Effect of acceleration on quantum systems


 Robert Freeman
 2 years ago
 Views:
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. 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,
From Einstein to KleinGordon Quantum Mechanics and Relativity
From Einstein to KleinGordon Quantum Mechanics and Relativity Aline Ribeiro Department of Mathematics University of Toronto March 24, 2002 Abstract We study the development from Einstein s relativistic
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for nonphysics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationThree Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009
Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,
More informationCoefficient of Potential and Capacitance
Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that
More informationOn Motion of Robot EndEffector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix
Malaysian Journal of Mathematical Sciences 8(2): 89204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot EndEffector using the Curvature
More information develop a theory that describes the wave properties of particles correctly
Quantum Mechanics Bohr's model: BUT: In 192526: by 1930s:  one of the first ones to use idea of matter waves to solve a problem  gives good explanation of spectrum of single electron atoms, like hydrogen
More informationTime Ordered Perturbation Theory
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Quantization of the Free Electromagnetic Field We have so far quantized the free scalar field and the free Dirac field.
More informationEuclidean quantum gravity revisited
Institute for Gravitation and the Cosmos, Pennsylvania State University 15 June 2009 Eastern Gravity Meeting, Rochester Institute of Technology Based on: Firstorder action and Euclidean quantum gravity,
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationChapter 22 The Hamiltonian and Lagrangian densities. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries
Chapter 22 The Hamiltonian and Lagrangian densities from my book: Understanding Relativistic Quantum Field Theory Hans de Vries January 2, 2009 2 Chapter Contents 22 The Hamiltonian and Lagrangian densities
More informationGenerally Covariant Quantum Mechanics
Chapter 15 Generally Covariant Quantum Mechanics by Myron W. Evans, Alpha Foundation s Institutute for Advance Study (AIAS). (emyrone@oal.com, www.aias.us, www.atomicprecision.com) Dedicated to the Late
More informationSpecial Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN
Special Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN United States Particle Accelerator School, University of California  Santa Cruz, Santa Rosa, CA 14 th 18 th January 2008 1 Outline Notions
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationTill now, almost all attention has been focussed on discussing the state of a quantum system.
Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationMixed states and pure states
Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements
More information4. The Infinite Square Well
4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the
More informationEffective actions for fluids from holography
Effective actions for fluids from holography Based on: arxiv:1405.4243 and arxiv:1504.07616 with Michal Heller and Natalia Pinzani Fokeeva Jan de Boer, Amsterdam Benasque, July 21, 2015 (see also arxiv:1504.07611
More informationThe Special Theory of Relativity explained to children
The Special Theory of Relativity explained to children (from 7 to 107 years old) CharlesMichel Marle marle@math.jussieu.fr Université Pierre et Marie Curie, Paris, France Albert Einstein Century International
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:3705:00 Copyright 2003 Dan
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationThe Essence of Gravitational Waves and Energy
The Essence of Gravitational Waves and Energy F. I. Cooperstock Department of Physics and Astronomy University of Victoria P.O. Box 3055, Victoria, B.C. V8W 3P6 (Canada) March 26, 2015 Abstract We discuss
More information6 J  vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J  vector electric current density (A/m2 ) M  vector magnetic current density (V/m 2 ) Some problems
More informationLorentzian Quantum Einstein Gravity
Lorentzian Quantum Einstein Gravity Stefan Rechenberger Uni Mainz 12.09.2011 Phys.Rev.Lett. 106 (2011) 251302 with Elisa Manrique and Frank Saueressig Stefan Rechenberger (Uni Mainz) Lorentzian Quantum
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationSpecific Intensity. I ν =
Specific Intensity Initial question: A number of active galactic nuclei display jets, that is, long, nearly linear, structures that can extend for hundreds of kiloparsecs. Many have two oppositelydirected
More informationHarmonic Oscillator Physics
Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the timeindependent Schrödinger equation: d ψx
More informationIntroduces the bra and ket notation and gives some examples of its use.
Chapter 7 ket and bra notation Introduces the bra and ket notation and gives some examples of its use. When you change the description of the world from the inutitive and everyday classical mechanics to
More informationThe career of a young theoretical physicist consists of treating the harmonic oscillator in everincreasing levels of abstraction.
2. Free Fields The career of a young theoretical physicist consists of treating the harmonic oscillator in everincreasing levels of abstraction. Sidney Coleman 2.1 Canonical Quantization In quantum mechanics,
More informationLecture 1: Microscopic Theory of Radiation
253a: QFT Fall 2009 Matthew Schwartz Lecture : Microscopic Theory of Radiation Blackbody Radiation Quantum Mechanics began on October 9, 900 with Max Planck s explanation of the blackbody radiation spectrum.
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationConstruction and asymptotics of relativistic diffusions on Lorentz manifolds
Construction and asymptotics of relativistic diffusions on Lorentz manifolds Jürgen Angst Section de mathématiques Université de Genève Rencontre de l ANR Geodycos UMPA, ÉNS Lyon, April 29 2010 Motivations
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationA Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle
A Theoretical Model for Mutual Interaction between Coaxial Cylindrical Coils Lukas Heinzle Page 1 of 15 Abstract: The wireless power transfer link between two coils is determined by the properties of the
More informationA new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space
A new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), 31 38 (Partially supported by DGICYT grant PB970784
More informationSpecial Theory of Relativity
June 1, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deeprooted superstition
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More information7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.
7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated
More information8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, AppI Li. 7 1 4 Ga. 4 7, 6 1,2
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationChapter 7: Polarization
Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces
More informationSound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8
References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationIntroduction to SME and Scattering Theory. Don Colladay. New College of Florida Sarasota, FL, 34243, U.S.A.
June 2012 Introduction to SME and Scattering Theory Don Colladay New College of Florida Sarasota, FL, 34243, U.S.A. This lecture was given at the IUCSS summer school during June of 2012. It contains a
More informationThe Schrödinger Equation
The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development
More informationThe Essentials of Quantum Mechanics
The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008Oct22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationLaws of Motion and Conservation Laws
Laws of Motion and Conservation Laws The first astrophysics we ll consider will be gravity, which we ll address in the next class. First, though, we need to set the stage by talking about some of the basic
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationAssessment Plan for Learning Outcomes for BA/BS in Physics
Department of Physics and Astronomy Goals and Learning Outcomes 1. Students know basic physics principles [BS, BA, MS] 1.1 Students can demonstrate an understanding of Newton s laws 1.2 Students can demonstrate
More informationMICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 00029939(XX)00000 MICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI
More informationTesting dark matter halos using rotation curves and lensing
Testing dark matter halos using rotation curves and lensing Darío Núñez Instituto de Ciencias Nucleares, UNAM Instituto Avanzado de Cosmología A. González, J. Cervantes, T. Matos Observational evidences
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationPhysics 403: Relativity Takehome Final Examination Due 07 May 2007
Physics 403: Relativity Takehome Final Examination Due 07 May 2007 1. Suppose that a beam of protons (rest mass m = 938 MeV/c 2 of total energy E = 300 GeV strikes a proton target at rest. Determine the
More information14.11. Geodesic Lines, Local GaussBonnet Theorem
14.11. Geodesic Lines, Local GaussBonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize
More informationAP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.
AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods
More informationGravitational selfforce in the ultrarelativistic regime Chad Galley, California Institute of Technology
Gravitational selfforce in the ultrarelativistic regime Chad Galley, California Institute of Technology with Rafael Porto (IAS) arxiv: 1302.4486 v2 soon! (with details) Capra16; Dublin, Ireland; July
More informationThe Early History of Quantum Mechanics
Chapter 2 The Early History of Quantum Mechanics In the early years of the twentieth century, Max Planck, Albert Einstein, Louis de Broglie, Neils Bohr, Werner Heisenberg, Erwin Schrödinger, Max Born,
More informationA unifying description of Dark Energy (& modified gravity) David Langlois (APC, Paris)
A unifying description of Dark Energy (& modified gravity) David Langlois (APC, Paris) Outline 1. ADM formulation & EFT formalism. Illustration: Horndeski s theories 3. Link with observations Based on
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationA Theory for the Cosmological Constant and its Explanation of the Gravitational Constant
A Theory for the Cosmological Constant and its Explanation of the Gravitational Constant H.M.Mok Radiation Health Unit, 3/F., Saiwanho Health Centre, Hong Kong SAR Govt, 8 Tai Hong St., Saiwanho, Hong
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANACHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationElectrostatic Fields: Coulomb s Law & the Electric Field Intensity
Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationDirichlet forms methods for error calculus and sensitivity analysis
Dirichlet forms methods for error calculus and sensitivity analysis Nicolas BOULEAU, Osaka university, november 2004 These lectures propose tools for studying sensitivity of models to scalar or functional
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationQuantum Mechanics I. Peter S. Riseborough. August 29, 2013
Quantum Mechanics I Peter S. Riseborough August 9, 3 Contents Principles of Classical Mechanics 9. Lagrangian Mechanics........................ 9.. Exercise............................. Solution.............................3
More informationModule 1: Quantum Mechanics  2
Quantum Mechanics  Assignment Question: Module 1 Quantum Mechanics Module 1: Quantum Mechanics  01. (a) What do you mean by wave function? Explain its physical interpretation. Write the normalization
More informationThe Role of Electric Polarization in Nonlinear optics
The Role of Electric Polarization in Nonlinear optics Sumith Doluweera Department of Physics University of Cincinnati Cincinnati, Ohio 45221 Abstract Nonlinear optics became a very active field of research
More informationThe Central Equation
The Central Equation Mervyn Roy May 6, 015 1 Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given
More informationLecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows
Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3. 1 Basics: equations of continuum mechanics  balance equations for mass and momentum  balance equations for the energy and the chemical
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is NonEuclidean Geometry? Most geometries on the plane R 2 are noneuclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationMeson cloud effects in the electromagnetic hadron structure
Meson cloud effects in the electromagnetic hadron structure Daniel Kupelwieser Thesis supervisor: Wolfgang Schweiger Collaborators: Elmar Biernat, Regina Kleinhappel Universität Graz Graz Jena monitoring
More informationAPPLICATIONS OF TENSOR ANALYSIS
APPLICATIONS OF TENSOR ANALYSIS (formerly titled: Applications of the Absolute Differential Calculus) by A J McCONNELL Dover Publications, Inc, Neiv York CONTENTS PART I ALGEBRAIC PRELIMINARIES/ CHAPTER
More informationarxiv:physics/0004029v1 [physics.edph] 14 Apr 2000
arxiv:physics/0004029v1 [physics.edph] 14 Apr 2000 Lagrangians and Hamiltonians for High School Students John W. Norbury Physics Department and Center for Science Education, University of WisconsinMilwaukee,
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationOperator methods in quantum mechanics
Chapter 3 Operator methods in quantum mechanics While the wave mechanical formulation has proved successful in describing the quantum mechanics of bound and unbound particles, some properties can not be
More information1 Complex Numbers in Quantum Mechanics
1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. However, they are not essential. To emphasize this, recall that forces, positions, momenta, potentials,
More informationThe de Sitter and antide Sitter Sightseeing Tour
Séminaire Poincaré 1 (2005) 1 12 Séminaire Poincaré The de Sitter and antide Sitter Sightseeing Tour Ugo Moschella Dipartimento di Fisica e Matematica Università dell Insubria, 22100 Como INFN, Sezione
More informationarxiv:1111.4354v2 [physics.accph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.accph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More informationLearning Vector Quantization: generalization ability and dynamics of competing prototypes
Learning Vector Quantization: generalization ability and dynamics of competing prototypes Aree Witoelar 1, Michael Biehl 1, and Barbara Hammer 2 1 University of Groningen, Mathematics and Computing Science
More informationDerivation of the relativistic momentum and relativistic equation of motion from Newton s second law and Minkowskian spacetime geometry
Apeiron, Vol. 15, No. 3, July 2008 206 Derivation of the relativistic momentum and relativistic equation of motion from Newton s second law and Minkowskian spacetime geometry Krzysztof Rȩbilas Zak lad
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
More information