Effect of acceleration on quantum systems


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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. nonpositive 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 noninertial 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 noninertial 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 τparticles instead of Minkowskiparticles [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 noninertiality 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 noninertial 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 NonRelativistic 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 noninteracting 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 nonrelativistic 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 BoseEinstein distribution with temperature T = ħa 2πk B c, called HawkingUnruh 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 BoseEinstein energy distribution 1.4, that has temperature T = ħa 2πk B c, called HawkingUnruh temperature. Proportionality between particle number spectrum and BoseEinstein distribution is rigorously established by the socalled 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 KleinGordon equation Consider M, a ndimensional 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 KleinGordon 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 KleinGordon equation if the field is confined into a box then V includes also a timelike surface Σ l that has nullcontribution 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 selfadjoint operators on H and evolve according to the Heisenberg equation Ȯ = i [H, O] where O is a selfadjoint 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 welldefined. 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, lightlike 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 KleinGordon 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 KleinGordon 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 KleinGordon 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,
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