (3.31b) when t - t + to, for some function f3(to). Of course, 1/7 1- -/. This demands that

U - y(1-3)U, (3.32a) V - y(l + )V, (3.32b) or equivalently UV -UV, (3.33a) V (1 + V (3.33b) U 1- U Equation (3.33a) means that the product UV - A(r) is a time independent function

______. _ of r only. Equation (3.33b) means that V/U depends exponentially9 on t as V B(r) e' t (3.34) with B(r) and k arbitrary. That k must be an r-independent constant can be seen from Equation (3.33b), which can be solved to give

l(to) = tanh kto. (3.35)

Since 3 is r-independent by construction, so is k. We now have U2 = (A/B)e - 2kt and V2 = ABe 2kt. For U to be a function of u only and V to be a function of v only, while maintaining simultaneously that A and B are functions of r only, it must be the case that A = cucve2kr* and B = cv/cu, for arbitrary constants cu, cv. The (U, V) coordinates now take the form

U = cue ku, (3.36a) V = cvekv, (3.36b)

and metric in the now reads

ds2= k2c f(r)e- 2krdUdV + r2dQ2 = fk dU (3.37)

To examine the metric near the horizon, we Taylor expand the metric functions h(r) and f(r) around r = rh using the notation of Section 3.2.1.2:

f(r) - 2(r - rh)+ f((rh)(r- rh, (3.38a) n=2 h(r) = 2(r - rh)+ E () (r- rh. (3.38b) n=2En=2 This yields

1 (r-rh +Sjf"(rh) + h" (rh) A (3.39) r-rro-rhh (r -ro) + near the horizon, where r is an integration constant. (A sensible choice is ro = rh + 1/(2j).) The metric is approximately

k 2 2 ds 2 r - (rhr-rh) K+O ((r- rh) K) dUdV + r dQ . (3.40)

9 For a sufficiently well-behaved real function a(x) to obey a(x+y) = b(y)a(x) for a given function b(y), also sufficientlywell-behaved, it must be true that a(x) = Aekz and b(y) = eky . This can be proven by writing a(x + y) = e . a(x) and Taylor expanding about y = 0.

73 This metric will only have a non-vanishing determinate at r = rh if k = . As promised, regularity has fixed the value of the free constant k. We fix the scale factors cu and cv by aesthetics, choosing cv = -cu = 1/r. This choice gives as close as meaningfully possible to a unit determinate at the horizon, such that the (T, R) coordinates are a close as possible to Minkowski coordinates. The final expressions for the Kruskal extension of the metric (3.4) are given by

U= _ 1-e -`(t-r.)K(tr*) (3.41a)

V= l e (t+r*) (3.41b) and ds2 = f(r)e-2xrdUdV + r2dfQ2 = f(r) dUdV+ r2dQ2, (3.42) K2UV with r given by Equation (3.26). Neither form of the metric given in Equation (3.42) is obviously nonsingular at the horizon, but we have shown by the constructions of this section that they are. The horizon at r = rh is now seen to occur at both U = 0 and V = 0. The horizon is a bifurcate Killing horizon with bifurcation surface given by the S d - 2 at U = V = 0 (t = 0, r = rh). The U = 0 surface also corresponds to t = +oo and is therefore referred to as the future horizon, while the V = 0 surface corresponds to t = -oo and is referred to as the past horizon. Since all of the metric components are t-independent, they depend on U and V only through the product UV. The metric is thus trivially invariant under translation of one coordinate along the hypersurface where the other coordinate is zero. This shows that translation along the horizon is a valid local isometry.

3.2.2.1 The Quantum Vacua We have constructed a coordinate system for an arbitrary static, spherically symmetric black hole in d dimensions that has all the familiar properties of the Kruskal extension of d = 4 Schwarzschild spacetime. In particular, the fact that the Lorentz boosts of the local Minkowski-like coordinates near the horizon generate t-translations in the original coordinates of Equation (3.4) shows that near the horizon, the original coordinates are more akin to Rindler coordinates than Minkowski ones. Equation (3.39) shows that the logarithmic divergence or r, which will ultimately translate into a logarithmic phase singularity in wave modes defined with respect to the global Killing vector 77,is a universal feature of horizons in this family of spacetimes. Fur- thermore, just as for Schwarzschild spacetime, the final form of the coordinates (3.41) shows that U --* U + Uo is an isometry along the past horizon V = 0, and thus O-u is a Killing vector there. Thus, it is meaningful to follow the logic of Unruh [41] and define a quantum vacuum state, the so-called Unruh state or -vacuum, for this spacetime l0 by defining

' 0 The quantum vacuum for a star dynamically collapsing to a form black hole, such that the metric of this collapsing spacetime settles down at late times to a stationary black hole metric of the form (3.4), is thought to be well represented at times long after the collapse by the -vacuum of

__ positive energy states as those that have positive frequency with respect to E on the past horizon, and letting this vacuum propagate outward and forward in time. The vacuum state defined in this way differs from the one obtained by defining positive energy with respect to the global Killing vector 7, which is called the Boulware state or r1-vacuum [30]. In fact, Unruh found that the -vacuum is a thermal ensemble of i-frequency states at temperature TH , in exact agreement with the earlier results of Hawking [29, 28]. The 7-vacuum has divergences in its energy-momentum tensor arising from horizon-skimming modes, despite appearing empty" to static observers. This divergence renders the Boulware state an unphysical candidate vacuum. The Unruh state, on the other hand, is well behaved despite its nonintuitive mode occupation spectrum. Unruh's arguments go thorough essentially unchanged using the spacetime (3.4) and its Kruskal extension (3.41). Thus, we find a temperature for the -vacuum of this spacetime given by TH. Furthermore, the 7-vacuum of this spacetime also has divergences that render it unphysical.

3.2.2.2 Euclidean Section

A relatively easy route to determining the thermal nature of a spacetime is to consider the Euclidean section of the complexified coordinates, following the method of [42]. This can be effectively accomplished by letting t -- i. Using the Taylor expansions 2 of equation (3.38) and defining x = (2/Y)(r- rh), the near-horizon metric becomes

ds2 dx2 + K22x2dr 2 + r2dQ2. (3.43)

This is a metric on the manifold Cone2 ® Sd-2, with the conical singularity occurring at the horizon. The singularity is removed if the coordinate Kir is periodic in 27r. The period of T- must then be given by 21. Since the inverse period of Euclidean time is interpretable as the temperature of a system, we see that the temperature of the black hole is given by TH. Note that without the condition f(rh) = h(rh) = 0, no coordinate x exists which gives a near-horizon conical metric like Equation (3.43). This demonstrates that, in general, the stated condition is necessary for the interpretation of a Euclidean black hole as a non-singular thermal system.

3.2.3 Wave Equation

Consider an interacting classical scalar field 0(x) living in a spacetime with the metric (3.4). This field has an action given by:

S = | ddx+{ _-V2 X-,_ ZAn( } . (3.44) the eternal black holes we have been studying. "Here, "empty" means all modes have zero occupation number.

75 2 The A,, are a set of arbitrary coupling constants. In particular, 2 m gives the mass of a weakly coupled excitation of this field. We will expand X in the eigenmodes of the free, classical wave equation such that

(x) = dpapp(X), (3.45) 2 V 0 = m2op. (3.46)

A sufficiently large set of quantum numbers p label the eigenbasis. The abstract formal expression dtp simply represents an appropriate measure over the modes under whichf dpp(x)p(y) - d(x - y) and f ddx /ip()q(x) = pq. Of course,the full theory will not necessarily obey Equation (3.46), but the expansion (3.45) is nevertheless a complete functional basis that may be applied to any field configuration, regardless of the field equations it solves. In a second-quantized quantum theory, the coefficients ap would be promoted to field operators. We will exploit the high degree of symmetry in the background spacetime and use separation of variables to examine the normal modes of this field. As remarked in Section 3.2.2, the wave equation arising from the action (3.44) will take a much simpler form in the tortoise coordinates described by Equations (3.26) and (3.27). For the sake of generality and to illustrate the nature of the useful cancellations inherent in certain coordinate systems, we now examine the wave modes in a coordinate system with metric ds2 =-f (r)dt2 + j(r)-ldr2 + r2dQ2, (3.47) where r = (r) is an arbitrary radial coordinate. The metric (3.4) will be recovered with the choice r = r, j = h, while the tortoise coordinates will be recovered from r= r, j = l/f. We factor a given basis function as

p _2r 1 R(t, r)Y(f). (3.48)

With this factorization, the eigenvalue equation becomes

V mp [m 1_tfj2_ fi 2 (r d-2(fj)1/4) [ar]2 2 Rfln[-]) ((fj)1/4) R 2r~~~

1 1 d-2 9i [(sin i) aiY]

+y2i l[Iji +lsin j] (sin Oi) 1 (3.49)

Clearly, all the variables are separable. We first turn our attention to the angular part. The radial part will be approached in Section 3.2.3.2.

______ ·__ -- ... 3.2.3.1 Spherical Harmonics

The angular part of Equation (3.49) is again separable, so we write

1 2 Y(Q) = E1 (0 )E2(0 ) ... Ed-2 (Od-2). (3.50)

We then get

d-2 i [(sin ')'-'i Y] d-2 0i [(sin i)'-' i] Y-E (3.51) i=l [rdl-2+sin 2 ji] (sin 9i) i-1 i=1 [--~jld-2+ sin 2 O] (sin i)'-1 By standard separation of variables methods, the above must be equal to a constant, which we name -1(1 + d - 3). Further, the angular separation clearly requires the following equations to hold:

Ol2e= - n2 , (3.52a) 2 a2[(sin 0 )026 2] 2 sn =- n 82 (3.52b) sin 02 03 [(sin 0 3)20 e ] sin2n2 O~e2 3 3 ' (3.52c) sin 2 03 sin2 03-03

2 i [(sin i)i-laiei] ni-i (3.52d) sini-l Oi sin 2 0i

ad-3 [(sin d-3)d-4ad-30 d-3] n2_ nd-4 - 2 -n_3(n2 d-3, (3.52e) sind-4 d-3 sin d3 d-3 - 2 ad-2 [(sin Od- )d-3ad_2ed_2] 2 - 1(1 + d - 3)EOd-2, (3.52f) sind-3 d-2 sin Od2 d-2 = where the ni are constants. Defining x = cos 9 i and

) ) F(x) = (sin Oi 2 oi = (1 X i 4i, j (3.53) we find each function obeys 2 (1-x2)0xF-2x0F + ( ( )+ ni 1 - x2' (3.54)

where i = 1 .. d-2, no - 0, and n_ 2 l( + d -3). This is solved by

F(x) = L 2 (V)(i-2,/4+n-; (3.55) (i 1) /4+ n 1/21 , where L is a Legendre function of the indicated order. In principle, Equation (3.54) is solved by any linear combination of the Legendre

77 P and Legendre Q functions. We will now show that the physical requirements of single-valuedness and finiteness of the scalar wavefunction restrict both the choice of Legendre function and the allowed values for the ni. In general, the Legendre functions have singularities unless they are P functions with integer parameters or Q functions with half-integer parameters. Looking at i = 1, we see that

E = eimlo, (3.56) where ml = n is restricted to be any integer by the condition of single-valuedness. We now have, for i = 2, e2 = Lm 1 (COS02). (3.57)

In order for this function to be finite over all values of 02 for integer ml, L must be a P function and n2 must obey n = m2 (m2 + 1) for m2 a positive integer obeying m2 > ml . This yields 02 = P2(cos 02). (3.58) For i = 3 we get 3 = (sin 3) 1/2Lm2- 2l/ (cos 03 ), (3.59) which requires that L must be a Q function and n3 must obey n2 = m3 (m3 + 2) for m3 a positive integer obeying m3 > m2. This yields

e 3 = (sin 03)-1/2Qm2+1/2(COS93). (3.60)

This pattern continues, giving for angle i

E = (zs+in(sin~ t ) 2 _/)-Lm-miL(i-2)/2 ) (o (3.61) where L = P for i even, L = Q for i odd, and mi a positive integer obeying mi > mi-1 . Finally, for i = d- 2 we find Oi (sin d-2)- "-4 rMd-3s+(d-4)/2 (COS d - 2 ) (.2 i= (sin 2) Ll+(d-4)/2 (cos ) (3.62) where L = P for d even, L = Q for d odd, and a positive integer obeying > md-3. This, of course, motivated the choice of form for the constant in Equation (3.52f).

So, the angular distribution can be parameterized by an eigenbasis labeled by the quantum numbers {, m}, where m = {m1,. . , md-3}. Thus, we write Y = Y/m(Q). Only the "highest" angular quantum number, , will appear in the radial equation, so it will be useful to calculate the degeneracy, Dd(l), of angular states with a given value of 1. The construction outlined above makes clear that

l Dd(l) = E Dd-l(n). (3.63) n=O

Starting from D4(1) = 21 + 1, Equation (3.63) can be iterated to find the proper

__.__· d Dd(l) 3 1 for = 0, 2 otherwise 4 21+1 5 ( +1)2 6 (21+ 3)(1+ 2)(1+ 1) 7 ~f(21 + 4)( + 3)( + 2)(1 + 1) 8 1-(2 + 5)( + 4)( + 3)( + 2)( + 1) 9 9(21+ 6)(1+ 5)(1+4)( +3)( + 2)(1 + 1)

Table 3.2: The angular state degeneracies for total angular quantum number 1, as determined by Equation (3.64), in a few chosen dimensions.

expression for higher dimensions1 2 . The exact expression1 3 for arbitrary d and is

Dd(1) = (21+ d - 3)( + d- 4)! (3.64) l!(d - 3)!

Some values of Dd(l) are tabulated in Table 3.2. We chose the normalization of the Ylm such that

2 6 Jdd QYlm()Yk(Q) = ilkcmn (3.65)

3.2.3.2 Radial Wave Equation

Separating off the time dependence by writing R(t, r) = eiwtR(r), the eigenvalue equation (3.49) now simplifies to

2=1 1 2 fjo9- ~fJ 2( d-(fj)l1/4 Or 2 R2f [ fj&- j (r(fj)/4) [j)4ar

Oft - (j /) f22r ([r]2)ar n (r 2 (fj)l/4) fl(l( +d-3)]R.d-3 (3.66)

There is no general solution to this equation, but one can be found for certain special cases, and approximate solutions can be found in others.

2 {1for 1=0 o 0f course, one could also start with D3 (1) = for 1ll> 0 l3We are aware of three different original proofs of this expression, each developed independently when the problem was presented by the author as a puzzle to a group of fellowstudents. The method of Guido Festuccia and Antonello Scardicchio counts the number of lattice points in a certain discrete (d - 3)-simplex [87]. The method of Ian Ellwood formulates the problem as a combinatoric "balls- and-buckets" problem [88]. The method of the author uses inspired guess work checked against Equation (3.63).

79 In Minkowskispace (f = j = 1, = r), Equation (3.66) becomes

- (I + (d - 4)/2)(1+ (d - 2)/2) + k2] R(r) = (367) r ~~~r2

where k2 = 2 m2. This is solved by R(r)/r(d- 2)/2 = k(d-2)/2 J (kr), where J(z) = d-3 hVh (2) 2 jl+(d-3)/2(z) is the 1th (d- 2)-spherical Bessel function, and j(z) is the uth ordinary, cylindrical Bessel function. This Bessel function will also be the solution to the full radial wave equations in regions of spacetime that are approximately flat, such as near spatial infinity in a black hole spacetime. Another situation of interest is the wave equation in tortoise coordinates (j = 1/f, = r, Or/ar, = \/f-h).The radial wave equation becomes

[ar* + w2 - fV(r*)] R = 0, (3.68)

with

1 [(I+ d-4) (1 + d-2)-- (d -2)(d -4) (-h)+(d-2) r(f h)+ 2 r*) 2 2 44 f (3.69) Using the fact that h and f both vanish when f does, we see that all the terms in V are nonsingular at a horizon, f = h = 0. Thus, the overall factor of f causes this effective radial scattering potential to vanish near the horizon. In fact, using f 2(r - rH) and h 2(r -rH), wefind

fV(r*) me 2~r* (1(I+d-3)+ (d-2)rH5 + ) (370) r} 2 which vanishes exponentially fast as the horizon at r - -oc is approached. A sketch of a typical effective radial scattering potential is given in Figure 3-4. Thus, the dynamics of the radial wave function near a horizon is identical to that of 1 + 1-dimensional flat spacetime with coordinates t and r,. The solutions are simply plane waves of the given frequency. Using the coordinates of Section 3.2.2, these can be written as R(t, r*) eu + eiwv , Uiiw/ + IVI- i/. (3.71) The identification of the near-horizon dynamics of this spacetime with that of standard Schwarzschild-type black hole physics is now essentially complete. Essentially all qualitative results for the case of f(r) = h(r) = 1 - 2GNm have a direct analog in this spacetime as long as the Killing horizons and event horizons are coincident.

3.2.3.3 Near-Horizon Action

Now that the dynamical modes of a scalar field in this spacetime are understood, we can execute a partial wave decomposition of the action (3.44). The field is expanded

__11 __·_·_____ fV(r,)

(r.=-oo) r a few rh r, r=rh

Figure 3-4: A sketch of a typical effective radial scattering potential. The potential for any metric qualitatively similar to the one sketched in Figure 3-3 will be qualitatively similar to the one sketched here for I > 0 and d > 3. The potential falls off exponentially for negative r and as is typically dominated by the centrifugal term at large r, . r, which falls off as 1/r2.

as 0 1 ERi tr)Y () (3.72)

The action becomes

S = Jdtdrdd -2Q {

2 fjl ( (fj)114 ar - E JRimYim [-r +n( fj,91 d-2 (fj)/4 r 2 r 2~~~~ 2,k,m,n rd-2(f j)l/4) _ fk(k d 3)] RknY ar ln ( + 2 ( [ar] 7,2 n fli E_ r(n-2)(d-2)/2(fj)n/4 E E [RimYm ... RinMn~nn ] } 11 I...nml...mn

dtdr 1 R - + fjO9- d i 2 (rd2 (fj)1/4)[r] _E -Rim- d-2 (f1/4-- rj)l/4

2- ([-ar In (rd-2 (fj)/4) _ fl1(l+ d-2 3) 2 [ar ,Or2 In (r )- f r Rlm -00 An Vf/ E nC~m'R .. RlnM (3.73) =- r(n-2)(d-2)/ 2 (fj)n/4 {} l1ml'· · n=2 11/...1n Ml ..Mn 81 where n- is the set of group theoretic constants obtained by integrating the local product of n properly normalized d-dimensional spherical harmonics over Sd- 2 . In particular, Equation (3.65) tells us that 2C{I1,M12} = 611 26mm2 The radial wavefunction, Rim, in unitless, just like a canonically normalized 1 + 1 dimensional scalar field. All the units of the original d-dimensional field are carried by the prefactor 1/r(d - 2 )/ 2 in Equation (3.72). The mass units of A are [An] = 2- (n- 2)(d- 2)/2. Thus, we can write the unit-carrying factors in the interaction series of Equation (3.73) as

r(n-2)(d-2)/2 A ,,r2-[A,,] An ' (3.74)

This rewriting emphasizes that interactions which are not perturbatively renormalizable- and are therefore governed by coefficients with negative mass dimension - fall off at large distances faster than 1/r2 in a partial wave decomposition. Super- renormalizable interactions fall off slower than 1/r 2 and marginal interactions fall off as 1/r 2. The scaling relative to 1/r 2 is important because the large-distance scaling of the centrifugal term in the kinetic part of the action is given by 1/r2 . If we now evaluate the action (3.73) for the tortoise coordinates as in Section 3.2.3.2, we get

S Jdtdr, {-i REm[--t + a - fV(r.)] RIm lm An -- r2 ,2-[~nEZ E nC/{ im} Rlnmn (3.75) n=3 1l... ln ml ...IMn

2 where V is defined as in Equation (3.69) with m = A2. Again, f(r) vanishes exponentially fast near the horizon in tortoise coordinates. So, near the horizon, the action becomes simply S d2XE -RimLmRim, (3.76) I,m where d2x = dtdr, and D = -_t + a2,. This is the action for an infinite collection of free scalar fields in 1 + 1 dimensions. This description of the dynamics becomes exact at the event horizon, however the notation and coordinates used in this section fail there.

3.3 Thermal Radiation

The theoretical derivation of the blackbody spectrum is standard fare for 3 + 1 dimensions with wave modes labeled by Cartesian coordinates, but the derivation for d-dimensional spherical coordinates is somewhat less well known. We will illustrate this derivation in this section. For purposes of comparison and introduction of some necessary mathematics, we will first study the spectrum of blackbody radiation in d

82 Cartesian coordinates. Some subtleties of the spherical case will be illuminated by comparison with the Cartesian case. Minkowski space will be used throughout this section.

3.3.1 Hypercubic Blackbody Cavity Consider a real scalar field '(x,I t), where I is some kind of p-dimensional polarization index representing p internal degrees of freedom14 . Further assume that the field is sufficiently weakly coupled that each polarization component can be treated as an independent field obeying an action similar to Equation (3.44) with all interaction 2 coefficients higher than A2 M set equal to zero. Then each field component obeys the classical equation of motion

V2qI(x) = M2q'(x). (3.77)

The solutions to Equation (3.77) may be expressed as a sum over modes labeled by by a wave vector ka obeying k2 = _M2:

0)k = A, sin (kaXa) + B/ cos (kaXa), (3.78) for arbitrary real coefficients A, and BI. We take the state to be labeled by the d -1 spatial components of ka and fix the frequency of each mode by w- k02= M 2 + kiki . We now confine the field to live in a cubic box of side length L by demanding Dirichlet boundary conditions at xi = {0, L} for i = 1... d- 1. This demands B = 0 and ki= 7r mi, (3.79) where mi is a spatial vector whose components are non-negative integers. The full space of modes available to the system is given by the tensor product of the space of all vectors of integers with the state space of the polarization index. The theory is then effectively quantized by stating that the energy in each such mode is a non- negative integer multiple of Wk. We denote the integer as nml. The total energy in the hypercube is then given by

p 00 00 U = E E ... E Wknm, (3.80) I=1 ml=O md-1=O where we understand that wk is given by

2d-1 Wk2 + rMZ 2m. (3.81) 14 Sufficiently simple external degrees of freedom will also be captured by the following discussion. For example, in a well-chosen gauge, the transverse polarization of an Abelian vector field behaves essentially like an internal index on a scalar field with p = d - 2.

83 The set of integers nmI defining the quantum state must obey the appropriate statistics 15 for the field I. That is, nmI is unrestricted if the field obeys Bose-Einstein statistics, but can only take the values 0 or 1 if the field obeys Fermi-Dirac statistics. So far, we have described a pure quantum state of the theory. At a finite temperature T and zero chemical potential, the system will be in a mixed state governed by the partition function Q= e k (3.82) where fi- 1/T. This can be evaluated to give

00 00 lnQ=-p. E . E ln(1-e-k), (3.83) ml =0 md-1l2° where = 1 for bosons and ~ = -1 for fermions. The overall factor of p occurs because the energy is independent of p, so each polarization mode contributes equally. The average occupation number of a given momentum mode in the thermal state is then given by p I= 1 p (3.84) 3 9 I=1 / e Wk efk- The total energy in the hypercube can now be found by combining the expressions (3.80) and (3.84), or by

&a 00 00 00 oo Zd PWk_ (U) = -- lnQ= - E Wk(nm) ..mi (3.85) ml =O md-1= ml =0 Md-1=0 ° -~

In principle, Equation (3.85) can be evaluated and the summand can be interpreted as the spectral energy density over the quantum numbers mi. Makingthis interpretation is problematic, however, in the L - o limit. The problem is illustrated by looking at the M = 0 case, such that Wk = I ml, and examining the scaling with L:

00 oo (U) = E .. 5 Iml E e7r31ml/L _ ml=0 md_1-0

00 00 r L-O plr 1 $r (3.86) 1=0Iml 1- Iml( -)2L L ml =0 md'~ which is nonsensical for several reasons. The total energy does not scale with the volume of the hypercube, L d -l. In fact, the total energy appears to either diverge or vanish as L- 2, depending on the statistics. The expansion itself is not even well

150f course, the spin-statistics theorem says that if OI is a scalar field, then it obeys Bose-Einstein statistics. However, we want to allow for the case where -bz is a single real component of larger multiplet, which may obey Fermi-Dirac statistics. In this case, the polarization index I would also run over spin.

------_____I··_ __I defined, since it is an expansion in the ratio of a divergent quantity to a infinitely large quantity. The major mistake in Equation (3.86) is that in the L - o limit, the mode density of mi states diverges, as can be seen from Equation (3.79). That is, the modes cease to be countable and discrete. As the limit is taken, we should pass from a state labeling in terms of quantum numbers mi to a labeling in terms of physical momenta ki, with a mode density determined by the differential limit of Equation (3.79). The sums over mi then become integrals over ki as L-+o d- 1 Ldli PWk (U) L-p d k (387) (27r)d-I ek - (3.87)

2 i where now Wk is understood as VM + kik . The factors of 2 in the denominator of the measure arise because the integrals over the ki run over both positive and negative values, whereas the mi were only summed over non-negative values. Equation (3.87) scales properly with the volume, so that even in the infinite volume limit we can define the spectral energy density over ki modes. For the current case of Cartesian coordinates, the failure of Equation (3.86) and its resolution by passing from quantum numbers to physical momenta are obvious and the discussion has been overly pedantic. Similar failures will be encountered when using spherical coordinates to study the spectral density of angular momenta in blackbody radiation. In that case, however, the nature of the problem and its solution will not be as obvious. Using the spherical symmetry of the infinite volume limit and defining k = /kii = ki, Equation (3.87) becomes

(U) =p Vol (Sd- 2) kk- 2VM 2 + (3.88) V (27r)d-1 J- This definesthe spectral energy density over the magnitude of the spatial momentum, via p f dkuk(k), as

Uk(k)= Vol (Sd- 2) kd- 2 M 2 + k 2 (3.89) 1 )= (23.) eg3vM+-k3-

Similarly, we can define the spectral energy density over the frequency 16 as

Vol (Sd2) W(W- 2 _ M2)(d-3)/2 u'(w) = p (2 7r)d-1 e, (3.90) where w runs over [M, oo]. The total energy density can now be evaluated using either Equation (3.89) or Equation (3.90). Simple analytic results can be found for the case M = 0, which will also apply when T M. In this case, w = k and Equations (3.89) and (3.90) match.

6 Note, dk = wdw 2 -2 V1W --M 85 They give d- 2) d Vol(S x d-1 = pT (2rd-(27r~d-1e dx. x_ (3.91) The integral can be evaluated by pulling the exponential into the numerator, performing a Taylor series in e-x, doing the integral, and resumming the Taylor series. The result for the general definite integral is given by b ex ( - dd-1 [d(aa -_ _ eX ~(d)(--xdog-1- - _ 01 l a e ~ / d a ll - a ea-e°b a=d)1 L -(=-)/ ane-a = 2-j I-' (d)(d-1)! ana- bneb (3.92) (i ~~~~)( ~n=O zd- _e-b where ((s) is the famous Riemann zeta function, which can be defined for real s > 1 as 00 =(s1E(3.93) n=1 This series arises in the evaluation of the integral with the ~ = 1, for bosons. For fermions, = -1, the corresponding series is

00 · =E n----1~~(_l)n-l (3.94) n=1 This series can be evaluated using C(s)- NW=(s) 2 + + + . 2.C.= (s), (3.95) so that N(s) = 1 - -) (s), (3.96) which is the origin of this factor in Equation (3.92). So, Equation (3.91) becomes

p = p - 2d_ 2r (-1) 7r(di)/2T (397)

3.3.2 Flux Versus Energy Density In the previous section, we considered the energy density p of a massless scalar field in a infinitely large hypercubic cavity at temperature T. We now want to calculate the energy flux (power per unit area) ·P emitted from a blackbody with this same temperature. Any number of textbook arguments can be followed to derive an expression for 4D. The key insight in all of them is that thermal equilibrium requires the condition of detailed balance on every mode of the system, which is a far more restrictive condition than simply total energy conservation.

___·_· At the end of the day, the calculation can be cast as finding the flux through a chosen spatial boundary. We choose a set of (d- 1)-dimensional spherical coordinates for the spatial momenta at some given point of the boundary using the same set of angular conventions introduced in Section 3.2.1. We orient the coordinates such that the vector normal to the boundary at the chosen point lies at the "north pole" of the coordinates. Then the calculation of the flux proceeds exactly as the calculation of the energy density, with two key differences. First, the polar angle 0 d-2 only ranges over [0, 7/2] instead of [0, 7r] because we are interested in the flux passing in only one direction through the surface. Second, there will be an additional factor of cos (0 d - 2) multiplying the phase space measure to account for the scalar product between the flux vector and the area element of the boundary. The only change to the mathematics of Section 3.3.1 is in the passing from Equa- tion (3.87) to Equation (3.88). We used there dd-lk = dd-2Qkd-2dk and f dd-2Q = Vol (d-2). For the flux calculation, instead of Vol (S d - 2) we will encounter

dd-2 Q cos (od- 2)e(7/2 - Od-2)

r2r r7r 7r 7r/2 d-3 d- 3 d- 2 d- - j2dj O' d2... do do cos ( 2) H (sin (i)) } J ~~~~~0 i=2 27r( d - 2 ) / 2 1 2) (d2) d- - (d ---2) (-2) d-d-2 2 Vol(S 3) = Vol(Bd 2)

= 1 F( ) Vol (Sd-2), (3.98) ~ V/7(d - 2) (-2 )

where (x) is the step function and B is the n-dimensional unit ball1 7: the compact subspace of IR bounded by Sn- 1 . The fact that the expression for the energy density becomes that for the flux when Vol (Sd-2) is replaced by Vol (Id-2) makes physical sense, since Bn is the projection of Sn onto iRn. We are left with the relationship of flux to energy density as

(-1 Vol (Bd- 2) F(~j-l)(d 2)P.Vol (3.99) v(d- 2)r (d )P Vol (d 2)P.( Thus, the d-dimensional Stefan-Boltzmann law is given by

( -2 (1- )/2 ( 1 Td (3.100) 1- -) 2d-2(d - 2)F (d-2) d/2

Thermodynamic relations like Equations (3.97) and (3.100) are independent of the mode labeling that was used to compute them. So, when attempting to rederive (3.100) using an angular mode labeling instead of the Cartesian one used in Section

17For example, lB2 is the unit disc of volume 7r bounded by the circle S1 of area 2r. Also, 3 is the unit 3-ball of volume 47r/3 bounded by the sphere S2 of area 47r.

87 3.3.1, it will be sufficient to rederive Equation (3.97) and then apply Equation (3.99). The case of d = 2 will be important to the core analysis of Section 3.4. Since Equations (3.99) and (3.100) are somewhat ambiguous for d = 2, we explicitly list the results for that case here:

PT2 (3.101) b 1 2 (12 ) (1-E,)/2PT12' 311

3.3.3 Spherical Blackbody Cavity

We will now repeat the calculation of Section 3.3.1 in a spherical cavity of radius R instead of a hypercubic cavity. We define spherical coordinates as in Section 3.2.1 with f = h = 1 and decompose q$I in partial waves as in Section 3.2.3. The field still obeys Equation (3.77), but instead of expanding classical solutions in the plane wave basis (3.78), we separate variables as in Section 3.2.3:

5'(x) = - eiwktRkl(r)Ylm(f1), (3.102) r 2 where the (d- 2)-spherical harmonics Y were defined in Section 3.2.3.1, the radial part of the field obeys Equation (3.67):

[oq (1 + (d - 4)/2)(1 + (d - 2)/2) + k2] Rkl(r) = 0, (3.103) /,2

and, again, k2 = w2- M2. As discussed in Section 3.2.3.2, Equation (3.103) is solved by

2 )/ 2 = k(d 2)/2 J(kr)- r j(d-3)/2(kr), (3.104) r(d Rkl(r)r~~~~~~~~~~~~~ d-3)/2Fk 2i where j,(z) is the vth cylindrical Bessel function. The boundary condition at r = R demands that J(kTZ) = 0. This quantizes the allowed values of k as

k = A7n1 ' (3.105)

where the pure numbers A, are defined by J(Al) = 0. The modes are now labeled by the quantum numbers {n, 1, m, I}. The quantization of the angular quantum numbers was discussed in Section 3.2.3.1. The new radial quantum number, n, takes all positive integer values. Upon second- quantization of the q1, the energy of a singly occupied mode is given by the frequency, wnl = M 2 Anl/7Z2. Analogously to Equation (3.80), the total energy in the cavity for a pure quantum state is given by p 00 U = E E wnlnlnmI, (3.106) I=1 n=l l,m

____il _ where the integers nlmI are the occupation numbers specifying the state. Except for the detailed labeling of modes, the partition function for a thermal state is computed exactly as it was in Equations (3.82) and (3.83). In fact, the expressions are somewhat simpler because the energy only depends on the two quantum numbers n and l: 00 00 lnQ= -p EEDd(l)l n (1- e-w'n). (3.107) n=1 1=0 The angular momentum degeneracy Dd(l) is given by Equation (3.64). The average occupation number of a given n, 1} mode in the thermal state is then given by 1 0 (nnl)=- aw InQ = pDa(l), , . (3.108) ,3 19U~n1 eIcwnl -

Likewise, the total energy in the cavity is given by

a 00 pDd()wnl (U)=-l n = E 2QE PD)Cl (3.109) (U-~'9 n=l 1-O ef~wnl -

This had better be proportional to the volume of the cavity Viz= Vol(Bd-l)7d-1. As in the case of the hypercubic cavity, we cannot simply take the 7 - oo limit of Equation (3.109), interpret the result as a spectral density unl (n, 1), and then sum over n, l} to get the total energy. For the reasons discussed in Section 3.3.1, such a procedure gives meaningless results. One's intuition from commonly interpreting the I quantum number as the actual, physical angular momentum of a wave mode would lead one to believe that in the large 7Rlimit, the n integral can be performed in some properly regulated way, leaving a quantity that can be appropriately interpreted as the angular momentum spectral density ul(l) which would obey

00 (U) = Eui(l). (3.110) 1=0 Unfortunately, this is not the case. Instead, the quantum number must be regulated due to a diverging density of states by passing to a new variable e = /R. This new e has units of linear momentum 19 and actually would be the physical linear momentum

8 Care should be taken not to confuse the symbols for occupation number and radial quantum number. Likewise, the symbol m is being used differently in this section than it was used in Section 3.3.1. 19 The quantum numbers {1, m} were never really the angular momentum operator eigenvalues to begin with. As defined in Section 3.2.3.1, they are more properly referred to as the canonical momenta conjugate to the angular coordinates. This is why there are d-2 of these quantum numbers. Angular momenta are the generators of the rotation group SO(d - 1), which is (d - 1)(d - 2) dimensional. Of course, a wonderful point of connection between group theory and mechanics occurs here in that the d- 2 conjugate momenta form a set consisting of the Casimir operator for the representation and a Cartan subalgebra of SO(d- 1), which is why these quantum numbers are all that is needed to label physical states.

89 if space were an Sd-2 of radius R. Also, with this definition we get

2fd-3 Dd(l) -= d- s d_ (3.111) (d - 3)!

We will approach the radial part of Equation (3.109) by eliminating the sum over n in favor of a sum over physical radial momentum k, which is simply related to A by Equation (3.105). This will require knowing the phase space density On/OA in the limit of large 7R. We will now evaluate On/OA by two independent methods.

3.3.3.1 Radial Mode Density

One approach to finding On/OA is to find an approximate explicit expression for Ani which is valid in the limit 1 - oo. We begin on this route with an asymptotic series expression for jn(Z) of the Hankel type (large argument, fixed order) [89, Section III.3.14.1, for example]: \ -E(-l) mL'(n+ + 2m) -jn(Z) = cos - 7r( 2+)) 2 mo(2m)!r(n + 1 - 2m) (2z)-2m si( ~r(n - 1))[ (-1)m F(n + + 2m + 1) (2z)2m1 - 2m (3.112) 2 (2m(2m + 1)!F(n )!(n + --z)2m2m -1)- 1) .m--0 2~~~~~ Thus, we have

(d-4)/2 R (d-2)/2 (2> =wz J (Z) = yjl+(d-3)/2(Z) 7r) (-1)m r(l + (d- 2)/2 + 2m) z 7r(21+d-2)) [e Z)-2m] 4 =o(2m)!r(l + (d- 2)/2- 2m) ( (-1)m r(l + (d- 2)/2 + 2m + 1) - sin z -7r(21 + d-2) (2Z)-2m- 1 4 [m=o (2m+1)!r(l+ (d-2)/2-2m-1) (3.113)

Since we will be writing = RI for large 7?, we can apply the following math fact: if b is an integer, then

F+1 - b) (a r(a + 1-b) - b) =(a+ b)(a+ b- 1)... (a + 2)(a+ 1)a(a- 1)... (a-b+ 2)(a-b+ 1) =[a(a + 1)- (b- 1)b][a(a+ 1)- (b-2)(b-1)] ... [a(a+ 1)-12][a(a + 1)]

z[a(a + 1)]b, (3.114)

_ 1_ where the last line holds when a > b. Then we have

(2> (d-4)//2 Rkl(z/k) 7r / 7 ~d2 2m 00 [I+ (d- 4)/2][ + (d- 2)/2]) cos z +2 (-1)M ( 2z M=0Y- (2m)! [ + (d 4)/2][ + (d 2)/2]) 2m+1 j T [Id- 2 0 (-l)m - - - sin z - 142 (2m + 1)! 2z I[1+ (d-4)/2][ + (d- 2)/2]) ( 2T 2]) ( =COS Z - - + ( 2z 1 2)/2]> (2 [Id2 2)COS] ([ + (d-4)/2][ + (d- - sin ( [I+ 2]sin 2z [I+ (d-4)/2][1 + (d-2)/2]\ =cos~z~~ -~ d (3.115) 2z j So,

J (z) zoo 2 ( 7 (d-2)/2 7 [, , d - 2 [I (d - 4)/2][I + (d - 2)/2] \ {-· N _, 2 I '~' V2 [ 2 j ' 2z (3.116) This vanishes when z = Anl obeys

[I+ (d- 4)/2][1 + (d- 2)/2] I - [I + + = '7r n - - 1 (3.117) A,,2 2 such that

On _ 1 1 [ + (d- 4)/2][1 + (d -2)/2]> A 7- 2A2 J [I + (d- 4)/2][1 + (d- 2)/2] 2 (3.118) 71< A

This last expression holds for large values of A, such that A21 >> 12, Thus, the mode hierarchy implicit here is A > 1 > 1, or equivalently 7R= A/k >>R/k >>1/k. This makes sense, since k and are physical momenta which should have finite values as R - o. Equation (3.118) can also be derived by the method of Ari Turner [90], which is to look at the WKB solution to (3.103). This is

jRnlV(7r)( ) - /4 exp i r Rnl(r) ';z V(r>-1/4 exp i/ V(x)dx} , (3.119) with 2 V(r) = k - [I + (d - 4)/2][1 + (d- 2)/2] (3.120) (r)- - 2~~~_

91 Taking an appropriate real solution, this will vanish when the phase is rn. So,

[Anl / [1+ (d- 4)/2][1+ (d- 2)/2 d (3.121)

krJ x2 which again yields

On = 11 1_I + (d-4)/2][1 + (d-2)/2] (3.122) aA r A2

3.3.3.2 Spectral Densities

We can now return to Equation (3.109). Using R -- oo, either of Equations (3.118) and (3.122) become On I On I £ an 1 an 1 2 (3.123) ,\ TRak 7r k2 Note that the square root sets either a lower limit on k or an upper limit on e, depending on the order of integration. We get

(U) 00dk OdRan2pd3Rd-37gk (U) =ek (d- 3)!(e k - i)

- dk] d£ (p d-3 k 7r-3)! d k 2 e,3wk -

=fo°°0ofk £e2 d- 3Wk r(d - 3)!Vol (I d- 1) k2edk -

=VZ L dkp Vol (Sd- 2) kd- 2wk (3.124) Jo (27r)d-1 efPWk- '(314 which is in perfect agreement20 with Equation (3.88). As in Section 3.3.1, this yields a spectral energy density over the radial momentum given by Equation (3.89) which can be exchanged for the spectral energy density over the frequency given in Equation (3.90). Either of these spectra could then be integrated to give the energy density of the thermal state. The methods of Section 3.3.2 could then be applied to give the blackbody flux. We still need to calculate the new physical quantity made available by analysis in spherical coordinates: the angular momentum spectral energy density. Reversing the

2 0The e integral in Equation (3.124) was performed using fl dxx v-_ = r[(+W)/21. We

- alsoused the identity r (n+2) r2 (n-') 2 = ~2n

------order of integration in Equation (3.124), we find

3 (U) _ _2p__ _ _ d dk 1 k. £d Wk V P=w(d - 3)!Vol (Bd-1) J k efke -

d - a ' / 2 M 2 2 ______d 00 d , - I) - 2 (3125) -r(d- 3)!Vol (d-1) J W2 M2 e - . (3.125)

Some remarks are in order regarding the form of Equation (3.125). Note that the d-dependence has dropped out of the k or w integrals. This is because these integrals describe the thermodynamics of only the t- r section of spacetime, which is independent of dimension. In fact, for low partial wave modes ( < M) or large masses, the frequency form of Equation (3.125) yields

00 pDd(e)d- dw 1I 2 ue(e) Vol ( ) A L- M e - (3.126) suggesting, via Equation (3.90), an infinite collection of massive 1 + 1 dimensional modes carrying an internal quantum number with degeneracy volpDd()l), dl) each ther- mally occupied according to standard 1 + 1 dimensional physics. This degeneracy factor is just the normal counting of angular momentum states, taken to the continuum limit. Thus Equation (3.126) is telling us that all the original {l, m} modes with e << M are all uniformly occupied like 1 + 1 dimensional blackbodies.

Another important limit of Equation (3.125) is high temperature (T > M, £). In this limit, the mass and angular momentum contribute to the w integral only through the lower integration limit, so we can use Equation (3.92) to get

uepT2£d- 3 _ 2 ~V-+M 2 /T (1T (1- ~)/2 3)3(d -3)! Vol ( d- l) 2 (3.127)

Equation (3.127) should be valid for both massive and massless fields near = 0, as long as T > M.

For M = 0, Equation (3.125) yields

U,=2p() T2d3~o2 3 ue(e) 7r(d- 3)! Vol (d-1)T ed I(3e) (3.128)318 where I(a) = dx _ -a (3.129)

Note that I+(O) = 7r2 /6 and I_(0) = 7r2/12. The function I(a) is plotted in Figure 3-5. For << T we again find Equation (3.127), evaluated at M = 0. Again factoring out the angular degeneracy factor, Equation (3.127) implies that for <

93 1.8

I- C3

Figure 3-5: The thermal integral Is(a) defined in Equation (3.129). The solid top curve indicates the bosonic function I+(a). The dashed bottom curve indicates the fermionic function I_(a). The dotted central curve indicates ~ = 0, corresponding to Boltzmann statistics. All three lines seem to converge towards el-a for a 1.

the original {l, m} modes contributes an amount

- -T/ e (3.130) (1)(2 -)/2 p.T2ee/T6 to the energy density p. At = 0, this agrees with Equation (3.101), showing once again that each partial wave in this regime behaves like a 1+ 1 dimensional blackbody. As was noted in Section 3.2.3.3, near a black hole event horizon all effects of masses, interactions, and angular momentum in the radial wave equation of a field are exponentially suppressed, with the suppression becoming exact at the horizon. In particular, this means that all modes of blackbody radiation near an event horizon behave just like the = 0 mode, which makes no contribution to the effective radial scattering potential. Thus, we expect that the energy density in every partial wave of d-dimensional blackbody radiation at the event horizon of a d-dimensional black hole is given exactly by the 1 + 1 dimensional result of Equation (3.101). Figure 3-5 shows that, at least for large ( > T), we have

ue(£) (d- T2d3e/T. (3.131) -r(d-3)! Vol (d 1) Equations (3.127) and (3.131) together should accurately reproduce the exact M = 0 spectrum given in Equation (3.128) except for e - T. It is interesting to see that apparently all the effects of statistics are in the small e modes. Also, the Fermi-Dirac

I_·___... factor appearing at small f is , the factor for d = 2. This again shows, as was noted with Equations (3.126) and (3.130), the special role of the 1 + 1 dimensional blackbody in the partial wave spectrum. Of course, when summed up over all values of £, the Fermi-Dirac factors must combine to give 1- 2/2d, in order to properly reproduce d-dimensional physics. It is not obvious from Equations (3.128) and (3.129) exactly how this happens, but the constructions of this section guarantee that it does.

3.4 Calculation

We now return to formulating an effective field theory outside of a black hole. The effective theory is formed by eliminating the dangerous horizon-skimming modes by hand - to avoid singularities in the energy-momentum tensor near the horizon - while simultaneously adding a compensating energy-momentum tensor - to avoid anomalies. Consider the partial wave decomposition of a scalar field in a static, spherically symmetric background spacetime. In suitable coordinates, the metric of the spacetime can be written as in Section 3.2:

ds= -f(r)dt2 2 + 1 ~~1dr 2 +r22d 22. (3.132) h(r)

The properties of this metric were discussed extensively in Section 3.2. In particular, we adopt here the physically motivated assumption that the zeros of f(r) and h(r) are coincident. In this scenario, we found that physics near the horizon can be described using an infinite collection of 1 + 1 dimensional fields, each propagating in a spacetime with a metric given by the "r-t" section of the full spacetime metric (3.132). We will also adopt this simplification. For the reasons discussed in Section 3.1.4, we impose the constraint that outgoing (horizon-skimming) modes vanish near the horizon as a boundary condition. We take this condition to be localized on a slab of width 2 straddling the horizon at r = rh with - 0 ultimately (see Figure 3-1). The energy-momentum tensor in this region then exhibits an anomaly of the form (3.2). For a metric of the form (3.132), the anomaly is purely time-like and can be written as 1 a VaTa - Ab- 1=aNia, (3.133) where the components of Na are

Nt = Nr = 0, (3.134a) 1 N = 192 (f'h' + f"h), (3.134b) Nt = 19 2in ( ah"). (3.134c) 192n-r

95 The parameter a is an arbitrary number with no effect on physics21 because Ab is independent of N'. The contribution to effective action for the metric gab due to matter fields that interact with this metric is given by

W[gab] -i in ( D[matter]eis[matter'9ab), (3.135) where S[matter, gab] is the classical action functional. Under general coordinate transformations the classical action S changes by A5S = - f ddxx/IgAbVaTb where T is the energy-momentum tensor and A is the variational parameter. General covariance of the full quantum theory requires 6xW -= 0. We write this as

2 -SA0vW= Jd X/7AbVa {TbH + TaE@+ + Tbe2 } = Jd2 xAt {r (NtrH)+ (z-To - TXt + Nt) ae+ + (/-Tit- Tx[ + Nt) a@-}

+ dxArv-g{(To - T~) ae++ (orre -Tx;) } (3.136) where E)+ = ( (r T rh- e) are scalar step functions and H = 1 - +- e_ is a scalar "top hat" function which is 1 in the region between rh ± E and zero elsewhere. The anomalous chiral physics is described by Txb via Equation (3.133). The energy- momentum tensors Tob and T b are the covariantly conserved energy-momentum tensors outside and inside the horizon, respectively. Constancy in time and Equation (3.133) together restrict the form of the Tb up to an arbitrary function of r, which is the trace T, and two constants of integration, K and Q: Ttt = -(K + Q)/f - B(r)/f - I(r)/f + Ta(r), (3.137a) Trr = (K + Q)/f + B(r)/f + I(r)/f, (3.137b) V/-LTr = - K + C (r -f-h-g h Trt, (3.137c) wherewe have defined C(r) - frrr /A(x)dx, -f B(r) f (x)Ar(x)dx,and I(r)

2 f rhTa (x)f (x)dx. A few remarks regarding the evaluation of Equation (3.137) are in order. A trace could arise from a number of physical sources, among them a conformal anomaly. We assume, however, that I/f = Tad is finite. Since we will be concerned with the rh rh conditions imposed by canceling potential divergences, finite terms will play no role. Moreover, the terms containing the components of Ab vanish at the horizon. Note

2 1Interestingly, for ac = 1 the antisymmetric part of Nab is equal to -eabR/(192ir), where R is the d = 2 Ricci scalar as given in Equation (3.10).

96 that for the diagonal terms in Equation (3.137), the limit r -- rh depends on whether rh is approached from above or below, since f flips signs as the horizon is crossed. The limit on 1/f is exactly antisymmetric, so lim -= ( lim -). (3.138) r-rh-*0-f r-rh-*0+f

Finally, A/ is a finite number at the horizon, given by /-5 in the notation of Section 3.2.1.2. We can now take the - 0 limit of Equation (3.136). The term Or9(N[trH)vanishes in this limit. Using the relation (3.138) to take all limits from above, and the small e expansions, aE_=6r 1 _ r±12_2 0a4- = -- [+1-r ±- o- ...* ] 6(r - rh), (3.139) the variation (3.136) becomes

6AW = d2xAt {[Ko - Ki 6(r - rh) - [Ko+ Ki- 2Kx-2Ntr]a(r - rh)+-.}

-J d2xAr { [KO+Qo+Ki+Qi 2Kx 2Qx] (r -rh)

-_E [Ko+QofKi-Qi] O(r- rh) +.. .. (3.140)

The ellipses represent higher order terms in with higher derivatives of 6-functions; the coefficients of these terms are simply repetitions of the ones given above. The delta functions in Equation (3.140) indicate that only the on-horizon values of the energy-momentum tensors will contribute to the possible loss of general covariance. The finite trace terms make no contribution in comparison to the divergent K + Q terms. Since At and Ar are independent arbitrary variational parameters, each of the four terms in square brackets in Equation (3.140) must vanish simultaneously, but only need do so at r = rh. These four conditions can be solved to give

Ko= K =Kx + , (3.141a) Q = Qi =Qx - P, (3.141b) where = Ntr| = . (3.142) rh 48ir' These conditions fix the 4 of the 6 constants Q and K. The total energy momentum tensor b - _ob T = To + + T XbH (3.143) becomes, in the limit - 0, Tb = TCb + Tb, (3.144)

97 where Tc~ is the conserved energy-momentum tensor which the matter in this theory would have without any quantum effects, and TD is a conserved tensor with K = -Q = ), a pure flux. As discussed in Section 3.3.2, a beam of massless blackbody radiation moving in the positive r direction at a temperature T has a flux of the form = T 2. Thus we see that the flux required to cancel the gravitational anomaly at the horizon has a form equivalent to blackbody radiation with a temperature given by T = /(27r). This is exactly the Hawking temperature for this spacetime, as discussed in Sections 3.2.2.1 and 3.2.2.2. Thus, the thermal flux required by black hole thermodynamics is capable of canceling the anomaly. If we fill each partial wave of the full d-dimensional theory so that each one behaves like a 1 + 1 dimensional blackbody source at the Hawking temperature, then we reproduce the core of the standard calculation of black hole emission. This is exactly what one requires such that when the partial waves are propagated outwards from the black hole and undergo mode-dependent scattering from effective potential due to spatial curvature outside the horizon, the resultant occupation density at infinity is that of a d-dimensional blackbody at the Hawking temperature, modulo greybody factors.

3.5 Commentary

In contrast to the preceding argument based on gravitational anomaly cancellation, it appears difficult to generalize the conformal anomaly derivation [77] to arbitrary dimensions using partial wave analysis. In that framework the connection between the anomaly and the Hawking flux is made through an integral over all of space. In our framework the connection between the anomaly and the Hawking flux is made through a boundary condition at the horizon, which is accurately described using 1 + 1 dimensional physics, irrespective of the true dimension. Comparing the fluxes for thermal radiation of massless bosons and fermions in 1 + 1 dimensions, we find, as in Equation (3.101), that the boson flux is twice that of the fermion flux. This same factor of two appears in the relative values of the conformal anomalies (central charge) and of the gravitational anomalies. There does not appear to be any comparably simple correspondence in higher dimensions. How- ever, when the 1 + 1 dimensional field in question is really a single partial wave of a higher dimensional system, the blackbody angular momentum spectrum discussed in Section 3.3.3.2 guarantees us that this factor of two arising from the 1 + 1 dimensional - anomaly adds up to an overall (1 - 2/ 2 d) 1, which is the proper statistical factor in d dimensions. In the context of an eternal black hole one can find a role for thermal radiation incoming to the black hole by imposing additional boundary conditions near the past horizon (V = 0, in the language of Section 3.2.2) that are symmetric with the ones we imposed above near the future horizon. This corresponds to the Hartle-Hawking state [91].

___ 3.6 Blackbody Spectrum from an Enhanced Sym- metry?

While the arguments advanced here show a pleasing consistency between the existence of Hawking radiation flux and gravitational anomaly cancellation, they do not in themselves suffice to show that the spectrum of radiation is thermal. One might hope to single out the thermal state by imposing an appropriate symmetry. Indeed, thermal states support a form of time-translation symmetry that makes sense even near the horizon, namely translation by discrete units 3 of imaginary time. Specifically, in Section 3.2.2.2 we generalized the Euclidean method of [42] to show that the Euclidean section of the black holes we are studying contains a conical singularity at the horizon unless the period of Euclidean time is taken to be /3 = 1/TH. If multivalued coordinates are allowed, however, the singularity can be removed by taking the period to be any integer multiple22 of this factor, /3 = n/TH, giving a discrete set of allowed temperatures for this spacetime, Tn = TH. Perhaps less objectionable is consideration of the Kruskal coordinate U. We argued in Section 3.2.2.1 that respecting translation in U as a good spacetime symmetry at the past horizon, V = 0, can be seen to give rise to Hawking radiation. We also noted, by construction, that t-translation is a global Killing vector. From the definition (3.41a), we see that under a time translation t - t + a, U -+ e-aU. If we want Hawking radiation modes to be properly analytic in the complex frequency plane, we should demand that that U is invariant under this transformation when a is a pure imaginary number. This demands that a = 2rin/n, for any integer n, which says that the black hole must be invariant under discrete translations of imaginary time of magnitude 3 = 2n/n. Again, this gives a discrete set of allowed temperatures, T. = 1TH While neither of these arguments is quite satisfying, they do make plausible the idea that an enhanced discrete translation symmetry can fix the thermal nature of the black hole system, and that temperatures of different magnitude can be accommo- dated as different units for the periodicity by T = 1/,3. If we assume that a symmetry of this form exists, then anomaly cancellation fixes the unit. One could certainly wish for a less formal, more physically enlightening perspective, however.

22 With this integer factor n, one must travel in a circle around the origin a distance 2rn in Euclidean time to return to one's starting point. This makes the black hole appear like an anyon [92] of spin 1/n. Such particle states are indeed allowed in the two dimensional r-x plane of Equation (3.43), but their physical significance is uncertain.

99 100

____II__ Chapter 4 Finale

4.1 Summary

In Chapter 2, we found that the addition of gravity to four-dimensional Yang-Mills theories adds a term to the one-loop 3 functions of the form

3 E 2 Agrav (gym, E) = 3-9E 2 , (4.1) which renders all Yang-Mills couplings asymptotically free. To some extent, this result simplifies the physics of the the early universe because, for what it is worth, gluon dynamics can be ignored and we only need to worry about quantum gravity. We believe this gravitational correction is only directly observable if the true scale of quantum gravity is unexpectedly low. We also found that the gravitational correction does not spoil coupling constant unification in a theory whose Yang-Mills couplings exhibited unification before gravity was added. The unification point remains unique and is shifted slightly in energy (approximately one part in 106 for realistic theories). We showed in Chapter 3 that the Hawking radiation from a rather generic class of spherically symmetric black holes in arbitrary dimensions can be understood as arising from a gravitational anomaly in the 1 + 1 dimensional effective quantum field theory that governs physics close to the horizon. This helps to raise the connection of anomalies to Hawking radiation above the level of an isolated curiosity. We did not prove definitively that anomaly cancellation requires the radiation to be thermal, but argued that the radiation has several features in common with blackbody radiation at the Hawking temperature.

4.2 Open Possibilities

As mentioned in Section 2.11, there are several questions raised by the calculation of the gravitational correction to the Yang-Mills function. For example, how is the calculation implemented in TeV-scale gravity theories and what are the experimental signatures thereof in colliders, cosmic rays, and atomic systems? Also, how the does

101 calculated correction effect the fitting of experimental data with uncertainties to the hypothesis of coupling constant unification? An obvious program is to extend the calculation to Yukawa couplings and scalar self-interactions. It would be rather astonishing if these couplings were also rendered asymptotically free by gravitation, since they can not be asymptotically free in four dimensions without gravity. Mass renormalization could also be studied, but this is not as interesting since we expect gravity-induced renormalization to be important only at scales much higher than any conventional particle physics masses. The derivation in Section 3.4 of Hawking radiation as a mechanism that cancels a gravitational anomaly also leaves some open questions. Obvious among these is whether it can be generalized to rotating or otherwise non-spherical black holes. Also, it remains to be seen if the Killing frequency spectrum of the 1 + 1 dimensional theory can be calculated from the anomaly such that the thermal nature of the radiation is completely elucidated. One might further wonder if this mechanism has any relevance to the black hole information paradox or whether it can be used to calculate black hole entropy. The answer to these last questions is probably "no" because Hawking radiation is a kinematic effect, while information and entropy issues necessarily involve dynamics. However, these questions do deserve a deeper investigation. Our study of black hole effective field theory was motivated by rather general concerns regarding an observer's ability to describe physics in terms of the degrees of freedom which he can experimentally probe. The essential difficulty in doing this was encountered most directly in a semiclassical black hole background, but the same problem should generically arise in any generally covariant quantum field theory. Per- haps the anomaly cancellation mechanism discovered here can be generalized to allow a generally covariant formulation of quantum field theory in which local quantum field theories formulated on local coordinate patches of spacetime are stitched together to form a global theory using anomaly-driven currents that act as a kind of connection. If such a theoretical framework existed it would, in a certain sense, make no essential distinction between fundamental and effective degrees of freedom.

4.3 Conclusion

We hope that we have shown in this thesis that much can be learned within the limited scope of quantum general relativity and semiclassical gravity if the logical boundaries of these models are properly respected. Furthermore, we believe that the work presented here points in a promising way towards features that should ultimately be reproducible in a more fundamental theory of quantum gravity.

102

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