The Unruh effect and its applications

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

共Published 1 July 2008兲

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共s兲: 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共dynamical Casimir兲 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

I. INTRODUCTION

It has been 30 years since the discovery of the Unruh effect共Unruh, 1976兲 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 共Fulling, 1973兲 关see also Unruh 共1977兲兴. 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 共Hawking, 1974,

1975兲 and cosmological horizons 共Gibbons and Hawk-ing, 1977兲. It is interesting that the Unruh effect was on

Feynman’s list of things to learn in his later years 共see Fig.1兲. 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 共also called Rindler observers兲, as-sociate a thermal bath of Rindler particles 共also called Fulling-Rindler particles兲 to the no-particle state of iner-tial observers共also called the Minkowski vacuum兲. 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 共1976兲 also provided an explanation for the conclusion obtained byDavies 共1975兲that an observer undergoing uniform acceleration a = const in Minkowski spacetime would see a fixed inertial mirror to emit thermal radia-tion with temperature aប/共2␲kc兲, 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,1we 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 共Narozhny et al., 2002,2004兲. 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共Fulling and Unruh, 2004兲, 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 共i兲 uniformly accelerated in the Minkowski vacuum and共ii兲 static in a thermal bath of Minkowski particles 共rather than of Rindler particles兲. There is no such equivalence in general, although this showed up by coincidence in some early examples in the literature 共see discussion in

Sec. III.A.4兲. 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共Unruh, 1976兲. 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 关see Fulling 共2005兲 and related references兴. 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共see Fig.2兲 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.

1

See, e.g., Sciama et al. 共1981兲; Birrell and Davies 共1982兲;

Takagi 共1986兲; Fulling and Ruijsenaars共1987兲; Ginzburg and Frolov共1987兲;Wald共1994兲;Padmanabhan共2005兲.

2

Throughout 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.

3_{This 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.

4_{The data in Fig.2should 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.

5

See, e.g.,Unruh and Wald共1982,1983兲;Wald共2001兲. 6

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 unitsប=c=G=k=1 and signature 共⫹ ⫺ ⫺ ⫺兲 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 共QFT兲 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 byBirrell and Davies共1982兲. A mathematically more satisfactory treatment can be found, e.g., inWald共1994兲.

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 k}

0⬎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 共D+1兲-dimensional spacetimes whose metric takes the form

ds2=关N共x兲兴2dt2− Gab共x兲dxadxb, 共2.1兲

where x =共t,x兲. The coefficient N共x兲 is called the lapse function共Arnowitt et al., 1962兲 and Gabis the metric on

the spacelike hypersurface of constant t.共All spacetimes considered have a metric of this form.兲 In this spacetime the minimally coupled7massive Klein-Gordon equation 共ⵜ␮ⵜ␮+ m2兲␾= 0, which arises as the Euler-Lagrange

equation from the Lagrangian density,

L =

冑

− g共ⵜ␮␾ⵜ␮␾− m2␾2兲/2, 共2.2兲

takes the form

⳵t共N−1

冑

G⳵t␾兲 −⳵a共N

冑

GGab⳵b␾兲 + N

冑

Gm2␾= 0,

共2.3兲 where the space indices a and b run from 1 to D.

Given two complex solutions fA共x兲 and fB共x兲 to the

Klein-Gordon equation, we define the Klein-Gordon current J_共f A,fB兲 ␮ _{共x兲 ⬅ f} A *_共x兲ⵜ␮_f B共x兲 − fB共x兲ⵜ␮f_A*共x兲. 共2.4兲

Then, one can show that ⵜ_␮J_共f

A,fB兲

␮ _{共x兲=0. Hence the}

quantity

共fA,fB兲KG⬅ i

冕

dDx

冑

Gn␮J共fA,fB兲

␮ _共2.5兲

is independent of t, where n_␮is the future-directed unit vector normal to the hypersurface⌺ of constant t. 关The integral here and throughout this subsection共Sec. II.A兲 is over the hypersurface ⌺.兴 We call this quantity the Klein-Gordon inner product of fAand fB. For the metric

共2.1兲 it takes the following form:

共fA,fB兲KG= i

冕

dDx

冑

GN−1共f_A*⳵tfB− fB⳵tf_A*兲. 共2.6兲

The conjugate momentum density ␲共x兲 is defined as ␲ ⬅⳵L/⳵␾˙ , where ␾˙ ⬅⳵t␾. For the metric共2.1兲 one finds

7

It is customary to allow the field to couple to the scalar curvature. Thus the general Klein-Gordon equation takes the form共ⵜ_␮ⵜ␮+␰R+m2_{兲␾=0. The minimally coupled scalar field} has␰=0 by definition.

␲共x兲 = N−1

冑

_{G␾˙ 共x兲. 共2.7兲}

Note that if we let pA共x兲 and pB共x兲 be the conjugate

momentum density for the solutions fA共x兲 and fB共x兲,

re-spectively, then the Klein-Gordon inner product can be expressed as

共fA,fB兲KG= i

冕

dDx关fA*共x兲pB共x兲 − pA*共x兲fB共x兲兴. 共2.8兲

We assume that the Klein-Gordon equation determines the classical field ␾共x兲 uniquely given a 共well-behaved兲 initial data 共␾,␲兲 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 ␾ˆ and ␲ˆ , respectively. One imposes the following

equal-time canonical commutation relations:

关␾ˆ 共t,x兲,␾ˆ 共t,x

⬘

兲兴 = 关␲ˆ共t,x兲,␲ˆ共t,x

⬘

兲兴 = 0, 共2.9兲

关␾ˆ 共t,x兲,␲ˆ共t,x

⬘

兲兴 = i␦D_共x,x

_⬘

_{兲, 共2.10兲}

where the delta function␦D_共x,x

_⬘

_{兲 is defined by}

冕

dD_xf共x兲␦D_共x,x

_⬘

_{兲 = f共x}

_⬘

_{兲. 共2.11兲}

Note here that there is no density factor

冑

G on the

left-hand side. For arbitrary complex-valued solutions fA共x兲

and fB共x兲 to the Klein-Gordon equation 共2.3兲 共with a

suitable integrability conditions兲 one finds

关共fA,␾ˆ 兲KG,共␾ˆ ,fB兲KG兴 = 共fA,fB兲KG, 共2.12兲

from the equal-time canonical commutation relations 共2.9兲 and 共2.10兲 by using Eq. 共2.7兲.

Now, assume that there is a complete set of solutions, 兵fi, f_i*其, to the Klein-Gordon equation 共2.3兲 satisfying

共fi,fj兲KG= −共f_i*,f_j*兲KG=␦ij, 共2.13兲

共f_i*_,f

j兲KG=共fi,f_j*兲KG= 0. 共2.14兲

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 f_i’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␾ˆ 共x兲 as

␾ˆ 共x兲 =

兺

i 关aˆifi共x兲 + aˆi †_f i *_{共x兲兴, 共2.15兲} one finds aˆi=共fi,␾ˆ 兲KG, aˆi † =共␾ˆ ,fi兲KG. 共2.16兲

One can readily show, by using Eqs.共2.12兲–共2.14兲, that

关aˆi,aˆj兴 = 关aˆi

†

,aˆ_j†兴 = 0, 关aˆi,aˆj

†_{兴 =␦}

ij. 共2.17兲

Conversely, if these commutation relations are satisfied, then the equal-time canonical commutation relations 共2.9兲 and 共2.10兲 follow. To prove this, one first shows that

any two complex-valued solutions fA共x兲 and fB共x兲 to the

Klein-Gordon equation 共2.3兲 satisfy Eq. 共2.12兲 by

ex-panding them in terms of fi共x兲 and f_i*共x兲 and using the

commutators 共2.17兲. Then, for example, by letting fA共t,x兲=f_B*共t,x兲 and pA共t,x兲=−p_B*共t,x兲, at a given time t

and evaluating the Klein-Gordon inner products in Eq. 共2.12兲 at time t, one obtains

冕

dDxdDx

⬘

f_B共t,x兲p_B共t,x

⬘

兲关␾ˆ 共t,x兲,␲ˆ_共t,x

⬘

_兲兴

= i

冕

dDxfB共t,x兲pB共t,x兲. 共2.18兲

Since fB共t,x兲 and pB共t,x兲 are arbitrary, one can conclude

that Eq. 共2.10兲 holds at time t. Equation 共2.9兲 can be

derived in a similar manner. 8

A 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共1984兲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 ofUnruh 共1976兲over the years.

The operators aˆiand aˆi†are called the annihilation and

creation operators, respectively. The vacuum state兩0典 is defined by requiring aˆi兩0典=0. The Fock space of states is

obtained by applying the creation operators aˆ_i† on the vacuum state 兩0典. We call the operator Nˆi= aˆi

†

aˆi共with no

summation on the right-hand side兲 the number operator in the mode i. However, note that, since it is not always easy to construct a共theoretical兲 detector model which is excited when the eigenvalue of Nˆ_ichanges from 1 to 0, say, the operator Nˆidoes not necessarily lead to a useful

particle concept.

Since the coefficient operators aˆi of the functions fi

annihilate the vacuum state 兩0典, the choice of the func-tions f_i satisfying Eqs. 共2.13兲 and 共2.14兲 determines the

vacuum state. For this reason we call the functions fithe

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 兵f_i共1兲其 and 兵f_I共2兲其 satisfy the Klein-Gordon inner-product relations 共2.13兲 and 共2.14兲, where

the lower-case letters i, j are replaced by the upper-case equivalents I, J for f_I共2兲. Since both sets are complete, the modes f_I共2兲can be expressed as linear combinations of f_i共1兲 and f

i

共1兲* _{and vice versa. Thus} f_I共2兲=

兺

i 关␣Iifi共1兲+␤Iif_i共1兲*兴, 共2.19兲 f_I共2兲*=

兺

i 关␣_Ii*_f i 共1兲*_+␤ Ii *_f i 共1兲_{兴. 共2.20兲} By noting that ␣Ii=共fi共1兲,fI共2兲兲KG=共fI共2兲,fi共1兲兲KG * _{, 共2.21兲} ␤Ii= −共fi 共1兲*_,f I 共2兲_兲 KG=共fI 共2兲*_,f i 共1兲_兲 KG, 共2.22兲

one can express f_i共1兲 as a linear combination of f_I共2兲 and

f I 共2兲* _as f_i共1兲=

兺

I 关␣_Ii*_f I 共2兲_−␤ Iif_I共2兲*兴, 共2.23兲 f_i共1兲*=

兺

I 关␣Iif_I共2兲*−␤_Ii*fI共2兲兴. 共2.24兲

The scalar field␾ˆ 共x兲 can be expanded using either of the two sets兵f_i共1兲其 and 兵f_I共2兲其:

␾ˆ 共x兲 =

兺

i 关aˆi共1兲fi共1兲+ aˆi共1兲†fi 共1兲*_{兴 =}

_兺

I 关aˆI共2兲fI共2兲+ aˆI共2兲†fI 共2兲*_兴. 共2.25兲 Using the expansion given by Eqs.共2.19兲 and 共2.20兲, and

comparing the coefficients of f_i共1兲and f_i共1兲*, we find

aˆ_i共1兲=

兺

I 共␣IiaˆI共2兲+␤Ii *_aˆ I 共2兲†_{兲, 共2.26兲}

and similarly by using Eqs. 共2.23兲 and 共2.24兲 we have aˆ_I共2兲=

兺

i 共␣_Ii*_aˆ i 共1兲_−␤ Ii *_aˆ i 共1兲†_{兲. 共2.27兲}

This transformation, which mixes annihilation and cre-ation operators, is called a Bogoliubov transformcre-ation, and the coefficients␣Iiand␤Iiare 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共Parker, 1968,Sexl and Urbantke, 1969兲.

The vacuum states 兩0_共1兲典 and 兩0_共2兲典 corresponding to the two sets of positive-frequency modes兵f_i共1兲其 and 兵f_I共2兲其, respectively, are distinct if␤Iido not vanish for all I and

i. For example, the expectation value of the number

op-erator Nˆ_i共1兲= aˆ_i共1兲†aˆ_i共1兲for the state 兩0_共1兲典 is zero by defini-tion but for the state 兩0_共2兲典 it can be calculated by using Eq.共2.26兲 as

具0_共2兲兩Ni共1兲兩0共2兲典 =

兺

I

兩␤Ii兩2. 共2.28兲

We similarly find for the number operator N_I共2兲= aˆ_I共2兲†aˆ_I共2兲, 具0共1兲兩NI共2兲兩0共1兲典 =

兺

i

兩␤Ii兩2. 共2.29兲

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 共2.1兲

with the lapse function N共x兲 and the metric G_ab being independent of t.9In such a case the equation for deter-mining the mode functions becomes

⳵t

2

fi= NG−1/2⳵a共NG1/2Gab⳵bfi兲 − N2m2fi. 共2.30兲

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共future-directed兲 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

9

In fact, if a globally hyperbolic spacetime is stationary, i.e., if the metric is t independent with 共⳵/⳵t兲␮ 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共Ashtekar and Magnon, 1975;Kay, 1978兲.

10

A Killing vector K␮ is a vector satisfying ⵜ_␮K_␯+ⵜ_␯K_␮ = K␣⳵_␣g_␮␯+ g_␣␯⳵_␮K␣+ g_␮␣⳵_␯K␣= 0. In a coordinate system such that K␮=共⳵/⳵␪兲␮, one has⳵g_␮␯/⳵␪=0. See, e.g.,Wald共1984兲.

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 k}

0⬎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 兩t兩⬍z in Minkowski spacetime 共here z is one of the space coordinates兲, the boost Killing vector z共⳵/⳵t兲+t共⳵/⳵z兲, 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 playKill-ing the role of time translation. Thus one can define the corresponding static vacuum state. As ob-served by Fulling 共1973兲, 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 共2.31兲 is of course a static globally hyperbolic spacetime. As mentioned above, the part of this spacetime defined by 兩t兩⬍z, called the right Rindler wedge, is also a static globally hyperbolic spacetime. The region with the con-dition兩t兩⬍−z is called the left Rindler wedge, and is also a static globally hyperbolic spacetime. The region with

t⬎兩z兩, 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⬍−兩z兩 is called the contracting degenerate Kasner universe. These regions are shown in Fig.3.

Minkowski spacetime is invariant under the boost

t哫 t cosh␤+ z sinh␤, 共2.32兲

z哫 t sinh␤+ z cosh␤, 共2.33兲

where ␤ is the boost parameter. These transformations are generated by the Killing vector z共⳵/⳵t兲+t共⳵/⳵z兲. The

boost invariance of Minkowski spacetime motivates the following coordinate transformation:

t =␳sinh␩, z =␳cosh␩, 共2.34兲 where ␳ and ␩ takes any real value. Then, the Killing vector is⳵/⳵␩, and the metric takes the form

ds2=␳2d␩2− d␳2− dx2− dy2. 共2.35兲 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. 共2.32兲 and 共2.33兲.

Each world line has a constant proper acceleration given by␳−1_{= const.}

The coordinates共␩,␳, x , y兲 cover only the regions with

z2⬎t2, i.e., the left and right Rindler wedges, as can be seen from Eq. 共2.34兲. To discuss QFT in the right

Rin-dler wedge, it is convenient to make a further coordinate transformation ␳= a−1_ea␰_{,␩= a␶, i.e.,}

t = a−1ea␰sinh a␶, z = a−1ea␰cosh a␶, 共2.36兲 where a is a positive constant共Rindler, 1966兲. Then, the

metric takes the form

ds2_{= e}2a␰_共d␶2_{− d␰}2_{兲 − dx}2_{− dy}2_{. 共2.37兲}

This coordinate system will be useful because the world line with ␰= 0 has a constant acceleration of a. The co-ordinates共␶¯ ,␰¯兲 for the left Rindler wedge are given by

t = a−1_ea␰¯_{sinh a␶¯, z = − a}−1_ea␰¯_{cosh a␶¯ . 共2.38兲}

The Killing vector z共⳵/⳵t兲+t共⳵/⳵z兲 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共Wald, 1994兲.

11

It has been shown byChmielowski共1994兲that two commut-ing global timelike Killcommut-ing vector fields define the same vacuum state.

t

z

EDK CDK RR LR V U

FIG. 3. The regions with 兩t兩⬍z, 兩t兩⬍−z, t⬎兩z兩, and t⬍−兩z兩, 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共⳵/⳵t兲+t共⳵/⳵z兲. The direction of increas-ing U = t − z and that of increasincreas-ing V = t + z are also indicated.

Since the right 共or left兲 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 共Coleman, 1973兲. 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⌽ˆ共t,z兲 sat-isfies

共⳵2_/⳵t2_−⳵2_/⳵z2_{兲⌽ˆ = 0. 共2.39兲}

This field can be expanded as ⌽ˆ共t,z兲 =

冕

0 ⬁ _dk

冑

4␲k共bˆ−ke −ik共t−z兲_{+ bˆ} +ke−ik共t+z兲 + bˆ_−k† eik共t−z兲+ bˆ_+k† eik共t+z兲兲. 共2.40兲

The annihilation and creation operators satisfy 关bˆ±k,bˆ±k⬘

† _{兴 =␦共k − k}

_⬘

_{兲 共2.41兲}

with all other commutators vanishing. By using the defi-nitions U = t − z, V = t + z, 共2.42兲 we can write ⌽ˆ共t,z兲 = ⌽ˆ−共U兲 + ⌽ˆ+共V兲, 共2.43兲 where ⌽ˆ+共V兲 =

冕

0 ⬁ dk关bˆ+kfk共V兲 + bˆ+k † f_k*_{共V兲兴 共2.44兲} with fk共V兲 = 共4␲k兲−1/2e−ikV, 共2.45兲

and similarly for⌽ˆ−共U兲. Since the left- and right-moving

sectors of the field⌽ˆ₊共V兲 and ⌽ˆ₋共U兲 do not interact with one another, we discuss only the left-moving sector ⌽ˆ+共V兲. 共Thus we discuss the Unruh effect for the theory

consisting only of the left-moving sector.兲 The Minkowski vacuum state 兩0_M典 is defined by bˆ_+k兩0_M典=0 for all k.

Using the metric in the right Rindler wedge given by Eq.共2.37兲, one finds a field equation of the same form as

Eq.共2.39兲:

共⳵2_/⳵␶2_−⳵2_/⳵␰2_{兲⌽ˆ = 0. 共2.46兲}

共This is a result of the conformal invariance of the mass-less scalar field theory in two dimensions.兲 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, 共2.47兲

V = t + z = a−1eav. 共2.48兲

The Lagrangian density is invariant under the coordi-nate transformation 共t,z兲哫共␶,␰兲. 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共t,z兲 replaced by 共␶,␰兲. Thus we have, for 0⬍V, ⌽ˆ+共V兲 =

冕

0 ⬁ d␻关aˆ_+␻Rg_␻共v兲 + aˆ_+␻R†g_␻*_{共v兲兴, 共2.49兲} where g_␻共v兲 = 共4␲␻兲−1/2_e−i␻v_{, 共2.50兲} and 关aˆ_+␻R _,aˆ +␻_⬘ R†_{兴 =␦共␻} −␻

⬘

兲 共2.51兲

with all other commutators vanishing. Note that the functions g_␻共v兲 are eigenfunctions of the boost generator

⳵/⳵␶.

The field ⌽ˆ₊共V兲 can be expressed in the left Rindler wedge with the condition V⬍0⬍U, using the left Rin-dler coordinates 共␶¯ ,␰¯兲 defined by Eq. 共2.38兲. Defining v¯

=␶¯ −␰¯, one obtains Eqs. 共2.49兲–共2.51兲 with v replaced by v¯ and with the annihilation and creation operators aˆ_+␻R and aˆ₊R†_␻replaced by a new set of operators aˆ₊L_␻and aˆ₊L†_␻. The variable v¯ is related to V by

V = − a−1e−av¯. 共2.52兲

The static vacuum state in the left and right Rindler wedges, the Rindler vacuum state 兩0_R典, is defined by

aˆ_+␻R 兩0_R典=aˆ_+␻L 兩0_R典=0 for all␻.

To understand the Unruh effect we need to find the Bogoliubov coefficients␣_␻kR,␤_␻kR,␣_␻kL , and␤_␻kL, where

␪共V兲g␻共v兲 =

冕

0 ⬁ _dk

冑

4␲k共␣␻k R e−ikV+␤_␻kReikV兲, 共2.53兲 ␪共− V兲g_␻共v¯兲 =

冕

0 ⬁ _dk

冑

4␲k共␣␻k L e−ikV+␤_␻kLeikV兲. 共2.54兲

Here ␪共x兲=0 if x⬍0 and ␪共x兲=1 if x⬎0, i.e., ␪ is the Heaviside function. To find ␣_␻kR we multiply Eq.共2.53兲

␣␻kR =

冑

4␲k

冕

0 ⬁_dV 2␲g␻共V兲e ikV =

冑

k ␻

冕

0 ⬁_dV 2␲共aV兲 −i␻/a_eikV_{. 共2.55兲}

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 ␣␻kR = ie␲␻/2a

冑

␻k

冉

a k

冊

−i␻/a

冕

0 ⬁_dx 2␲x −i␻/a_e−x_dx = ie ␲␻/2a 2␲

冑

␻k

冉

a k

冊

−i␻/a ⌫共1 − i␻/a兲. 共2.56兲 To find the coefficients␤_␻kR we replace eikVin Eq.共2.55兲

by e−ikV. Then, the appropriate substitution is V = −ix / k. As a result we obtain ␤␻kR = − ie−␲␻/2a 2␲

冑

␻k

冉

a k

冊

−i␻/a ⌫共1 − i␻/a兲. 共2.57兲 A similar calculation leads to

␣_␻kL _{= −} ie␲␻/2a 2␲

冑

␻k

冉

a k

冊

i␻/a ⌫共1 + i␻/a兲, 共2.58兲 ␤␻kL = ie−␲␻/2a 2␲

冑

␻k

冉

a k

冊

i␻/a ⌫共1 + i␻/a兲. 共2.59兲

We find that these coefficients obey the following rela-tions crucial to the derivation of the Unruh effect:

␤_␻kL _{= − e}−␲␻/a_␣ ␻k R*_{, ␤} ␻k R _{= − e}−␲␻/a_␣ ␻k L*_{. 共2.60兲}

By substituting these relations into Eqs.共2.53兲 and 共2.54兲

we find that the following functions are linear combina-tions of positive-frequency modes e−ikV in Minkowski spacetime:

G_␻共V兲 =␪共V兲g_␻共v兲 +␪共− V兲e−␲␻/ag_␻*_{共v¯兲, 共2.61兲} G¯_␻共V兲 =␪共− V兲g_␻共v¯兲 +␪共V兲e−␲␻/ag_␻*_{共v兲. 共2.62兲}

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_␻共v兲=共4␲␻兲−1/2_共V兲−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 共4␲␻兲−1/2_e−␲␻/a_共−V兲−i␻/a _for V⬍0. This was the original argument byUnruh 共1976兲.

Equations共2.61兲 and 共2.62兲 can be inverted as

␪共V兲g␻共v兲 ⬀ G␻共V兲 − e−␲␻/aG¯_␻*共V兲, 共2.63兲

␪共− V兲g␻共v¯兲 ⬀ G¯␻共V兲 − e−␲␻/aG_␻*共V兲. 共2.64兲

By substituting these equations into ⌽ˆ+共V兲 =

冕

0 ⬁

d␻兵␪共V兲关aˆ₊R_␻g_␻共v兲 + aˆ₊R†_␻g_␻*_共v兲兴

+␪共− V兲关aˆ₊L_␻g_␻共v¯兲 + aˆ₊L†_␻g_␻*_{共v¯兲兴其, 共2.65兲}

we find that the integrand is proportional to

G_␻共V兲关aˆ_+␻R − e−␲␻/aaˆ_+␻L†兴

+ G¯_␻共V兲关aˆ_+␻L − e−␲␻/aaˆ_+␻R†兴 + H.c.

Since the functions G_␻共V兲 and G¯_␻共V兲 are positive-frequency solutions共with respect to the usual time trans-lation兲 in Minkowski spacetime, the operators aˆ_+␻R

− e−␲␻/aaˆ_+␻L† and aˆ_+␻L − e−␲␻/aaˆ_+␻R† annihilate the Minkowski vacuum state 兩0M典. Thus

共aˆ+R␻− e−␲␻/aaˆ+L†␻兲兩0M典 = 0, 共2.66兲

共aˆ_+␻L

− e−␲␻/aaˆ_+␻R†兲兩0M典 = 0. 共2.67兲

These relations uniquely determine the Minkowski vacuum兩0_M典 as explained below.

To explain how the state兩0_M典 is formally expressed in the Fock space on the Rindler vacuum state兩0R典 and to

show that the state 兩0M典 is a thermal state when it is

probed only in the right共or left兲 Rindler wedge, we use the approximation where the Rindler energy levels ␻ are discrete.13 Thus we write␻_iin place of␻and let

关aˆ+␻i R _,aˆ +␻j R†_{兴 = 关aˆ} +␻i L _,aˆ +␻j L†_{兴 =␦} ij 共2.68兲

with all other commutators among aˆ_+␻

i R _{, aˆ}

+␻_i

L _{and their}

Hermitian conjugates vanishing. Using the discrete ver-sion of Eqs.共2.66兲 and the commutators 共2.68兲, we find

具0M兩aˆ+␻i R†_aˆ +␻i R _兩0 M典 = e−2␲␻i/a具0M兩aˆ+␻i L†_aˆ +␻i L _兩0 M典 + e−2␲␻i/a. 共2.69兲 The same relation with aˆ_+␻

i R _{and aˆ} +␻i R† _{replaced by aˆ} +␻i L _and aˆ_+␻ i

L†_{, respectively, and vice versa can be found using Eq.}

共2.67兲. By solving these two relations as simultaneous

equations, we find

具0M兩aˆ+␻R†_iaˆ+␻R _i兩0M典 = 具0M兩aˆL†+␻_iaˆ+␻L _i兩0M典 = 共e2␲␻i/a− 1兲−1.

共2.70兲 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.

12

A cutoff of this kind is always understood in these calcula-tions in field theory, as exemplified by the definition ␦共k兲 =兰共dx/2␲兲eikx−␧兩x兩=共2␲i兲−1关共k−i␧兲−1−共k+i␧兲−1兴.

13

We comment on how one can discuss thermal states in field theory without discretization in Sec. II.I.

Equation 共2.70兲 can be expressed without discretiza-tion. Define aˆ_+fR⬅

冕

0 ⬁ d␻f共␻兲aˆ₊R_␻, 共2.71兲 where兰₀⬁d␻兩f共␻兲兩2_{= 1. Then,} 具0M兩aˆ+f R†_aˆ +f R_兩0 M典 =

冕

0 ⬁ d␻ 兩f共␻兲兩 2 e2␲␻/a− 1. 共2.72兲

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 aˆ_+fR†aˆ_+fR and aˆ_+fL†aˆ_+fL 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. 共2.66兲 and 共2.67兲. First we note that these equations

im-ply 共aˆ+␻i R†_aˆ +␻i R _{− aˆ} +␻i L†_aˆ +␻i L _兲兩0 M典 = 0. 共2.73兲

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 兩0M典 ⬀

兿

i n

兺

_i=0 ⬁ _K n_i ni! 共aˆ_+␻ i R† aˆ_+␻ i L†_兲n_i兩0 R典. 共2.74兲

One can readily find the recursion formula satisfied by

K_n

i using the discrete version of Eqs. 共2.66兲 and 共2.67兲.

The result is Kn_i+1− e−␲␻i/aKn_i= 0. 共2.75兲 Hence Kn_i= e−␲ni␻i/aK0and 兩0M典 =

兿

i

冉

Ci

兺

n_i=0 ⬁ e−␲ni␻i/a兩n i,R典丢兩ni,L典

冊

, 共2.76兲

where Ci=

冑

1 − exp共−2␲␻i/ a兲. Here the state with ni

left-moving particles with Rindler energy ␻i in each of the

left and right Rindler wedges is denoted兩ni, R典丢兩ni, L典,

i.e.,

兿

i 兩ni,R典丢兩ni,L典 ⬅

再

兿

i 1 n_i!共aˆ+␻i R†_aˆ +␻i L†_兲n_i

冎

_兩0 R典. 共2.77兲

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.,

␳ˆ_R=

兿

i

冉

C_i2

兺

n_i=0 ⬁ exp共− 2␲n_i␻_i/a兲兩n_i,R典具n_i,R兩

冊

. 共2.78兲 This is the density matrix for the system of free bosons with temperature T = a / 2␲. Thus the Minkowski vacuum state 兩0M典 for the left-moving particles restricted to the

left 共or right兲 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 共Belinskii et al., 1997; Fedotov et al. 1999;Oriti, 2000;Narozhny et al., 2002,2004兲 we review

the Unruh effect in this theory 共Fulling 1973; Unruh 1976兲, drawing attention to some aspects that appear to

have caused the skepticism. 关See Fulling and Unruh 共2004兲 for an explanation as to why this skepticism is unfounded.兴

The free quantized massive scalar field ⌽ˆ共t,z,x_⬜兲,

x_⬜⬅共x,y兲, can be expanded as

⌽ˆ共t,z,x_⬜兲 =

冕

d3k共aˆ_k zk⬜ M fk_zk_⬜+ aˆk_zk_⬜ M† f_k zk⬜ * _{兲, 共2.79兲}

where the positive-frequency mode functions are

fk_zk_⬜共t,z,x⬜兲 = 关共2␲兲32k0兴−1/2e−ik0t+ikzz+ik⬜·x⬜ 共2.80兲

with k_⬜⬅共k_x, k_y兲 and k₀⬅

冑

k_z2+ k_⬜2+ m2_{. The}

Klein-Gordon inner product can be calculated as

共fk_zk_⬜,fk_z⬘k_⬜⬘兲KG=␦共kz− kz

⬘

兲␦2共k⬜− k⬜

⬘

兲, 共2.81兲

共f_k

zk⬜

* _,f

k_z⬘k_⬜⬘兲KG= 0. 共2.82兲

Hence quantizing the scalar field⌽ˆ共t,z,x_⬜兲 by imposing the equal-time commutation relations 共2.9兲 and 共2.10兲,

we find 关aˆk_zk_⬜ M ,aˆ_k z ⬘k_⬜⬘ M† _{兴 =␦共k} z− kz

⬘

兲␦2共k⬜− k⬜

⬘

兲 共2.83兲

with all other commutators among annihilation and cre-ation operators vanishing.

The field equation in the right Rindler wedge with the metric 共2.37兲 can readily be found from Eq. 共2.30兲 by

letting N = ea␰ _{and the metric of the hypersurfaces with}

constant ␶ be diagonal with G_␰␰= e2a␰and Gxx= Gyy= 1.

Thus ⳵2_⌽ˆ ⳵␶2 =

冋

⳵2 ⳵␰2+ e 2a␰

冉

⳵2 ⳵x2+ ⳵2 ⳵y2

冊

− m 2_e2a␰

册

_{⌽ˆ. 共2.84兲}

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}

v_␻k ⬜ R = 1 2␲

冑

2␻g␻k⬜共␰兲e −i␻␶+ik_⬜·x_{⬜ 共2.85兲}

with the function g_␻k

⬜共␰兲 satisfying

冋

− d 2 d␰2+ e 2a␰_共k ⬜ 2 + m2兲

册

g_␻k ⬜共␰兲 =␻ 2_g ␻k_⬜共␰兲. 共2.86兲

This equation is analogous to a time-independent Schrödinger equation with an exponential potential. Thus the physically relevant solutions g_␻k

⬜共␰兲 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

⬜共␰兲 to satisfy, for␰⬍0 and 兩␰兩Ⰷ1, g_␻k ⬜共␰兲 ⬇ 1

冑

2␲共e i关␻␰+␥共␻兲兴_{+ e}−i关␻␰+␥共␻兲兴_{兲, 共2.87兲}

where␥共␻兲 is a real constant. This choice of normaliza-tion implies关see, e.g.,Fulling共1989兲兴

冕

−⬁ ⬁ d␰g_␻k ⬜ * _共␰兲g ␻_⬘k_⬜共␰兲 =␦共␻−␻

⬘

兲. 共2.88兲

We present the derivation of this formula in the Appen-dix for completeness. As a result we have

共v_␻kR_⬜ ,v_␻⬘k ⬜ ⬘ R _兲 KG=␦共␻−␻

⬘

兲␦2共k⬜− k⬜

⬘

兲, 共2.89兲 共v_␻k ⬜ R* _,v ␻_⬘k_⬜⬘ R _兲 KG= 0. 共2.90兲

The Klein-Gordon inner product here is defined taking the hypersurface⌺ in Eq. 共2.5兲 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 共and on the plane t=z=0 for definiteness兲. The functions g_␻k

⬜共␰兲 satisfying the differential equation 共2.86兲 and normalization condition 共2.87兲 are

g_␻k ⬜共␰兲 =

冋

2␻sinh共␲␻/a兲 ␲2_a

册

1/2 Ki␻/a

冉

␬ ae a␰

冊

_共2.91兲 with␬⬅共k_⬜2 + m2_兲1/2_{, where K}

␯共x兲 is the modified Bessel

function共Gradshteyn and Ryzhik, 1980兲. Hence v_␻k ⬜ R =

冋

sinh共␲␻/a兲 4␲4a

册

1/2 Ki␻/a

冉

␬ ae a␰

冊

_eik_⬜·x_⬜−i␻␶_{. 共2.92兲}

We present the derivation of this result in the Appendix as well. Thus we can expand the field ⌽ˆ in the right Rindler wedge as ⌽ˆ共␶,␰,x_⬜兲 =

冕

−⬁ ⬁ d␻

冕

d2k_⬜共aˆ_␻k ⬜ R _v ␻k_⬜ R _{+ aˆ} ␻k_⬜ R† _v ␻k_⬜ R* _兲. 共2.93兲 Then, according to the general results presented in Sec. II.A, we have

关aˆ_␻kR_⬜,aˆ

␻_⬘k_⬜⬘

R†

兴 =␦共␻−␻

⬘

兲␦2共k_⬜− k_⬜

⬘

兲 共2.94兲 with all other commutators among aˆ_␻k

⬜

R _{and aˆ}

␻k_⬜

R†

vanish-ing.

Quantization of the field⌽ˆ in the left Rindler wedge proceeds in exactly the same way. The positive-frequency modes v_␻k

⬜

L _{共␶¯ ,␰¯ ,x}

⬜兲 are obtained from v_␻k

⬜

R _{共␶,␰, x}

⬜兲 simply by replacing ␶ and ␰ by ␶¯ and ␰¯,

respectively. The coefficient operators aˆ_␻k ⬜

L _{and aˆ}

␻k_⬜

L†

sat-isfy the commutation relations 关aˆ␻kL_⬜,aˆ␻_⬘k_⬜⬘

L†

兴 =␦共␻−␻

⬘

兲␦2共k_⬜− k_⬜

⬘

兲 共2.95兲 with all other commutators vanishing. Thus one can ex-pand the field ⌽ˆ in the left and right Rindler wedges as

⌽ˆ =

冕

0 +⬁ d␻

冕

d2k_⬜关aˆ_␻k ⬜ R v_␻k ⬜ R _共␶ ,␰,x_⬜兲 + aˆ_␻k ⬜ R† v_␻k ⬜ R* _共␶,␰,x ⬜兲 + aˆ␻kL_⬜v␻kL_⬜共␶¯,␰¯,x⬜兲 + aˆ_␻k ⬜ L† _v ␻k_⬜ L* _{共␶¯,␰¯,x} ⬜兲兴. 共2.96兲

The Rindler vacuum state 兩0R典 is defined by requiring

that aˆ_␻k ⬜

R _兩0

R典=aˆ␻kL_⬜兩0R典=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关see Eqs. 共2.112兲–共2.114兲兴. After this extension

we see that Eq. 共2.96兲 gives another valid mode

expan-sion of the field⌽ˆ in Minkowski spacetime.14In particu-lar, in Sec. II.F the two-point function calculated using this expansion in the state兩0M典 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⌽ˆ 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 =

冕

−⬁ ⬁ _dk z

冑

4␲k₀关␣␻kzk⬜ R e−ik0t+ikzz +␤_␻k zk⬜ R eik0t−ikzz兴e ik_⬜·x_⬜ 2␲ , 共2.97兲 14

v_␻k ⬜ L =

冕

−⬁ ⬁ _dk z

冑

4␲k₀关␣␻kzk⬜ L e−ik0t+ikzz +␤_␻k zk⬜ L eik0t−ikzz兴e ik_⬜·x_⬜ 2␲ . 共2.98兲

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共␶,␰兲 and 共t,z兲 given by Eq. 共2.36兲 is the same as that between 共␶¯ ,␰¯兲 and 共t,−z兲

given by Eq.共2.38兲. Hence v_␻k

⬜

L _{is obtained from v}

␻k_⬜

R _by

letting z哫−z. From this observation we find the follow-ing relations: ␣␻k_zk_⬜ L =␣_␻−k zk⬜ R , ␤_␻k zk⬜ L =␤_␻−k zk⬜ R . 共2.99兲

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

共aˆ_␻k ⬜ R − e−␲␻/aaˆ_␻−k ⬜ L† _兲兩0 M典 = 0, 共2.100兲 共aˆ_␻kL_⬜ − e−␲␻/aaˆ_␻−k ⬜ R† _兲兩0 M典 = 0. 共2.101兲

关See the corresponding Eqs. 共2.66兲 and 共2.67兲 in the

two-dimensional model.兴 These relations in turn will result if the following modes are purely positive frequency in Minkowski spacetime: w−␻k_⬜⬅ v_␻k ⬜ R _{+ e}−␲␻/a_v ␻−k_⬜ L*

冑

1 − e−2␲␻/a , 共2.102兲 w_+␻k ⬜⬅ v_␻k ⬜ L _{+ e}−␲␻/a_v ␻−k_⬜ R*

冑

1 − e−2␲␻/a . 共2.103兲

关See the corresponding Eqs. 共2.61兲 and 共2.62兲 in the

two-dimensional model.兴 This fact in turn will follow if

␤_␻kR _zk_⬜= − e−␲␻/a␣_␻k zk⬜ L* _{, ␤} ␻kzk⬜ L _{= − e}−␲␻/a_␣ ␻kzk⬜ R* _. 共2.104兲 关See the corresponding Eq. 共2.60兲.兴 We show Eq. 共2.104兲

by explicit evaluation of the Bogoliubov coefficients, which were originally computed byFulling共1973兲.

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 _→

冕

−⬁ ⬁ _dk z

冑

4␲k0 关␣_␻kR_zk_⬜e−i共k0−kz兲V/2 +␤_␻k zk⬜ R _ei共k₀−k_z兲V/2_兴e ik_⬜·x_⬜ 2␲ . 共2.105兲

On the other hand, using the small-argument approxi-mation共A10兲 for the modified Bessel function, we have

for␰→−⬁

v_␻k

⬜

R _→ i

4␲关a sinh共␲␻/a兲兴

−1/2_eik_⬜·x_⬜

⫻

冉

共␬/2a兲

i␻/a_e−i␻u

⌫共1 + i␻/a兲 −

共␬/2a兲−i␻/a_e−i␻v

⌫共1 − i␻/a兲

冊

,

共2.106兲 where␬=共k_⬜2+ m2_兲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 ␣_␻k

zk⬜

R _{is obtained by multiplying}

Eq.共2.106兲 by ei共k0−kz兲V/2_{and integrating over V as}

␣␻kR_zk_⬜= −

i共␬/2a兲−i␻/a共k0− kz兲

4

冑

␲ak0sinh共␲␻/a兲⌫共1 − i␻/a兲

⫻

冕

0 ⬁ dV共aV兲−i␻/aei共k0−kz兲V/2 = e ␲␻/2a

冑

4␲k0a sinh共␲␻/a兲

冉

k0+ kz k0− kz

冊

−i␻/2a , 共2.107兲 where␬=

冑

共k₀− k_z兲共k₀+ k_z兲. Note that we have implicitly chosen a particular共and natural兲 extension of the modes

v_␻k

⬜

R

to the whole of Minkowski spacetime.关Otherwise it should not be possible to find the coefficients Bogoliu-bov coefficients␣_␻k

zk⬜

R _{and ␤}

␻k_zk_⬜

R _{in Eq.共2.97兲.兴 In}

par-ticular, we have excluded any delta-function contribu-tion at V = 0.

By multiplying Eq.共2.106兲 by e−i共k0−kz兲V/2_and

integrat-ing over V we find

␤_␻kR_zk_⬜= − e−␲␻/2a

冑

4␲k₀a sinh共␲␻/a兲

冉

k₀+ k_z k₀− k_z

冊

−i␻/2a . 共2.108兲 Introducing the rapidity␽共kz兲 defined as

␽共kz兲 ⬅ 1 2 ln

冉

k0+ kz k0− kz

冊

, 共2.109兲

and using Eq.共2.99兲, we have

␣␻k_zk_⬜ R =␣_␻−k zk⬜ L = e −i␽共kz兲␻/a

冑

2␲k₀a共1 − e−2␲␻/a兲, 共2.110兲 ␤␻k_zk_⬜ R =␤_␻−k zk⬜ L = − e

−␲␻/a_e−i␽共kz兲␻/a

冑

2␲k₀a共1 − e−2␲␻/a兲. 共2.111兲

Hence Eq.共2.104兲 is satisfied and as a result the vacuum

state兩0_M典 restricted to the left 共or right兲 Rindler wedge is a thermal state with temperature T = a / 2␲ with the boost generator normalized on t2_{− z}2_{= 1 / a}2 _{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.共2.102兲 and 共2.103兲 are