Effect of acceleration on quantum systems

Transcription

1 University of Naples Federico II Faculty of Mathematical, Physical and Natural Sciences Master studies in Physics Degree Thesis Effect of acceleration on quantum systems Academic Year Supervisor Prof. Rodolfo Figari Candidate Nicola Vona matr. 358/47

2 Contents Notation Introduction III IV 1 First evidence Uniformly accelerated observer Time dependent Doppler effect Field consequences Conclusion Quantum Fields and General Relativity Quantum Theory of Scalar Field in Minkowski Spacetime Quantum Theory of Scalar Field in Curved Spacetime The Unruh effect Minkowsi and Rindler particles Bogolubov transformations Thermalization theorem Physical interpretation Hawking effect Dynamical Casimir effect Moving box Particles in a asymptotically stationary spacetime Wave equation for the field in the box Three remarkable cases Null acceleration Instantaneous acceleration I

3 CONTENTS II Exponential acceleration Two non relativistic models Harmonic oscillator Moving delta Conclusions Conclusions 74

4 Notation The spacetime metric signature is n 2, where n is the spacetime dimension. The metric determinant is indicated as g. For tensors on spacetime is used abstract index notation, as in [20]. These indexes are identified with first letters of latin alphabet. Components in a particular frame are identified with greek letters; if spatial and temporal components are distinguished, the first ones are identified with central letters of latin alphabet and the second one with zero. Units in which c = ħ = G = 1 are used, except that in the first and last chapter. III

5 Introduction Since the overtaking of Aristotelian theory, contrast between corpuscular and wave theory of light and matter has animated scientific debate. Newton and Huygens proposed for light the two descriptions in the XVII century; wave theory was established after Young diffraction experiment 1801 and was confirmed by Maxwell electromagnetism theory The photoelectric effect 1900 proposed again the problem, suggesting that the combination of the two theories was necessary. For the matter the same contradiction came out at the early 900, when some experiments seemed to support the wave theory ex. Davisson and Germer experiment 1927, while some others the corpuscular one ex. Wilson s observations with cloud chamber The Quantum Mechanics development solved for matter this paradoxical situation, describing the two aspects in a unified way. In particular Quantum Mechanics describes every system with a classical field that is propagated as a wave. Measurement previsions are evaluated from the field, so they will always have wave features. Applying the theory to a system made by a particle you have a description of this particle with both corpuscular and wave features. Quantum Mechanics, in his first formulation, has two main problems. The first one is that it is not relativistic. Dirac dealt with this aspect in the 20s. He tried to include in the quantum equation of evolution the relativistic relation between energy and momentum. The resulting equation gives previsions in great agreement with experimental data, but it leads to some insurmountable contradictions, ex. non-positive probability distributions. This theory can not be therefore interpreted as a quantum theory of relativistic particles. The second problem is that Quantum Mechanics describes light with classical electromagnetic field, that has wave features, IV

6 INTRODUCTION V but not corpuscular ones. These two problems are solved by Quantum Field Theory, developed in the 50s. Quantum Mechanics describes directly corpuscular features of microscopic systems. On the contrary in Quantum Field Theory there is not an obvious link between field and particles. This link must hence be defined separately, by a paradigm that specifies how extract information on particles from the field. In the common formulation of the theory you solve the field equation in Fourier representation and quantize the resulting system of harmonic oscillators. So you are naturally taken to identify particles with quanta of oscillation normal modes. This interpretation is supported by the fact that states with defined quanta number have defined energy proportional to the number and by the fact that you can construct a Fock space for field states see chapter 2. Fourier transform creates a link between particles and plane waves, that are positive frequency solutions of the field equation with respect to Minkowski time. Plane waves viewed by a different inertial reference frame have a different frequency because of Doppler shift, but normal modes number is the same, so particle content of a state is the same for each inertial observer. The particle notion is therefore the same in each inertial frame, according to special relativity. One plane wave viewed by a non-inertial reference frame, such as an uniformly accelerated frame, becomes a superposition of plane waves because of time dependent Doppler shift. So passing from an inertial frame to a non-inertial frame the number of normal modes changes. This observation suggests that particle content of a field state is observer dependent. This result was obtained exactly by Fulling in 1973 [6]. He demonstrated that positive frequency functions with respect to Minkowski time defines an acceptable particle definition because Minkowski spacetime is symmetric with respect to temporal translation. Therefore you have an acceptable particle definition for each set of functions that are positive frequency with respect to a time τ if the spacetime is symmetric with respect to translations in the τ direction. Two particle definitions corresponding to two different temporal symmetries are in general different; this means that the vacuum state of one definition is not empty with respect to the other definition. In Fulling s work alternative particle definitions are considered only in a

7 INTRODUCTION VI mathematical point of view, but in 1976 Unruh demonstrated that a particle detector moving along a trajectory with the proper time τ detects τ-particles instead of Minkowski-particles [18]. This shows that Fulling particles are not only a mathematical structure, but they are physically real. In the same work Unruh considers the particular case of Minkowski spacetime seen by an uniformly accelerated observer Rindler spacetime. He showed that Minkowski vacuum, written in terms of accelerated particles, corresponds to a thermal state with temperature proportional to acceleration. Usually this situation is described as particle creation from vacuum due to observer s acceleration Unruh effect. The mentioned works are inserted in a wider sight, known as Quantum Field Theory in Curved Spacetime. This is the study of quantum fields when is present a gravitational field described by a curved spacetime, as in general relativity. In this sight the non-inertiality of a reference frame is expressed by an apparent gravitational field, that is a deviation of the metric from the Minkowski one. Studying the quantum field in a non-inertial frame is therefore equivalent to studying it when is present a fixed gravitational field. Historically the effect of acceleration on a quantum system was studied only in Quantum Field Theory because it involves particle creation. Nevertheless you can t think that acceleration has consequences only in relativistic conditions, on the contrary you can expect an effect similar to Unruh effect also in Non-Relativistic Quantum Mechanics. To see this effect you have to define an interpretative picture that accounts for particle creation. For example you can consider a system with a non-interacting particle sea in a bound state and you can interpret this state as the vacuum state. Particle creation is now the ionization of this state. In particular you can consider a system for which only the fundamental state is a bound state, while all the excited states are scattering states. In this case particle creation corresponds to spatial deconfinement. In this picture to the Unruh effect corresponds the system ionization due to acceleration. The effect of acceleration in Quantum Field Theory is presented in this thesis. The same effect is considered in non relativistic conditions too, using two explicit models. The thesis is structured as follows: Chapter 1 A simplified derivation of the Unruh effect is presented. This derivation is based on time dependent Doppler effect and is not rigor-

8 INTRODUCTION VII ous, but is useful to underline the physical origin of the phenomenon. Chapter 2 Quantum Field Theory in Curved Spacetime is presented as an extension of Quantum Field Theory in Minkowski spacetime. This theory provides the methods necessary to study the Unruh effect in a rigorous way. Chapter 3 The theory established in chapter 2 is applied to the case of Minkowski spacetime, finding the connection between the particle definition of an inertial observer and of an accelerated observer. The vacuum state of the inertial observer is then expressed as thermic state of the accelerated observer. Finally the analogy between Rindler and Schwarzschild metrics is used to present an analogy between Unruh effect and Hawking effect black holes evaporation. Chapter 4 A particular case of dynamical Casimir effect is addressed. In this case the field is confined in a box that undergoes a phase of acceleration. This configuration appears more realistic than the configuration considered in the Unruh effect, moreover is similar to the scattering problem, widely studied in physics. Chapter 5 Two non-relativistic models are proposed to study the effect of the acceleration in these conditions. The first one is solved analytically, the second one numerically.

9 Chapter 1 A first evidence of the thermal effect of acceleration Quantum Field Theory is developed on the basis of special relativity, so is expressed from the point of view of an inertial observer. How does the theory change for an accelerated observer? To answer this question Quantum Field Theory in Curved Spacetime is necessary, but the development of this theory is quite difficult. In order to achieve an intuitive idea of the effect of acceleration, in this chapter a simplified analysis of the problem is presented [1]. This analysis is not rigorous, but is explanatory of the physical origin of the phenomenon. 1.1 Uniformly accelerated observer Consider a flat spacetime with only one spatial dimension and an observer with constant speed in this spacetime. We will call this observer inertial or Minkowskian or M and we will denote his coordinates with {t, z}. Consider another observer, called of Rindler or R, uniformly accelerated along the positive z axis of the inertial observer. Uniformly accelerated means that the observer has the same acceleration at every time with respect to the reference frame in which the observer is at rest in that moment. In this frame the acceleration is a = dv/dt > 0, while the acceleration in the Minkowskian frame is given be the Lorentz transformation: dv 1 dt = a v2 c 2 1 3/2

10 CHAPTER 1. FIRST EVIDENCE 2 In order to obtain the speed of the R observer in the M frame you should integrate this equation with vt = 0 = 0, but it is simpler to evaluate it in terms of the proper time τ: dv dt = dv dτ dτ dt = dv 1/2 1 v2 dτ c 2 dv 1 dτ = a v2 c 2 aτ vτ = c tanh c where it was used the relation dt = dτ/ 1 v 2 /c 2. The R observer trajectory in the M frame is given by the integral of the equation: dz dt = vt but it is still simpler to consider this trajectory in terms of the proper time τ, given by: 1/2 dt dτ = 1 v2 τ dz dτ c 2 dτ dt = vtτ therefore dt dτ = cosh aτ c dz dτ = c sinh aτ c tτ = τ 0 zτ = c aτ cosh c τ 0 dτ = c aτ a sinh c aτ sinh c dτ = c2 a cosh aτ c Finally the trajectory of the R reference origin in the Minkowski frame, parametrized with the proper time, is: tτ = c aτ a sinh c zτ = c2 aτ 1.1 a cosh c

11 CHAPTER 1. FIRST EVIDENCE Time dependent Doppler effect Consider a plane wave in the M frame, with wave vector k// e z and frequency ω k = kc: At, z = A 0 e iϕ ±t,z with ϕ ± t, z = kz ± ω k t The observer in the origin of the M reference frame sees the wave At = A 0 e ±iω kt, while the observer in the origin of the R frame moves along the trajectory 1.1 and sees the wave: [ c Aτ = A tτ, zτ = A 0 exp iω k cosh aτ a c ± sinh aτ c ] so [ Aτ = A 0 exp ±i ω kc aτ e± c a Therefore the R observer doesn t see a plane wave, but a superposition of plane waves time dependent Doppler effect: ] Aτ = dω ÃΩ e iωτ where Ω is the frequency of R frame plane waves and for waves moving toward z ÃΩ = 1 2π dτ A 0 exp Intensity of each plane wave seen by R is ÃΩ 2 = A2 0 2π 2 [ i c a ω k e a c τ ] e iωτ dτ exp [i c a ω k e a c τ ] 2 e iωτ Introducing the variable y = e aτ /c in the integral we have dτ exp [ i c a ω k e a c τ ] e iωτ = = c a 0 0 e i c a ω k y y i c a Ω c a y 1 dy = cos c a ω k y + i sin c a ω k y y i c a Ω 1 dy This integral converges only for 0 < Re i c a Ω < 1, but can be regularized considering Ω Ω i a c ε, with 0 < ε < 1, and taking the limit for ε 0.

12 CHAPTER 1. FIRST EVIDENCE 4 In such way we have dτ exp Using the relation Γix 2 = [ i c a ω k e a c τ ] e iωτ = c c a a ω i c a k Ω e π 2 π x sinhπx we obtain c a Ω Γ i c a Ω 1.2 ÃΩ 2 = ca2 0 2πaΩ 1 e 2π c a Ω 1 Time dependent Doppler effect therefore results in the Planck factor e ħω k B T 1 1, typical of a Bose-Einstein distribution with temperature T = ħa 2πk B c, called Hawking-Unruh temperature. Note that this temperature is very small for experimental practicable accelerations: substituting the constants with their MKS numerical values we have T = s2 m K a. 1.3 Field consequences In the previous section we studied the case of a single plane wave i.e. a single frequency, finding that in an accelerated frame it becomes a superposition of plane waves. When you quantize the scalar field you identify plane waves with single particle states, therefore Doppler effect turns a single particle state of the inertial frame into a superposition of single particle states of the accelerated frame. Consider a massless real scalar field, in one dimension z, quantized in the whole space: φt, z = ħc dk 2 2πω k a k e iωkt + a k eiω kt e ikz with ω k = kc The excitation energy operator of the field is W = H E 0 = ħ 2π dk ω k N k 1.3 where N k = a k a k is the field quantum number with momentum k operator and E 0 is the energy mean value on the vacuum state.

13 CHAPTER 1. FIRST EVIDENCE 5 The inertial observer in the origin sees the field φt = φt, z = 0 = dk ħc 2 2πω k a k e iω kt + a k eiω kt The accelerated observer sees the field φτ = φtτ, zτ, obtained substituting 1.1 page 2 in 1.3: φτ = dk ħc 2 2πω k a k exp [i c a ω kε k e ε k aτ c ] + +a k exp [ i c a ω kε k e ε k aτ c ] with ε k = k k = sign k. The number operator N p relative to this observer is defined by N p = b pb p, where b p operators come from the expansion of the field φτ with respect to the plane waves of the accelerated frame: φτ = dp ħc 2 2πΩ p b p e iωpτ + b p eiωpτ In order to find b p operators we consider the Fourier transform of the field φτ: then φτ = gω = 1 2π dω gω e iωτ = b p = 0 dτ φτ e iωτ dω gω e iωτ + g Ω e iωτ πω p 2ħ gω p Suppose that the field is in the vacuum state of the inertial observer, denoted with 0 M. The mean value of N p, quantum number with momentum k operator, in the accelerated frame is: M 0 N p 0 M = = Ωp 8πħ = M 0 b pb p 0 M = πωp 2ħ 0 M g Ω p gω p 0 M = dτ dτ e iωpτ e iωpτ M 0 φ τφτ 0 M = = dk c2 8π Ω p ω k I k Ω p 2

14 CHAPTER 1. FIRST EVIDENCE 6 with I k Ω = dτ e iωτ exp [ i c ] a ω kε k e ε k aτ c = = c a e π 2 c a Ω c a ω c iεk a k Ω Γ iε k c a Ω where we considered that just the term proportional to M 0 a k a k 0 M = δ kk gives a contribution and solved the integral as in 1.2. So we have M 0 N p 0 M = c4 c 8πa 2 e π a Ωp dk Ω p ω k Γ c iεk a Ω p 2 By the property Γ z = Γz, from which it follows that Γi x 2 = Γ i x 2, we have M 0 N p 0 M = c3 c 4πa 2 e π a Ωp Ω p Γ i c a Ω p 2 0 dω k 1 ω k The integral with respect to ω k diverges, so the accelerated observer sees an infinite number of quanta with momentum p in the field state 0 M. Therefore let us calculate the fraction of excitation energy relative to the momentum p on the total excitation energy: e p = W p W = 0 N M p 0 M Ω p 0 N M p 0 M Ω p dp/2π = 1 c 2π Z e π a Ωp Ω 2 p Γ i c a Ω p 2 where Z = 2 c By the property Γix 2 = 0 dω e π c a Ω Ω 2 Γ i c a Ω 2 π x sinhπx we have Z = 4πa c 2 0 dω Ω e 2π c a Ω 1 = πa3 6c 4 e p = 1 Z Ω p 1 e 2π c a Ωp 1 1.4

15 CHAPTER 1. FIRST EVIDENCE Conclusion In this chapter it has been showed that plane waves of the inertial frame seen by an accelerated observer become superpositions of plane waves by virtue of time dependent Doppler effect. In quantum theory this means that single particle states of the two observers are different. In particular inertial vacuum state of the field for the accelerated observer is full of particles with the Bose-Einstein energy distribution 1.4, that has temperature T = ħa 2πk B c, called Hawking-Unruh temperature. Proportionality between particle number spectrum and Bose-Einstein distribution is rigorously established by the so-called thermalization theorem, that we will present in chapter 3.

16 Chapter 2 Quantum Fields and General Relativity The previous chapter shows that particle content of a field state is observer dependent. In order to fully understand this statement you have to analyze the construction of Quantum Field Theory in an arbitrary spacetime and therefore also with respect to an arbitrary observer. In this chapter we introduce the fundamental ideas of the Quantum Field Theory in a Curved Spacetime [3, 21]. This theory studies quantum fields propagating in a classical gravitational field and has been developed to describe that phenomena for which both quantum and relativistic aspects are important, but for which the quantum nature of gravity is unessential and therefore negligible. In this case gravitation can be described by a classical curved spacetime, as prescribed by general relativity. To study the Unruh effect we will consider only flat spacetime, but it is convenient that we start from the general case, coming back to the flat case in a second moment. 2.1 Quantum Theory of Scalar Field in Minkowski Spacetime In this section we review the Quantum Field Theory in Minkowski Spacetime pointing out its main steps. In this way we set up a scheme useful to generalize the theory. 8

17 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 9 Classical Klein-Gordon equation Consider M, a n-dimensional Minkowski spacetime with metric η µν with positive signature and denote with x = t, x = x 0, x its points. Consider a scalar field φ : M R, satisfying the Klein-Gordon equation 1 m 2 φx = where η µν µ ν i 2 i. This equation cam be obtained from the lagrangian density L x = 1 2 η µν µ φ ν φ + m 2 φ 2 = 1 2 φ2 φ 2 m 2 φ with φ = t φ, considering the action S = M L x d n x 2.3 and demanding that δs = 0 for field variations null at initial and final instant. We can consider the hamiltonian density H, that is the Legendre transform of L : H = π φ L where π is the canonical momentum density. π = L φ The system energy corresponds to the hamiltonian operator H = H x d n 1 x In our case π = φ H = 1 2 φ2 + φ 2 + m 2 φ Note that hamiltonian formalism distinguishes space and time, breaking the theory covariance. This means that with respect to an other reference frame π and H are different. Equation 2.1 is linear, so its solution space is a vector space. Con- 1 If you consider the whole space then the boundary condition is φ 0 sufficiently x rapidly to have finite 2.5 norm; if you consider a box then the boundary condition is periodic.

18 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 10 sidering complex solutions too we are able to write down complete sets of solutions. Define on this space the scalar product where φ 1, φ 2 = φ 1 x i t φ 2x d n 1 x 2.5 t φ 1 x i t φ 2x φ 1 x [ i t φ 2x ] [ i t φ 1 x ] φ 2 x and the integral is evaluated on the spacelike hyperplane of equation x 0 = t, with fixed t. This product is independent of t choice 2 and has the properties: φ 1, φ 2 = φ 2, φ 1 αφ 1, φ 2 = αφ 1, φ 2 φ 1, αφ 2 = α φ 1, φ 2 φ 1, φ 2 = φ 2, φ 1 φ 1, φ 2 = φ 2, φ 1 φ 1, φ 2 = φ 2, φ 1 = φ 1, φ 2 φ, φ = 0 Consider the set of solutions such as 0 < φ, φ <. On this set the product 2.5 defines a norm. Consider the Hilbert space H obtained by the completion of this set with respect to the norm only just defined. We call orthonormal basis on this space a set of solutions {u k } k R n 1 {u k } k is a complete set of H even in generalized sense such that u k, u k = δ n 1 k k, therefore u k, u k = δn 1 k k u k, u k = 0 k, k The solutions u k are also called positive frequency modes with respect to the scalar product 2.5. You can expand each solution ψ with respect to an orthonormal basis 2 Denote with Σ t the hyperplane of equation x 0 = t and with n µ its future oriented normal versor that is parallel to x 0 axis. Therefore t = n µ µ and φ 1, φ 2 t = R φ Σ t 1x i µφ 2x dσ µ. Denote with V the volume between the two hyperplanes Σ t and Σ t, so φ1, φ2 t φ 1, φ 2 t = R V φ1x i µφ 2x dσ µ = R V µ [φ 1x i µφ 2x] d n x = 0 where we used Gauss theorem and Klein-Gordon equation if the field is confined into a box then V includes also a timelike surface Σ l that has null-contribution to the integral by virtue of the bounding conditions.

19 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 11 with the expression ψx = d n 1 k a k u k x + b k u k x In particular, for a real solution φ a k = ψ, u k b k = ψ, u k φx = d n 1 k a k u k x + a k u k x One orthonormal basis is constituted by plane waves: 1 u k x = 2ω k 2π n 1 ei k x iω k t where ω k = k 2 + m Canonical Quantization Canonical quantization is performed substituting the classical solution φ with an operator on the Hilbert space H, still denoted with φ, and imposing the equal time canonical commutation relations: [φt, x, φt, x ] = 0 [πt, x, πt, x ] = 0 [φt, x, πt, x ] = iδ n 1 x x 2.7 These commutation relations can be satisfied only if H has infinite dimensions. 3 We use Heisenberg representation, so a system state is described by a vector in H. All physical quantities of the system are represented by self-adjoint operators on H and evolve according to the Heisenberg equation Ȯ = i [H, O] where O is a self-adjoint operator that doesn t explicitly depend on time. Particles So far the theory development has not involved any particle concept, that therefore is not a fundamental block of the theory, but rather a major interpretative key. To understand its importance is sufficient to consider that 3 If H has finite dimensions, therefore for each pair of operators Tr AB = Tr BA, so Tr [A, B] = 0, that is not compatible with 2.7.

20 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 12 experiments regarding Quantum Field Theory are usually called Particle Physics! Let us introduce this important concept. Expand the field φ with respect to plane waves 2.6: φx = d n 1 k a k u k x + a u k k x 2.8 Substituting this expansion into 2.7 we have [a k, a k ] = 0 [a k, a k ] = 0 [a k, a k ] = δn 1 k k 2.9 These operators can be used to write H as a Fock space F. Consider the state 0, called vacuum, defined by the relation a k 0 = 0 k From vacuum we can construct the single particle states 1 k = a k 0. Similarly we can define the n k -particles states n k = a k n k n k! 0 where the multiplicative factor corresponds to the normalization n k n k = δ n 1 k k 2.10 For this states this relations are valid: a n k k = n + 1 n + 1 k a k n k = n n 1 k Finally many particle states are n k1, n k2,..., n kj = a k1 n k1 a k2 n k2 a kj n kj n k1! n k2! n kj! 1/2 0 These vectors are a basis for H and are eigenstates for the number operators,

21 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 13 defined by N k = a a k k N = d n 1 k N k in fact N k n k1, n k2,..., n kj = n ki δ n 1 k k i n k1, n k2,..., n kj i N n k1, n k2,..., n kj = n ki n k1, n k2,..., n kj i Substituting the expansion 2.8 into 2.4 we find the energy expression in terms of operators a k : H = d n 1 k ω k 2 a k a k + a k a k = = d n 1 k N k δn 1 0 ω k 2.11 Therefore [ H, N k ] = 0 [ H, N ] = 0 so states with defined number are energy eigenstates. Increasing n k by one the energy of the state increases by ω k, therefore we interpret n k as number of field quanta contained in the state and the operators a and a k k respectively as creation and annihilation operators of one field quantum. The functions u k, used to define particles, are called single particle solutions. Energy regularization From 2.11 we can see that the energy of any state with defined number is infinite, because of the divergent factor proportional to identity. This is the first of the large set of infinite quantities that appears in Quantum Field Theory. These quantities need to be regularized to gain physical significance. In our case regularization is very simple and takes advantage of the fact that energy is not absolutely measurable: what we measure are energy differences. Mean energy of vacuum state is E 0 = 0 H 0 = δ n 1 0 d n 1 k ω k 2

22 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 14 This quantity diverges because of two reasons: the factor δ n 1 0 and the integral of ω k. The first comes from quantization in the whole space and is related to infiniteness of quantization volume. Indeed vacuum energy per unity of volume is [14] E 0 ε 0 = lim V V = d n 1 k ω k 2 The second reason of divergence is more relevant: it is the sum of zero point energy of each normal mode. Regularization is made considering only excitation energy W, that is energy difference respect to vacuum energy: W = H 0 H 0 = d n 1 k N k ω k In a more formal way W is defined by normal ordering: : a k a k := a k a k : a k a k := a k a k W =: H : Summary Elements used to develop quantum theory of scalar field in Minkowski spacetime and its particle interpretation are: 1. Classical field dynamics: S and L 2. Space and time division: π and H 3. Time independent scalar product on solution space 4. Quantization: canonical commutation relations 5. Single particle solutions: orthonormal basis of classical solutions that define quantum states with defined energy 6. Energy regularization method

23 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY Quantum Theory of Scalar Field in Curved Spacetime In this section the construction of Quantum Field Theory in Curved Spacetime is presented following the scheme drawn for quantization in Minkowski spacetime. We will see that there are some problems with the particle interpretation of the theory because particle concept is not always well-defined. Spacetime structure Let us suppose that spacetime: 4 is a differentiable metric manifold M with n dimensions and lorentzian metric g ab. In this case you can identify three families of curves: timelike possible observer trajectories, light-like light rays and spacelike linking events without any causal relation; on the contrary curves that are not spacelike are called causal curves is such that classical Klein-Gordon equation has a unique solution if you assign initial condition on a suitable region is such that you can divide space and time in at least one manner necessary to hamiltonian quantization Let us inspect the last two hypotheses, that are connected. Firstly consider some definitions. A spacetime is said to be time orientable if for each event you can choice which half of light cone is the future and which is the past in a continuous way through the spacetime. A region Σ M is said to be achronal if its points are not connectable by a timelike curve see fig. 2.1a; we define domain of dependence of Σ the set DΣ of points p of M such that every inextendible causal curve through p intersects Σ see fig. 2.1b. If DΣ = M, then Σ is said to be a Cauchy surface for M. It is possible to show that a Cauchy surface is a n 1 dimensional spacelike surface. 5 A time orientable spacetime that admits a Cauchy surface is said to be globally hyperbolic. The following theorem holds: 4 For a construction of the theory under less strict conditions see for ex. [2, 13] 5 For a definition of extendibility of causal curves and a general treatment of causal structure of spacetime see [20]

24 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 16 a b Figure 2.1. a Causal cones of points belonging to an achronal surface don t intersect the same surface elsewhere. b All causal curves through points belonging to the domain of dependence of Σ intersect Σ. Theorem [8, 5] If M, g ab is globally hyperbolic, then exists a function F : M R such that its level hypersurfaces Σ t = {x M : F x = t} are Cauchy surfaces and M = t Σ t. So if spacetime is globally hyperbolic then you can divide space and time, and therefore use hamiltonian formalism. Furthermore in this case causal curves intersect each surface Σ t in one point, so you can reasonably expect that classical dynamics is well defined if you assign initial data on surface Σ t0. This is confirmed by: Theorem [11] If M, g ab is globally hyperbolic and you assign the functions φ 0 and φ 0 on a Cauchy surface Σ t0, then there exists a unique solution φ to Klein-Gordon covariant equation 2.14, defined on all of M and such that φ Σt0 = φ 0 and n a a φ Σt0 = φ 0, where n a is the future oriented versor normal to Σ t0. The solution φ varies continuously with the initial data. Moreover if you assign initial data only on a subset σ Σ t0, then there exists a unique solution φ restricted to the domain of dependence of σ that depends only on initial data on σ. An analogous theorem holds for Green equation related to Klein-Gordon covariant equation. In the following we will always consider globally hyperbolic spacetimes. Let us set a last hypothesis: metric is assigned and not dependent on field, therefore we are studying scalar field dynamics in a gravitational frame,

25 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 17 disregarding the influence of scalar field on gravitational field. This influence is very important in astrophysics [21] and is related to quantization in general relativity; we will return on this aspect later see page 22. Klein-Gordon covariant equation Let us extend the action to a generic spacetime. Properties requested to action S = M L x dn x are: S is a real functional and has at least one extreme so dynamics can come from action principle L is a local function L depends only on the field and its first derivatives so dynamics is determined by initial value of the field and its first derivatives If spacetime is flat, then S is invariant under Poincarè transformations S is covariant, that is not depending on coordinate system so field theory is compatible with general relativity S has the same symmetries of field From this requirements you can extract, for the scalar field, a prescription [14] that allows to achieve a covariant action S from a Minkowski spacetime action S 0, such that S = S 0 if g ab = η ab : Substitute g ab to η ab Substitute covariant derivative µ to partial derivatives µ when applied to scalar field they are equal Substitute invariant volume element gd n x to volume element dtd n 1 x Add a term proportional to curvature and φ 2 Applying this procedure to action we have: S = 1 2 M g ab a φ b φ + m 2 φ 2 + ζrφ 2 g d n x 2.12

26 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 18 where R is the scalar curvature and ζ is the coupling constant between field and curvature. The most common values of ζ are ζ = 0, called minimal coupling, and ζ = 1 n 2 4 n 1, called conformal coupling. In the conformal coupling case, if mass is zero, S is invariant under conformal transformations [7]. Through the expression S = M L x dn x we define the lagrangian density g L = 2 from which we have the equation of motion g ab a φ b φ + m 2 φ 2 + ζrφ m 2 ζrx φx = where 6 φ = g µν µ ν φ = 1 g µ g g µν ν φ. Note that also in this case equation of motion is linear with respect to φ cf Hamiltonian formalism: space and time division Consider a function F : M R, such that spacetime can be written as a collection of its level hypersurfaces Σ t, with t as parameter, as seen on page 16 in general F choice is not unique. We can associate to this function a vector field of time evolution t a, satisfying the equation t a a F = 1. Introduce a coordinate system t, x i, where tx = F x and x i coordinates on Σ t, such that t a a x i = 0. In this system t a = t a, so φ = 0 φ = t φ = t a a φ. From 2.13 we have the canonical momentum density π = therefore the hamiltonian density is L 0 φ = g g 0ν ν φ = g 0 φ H = π 0 φ L = g g iµ µ φ i φ g 0µ µ φ 0 φ + m 2 φ 2 + ζrφ 2 = 2 g = i φ i φ 0 φ 0 φ + m 2 φ 2 + ζrφ Note that π and H depend on time choice, that is F choice, that in gen- 6 It comes from µω ν = µω ν 1 2 gρσ ω ρ µg νσ + νg µσ σg µν, while µf = µf, with ω covector on M and f function on M.

27 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 19 eral is not unique. Therefore hamiltonian theory depends on chosen observer system. We can make space and time division more explicit by considering apart directions that are orthogonal and tangential to space hypersurfaces Σ t : write t a as t a = t a +ta / = Nna +t a /, where na = a F a F is the future oriented versor normal to Σ t. Denote with h ab the riemannian metric induced by g ab on Σ t, defined as the symmetric tensor that is null on vectors orthogonal to Σ t and works as g ab on tangent vectors: h ab n a = 0 h ab δ a c + n a n c = g ab δ a c + n a n c + sign in projector orthogonal to n a comes from the fact that n a is timelike, so it has negative norm from which Substituting into 2.12 S = 1 2 M h ab = g ab + n a n b [ n a a φ 2 h ab a φ b φ m 2 φ 2 ζrφ 2] h Ndt d n 1 x From t a = Nn a + t a / follows that n a a φ = 1 N φ t a / aφ Therefore lagrangian and canonical momentum densities are L = h 2 Scalar product [ ] 1 N φ t/ a aφ 2 Nh ab a φ b φ + m 2 φ 2 + ζrφ 2 π = L φ = h 1 N φ t a / aφ = h n a a φ 2.16 Let us generalize the scalar product 2.5 with: φ 1, φ 2 = = Σ t Σ t φ 1 x i a φ 2x dσ a = φ 1 x i n a a φ 2x h d n 1 x 2.17

28 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 20 All properties stated for product 2.5 stand also for this one, except for the fact that now plane waves 2.6 are not, in general, solutions. Note that this product takes the same value if you choice differently both t and space and time division. 7 Canonical quantization Canonical quantization is still performed substituting an operator on the Hilbert space H to the classical solution φ, imposing the equal time canonical commutation relations2.7 and using Heisenberg representation. Particles Particle definition that we adopt corresponds to the identification of particles and field quanta. This means that we use as single particle solutions a basis {u k } whose operators a k and a k satisfy commutation relations 2.9. These relations are necessary to write H as a Fock space F. Moreover Fock states must have definite energy. Let us study what conditions come from requests put on solutions {u k }. Consider that {u k } k R 3 φx = is a basis for H, therefore d n 1 k a k u k x + a u k k x 2.18 Define au k = φ, u k 2.19 annihilation operator associated to the classical solution u k [12]. Consider [ au k, au k ] = [ φ, u k, φ, u k ] = [ = φx hx n a x a u x k, Σ t Σ t φx hx n b x b u k ] x d n 1 x d n 1 x 2.20 Scalar product doesn t depend on t, so we can set t = t. Using 2.16 we can recast 2.20 in terms of equal time canonical commutation relations 7 Proof is the same that the one in note 2 on page 10

29 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY , from which follows that [ au k, au k ] = u k x i n a a u k x h d n 1 x = u k, u k Σ t By the substitutions { u k u k, u k u k } { u k u k, u k u k } we obtain also the commutators [ au k, au k ] = [ φ, u k, φ, u k ] = u k, u k [ au k, au k ] = [ φ, u k, φ, u k ] = u k, u k Summarizing, from classical solutions {u k } we have the annihilation operators au k satisfying the commutation relations [ au k, au k ] = u k, u k [ au k, au k ] = u k, u k [ au k, au k ] = u k, u k In order to obtain the commutation relations 2.9, necessary to write H as a Fock space F, we must therefore use as single particle solutions a basis of solutions that are orthonormal with respect to the product In this case a k = au k cf and The second request is that Fock states have defined energy. This means that hamiltonian operator H = Σ t H d n 1 x commutes with number operator N. Substituting the expansion 2.18 into 2.15 we obtain H = d n 1 k d n 1 k [ Re B k k a k a k B k k δ n 1 k k A k k a k a k A k k a k a k ] where A k k = g Σ t g µi i u k µ u k g µ0 0 u k µ u k + m 2 + ζru k u k d n 1 x

30 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 22 B k k = g Σ t g µi i u k µ u k gµ0 0 u k µ u + k m2 + ζru k u k d n 1 x In order to have [H, N ] = 0 it must be A k k = 0. In Minkowski spacetime case we used plane waves as solutions u k. Plane waves are harmonic functions with respect to both time and space. In particular time harmonicity is fundamental to write the hamiltonian as in 2.11, while you can think that spatial dependence is determined by Klein-Gordon equation, as a consequence of time dependance. Similarly we can expect that our requests are satisfied considering orthonormal solutions that are time harmonic. This expectation is validate if spacetime is stationary, i.e. it admits a timelike Killing 8 vector field ξ a. Actually if you use the field ξ a as vector field t a of time evolution, 9 then sufficient condition [21] so that solutions {u k } define single particle states satisfying our requests is that they are eigenfunctions of the operator ξ a a = t a a = t : t u k = iω k u k con ω k 0 This is a sufficient but not necessary condition, therefore in a generic spacetime there isn t any preferred set of solutions u k, so there isn t an obvious particle definition. Only if spacetime is stationary there is a natural particle definition that come from time symmetry. Finally if there are more than one timelike Killing vectors, a different particle notion comes from each of them [6]. To each particle notion corresponds a different vacuum notion. 10 This arbitrariness is the main feature of Quantum Field Theory in Curved Spacetime. Energy regularization Given the basis {u k } of single particle solutions, regularization can be performed as described on page 13. Naturally to different sets of functions u k 8 A vector field ξ a is said to be a Killing vector field for the metric g ab if it satisfies the equation ξ ρ x ρ gµν + ξ ρ gµρ x + ξ ρ ν gρν x = µ With respect to coordinates such that ξ a = x 0 a, that is ξ ρ = δ ρ 0, this equation becomes g x 0 µν = 0. Therefore metric doesn t depend on x 0, so system is symmetric for translation along this coordinate. For further informations see [20]. 9 In this case g ab doesn t depend on time. 10 Since we set t a = ξ a, different particle definitions can be seen as corresponding to different space and time divisions.

31 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 23 correspond different notions of particle and vacuum, and so different regularizations. From the hypothesis that scalar field is not a source for gravitational field follows that only energy differences between different states are observable. On the contrary, to study gravity production by the field you use the semiclassical Einstein equation [21] G ab = 8π T ab where G ab is the Einstein tensor and T ab is the energy-momentum tensor. The tensor T ab is divergent and need to be regularized; the simplest procedure consists of postulating that vacuum state doesn t give rise to gravitational field and subtracting from T ab the mean value on vacuum T ab. Now to different regularizations correspond different produced gravitational fields, with observable effects. Here starts a new chapter of Quantum Field Theory in Curved Spacetime, that we don t discuss and that researches the physical vacuum, that is the vacuum that effectively produce no gravitational field, or researches a covariant regularization [21, 3, 7, 14]. Bogolubov transformations We have seen that particle and vacuum definitions are not unique and depend on the choice of functions u k. relative to two different sets {u k } and {v p } are linked. 11 with respect to the two bases: φx = d n 1 k a k u k x + a u k k x = So we can ask how two definitions Expand the field d n 1 p b p v p x + b p v p x 2.22 We can also expand the functions v p with respect to the u k and vice versa, obtaining the Bogolubov transformations: u k = d n 1 p α k p v p x + β k p v p x v p = d n 1 k α k p u k x β k p u k x These two sets are analogous to plane waves of Minkowski and Rindler considered on page 3 and the transformations that we are looking for are analogous to what considered on section 1.3.

32 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 24 with α k p = u k, v p = v p, u k β k p = u k, v p = v p, u k 2.25 Substituting 2.23 into 2.22 we get b p = d n 1 k α k p a k + β k p a k while substituting 2.24 into 2.22 a k = d n 1 p α k p b p β k p b p furthermore substituting 2.24 into v p, v p = δ n 1 p p and into v p, v p = 0 we have the properties d n 1 k α k p α k p β k p β k p = δ n 1 p p d n 1 k β k p α k p α k p β k p = 0 Operators a k and b p lead to different Fock spaces, with different vacuum states, defined by: a k 0 a = 0 k b p 0 b = 0 p Consider a k 0 b = d n 1 p α k p b p β k p b p 0 b = d n 1 p β k p 1 p b from which 0 N a b k 0 b = d n 1 p β k p If coefficients β k p are not zero, vacuum 0 b, obtained from functions v p, can be not-annihilated by operators a k that correspond to functions u k ; moreover 0 b contains a non-zero number of quanta of modes u k. Similarly b p 0 a = d n 1 k α k p a k + β k p a k 0 a = d n 1 k β k p 1 k a

33 CHAPTER 2. QUANTUM FIELDS AND GENERAL RELATIVITY 25 from which 0 N b a p 0 = d n 1 a k β k p

34 Chapter 3 The Unruh effect We have seen that in a general spacetime particle and vacuum notions are not well defined. Only if spacetime is stationary there is a natural particle definition, that rises from system symmetry. In this chapter we will reconsider the field quantization in Minkowski spacetime in light of the general theory developed in the previous chapter. We will see which particle definitions come from time symmetries of the system and what physical interpretation we can give them. 3.1 Minkowsi and Rindler particles Let us derive a single particle states definition from the fact that spacetime is stationary, using the sufficient condition of page 22. Minkowski particles Consider the s + 1 dimensional Minkowski spacetime and mark with {t, x} a global inertial reference frame, called M. With respect to this frame the metric is g µν = η µν = δ 0 µ δ 0 ν + i δi µ δ i ν. A Killing vector field ξ a satisfies the equation 2.21 ξ ρ x ρ g ξ ρ µν + g µρ x ν + g ρν ξ ρ x µ =

35 CHAPTER 3. THE UNRUH EFFECT 27 that with respect to coordinates M is ξ 0 t = 0 ξ 0 x i = ξi t ξ i x j = ξj x i 3.2 The fields µ a satisfy this system, so they are Killing fields. means that translations are isometries for Minkowski spacetime. This just Among this fields t a is timelike, so it can be employed to define a particle notion. Using frame M we implicitly used t a as vector of time evolution, therefore we have only to find solutions of Klein-Gordon 1 equation that are eigenfunctions of t a. So this functions are such that 2 t + i 2 i m 2 u k = 0 t u k = iω k u k This equations are satisfied by plane waves 2.6, so the prescription for the particle definition in a stationary spacetime applied to Minkowski spacetime corresponds to the ordinary quantization, seen on section 2.1. Field expansion with respect to solutions u k is φx = d s k a k u k x + a u k k x while vacuum is defined by a k 0 M = 0 k 1 In our case curvature is zero, so there isn t the term ζr cf. eq

36 CHAPTER 3. THE UNRUH EFFECT 28 Rindler particles Translations are not the only isometries of Minkowski spacetime, for example Lorentz boosts are isometries too. A boost along the x direction is: t = γ t v x x = γ x v t y = y γ = 1 1 v 2 where, to make notation simpler, first spatial coordinate marked as x is distinguished from the others bold marked. Transformation is simpler if written in terms of rapidity ν { t = t cosh ν x sinh ν x = t sinh ν + x cosh ν with { γ = cosh ν v γ = sinh ν 3.3 For small ν we can truncate the transformation to the first order: { t = t νx x = x νt the infinitesimal generator of this transformation is therefore the vector field b a = α x t a + t x a with constant α. This field satisfies the system 3.2, so it is a Killing field obvious because it generates an isometry. The b a norm is b 2 = b a b a = g µν b µ b ν = b b i 2 = α 2 t 2 x 2 and is timelike if x > t, so in regions I and II of fig In order to construct a particle definition, as we have done with t, we need a timelike field on the whole spacetime. Nevertheless region I or II is globally hyperbolic and, even if not geodetically complete 2, it can be treated 2 A set is said to be geodetically complete if all geodesics passing trough every point of the set are entirely comprised into the set. In general relativity it is understood that we refer to timelike and light-like geodesics, corresponding respectively to free particles and light rays, so completeness means that objects can t leave the spacetime. i

37 CHAPTER 3. THE UNRUH EFFECT 29 t Regione II Regione I x Figure 3.1 as a spacetime is its own right [7]. Therefore we can consider a quantum theory for region I or II alone, using b a as time evolution field b a in II because is future directed. Fock states we will obtain are said Rindler particles. Region I: x 0,, t x, x. Introduce the time function τt, x such that b a a τ = 1 that is x t τ + t x τ = 1 α You can easily verify that this condition is satisfied by the function τt, x = 1 x + t 2α log x t τ, Level hypersurfaces of this function are Cauchy surfaces for the spacetime [21], so we can use τ as time function. The versor orthogonal to these surfaces is n a = ba b a. To complete the coordinate system consider a set of spatial coordinates

38 CHAPTER 3. THE UNRUH EFFECT 30 ξ i t, x ξ ξ, υ such that b a a ξ i = 0 i cioè x t ξ i + t x ξ i = Consider therefore [14] ξt, x = 1 α + x 2 t 2 ξ 1 α, υ = y Coordinates {τ, ξ, υ} are called Rindler coordinates 3 and are the coordinates of a reference composed by observers moving with uniform acceleration α [14]. In this frame line element is ds 2 = dt 2 + d x 2 = αξ dτ 2 + d ξ furthermore b 2 = αξ + 1 2, so b a a = τ n a a = 1 αξ + 1 τ The scalar product 2.17 now is φ 1, φ 2 = τ d s ξ φ1 τ, ξ i αξ + 1 τ φ 2τ, ξ 3.6 In order to get to particle definition that derive from b a, we have to consider the solutions v p I of Klein-Gordon equation that are eigenfunctions of b a a : [ 1 αξ τ + ] α αξ + 1 ξ + ξ y m 2 v p I = 0 b a a v I p = iω vi p that is τ v I p = iω vi p 3 Inverse transformations are cf. 1.1 t = αξ + 1 α sinh ατ x = αξ + 1 α cosh ατ

39 CHAPTER 3. THE UNRUH EFFECT 31 Set 4 v I p τ, ξ = 1 2Ω 2π s 1 e iω τ+ip y ψ I p ξ now p stands for Ω > 0 and p R s 1 so that eigenvalue equation is satisfied, while ψ p I ξ satisfies [αξ d2 dξ 2 + ααξ + 1 d ] dξ αξ + 12 m 2 p + Ω 2 ψ p I ξ = where m 2 p = m 2 + p 2. Imposing on v I p the normalization we find the normalization on ψ I p ξ: v I p, vi p = δs 1 p p δω Ω 1 α ψω,p I ξ ψi Ω,p ξ αξ + 1 dξ = δω Ω 3.8 Introduce the variable ζ = αξ+1 α, so ζ 0, and equations 3.7 and 3.8 become [ ζ 2 d2 dζ 2 + ζ d ] dζ ζ2 m 2 p + Ω2 α 2 0 ˆψ Ω,p I ζ ˆψ I Ω,p ζ αζ ˆψ p I ζ = dζ = δω Ω 3.10 where ˆψ p Iζ = ψi p ξζ. Equation 3.9 is a Bessel equation whose solutions, normalized according to 3.10, are [6] ˆψ I p ζ = 1 π 2Ω α sinh πω α K Ω m i pζ α where K is the modified Bessel function [9]. Coming back to the variable ξ: ψ I p ξ = 1 π 2Ω α sinh πω α K Ω i α m p αξ+1 α It can be proved that the set {v p I, vi p } is complete on region I [17]. 4 Note that v I p is a plane wave with respect to τ, circumstance that justifies what we did in chapter 1.

40 CHAPTER 3. THE UNRUH EFFECT 32 Region II: x, 0, t x, x. In order to have a future-oriented timelike vector consider b a ; time function τt, x now satisfies b a a τ = 1 cioè x t τ + t x τ = 1 α Obviously this condition is satisfied by the function τt, x = 1 x + t 2α log x t Spatial coordinates still satisfy 3.4, but now set ξt, x = 1 α x 2 t 2 τ, ξ, 1 α on region II. Now inverse trans- so that ξ > 1 α on regione I and ξ < 1 α formations are t = αξ + 1 α sinh ατ x = αξ + 1 α cosh ατ We have b a a = τ n a = ba b a n a a = 1 αξ + 1 τ so the scalar product 2.17 is still expressed by 3.6. Line element is still given by 3.5, so Klein-Gordon equation is the same as in region I; consider its solutions v II p that are eigenfunctions of ba a : [ 1 αξ τ + b a a v II p ] α αξ + 1 ξ + ξ y m 2 v p II = 0 = iω vii p that is τ v p II = iω vii p Therefore v II p τ, ξ = 1 2Ω 2π s 1 eiω τ+ip y ψ II p ξ where ψ II p is a solution of 3.7. Normalization v II p, vii p = δs 1 p p δω Ω

41 CHAPTER 3. THE UNRUH EFFECT 33 on ψ II p becomes 0 ˆψ Ω,p II ζ ˆψ II Ω,p ζ αζ dζ = δω Ω so for ˆψ II p ζ equations 3.9 and 3.10 hold, therefore The set {v II p, vii p ˆψ II p ζ = ˆψ I p ζ ˆψ p ζ } is complete on region II [17]. We can compare Minkowski and Rindler particle notions only if they are defined on the same spacetime, so we need some Rindler solutions definined on the whole spacetime. Consider therefore the union of region I and II and define { + 1 on region I λ = 1 on region II Using the vector λb a we define the coordinates 5 τt, x = λ 1 x + t 2α log x t τ, ξt, x = 1 α + λ x 2 t 2 ξ, 3.11 Coordinate curves are shaped as on fig Functions v p,σ τ, ξ = θσλ 1 2Ω 2π s 1 e iσω τ+ip y ˆψ p σζ 3.12 with σ = ±1, λ λξ and ζ = αξ+1 α, are solutions of Klein-Gordon equation and eigenfunctions of λb a. This solutions can be analytically extended to the whole Minkowski spacetime 6 and constitutes a complete set [3]. Functions v p,σ are called Rindler normal modes. 5 ξ 1 α, on region I and ξ, 1 α on region II 6 Using the fact that surfaces with constant τ, which lie in I II, are Cauchy surfaces for the whole spacetime. This means that if you assign the value of solution of Klein-Gordon equation on these surfaces then you fixed its value on the whole spacetime.

42 CHAPTER 3. THE UNRUH EFFECT 34 t = < 0 1 = > 0 0 = x Figure 3.2. Two curves with varying τ and fixed opposite values of ξ are marked in blue; a curve with fixed τ is marked in green. Expand the field with respect to solutions v p,σ : φx = 0 dω d s 1 p b p,σ v p,σ x + b p,σ v p,σ x σ=±1 Field quanta created by operators b p,σ vacuum is defined by b p,σ 0 R = 0 are called Rindler particles. The p, σ 3.2 Bogolubov transformations The relation between the two particle definitions is expressed by Bogolubov coefficients Consider α k p,σ = u k, v p,σ = τ d s ξ i u k x αξ + 1 τ v p,σ x

43 CHAPTER 3. THE UNRUH EFFECT 35 It is convenient to use the surface τ = 0, where t = 0, x = αξ+1 α τ = λb a a = λx t + t x = λ αξ+1 α t; substituting we have and α k p,σ = δs 1 k p 2 2πωΩ where ω ω k. Similarly dζ θσλ e ik 1ζ αζ σω + λωζ ˆψ p σζ 3.13 β k p,σ = v p,σ, u k = δs 1 k + p 2 2πωΩ dζ θσλ eik 1ζ αζ σω λωζ ˆψ p σζ 3.14 The integrals in these expressions fundamentally are Fourier transformations of a Bessel function and are tabled. Note that α k p,σ δ s 1 k p β k p,σ δ s 1 k + p so Bogolubov coefficients are distributions with respect to their indexes. Therefore expressions containing products of Bogolubov coefficients, as 2.26 and 2.27, are not well defined. From this it is possible to show that a unitary transformation that links the vacua of the two observers doesn t exist 7 [17]. This is only an apparent problem, indeed Bogolubov coefficients are distributions because in our case states with defined number are not normalizable cf. 2.10, for example M 0 a k a k 0 M = δ s 0 This states belong to the Fock space F only in generalized sense. This situation is very common in Quantum Field Theory and is usually faced considering a field confined into a finite box and then taking the limit as the box side approaches infinity. In Rindler frame the solution of Klein-Gordon equation in a box is not simple, so we consider wave packets in the whole spacetime. Wave packets are normalizable states so they belong to the Fock space F in proper sense. In order to obtain wave packets, consider the functions f l k, with l Z s, such that f l k f l k d s k = δ l l and l f l k f l k = δ s k k 7 Minkowski and Rindler field representations are unitary inequivalent.

44 CHAPTER 3. THE UNRUH EFFECT 36 These properties allow to derive from a set of functions {u k x} k R s that is complete for F in generalized sense, a set {ǔ l x} l Z s that is complete in proper sense, through the relation ǔ l x = f l k u k x d s k ǔ l, ǔ l = δ l l You can found a possible set of functions f l k in [10]. The field φ can be expanded with respect to the ǔ l, giving φx = ǎ l ǔ l x + ǎ ǔ l x l l ǎ l = f l k a k d s k 3.15 Operators ǎ l satisfy the canonical commutation relations, moreover they define normalizable states, for example 0 ǎ k ǎ k 0 = 1 Denote with ǔ l x wave packets obtained from plane waves 2.6 and with ǎ l the relative annihilation operators. Denote with ˇv m x wave packets obtained from Rindler modes 3.12 and with ˇb m the relative annihilation operators. Bogolubov transformations between these two representation are regular and are linked to the previous by ˇα l m = ǔ l, ˇv m = d s k d s p, σ f l k f m p,σ α k p,σ 3.16 ˇβ l m = ˇv m, ǔ l = d s k d s p, σ f l k f m p,σ β k p,σ where d s p, σ 0 dω d s 1 p σ=±1. Assuming that wave packets are always used, we can write effective versions of 3.13 and 3.14 [17], that are valid only if used through 3.16: α k p,σ α eff k p,σ = 1 2π Ω αω δs 1 k p e πω 2α β k p,σ β eff k p,σ = 1 Ω 2π αω δs 1 k + p e πω 2α Γi Ω α Γi Ω α ω + k1 ω k 1 ω + k1 ω k 1 iσ Ω 2α iσ Ω 2α 3.17

45 CHAPTER 3. THE UNRUH EFFECT 37 Substituting 3.16 and 3.17 into ˇb m = l ˇα l m ǎ l + ˇβ l m ǎ l and using 3.15 we find ˇb m = d s p, σ f m p,σ beff p,σ 3.18 where b eff p,σ = d s k α eff k p,σ a k + β eff k p,σ a k From 3.17 and the relation Γix 2 = where π x sinhπx we have b eff p,σ N = Ω α + 1 d Ω,p,σ + N Ω α d Ω, p, σ 3.19 N Ω α = 1 e 2πΩ α 1 1 ω1 + k P Ω,p,σ k 1 = 1 2πω1 ω 1 k 1 d Ω,p,σ = iσ Ω 2α dk 1 P Ω,p,σ k 1 a k1,p ω 1 Operators d p,σ are linear combinations of operators a k, so k p 2 + m 2 d p,σ 0 M = 0 p, σ from which b eff p,σ 0 M = N Ω α d Ω, p, σ 0 M 3.20 With a direct calculation you can show that operators d p,σ satisfies canonical commutation relations: [d p,σ, d p,σ ] = 0 [d p,σ, d p,σ ] = 0 [d p,σ, d p,σ ] = δ σσ δ s p p 3.21 Let us evaluate the mean value of the number operator of Rindler quanta on Minkowski vacuum with wave packets: M ˇ0 Ň m ˇ0 M = M ˇ0 ˇb mˇb m ˇ0 M The state ˇ0 M is defined by ˇb m ˇ0 M = 0 m

46 CHAPTER 3. THE UNRUH EFFECT 38 that is satisfied by 0 M cf. 3.15, so ˇ0 M = 0 M. 8 From 3.18 it follows that ˇ0 Ň M m ˇ0 M = d s p, σ d s p, σ f m p,σ f m p,σ M 0 b eff p,σ b eff p,σ 0 M From 3.20 and 3.21 we obtain 0 beff M p,σ b eff p,σ 0 M = = N Ω α NΩ α M 0 d Ω, p, σd Ω, p, σ 0 M = = N Ω α δ σσ δs p p In conclusion 0 Ň M m 0 M = d s p, σ f m p,σ 2 1 e 2πΩ α 1 In particular, using functions f m that are different from zero only on a small region around a value p m, σ m, we have M 0 Ň p m,σ m 0 M 1 e 2π Ω m α 1 Mean value on Minkowski vacuum of the number operator of Rindler quanta with momentum p m is a Bose-Einstein distribution, at temperature α T = 2πk B, called Hawking-Unruh temperature. This temperature expressed in MKS units is T = ħα 2πk B c. 3.3 Thermalization theorem The result just presented is the principal expression of the Unruh effect, and shows that Minkowski vacuum for an accelerated observer is equivalent to a thermal state. We can make more explicit the thermal nature of the relation between the two frames writing the Minkowski vacuum in terms of states with defined Rindler particle number. Consider the inverse of 3.19 d Ω,p,σ = 8 Similarly ˇ0 R = 0 R. N Ω α + 1 beff Ω,p,σ N Ω α beff Ω, p, σ

47 n + 1 c n+,n +1 = e A n+ c n+ 1,n CHAPTER 3. THE UNRUH EFFECT 39 Substituting into d p,σ 0 M = 0 we have Considering wave packets it becomes b eff Ω,p,σ e π Ω α b eff Ω, p, σ 0 M = ˇbΩ m,p m,σ m e π Ω m α ˇb Ω m, p m, σ m 0 M = 0 To simplify notation consider a single normal mode, on regions I and II with opposite p m, so ˇbσ e A ˇb σ 0 M = with A = π Ω m α. Expand the state 0 M with respect to Rindler states 0 M = c n+,n n +, n R n +, n R = ˇb + n + ˇb n n +,n n+! n! 0 R Substituting into 3.23 we get n+ + 1 c n+ +1,n e A n c n+,n 1 n +,n n +,n n + 1 c n+,n +1 e A n+ c n+ 1,n n +, n R = 0 n +, n R = 0 from which we have the recursive relations { n+ + 1 c n+ +1,n = e A n c n+,n 1 This system is satisfied by where B is a constant; substituting c n+,n = B e n +A δ n+,n M = B n e na ˇb + n ˇb n n! 0 R = B exp e A ˇb ˇb + 0 R 3.25

48 CHAPTER 3. THE UNRUH EFFECT 40 The constant B is determined by the normalization: M 0 0 M = B n e 2nA = B e2a e 2A 1 = 1 So B = 1 e 2A and 0 M = 1 e 2A exp e A ˇb ˇb + 0 R Considering all normal modes we have the correct expression: 0 M = 1 e 2πΩ m α exp m m e πω m α ˇb Ω m,p m,σ m ˇb Ω m, p m, σ m 0 R Minkowski vacuum can be written as a Rindler state for which there is entanglement 9 between regions I and II: if you find n m particles with momentum Ω m, p m in region I, then you will find n m particles with momentum Ω m, p m in region II. Note that accelerated observers in region I and that in region II can t communicate 10, so there isn t any causal relation among them see fig Measures carried out by accelerated observers in region I can t be influenced by events in region II. Indicate with O + an observable relative to an observer in region I; O + depends only on events in region I, so 0 O M + 0 M = Tr ρ O + = Tr + ρ + O where ρ = 0 M M 0 Tr O = n +,n R n +, n O n +, n R while Tr + and ρ + are the partial trace and the reduced density matrix relative to region I. Writing this relation we have considered the field on 9 An entangled state is a state of a compound quantum system that you can t write as a tensor product of states of constituent systems [15]. In our case the system is composed by field in region I and in region II. The entangled nature of state 0 M clearly emerges in the first equality of 3.25, where n is the same for + and factor. 10 For these regions lines t = x and t = x are horizons.

49 CHAPTER 3. THE UNRUH EFFECT 41 t x Figure 3.3. The blue line is the trajectory of a Rindler observer and the yellow area is the set of events with that, at a certain time, he can exchange signals. You can see that region II is unreachable at any time. whole spacetime as a compound system: Therefore 0 R = 0 I 0 II n +, n R = n + I n II Tr + O = n + I n + O n + I ρ + = Tr ρ = n II n ρ n II To avoid an heavy notation consider again a single normal mode and use 3.25 for 0 M, from which it follows ρ + = B n e 2nA n! ˇb + n 0 I I 0 ˇb + n = B e A n e 2An+ 1 2 n I I n = = B e A n e 2π α ȟi n I I n = B e A e 2π α ȟi

50 CHAPTER 3. THE UNRUH EFFECT 42 where ȟi = ˇb ˇb Ω. Consider Tr + e 2π α ȟi = n I n e 2π α ȟi n I = e A n e 2nA = e A 1 e 2A = 1 B e A therefore ρ + = 1 Z e 2π α ȟi Z = Tr + e 2π α ȟi Considering all normal modes ρ + = 1 Z e 2π α ȞI Z = Tr + e 2π α ȞI 3.27 with Ȟ I = m ˇb ˇb Ω m,p m,+ Ω m,p m, Ω m The equalities 3.27 and 3.26 constitute the so-called thermalization theorem, that states that Minkowski vacuum for accelerated observers in region I is equivalent to a canonical ensemble at temperature T = Rindler states. α 2πk B of 3.4 Physical interpretation Theory developed in chapter 2 says that Rindler quantization has the same properties of Minkowski quantization, so this two procedures are mathematically equivalent. This doesn t mean, however, that they are physically equivalent. Minkowski quanta are interpreted as particles and this identification is widely validated by experiments. This is not the case for Rindler quanta, because proportionality constant between observer acceleration and Hawking-Unruh temperature is very small s2 m K, so its effects are to date not experimentally detectable. The only way to acknowledge the physical significance of Rindler quantization consists therefore in the study of a detector model. You can say that Rindler quanta are particles detected by an accelerated observer if the detector model is sensitive to Minkowski particles if in inertial motion and is sensitive to Rindler quanta if in accelerated motion. Consider a particle detector at rest in a static four-dimensional s = 3

51 CHAPTER 3. THE UNRUH EFFECT 43 spacetime, 11 so it is moving along a timelike Killing field orbit. Consider that the detector is a two-level system and indicate with {, } an energy eigenstate basis, with eigenvalues respectively Ω and 0. Consider the operator A defined by A = A = 0 A = 0 Define the detector free hamiltonian by H riv = Ω A A and the filed interaction hamiltonian by H int = ɛt Σ t φx A = F x A + F x A h d 3 x where t is the Killing time, ɛt C0 R is a coupling constant that turns on and off the detector in a smooth way and at finite time, F x C0 R3 expresses the detector spatial sensitivity and Σ t is the space at time t. The hamiltonian of the detector-field system is H = H KG + H riv + H int where H KG is the Klein-Gordon hamiltonian of the free field, written with respect to Killing time. We are interested in the case in which at the beginning the detector is in the state and the field is in the state with n particles, all in the normal mode χx. It is possible to show that the system state, after the detector is turned off, at the lowest order in ɛ is [19] n i n Q n 1 with Q = M e iωt ɛt F x χx g d 4 x 11 The n-dimensional case can be found in [17].

52 CHAPTER 3. THE UNRUH EFFECT 44 so the probability of finding the detector in the state is P = n Q 2 We see that the trigger probability for the detector is proportional to the number of field quanta and to the factor Q 2, that expresses the overlapping of detector and the considered normal mode. We also see that the transition of the detector corresponds to the absorption of a field quantum. Therefore we infer that this model behaves as a detector sensitive to field quanta relative to the Killing field along whose orbits the detector is moving. In the case of Minkowski spacetime it can be used to describe both an inertial and an accelerated detector; in the first case it responds to Minkowski quanta, in the second case to Rindler quanta. We can therefore conclude that Rindler quanta are particles perceived by accelerated detectors. 3.5 Hawking effect The Hawking effect consists in the emission of a particle thermal spectrum from a black hole. This result has had great importance and has brought great interest to quantum theory in curved spacetime. Hawking effect is analogous to Unruh effect and here it is presented a derivation in the case of two-dimensional spacetime s = 1 and massless scalar field. Two-dimensional Minkowski spacetime In order to simplify the comparison between Unruh and Hawking effect, let us recast the results obtained for two-dimensional Minkowski spacetime in suitable coordinates. Minkowski metric is ds 2 = dt 2 + dz 2 where z is the spatial coordinate. From the Killing field t a follows the definition of Minkowski particles, to which it corresponds the vacuum 0 M

53 CHAPTER 3. THE UNRUH EFFECT 45 and the expansion φx = dk a k u k x + a k u k x 3.28 with Introduce the coordinate 1 u k x = 2π 2 k e ikz i k t { ū = t z v = t + z ds 2 = dūd v Therefore u k x = 1 2π 2ω e iωū per k > 0 u k x = 1 2π 2ω e iω v per k < 0 where ω = k. Substituting into φ expansion we get φx = 0 dω a ω e iωū + a ω e iω v + h.c Coordinates relative to a frame with uniform acceleration α are the Rindler coordinates, linked to Minkowski coordinates by coordinates the metric is ds 2 = αξ dτ 2 + dξ 2 In this The field τ a is a Killing field and from it follows the definition of Rindler particles, to which it corresponds the vacuum 0 R. Our case, with null mass and s = 1, is very simple instead of using solutions 3.12, it is convenient to directly solve the Klein-Gordon equation. Indeed defining ξ = 1 α logαξ + 1 metric is ds 2 = e 2αξ dτ 2 + dξ 2 so it is conformally equivalent to Minkowski metric. Klein-Gordon equation

54 CHAPTER 3. THE UNRUH EFFECT 46 with m = 0 in a flat spacetime R = 0 is conformally invariant cf. page 18, so we get the solutions 1 v p x = 2π 2 p e ipξ i p τ to which it corresponds the expansion φx = dp b p v p x + b p vpx 3.30 Define { u = τ ξ v = τ + ξ ds 2 = e αv u dudv Therefore φx = 0 dω b Ω e iωu + b Ω e iωv + h.c with Ω = p. Coordinates u, v are linked to the ū, v by the transformations ū = 1 α e αu v = α eαv The connection between operators a k and b p is given by Bogolubov coefficients Two-dimensional Schwarzschild spacetime Consider a non-rotating black hole of mass M. Its gravitational field out of event horizon r > r s = 2M is described by Schwarzschild metric ds 2 = 1 rs r dt r s dr 2 r We have used coordinates that correspond to observers at fixed distance from black hole, called stationary observers. As r spacetime is no longer influenced by black hole and coordinates t, r become Minkowski coordinates. The vector t a is a timelike Killing vector, so it can be used to quantize the field, leading to the particle notion of stationary observers. Indicate with b p annihilation operators of these particles and with 0 S their vacuum.

55 CHAPTER 3. THE UNRUH EFFECT 47 Consider r r such that 12 dr = 1 rs r dr from which ds 2 = 1 rs r dt 2 + dr Therefore two-dimensional Schwarzschild metric is conformally equivalent to Minkowski metric, so the solutions of Klein-Gordon equation are 1 v p x = 2π 2 p e ipr i p t to which it corresponds a φx expansion alike It is again convenient to consider the coordinates { u = t r v = t + r ds 2 = 1 rs r dudv with respect to which the φx expansion become alike Schwarzschild metric can be used only out of event horizon, but it is well-known that a free-falling observer gets to the horizon in a finite proper time, so we have to consider also the black hole interior. So introduce the coordinates { ū = 2rs e u 2rs v = 2r s e v 2rs [ ds 2 rs = rs r e1 r ] dūd v 3.34 Now the horizon is at ū = 0, v = 0, where metric is regular and so can be extended to the black hole interior. 13 Define { ū = t z v = t + z [ ds 2 rs = rs r e1 r ] d t 2 + d z 2 It can be showed that coordinates t, z correspond to free-falling observers 12 Integrating you get «r r r = r r s + r s log 1 r s 13 Kruskal metric, that comes from this extension, describes both the exterior and the interior of the black hole, but it describes also some other regions white hole and parallel universe that seem to be not realizable in nature [20].

56 CHAPTER 3. THE UNRUH EFFECT 48 [20] and that the time t leads to a well-defined particle notion [3]. This notion is based on modes 1 u k x = 2π 2 k e ik z i k t obtained from conformal equivalence between Kruskal and Minkowski metrics. These modes conduct to a φx expansion identical to 3.28; this expansion becomes identical to 3.29 if expressed in terms of coordinates t, z. Indicate the vacuum relative to these particles with 0 K. Note that transformations 3.34 reduce to 3.32 setting α = 1 2r s, so we can say that the link between particles of stationary and free-falling observers is given by Bogolubov coefficients 3.17, from which follows a particle thermal spectrum at Hawking temperature T H = 1 8πMk B. It is convenient to remark that free-falling observers move with null acceleration and correspond to Minkowski observers, while stationary observers have to constantly accelerate in order to remain at fixed distance from the horizon so they correspond to Rindler observers. Particle emission Until now we have identified two possible particle definitions relative to two different frames and the link between them. Let us use these results to describe a physically realizable situation: a black hole created by the gravitational collapse of a spheric massive body and a far observer. Firstly choose what is the right frame that corresponds to a far observer. As r black hole influence becomes negligible and spacetime reduces to Minkowski spacetime; this condition is satisfied by coordinates t, r and not satisfied by coordinates t, z, so a far observer is a stationary observer. The inferring of the right vacuum that corresponds to a physically realizable situation is more difficult. We need to impose some hypotheses: consider that at the beginning the body is so rarefied that spacetime is flat and the field is in Minkowski vacuum state. It can be showed that with this initial condition the gravitational collapse leads the field to the state 0 K, relative to free-falling observers [3]. In conclusion the physically realizable situation corresponds to the field in the state 0 K seen by stationary observers. This situation is analogous to the Unruh effect with the field in the state 0 M seen by accelerated

57 CHAPTER 3. THE UNRUH EFFECT 49 observers, that results in a thermal distribution of quanta. For this reason Hawking effect is stated as black hole emission of a thermal particle spectrum at temperature T H = 1 8πMk B.

58 Chapter 4 A formulation in terms of scattering: links with dynamical Casimir effect Dynamical Casimir effect is the particle creation from the vacuum state of a field confined in a box due to moving box walls. This happens because the time dependent borders reshape the field modes. Examples of time dependent borders are oscillating walls and accelerated boxes. In the last case dynamical Casimir effect is closely connected to Unruh effect, indeed an observer that moves with the box sees its border unmoving and can t ascribe particle creation to dynamical Casimir effect. Vacuum and particle are observer dependent notions, so it is not surprising that not equivalent observers identify different causes for the same phenomenon. In this chapter we consider a box at the beginning at rest in an inertial frame in a flat spacetime. Successively the box is sharply accelerated to a non relativistic speed, then maintained constant. The problem is faced in the comoving frame, in which the box is at rest and acceleration doesn t modify the border but the metric and so the Klein-Gordon equation [16]. 50

59 CHAPTER 4. DYNAMICAL CASIMIR EFFECT Moving box Consider a box that can move along the X direction in a Minkowski reference frame {T, X, Y, Z}. In this frame line the element is ds 2 = dt 2 + dx 2 + dy 2 + dz 2 Consider now the frame {t, x, y, z}, linked to the previous by the transformations: T = t X = x + ft Y = y Z = z therefore with v = df dt. In this frame line the element is dt = dt dy = dy dz = dz dx = dx + vtdt ds 2 = 1 v 2 dt 2 + 2v dx dt + dx 2 + dy 2 + dz The world line of an observer static in this frame is with respect to the two coordinates: so this observer has four-velocity indeed u = dx ds = x = t, x 0 = T, x 0 + ft, y 0, z 0 1, 1 0 =, 1 v 2 1 v 2 v, 0, 0 1 v 2 dx = 0, dy = 0, dz = 0 ds = 1 v 2 dt dt = dt ds = 1 v 2 dt Therefore this reference is moving with speed v along the X axis, so it can be used as box comoving frame. Note that this coordinates are generic coordinates, in the sense of general relativity, and are not related to the previous by Lorentz transformations. In the case of constant v they reduce to Galilei transformations, that is coherent with the hypothesis of non relativistic speed.

60 CHAPTER 4. DYNAMICAL CASIMIR EFFECT 52 From 4.1 we obtain that the metric with respect to comoving coordinates is 1 v 2 v 0 0 g µν = v with v = vt. We are interested in the case of a box at the beginning at rest, that accelerates sharply to a non relativistic speed v 0 and then moves with this speed. Therefore vt t 0 vt t v 0 vt 1 t 4.2 Particles in a asymptotically stationary spacetime In section 2.2 we saw that in a generic spacetime there isn t a natural particle definition; on the contrary if spacetime is stationary its symmetry provides this concept. This criterion can be extended to the case of asymptotically stationary spacetime, that is stationary only in remote past and in remote future, defining particles separately in these two regions. We indicate these regions respectively as in region R in and out region R out. Consider the timelike vector field t a and substitute it into the Killing equation 3.1 finding t g µν = 0 The metric 4.2 satisfies this equation only if v t = 0 that is true in in and out regions. So the field t a can be used to define the particles related to the comoving frame in the remote past and in the remote future. Note that in these regions the speed is constant, so the comoving frame is inertial and its particles coincides with Minkowski particles. Indicate with φ in k x the solutions of Klein-Gordon equation that are positive frequency with respect to t a as t R in. The scalar-field operator

61 CHAPTER 4. DYNAMICAL CASIMIR EFFECT 53 can therefore be written as φx = [ ] a k φ in k + a φ in k k k where a k are the annihilation operators relative to particles of region R in. Indicate with 0 in the vacuum relative to these operators. Consider now φ out x, that are the positive frequency solutions of Klein- p Gordon equation with respect to t a as t R out. We have the expansion φx = p [ b p φ out p ] + b p φout p operators b p are the annihilators of particles of region R out, to which it corresponds the vacuum 0 out. The relation between these two particle definitions is expressed by Bogolubov coefficients 2.25 and by the transformation 2.23 α k p = φ in k, φ out p β k p = φ out p φ in k = p α k p φ out p, φin k + β k p φ out p 4.3 In particular the mean value on the vacuum state relative to one region of the number operator of quanta relative to the other region is linked to β k p by 2.26 and by Wave equation for the field in the box Let us explicitly solve the field equation in the box, in order to obtain the normal modes φ in x and φ out k p x. Consider a reflective box, with two walls orthogonal to the direction of motion X at a fixed proper distance L and with area A sufficiently large that we can ignore the other walls. Consider that in this box it is confined a massless scalar field φ. The field is described by the Klein-Gordon equation 2.14, that for a

62 CHAPTER 4. DYNAMICAL CASIMIR EFFECT 54 massless field in a flat spacetime is g µν µ ν φ = 0 where a is the covariant derivative relative to the metric g ab ; in terms of partial derivative we have 1 g µ g g µν ν φ = The inverse of metric 4.2 is so equation 4.4 becomes 1 v 0 0 g µν = v 1 v [ 2 t + 2v t x + v x + 1 v 2 x 2 + y 2 + z 2 ] φ = Consider now only the region R in, where v = 0 and equation 4.5 reduces to d Alembert equation, so its positive frequency solutions are φ in 1 k x = AL 2ω k e iω k t [ sin nπ L x e ik x] with t R in where n Z, ω k = nπ 2 L + k 2 where we have bold marked the components that are orthogonal to the motion direction. In region R out v = v 0 and 4.5 becomes [ 2 t + 2v 0 t x + 1 v0 2 x 2 + y 2 + z 2 ] φ = 0 whose positive frequency solutions are φ out p x = 1 AL 2Ω p e iω pt [ sin mπ L x e ip x] with t R out where m Z, Ω p = mπ 2 L + p 2 mπ v 0 L

63 CHAPTER 4. DYNAMICAL CASIMIR EFFECT 55 In the expression of Bogolubov coefficients the functions that you are comparing are evaluated at the same time, so we can t compare φ in x and k φ out x using the expressions that we have found because they are valid in p separate regions. Therefore let us solve the equation 4.5 for all values of t using a perturbative technique. If the speed v is non relativistic v 1, we can stop ourselves to the first order in v, so [ + ˆV ]φ = 0 ˆV = 2vt t x + v x 4.6 where = 2 t + 2 x + 2 y + 2 z. We can perturbatively solve the equation 4.6 considering φ = ˆV φ where φ 0 is a solution of the unperturbed equation φ 0 = 0. In particular consider φ 0 k,n = 1 AL 2ωk,n e iω k,nt [ sin nπ L x e ik x] 4.8 where n Z, ω k,n = nπ 2 L + k 2 The solution of 4.7 can be written as φ k,n = φ 0 k,n + d 4 x Gx x jx j = ˆV φ 0 k,n 4.9 with G retarded Green function, so G respect causality and satisfies x Gx x = δ 4 x x Dirichlet condition on φ k,n x is satisfied by φ 0 k,n x, so from 4.9 we get the condition G x=0,l = 0, from which it can be showed that Gx x = θt t 2Lπ 2 l=1 d 2 p e ip x x sin[ ω p,n t t ] ω p,n sin lπx L lπx sin L Substituting into 4.9 we get an expression immediately integrable with respect to d 2 x = dy dz, obtaining the factor 2π 2 δ 2 p + k that enable

64 CHAPTER 4. DYNAMICAL CASIMIR EFFECT 56 us to evaluate also the integral in d 2 p, therefore: φ k,n = φ 0 k,n 2πn L 2 e ik x 2ALωk,n l=1 vt 2iω k,n vt [ I l,n sin lπx L θt t e iω k,nt 2iω k,l e iω k,lt t ] e iω k,lt t dt 4.10 where I l,n = L 0 dx sin lπx L nπx cos L 4.11 We are interested in the asymptotic inspection of the problem, so we consider only the initial and final states. If the perturbation acts only in a small interval around t = 0, these states correspond respectively to t 0 and t 0, so: φ finale k,n t, x = φ 0 2πn k,n t, x vt 2iω k,n vt φ iniziale k,n t, x = φ 0 k,n t, x e ik x 1 L 2 2ALωk,n L l=1 e ] iω k,n+ω k,l t e iωk,lt e iω k,n ω k,l t e iω k,lt dt [ I l,n sin lπx 2iω k,l 4.12 Note that for t 0 it is φ in k,n = φ0 k,n, so this relation can be used to write φ in k,n with t 0 that is t R out, from which we can evaluate the Bogolubov coefficients.

65 CHAPTER 4. DYNAMICAL CASIMIR EFFECT Three remarkable cases In this section we consider that the field initially is in the state 0 in and verify if the acceleration phase results in particle creation using the condition β k,n p,m 0 see page 24. In the evaluation of the scalar product of normal modes relative to R in and R out we set t 0 cf and use 4.12 to write φ in k,n with t 0. Consider three remarkable cases: Null acceleration Consider a uniformly moving box, 1 that is vt = v 0 1. So 4.12 becomes: e ik x 2ALωk,n ω k,l l=1 δω ] k,n + ω k,l e iωk,lt δω k,n ω k,l e iω k,lt φ finale k,n t, x =φ 0 2πn k,n t, x + L 2 [ I l,n ω k,n v 0 sin lπx L The first δ is zero because the ω are both positive; the second δ is equivalent to δ l n, but I n,n = 0 cf. 4.11, so φ finale k,n = φ 0 k,n From this it follows that φ in k,n x = 1 AL 2ω k e iω k t [ sin nπ L x e ik x] t so for t R out it is φ in k,n = φout k,n so the coefficients β are all zero. In the case of a uniformly moving box there is no particle creation. This is coherent with the fact that the observer moving with the box is inertial. 1 Until now we have set v = 0 as t R in, but if v is constant it is the same.

66 CHAPTER 4. DYNAMICAL CASIMIR EFFECT Instantaneous acceleration Consider vt = θt v 0, with v 0 1. Now 4.12 is: φ finale k,n t, x = φ 0 2πn k,n t, x L 2 v 0 δt 2iω k,n θt [ I l,n sin lπx e ik x 1 2ALωk,n L l=1 e ] iω k,n+ω k,l t e iωk,lt e iω k,n ω k,l t e iω k,lt dt 2iω k,l We are interested in coefficients β, so consider only the part proportional to negative frequency modes e iωk,lt. From the comparison with 4.3 we get: nπ v 0 I l,n β k,n k,l = ω k,l L 2 2ALω k,n Evaluating the integral you obtain [16] [ ] dt 2ω k,n θt + iδt e iω k,n+ω k,l t β k,n k,l = nπ v 0 I l,n ω k,n ω k,l L 2 2ALω k,n ω k,l ω k,n + ω k,l Exponential acceleration Consider a more realistic case, in which the acceleration is less sharp than that in the previous case. Set [ vt = v 0 e a 2 t e a2 t 2 θt ] with v 0 1 and a 1 strong acceleration. Proceeding as in the previous case we find β k,n k,l = nπ v 0 I l,n ω k,l L dt [2ω 2 k,n e a2 t 2 + 2ω k,n 1 e a 2 t 2 θt + 2ALω k,n + 2ia 2 t e a2 t 2 θt 1 + i ] 1 e a2 t 2 δt e iω k,n+ω k,l t Integrating by parts and performing some algebra we arrive to β k,n k,l = β 0 k,n k,l D k,n k,l

67 CHAPTER 4. DYNAMICAL CASIMIR EFFECT 59 where βk,n 0 k,l is the result of the previous case, while D k,n k,l = 1 + i π 2a ω k,n + ω k,l e 1 4a 2 ω k,n +ω k,l [1 ] 2 + Erf i 2a ω k,n + ω k,l and Erf is the error function. It is a 1 so we can expand with respect to 1/a obtaining β k,n k,l [ nπ v 0 I l,n ω k,n ω k,l a 1 L 1 + i ] π 2 2ALω k,n ω k,l ω k,n + ω k,l 2a ω k,n + ω k,l The box acceleration phase leads to particle creation with a spectrum that preserves informations on the acceleration phase itself.

68 Chapter 5 Two non relativistic models The simplified analysis of chapter 1 shows that the thermal nature of the spectrum of particles created by acceleration is direct consequence of relativistic kinematics. However this doesn t mean that the Unruh effect is a relativistic effect, indeed we reasonably believe that acceleration leads to observable consequences in non relativistic conditions too. In the Unruh effect derivation it is very important the possibility of separate space and time in several ways, to which they correspond different particle notions. This make no sense in non relativistic theory, so we can t obtain informations on the effect of acceleration in classic theory by a limit process on relativistic results. It is therefore necessary to study this effect directly in classic theory. In order to make available a mechanism of creation and annihilation of particles also in non relativistic quantum theory, where the particle number is constant, we need to set a suitable prescription. In this chapter we do it for two one-dimensional models. 5.1 Harmonic oscillator Model structure Let us start by setting out the interpretative paradigm that we use to describe the particle creation. Consider a particle subjected to a harmonic oscillator potential in a inertial frame. Interpret the fundamental state of this system as the vacuum and say that the particle is created if it jumps to 60

69 CHAPTER 5. TWO NON RELATIVISTIC MODELS 61 an excited state. 1 Consider an inertial frame {T, X} called laboratory frame. In order to describe particles relative to an accelerated observer consider the frame {τ, ξ} with axes parallel to the laboratory ones and origin that moves respect the laboratory frame with trajectory RT = α 2 T 2 This frame has uniform acceleration α and at T = 0 it has null speed and origin coincident with the laboratory one. If we compare the laboratory frame with the accelerated frame then the result relies not only on acceleration but also on the effect of translation. To neglect this effect and focus just on acceleration consider the so-called comovent frame, that is the set of inertial reference frames with axes parallel to the laboratory ones and that for each time they have the same origin and speed than the accelerated frame. The origin of the frame {t, x T0 } relative to the time T = T 0 with respect to the laboratory frame has trajectory r T0 T = RT 0 + ṘT 0 T T 0 Considering T 0 = τ we have that the coordinates of accelerated and comovent frames coincide, that is { ξ = xτ τ = t We will compare the vacuum relative to the accelerated frame {τ, ξ} with the states with defined particle number relative to the comovent frame. With this purpose let us consider one harmonic oscillator in each inertial frame that constitutes the comovent frame. At each time τ we will expand the accelerated vacuum with respect to the energy eigenvalues relative to the inertial frame of that τ. 1 To describe a many-particle state we have to consider N non-interacting particles. A state of this system is the product of N one-particle states so we consider only the case of a single particle.

70 CHAPTER 5. TWO NON RELATIVISTIC MODELS 62 Comovent oscillator Consider the inertial frame {t, x τ } relative to the fixed time τ. Consider a particle subjected to a harmonic oscillator potential in this frame. The hamiltonian of this system is H τ p τ, x τ = p2 τ 2m + mω2 2 x2 τ Let us simplify the notation omitting from now on the subscript τ. Define a = mω 2ħ x + ip mω from [x, p] = iħ follows [a, a ] = 1, furthermore H = ħω a a Consider the eigenstates n of a a, defined by It can be showed that a a n = n n a n = n n a n = n + 1 n + 1 H n = E n n E n = ħω n We interpret the fundamental state 0 as vacuum; this state is defined by the relation Multiplying by x we have a 0 = 0 x x + ip 0 = 0 mω using x x ψ = x x ψ and x p ψ = iħ d dx x ψ we obtain x + x 2 d 0 x 0 = 0 con x 2 0 = ħ dx mω

71 CHAPTER 5. TWO NON RELATIVISTIC MODELS 63 whose normalized solution is x 0 = 1 4 πx 2 0 exp x2 2x 2 0 Similarly x n = 1 2 n n! H n x x 0 x 0 where H n x are the Hermite polynomials, defined by from which it can be showed that H n x = 1 n x2 dn e dx n e x2 H x x n x 0 = x0 n 2 d n exp x 2 0 dx n exp x2 x 2 = 0 x = x 0 n 2 d exp 2x 2 0 dx x 2 x 2 exp x2 0 2x 2 0 Restoring the subscript τ, the energy eigenstates relative to this frame are x τ τ 0 τ = 1 4 πx 2 0 exp x2 τ 2x x τ τ n τ = 1 2 n n! H n xτ x x 0 τ τ 0 τ 5.2 Accelerated oscillator Consider the accelerated frame {τ, ξ}. A harmonic oscillator rigid with this frame has the hamiltonian H α κ, ξ = κ2 2m + mω2 2 ξ2 + mα ξ where mα ξ is the apparent potential due to acceleration. that {τ, ξ} {τ, x τ } and omitting again the subscript τ we get H α κ, x = κ2 2m + mω2 2 x2 + mα x Remembering

72 CHAPTER 5. TWO NON RELATIVISTIC MODELS 64 Note that mω 2 2 x2 + mα x = mω2 2 x 2 + 2α [ ω 2 x = mω2 x + α 2 α 2 ] 2 ω 2 ω 4 from which where H α κ, ζ = κ2 2m + mω2 2 ζ2 mα2 2ω 2 ζ = x + α ω 2 Define b = mω 2ħ ζ + iκ mω from [x, κ] = iħ follows [ζ, κ] = iħ, from which [b, b ] = 1, therefore H α = ħω b b + 1 mα2 2 2ω 2 Indicate the eigenvectors of b b with n α, then It can be showed that b b n α = n n α b n α = n n α H α n α = E a n n α b n α = n + 1 n + 1 α En a = ħω n + 1 mα2 2 2ω 2 The fundamental state 0 α corresponds to the accelerated vacuum and is defined by b 0 α = 0 Proceeding as in the previous case we obtain ζ + x 2 d 0 α dζ ζ 0 = 0 α

73 CHAPTER 5. TWO NON RELATIVISTIC MODELS 65 whose normalized solution 2 is α ζ 0 α = 1 4 πx 2 0 exp ζ2 2x 2 0 Turning back to the original variable it becomes x τ τ 0 α = 1 4 πx 2 0 exp 1 x 2x 2 τ + α 2 0 ω Similarly α ζ n α = 1 2 n n! H n ζ x 0 α ζ 0 α 5.4 Transformation Let us compare the vacuum relative to the accelerated observer with the eigenstates of the comovent oscillators. Consider the expansion 0 α = n n τ τ n 0 α Using 5.1, 5.2, 5.3 and the notation x x τ, evaluate the product n 0 = dx τ α τ n x τ τ x 0 α = exp α2 4 x = 2 0 ω4 π x n n! that by the variables change y = x x 0 exp α2 n 0 = τ α π 2 n n! 4 x 2 0 ω4 [ dx H x n x 0 exp 1 x 2 0 becomes Using the relation see [9], num The normalization is given by Z dy H n y exp [ dy e y y 0 2 H n y = π 2 n y n 0 dx α x 0 α 2 = Z dζ α ζ 0 α 2 = 1 x + α ] 2 2ω 2 y + α ] 2 2 x 0 ω 2

74 CHAPTER 5. TWO NON RELATIVISTIC MODELS 66 0,3 0,2 1,0 0,1 0,0 0,0 0,5 F 1/n 0,5 1,0 0,0 Figure 5.1. Plot of the probability τ n 0 α 2 as a function of 1/n and F. You can see that the probability of measuring high energies small 1/n is small and it increases with F, taht is with α. we obtain n 0 = τ α 1n e mα 2 4ħω 3 en log q mα 2 2ħω 3 n! from which it follows that 0 α = n 1 n D n! e nc n τ C = log mα 2 2ħω 3 mα D = e 2 4ħω 3 Therefore the vacuum relative to the accelerated frame is seen by comovent observers as a superposition of eigenstates different from their vacuum. The probability that the state 0 α appears as n τ is 3 τ n 0 α 2 F en log F = e n! F = mα2 2ħω 3 3 To have an idea of the order of magnitude of F you can think that for an electron, using the estimate E mc 2 ħω and using for the acceleration atomic values, you have F

75 CHAPTER 5. TWO NON RELATIVISTIC MODELS 67 so the probability of measuring high energies is small, but it increases if the acceleration increases cf. fig The mean value of the energy relative to the comovent system on the state 0 α is Consider 0 H α τ 0 α = ħω 0 α a τ a τ 0 α α 0 a τ a τ 0 α = n α 0 a τ a τ n τ τ n 0 α = n = n n τ n 0 α 2 = e F n en log F n! = F from which α 0 H τ 0 α = ħω 2 + mα2 2ω 2 Therefore also in non relativistic conditions the acceleration results in particle creation, but in this case it is not possible to identify any thermal distribution. Note that in the relativistic case the comparison occurs between accelerated and laboratory observers, while in this case we have compared accelerated and comovent observers. The model that corresponds to the relativistic situation has been solved, but it is omitted because it doesn t supply any further informations compared to the discussed model. Also in this case the acceleration leads to particle creation, whose spectrum is not thermal.

76 CHAPTER 5. TWO NON RELATIVISTIC MODELS Moving delta Model structure The harmonic oscillator model has a problem: its stationary states are all proper eigenstates, so even when the particle get out from vacuum it is always spatially confined. More realistic in this sense is the delta potential [4], that has only one bound state with negative energy and then for each positive energy it has a generalized eigenstate. In this case, still interpreting the fundamental state as vacuum, when the particle is created it passes from a spatially confined state to a scattering one. This model has in its turn a drawback: in the accelerated frame the apparent potential transforms the energy spectrum so that the bound state vanishes. Therefore it is not possible to represent the vacuum relative to the accelerated observer in such a way. So we use this model to study a case analogous to the dynamical Casimir effect discussed on chapter 4. Consider a frame {t, x} initially at rest. Suppose that from time t = 0 to time t = 1 the frame accelerates in a smooth way and then it returns inertial. Suppose also that at the beginning the field is in its vacuum state. We want to know if after the acceleration phase the field is still in its vacuum state. To answer to this question we will evaluate the transition probability between the field state after the acceleration and the vacuum state. Initial state Consider a particle subjected to an attractive delta potential in the frame {t, x}. The hamiltonian of this system is Hp, x = p2 γ δx + mαtx 2m with γ > 0 and αt = 0 as t / 0, 1. Set in t < 0 and out t > 1, therefore H in p, x = H out p, x = p2 2m γ δx This hamiltonian has one generalized eigenfunction for each eigenvalue E 0 and only the bound state φx = γ0 γ 02 2 e x γ 0 = 2m ħ 2 γ

77 CHAPTER 5. TWO NON RELATIVISTIC MODELS 69 1,0 v 0,5 0,8 0,4 0,6 0,3 0,4 0,2 0,2 0,1-1,0-0,5 0,0 0,5 1,0 1,5 2,0 a t -1,0-0,5 0,0 0,5 1,0 1,5 2,0 b t Figure 5.2. a Acceleration of the observer. b Speed of the observer with respect to the inertial frame in which it is at rest at the beginning. relative to the energy E 0 = mγ2 2ħ 2 The prescription given for the interpretation of the model corresponds to x 0 in = x 0 out = x φ = φx We will consider that the field when t 0 is in the vacuum state 0 in. Transition probability The system evolution is given by the Schrodinger equation iħ t ψx, t = In our case the initial condition is ħ2 2m 2 x γ δx + mαtx ψx, t 5.5 ψx, 0 = φx The transition probability between the field state after the acceleration and the vacuum state is with t > 1. P = out 0 ψ t 2 = dx φx ψx, t The equation 5.5 has been numerically solved using the commercial software Mathematica. 2

78 CHAPTER 5. TWO NON RELATIVISTIC MODELS ,5 t 1 0, x Figure 5.3. Level plot of ψx, t 2. The end of the acceleration phase is marked with a red line. For the acceleration it has been used the function fig. 5.2a: αt = 1 cos2π t 2 θt θ1 t where θt is the Heaviside step function, with the convention { 0 t < 0 θt = 1 t 0 The delta potential has been approximated by the function V x = θ ε 2 + x θ ε 2 x ε with ε = 10 3 We have used the parameters ħ = m = γ = 1; the evolution has been carried out for t 0, 2. We have considered the x interval 5, 5; for simplicity we have used periodic boundary conditions, neglecting the boundary effects.

79 CHAPTER 5. TWO NON RELATIVISTIC MODELS t = t = 0, t = 0, t = 0, t = 0, t = t = 1, t = 1, t = 1, t = 1, t = 2 Figure 5.4. Plot of ψx, t 2 as a function of x for different values of t. The first two rows are subjected to acceleration. As a comparison the initial condition is marked in red.

80 CHAPTER 5. TWO NON RELATIVISTIC MODELS 72 P t Figure 5.5. Transition probability between the field state and the vacuum as a function of time. The dots represents the evaluated data. The result of the evaluation is showed in figure 5.3 as a level plot of ψx, t 2. Note that for t between zero and one, when there is acceleration, the probability is trailed towards the left. On the contrary for t greater than one, when acceleration is over, the probability near the origin remains confined, while the exterior probability scatters. In figure 5.4 is showed the same result as a plot of ψx, t 2 as a function of x for different values of t. In the first two rows there is acceleration and the whole figure moves towards the left; in the following rows the right part of the figure remains almost unchanged while the left part starts going away from the center. It has been numerically evaluated also the value of P = out 0 ψ t 2 at different times, plotted in figure 5.5. We can see that the ionization probability 1 P increases during the acceleration phase and then it remains constant. Therefore during the acceleration phase there is particle creation, that cease when the frame returns inertial. Note that the potential in the equation 5.5 can be turned to a moving delta by a unitary transformation. 4 For this case some explicit expressions of the evolution are known [4]. These expression are formulated in terms of Volterra equations and they allows for a recursive strategy for the analytic solution of the problem. This method is not simple, but if implemented it 4 The transformation is carried out passing from the frame rigid with the observer to the inertial frame with respect to which the observer is at rest at the beginning.

81 CHAPTER 5. TWO NON RELATIVISTIC MODELS 73 permits to inspect what features of the acceleration phase affect the particle generation mechanism, therefore it constitutes a possible progress of the model. 5.3 Conclusions The two simple models that we have considered, even if does not provide any quantitative information, lead us to the conclusion that even in non relativistic conditions the acceleration results in particle creation.

82 Conclusions In this thesis we examined the Unruh effect. Firstly we followed a very simple process that brought to light the link between Unruh effect and dynamical Doppler effect. Then the Quantum Field Theory in Curved Spacetime was presented. This theory asserts that particle notion is observer dependent and that it is possible to build a certainly valid particle definition by each time symmetry of the spacetime. Using methods typical of this theory, a complete treatment of the Unruh effect was presented. We obtained that the Minkowski vacuum for an observer moving with uniform acceleration α is equivalent to a canonical ensemble at temperature T = with defined particle number. ħα 2πk B c of states Moreover an analogy between the Unruh effect and the Hawking effect particle creation out of a black hole has been presented. This analogy is based upon the fact that an observer to remain at fixed distance from a black hole needs to constantly accelerate. It was considered also the case in which the field is confined in a box that undergoes an acceleration phase. Acceleration results in particle creation in this case too, and the energy spectrum of created particles depends on the detail of the acceleration phase. Finally they were considered two non relativistic models of the quantum field, that exhibit ionization as a consequence of acceleration, but that don t give rise to a thermal state, differently from the relativistic case. In conclusion it has been showed that in different conditions acceleration always leads to particle creation, but the resulting spectrum is thermal only in the case of field in the whole space seen by a uniformly accelerated observer. This situation implies that particle creation is an universal consequence of acceleration, while the thermalization is a feature characteristic of the case of uniformly accelerated observer in relativistic conditions. 74

83 Bibliography [1] P. M. Alsing and P. W. Milonni. Simplified derivation of the Hawking- Unruh temperature for an accelerated observer in vacuum. American Journal of Physics, 72: , December [2] S. J. Avis, C. J. Isham, and D. Storey. Quantum field theory in anti-de sitter space-time. Phys. Rev. D, 1810: , Nov [3] N. D. Birrell and P. C. W. Davies. Quantum fields in curved space. Cambridge: University Press, 1982, [4] Luis L. Bonilla, editor. Inverse Problems and Imaging, Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 15-21, Springer-Verlag Berlin Heidelberg, [5] J. Dieckmann. Cauchy surfaces in a globally hyperbolic space-time. Journal of Mathematical Physics, 29: , March [6] S. A. Fulling. Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time. Phys. Rev. D, 7: , May [7] S. A. Fulling. Aspects of Quantum Field Theory in Curved Spacetime. Aspects of Quantum Field Theory in Curved Spacetime, by Stephen A. Fulling, pp ISBN Cambridge, UK: Cambridge University Press, September 1989., September [8] R. Geroch. Domain of Dependence. Journal of Mathematical Physics, 11: , February [9] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series and products. New York: Academic Press, 1965, 4th ed., edited by Geronimus, Yu.V. 4th ed.; Tseytlin, M.Yu. 4th ed.,

84 BIBLIOGRAPHY 76 [10] S. W. Hawking. Particle creation by black holes. Communications in Mathematical Physics, 43: , August [11] S. W. Hawking and G. F. R. Ellis. The large scale structure of spacetime. Cambridge Monographs on Mathematical Physics, London: Cambridge University Press, 1973, [12] T. Jacobson. Introduction to Quantum Fields in Curved Spacetime and the Hawking Effect. ArXiv General Relativity and Quantum Cosmology e-prints, August [13] B. S. Kay. The Principle of Locality and Quantum Field Theory on non Globally Hyperbolic Curved Spacetimes. Reviews in Mathematical Physics, 4: , [14] Viatcheslav Mukhanov and Sergei Winitzki. Introduction to Quantum Effects in Gravity. Cambridge University Press, June [15] J. J. Sakurai. Modern Quantum Mechanics, Revised Edition. Addison- Wesley, [16] F. Sorge. Dynamical Casimir Effect in a Kicked Box. International Journal of Modern Physics A, 21: , [17] S. Takagi. Vacuum Noise and Stress Induced by Uniform Acceleration Hawking-Unruh Effect in Rindler Manifold of Arbitrary Dimension. Progress of Theoretical Physics Supplement, 88:1 142, [18] W. G. Unruh. Notes on black-hole evaporation. Phys. Rev. D, 14: , August [19] William G. Unruh and Robert M. Wald. What happens when an accelerating observer detects a rindler particle. Physical Review D, 296, [20] Robert M. Wald. General Relativity. University Of Chicago Press, June [21] Robert M. Wald. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics Chicago Lectures in Physics. University Of Chicago Press, November 1994.