The Unruh effect and its applications Luís C. B. Crispino* Faculdade de Física, Universidade Federal do Pará, Campus Universitário do Guamá, 66075-900 Belém, Pará, Brazil Atsushi Higuchi† Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom George E. A. Matsas‡ Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 São Paulo, SP, Brazil �Published 1 July 2008� It has been 30 years since the discovery of the Unruh effect. It has played a crucial role in our understanding that the particle content of a field theory is observer dependent. This effect is important in its own right and as a way to understand the phenomenon of particle emission from black holes and cosmological horizons. The Unruh effect is reviewed here with particular emphasis on its applications. A number of recent developments are also commented on and some controversies are discussed. Effort is also made to clarify what seem to be common misconceptions. DOI: 10.1103/RevModPhys.80.787 PACS number�s�: 03.70.�k, 04.70.Dy, 04.62.�v CONTENTS I. Introduction 788 II. The Unruh Effect 789 A. Free scalar field in curved spacetime 789 B. Rindler wedges 792 C. Two-dimensional example 793 D. Massive scalar field in Rindler wedges 795 E. Bogoliubov coefficients and the Unruh effect 796 F. Completeness of the Rindler modes in Minkowski spacetime 798 G. Unruh effect and quantum field theory in the expanding degenerate Kasner universe 799 H. Unruh effect and classical field theory 800 I. Unruh effect for interacting theories and in other spacetimes 801 III. Applications 804 A. Unruh-DeWitt detectors 804 1. Uniformly accelerated detectors in Minkowski vacuum: Inertial observer perspective 805 2. Uniformly accelerated detectors in Minkowski vacuum: Rindler observer perspective 806 3. Rindler particles with frequency ��m 808 4. Static detectors in a thermal bath of Minkowski particles 809 5. Discussion on whether or not uniformly accelerated sources radiate 810 6. Other results concerning the Unruh-DeWitt detector 810 7. Circularly moving detectors with constant velocity in the Minkowski vacuum 812 B. Weak decay of noninertial protons 813 1. Inertial observer perspective 814 2. Rindler observer perspective 816 C. Bremsstrahlung 818 1. Inertial observer perspective 818 2. Rindler observer perspective 819 IV. Experimental Proposals 822 A. Electrons in particle accelerators 823 B. Electrons in Penning traps 825 C. Atoms in microwave cavities 825 D. Backreaction radiation in ultraintense lasers and related topics 826 E. Thermal spectra in hadronic collisions 827 F. Unruh and Moore �dynamical Casimir� effects 828 V. Recent Developments 828 A. Entanglement and Rindler observers 828 B. Decoherence of accelerated detectors 829 C. Generalized second law of thermodynamics and the “self-accelerating box paradox” 829 D. Entropy and Rindler observers 830 E. Einstein equations as an equation of state? 830 F. Miscellaneous topics 830 VI. Concluding Remarks 831 Acknowledgments 831 Appendix: Derivation of the Positive-Frequency Solutions in the Right Rindler Wedge 831 References 832 *crispino@ufpa.br †ah28@york.ac.uk ‡matsas@ift.unesp.br REVIEWS OF MODERN PHYSICS, VOLUME 80, JULY–SEPTEMBER 2008 0034-6861/2008/80�3�/787�52� ©2008 The American Physical Society787 http://dx.doi.org/10.1103/RevModPhys.80.787 I. INTRODUCTION It has been 30 years since the discovery of the Unruh effect �Unruh, 1976� which can be also found under the name of Davies-Unruh, Fulling-Davies-Unruh, and Unruh-Davies-DeWitt-Fulling effect. This is a conceptu- ally subtle quantum field theory result, which has played a crucial role in our understanding that the particle con- tent of a field theory is observer dependent in a sense to be made precise later �Fulling, 1973� �see also Unruh �1977��. This effect is important in its own right and as a tool to investigate other phenomena such as the thermal emission of particles from black holes �Hawking, 1974, 1975� and cosmological horizons �Gibbons and Hawk- ing, 1977�. It is interesting that the Unruh effect was on Feynman’s list of things to learn in his later years �see Fig. 1�. In short, the Unruh effect expresses the fact that uniformly accelerated observers in Minkowski space- time, i.e., linearly accelerated observers with constant proper acceleration �also called Rindler observers�, as- sociate a thermal bath of Rindler particles �also called Fulling-Rindler particles� to the no-particle state of iner- tial observers �also called the Minkowski vacuum�. Rin- dler particles are associated with positive-energy modes as defined by Rindler observers in contrast to Minkowski particles, which are associated with positive- energy modes as defined by inertial observers. Unruh �1976� also provided an explanation for the conclusion obtained by Davies �1975� that an observer undergoing uniform acceleration a=const in Minkowski spacetime would see a fixed inertial mirror to emit thermal radia- tion with temperature a� / �2�kc�, and the reason why this is not in contradiction with energy conservation. Al- though there are some accounts in the literature discuss- ing the Unruh effect,1 we believe that this review will be a useful contribution for the reasons listed below. First, some have recently questioned the existence of the Unruh effect �Narozhny et al., 2002, 2004�. We be- lieve there are two main sources of confusion, which need to be clarified in order to address these objections. One is that it has not been made clear that the heuristic expression of the Minkowski vacuum as a superposition of Rindler states makes sense outside as well as inside the two Rindler wedges. Although this point is not cen- tral to the Unruh effect �Fulling and Unruh, 2004�, it will be useful to point out that this heuristic expression in fact makes sense in the whole of Minkowski spacetime. Another common source of confusion is that the Unruh effect is sometimes tacitly assumed to be the equiva- lence of the excitation rate of a detector when it is �i� uniformly accelerated in the Minkowski vacuum and �ii� static in a thermal bath of Minkowski particles �rather than of Rindler particles�. There is no such equivalence in general, although this showed up by coincidence in some early examples in the literature �see discussion in Sec. III.A.4�. We emphasize that this point does not con- tradict the fact that the detailed balance relation satis- fied by static detectors in a thermal bath of Minkowski particles is in general also valid for uniformly acceler- ated ones in the Minkowski vacuum �Unruh, 1976�. The identification of the Unruh effect with the behavior of accelerated detectors seems to have generated some- times unnecessary confusion. It is worthwhile to empha- size that the Unruh effect is a quantum field theory re- sult, which does not depend on the introduction of the detector concept. In this sense, it is better to see the detailed balance relation satisfied by uniformly acceler- ated detectors as a natural consequence or application rather than a definition of the Unruh effect. In order to exemplify the meaning of the Unruh effect as the equivalence between the Minkowski vacuum and a ther- mal bath of Rindler particles, we collect and discuss a number of illustrative physical applications. The Unruh effect has also been connected with the long-standing discussion about whether or not sources2 uniformly accelerated from the null past infinity to the null future infinity radiate with respect to inertial ob- servers. Although some aspects of this issue are worth investigating and the corresponding discussion can be enriched by considering the Unruh effect, it is useful to keep in mind that the Unruh effect does not depend on a consensus on this issue which seems to be mostly se- mantic �see Fulling �2005� and related references�. We comment on this issue in Sec. III.A.5. Second, there have been several recent proposals to detect the Unruh effect in the laboratory and it is useful to review and assess them. We emphasize that, although it is fine to interpret laboratory observables from the point of view of uniformly accelerated observers, the Unruh effect itself does not need experimental confir- mation any more than free quantum field theory does.3 Finally, there has been an increasing interest in the Unruh effect �see Fig. 2� because of its connection with a number of contemporary research topics.4 The thermo- dynamics of black holes and the corresponding informa- tion puzzle is one of them. It will be beneficial therefore to review the literature on the generalized second law,5 quantum information,6 and related topics with the Un- ruh effect as the central theme. 1See, e.g., Sciama et al. �1981�; Birrell and Davies �1982�; Takagi �1986�; Fulling and Ruijsenaars �1987�; Ginzburg and Frolov �1987�; Wald �1994�; Padmanabhan �2005�. 2Throughout this review we will use the word “sources” to mean scalar sources, particle detectors, or electric charges, de- pending on the context where it appears. 3This statement should be understood in the sense that we are dealing with mathematical constructions that stand on their own. The assertions follow from the definitions and so do not need to be experimentally verified. The fact that “Rindler and Minkowski perspectives” give consistent physical predictions is a consequence of the validity of these constructions. 4The data in Fig. 2 should not be used to infer any relative or absolute measure of the importance of the Unruh effect. They have been introduced only as an illustration on the increasing interest in this issue. 5See, e.g., Unruh and Wald �1982, 1983�; Wald �2001�. 6See, e.g., Bousso �2002�; Peres and Terno �2004�. 788 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 The review is organized as follows. In Sec. II we re- view the derivation of the Unruh effect, emphasizing the fact that the quantum field expanded by the Rindler modes can be used in the whole of Minkowski space- time, partly to respond to the recent criticisms men- tioned above. We also touch upon more rigorous ap- proaches such as the Bisognano-Wichmann theorem in algebraic field theory and general theorems on field theories in spacetimes with bifurcate Killing horizons. A discussion of the Unruh effect in interacting field theo- ries is also included. In Sec. III we present some typical examples which illustrate the physical content of the Unruh effect. We start by reviewing the behavior of ac- celerated detectors which can be also used to describe the physics of accelerated atomic systems. Then, we ana- lyze the weak decay of accelerated protons and the bremsstrahlung from accelerated charged particles. Sec- tion IV discusses some experimental proposals for labo- ratory signatures of the Unruh effect in particle accel- erators, in microwave cavities, in the presence of ultraintense lasers, in the vicinity of accelerated bound- aries, and in hadronic processes. In Sec. V we comment on some recent developments concerning the Unruh ef- fect, which include the possible reduction in fidelity of teleportation when one party is accelerated, the deco- herence of accelerated systems, and the possible ob- server dependence of the entropy concept. We conclude the review with a summary in Sec. VI. Throughout this review we use natural units �=c=G=k=1 and signature �� � � �� unless stated otherwise. It would be impossible to give a completely balanced account of so much work in the literature concerning the Unruh effect. This review reflects our own experience with the Unruh effect, and we are concerned that we may have overlooked some important related papers. However, we hope at least to have included a sufficient number of papers to allow the readers to trace back to most related results. II. THE UNRUH EFFECT A. Free scalar field in curved spacetime Even though the Unruh effect is about quantum field theory �QFT� in flat spacetime, it is useful to review briefly the general framework of noninteracting QFT in curved spacetime. We treat only the simplest theory, i.e., the theory of a Hermitian scalar field satisfying the Klein-Gordon equation. We present it in a schematic and heuristic way as done by Birrell and Davies �1982�. A mathematically more satisfactory treatment can be found, e.g., in Wald �1994�. We first remind the reader of some important features of QFT in Minkowski spacetime. In this spacetime the scalar field is expanded in terms of the energy- momentum eigenfunctions, and the vacuum state is de- fined as the state annihilated by all annihilation opera- tors, i.e., the coefficient operators of the positive- frequency eigenfunctions defined to be those proportional to e−ik0t with k0�0, where t is the time pa- rameter. The coefficient operators of the negative- frequency modes proportional to eik0t are the creation operators, and the states obtained by applying creation operators on the vacuum state are identified with states containing particles. Note that the time-translation sym- metry, which enables one to define positive- and negative-frequency solutions to the Klein-Gordon equa- tion, plays a crucial role in the definition of the vacuum state and the Fock space of particles. Therefore, in a general curved spacetime with no isometries, there is no reason to expect the existence of a preferred vacuum state or a useful concept of particles. For simplicity we specialize to �D+1�-dimensional spacetimes whose metric takes the form ds2 = �N�x��2dt2 − Gab�x�dxadxb, �2.1� where x= �t ,x�. The coefficient N�x� is called the lapse function �Arnowitt et al., 1962� and Gab is the metric on the spacelike hypersurface of constant t. �All spacetimes considered have a metric of this form.� In this spacetime the minimally coupled7 massive Klein-Gordon equation �����+m2� =0, which arises as the Euler-Lagrange equation from the Lagrangian density, L = �− g��� �� − m2 2�/2, �2.2� takes the form �t�N−1�G�t � − �a�N�GGab�b � + N�Gm2 = 0, �2.3� where the space indices a and b run from 1 to D. Given two complex solutions fA�x� and fB�x� to the Klein-Gordon equation, we define the Klein-Gordon current J�fA,fB� � �x� � fA * �x���fB�x� − fB�x���fA * �x� . �2.4� Then, one can show that ��J�fA,fB� � �x�=0. Hence the quantity �fA,fB�KG � i� dDx �Gn�J�fA,fB� � �2.5� is independent of t, where n� is the future-directed unit vector normal to the hypersurface of constant t. �The integral here and throughout this subsection �Sec. II.A� is over the hypersurface .� We call this quantity the Klein-Gordon inner product of fA and fB. For the metric �2.1� it takes the following form: �fA,fB�KG = i� dDx �GN−1�fA * �tfB − fB�tfA * � . �2.6� The conjugate momentum density ��x� is defined as � ��L /� ̇, where ̇��t . For the metric �2.1� one finds 7It is customary to allow the field to couple to the scalar curvature. Thus the general Klein-Gordon equation takes the form �����+�R+m2� =0. The minimally coupled scalar field has �=0 by definition. 789Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 ��x� = N−1�G ̇�x� . �2.7� Note that if we let pA�x� and pB�x� be the conjugate momentum density for the solutions fA�x� and fB�x�, re- spectively, then the Klein-Gordon inner product can be expressed as �fA,fB�KG = i� dDx�fA * �x�pB�x� − pA * �x�fB�x�� . �2.8� We assume that the Klein-Gordon equation determines the classical field �x� uniquely given a �well-behaved� initial data � ,�� on a hypersurface of constant t. This property is known to hold if the spacetime is globally hyperbolic with t=const hypersurfaces as the spacelike Cauchy surfaces.8 The quantization of the field proceeds as follows. We denote the field operators corresponding to and � by ̂ and �̂, respectively. One imposes the following equal-time canonical commutation relations: � ̂�t,x�, ̂�t,x��� = ��̂�t,x�,�̂�t,x��� = 0, �2.9� � ̂�t,x�,�̂�t,x��� = i�D�x,x�� , �2.10� where the delta function �D�x ,x�� is defined by � dDxf�x��D�x,x�� = f�x�� . �2.11� Note here that there is no density factor �G on the left- hand side. For arbitrary complex-valued solutions fA�x� and fB�x� to the Klein-Gordon equation �2.3� �with a suitable integrability conditions� one finds ��fA, ̂�KG,� ̂,fB�KG� = �fA,fB�KG, �2.12� from the equal-time canonical commutation relations �2.9� and �2.10� by using Eq. �2.7�. Now, assume that there is a complete set of solutions, �fi , fi * , to the Klein-Gordon equation �2.3� satisfying �fi,fj�KG = − �fi *,fj *�KG = �ij, �2.13� �fi *,fj�KG = �fi,fj *�KG = 0. �2.14� We assume here that the indices labeling the solutions are discrete for simplicity of the discussion but its exten- sion to the cases with continuous labels is straightfor- ward. In Minkowski spacetime one chooses the positive- frequency modes as fi’s and, consequently, the negative- frequency modes as f i *’s. In a general globally hyperbolic curved spacetime without isometries, there are infinitely many ways of choosing the functions fi’s. Expanding the quantum field ̂�x� as ̂�x� = i �âifi�x� + âi †fi *�x�� , �2.15� one finds âi = �fi, ̂�KG, âi † = � ̂,fi�KG. �2.16� One can readily show, by using Eqs. �2.12�–�2.14�, that �âi, âj� = �âi †, âj †� = 0, �âi, âj †� = �ij. �2.17� Conversely, if these commutation relations are satisfied, then the equal-time canonical commutation relations �2.9� and �2.10� follow. To prove this, one first shows that any two complex-valued solutions fA�x� and fB�x� to the Klein-Gordon equation �2.3� satisfy Eq. �2.12� by ex- panding them in terms of fi�x� and f i *�x� and using the commutators �2.17�. Then, for example, by letting fA�t ,x�= f B * �t ,x� and pA�t ,x�=−p B * �t ,x�, at a given time t and evaluating the Klein-Gordon inner products in Eq. �2.12� at time t, one obtains � dDxdDx�fB�t,x�pB�t,x��� ̂�t,x�,�̂�t,x��� = i� dDxfB�t,x�pB�t,x� . �2.18� Since fB�t ,x� and pB�t ,x� are arbitrary, one can conclude that Eq. �2.10� holds at time t. Equation �2.9� can be derived in a similar manner. 8A Cauchy surface is a closed hypersurface which is inter- sected by each inextendible timelike curve once and only once. A spacetime is said to be globally hyperbolic if it possesses a Cauchy surface; see, e.g., Wald �1984� for more details. FIG. 1. Part of Feynman’s blackboard at California Institute of Technology at the time of his death in 1988. At the right-hand side one can find accel. temp. as one of the issues to learn. 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 06 10 20 30 40 50 Citation Number FIG. 2. Histogram depicting the number of citations of Unruh �1976� over the years. 790 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 The operators âi and âi † are called the annihilation and creation operators, respectively. The vacuum state �0� is defined by requiring âi�0�=0. The Fock space of states is obtained by applying the creation operators âi † on the vacuum state �0�. We call the operator N̂i= âi †âi �with no summation on the right-hand side� the number operator in the mode i. However, note that, since it is not always easy to construct a �theoretical� detector model which is excited when the eigenvalue of N̂i changes from 1 to 0, say, the operator N̂i does not necessarily lead to a useful particle concept. Since the coefficient operators âi of the functions fi annihilate the vacuum state �0�, the choice of the func- tions fi satisfying Eqs. �2.13� and �2.14� determines the vacuum state. For this reason we call the functions fi the positive-frequency modes and their complex conjugates f i * the negative-frequency modes in analogy with the case in Minkowski spacetime. Thus the choice of the positive-frequency modes determines the vacuum state. In a general curved spacetime there is no privileged choice of the positive-frequency modes, and, conse- quently, there is no privileged vacuum state unlike in Minkowski spacetime, as we mentioned before. Now, suppose that two complete sets of positive- frequency modes �fi �1� and �fI �2� satisfy the Klein- Gordon inner-product relations �2.13� and �2.14�, where the lower-case letters i, j are replaced by the upper-case equivalents I, J for fI �2�. Since both sets are complete, the modes fI �2� can be expressed as linear combinations of fi �1� and f i �1�* and vice versa. Thus fI �2� = i � Iifi �1� + �Iifi �1�*� , �2.19� fI �2�* = i � Ii * fi �1�* + �Ii * fi �1�� . �2.20� By noting that Ii = �fi �1�,fI �2��KG = �fI �2�,fi �1��KG * , �2.21� �Ii = − �fi �1�*,fI �2��KG = �fI �2�*,fi �1��KG, �2.22� one can express fi �1� as a linear combination of fI �2� and f I �2�* as fi �1� = I � Ii * fI �2� − �IifI �2�*� , �2.23� fi �1�* = I � IifI �2�* − �Ii * fI �2�� . �2.24� The scalar field ̂�x� can be expanded using either of the two sets �fi �1� and �fI �2� : ̂�x� = i �âi �1�fi �1� + âi �1�†fi �1�*� = I �âI �2�fI �2� + âI �2�†fI �2�*� . �2.25� Using the expansion given by Eqs. �2.19� and �2.20�, and comparing the coefficients of fi �1� and f i �1�*, we find âi �1� = I � IiâI �2� + �Ii * âI �2�†� , �2.26� and similarly by using Eqs. �2.23� and �2.24� we have âI �2� = i � Ii * âi �1� − �Ii * âi �1�†� . �2.27� This transformation, which mixes annihilation and cre- ation operators, is called a Bogoliubov transformation, and the coefficients Ii and �Ii are called the Bogoliubov coefficients. The Bogoliubov transformation found its first major application to QFT in curved spacetime in the derivation of particle creation in expanding uni- verses �Parker, 1968, Sexl and Urbantke, 1969�. The vacuum states �0�1�� and �0�2�� corresponding to the two sets of positive-frequency modes �fi �1� and �fI �2� , respectively, are distinct if �Ii do not vanish for all I and i. For example, the expectation value of the number op- erator N̂i �1�= âi �1�†âi �1� for the state �0�1�� is zero by defini- tion but for the state �0�2�� it can be calculated by using Eq. �2.26� as 0�2��Ni �1��0�2�� = I ��Ii�2. �2.28� We similarly find for the number operator NI �2�= âI �2�†âI �2�, 0�1��NI �2��0�1�� = i ��Ii�2. �2.29� Although the choice of the vacuum state is not unique in general, there is a natural vacuum state if the spacetime is static, i.e., if the spacetime metric is of the form �2.1� with the lapse function N�x� and the metric Gab being independent of t.9 In such a case the equation for deter- mining the mode functions becomes �t 2fi = NG−1/2�a�NG1/2Gab�bfi� − N2m2fi. �2.30� Then, it is natural to let the positive-frequency solutions fi have a t dependence of the form e−i�it, where �i are positive constants interpreted as the energy of the par- ticle with respect to the �future-directed� Killing vector10 � /�t. If the spacetime is globally hyperbolic and static, then this choice of positive-frequency modes leads to a well-defined and natural vacuum state that preserves the 9In fact, if a globally hyperbolic spacetime is stationary, i.e., if the metric is t independent with �� /�t�� everywhere timelike but with the cross term gti, i� t, not necessarily vanishing, one has a natural vacuum state in this spacetime under certain con- ditions �Ashtekar and Magnon, 1975; Kay, 1978�. 10A Killing vector K� is a vector satisfying ��K�+��K� =K � g��+g ���K +g� ��K =0. In a coordinate system such that K�= �� /����, one has �g�� /��=0. See, e.g., Wald �1984�. 791Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 time-translation symmetry. We call this state the static vacuum. Minkowski spacetime has global timelike Killing vec- tor fields, which generate time translations in various inertial frames. The sets of positive-frequency modes corresponding to these Killing vectors are the same and are the usual positive-frequency modes proportional to e−ik0t with k0�0, where t is the time parameter with re- spect to one of the inertial frames. Thus all these Killing vector fields define the same vacuum state.11 Now, in the region defined by �t��z in Minkowski spacetime �here z is one of the space coordinates�, the boost Killing vector z�� /�t�+ t�� /�z�, i.e., the vector with t and z components being z and t, respectively, is time- like and future directed. Hence this region, called the right Rindler wedge, is a static spacetime with this Kill- ing vector playing the role of time translation. Thus one can define the corresponding static vacuum state. As ob- served by Fulling �1973�, this vacuum state is not the same as the state obtained by restricting the usual Minkowski vacuum to this region. This observation is crucial in understanding the Unruh effect, as explained in the next subsections. B. Rindler wedges As we have seen in the previous section, one has a natural static vacuum state in a static globally hyperbolic spacetime. Minkowski spacetime with the metric ds2 = dt2 − dx2 − dy2 − dz2 �2.31� is of course a static globally hyperbolic spacetime. As mentioned above, the part of this spacetime defined by �t��z, called the right Rindler wedge, is also a static globally hyperbolic spacetime. The region with the con- dition �t��−z is called the left Rindler wedge, and is also a static globally hyperbolic spacetime. The region with t� �z�, also a globally hyperbolic spacetime though not a static one, is called the expanding degenerate Kasner universe and the globally hyperbolic spacetime with the condition t�−�z� is called the contracting degenerate Kasner universe. These regions are shown in Fig. 3. Minkowski spacetime is invariant under the boost t � t cosh � + z sinh � , �2.32� z � t sinh � + z cosh � , �2.33� where � is the boost parameter. These transformations are generated by the Killing vector z�� /�t�+ t�� /�z�. The boost invariance of Minkowski spacetime motivates the following coordinate transformation: t = � sinh �, z = � cosh � , �2.34� where � and � takes any real value. Then, the Killing vector is � /��, and the metric takes the form ds2 = �2d�2 − d�2 − dx2 − dy2. �2.35� This metric is independent of � as expected. The world lines with fixed values of �, x, y are trajectories of the boost transformation given by Eqs. �2.32� and �2.33�. Each world line has a constant proper acceleration given by �−1=const. The coordinates �� ,� ,x ,y� cover only the regions with z2� t2, i.e., the left and right Rindler wedges, as can be seen from Eq. �2.34�. To discuss QFT in the right Rin- dler wedge, it is convenient to make a further coordinate transformation �=a−1ea�, �=a�, i.e., t = a−1ea� sinh a�, z = a−1ea� cosh a� , �2.36� where a is a positive constant �Rindler, 1966�. Then, the metric takes the form ds2 = e2a��d�2 − d�2� − dx2 − dy2. �2.37� This coordinate system will be useful because the world line with �=0 has a constant acceleration of a. The co- ordinates ��̄ , �̄� for the left Rindler wedge are given by t = a−1ea�̄ sinh a�̄, z = − a−1ea�̄ cosh a�̄ . �2.38� The Killing vector z�� /�t�+ t�� /�z� is timelike in the two Rindler wedges and spacelike in the degenerate Kasner universes. It becomes null on the hypersurfaces t= ±z dividing Minkowski spacetime into the four re- gions. These hypersurfaces are examples of Killing hori- zons, which are defined as null hypersurfaces to which the Killing field is normal �Wald, 1994�. 11It has been shown by Chmielowski �1994� that two commut- ing global timelike Killing vector fields define the same vacuum state. t z EDK CDK RRLR VU FIG. 3. The regions with �t��z, �t��−z, t� �z�, and t�−�z�, denoted RR, LR, EDK, and CDK, respectively, are the left Rindler wedge, right Rindler wedge, expanding degenerate Kasner universe, and contracting degenerate Kasner universe, respectively. The curves with arrows are integral curves of the boost Killing vector z�� /�t�+ t�� /�z�. The direction of increas- ing U= t−z and that of increasing V= t+z are also indicated. 792 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 Since the right �or left� Rindler wedge is a static spacetime in its own right, it has a natural static vacuum state as noted before. The Unruh effect is defined here as the fact that the usual vacuum state for QFT in Minkowski spacetime restricted to the right Rindler wedge is a thermal state with � playing the role of time, and similarly for the left Rindler wedge. The correlation between the right and left Rindler wedges in the usual Minkowski vacuum state plays an important role in the Unruh effect. C. Two-dimensional example The two-dimensional massless scalar field in Minkowski spacetime is problematic because of infrared divergences �Coleman, 1973�. Nevertheless, this theory is a good model for explaining the Unruh effect, and it is not necessary to deal with infrared divergences for this purpose. It also turns out that the Unruh effect in scalar field theory in higher dimensions can be derived in es- sentially the same manner as in this model. The massless scalar field in two dimensions �̂�t ,z� sat- isfies ��2/�t2 − �2/�z2��̂ = 0. �2.39� This field can be expanded as �̂�t,z� = � 0 � dk �4�k �b̂−ke−ik�t−z� + b̂+ke−ik�t+z� + b̂−k † eik�t−z� + b̂+k † eik�t+z�� . �2.40� The annihilation and creation operators satisfy �b̂±k,b̂±k� † � = ��k − k�� �2.41� with all other commutators vanishing. By using the defi- nitions U = t − z, V = t + z , �2.42� we can write �̂�t,z� = �̂−�U� + �̂+�V� , �2.43� where �̂+�V� = � 0 � dk�b̂+kfk�V� + b̂+k † fk *�V�� �2.44� with fk�V� = �4�k�−1/2e−ikV, �2.45� and similarly for �̂−�U�. Since the left- and right-moving sectors of the field �̂+�V� and �̂−�U� do not interact with one another, we discuss only the left-moving sector �̂+�V�. �Thus we discuss the Unruh effect for the theory consisting only of the left-moving sector.� The Minkowski vacuum state �0M� is defined by b̂+k�0M�=0 for all k. Using the metric in the right Rindler wedge given by Eq. �2.37�, one finds a field equation of the same form as Eq. �2.39�: ��2/��2 − �2/��2��̂ = 0. �2.46� �This is a result of the conformal invariance of the mass- less scalar field theory in two dimensions.� The solutions to this differential equation can be classified again into the left- and right-moving modes which depend only on v=�+� and u=�−�, respectively. These variables are re- lated to U and V as follows: U = t − z = − a−1e−au, �2.47� V = t + z = a−1eav. �2.48� The Lagrangian density is invariant under the coordi- nate transformation �t ,z�� �� ,��. As a result, by going through the quantization procedure laid out in Sec. II.A one finds exactly the same theory as in the whole of Minkowski spacetime with �t ,z� replaced by �� ,��. Thus we have, for 0�V, �̂+�V� = � 0 � d��â+� R g��v� + â+� R†g � * �v�� , �2.49� where g��v� = �4���−1/2e−i�v, �2.50� and �â+� R , â+�� R† � = ��� − ��� �2.51� with all other commutators vanishing. Note that the functions g��v� are eigenfunctions of the boost generator � /��. The field �̂+�V� can be expressed in the left Rindler wedge with the condition V�0�U, using the left Rin- dler coordinates ��̄ , �̄� defined by Eq. �2.38�. Defining v̄ = �̄− �̄, one obtains Eqs. �2.49�–�2.51� with v replaced by v̄ and with the annihilation and creation operators â+� R and â+� R† replaced by a new set of operators â+� L and â+� L†. The variable v̄ is related to V by V = − a−1e−av̄. �2.52� The static vacuum state in the left and right Rindler wedges, the Rindler vacuum state �0R�, is defined by â+� R �0R�= â+� L �0R�=0 for all �. To understand the Unruh effect we need to find the Bogoliubov coefficients �k R , ��k R , �k L , and ��k L , where ��V�g��v� = � 0 � dk �4�k � �k R e−ikV + ��k R eikV� , �2.53� ��− V�g��v̄� = � 0 � dk �4�k � �k L e−ikV + ��k L eikV� . �2.54� Here ��x�=0 if x�0 and ��x�=1 if x�0, i.e., � is the Heaviside function. To find �k R we multiply Eq. �2.53� by eikV /2�, k�0, and integrate over V. Thus we find 793Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 �k R = �4�k� 0 � dV 2� g��V�eikV =�k � � 0 � dV 2� �aV�−i�/aeikV. �2.55� We introduce a cutoff for this integral for large V by letting V→V+ i�, �→0+.12 Then, changing the integra- tion path to the positive imaginary axis by letting V = ix /k, we find �k R = ie��/2a ��k � a k �−i�/a� 0 � dx 2� x−i�/ae−xdx = ie��/2a 2���k � a k �−i�/a ��1 − i�/a� . �2.56� To find the coefficients ��k R we replace eikV in Eq. �2.55� by e−ikV. Then, the appropriate substitution is V=−ix /k. As a result we obtain ��k R = − ie−��/2a 2���k � a k �−i�/a ��1 − i�/a� . �2.57� A similar calculation leads to �k L = − ie��/2a 2���k � a k �i�/a ��1 + i�/a� , �2.58� ��k L = ie−��/2a 2���k � a k �i�/a ��1 + i�/a� . �2.59� We find that these coefficients obey the following rela- tions crucial to the derivation of the Unruh effect: ��k L = − e−��/a �k R*, ��k R = − e−��/a �k L*. �2.60� By substituting these relations into Eqs. �2.53� and �2.54� we find that the following functions are linear combina- tions of positive-frequency modes e−ikV in Minkowski spacetime: G��V� = ��V�g��v� + ��− V�e−��/ag � * �v̄� , �2.61� Ḡ��V� = ��− V�g��v̄� + ��V�e−��/ag � * �v� . �2.62� One can show that these functions are purely positive- frequency solutions in Minkowski spacetime by analyt- icity argument as well: since a positive-frequency solu- tion is analytic in the lower half plane on the complex V plane, the solution g��v�= �4���−1/2�V�−i�/a, V�0, should be continued to the negative real line avoiding the singularity at V=0 around a small circle in the lower half plane, thus leading to �4���−1/2e−��/a�−V�−i�/a for V�0. This was the original argument by Unruh �1976�. Equations �2.61� and �2.62� can be inverted as ��V�g��v� � G��V� − e−��/aḠ � * �V� , �2.63� ��− V�g��v̄� � Ḡ��V� − e−��/aG � * �V� . �2.64� By substituting these equations into �̂+�V� = � 0 � d����V��â+� R g��v� + â+� R†g � * �v�� + ��− V��â+� L g��v̄� + â+� L†g � * �v̄�� , �2.65� we find that the integrand is proportional to G��V��â+� R − e−��/aâ+� L†� + Ḡ��V��â+� L − e−��/aâ+� R†� + H.c. Since the functions G��V� and Ḡ��V� are positive- frequency solutions �with respect to the usual time trans- lation� in Minkowski spacetime, the operators â+� R −e−��/aâ+� L† and â+� L −e−��/aâ+� R† annihilate the Minkowski vacuum state �0M�. Thus �â+� R − e−��/aâ+� L†��0M� = 0, �2.66� �â+� L − e−��/aâ+� R†��0M� = 0. �2.67� These relations uniquely determine the Minkowski vacuum �0M� as explained below. To explain how the state �0M� is formally expressed in the Fock space on the Rindler vacuum state �0R� and to show that the state �0M� is a thermal state when it is probed only in the right �or left� Rindler wedge, we use the approximation where the Rindler energy levels � are discrete.13 Thus we write �i in place of � and let �â+�i R , â+�j R† � = �â+�i L , â+�j L† � = �ij �2.68� with all other commutators among â+�i R , â+�i L and their Hermitian conjugates vanishing. Using the discrete ver- sion of Eqs. �2.66� and the commutators �2.68�, we find 0M�â+�i R† â+�i R �0M� = e−2��i/a 0M�â+�i L† â+�i L �0M� + e−2��i/a. �2.69� The same relation with â+�i R and â+�i R† replaced by â+�i L and â+�i L† , respectively, and vice versa can be found using Eq. �2.67�. By solving these two relations as simultaneous equations, we find 0M�â+�i R† â+�i R �0M� = 0M�â+�i L† â+�i L �0M� = �e2��i/a − 1�−1. �2.70� Hence the expectation value of the Rindler-particle number is that of a Bose-Einstein particle in a thermal bath of temperature T=a /2�. This indicates that the Minkowski vacuum can be expressed as a thermal state in the Rindler wedge with the boost generator as the Hamiltonian. 12A cutoff of this kind is always understood in these calcula- tions in field theory, as exemplified by the definition ��k� =��dx /2��eikx−��x�= �2�i�−1��k− i��−1− �k+ i��−1�. 13We comment on how one can discuss thermal states in field theory without discretization in Sec. II.I. 794 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 Equation �2.70� can be expressed without discretiza- tion. Define â+f R � � 0 � d�f���â+� R , �2.71� where �0 �d��f����2=1. Then, 0M�â+f R†â+f R �0M� = � 0 � d� �f����2 e2��/a − 1 . �2.72� Exactly the same formula applies to the left Rindler number operator. It should be emphasized that showing the correct properties of the expectation value of the number op- erators â+f R†â+f R and â+f L†â+f L is not enough to conclude that the Minkowski vacuum state restricted to the right or left Rindler wedge is a thermal state. It is necessary to show that the probability of each right or left Rindler- energy eigenstate corresponds to the grand canonical ensemble if the other Rindler wedge is disregarded. One can show this fact by using the discrete version of Eqs. �2.66� and �2.67�. First we note that these equations im- ply �â+�i R† â+�i R − â+�i L† â+�i L ��0M� = 0. �2.73� Thus the number of the left Rindler particles is the same as that of the right Rindler particles for each �i. This implies that we can write �0M� � � i ni=0 � Kni ni! �â+�i R† â+�i L† �ni�0R� . �2.74� One can readily find the recursion formula satisfied by Kni using the discrete version of Eqs. �2.66� and �2.67�. The result is Kni+1 − e−��i/aKni = 0. �2.75� Hence Kni =e−�ni�i/aK0 and �0M� = � i �Ci ni=0 � e−�ni�i/a�ni,R� � �ni,L�� , �2.76� where Ci=�1−exp�−2��i /a�. Here the state with ni left- moving particles with Rindler energy �i in each of the left and right Rindler wedges is denoted �ni ,R� � �ni ,L�, i.e., � i �ni,R� � �ni,L� � �� i 1 ni! �â+�i R† â+�i L† �ni��0R� . �2.77� If one probes only the right Rindler wedge, then the Minkowski vacuum is described by the density matrix obtained by tracing out the left Rindler states, i.e., �̂R = � i �Ci 2 ni=0 � exp�− 2�ni�i/a��ni,R� ni,R�� . �2.78� This is the density matrix for the system of free bosons with temperature T=a /2�. Thus the Minkowski vacuum state �0M� for the left-moving particles restricted to the left �or right� Rindler wedge is the thermal state with temperature T=a /2� with the boost generator normal- ized on z2− t2=1/a2 as the Hamiltonian. This is the Un- ruh effect for the left-moving sector. It is clear that the Unruh effect for the right-moving sector can be derived in a similar manner. D. Massive scalar field in Rindler wedges The Unruh effect for scalar field theory in four- dimensional Minkowski spacetime can be derived in the same way as for the two-dimensional example. Never- theless, in view of the skepticism on the Unruh effect expressed recently �Belinskii et al., 1997; Fedotov et al. 1999; Oriti, 2000; Narozhny et al., 2002, 2004� we review the Unruh effect in this theory �Fulling 1973; Unruh 1976�, drawing attention to some aspects that appear to have caused the skepticism. �See Fulling and Unruh �2004� for an explanation as to why this skepticism is unfounded.� The free quantized massive scalar field �̂�t ,z ,x��, x���x ,y�, can be expanded as �̂�t,z,x�� =� d3k�âkzk� M fkzk� + âkzk� M† fkzk� * � , �2.79� where the positive-frequency mode functions are fkzk� �t,z,x�� = ��2��32k0�−1/2e−ik0t+ikzz+ik�·x� �2.80� with k���kx ,ky� and k0��kz 2+k� 2 +m2. The Klein- Gordon inner product can be calculated as �fkzk� ,fkz�k �� �KG = ��kz − kz��� 2�k� − k�� � , �2.81� �fkzk� * ,fkz�k �� �KG = 0. �2.82� Hence quantizing the scalar field �̂�t ,z ,x�� by imposing the equal-time commutation relations �2.9� and �2.10�, we find �âkzk� M , âkz�k �� M† � = ��kz − kz��� 2�k� − k�� � �2.83� with all other commutators among annihilation and cre- ation operators vanishing. The field equation in the right Rindler wedge with the metric �2.37� can readily be found from Eq. �2.30� by letting N=ea� and the metric of the hypersurfaces with constant � be diagonal with G��=e2a� and Gxx=Gyy=1. Thus �2�̂ ��2 = � �2 ��2 + e2a�� �2 �x2 + �2 �y2� − m2e2a���̂ . �2.84� The positive-frequency solutions are chosen to be pro- portional to e−i��, where � is a positive constant. This choice corresponds to the static vacuum state with re- spect to the � translation, i.e., the Rindler vacuum state. It is also clear that one may assume that they are pro- portional to eik�·x�. Thus we write the positive-frequency modes as 795Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 v�k� R = 1 2��2� g�k� ���e−i��+ik�·x� �2.85� with the function g�k� ��� satisfying �− d2 d�2 + e2a��k� 2 + m2��g�k� ��� = �2g�k� ��� . �2.86� This equation is analogous to a time-independent Schrödinger equation with an exponential potential. Thus the physically relevant solutions g�k� ��� tend to zero as �→ +� and oscillate like e±i�� as �→−�. Note in particular that there is no distinction between the left- and right-moving modes. We choose g�k� ��� to satisfy, for ��0 and ����1, g�k� ��� � 1 �2� �ei���+����� + e−i���+������ , �2.87� where ���� is a real constant. This choice of normaliza- tion implies �see, e.g., Fulling �1989�� � −� � d�g �k� * ���g��k� ��� = ��� − ��� . �2.88� We present the derivation of this formula in the Appen- dix for completeness. As a result we have �v�k� R ,v��k �� R �KG = ��� − ����2�k� − k�� � , �2.89� �v �k� R* ,v��k �� R �KG = 0. �2.90� The Klein-Gordon inner product here is defined taking the hypersurface in Eq. �2.5� to be a �=const Cauchy surface of the right Rindler wedge. It can also be defined taking to be the entire t=0 hypersurface of the Minkowski spacetime by defining v�k� R =0 in the left Rin- dler wedge �and on the plane t=z=0 for definiteness�. The functions g�k� ��� satisfying the differential equation �2.86� and normalization condition �2.87� are g�k� ��� = �2� sinh���/a� �2a �1/2 Ki�/a��a ea�� �2.91� with ���k� 2 +m2�1/2, where K��x� is the modified Bessel function �Gradshteyn and Ryzhik, 1980�. Hence v�k� R = � sinh���/a� 4�4a �1/2 Ki�/a��a ea��eik�·x�−i��. �2.92� We present the derivation of this result in the Appendix as well. Thus we can expand the field �̂ in the right Rindler wedge as �̂��,�,x�� = � −� � d�� d2k��â�k� R v�k� R + â�k� R† v �k� R* � . �2.93� Then, according to the general results presented in Sec. II.A, we have �â�k� R , â ��k �� R† � = ��� − ����2�k� − k�� � �2.94� with all other commutators among â�k� R and â�k� R† vanish- ing. Quantization of the field �̂ in the left Rindler wedge proceeds in exactly the same way. The positive- frequency modes v�k� L ��̄ , �̄ ,x�� are obtained from v�k� R �� ,� ,x�� simply by replacing � and � by �̄ and �̄, respectively. The coefficient operators â�k� L and â�k� L† sat- isfy the commutation relations �â�k� L , â ��k �� L† � = ��� − ����2�k� − k�� � �2.95� with all other commutators vanishing. Thus one can ex- pand the field �̂ in the left and right Rindler wedges as �̂ = � 0 +� d�� d2k��â�k� R v�k� R ��,�,x�� + â�k� R† v �k� R* ��,�,x�� + â�k� L v�k� L ��̄, �̄,x�� + â�k� L† v �k� L* ��̄, �̄,x��� . �2.96� The Rindler vacuum state �0R� is defined by requiring that â�k� R �0R�= â�k� L �0R�=0 for all � and k�. As it stands, this expansion makes sense only in the Rindler wedges. However, it will be shown that the modes v�k� R and v�k� L can naturally be extended to the whole of Minkowski spacetime �see Eqs. �2.112�–�2.114��. After this extension we see that Eq. �2.96� gives another valid mode expan- sion of the field �̂ in Minkowski spacetime.14 In particu- lar, in Sec. II.F the two-point function calculated using this expansion in the state �0M� will be shown to give the standard result in Minkowski spacetime. E. Bogoliubov coefficients and the Unruh effect In this section we find the Bogoliubov coefficients be- tween the two expansions of the massive scalar field �̂ in Minkowski spacetime and derive the Unruh effect, i.e., the fact that the Minkowski vacuum state is a thermal state with temperature T=a /2� on the right or left Rin- dler wedge. It is clear that the Bogoliubov coefficients between modes with different k� are zero. Thus we can write in general v�k� R = � −� � dkz �4�k0 � �kzk� R e−ik0t+ikzz + ��kzk� R eik0t−ikzz� eik�·x� 2� , �2.97� 14This point is emphasized in Birrell and Davies �1982�. 796 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 v�k� L = � −� � dkz �4�k0 � �kzk� L e−ik0t+ikzz + ��kzk� L eik0t−ikzz� eik�·x� 2� . �2.98� We assume here that the modes v�k� R and v�k� L have been suitably extended to the whole of Minkowski spacetime. The relation between �� ,�� and �t ,z� given by Eq. �2.36� is the same as that between ��̄ , �̄� and �t ,−z� given by Eq. �2.38�. Hence v�k� L is obtained from v�k� R by letting z�−z. From this observation we find the follow- ing relations: �kzk� L = �−kzk� R , ��kzk� L = ��−kzk� R . �2.99� These Bogoliubov coefficients will be found explicitly later, but it is clear from the discussion of the massless scalar field theory in two dimensions that the Unruh ef- fect will follow if �â�k� R − e−��/aâ�−k� L† ��0M� = 0, �2.100� �â�k� L − e−��/aâ�−k� R† ��0M� = 0. �2.101� �See the corresponding Eqs. �2.66� and �2.67� in the two- dimensional model.� These relations in turn will result if the following modes are purely positive frequency in Minkowski spacetime: w−�k� � v�k� R + e−��/av �−k� L* �1 − e−2��/a , �2.102� w+�k� � v�k� L + e−��/av �−k� R* �1 − e−2��/a . �2.103� �See the corresponding Eqs. �2.61� and �2.62� in the two- dimensional model.� This fact in turn will follow if ��kzk� R = − e−��/a �kzk� L* , ��kzk� L = − e−��/a �kzk� R* . �2.104� �See the corresponding Eq. �2.60�.� We show Eq. �2.104� by explicit evaluation of the Bogoliubov coefficients, which were originally computed by Fulling �1973�. To calculate the Bogoliubov coefficients it is conve- nient to examine the behavior of the solutions on the future Killing horizon, t=z, t�0. There we have v�k� R → � −� � dkz �4�k0 � �kzk� R e−i�k0−kz�V/2 + ��kzk� R ei�k0−kz�V/2� eik�·x� 2� . �2.105� On the other hand, using the small-argument approxi- mation �A10� for the modified Bessel function, we have for �→−� v�k� R → i 4� �a sinh���/a��−1/2eik�·x� � � ��/2a�i�/ae−i�u ��1 + i�/a� − ��/2a�−i�/ae−i�v ��1 − i�/a� � , �2.106� where �= �k� 2 +m2�1/2. The first term inside the parenthe- ses in this equation oscillates infinitely many times as u →�, where the future Killing horizon is, and is bounded. Such a term should be regarded as zero. Hence the Bo- goliubov coefficient �kzk� R is obtained by multiplying Eq. �2.106� by ei�k0−kz�V/2 and integrating over V as �kzk� R = − i��/2a�−i�/a�k0 − kz� 4��ak0 sinh���/a���1 − i�/a� � � 0 � dV�aV�−i�/aei�k0−kz�V/2 = e��/2a �4�k0a sinh���/a� �k0 + kz k0 − kz �−i�/2a , �2.107� where �=��k0−kz��k0+kz�. Note that we have implicitly chosen a particular �and natural� extension of the modes v�k� R to the whole of Minkowski spacetime. �Otherwise it should not be possible to find the coefficients Bogoliu- bov coefficients �kzk� R and ��kzk� R in Eq. �2.97�.� In par- ticular, we have excluded any delta-function contribu- tion at V=0. By multiplying Eq. �2.106� by e−i�k0−kz�V/2 and integrat- ing over V we find ��kzk� R = − e−��/2a �4�k0a sinh���/a� �k0 + kz k0 − kz �−i�/2a . �2.108� Introducing the rapidity ��kz� defined as ��kz� � 1 2 ln�k0 + kz k0 − kz � , �2.109� and using Eq. �2.99�, we have �kzk� R = �−kzk� L = e−i��kz��/a �2�k0a�1 − e−2��/a� , �2.110� ��kzk� R = ��−kzk� L = − e−��/ae−i��kz��/a �2�k0a�1 − e−2��/a� . �2.111� Hence Eq. �2.104� is satisfied and as a result the vacuum state �0M� restricted to the left �or right� Rindler wedge is a thermal state with temperature T=a /2� with the boost generator normalized on t2−z2=1/a2 as the Hamiltonian. Although we have now established the Unruh effect, it is useful to examine the modes natural to the Rindler wedges further for later discussion. The purely positive- frequency modes in Minkowski spacetime defined by Eqs. �2.102� and �2.103� are 797Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 w±�k� = � −� � dkz �8a�k0 e±i��kz��/ae−ik0t+ikzzeik�·x� 2� . �2.112� The modes v�k� R and v�k� L , which vanish in the left and right Rindler wedges, respectively, are expressed in terms of these modes as v�k� R = w−�k� − e−��/aw+�−k� * �1 − e−2��/a , �2.113� v�k� L = w+�k� − e−��/aw−�−k� * �1 − e−2��/a . �2.114� These formulas and Eq. �2.112� give the modes v�k� R and v�k� L as distributions in the whole of Minkowski space- time. One can verify that the modes w±�k� satisfy �w±�k� ,w±��k �� �KG = ��� − ����2�k� − k�� � , �2.115� �w±�k� * ,w ±��k �� * �KG = − ��� − ����2�k� − k�� � �2.116� with all other Klein-Gordon inner products among w±�k� and their complex conjugates vanishing. Here the Klein-Gordon inner product is defined as an integral over the t=0 hypersurface in Minkowski spacetime. The following formula, which can be shown by using d��kz�=dkz /k0, is useful in calculating these Klein- Gordon inner products: � −� � dkz 2�ak0 ei��kz���−���/a = ��� − ��� . �2.117� It is worth emphasizing that these modes form a com- plete set of solutions to the Klein-Gordon equation, not only in the left and right Rindler wedges but also in the whole of Minkowski spacetime. This fact can be seen by inverting the relation �2.112�: 1 �2k0�2��3 e−ik0t+ikzz+ik�·x� = 1 �2�ak0 � 0 � d��ei��kz��/aw−�k� + e−i��kz��/aw+�k� � . �2.118� One may object to this conclusion as do Belinskii et al. �1997� because the modes w±�k� were originally defined only on the left and right Rindler wedges, which are open regions; in particular, these modes are not defined on the plane t=z=0. However, the formula �2.112� gives the positive-frequency modes w±�k� in terms of the mo- mentum eigenfunctions unambiguously as a distribution over the whole of Minkowski spacetime. In other words, if f�t ,x� is a compactly supported smooth function on Minkowski spacetime, whose support may intersect the plane t=z=0, the mode functions w±�k� smeared with f is well defined and unique. That is, f̂ R�±�,k�� � � d4xw±�k� * �t,z,x��f�t,z,x�� = � −� � dkz �2�ak0 e�i��kz��/af̂M�kz,k�� , �2.119� where f̂ M�kz,k�� � 1 ��2��32k0 � d4x eik0t−ik·xf�t,x� . �2.120� We have used w±�k� * rather than w±�k� here for later convenience. Note that the function f̂ M�kz ,k�� tends to 0 as kz→ ±� faster than any powers of �kz�−1 due to the smoothness of f�t ,x�. This implies that the integral in Eq. �2.119� is absolutely convergent. �In fact Eq. �2.119� should be taken as the definition of the modes w±�k� * as distributions over the full Minkowski spacetime.� Since the modes w±�k� and w±�k� * form a complete set of solutions in Minkowski spacetime, the Rindler modes v�k� R and v�k� L and their complex conjugates form a com- plete set as is clear from Eqs. �2.102� and �2.103�. Re- lated comments will be made in the next sections. F. Completeness of the Rindler modes in Minkowski spacetime In the previous section we commented that the Rin- dler modes form a complete set of solutions to the Klein-Gordon equation in Minkowski spacetime. To em- phasize this point again we show here that the Wight- man two-point function is correctly reproduced every- where in Minkowski spacetime even if we use the Rindler modes. It is our hope that the calculation here will dispel any suspicion that the Rindler modes may be incomplete due to the singularity on the hypersurfaces t= ±z. The two-point function in the Minkowski vacuum state is well known to be ��x ;x�� = 0M��̂�x��̂�x���0M� =� dkzd2k� 2k0�2��3e−ik·�x−x��, �2.121� where x= �t ,z ,x�� and similarly for x�. To calculate the two-point functions with the Rindler modes we use the expansion �2.96� with the Rindler modes v�k� L and v�k� R given by Eqs. �2.112�–�2.114�. Using Eqs. �2.100� and �2.101� we see that the Rindler annihilation operators can be written as â�k� R = b̂−�k� + e−��/ab̂+�−k� † �1 − e−2��/a , �2.122� 798 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 â�k� L = b̂+�k� + e−��/ab̂−�−k� † �1 − e−2��/a , �2.123� where the operators b±�k� annihilate the Minkowski vacuum �0M� and have the following standard commuta- tion relations: �b±�k� ,b±��k �� † � = ��� − ����2�k� − k�� � �2.124� with all other commutators vanishing. Equations �2.122� and �2.123� can be used to find the following expectation values: 0M�a�k� R† a��k �� R �0M� = 0M�a�k� L† a��k �� L �0M� = �e2��/a − 1�−1��� − ����2�k� − k�� � , �2.125� 0M�a�k� R a ��k �� R† �0M� = 0M�a�k� L a ��k �� L† �0M� = �1 − e−2��/a�−1��� − ����2�k� − k�� � , �2.126� 0M�a�k� L a��k �� R �0M� = 0M�a�k� L† a ��k �� R† �0M� = �e��/a − e−��/a�−1��� − ����2�k� + k�� � . �2.127� The vacuum expectation values of the other products of two creation and annihilation operators vanish. Then, the two-point function of the field �̂�x� given in Eq. �2.96� is ��x ;x�� = � 0 � d�� d2k���v�k� R �x�v �k� R* �x�� + v�k� L �x�v �k� L* �x����1 − e−2��/a�−1 + �v �k� R* �x�v�k� R �x�� + v �k� L* �x�v�k� L �x��� ��e2��/a − 1�−1 + 2�v�k� R �x�v�−k� L �x�� + v �k� R* �x�v �−k� L* �x����e��/a − e−��/a� + 2�v�k� L �x�v�−k� R �x�� + v �k� L* �x�v �−k� R* �x����e��/a − e−��/a� . �2.128� This expression can be simplified using Eqs. �2.113� and �2.114� as ��x ;x�� = � 0 � d�� d2k��w+�k� �x�w+�k� * �x�� + w−�k� �x�w−�k� * �x��� , �2.129� where w±�k� are given by Eq. �2.112�. Thus ��x ;x�� = 1 32�4a � −� � d�� −� � dkz k0 � −� � d��kz�� �� d2k�ei���kz�−��kz����/a �e−ik0t+ik0�t�+ikzz−ikz�z�+ik�·�x�−x�� �. �2.130� The � and ��kz�� integration can readily be performed, and we find that the two-point function indeed takes the form given by Eq. �2.121�. The expression �2.129� is undefined if either of the two points is on the hyperplane t= ±z. However, since the two-point function ��x ;x�� is defined as a distribution, it is well defined on the whole of Minkowski spacetime if the following integral exists for all compactly supported functions f�x� and g�x�: F�f,g� � � d4x d4x�f *�x�g�x����x ;x�� . �2.131� We find using Eq. �2.129� F�f,g� = � 0 � d�� d2k��f̂ R*�− �,k��ĝR�− �,k�� + f̂ R*�+ �,k��ĝR�+ �,k��� , �2.132� where f̂ R is defined by Eq. �2.119�, and ĝR is defined similarly. It can readily be shown that this agrees with the standard expression for the smeared two-point func- tion, F�f,g� =� d3kf̂ M*�k�ĝM�k� , �2.133� where f̂ M is defined by Eq. �2.120� and the Fourier trans- form ĝM is defined similarly. G. Unruh effect and quantum field theory in the expanding degenerate Kasner universe In this section we review the relation between the modes in the Rindler wedges and those in the expanding degenerate Kasner universe. It is well known that there is a choice of the positive-frequency modes in the degen- erate Kasner universes that gives a state identical to the 799Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 Minkowski vacuum �Fulling et al., 1974�. We first show that these positive-frequency modes are in fact the modes w±�k� in the Rindler wedges �Gerlach, 1988�. Then, we show that the Rindler vacuum state �0R� is identical to one state in the expanding degenerate Kas- ner universe �Fulling et al., 1974; Birrell and Davies, 1982�. We introduce the following coordinate transforma- tion: t = T cosh a , z = T sinh a . �2.134� With T�0 the coordinate system �T , ,x�� covers the region with the condition t� �z�, i.e., the expanding de- generate Kasner universe. Then, the Minkowski metric becomes ds2 = dT2 − a2T2d 2 − dx2 − dy2. �2.135� The hyperplanes of constant T are spacelike, and the variable T plays the role of time. Hence the T=const space expands in the direction linearly. We note that t + z t − z = e2a , �2.136� and �t2−z2�1/2=T. We change the integration variable in the expression �2.112� for modes w±�k� from kz to the rapidity � =��kz� �see Eq. �2.109��. Then, we have k0 = � cosh �, kz = � sinh � , �2.137� where �= �k� 2 +m2�1/2 as before. Thus we obtain, using Eq. �2.136� after shifting of the integration variable as �→�+a , w±�k� = eik�·x�±i� 2� � −� � d� �8a� e±i��/a �exp�− i�T cosh �� . �2.138� The � integral is the same for both signs of e±i��/a be- cause the imaginary part of the integrand is odd in �. Adopting the minus sign and using the formula �Grad- shteyn and Ryzhik, 1980� H� �2��x� = − ei��/2 �i � −� � e−ix cosh t−�tdt �2.139� with �= i� /a, we find w±�k� = − i eik�·x�±i� 2��8a e��/2aHi�/a �2� ��T� . �2.140� These modes are well known to form a complete set of positive-frequency modes which correspond to the Minkowski vacuum state �Fulling et al., 1974�. Now, from Eq. �2.113� we find that the positive- frequency modes with respect to the boost generator in the right Rindler wedge corresponding to the Rindler vacuum state take the following form in the expanding degenerate Kasner universe: v�k� R = − i e−i� eik�·x� 2��8a�e��/a − e−��/a� �e��/aHi�/a �2� ��T� + �Hi�/a �2� ��T��* . �2.141� Then, recalling the fact that �H� �2��x��*=H−� �1��x� if � is purely imaginary and if x is real and using the formulas �Gradshteyn and Ryzhik, 1980� e��iH−� �2��z� = H� �2��z� , �2.142� H� �1��z� + H� �2��z� = 2J��z� �2.143� with �=−i� /a in Eq. �2.141�, we find v�k� R = − i e−i� eik�·x� 2��4a sinh���/a� J−i�/a��T� . �2.144� In exactly the same manner we find that the left Rindler modes v�k� L are given by Eq. �2.144� with e−i� replaced by ei� in the expanding degenerate Kasner universe. These modes have been identified as the positive- frequency modes corresponding to a state which is in- equivalent to the Minkowski vacuum �Fulling et al., 1974; Birrell and Davies, 1982�. Thus the Rindler vacuum state �0R� is in fact one of the states in the ex- panding degenerate Kasner universes given in the litera- ture. H. Unruh effect and classical field theory Although the Unruh effect, like the Hawking effect, is a quantum effect, its derivation does not involve any loop calculations. It is also the result of properties of classical solutions to the field equation. These observa- tions naturally lead to the following question: Are there any aspects of the Unruh effect that can be described entirely in the framework of classical field theory? In this context, it is useful to note that, although the Unruh temperature T=�a / �2�c� �at �=0� is proportional to �, since the energy of a particle can be written as E=��, where � is the angular frequency, the Boltzmann factor exp�−E /T�=exp�−2��c /a� is independent of �. This is consistent with the fact that the Bogoliubov transforma- tion encoding the Unruh effect is derived using only classical solutions. It is indeed possible to define some quantities in classical field theory which exhibit what one may call the classical Unruh effect �Higuchi and Matsas, 1993� as described here.15 We consider the classical scalar field in Minkowski spacetime satisfying ��+m2� =0. The energy- momentum tensor is 15However, we find the claim by Barut and Dowling �1990� that the Unruh effect can be explained without invoking a thermal bath rather misleading. If one were to describe physics in the Rindler wedge with the boost generator as the Hamil- tonian, then the thermal bath with the temperature a /2� would be a necessary ingredient. 800 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 T�� = �� �� − g���� � − m2 2�/2. �2.145� Now, if X� is a Killing vector, then the current J�X� � de- fined by J�X� � = X�T �� �2.146� is conserved because of the Killing equation and the equation ��T��=0. Hence the energy associated with the Killing vector X� defined by EX =� d n�J�X� � �2.147� is conserved, where is a Cauchy hypersurface and n� is the future-directed unit vector normal to . If T� is the time-translation vector, then the energy ET with X� =T� is the ordinary energy. If R�=a�z�� /�t��+ t�� /�z���, i.e., the boost Killing vector �normalized at �=0�, then ER with X�=R� is the Rindler energy. It is convenient for our purposes to rewrite the energy EX as EX = �i/2�� ,X��� �KG. �2.148� This can be obtained using the equality ���X � � − X � �� + 2X T � = � � �X �� − X�� �� . �2.149� Now, one can divide the scalar field into the positive- and negative-frequency parts with respect to the time- translation Killing vector as �x� = �T+��x� + �T−��x� , �2.150� where the negative-frequency part is the complex conju- gate of the positive-frequency part, �−T��x�= � �+T��x��*, and where the positive-frequency part is given as �T+��x� =� d3k �2k0�2��3/2 cT�k�e−ik0t+ik�·x�, �2.151� for some function cT�k�. Then, since T���=�t, we find the energy by using Eq. �2.148� as ET =� d3kk0�cT�k��2. �2.152� It is natural to define the quantity NT by dividing the integrand k0�cT�k��2 by k0 as NT =� d3k�cT�k��2, �2.153� because the expected quantum-mechanical particle number is NT /�. We call NT the classical Minkowski par- ticle number. It is clear that NT = � �T+�, �T+��KG. �2.154� Now, if the field vanishes in the left Rindler wedge, then it can be expanded in terms of the right Rindler modes v�k� R . Thus we have �x� = �R+��x� + �R−��x� , �2.155� where the positive-frequency part with respect to the boost Killing vector R� is defined by �R+��x� = � 0 � d�� d2k�cR��,k��v�k� R �2.156� for some function cR�� ,k��, and the negative-frequency part is �R−��x�= � �R+��x��*. The Rindler energy is found by letting X�=R� in Eq. �2.148� as ER = � 0 � d�� d2k���cR��,k���2. �2.157� We can define the classical Rindler particle number as NR � � 0 � d�� d2k��cR��,k���2. �2.158� Then we have NR = � �R+�, �R+��KG. �2.159� It is possible to express the Minkowski particle num- ber NT in terms of cR�� ,k�� as follows. From Eq. �2.113� we find = � 0 � d�� d2k��cR��,k��v�k� R + cR * ��,k��v �k� R* � = �T+� + �T−�, �2.160� where �T+� = � 0 � d�� d2k�� cR��,k�� �1 − e−2��/a w−�k� − e−��/acR * ��,k�� �1 − e−2��/a w+�k� � . �2.161� Then, using Eq. �2.115�, we obtain the classical Minkowski particle number as NT = � �T+�, �T+��KG = � 0 � d�� d2k��cR��,k���2 coth��� a � . �2.162� Comparing this expression with that for the classical Rindler particle number �2.158�, we find that the Fourier components with respect to the Rindler time � of the classical Minkowski particle number is enhanced by a factor of coth��� /a� in comparison to those of the clas- sical Rindler particle number. We refer the reader to Higuchi and Matsas �1993� for the interpretation of this formula in the context of the Unruh effect. I. Unruh effect for interacting theories and in other spacetimes In this section we mention some works which extend the Unruh effect to interacting field theory and other spacetimes. We first discuss the work of Bisognano and Wichmann �1975, 1976�, who derived the Unruh effect for �interact- 801Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 ing� quantum field theory satisfying Wightman axioms �Wightman, 1956; Streater and Wightman, 1964; Jost, 1965�. The Unruh effect was not presented as the main result in their work, and it was only several years after its publication that its connection to the Unruh effect was discovered by Sewell, who also extended their deri- vation of the Unruh effect to the class of Schwarzschild and de Sitter spacetimes �Sewell, 1982�. In order to discuss the work of Bisognano and Wich- mann, it is necessary to review a mathematically more satisfactory way to define a thermal state in quantum field theory, which is called the KMS condition �Haag et al., 1967�. �The initials KMS stand for Kubo �1957� and Martin and Schwinger �1959�.� We first explain the KMS condition for a quantum system with a finite number of energy levels with a Hamiltonian Ĥ and a complete set of eigenstates �n� with energy En. The expectation value of an operator  in a thermal state with inverse tem- perature �=1/T is Â�� = n e−�En n�Â�n� m e−�Em = Tr�e−�ĤÂ� Tr�e−�Ĥ� . �2.163� Let H be the Hilbert space spanned by �n�. This thermal state is realized as a pure state in the Hilbert space H � H as ��� = n e−�En/2�n� � �n� � m e−�Em , �2.164� if the operators  on H are identified with Â�e�= Î � Â, where Î is the identity operator. That is, ��Â�e���� = Â��. �2.165� The time-evolution operator is taken to be exp�− iĤ�e��� = exp�iĤ�� � exp�− iĤ�� . �2.166� Now, we define an antiunitary involution Ĵ�e� by Ĵ�e� �n� � �m� = *�m� � �n� , �2.167� where is any c number. Then, the operator Ĵ�e� com- mutes with the time-evolution operator: Ĵ�e� exp�− iĤ�e��� = exp�− iĤ�e���Ĵ�e�, ∀ � � R . �2.168� One can also show by an explicit calculation that, for any operator  given by a matrix as Â�n�= m�m�Amn, exp�− Ĥ�e��/2�Â�e���� = Ĵ�e�Â�e�†��� . �2.169� It can be seen that, in our model with a finite number of energy levels, Eq. �2.169� implies that the state ��� must be given by Eq. �2.164� up to an overall phase factor. In algebraic field theory a state16 that allows a Hilbert space representation satisfying the conditions �2.168� and �2.169� is called a KMS state at inverse temperature �. Thus the Unruh effect in algebraic field theory is the statement that the Minkowski vacuum restricted to the right Rindler wedge is a KMS state at inverse tempera- ture �=2� /a if the time evolution is identified with a boost, which is the � translation in the Rindler coordi- nates �2.36�. Remarkably, the doubling of the Hilbert space and the involution Ĵ�e� in the above construction, which might look somewhat artificial in the context of statistical mechanics, naturally arise here. Thus given the QFT in the right Rindler wedge with a boost generator as the Hamiltonian we “extend” it by including the left Rindler wedge and operators acting there. The extended boost generator automatically takes the form �2.166� since the corresponding Killing vector field is past- directed in the left Rindler wedge. In the two-dimensional model �with only the left mov- ers� the involution Ĵ�e� is defined by requiring Ĵ�e��0M� = �0M� , �2.170� Ĵ�e�â+� R Ĵ�e� = â+� L , �2.171� J�e�â+� R†J�e� = â+� L† . �2.172� Note that �Ĵ�e��2= Î � Î. The involution Ĵ�e� is in fact the PCT transformation, i.e., the antiunitary transformation �̂�t ,z���̂�−t ,−z� in this two-dimensional model. For the four-dimensional scalar field it is the � rotation about the z axis times the PCT transformation �see Bisognano and Wichmann �1975��. With these defini- tions one can verify that Eqs. �2.66� and �2.67� imply Eq. �2.169�. The commutation relation �2.168� follows from the fact that the Lorentz boost commutes the PCT transformation. The derivation of the Unruh effect by Bisognano and Wichmann �1975� using the algebraic approach was for any interacting scalar field satisfying the Wightman axi- oms. They also generalized this result to quantum fields of arbitrary spins �Bisognano and Wichmann, 1976�. They showed that the Minkowski vacuum restricted to the right or left Rindler wedge is a KMS state as ex- plained above. For the four-dimensional scalar field theory, for example, if exp�−iK̂ � is the boost operator corresponding to t � t� � � t cosh a + z sinh a , �2.173� z � z� � � t sinh a + z cosh a , �2.174� then, for = i� /a, one has �t ,z�� �−t ,−z�. Bisognano and Wichmann proved that this fact translates to 16In algebraic field theory a state means a density matrix in general. 802 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 exp�− K̂�/a��̂�t�1�,z�1�,x� �1�� ¯ �̂�t�n�,z�n�,x� �n���0M� = �̂�− t�1�,− z�1�,x� �1�� ¯ �̂�− t�n�,− z�n�,x� �n���0M� , �2.175� where �0M� is a unique Poincaré invariant vacuum, which is assumed to exist, if �t�i� ,z�i� ,x� �i��, i=1,2 , . . . ,n, are spa- tially separated points in the right Rindler wedge.17 Then, they converted the relation �2.175� to the KMS condition �2.169� with Ĥ�e�=K̂, �=2� /a, and Ĵ�e� the PCT operator times the � rotation about the z axis for operators Â�e� acting in the left Rindler wedge by a re- sult similar to the Reeh-Schlieder theorem �Reeh and Schlieder, 1961�.18 �See Kay �1985� for a discussion of the Bisognano-Wichmann theorem in the context of free field theory.� We describe how Eq. �2.175� can be derived in the simplest case with n=1 and with free �four-dimensional� scalar field. Using K̂�0M�=0 and âkzk� M �0M�=0, we have for a real parameter exp�i K̂��̂�t,z,x���0M� = �̂�t� �,z� �,x���0M� =� d3k ��2��32k0 ei�k0z−kzt�sinh a−ik�·x� �ei�k0t−kzz�cosh aâkzk� M† �0M� , �2.176� where t� � and z� � are defined by Eqs. �2.173� and �2.174�, respectively. It can be shown that the variable can be analytically continued from 0 to i� /a if z� �t�, i.e., if the point �t ,z ,x�� is in the right Rindler wedge.19 Thus exp�− K̂�/a��̂�t,z,x���0M� = �̂�− t,− z,x���0M� , �2.177� if the point �t ,z ,x�� is in the right Rindler wedge. This is Eq. �2.175� with n=1 for a free field. Noting that the point �−t ,−z ,x�� is in the left Rindler wedge and using the expansion �2.96� of the field �̂ in terms of the Rin- dler modes, one can deduce from Eq. �2.177� the rela- tions �2.100� and �2.101�, which were crucial in showing the Unruh effect. Unruh and Weiss �1984� derived the Unruh effect for the scalar field theory with arbitrary potential term V��̂� in the path integral approach. �They also discussed the Unruh effect for spinors. See also Gibbons and Perry �1976�.� Here we present their argument, for the two- dimensional scalar field for simplicity of notation, in a slightly modified manner. What needs to be shown is that 0M�T��̂�x��̂�x����0M� = Tr�e−�K̂T��̂�x��̂�x��� Tr�e−�K̂� , �2.178� where the trace is over all states, K̂ is the boost operator defined above, and �=2� /a. The argument for a similar equality involving an n-point function with arbitrary n is almost identical. The Lagrangian density for the scalar field with poten- tial V� � is L = ��� /�t�2 − �� /�z�2�/2 − V� � . �2.179� In the Rindler coordinates given by Eq. �2.34� with � =a�, i.e., t = � sinh a�, z = � cosh a� , �2.180� this Lagrangian density is given by L = a�� 1 2a2�2� � �� �2 − 1 2 � � �� �2 − V� �� . �2.181� Define the Euclidean action by letting �=−i�e as SE R��� � − � 0 � d�e� 0 � d�L��=−i�e� = � 0 � ad�e� 0 � d�� ��1 2 � � �� �2 + 1 2�2�1 a � ��e �2 + V� �� , �2.182� where ��e+� ,��= ��e ,��. It is well known �see, e.g., Bernard �1974�� that the right-hand side of Eq. �2.178� for an arbitrary value of � is obtained by the analytic continuation �e= i� of the following expression: D��xe,xe�� � � ��e=0�= ��e=�� �D � �xe� �xe��exp�− SE R���� � ��e=0�= ��e=�� �D �exp�− SE R���� , �2.183� where xe= �te ,ze� is obtained from Eq. �2.180� as 17Bisognano and Wichmann showed that the rigorous version of Eq. �2.175� makes sense, i.e., that states obtained by multi- plying �0M� by a finite number of operators of the form �d4xf�x��̂�x�, where f�x� has support in the right Rindler wedge, is in the domain of exp�−�K̂� for 0!�!� /a. 18This theorem states that any state in the Hilbert space of the scalar field theory can be approximated by applying poly- nomials of operators of the form �d4xf�x��̂�x� on the vacuum state �0M�, where f�x� have support in a finite spacetime region. 19To be precise, one needs to consider the inner product of the state in Eq. �2.176� with a normalized one-particle state. Note that the modulus of ei�k0z−kzz�sinh a is always less than or equal to 1 if is between 0 and i� /a. This fact is essential in showing that this analytic continuation is possible. 803Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 te = � sin a�e, ze = � cos a�e. �2.184� These equations show that the Euclideanized right Rin- dler wedge is the two-dimensional Euclidean space ex- pressed in polar coordinates if 0!a�e!2�, i.e., if � =2� /a. Thus one has SE R�2�/a� = SE � � −� � dte� −� � dze�� � �te �2 + � � �ze �2 + V� �� . �2.185� Hence D2�/a�xe;xe�� = � �D � �xe� �xe��exp�− SE� � �D �exp�− SE� . �2.186� It is well known that the time-ordered two-point func- tion in �two-dimensional� Minkowski spacetime, i.e., the left-hand side of Eq. �2.178�, is obtained from the right- hand side of Eq. �2.186� by the analytic continuation te = it. Since both sides of Eq. �2.178� are obtained by the analytic continuation of the same function D2�/a�xe ;xe�� with xe= �te ,ze�= �it ,z�, Eq. �2.178� holds. The analog of the Unruh effect in Schwarzschild spacetime was first derived by Hartle and Hawking �1976� using analytic properties of the time-ordered two- point function for scalar and other free fields. They showed that the physically acceptable20 vacuum state in- variant under the time translation in the Kruskal exten- sion �Kruskal, 1960� of Schwarzschild spacetime with mass M is a thermal state of temperature 1/8�M. This result has close connection to the Hawking effect �Hawking, 1974�. A similar method was used by Gib- bons and Hawking �1977� to show that the physically acceptable de Sitter–invariant vacuum state of the free scalar field in de Sitter spacetime with Hubble constant H is a thermal state of temperature H /2� of the theory inside the cosmological horizon with the de Sitter boost generator fixing the horizon as the Hamiltonian. Narn- hofer et al. �1996� found that an accelerated detector with acceleration a in de Sitter spacetime responds as if it was in a thermal bath of temperature �H2+a2�1/2 /2�, and Deser and Levin �1997� obtained a similar result in anti–de Sitter spacetime. Interestingly, they found that the temperature is equal to the Unruh temperature cor- responding to the acceleration of the detector in five- dimensional Minkowski spacetime in which �anti–�de Sitter spacetime is embedded. Jacobson �1998� gave a simple explanation of these results, and Buchholz and Schlemmer �2007� discussed them in the context of their definition of a local temperature. For some work related to the response rate of the Unruh-DeWitt detector in de Sitter spacetime see, e.g., Higuchi �1987� and Garbrecht and Prokopec �2004a, 2004b�. The Bisognano-Wichmann result was also extended to Schwarzschild and de Sitter spacetimes by Sewell as mentioned before. Kay and Wald �1991� proved the analog of the Unruh effect in a class of spacetimes with bifurcate Killing ho- rizons �Boyer, 1969� adopting the viewpoint that Had- amard states are the only physical states for the free scalar field theory. They showed that the Wightman two- point function ��x ;x�� on the horizon satisfies �U�U���U,s ;U�,s�� = − 1 4� 1 �U − U� − i��2� 2�s,s�� �2.187� with x= �U ,s�, where s parametrizes the null geodesics on the Killing horizon and U is an affine parameter on each geodesic, for a Hadamard state invariant under the Killing symmetry. This formula allowed them to show that if such a state exists, it must be unique. Then they applied essentially the same argument as for the mass- less scalar field theory in the two-dimensional Rindler wedges to derive the Unruh-like effect. �See also Kay �1993, 2001� for further developments and an account of this result.� III. APPLICATIONS In this section we review works using the Unruh effect to examine some selected phenomena. We begin by dis- cussing each phenomenon using plain quantum field theory adapted to inertial observers, and then we show how the same observables can be recalculated from the point of view of Rindler observers with the help of the Unruh effect. The first example is connected with the excitation of accelerated detectors and atoms, the sec- ond one with the weak decay of noninertial protons, and the third one with the interpretation of radiation emit- ted by charges from the point of view of uniformly ac- celerated observers. In particular we clarify the tradi- tional question whether or not uniformly accelerated charges emit radiation from the point of view of coac- celerated observers. A. Unruh-DeWitt detectors Models of photon detectors have been discussed for some time in quantum optics �Glauber 1963�. Unruh �1976� introduced a detector model consisting of a small box containing a nonrelativistic particle satisfying the Schrödinger equation. The system is said to have de- tected a quantum if the particle in the box jumps from the ground state to some excited state. In the same pa- per, Unruh also discussed a relativistic detector model �see also Sanchez �1981� for a similar model�. Here we consider the detector model introduced by DeWitt �1979�, which consists of a two-level point monopole. We generically call two-level point monopoles Unruh- 20The condition for “physical acceptability” here is essentially the so-called Hadamard condition. See Fulling et al. �1978� and Wald �1978� for an early use of this condition. 804 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 DeWitt detectors following the literature. A discussion on particle detectors with finite spatial extent can be found in Grove and Ottewill �1983�. Particle detectors have often been used to probe the Unruh thermal bath. Sometimes, however, distinct de- tector designs may lead to contrasting conclusions about the same given feature of the bath. For instance, Hinton �1983�, Hinton et al. �1983�, Israel and Nester �1983�, and Sanchez �1985�, have argued that the Unruh thermal bath is anisotropic while Gerlach �1983�, Grove and Ot- tewill �1985�, and Kolbenstvedt �1987�, have argued the opposite. It is not surprising that, in general, direction- ally sensitive detectors will respond differently if they are given distinct orientations. Nevertheless, the Unruh thermal bath is as isotropic as a thermal bath in equilib- rium in a general static spacetime can be in the sense that Killing observers will see no net energy flux, etc., in any space direction, as is well known. In general, the temperature ��−1�i measured by a Killing observer fol- lowing a curve i generated by a Killing vector will be position dependent. Two Killing observers following curves i=1,2 will have their temperatures related as ��−1�1/��−1�2 = ��� � ���2/�� � ���1�1/2, where � is the Killing vector tangent to the world line of the corresponding observer �Tolman, 1934�. We consider a two-level Unruh-DeWitt detector in Minkowski spacetime. The detector is represented by a Hermitian operator m̂ acting on a two-dimensional Hil- bert space. The excited state �E� and the unexcited state �E0� are assumed to be eigenstates of the detector’s Hamiltonian Ĥ: Ĥ�E� = E�E�, Ĥ�E0� = E0�E0� �3.1� with eigenvalues E and E0, respectively �E�E0�. The monopole is time evolved as usual: m̂��� � eiĤ�m̂0e−iĤ�, �3.2� where � is the detector’s proper time. The matrix ele- ment q� E�m̂0�E0� depends on the detector design.21 Now we couple our Unruh-DeWitt detector to a real massive scalar field �̂�x� satisfying the Klein-Gordon equation ��̂+m2�̂=0 through the interaction action ŜI = � −� � d�m̂����̂�x���� , �3.3� where x���� is the detector’s world line. Next, we analyze the response of the detector from the point of view of inertial and that of Rindler observers separately. Re- lated investigations for detectors coupled with electro- magnetic and Dirac fields can be found in Boyer �1980, 1984� and Iyer and Kumar �1980�, respectively. 1. Uniformly accelerated detectors in Minkowski vacuum: Inertial observer perspective In Cartesian coordinates x�= �t ,x ,y ,z� of Minkowski spacetime the world line x�=x���� of a uniformly accel- erated detector along the z axis with proper acceleration a is given by t��� = a−1 sinh a�, z��� = a−1 cosh a� , �3.4� and x��� ,y���=const �see Fig. 4�. We expand �̂�x� in terms of positive- and negative- energy eigenstates of the Hamiltonian Ĥ= i� /�t, associ- ated with inertial observers, as �see Sec. II.D� �̂�x� =� d3k�ukâk M + H.c.� , �3.5� where uk = �2��2��3�−1/2e−ik�x� �3.6� with k�= �� ,k�, �=�k2+m2 and �âk M, âk� M†� = �3�k − k�� . The proper excitation rate, i.e., the excitation prob- ability divided by the total detector proper time T, asso- ciated with the uniformly accelerated detector in the in- ertial vacuum is given by22 21Two-level point monopoles have also been used to model the excitation and deexcitation of atoms �Audretsch and Müller, 1994b; Zhu and Yu, 2007�. 22Often, the excitation rate is alternatively expressed in terms of the golden rule �3.52�. t = const z = co ns t FIG. 4. The world line of a uniformly accelerated detector moving along the z axis in the Minkowski spacetime covered with Cartesian coordinates. 805Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 excR = T−1� d3k�excAk em�2, �3.7� where the excitation amplitude is �up to an arbitrary phase� excAk em = i E� � kM�ŜI�0M� � �E0� = q �16�3��1/2� −� � d� exp�i�E�� � exp��i�/a�sinh a� − �ikz/a�cosh a�� �3.8� with �E�E−E0. We have adopted here the subscript M to label states defined by inertial observers in Minkowski spacetime. �Note that we are using the con- vention that space components of the momentum k� are given with lower indices. That is, kx, ky, and kz, are the x, y, and z components, respectively, of the contravari- ant vector k�.� We note that because Eq. �3.3� is linear in �̂�x����, the detector excitation is accompanied by the emission of a particle23 with momentum k. By using Eq. �3.8� in Eq. �3.7�, we obtain excR �� d2k�R�, �3.9� where k�= �kx ,ky� denotes the transverse momentum with respect to the direction of the acceleration, R� is given by R� = �q�2 16�3T � −� � dkz � � −� � d��� −� � d��ei�E���−��� � ei��sinh a��−sinh a���/ae−ikz�cosh a��−cosh a���/a = �q�2 16�3T � −� � dkz � � −� � d�� −� � d"ei�E" � e�2i/a�sinh a"�� cosh a�−kz sinh a��, and we have defined �����+��� /2 and "���−��. Be- cause the interaction is kept turned on for an arbitrarily long time interval, the total time T diverges. To obtain explicitly the excitation rate per unit time, the total time T must be canceled by factoring out the divergent part �−� � d� from the integrals above. To this end, we first note that the momentum of the emitted particle is boosted due to the nonzero velocity of the detector, which is � dependent. Hence it is expected that the integrand can be made � independent by boosting back the momentum variables. Motivated by this physical picture, we intro- duce a new momentum variable as kz � kz� � kz cosh a� − � sinh a� , �3.10� which can be inverted as kz� � kz = k�z cosh a� + �� sinh a� . �3.11� Here �����k�z�2+k� 2 +m2�1/2 can be expressed as �� = � cosh a� − kz sinh a� , where k����kx�2+ �ky�2. It can be shown that dk�z /�� =dkz /�. This transformation allows us to factor out T =�−� � d�, and we obtain R� = �q�2 16�3� −� � dk�z �� � −� � d"ei�E"e�2i��/a�sinh�a"/2�. �3.12� By making now a further change of variables as kz� � � ��� + k�z�/�k� 2 + m2, " � # � exp�a"/2� , �3.13� we obtain R� = �q�2 8�3a � 1 � d � 0 � d##2i�E/a−1 �exp�i�k� 2 + m2�1/2�# − #−1�� + −1�/�2a�� . Now, we make the change of variables � ,# � �� ,$ with �= # and $= /#, and write R� = �q�2 16�3a��0 � d� �1−i�E/aei�k� 2 + m2�1/2��−�−1�/�2a��2 = �q�2e−��E/a 4�3a �Ki�E/a��k� 2 + m2�1/2/a� 2, �3.14� where K��x� is the modified Bessel function �Gradshteyn and Ryzhik, 1980�. �We recall here that the function Ki#�x� is real if x and # are real.� Finally, we obtain for the proper excitation rate �3.9� excR = �q�2ae−��E/a 2�2 � 0 � d���Ki�E/a���2 + �m/a�2� 2. �3.15� The angular distribution of the corresponding emitted particles in the massless case can be found in Kolben- stvedt �1988�. Next, we reproduce this detector response from the point of view of Rindler observers and discuss it. 2. Uniformly accelerated detectors in Minkowski vacuum: Rindler observer perspective The spacetime appropriate for investigating the exci- tation rate of our detector with proper acceleration a from the point of view of uniformly accelerated observ- ers is the Rindler wedge. We choose the right Rindler wedge �z� �t�� to work with, where we recall that it has a global timelike isometry associated with the Killing field z� /�t+ t� /�z. By covering it with Rindler coordinates �� ,� ,y ,z� �−��� ,� ,x ,y� +��, which are related with �t ,x ,y ,z� by Eq. �2.36�, we obtain the line element of the Rindler wedge as written in Eq. �2.37�. 23This combination, i.e., excitation with particle emission, can be also observed in the anomalous Doppler effect where atoms move in media with refractive index n with velocity v�1/n �Frolov and Ginzburg, 1986�. 806 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 The world lines of the Rindler observers are given by � ,x ,y=const and are hyperbolas in the two-dimensional diagram of Minkowski spacetime with x and y sup- pressed �see Fig. 5�. The corresponding four-velocity and four-acceleration are u�=e−a��1,0 ,0 ,0� and a� =e−2a��0,a ,0 ,0�, respectively, where a�=u���u � �see, e.g., Wald �1984��. Thus the proper acceleration of the Rindler observers is �−a�a�=ae−a�=const. Our uni- formly accelerated detector with proper acceleration a will lie at �=0 �for some x ,y=const�. Next, we expand �̂�x� in terms of positive- and negative-energy eigenstates of the Hamiltonian Ĥ = i� /��, associated with the Rindler observers, as �see Sec. II.D� �̂�x� =� d�d2k��v�k� R â�k� R + H.c.� , �3.16� where v�k� R = � sinh���/a� 4�4a �1/2 Ki�/a��k� 2 + m2 ae−a� �eik�·x�−i�� �3.17� are Klein-Gordon orthonormalized, and we recall that the creation and annihilation operators of Rindler par- ticles satisfy the commutation relations �â�k� R , â ��k�� R† � = ��� − ����2�k� − k��� . �3.18� The Rindler vacuum �0R� is defined by â�k� R �0R�=0. A detector lying at rest within a uniformly accelerated cav- ity prepared in the Rindler vacuum is not excited �Levin et al., 1992�.24 We emphasize that the quantum numbers �� ,k� associated with the timelike and spacelike global Killing fields � /�� and � /�x, � /�y, respectively, are inde- pendent of each other �see Sec. III.A.3�. Before we analyze the behavior of the detector in the Minkowski vacuum, we formally consider the detector’s excitation probability with simultaneous emission of a Rindler particle in the Rindler vacuum. The amplitude associated with this process in first order of perturbation is excA�k� em � i E� � �k�R�ŜI�0R� � �E0� , �3.19� where we recall that we use Eq. �3.16� in ŜI as given in Eq. �3.3�. The differential probability associated with this amplitude is dWem = �excA�k� em �2d2k�d� . �3.20� Now, we take into account the fact that due to the Un- ruh effect the Minkowski vacuum corresponds to a ther- mal bath of Rindler particles. We emphasize that the Minkowski vacuum is indistinguishable from the ther- mal bath built on the Rindler vacuum as long as the detector stays in the Rindler wedge since the Minkowski vacuum is a linear combination of products of the left and right Rindler states. For this reason, the detector’s excitation rate with simultaneous emission of a Rindler particle into the Minkowski vacuum is given by Eq. �3.20� combined with the proper thermal factor �see Eq. �4.9� in Higuchi et al. �1992a� for more detail�: excRem = T−1� dWem�1 + n���� , �3.21� where n��� = 1/�exp���� − 1� �3.22� is the Rindler scalar particle number density in the mo- mentum space. Here �−1=a /2� is the Unruh tempera- ture as measured by Rindler observers at �=0. The first and second terms in the square brackets in Eq. �3.21� are associated with spontaneous and induced emission, re- spectively. Similarly, one can calculate the detector’s excitation rate with simultaneous absorption of a Rindler particle from the Unruh thermal bath as excRabs = T−1� dWabsn��� , �3.23� where dWabs � �excA�k� abs �2d2k�d� �3.24� and 24An account on vacuum states in static spacetimes with ho- rizons can be found in Fulling �1977�. ττττ = const ξξ ξξ = co n st FIG. 5. The world line of a uniformly accelerated detector moving along the z axis in Minkowski spacetime covered with Rindler coordinates. 807Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 excA�k� abs � i E� � 0R�ŜI��k�R� � �E0� �3.25� is the excitation amplitude with absorption of a Rindler particle ��k�R�. The excitation amplitudes �3.19� and �3.25� can be shown to be excA�k� em�abs� = q� −� � d� exp�ik� · x� + i��E + �− ���� � � sinh���/a� 4�4a �1/2 Ki�/a� �k� 2 + m2�1/2ea� a � �3.26� up to some multiplicative phase. It is easy to verify in this case that excA�k� em =0, as expected, since uniformly accelerated detectors are static according to Rindler ob- servers. Hence according to these observers the only contribution to the detector response comes from the absorption of Rindler particles from the Unruh thermal bath. Now, since in first order of perturbation there is no interference, the total detector excitation rate in the Minkowski vacuum is excR = excRem + excRabs. �3.27� By using Eq. �3.26� to calculate Eq. �3.27�, we get Eq. �3.15�, as expected. Of course, inertial and Rindler ob- servers must agree on the value of scalar observables, such as the proper excitation rate of a given detector, although they can differ in how they describe the phe- nomenon. Because inertial and Rindler observers would expand the quantum fields with different sets of normal modes, they would end up extracting different particle contents from the same field theory. As a result, it is natural for inertial and Rindler observers to describe the detector excitation as being accompanied by the emis- sion of a Minkowski particle and by the absorption of a Rindler particle from the Unruh thermal bath �Unruh and Wald, 1984�, respectively. This conclusion can be generalized for detectors confined in the Rindler wedge following general world lines by saying that the detector excitation which is associated with the emission of a Minkowski particle as described by inertial observers corresponds in this case to the absorption or emission of a Rindler particle from or to the Unruh thermal bath according to Rindler observers �Matsas, 1996�. We comment on one possible source of confusion con- cerning the Unruh-DeWitt detector. A naive �and wrong� application of the equivalence principle might lead to the conclusion that an inertial detector which has the same velocity as an accelerated one at a certain time would detect Unruh radiation. This is of course not the case: no detector in an inertial motion detects any Un- ruh radiation. Before proceeding further, we note for later purposes that in the particular case with m=0, Eq. �3.15� takes the form excRm=0 = �q�2 2� �E e��E − 1 . �3.28� 3. Rindler particles with frequency ��m Here we discuss the existence of Rindler particles with frequencies ��m, which was crucial in the discus- sion above �notice that the range of the � integrations in Eqs. �3.21� and �3.23� is 0��� +��. The standard theory of quantum fields uses the fact that Minkowski spacetime is invariant under time and space translations. The linear three-momentum �kx ,ky ,kz� associated with the translational isometries on the spacelike hypersur- faces t=const constitutes a suitable set of quantum num- bers to label free particles. In this simple case, the dis- persion relation E���=��pc�2+m2c4 imposes a simple constraint between the particle mass m, momentum p, and energy E, and thus free particles with well-defined linear momenta must have total energy E%mc2. More- over, in the classical context of general relativity, the detection in loco of point particles satisfying E�mc2 by direct capture is ruled out by the fact that an observer with four-velocity u� intercepting a particle with four- momentum p�=mv� assigns to the particle an energy E=mv�u�%mc2. On the other hand, it is well known that the field quantization carried over arbitrary spacetime does not lead in general to any dispersion relation connecting the frequency with other quantum numbers, avoiding the flat spacetime constraint E%mc2. This can be under- stood by recalling that, strictly speaking, the concept of point particle has no place in quantum field theory. This raises the following question: What is the probability density associated with the detection of particles with E�mc2, i.e., ��m, at different space points of the Rin- dler wedge? By answering this question, we can also ex- tract some information about the particle distribution of the Hawking radiation near the event horizon of black holes. Indeed, much insight into the Hawking effect can be obtained in the simplified context provided by the Rindler wedge as we see next. �We refer the reader to Castiñeiras et al. �2002� for more detail.� We start by considering the line element of a two- dimensional Schwarzschild spacetime: ds2 = �1 − 2M/r�dt2 − �1 − 2M/r�−1dr2. �3.29� This can be seen as describing a two-dimensional black hole25 with mass M. Close to the horizon, r�2M, it can be written as ds2 = ��/4M�2dt2 − d�2, �3.30� where ��r���8M�r−2M�. �Note that in these coordi- nates the horizon is at �=0.� One can identify Eq. �3.30� with the line element of the Rindler wedge �2.37� with x ,y=const by letting t=4Ma� and �=ea� /a provided that 0��� +� and −�� t� +�. 25The vacuum expectation value of the energy-momentum tensor for a massless scalar field in this spacetime was analyzed by Davies et al. �1976� �see also Davies �1976� and Davies and Fulling �1977a��. See Christensen and Fulling �1977� for discus- sion and Candelas and Dowker �1979�. 808 Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 From here to the end of this section we consider the spacetime of the Rindler wedge with line element �3.30�, where 0��� +� and −�� t� +�. Now, we choose a fiducial observer at �=�0=4M, whose proper time is t �see Eq. �3.30��, with respect to whom the particle’s en- ergy is to be measured. The total probability P���d� of detecting a particle at some point �=�d with energy � per �detector� proper time sd tot is defined as ����d� �P���d� /sd tot. Then, the normalized probability density is dP� d�d � ����d��� 0 +� ����d��d�d��−1 , �3.31� where �dP� /d�d�d�d is the probability that a particle with energy � is found between �d and �d+d�d. Observ- ers far away from the horizon will be able to interact only with the “tail” of the “wave functions” associated with particles with small � /m. The smaller the � /m, the more difficult it is to detect these particles. Now, in order to interpret Eq. �3.31� in the frame work of general relativity, we first consider a row of de- tectors, each of them lying at different �d, and define the average detection position �d� � � 0 +� d�d�ddP�/d�d. �3.32� By using Eq. �3.31�, this can be shown to be �see Fig. 6� �d� = � tanh�4�M���64M2�2 + 1� 64mM� � �M�/m ��� a� , �3.33� where a�1/4M is the proper acceleration of the fiducial observer. On the other hand, from general relativity, a classical particle with mass m lying at rest at some point �p has, according to our fiducial observer at �0=4M, en- ergy �=m�p /4M. By considering that the particle may have some kinetic energy in addition, the total energy would be �%m�p /4M. From this equation, we obtain �p ! 4M�/m � �p max. �3.34� This is expected to agree with �d�, i.e., �d�!�p max, at least in the “high-frequency” regime ��a �where the quantum and classical behaviors may be compared�. This conclusion is in agreement with Eqs. �3.33� and �3.34� as seen in Fig. 7. In summary, the smaller the � /m ratio, the more likely the observer is to detect the par- ticle closer to the horizon. 4. Static detectors in a thermal bath of Minkowski particles Now, we show explicitly that the response rate �3.15� does not correspond to the one obtained when the de- tector lies at rest in a plain thermal bath of Minkowski particles heated up to the Unruh temperature �−1 =a / �2��. In the latter case, the excitation rate is ob- tained by replacing Eq. �3.7� by excR� � T−1� d3k��excAk em�2�1 + n���� + �excAk abs�2n���� , �3.35� where n���=1/ �exp����−1� and excAk em � i E� � kM�ŜI�0M� � �E0� , excAk abs � i E� � 0M�ŜI�kM� � �E0� are the excitation amplitudes with emission and absorp- tion of Minkowski particles �kM�, respectively. In this case, the excitation amplitudes can be shown to be �up to an arbitrary multiplicative phase� excAk em�abs� = q�„� + �− ��E…/�4�� , �3.36� where we have assumed with no loss of generality that the detector is at the origin x=0. Clearly, excAk em=0. Hence the only contribution to the detector response is associated with the absorption of a Minkowski particle. By substituting Eq. �3.36� into Eq. �3.35�, we obtain 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 (d P ω /d ρ d )/ m mρd ω/m=0.5 ω/m=1.0 ω/m=1.5 FIG. 6. The probability density dP� /d�d for different � /m ratios, where Mm=1/4. Note that the smaller the � /m ratio, the closer to the horizon �on average� the particle lies, where the “gravitational potential” is lower. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.02 0.04 0.06 0.08 0.1 ω/m mρp max m<ρd> FIG. 7. �d� is shown to be smaller than �p max�4M� /m in the high-frequency regime �� �4M�−1 �i.e., at the right of the ver- tical broken line� as expected. We have let mM=10 but this agreement is verified for any mM. 809Crispino, Higuchi, and Matsas: The Unruh effect and its applications Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008 excR� = �q�2�E 2� ���E − m� e��E − 1 . �3.37� The presence of the step function ���E−m� expresses the fact that the detector can only be excited if its energy gap is large enough to absorb massive scalar particles from the thermal bath. Clearly excR� in Eq. �3.37� with �−1=a / �2�� and excR in Eq. �3.15� are distinct. Indeed, there is no a priori reason why they should be the same. Incidentally, in the case m=0, Eqs. �3.37� and �3.28� are equal. However, this is a coincidence, which has to do with the particular design of the detector and not with the Unruh effect. As we have seen, what the Unr