A New Regularization Procedure for Calculating the Casimir Energy

A new regularization procedure for calculating the Casimir energy

Bahman Ghadirian

A thesis submitted as part of the requirement for the degree of

Doctor of Philosophy

University of Western Sydney

2008

Acknowledgements

I would like to thank my supervisor, Dr Reynaldo Castillo, for the opportunity to work with him and for his invaluable help and supervision throughout this project.

I would also like to thank my family specially my wife Roya for her continued support and encouragement throughout the time of my candidature.

Statement of Authentication

The work presented in this thesis is, to the best of my knowledge and belief, original except as acknowledged in the text. I hereby declare that I have not submitted this

material, either in full or in part, for a degree at this or any other institution.

……………………………………………………………………………

Contents

1. Introduction...... 1

2. Linear Response Theory ...... 14 2.1. Quantum mechanics ...... 15 2.1.1. Wave functions...... 15 2.1.2. Eigenvalues and Eigenfunctions...... 17 2.1.3. Linear operator ...... 17 2.1.4. Hermitian and Unitary operators...... 19 2.1.5. Hilbert space...... 19 2.1.6. Inner product and Norm...... 20 2.1.7. Expectation value ...... 21 2.1.8. Classical equation of motion...... 22 2.1.9. Dirac’s compact notation...... 24 2.1.10. Transformation...... 25 2.1.11. Time-evolutions...... 26 2.2. Thermodynamics...... 29 2.2.1. The first law of thermodynamics...... 29 2.2.2. Entropy...... 30 2.2.3. Reversible and irreversible processes...... 33 2.2.4. The second law of thermodynamics ...... 34 2.2.5. The free energy functions ...... 35 2.2.6. The Gibbs distribution and its free energy...... 36 2.3. Linear response theory...... 39 2.3.1. Evolution operator...... 39 2.3.2. External perturbation ...... 41 2.3.3. Generalized susceptibility...... 43 2.3.4. Thermal average...... 46 2.3.5. Correlation function...... 47 2.3.6. Spectral function...... 49 2.3.7. Different forms of the Green functions...... 50 2.3.8. Connections between various Green functions...... 56 2.3.9. Kramers-Kronig relation...... 57 2.3.10. Dissipation...... 58 2.3.11. Fluctuation-Dissipation theorem ...... 59

3. Kubo-Martin-Schwinger (KMS) State ...... 61 3.1. K.M.S Condition...... 62 3.2. Many-body problem...... 65 3.2.1. Microscopic property; Green’s function...... 65

4. Field Quantization...... 77 4.1. Canonical commutation relation ...... 79 4.2. Scalar field quantization ...... 80 4.3. Complex scalar field quantization...... 81 4.4. Fermion (Dirac) field quantization...... 83 4.5. Electromagnetic field quantization...... 88 4.6. Propagators ...... 92

- i - 4.6.1. Propagator of a scalar field ...... 93 4.6.2. Propagator of a Dirac field...... 96 4.6.3. Propagator of the electromagnetic field...... 98 4.7. S-matrix ...... 100 4.8. Wick’s theorem...... 103 4.9. Divergent results in quantum field theory ...... 105 4.9.1. Regularization ...... 106 4.9.2. Renormalization ...... 108

5. Casimir Effect ...... 110 5.1. Casimir effect in various field of physics...... 111 5.2. Vacuum fluctuation...... 112 5.2.1. An alternative look at the vacuum fluctuations...... 112 5.2.2. Quantum Field Theory and vacuum fluctuation...... 115 5.3. A simple model...... 118 5.4. Field energy ...... 121 5.5. Green’s function method for calculating the Casimir Force...... 124 5.5.1. Casimir force on conducting parallel-plates ...... 125 5.5.2. Casimir force on a conducting spherical shell ...... 128 5.5.3. Casimir force on a conducting cylindrical shell...... 138 5.5.4. Comparison of the Casimir force for different geometrical boundaries145 5.6. Casimir force as a limiting case of Van der Waals interaction...... 146 5.7. Zeta-function method for calculation of the Casimir force ...... 154 5.7.1. Riemann zeta-function...... 155 5.7.2. Generalized zeta-function ...... 156 5.7.3. The heat equation...... 157 5.7.4. Zeta function expansion...... 161 5.7.5. Zeta functional approach to the vacuum energy ...... 161 5.7.6. An advanced look at the Casimir energy ...... 164 5.7.7. Various behaviours of the Casimir effect with different geometrical boundaries...... 166 5.7.8. One-loop effective action...... 168 5.8. Casimir effect in the Kaluza-Klein model...... 171

6. A New Approach to the Casimir Energy for a Massive Scalar Field...... 178 6.1. Casimir energy for a scalar field...... 179 6.2. Casimir energy for a massive scalar field...... 182 6.3. A new technique...... 184 6.3.1. Casimir energy and renormalization...... 191

Conclusion...... 192

References ...... 197 Bibliography...... 202

Appendices ...... 204 Appendix A: Bessel’s functions ...... 204 Appendix B: Heat Kernel Expansion...... 207

- ii - Abstract

This thesis deals with the concepts of a very interesting phenomenon in quantum physics, the Casimir effect. Here the effect is investigated in detail and its importance to other areas of physics is analysed. The Casimir effect is produced by disturbing the vacuum energy when material boundaries or background fields are introduced in the vacuum. The usual approach to this effect is the vacuum fluctuation that has been studied in the past in relation to the discussion of the zero-point energy as a result of the field resemblance to the quantum harmonic oscillators, where residual ground state energy must be considered. In this thesis a new method to study vacuum fluctuations is presented. This new approach to the problem which is more classical is based on the Heisenberg uncertainty principle and the very important fluctuation-dissipation theorem. The other aim of the thesis is to implement a new algorithm for regularizing the Casimir energy for a massive scalar field. Unlike the previous works on this problem by other authors that give approximate results, this attempt will produce precise results. My method is based on a new regularization procedure that allows us to employ the very reliable dimensional regularization scheme in place of a more mathematically complicated zeta-function regularization procedure. In order to achieve this goal I will deal with the problem by using the Euler-Maclaurin summation formula. The result will be a regularized Casimir energy for the case of a massive scalar field. This model may be used for the other geometrical boundaries and different fields.

- iii - 1

Chapter I

1. Introduction

In this thesis, research has been done to study one of the most interesting results in quantum field theory, the so called Casimir effect. Since original discussion by Casimir, many works on this effect have been published. The most important approach to the problem is the Green’s function formulation, where the zero-point energy of the vacuum is identified by the vacuum expectation value of the stress- energy tensor. Also, separately the zeta function technique gives us an appropriate method to formulate the Casimir energy for various geometrical boundaries with different kind of fields, such as scalar, electromagnetic or fermionic fields.

The Casimir effect is a consequence of the vacuum or zero-point fluctuation, this results in the apparition of an observable force between material bodies that can be measured. This effect was suggested by H. B. G. Casimir in 1948. In his work,

Casimir predicted that there must be an attractive force between two parallel conducting plates in a vacuum due to vacuum fluctuation of the quantum field. The value of this force was calculated by himself as proportional to the inverse of the forth power of the distance between those bodies. The Casimir force between material bodies is thought to be the manifestation of the Van der Waals

2 intermolecular interaction of those bodies, and even when this force (Van der Waals force) is a classical force, its mechanism can be explained by quantum field theory.

In general we know that deviation from the mean value of a physical quantity in a system at equilibrium, which can be either macroscopic or microscopic, is a measurement of the fluctuation of that quantity. It can be shown that there is a relationship between the fluctuation of any system and the dissipation of the energy in that system. This relationship, which is called fluctuation-dissipation theorem, is formulated by the introduction of a physical quantity, which is called general susceptibility. Now if we consider a quantum field, as for example the electromagnetic field, the fluctuation of this field is related to the dissipation of the energy in the system, and here the general susceptibility is characterized by components of a three dimensional tensor of rank two related to the Green’s function.

Also according to Heisenberg uncertainty principle, the accuracy in measurement of some physical quantities is confined to a certain relationship between those quantities, as for example the one that relates the momentum of the system to the position, or the energy of the system to time, where in latter case when we measure the energy of the system more accurate, the measuring of the time gets longer and longer; the more accurate measured energy, the longer time elapse. Thus, if we consider the energy of a vacuum equal to zero, the time consumed measuring this zero energy will tend to infinity, which is not acceptable for a physical quantity.

Therefore, according to the uncertainty principle, measuring a precise value for the energy of a system leads us to lose accuracy in measuring the time. Thus the uncertainty principle as well as the fluctuation-dissipation theorem for this system predicts that there must be a field fluctuation in the vacuum, since otherwise we

3 would get zero energy density, which is a certain value for energy that is in contradiction with the uncertainty principle. The vacuum fluctuation of a quantum fields leads us to measuring the energy of the vacuum, which is called zero-point energy. According to the field theory the ground state energy of a fluctuating field is

1 identified by 2 Zω , and the sum of the zero-point energies over the different modes as the vacuum expectation value of the field energy may be easily given. Also this agrees with the Bohr’s suggestion that the cause of the intermolecular forces which are responsible for the Van der Waals interaction, is the zero-point energy.

Other approaches to the fluctuation-dissipation relation are the linear response theory and the Kubo-Martin-Schwinger (KMS) formula. The KMS formula will be explained shortly; first we talk about linear response theory. We start with the definition of thermodynamic equilibrium. First we consider a measurable quantity or an observable as a linear self-adjoint operator mapping an appropriate Hilbert space to itself. The eigenvectors of these self-adjoint observables form a normalized and mutually orthogonal vector space, are called the eigenspace. The pure state or vectors of this space are represented by wave functions. There is a relationship between an observable and its eigenvector, which is characterized by a probability value, named the eigenvalue of that operator, these is the probability that the system can be found in a particular pure state. The above-mentioned relationship in this case relates the pure states to a particular operator that is called probability operator, which describes the state of the system. To express the linear response theory we need to know basically what an equilibrium state or a Gibbs state is. For this purpose we introduce the definition of the entropy as the level of the mixture of pure states, followed by the second law of thermodynamics, which expresses that the entropy of a system increases in the course of time, until it reaches a maximum, the state with maximum

4 entropy is called equilibrium or Gibbs state, and it reads as an exponential function of (F-H), where F is the free energy and H the Hamiltonian of the system.

To proceed on the discussion of the linear response theory we look at some quantum evaluation of a system with time. First we look at the Heisenberg and

Schrödinger equations of motion, and then we will talk about the interaction picture of the equation of motion, where the Hamiltonian of the system is divided into a time independent part (which is the Hamiltonian of the system) and a time evolving perturbation part Vt. Now if we perturb a Gibbs state, by considering a perturbation

Vt in the Hamiltonian, and applying the equation of the motion in the interaction picture, we relate the linear response theory to the corresponding perturbation.

Employing the linear response theory, we can calculate the expectation value of an observable M which gives us an expression for the response function.

The dynamic of a quantum system can be described by a time evolution of an unitary operator eitH. This operator has the same form as the equilibrium or Gibbs weight factor e-H. There is a relationship between equilibrium state (A) and evolution t(A) of a system which is characterized by the KMS condition. The KMS condition is defined on any pair of observable A and B of the system, where the equilibrium state (At(B)) is identified by (t(B)A) by an analytic continuation in a sufficient large strip of the complex plane around the real axis. The KMS state is indeed a state that acts like a Gibbs state, when there is a time evolution for the system, and using this state, the thermodynamic variables such as the number of particles, the energy, chemical potential and etc. can be formulated. The linear response theory and KMS formula help us to formulate the fluctuation-dissipation theorem of Callen and Welton with a new and easy approach. Therefore as a consequence the KMS state can be used to study the vacuum fluctuation of a field.

5 That is so since in the new concept of vacuum, which is called physical vacuum, the field which the vacuum consists of, must have fluctuations, therefore the physical vacuum is thought as a gas of virtual particles like photons. Thus the quantum statistical mechanics, where the linear response theory is derived and many-particle theory of Kubo-Martin-Schwinger are very helpful in a detailed understanding of the

Casimir effect in this thesis.

A formulation to the vacuum fluctuation comes from a very important physical entity, which is called Green’s function of the system. The Green’s function is used in many areas of physics; one of its most important applications is in quantum field theory. The Green function as the vacuum expectation value of the time-ordered product for the quantum field appears in many places. For example when we have an equation of the motion for a specific quantum field, using the Green’s function leads us to the solution, by substituting this term in place of the field equation, and replacing the source in the right hand side of the equation by an appropriate delta function. The solution for the quantum field therefore will be given by a convolution equation.

The electromagnetic field fluctuation will be formulated, using the Green’s function. In this sense the quantum field will be replaced by the electromagnetic field potential in the Green’s function of the photon. The expression for the generalized susceptibilities appearing in the fluctuation-dissipation theorem, now will be the retarded Green’s function for the photon.

The Casimir force between bodies having different dielectric constants can be interpreted as the Van der Waals potentials between the molecules of those bodies, when the retarded effect becomes important this potential is in a range of (1/r7) and when we are dealing with short range effect of this potential it is of (1/r6). Since the

6 range of the Van der Waals interaction is large compared with interatomic distances, its influence on the thermodynamic properties of the body is important. In the calculation of the Van der Waals force the long-wavelength electromagnetic field is used, and in this concept not only the thermal fluctuation, but also the zero-point oscillation of the field is included. One distinguishable property of this interaction is that the free energy is not additive. It follows that the free energy depends on the shape of the body. This property becomes important when the characteristic dimensions, although large compared with the interatomic distance, are sufficiently small. In the calculation of the Van der Waals interaction, which results in an expression for the Casimir force, we consider a change in the free energy of the medium due to a small change in the permittivity of that medium. This change can be given as a small variation in the Hamiltonian of the system. Using the free-photon

Green’s function, and the Matsubara operator as the field operator, the diagram technique of quantum field theory gives us an evaluation of the Van der Waals stress tensor. By applying this stress tensor to solid bodies whose surfaces are very close to each other, we arrive at an expression of the forces between those solids. The interaction between the bodies in this case is regarded as coming from the fluctuation of the electromagnetic field, with large wavelengths of the order of the distance between those bodies. In the case of a small electromagnetic field wavelength we get an expression for the force, which can be reduced to a very simple form when we consider the bodies made of metals. This force is exactly the same as the famous description for the Casimir force. Therefore it can be concluded that the Casimir force can be regarded as a limiting case and macroscopic manifestation of the Van der Waals molecular interaction.

7 It can be shown that the zero-point energy can be identified with the vacuum expectation value of the field energy or the zero-zero components of the stress tensor, but the equation relating these quantities will be divergent in the vacuum. The way we will treat this relationship is to consider the change in the zero-point energy when a material boundary is present or when we have a background field that restrict the modes of the field in some direction. Therefore we can calculate the Casimir force, in fact it is given by the normal-normal component of the stress tensor for the unit area of the bodies

f= dxdy T . (1.1) — zz

According to the above discussion, the Casimir force can be calculated for various geometrical boundaries and different quantum fields, starting from the zero- point energy and the stress tensor. The vacuum expectation value of the stress tensor can be obtained from the Green’s function, and considering the boundary condition for each particular field after some calculation gives us the Casimir force for that geometrical boundary.

The Casimir force calculation for two parallel plates independent of the degree of the conductivity, shows us that this force will be an attractive force in view of its connection to the Van der Waals interactions. Casimir was hoping that there is a similar attractive force between spherical shells and he suggested a stabilizing stress, called Poincare stress for an electron’s model by assuming the electron as a conducting spherical shell with charge e, therefore there will be a balance between the Coulomb repulsive force and the Casimir attractive force. But it was Boyer who showed in his work that there is no such an attractive force for spherical shells in fact the Casimir force in this case will be repulsive. According to this fact, it was thought that for an intermediate geometrical boundary such as a cylindrical shell, the force

8 should be zero. But the Casimir effect for this boundary again resulted in an attractive force.

The Casimir effect has been studied in many areas, such as quantum fluctuation in the curved space of general relativity to the quark and gluon fluctuation in a bag approximation in QCD. In the advanced view of this effect, it can be thought as the energy distortion of the vacuum due to presence of the boundaries or some background field such as gravity.

The Casimir effect can also be investigated in the context of the heat-kernel and zeta function technique. The Casimir energy in this method is thought as proportional to the principal part of a zeta function, applied to an arbitrary manifold in both flat and curved space-time, with or without boundaries

1» 1 ÿ ECasimir =2Z cµPP ζ3 ( − 2 + ε ) ⁄ , (1.2)

where is a normalization scale. The zeta function will be related to the heat-kernel expansion, whose trace can be related to two kinds of coefficients. The first one is a function of the gravitational field, and the second kind are functions of the extrinsic curvature, intrinsic curvature, and the nature of the imposed boundary. Therefore we conclude that the Casimir energy is actually characterized by the geometry of the boundary itself, where the sign of the energy gives us the attractive or repulsive behaviour of the force. It can be shown that this sign is related to the some particular coefficient of the heat-kernel expansion. This explains the different behaviour of the force.

The vacuum energy as the sum over the different modes of eigen-frequency value of the Hamiltonian is indeed divergent and the most elegant mathematical method used for extracting a finite value for this energy is called zeta function regularization. As it was explained before the zeta function can be expressed in terms

9 of the heat-kernel expansion. In this regularization the zeta function will be written in terms of a heat equation corresponding to a Laplace type differential operator acting on a vector bundle over a manifold, where the trace of the heat equation is equal to the volume integral of the heat-kernel. The heat-kernel itself will be expressed by an expansion, which has different coefficients for the manifolds with or without a boundary. As it is expected from any regularization model we will get some divergent part depending on the regularization parameter. The way of treating this problem is a physical procedure that is called the renormalization. The zeta-function regularization is a very useful tool that identifies a relationship between the Casimir energy and the one-loop effective action in quantum field theory.

It also seems that the Casimir energy depends on the topology of the space

[50, 4]. For example for a space with non-trivial topology such as a circle, which is one-dimensional, we will get non-zero energy. This is the starting point of the explanation for the mechanism of the compactification of the space-time in Kaluza-

Klein theory. This prescription for the simultaneous compactification can be further used in much advanced theory such as super-string theory.

The dependency of the Casimir energy on the topology can be investigated by applying the famous Gauss-Bonnet theorem. Although the relationship between the

Casimir energy and the curvature of the surface is completely geometrical, but according to the Gauss-Bonnet formula the Gaussian curvature of the manifold is directly related to a topological number characterized by 2 times the Euler characteristic of the manifold. This amazing theorem, which relates two completely independent branches of mathematics, helps us to formulate the Casimir energy in terms of the topological number of the corresponding manifold. This is a potential area of research in the Casimir effect, but we will not talk about that in this thesis.

10 In this thesis I will concentrate on formulating a new method of the Casimir energy regularization for a massive scalar field. In this new approach I develop a model for regularization purposes, where the dimensional regularization procedure will be employed. The main concept is that I actually follow the similar way of calculating the scalar field, by starting from vacuum energy and adapting some boundary conditions in the case of the Casimir energy in presence of a material boundary. As we will see we reach a point where we have a contribution of a sum over discrete modes of the fluctuation; this sum, in the case of a massless scalar field will be treated easily by a Riemann zeta function, but for a massive scalar field we need more work on it. As we will see an elegant procedure for regularizing an integral is called dimensional regularization method. As it is realised this is a procedure applicable on a divergent integral, but in our case we have a divergent sum. The way I overcome this difficulty is by using a useful mathematical formula for expanding the sum, which is called Euler-Maclaurin sum formula. By employing this formula we convert the divergent sum into integrals form that later will be regularized by a dimensional regularization procedure. An appropriate renormalization scheme to get ride of a regulator depending divergent part, gives us an expression for the Casimir energy of a massive scalar field. This will be a new result with an exact value for the contributing massless scalar field plus the additional correction part for the case of massive scalar field.

This thesis is organized as follow; in chapter 2, I will discuss the linear response theory. As it was mentioned before in the linear response theory we study the perturbation of a system from equilibrium, when an external force is applied to the system. This is actually the fundamental part of our discussion of the Casimir effect, since this effect is all about the vacuum fluctuation of a field, and in this

11 theory we study how the energy dissipated into a system results in a fluctuations of physical quantities in that system. For a good understanding of this theory, a detailed background on quantum mechanics and thermodynamics is essential. Concepts such as Heisenberg’s, Schrödinger and interaction pictures are needed, as well as concepts like Gibbs state and many others in thermodynamics, must be completely clear.

Therefore in this chapter, a detailed study in quantum mechanics and thermodynamics will be given.

Chapter 3 develops the alternative approach of the fluctuation-dissipation theorem by an analytic continuation of the time in the complex plane suggested by

Kubo, Martin and Schwinger. This continuation is indeed a consequence of exploiting the fact that the Gibbs weight factor, which is an exponential function of energy, e−βH is in fact in the same form as the unitary operator of the time evolution eiHt / Z of a system, which is again an exponential function of the energy. This was discussed before in this chapter as the KMS state, where a non-equilibrium situation with a time evolution of the system is discussed in the context of Gibbs state. The expansion of the time in the complex plane gives us a description for the fluctuation- dissipation theorem. This expansion makes up the famous KMS condition, which has many applications in many-particle physics especially it helps us to formulate the vacuum fluctuation, and subsequently Casimir energy.

Next chapter: chapter 4 is devoted to the quantum field theory. The Casimir effect as a result of fluctuation of the quantum fields in vacuum state is studied. The basic outcome of the quantization of a field, that is considering the field in each point of space-time as a set of harmonic oscillators, is the foundation of zero-point energy and therefore Casimir energy. To the fundamental idea of the field and its quantization, we add the creation and annihilation of the particles and anti-particles,

12 which describes the vacuum fluctuation, and also the interaction of the different fields. In this chapter we review the quantization of the different kind of field and then the corresponding Green’s function or propagator for each of them, since they are used in the formulation of the Casimir effect. The S-matrix and Wick’s theorem will be considered later in the Van der Waals interaction formulation. The general recipe for the regularization and renormalization will be reviewed, as the computation of the vacuum and Casimir energies involve many divergent expressions.

I focus on the Casimir effect in chapter 5. I will start from scratch by talking about vacuum fluctuations. The usual approach to the vacuum fluctuation comes from quantum field theory, when any quantum field is considered as a harmonic oscillator in each point of space-time. In this chapter I bring a new look at this problem by starting from the Heisenberg uncertainty principle and the fluctuation- dissipation theorem, and then I recover the concept of Vacuum fluctuation by employing two theorems. This is a new and original approach.

The calculation of the Casimir effect for different geometrical boundaries by using the Green’s function will be given and it will be shown that the Casimir force is actually a macroscopic manifestation of the Van der Waals interaction. Next I will talk about a more advanced model for investigating this effect with the zeta-function method. In these sections I clarify some basic idea for the zeta functions; we will see later that the zeta function can be written in terms of the heat function of a Laplace type operator. The expansion of this function will be discussed and the modern approach to the Casimir energy will be investigated. The result of this model can be seen by relating the Casimir energy with one-loop effective action. And finally I will show how the Casimir effect is a true nominate to the mechanism of the dimensional

13 spontaneous compactification in Kaluza-Kline theory. This beautiful subject leads the discussion to the super-string theory, where we have similar dimensional compactification not for one, but sometimes for seven additional dimensions.

Finally in the last chapter; chapter 6, I develop a new technique for confronting the problem of regularization of the Casimir energy for a massive scalar field in a cavity of two parallel planes with Dirichlet boundary conditions. Although there are methods to determine this energy specially given by Ambjorn and Wolfram in [6], but they are associated with an integral in the end, which in terms can be converted into a sum of Bessel’s modified functions, and therefore an approximate result is achieved. In my new technique I regularized the energy by an integral dimensional regularization method with the aid of Euler-Maclaurin sum formula. A proper renormalization scheme gives us a new but consistent result.

I will bring the additional but necessary information about the mathematical functions that are called Bessel’s functions in the appendix A. This function has many applications throughout my thesis especially in the calculation of the Casimir energy. In appendix B we see how the expansion of the heat kernel that are very important in determining of the behaviour of the Casimir force, can be done.

Chapter II

2. Linear Response Theory

The linear response theory as the starting point of our study is given in this chapter. This is a theory that investigates the response of a system to perturbations, when the system is at equilibrium. This is called a linear theory because only the first order term of the response to the perturbations will be considered [5, 33]. There are some non-linear response theories where the perturbation is a second order, third order and etc. which will not be considered in this thesis. We will see later that linear response theory results in an important theorem that is called fluctuation-dissipation theorem [1, 2, 33]. This is the theory the idea of our work on vacuum fluctuation and consequently Casimir effect originated from. In this chapter we will discuss concepts such as the thermodynamic equilibrium, Gibbs state and different quantum equation of motion, where some basic definitions in the thermodynamic and quantum mechanics need to be analysed.

15 2.1. Quantum mechanics

2.1.1. Wave functions

The foundation of the quantum mechanics lies on the concept of the wave function that depends on the space coordinates and the time [16]. Each state of a quantum system can be described by this function (q), where q is space and time coordinates. This quantity has three basic properties:

1. It can interfere with itself, so it can be used in experiments involving diffraction.

2. The magnitude of this function will be proportional to the likeliness of the existence of the particle or the photon in particular coordinates.

3. The wave function describes the behaviour of a single particle not the statistical distribution of a number of such quanta. Therefore a wave function will interfere with itself rather than with other quanta

The square of the modulus of the wave function which is the product of and its complex conjugate *

2 P()()()() r,*,,, t=ψ r t ψ r t = ψ r t (2.1)

gives us the position probability density. This is the probability of finding a particle in the volume element dr at time t.

The sum of the probability of finding the particle over all space must be equal to the unity and this gives us one important property of the wave function which is called normalization of this function

2 — ψ ()r, t d3 r = 1 (2.2)

Consider the eigenfunction E(r) corresponding to the total-energy operator for a particular quantum state shown by the wave equation

2 »Z 2 ÿ …− ∇ +V()()() r ŸψEE r = E ψ r . (2.3) 2m ⁄

∗ r Now from the above equation it can be shown that ψ E '( ), which is the complex conjugate of the E(r) also satisfies the same wave equation, where we are considering a new eigenvalue E’

2 »Z 2 ÿ * * …− ∇ +V()()() r Ÿψ'' r = E ψ r . (2.4) 2m ⁄ EE

* r (r) Multiplying the equation (2.3) by ψ E '( ) and equation (2.4) by E and integrating the difference between these two equation over the volume L3 of the cube gives us

2 Z * 2 2 * 3 ' * 3 −()ψ'' ∇ ψEE − ψ ∇ ψd r =() E − E ψ' ψ E d r. (2.5) 2m —EEE —

Using the Green’s theorem for integrals in the left hand side we will evaluate this integral over the surface of this cube

* 2 2 * 3 * * ψ''''∇ ψEEEE − ψ ∇ ψd r = ψ ∇ ψ − ψ ∇ ψ dA (2.6) —( EEEE) —A ( )n

where the subscript n for the integrand in the right hand side indicates that this integrand is a vector normal to the surface of the cube. Since each wave function and its normal derivative have the same values, the vectors normal to the opposite face of

A cancel each other and the right hand side of (2.6) will vanish, therefore for the case of E ≠ E ' the integral in the right hand side of (2.5) will be equal to zero

ψ* ψ d 3 r = 0 (2.7) — E' E

This condition for the wave function E and its conjugated is called orthogonality of the wave vectors. Since according to equations (2.2) and (2.7) wave functions, each of which is normalized and orthogonal to each other we say that the wave functions are orthonormal. This relation leads us to:

ψ*r ψ r d 3 r = δ (2.8) — E'( ) E ( ) EE'

E=E’ where δ EE ' is the Kronecker symbol, which is equal to unity when and is zero otherwise.

2.1.2. Eigenvalues and Eigenfunctions

Suppose we have a physical quantity that characterizes the state of a quantum system, the value of this quantity in a particular state of the system is called the eigenvalue of that quantity shown by , where the suffix takes the values

0,1,2,… . The wave function of the system in this state can be denoted by .Now the wave function is called the eigenfunction of that system. The set of eigenvalues is called the spectrum of the eigenvalues of that physical quantity, which can be either a continuous spectrum similar to the value in the classical mechanics, where these values run over a continuous spectrum, or it can be discrete values, when eigenvalues form a discrete set of the values.

2.1.3. Linear operator

Dynamical variables such as the coordinates, momentum components, and energy of a particle in the Schrödinger picture of the quantum mechanics, can be

18 represented by linear operators. These operators can be given in a simple representation such as a multiplication operator for the position r, or they can be a differential operator such as − iZ∇ for the momentum. For example a comparison of the Schrödinger equation for a free particle representing by the wave function (r,t)

∂ψ Z2 iZ = − ∇2ψ (2.9) ∂t2 m

and energy equation

P2 E = , (2.10) 2m gives us an expression of the differential operator for the energy and the momentum

∂ E → iZ and P→ − iZ ∇. (2.11) ∂t

Associated to each linear operator , there is an equation of linear eigenvalue

Ωυµ = ω µ υ µ , (2.12)

where υ µ is the eigenfunction of corresponding to the eigenvalue .

If dynamical variables appear explicitly in the equation of the motion of the system without any further mathematical operation such as multiplication and differentiation of the wave equation as it is done in the Schrödinger’s theory of the quantum mechanics, we have a resemblance between the classical mechanics and a different quantum formulism, which is called matrix or Heisenberg picture in quantum mechanics. In this case the quantum-dynamical variables do not obey the commutative law of multiplication, so they can be represented by matrices, and will be simply called operators.

19 2.1.4. Hermitian and Unitary operators

Since operators are matrices we generalized the concept of the hermitian adjoint in matrix calculus to the operator. A hermitian adjoint matrix A, which is shown by A† is a matrix whose rows and columns have been interchanged and a complex conjugation of each element have been taken

†† AAkl= lk (2.13)

It can be shown that the hermitian adjoint of the product of a series of matrices can be obtained by reversing the order in products of the adjoints of each matrix.

()ABC † = CBA††† (2.14)

An operator or a matrix is called self-adjoint or hermitian, when it is equal to its hermitian adjoint

AA= † (2.15)

If the hermitian adjoint of a matrix is equal to its inverse we say that matrix is unitary

B†= B− 1or BB † = B † B = 1 (2.16)

From the above discussion and since and unitary matrix is non-singular, it is obvious that the unitary matrices of finite rank must be square.

2.1.5. Hilbert space

Suppose we have a vector space of infinite dimension whose vectors are regarded as the state function of a quantum system. In this space, vectors are called the state vectors, and they are shown by column matrices. We consider different

20 representation of this system by choosing various axes orientation in the space, since we have different values for the vector’s components related to the various axes orientation and this constitute the different representation of the state. Each unitary transformation corresponds to change from a particular representation to another one, therefore they will give us a tool for rotating the coordinate axes in this space, without making any change in the vector states. This space is called Hilbert space and it is a very useful concept in the matrix theory of the quantum mechanics. There is another transformation, which is called generalized rotation of the state vectors, where in this case the rotating objects will be the state vectors themselves in the

Hilbert space. The generalized rotation involve not only the rotation of the vectors but also contracting and stretching of the components, therefore it gives a new state vector when we apply for example the operator to the state vector , and we will get the new vector .

2.1.6. Inner product and Norm

The inner product of two state vectors and which is a number is defined as

ψ, ψ= ψ† ψ = ψ∗ r ψ r d 3 r (2.17) ( α β) α β— α( ) β ( )

The inner product of two state vectors gives us a way to examine the orthogonality of those two state vectors simply by considering the value of the inner product equal to zero, in this case we say that they are orthogonal.

When the subscripts and in the equation (2.17) are the same, we have the definition of the norm of a state vector, and it is the square of the length of the corresponding state vector in Hilbert space.

2.1.7. Expectation value

We start by defining the expectation vale of the position vector r, whose components are the average components of the position of the particle. In this definition we use the concept of the position probability density P(r,t) from equation

(2.1). The expectation value is considered as the result of a single measurement or the average of the results of a large number of measurements, and it can be identified by

r=— rPrtdr( ,,,) 3 = —ψ∗ ( rtr) ψ ( rtdr) 3 (2.18)

The expectation value for the position is a function of the time since the state vectors are a function of the time and space and the integration is done over the space. A similar expectation value can be calculated using the position probability density for other physical quantities such as energy and momentum of the system. Now we can write the expression for the expectation value of the energy in the form

P2 EV= + (2.19) 2m

where V is the corresponding potential energy of the system. From (2.11) the above equation can be written in terms of the differential operators as follows:

∂ Z2 iZ = − ∇2 + V (2.20) ∂t2 m

From the equation and definition of the expectation value for a physical quantity we conclude the expressions for the energy and momentum expectation values are:

∂ψ E=ψ∗ iZZ d3 r P = ψ∗ () − i ∇ ψ d3 r (2.21) —∂t —

2.1.8. Classical equation of motion

In classical dynamics an equation of motion of a dynamical system that possesses f degrees of freedom can be characterized by the Lagrangian of the system

L that is a function of the generalized coordinates qi, the generalized velocities q"i ,

where i=1,2,…,f and the time L(q1,…,qf, q"1 ,…, q" f ,t) and also by action of the system

t2 A dtL q t q" t t (2.22) = — (,,)i()() i t1

Now we study the evolution of the system between t1 and t2. According to the principle of least action among all the trajectory that join qi(t1) to qi(t2), the system follows the trajectory that makes the action of the system stationary

t2 δA= δ — Ldt = 0 (2.23) t1

We define an infinitesimal variation around a particular trajectory as

d qt()()()()()→ qt +δ qt, qt"" → qt + δ qt() (2.24) i i i i idt i

The corresponding variation on the action will be

t 2 »∂L ∂ L d ∂ L ÿ δA= dt… δ qi + δ q i + Ÿ (2.25) — ∂q ∂ q" dt ∂ t t1 i i ⁄

The last term in the above integration vanishes, since the system is conservative and the Lagrangian does not depend explicitly to the time. Integrating by part gives us

t t2 »∂L d ∂ L ÿ ∂ L 2 A dt q q δ=— … − Ÿ δi + δ i (2.26) t ∂qi dt ∂ q"" i ∂ q i 1 ⁄ t1

23 The end points of the trajectory will not vary, then last term vanish

δqi( t1) = δ q i ( t 2 ) = 0 (2.27)

The principle of least action states that near the classical trajectory the variation of the action should be zero, and then we arrive at the famous classical equation of

Euler- Lagrange:

d≈∂ L ’ ∂ L ∆ ÷ = (2.28) dt«∂ q"i ◊ ∂ q i

Now we define the Hamiltonian function as

H( q1 ,..., qf , p 1 ,..., p f ,) t=ƒ p i q" i − L (2.29) i=1

where pi is the momentum conjugate to qi

∂L pi ≡ (2.30) ∂q"i

The Hamiltonian equation of motion is obtained by a variation of Hamiltonian H:

∂HH ∂ q""i= p i = − . (2.31) ∂pi ∂ q i

For any Function F of coordinates, momentum and time, the time dependency can be expressed as

d∂ Ff ≈ ∂ F ∂ H ∂ H ∂ F ’ F( q1 ,..., qf , p 1 ,..., p f ,) t = +ƒ∆ − ÷ (2.32) dt∂ ti=1 « ∂ qi ∂ p i ∂ q i ∂ p i ◊

where equation (2.31) has been employed. Now we define the Poisson bracket of any two functions A and B as

f ≈∂ABBA ∂ ∂ ∂ ’ {}AB, =ƒ∆ − ÷. (2.33) i=1 «∂qi ∂ p i ∂ q i ∂ p i ◊

It can be shown that the Poison bracket satisfies following properties:

{A, B}= −{B, A} { , cA }= 0 c is a number

{( AABABAB1,.,, 2) } ={ 1} +{ 2 }

{}{}{}AABABAAAB1 2,,,= 1 2 + 1 2 (2.34) {}ABCBCACAB,{} ,+{} ,{} , +{} ,{} , = 0

Considering the definition of the Poisson bracket the equation(2.32) is reduced to

dF∂ F = +{}FH,, (2.35) dt∂ t

this is the classical equation of the motion for a function of the dynamical variables.

2.1.9. Dirac’s compact notation

In this section we introduce the Dirac’s notation for a state function which is a compact form for this quantity, which it may be used in the rest of this thesis. First we introduce the notation for a state function or state vector by a ket vector α ,

+ and its hermitian adjoint state by α . In this notation the inner product will be represented by (2.17),

† ψα ψ β = α β (2.36)

The matrix element of a dynamical variable or an operator between states and is written as

Ω =ψ∗ r Ω ψ r d3 r = α Ω β (2.37) αβ— α( ) β ( )

and the matrix element of the hermitian adjoint operator will be represented in this notation according to definition of a hermitian adjoint operator by

† ∗ ∗ Ωβα = Ω αβ =α Ω β (2.38)

The eigenvalue of a coordinate eigenstate r , is identified by r, then the following identity can be verified using Dirac’s notation:

ψα (r) = r α (2.39)

2.1.10. Transformation

In quantum mechanics we work on three different quantities, each of one is represented by a different kind of matrix. The first of these objects specify the state of a quantum system at a particular time, and it is in fact the state function or vector in Hilbert space shown by or α , this quantity can be represented by a column matrix. The second is actually a rectangular matrix called the unitary matrix this object rotates the axes of the Hilbert space resulting in a different representation for the quantum system, without any changes in the state vector, the elements of which are shown by rk , this matrix transforms a vector state α from a particular representation to another. The last is a dynamical variable, which is indeed an operator, and it is represented by a square matrix. A unitary matrix which rotates the axes in the Hilbert space, gives the elements of this variable in various representations, for example it transforms this object from a coordinate representation to the energy representation. If elements of a dynamical variable are calculated relative to a particular state vector , these matrix elements are invariant

26 with respect to the unitary transformation, and they are shown by α Ω β . A vector state α can be acted by an operator , and the result would be another vector state Ω α , this transformation is actually a generalized rotation in the Hilbert space.

In a particular case when the vector state is the eigenstate of , the transformation will be a contraction or an extension of the vector in Hilbert space without any change in its direction.

2.1.11. Time-evolutions

There are three different views for time evolution of the state functions, or the dynamical variables, or both of them by means of a unitary transformatione −iHt / Z . In the first view which is called Schrödinger picture the state vectors are variable and dynamical variables are constant in time; this picture is identified by the time- dependent Schrödinger equation

d iZ α() t= H α () t (2.40) dt SS

If the Hamiltonian H is time-independent the above equation possesses a simple solution of

−iHt / Z αSS(t) = e α (0) (2.41)

where e −iHt / Z is an infinite sum of H, therefore it is an operator or a square matrix,

which transforms the state vector α S (0) into state vector or ket α S (t) , so it performs a generalized rotation. The above operator is also a unitary operator, and it conserves the length of the vector.

27 The second view for the time-evolution of a quantum system is called the

Heisenberg picture, in this picture the state vectors are considered as quantities which are constant in time and the dynamical variables are time-variant. In this picture the time-independent state vectors are defined as

iHt / Z αHSS(t) ≡ α(0) = e α ( t) (2.42)

and time-dependent dynamical variables are

iHt//ZZ− iHt ΩHS ≡e Ω e . (2.43)

The equation of motion in the Heisenberg picture is given by

dΩ ∂Ω 1 HH= +[] Ω , H (2.44) dt∂ t iZ H

where the bracket in the right hand side is called a commutation bracket, and it is defined by any two operators A and B as

[ A, B] ≡ AB − BA . (2.45)

The equation of motion (2.44) is very similar to a classical equation of motion given by equation (2.35) except that the Poisson bracket in the classical equation of motion has been replaced by commutation bracket in the Heisenberg picture. Therefore this resemblance suggests that a transition from a classical equation of motion to its quantum analogous is possible by dividing the commutation bracket by iZ , for the

Poisson bracket,

1 {}ABAB,,→ []. (2.46) iZ

28 There is another description of the equation of motion which is known as

Interaction picture, where time dependency will be assigned to both state and dynamical variables. In this picture the Hamiltonian will be divided into two parts

HHH=0 + ', (2.47)

where the first part, the Hamiltonian, is time-independent; the additional part H’ can be for example considered as an interaction part or the potential part of a total

Hamiltonian. This picture is defined by the following two equations:

iH0 S t / Z αIS(t) ≡ e α ( t) (2.48)

and

iH0SS t//ZZ− iH 0 t ΩIS(t) ≡ e Ω e (2.49)

The equation of motion in this picture can be obtained by differentiating the above equations which gives us

d iZ α() t= H′ α () t (2.50) dt III

and

dΩII ∂Ω 1 = +[] ΩII, H0 (2.51) dt∂ t iZ

The change of the state vector and the dynamical variable in time in this picture is related to the second part and first part of the Hamiltonian(2.47), respectively.

29 2.2. Thermodynamics

In the following sub-sections some basic definition in thermodynamic will be given, in fact this is an introduction to thermodynamics, where a knowledge on subjects such as the first and second laws of thermodynamics, entropy, reversible and irreversible processes, thermodynamic equilibrium and etc. are necessary for working on and understanding the linear response theory which will be discussed in this chapter.

2.2.1. The first law of thermodynamics

From an experimental point of view it is evident that a cyclic integral of ČQ -

ČW is zero, where the first contribution is the heat and the second one is the work:

— (ČQ–ČW) = 0 (2.52)

which indicates an exactness property of this differential. Therefore this will be equal to the differential of a new quantity we call this new function the internal energy, so the differential of the energy is given by

dE = ČQ – ČW. (2.53)

Equations (2.52) and (2.53) are both mathematical expressions for the first law of thermodynamics, and these indicate that the energy flows in a system either by transferring the heat or doing work on the system.

This law can also be explained quantum mechanically. From quantum mechanics, it can be shown that the expectation value of an observable M, is

M= ƒ wj(ψ j, M ψ j ) (2.54) j

30 where wj is the eigenvalue of a probability operator W associated to the state function

Wψj= w jψ j (2.55)

therefore according to the definition of the inner product and keeping in mind that the diagonal elements of a non-diagonal operator are in fact the expectation values of that operator we reach to the following relationship

M= trWM . (2.56)

Now consider the energy operator H, which is the Hamiltonian of the system, the expectation value of this operator is denoted as the internal energy E, and then we will get

E= H = trWH . (2.57)

Since the internal energy of a system will change either by changing the state of the system or by variation of the Hamiltonian, we get:

δE= tr δ WH + trW δ H . (2.58)

Now the first part of above equation is heatδQ , and the second contribution is the workδW , so we reach again to the first law of thermodynamics expressed in equation (2.53).

2.2.2. Entropy

Suppose we have a closed system which is in statistical equilibrium, the statistical distribution function in this system denoted by wn, may be written as a

31 function of the energy wn =w(En ). The number of quantum states with energies less than or equal to E is denoted by (E), and the number of states with energy between

E and E+dE will be

dΓ( E) dE , (2.59) dE

The probability distribution of the various values of the energy can be calculated by

dΓ() E W()() E= w E . (2.60) dE

Now if the maximum value of the probability is associated to the energy E , then according to the normalization condition

WEE( )∆ = 1 (2.61)

and using (1.55), we will get

w( E) ∆Γ =1, (2.62)

where

dΓ( E ) ∆Γ = ∆E (2.63) dE

Equation (2.62) for a quantum system may be written for classical situations where ∆Γ will be replaced by the volume of the part of the phase space ∆p∆q , then we will have

ρ (E) ∆ p ∆ q =1 (2.64)

32 On the other hand from quantum mechanics it is known that relation between

∆Γ and ∆p∆q can be written as

∆Γ = ∆p ∆ q /() 2π Z s , (2.65)

where s is the number of the degrees of the freedom of the system. Now we define a very useful function in thermodynamic as

S =log ∆Γ , (2.66)

which is called the entropy of the system, the classical counterpart of this function can be written, using (2.65)

∆p ∆ q S = log (2.67) ()2π Z s

From quantum statistical mechanics it can be shown that, the logarithm of the distribution function for a system must be in the form

log w( En) =α + β E n (2.68)

by the linear properties of this expression we have a similar relation for the logarithm of the distribution function of E ,

logw ( E ) =α + β E (2.69)

therefore this is the mean value log w(En ) . Now the entropy (2.66) can be written according to (2.62) and above discussion as

S= − log w( En ) (2.70)

33 working on the mean value in the above expression gives us another form for the definition of entropy

S= −ƒ wnlog w n (2.71) n

In a general operator form, the above formulation for the entropy may be written as

S( W )= − tr ( W log W ). (2.72)

In the above expression W is actually a mixed state of mixture of pure states with probabilities wn, and the entropy S (W) is a measure of the degree of mixture. Also from the classical point of view the change of the entropy of a system can be defined by the following equation:

dQ dS = rev , (2.73) T

where the subscript for dQ indicates a reversible process, which will be explained in the next sub-section and T is the absolute temperature.

2.2.3. Reversible and irreversible processes

Any physical system in the course of time will reach to a state of equilibrium.

This equilibrium can be disturbed if an external parameter such as changing the temperature, applying an electromagnetic field, mechanical stress and etc. is exerted on the system. If the perturbation of the system by the external parameters take place in a slowly manner the disturbed system will reach to a new equilibrium state, we call this process a reversible one. If the external disturbing parameters vary rapidly

34 the system can not attain the new equilibrium state and it will remain away from equilibrium and we call this process as an irreversible process.

2.2.4. The second law of thermodynamics

Assume we have a closed system, which is not in a state of equilibrium, a change in the environment will result in the time evolution of the system, to reach a complete equilibrium. If the state of the system is described by the distribution of the energy in that system, the flow of the system through the variable states corresponds to larger probabilities of the energy distribution. Since the probability is related to the eS, we conclude that the exponent, which is the entropy of the system will be additive in time. We therefore conclude that, a closed system in a non-equilibrium state will undergo a time evolution, whereas the system passes from the state with lower entropy to the state with higher entropy, until it possesses a maximum value of entropy corresponding to statistical equilibrium. This statement that the entropy of the system’s state increases in the course of the time is called the law of increase of entropy or the second law of thermodynamics. The increment of the entropy of a system is associated with the transformation from a more ordered state to a more chaotic one, so all processes in the universe result in a chaotic universe rather than an ordered one. The entropy of a closed system will either increase or in the limit it remains constant, but it never decreases, in the situation when it remains constant, we can experience the processes in opposite direction, then we have a reversible process, and when the entropy is increasing the process is irreversible, because otherwise we will get a decreasing entropy for this process, which is a contradiction of the second law of thermodynamics.

35 2.2.5. The free energy functions

From the first law of thermodynamics and taking into account the definition of the entropy (2.73) the work done on a system by a reversible isothermal change is given by

dF= dE − dQ = dE − TdS, (2.74)

since the change is isothermal we have

dF= d( E − TS ), (2.75)

where

FETS= − , (2.76)

this new function is called Helmholtz free energy. By differentiating the right hand side of (2.75)

dF= dE − TdS − SdT, (2.77)

and considering the relation

dE= TdS − PdV , (2.78)

we reach to a relationship for the differential of the Helmholtz free energy:

dF= − SdT − PdV. (2.79)

There is another free energy, which can be obtained by an appropriate change of variables in the above discussion, we start by substituting

36 PdV= d() PV − VdP (2.80)

in equation (2.79), and defining the enthalpy of the system by

H= E + PV (2.81)

we obtain

dF()()+ PV = dE − TS + PV = − SdT + VdP (2.82)

Substituting (2.81) in (2.82), we clearly have

d() H− TS = − SdT + VdP (2.83)

where we define the expression inside the parentheses, as Gibbs free energy

GHTS= − (2.84)

so the differential expression for Gibbs free energy will be

dG= − SdT + VdP. (2.85)

2.2.6. The Gibbs distribution and its free energy

The classical statistical distribution for a closed function, which is called the microcanonical distribution, is given by

ρ=CEE × δ () − 0 (2.86)

where C is a constant. The analogue distribution function for a subsystem in quantum statistic is

dw= C ×δ () E − E0 ∏ d Γa (2.87) a

37 where E0 is the total energy of the closed system, dΓa is the number of the quantum states of the subsystems, and dw, is the probability of finding the system in any of the dΓ states, where

dΓ =∏ d Γa (2.88) a

Now we consider the body and the medium, and identifying their energy and number of the states by E, d and E’, d’ respectively, then (2.87) may be written in the form

dw= C ×δ () E + E′′ − E0 d Γ d Γ (2.89)

Now the probability of a quantum state with energy En in the system is calculated by considering d = 1, and integrating the (2.89)

w= C ×δ E + E′′ − E d Γ , (2.90) n— ( n 0 )

From (2.66) we have

Γ′ = eSE′′() (2.91)

eSE′′() dΓ′′ = dE (2.92) ∆E′

After substituting the above relation in (2.90) and performing the integration we reach to

≈eS′ ’ wn = C ×∆ ÷ . (2.93) ∆E′ « ◊EEE′=0 − n

Now if we expand the S’(E0 – En) in power of En , and consider only the linear part of this expansion we will get

dS′( E0 ) SEESEE′′()()0−n = 0 − n (2.94) dE0

in the last part the derivative relative to the energy is equal to 1/T,

SEESEET′′( 0−n) =( 0 ) − n / (2.95)

In (2.93), E’ can be considered as a constant, since En is small in comparison with

E0 and in a small variation of E’ we put E’ = E0 ; which can be a constant independent of En (total energy E0 for this closed system is constant). According to this discussion and substituting (2.95), in (2.93) we obtain the expression for Gibbs distribution function

−ETn / wn = Ae . (2.96)

Calculating the free energy in this distribution, we recall the equation (2.70) and substitute the (2.96) in that equation we obtain

E SA= −log + (2.97) T

So we have

ETSF− log A = = (2.98) TT

by mean of equation (2.98),we get another description for Gibbs distribution function

()/FET− n wn = e (2.99)

normalizing the above function:

FT/ −ETn / ƒwn = e ƒ e =1 (2.100) n n

which can be written in the form

F= − Tlogƒ e−ETn / . (2.101) n

This is the relationship for the free energy in the Gibbs distribution. The term under the sum is called the partition function or the sum over the states, and it can be expressed in terms of the trace of an operator, which in this case is the Hamiltonian

Z≡ƒ e−ETn / = trexp( − H / T ) (2.102) n

So the free energy is calculated according to

F= − Tlog tre−HT/ (2.103)

2.3. Linear response theory

2.3.1. Evolution operator

In this sub-section we will calculate the evolution operator in the interaction representation that will be used in the rest of the section. For this purpose suppose we have a system evolving under the Hamiltonian

HHH=0 + ext , (2.104) where H0 is the unperturbed Hamiltonian and Hext is the corresponding external perturbation on the system. Now the Schrödinger’s equation for this system will be

d iZ ψ() t=()() H + H ψ t (2.105) dt 0 ext

now we define

ψ (t) = U0 ( t) U( t) i . (2.106)

Using the Schrödinger equation and converting that to operator equation gives us an

expression for U 0 (t), which is

−iH0 t / Z U0 ( t) = e , (2.107)

substituting (2.106) in (2.105) gives us

dU( t) iZ = H()() t U t , (2.108) dt ext

where

† Hext( t) = U0( t) H ext U 0 ( t). (2.109)

By integrating equation (2.108) and keeping in mind that (tU ) →1 ast → −∞ , we obtain

i t U t dt H t U t (2.110) () =1 − — 1 1()() 1 1 . Z −∞

Continuing the above procedure by putting the similar equation as (2.110) for U(t2) we will get the following relation

n ∞ ()−i t t t Ut=1 + dtdt ... dtTHtHtHt» ... ÿ . () ƒ —1 — 2 — n ext()()() 1 ext 2 extn ⁄ n=1 Zn! −∞ −∞ −∞

(2.111)

T in this equation indicates a time ordering of n operator that means each operator appears to the right of all other operators at later times. The compact form of the above equation may be written as

»≈i t ’ ÿ U t Texp dt′′ H t . (2.112) () =…∆ − — ext () ÷ Ÿ …«Z −∞ ◊ ⁄Ÿ

2.3.2. External perturbation

Consider a quantum system which is in the equilibrium state, any external disturbance to this system causes an evolution of any observable of the system.

According to the concept of the interaction picture expressed in the first section, the evolution of the observable is related to the Hamiltonian of the system alone (without the perturbation), whereas the evolution of the state will be related to the

Hamiltonian associated to the external perturbation. Now we consider the expectation value of an observable, identified by the operator O(x,t) in the ground state of the system’s Hamiltonian,

0O( x , t) 0 (2.113)

when the weak perturbation is applied to that system the expectation value becomes

0U−1 ( t) O( x , t) U( t) 0 (2.114)

where U(t) is the evolution operator expressed in (2.112). By expanding the evolution operator in the form of (2.111), up to the linear order in the perturbation

Hext the variation of the expectation value (2.113) is

i t O x t dt′′» H t O x t ÿ (2.115) δ 0() , 0= — 0 ext ()() , , ⁄ 0 Z −∞

This is the linear response of the system to the external perturbation, and as it can be seen this response, which is the change in the ground state of the expectation value of an observable is in fact the expectation value in the ground state of the commutator

42 of the observable and the perturbation. This formulation may be obtain using another method, to show that we consider the equation of motion in Schrödinger picture for the total Hamiltonian (2.104)

dO i =[]OHH, + (2.116) dt Z 0 ext

with

−1 O( t) = Ut OU t (2.117)

where the evolution operator Ut is defined by

i − tH Z Ut = e . (2.118)

applying the time evolution (2.118) to both side of equation (2.116) results

dO( t) i = »O()() t, H t ÿ (2.119) dt Z ext ⁄

this is the interaction picture of the equation of motion given in section 1. Now suppose we are perturbing the Gibbs state, since this is a stationary state we have

O( t) → G( t) = G = e(FHT− )/ when t → −∞ (2.120)

By integrating the equation(2.119), and considering the initial condition(2.120), we arrive at

t i O t G dt′′′» O t H t ÿ (2.121) () = + — ()(), ext ⁄ −∞ Z

which is the same as (2.115) expressing the linear response to a weak perturbation.

43 2.3.3. Generalized susceptibility

The observable O(x,t), and the external parameter Hext(t) can in general be linearly related to each other according to

H t d3 xO x t f x t (2.122) ext ( ) = — ( ,,) ( )

where f(x,t) is called the generalized force. Now we assume that the operator O(x,t) is normal ordered with respect to the ground state 0 ,

0O( x , t) 0= 0 (2.123)

Exerting the condition (2.122) in equation (2.115) gives us a new expression for the linear response in terms of the generalized force f(x,t), which is

i t 0(,)0Oxt= dtdx′′′′′′3 0» OxtOxt ,,,0 ÿ fxt ,. (2.124) f — — ()() ⁄ () Z −∞

It follows that the response is linear in the force, the proportionality in this relationship is called the generalized susceptibility χ(xt, ′tx ′) which is defined by

t 0Oxt() , 0≡χ . f = — dx3 ′′′′′′ — dt χ ()() xtxtfxt , , . (2.125) −∞

Now we define a very useful concept in mathematics, which has many applications in the physics, especially in quantum field theory. This is a mathematical tool that, can be employed to solve the differential equation, this is called the Green’s function

DO ( x, x′′) = − i 0 T » O( x) O( x ) ⁄ÿ 0 (2.126)

44 where the operation has been done in a time ordered manner, that means the operator with the later time must be put to the left of the operator with the earlier time.

ŒÀA( x) B( x′′)...... t> t T » A()() x B x′ ⁄ÿ ≡ Ã (2.127) ÕŒB()() x′′ A x...... t> t

The retarded Green function (propagator) of the observable O(x,t) is

ret DO ( xx,′′′′) = − ittθ ( − ) 0 » OxtOxt( ,) ,( ,) ⁄ÿ 0 , (2.128)

θ (t − t′) is heaviside step function, which is equal to one when t is later than t’, otherwise it will be equal to zero. Comparing the equation (2.124) and (2.125) we will see that the susceptibility is actually the retarded Green function. Considering the Fourier transform of f(x,t), it follows that

∞ dω f() x,, t= — eiω t f() x ω (2.129) −∞ 2π then equation (2.124) becomes

0O( x , t) 0 = ∞ dω 0 ≈− i ’ eiω t dx3 ′′′′′ dt e i ω t′ 0» OxtOxtt()() , , ,+ ÿ 0 fx() ,ω — — —−∞ ∆ ÷ ⁄ −∞ 2π «Z ◊

(2.130)

in which the frequency dependent observable O(x,) will be

0 3 ≈−i ’ 0Ox() ,ω 0=— dx′′′′′∆ ÷ — dt 0» OxtOxtt()() , , , + ÿ⁄ 0 fx() ,ω «Z ◊−∞

(2.131)

45 The frequency dependent susceptibility will be obtained again as the coefficient of proportionality between the frequency dependent response and the force in (2.131)

0 i iωτ χ()x, x′′ ; ω= − — d τ e 0 » O()() x ,0 , O x , τ ⁄ÿ 0 (2.132) Z −∞

This statement is called the Kubo formula. According to (2.128) the retarded Green

ret function DO ,( xx ′ ) may be used in the above equation to obtain another form of the susceptibility function

1 ∞ x x′′ d eiω t D ret x x (2.133) χ(),;, ω= — τ O () Z −∞

The frequency dependent response, force, and the Green function are related through the relationship

1 0O() p ,ω 0= Dret ()() p , ω f p , ω (2.134) Z O

but according to (2.133), since the generalized susceptibility is actually the retarded

Green function, the above equation may be reduced to

0O( p ,ω) 0 1 χ()p,, ω= = Dret () p ω (2.135) f() p,ω Z O

which shows a proportionality between the response and force that is represented by the susceptibility.

46 2.3.4. Thermal average

So far we have talked about the systems with zero temperature that is T=0, but in reality all the physical systems are found in a state with a finite temperature, therefore we have to expand our discussion in the cases when the temperature is not zero, T>0. first we define a temperature dependent operator (Matsubara operator) A(τ ), from a Schrödinger operator A, by

A(τ ) = eHHτ//ZZ Ae− τ (2.136)

that is an analytic continuation to the imaginary time of the Heisenberg operator A(t)

A( t) = eiHt//ZZ Ae− iHt (2.137)

Hence the evolution operator

U( t) = e−iHt / Z (2.138) will become

−Hτ / Z Ue (τ ) = e (2.139)

In these formulations we include the chemical potential in the Hamiltonian H, that is H → H − µN , where N is a number operator. Therefore we can express a very important conclusion from the above discussion that is any quantum system at temperature T, which is in equilibrium, can be regarded as the imaginary time evolution of that system.

Now the thermal expectation value or the thermal average of an observable

A(x,t), is given by

Tr( A( x, t) ρG ) A() x, t = (2.140) TrρG

47 where G , is the Gibbs distribution

−β()HN − µ ρG ≡ e , (2.141)

is in here 1/T. The set of eigenvalues Eλ and N λ correspond to the Hamiltonian and the number operator respectively, where the set of the eigenstates of these two operator are{λ }. Thus the thermal average or expectation value is

ƒ λA( x, t) λ e−β(ENλ − µ λ ) A x, t = λ (2.142) () −β()EN − µ ƒe λ λ λ

2.3.5. Correlation function

Suppose we have a quantum system, which is in equilibrium; that means the quantum dynamical variables characterized by some operators are found in their mean value, an external perturbation applied to this system causes the quantum operators deviate from their average values; if the external perturbation is weak, the response of the system to this perturbation will be linear, and as it was previously discussed this response is proportional to the external force; any deviation from mean values of the physical quantities (classical or quantum) is called a fluctuation of that quantity. Now if the system is left untouched for a while, it will reach again a partial

(locally) equilibrium. In this situation the values of the physical quantity at a given point x and time t, affects the values of this quantity in different point of space x’ at a later instant t’; we say there is a correlation between values of this quantity. We define the correlation function as a starting point of computing of the time- ordered

(Feynman), retarded, and temperature Green functions. For this reason we will investigate a connection between these functions in the next sub-section. That will be

48 related to the concept of spectral function which describes the fluctuation of the system. Now we will proceed with this definition of the correlation function for fermion and it will be expanded for the bosons as well.

The fermion correlator G> (x − ′, tx ) is actually a thermal average of the product of a fermion field operator and its hermitian adjoint given by

Tr( e−β H e iHt//ZZψ( x) e − iHt ψ + ( x′)) G()()() x− x′′, t =ψ x , t ψ + x ,0 = > Tre−β H

(2.143)

here we have applied the time evolution of the operators, but in the second operator since the corresponding time is the time zero, the exponentials are equal to unity. Now we can write the correlator as

ƒ λ ψ+( x′′′) e −β H λ λ e iHt//ZZ ψ( x) e − iHt λ ψ()()x, t ψ + x′ ,0 = λλ′ ƒe−β Eλ λ −β E + i() E′ − E t / Z ƒeλ′ λ ψ() x′′′ λ λ ψ() x λ e λ λ = λλ′ ƒe−β Eλ λ

(2.144) we have used the cyclic property of the operators, and keeping in mind the normality of the eigenstates

ƒ λ λ =1. (2.145) λ

In the case of the boson correlators, we will have instead of fermion operators, bosons operators, hence the correlation function will be

D> ( x− x′′, t) = O( x , t) O( x ,0) . (2.146)

49 2.3.6. Spectral function

The time Fourier transformation of the correlation function is called the spectral function or spectral density of the fluctuating of a physical quantity. This function as it was mentioned is used to describe the fluctuation of a quantum system, since the correlation function expresses the relation between different values of a quantum operator, which can be a consequence of the fluctuation of the system. It will be shown in the next sub-section that every form of the Green function or propagator can be expressed in terms of the spectral density. We may write

∞ J x− x′′;ω = ψ x , t ψ + x ,0 eiω t dt = 1 ()()()— −∞

−β E + ≈EE′ − ’ ƒeλ′ λ ψ() x′′′ λ λ ψ() x λ2 πδ∆λ λ + ω ÷ λλ′ «Z ◊

(2.147)

where

−β Eλ ZG = ƒ e (2.148) λ

is the grand partition function. J1 (x − x′;ω) is called the spectral function of the correlation function.

We may also define the correlation function as

+ G< ( x− x′′, t) = ψ( x ,0) ψ ( x , t) (2.149)

the Fourier transformation of this function is again the spectral function

∞ J x− x′′;ω = ψ+ x ,0 ψ x , t eiω t dt 2 ()()()— −∞

−β E + ≈EE′ − ’ eλ λ ψ() x′′′ λ λ ψ() x λ2 πδ∆λ λ + ω ÷ ƒ Z = λλ′ « ◊ ZG

(2.150)

The Dirac delta function gives us an opportunity to write the J 2 (x − x′;ω) in terms

of J1 (x − x′;ω), since

−β()E ′ +Z ω + ≈EE′ − ’ ƒeλ λ ψ() x′′′ λ λ ψ() x λ2 πδ∆λ λ + ω ÷ λλ′ «Z ◊ J2 () x− x′;ω = ZG −βZ ω =e J1 () x − x′;ω

(2.151)

Hence the spectral function of first kind and second kind are related by an exponential factor as the coefficient of proportionality.

2.3.7. Different forms of the Green functions

In this sub-section we will talk about a different formulation of the Green’s function, which is widely used in quantum systems. For example in a quantum system withT ≠ 0, we are facing a situation that a continuation to imaginary times in time-ordered Green function is used to compute the temperature Green function, which is an imaginary time-ordered Green function. As it was mentioned earlier and

51 we will show later, all different forms for the Green’s function; retarded, time- ordered, and temperature Green function can be calculated using the spectral

function J1 (k,ω), where k is the wave vector, and since the very important theorem in linear response theory which is fluctuation-dissipation theorem is actually stated according to the spectral function (density), we are actually establishing a bridge between different forms of the Green functions and this theorem. Also the interconnections between these Green functions will be obtained in the next sub-

section. First we calculate the space Fourier transformation of the J1 ( , xx ′;ω) as

J1 ( k,ω) =

−β E 2 ≈EEPP′ − ’≈3 ()3 ′ − ’ ƒe λ′ λ′ ψ()0 λ 2 πδ∆λ λ+ ω ÷∆() 2 π δ λ λ + k ÷ λλ′ «ZZ ◊« ◊

(2.152)

where P is the linear momentum operator, and it gives us a space translation

ψ( x) = eiP././ xZZ ψ (0) e− iP x (2.153)

P and P ’ are the total linear momentums of the states λ and λ′ . The second kind of the spectral function in this space will be again calculated as

−βZ ω J2( k,,ω) = e J 1 ( k ω) (2.154)

Now according to the definition of the retarded Green function for fermions given by

GR ()()()() x, t ; x′′ ,0= − iθ t{ ψ x , t , ψ + x ,0 } (2.155) = −iθ()()() t() ψ x, t ψ+ x′′ ,0 + ψ + ()() x ,0 ψ x , t

and the concept of the spectral functions (2.152) and (2.154), we can calculate the retarded Green function at T ≠ 0

∞ dω′ ′ J( k,ω′) GR () k,ω =— () 1 + e−βZ ω 1 (2.156) −∞ 2π ω− ω′ + i η

where (η → 0 + ). As it can be seen the retarded Green function is expressed in terms of the spectral function. Applying a very useful property in the complex analysis, in which separates the real part and imaginary part in the right hand side of (2.156), we get

1 1 1 x lim= P −iπδ () x and P = lim2 2 (2.157) η→0+ x+ iη x xη →0+ x +η

P in this distribution is called the principal part. Using this relationship gives an equation for the spectral function in terms of the imaginary part of the retarded green function

≈2 ’ R J1 () k,ω= −∆ ÷ Im G() k , ω (2.158) «1+ e−βZ ω ◊

Similarly we define the retarded Green function for bosons as

R Dxxt( −′′,) = − itθ ( ) » OxtOx( ,) ,( ,0) ⁄ÿ (2.159)

in here unlike fermions retarded Green function, in which was defined as the thermal average of the anti-commutator of the fermion operators and its hermitian adjoint, we have a thermal average value for the commutator of the boson operators. In this case according to Bose statistics the equation (2.158) will be slightly different

≈2 ’ R J1 () k,ω= −∆ ÷ Im D() k , ω (2.160) «1− e−βZ ω ◊

where we have deduced this relationship from the retarded Green function of bosons, which is

∞ dω′ ′ J( k,ω′) DR () k,ω =— () 1 − e−βZ ω 1 (2.161) −∞ 2π ω− ω′ + i η

this is a result of different statistic for bosons.

The time-ordered or Feynman propagator for fermions is defined as

+ GF ()()() x− x′′, t = − i T(ψ x , t ψ x ,0 ) (2.162) = −iθ()()() t ψ x, t ψ+ x′′ ,0 + i θ()()() − t ψ + x ,0 ψ x , t

the time Fourier transformation of each part in the right hand side gives us an

expression for the spectral functions J1 (x − x′,ω′) and J 2 (x − x′,ω′) respectively, and consequently by using the time and space Fourier transformation we obtain

∞ dω′ À1 e−βZ ω′ ¤ G k J k ′ (2.163) F (),,ω=— 1 () ω Ã + ‹ −∞ 2πÕ ω− ω′′ +i η ω − ω − i η ›

which is again a function of spectral function J1 (k,ω′). Similarly again we define the time-ordered propagator or Green function for bosons as

DF ( xxt−′′,) = − iTOxtOx( ( ,) ( ,0)) (2.164) = −itOxtOxθ()()()′′, ,0 + i θ ()()() − tOx ,0 Oxt ,

and obtain an equation for the boson time-ordered Green function analogue to

(2.163)

∞ dω′ À1 e−βZ ω′ ¤ D k J k ′ F (),,ω=— 1 () ω Ã − ‹ . (2.165) −∞ 2πÕ ω− ω′′ +i η ω − ω − i η ›

Next we work on the temperature Green function for fermions, which is in fact the thermal average value of the imaginary time-ordered of the product of the fermion operator with its hermitian adjoint

+ GT ()()() x− x′′,τ = − Tτ ( ψ x , τ ψ x ,0 ) + ′ (2.166) Tr() Tτ ()ψ()() x, τ ψ x ,0 ρG = − TrρG

with the range 0 ≤ τ ≤ βZ , since this Green function obeys periodic condition for the boson operators and anti-periodic condition for the fermion operators in imaginary time

GTT( x,;,,;,τ+ βZ x′′′′ τ) = ± G( x τ x τ ) (2.167)

we have the positive sign for bosons and negative sign for fermions. Accordingly a

Fourier series expansion of the Green function in imaginary time, for frequency gives the following conditions to satisfy the above discussion for periodic and anti-periodic boundary conditions

2π ω = n , (in the case of bosons) and n βZ (2.168) 2π ≈ 1 ’ ωn =∆n + ÷ , (in the case of fermions). βZ «2 ◊

We may write the imaginary time ordering of the fermion operators (2.166) as

+ −Tτ (ψ()() x, τ ψ x′ ,0 ) =

−θ()()() τ ψx, τ ψ+ x′′ ,0 + θ()()() − τ ψ + x ,0 ψ x , τ

(2.169)

Now we compute the Fourier expansion of the temperature propagator for the frequency

1 β Z G x x′′ d e−iωn τ T x+ x T()−,ω n = — ττ () ψ()() , τ ψ ,0 (2.170) βZ 0

55 in this expansion the limit of the integration has been modified to zero and βZ , since there is an anti-periodic boundary condition in the imaginary time for fermions. Now we can write the thermal average in the form of (2.144) except that the time t in this relationship will be replaced by the temperature variable (imaginary time) . In this situation the equation (2.170) will become

−βEEλ − β λ′ 1 + Àe+ e ¤ GT() x− x′′′′,ω n = − ƒ λψ() x λ λ ψ() x λ Ã ‹ βZZG λλ Õ Eλ′ − E λ − iZωn ›

(2.171)

2π ≈ 1 ’ where ω n = ∆n + ÷ . We introduce a space Fourier transformation and use the βZ « 2 ◊

(2.147), (2.150), and (2.151) to write the temperature Green function in terms of spectral function

∞ 1 dω′ ′ J( k,ω′) G k e−βZ ω 1 (2.172) T(),ω n = −— () 1 + βZ −∞ 2 π−i ωn − ω′

The counterpart for bosonic temperature Green function can be written according to

DT ( x− x′′,τ) = Tτ ( O( x , τ ) O( x ,0)) (2.173)

∞ 1 dω′ ′ J( k,ω′) D k e−βZ ω 1 T(),ω n = −— () 1 − (2.174) βZ −∞ 2 π−i ωn − ω′ where in this case the temperature Green function obeys periodic boundary condition

2π in imaginary time, and the corresponding frequency will be ω = n . n βZ

56 2.3.8. Connections between various Green functions

Consider the equation (2.156) and (2.163) in both cases we can express the real part of these equation as

∞ ′ J( k,ω′) dω′ GR k G k P e−βZ ω 1 (2.175) Re()() ,ω= ReF , ω =— () 1 + −∞ ω− ω′ 2 π

And the imaginary part will be

R 1 −βZ ω ImG() k ,ω= −() 1 + e J1 () k , ω 2

1 −βZ ω ImGF () k ,ω= −() 1 − e J1 () k , ω (2.176) 2

Therefore, using the definition of the coth function we will get

R ≈βZ ω ’ ImG() k ,ω= coth∆ ÷ Im GF () k , ω (2.177) «2 ◊

Equations (2.175) and (2.177) give us the connection between the retarded and the time-ordered (Feynman) propagator.

By comparison between (2.156) and (2.172) we see that

R G( k,,− iωn) = − βZ G T( k ω n ) (2.178)

In this case we have an analytic continuation from –in to +i of the complex

R plane including the imaginary axes for − βZGT (k,ω n ) into G (k,ω). Similar connections can be found between retarded, time-ordered, and temperature Green functions in the case of the boson, except that we have

R ReD( k ,ω) = Re DF ( k , ω) (2.179) R ≈βZ ω ’ ImD() k ,ω= tanh∆ ÷ Im DF () k , ω «2 ◊

2.3.9. Kramers-Kronig relation

As it was mentioned earlier, the real part and imaginary part of a function of the form (2.157) can be separated by employing the principle part and delta function as the real part and imaginary part of the equation. Returning to the equation of the retarded Green function (2.156) we will separate the real and imaginary part of this equation. First we look at the principle part of this equation which is

∞ dω′ ′ J( k,ω′) ReGR () k ,ω = P— () 1 + e−βZ ω 1 (2.180) −∞ 2π ω− ω′

On the other hand the imaginary part of this function is

R 1−βZ ω′ 1 −βZ ω ImG() k ,ω= −() 1 + e J1 ()() k , ω′′ δ ω − ω = −() 1 + e J1 () k , ω 2 2

(2.181) substituting right hand side of (2.181) as the imaginary part of the retarded Green function into (2.180), gives us a relationship between the real part and imaginary part of the retarded Green function, which is called the Kramers-Kronig relation

∞ dω′ ImGR ( k ,ω′) ReGR () k ,ω = − P — (2.182) −∞ π ω− ω′

A similar relation can be obtained for the retarded Green function in the case of bosons, except that we have to replace the ImG R (k,ω) by Im D R (k,ω′).

58 2.3.10. Dissipation

In this section we will show that the dissipation of energy in a system as a result of the external perturbation by the force f (p,ω), which is converted into heat after absorption, is obtained by the imaginary part of the generalized susceptibility χ ′′(p,ω). To proceed in this discussion we keep in mind that the generalized susceptibility is in general a complex quantity

χ( p,,, ω) = χ′( p ω) + i χ ′′( p ω) (2.183)

from the following property of the susceptibility

χ( p,,− ω) = χ ∗( p ω) (2.184)

and according to the response function (2.135)

0O( p ,ω) 0= χ( p , ω) f( p , ω) (2.185)

for a real force expressed by

−iω t1 − i ω t ∗ i ω t fpt()(),= Re fpe0 ,ω =() fpe 0()() , ω + fpe 0 , ω (2.186) 2

Then for the equation (2.184), we will get expression for the response

1 0O() p ,ω 0=» χ()()()() p , ω f p , ω e−iω t + χ p , − ω f ∗ p , ω e iω t ÿ 2 0 0 ⁄

(2.187)

where the bracket gives us a real value. Now we look at the time variation of the energy of the system that is the time derivative of the Hamiltonian. The variation of the energy of the system by the exertion of an external perturbation indicates that

59 there is an absorption of the energy, which consequently will be a change in the heat

Q of the system. This variation is

dE∂ H = (2.188) dt∂ t

But the only part of the Hamiltonian, which is time dependent, is the perturbation

H ext (t), therefore according to the equation (2.122) we will get

dE −O( x,, t) df( x t) = (2.189) dt dt

Substituting (2.186) and (2.187) in the above equation and averaging over time gives

12 1 2 Q= iω» χ ∗()()() p,,,,, ω − χ p ω ÿ f p ω = ωχ′′()() p ω f p ω 4 ⁄ 0 2 0

(2.190)

As it can be seen the dissipated energy or the heat added to the system is related to the imaginary part of the generalized susceptibility which is a very important result, and we will use this outcome in the next sub-section to formulate the fluctuation- dissipation theorem.

2.3.11. Fluctuation-Dissipation theorem

From our discussion on the response function in sub-section 2.3.3, we see that, this function is in fact related to the bosonic retarded Green function of the operators of the observable, and as we can see the retarded Green function for these operators can be calculated according to the generalized susceptibility by (2.135), which tells us that the generalized susceptibility is proportional to the retarded Green function, where the proportionality is inverse of the plank’s constant

60 1 χ()k,, ω= Dret () k ω (2.191) Z O

On the other hand from equation (2.160) the imaginary part of the retarded Green function is related to the spectral density of the fluctuation of the observable O(x,t) by

≈2 ’ R J1 () k,ω= −∆ ÷ Im D() k , ω , (2.192) «1− e−βZ ω ◊

but the spectral function actually describes the fluctuation of the observable in a quantum system. Now if we use (2.191) and we denote the imaginary part of

ret DO (p,ω) or χ(p,ω) by χ ′′(k,ω), we can write the fluctuation-dissipation theorem as

Z −βZ ω χ′′()k, ω= −() 1 − e J1 () k , ω (2.193) 2

that indicates a connection between the dissipation of the energy in the system,

χ ′′(k,ω) (which will be later converted to the heat in that system) and the fluctuation

of the observable in that system J1 (k,ω). This formula was obtained by H. B. Callen and T. A. Welton in 1951.

Chapter III

3. Kubo-Martin-Schwinger (KMS) State

In this chapter we will discuss one of the most important results obtained from an analytic continuation of the time in the complex plane suggested by Kubo,

Martin and Schwinger. This continuation is indeed a consequence of the fact that the

Gibbs weight factor, which is an exponential function of energy, e−βH is similar to the unitary operator of the time evolution eiHt / Z of a system, which is again an exponential function of the energy [7, 15, 29]. The expansion of the time in the complex plane gives us a description for the fluctuation-dissipation theorem.

Schwinger and Martin applied this mathematical technique to formulate the many- particle problem by employing the temperature Green’s function, which is calculated by using the imaginary time ordering of the field operators [42]. The abovementioned method has been used in vast areas of the many-body problems, such as electron gas in a crystal lattice, superconductivity, etc.

62 3.1. K.M.S Condition

The dynamic of a quantum system can be described by a time evolution of an observable A of that system:

iHt//ZZ− iHt αt ()A= e Ae , (3.1)

and the equilibrium state of this system is given by the Gibbs condition

Tr() e−β H A ϕ()A = . (3.2) Tr() e−β H

The KMS condition can be defined as a direct relationship between the equilibrium

ϕ(A) and the time evolution α t (A)of the system, in a way that if for any pair of the quantum operator A and B, we consider the function (tf ) as

f( t )= ϕ ( A αt ( B )) , (3.3)

an analytic continuation in a sufficient large strip of the complex plane around the real axis will be given by

f()(()) t+ iZβ = ϕ αt B A (3.4)

Equations (3.3) and (3.4) describe the KMS condition. Another equivalent approach to the above result may be obtained in the following way; define for z ∈ C

A( z) = e−iHz//ZZ Ae iHz (3.5)

now consider the equation

A( z) e−βHHHH= e − β e β A( z) e − β , (3.6)

since the time translation operator and Gibbs state are both exponential function of the energy, therefore using (3.5) in the (3.6) we get

63 A( z) G= GA( z − iZβ ) (3.7)

If now we multiply the right hand side by B, we will get

A( z) GB= GA( z − iZβ ) B , (3.8)

in the left hand side after applying the trace cyclic property we reach to the Kubo-

Martin-Schwinger formula or KMS condition

BA( z) = A( z − iZβ ) B , (3.9)

where we have used the expectation value in the Gibbs state

A= GA . (3.10)

One of the most interesting result obtained from the KMS formula, is the

Callen-Welton fluctuation-dissipation theorem. Here the KMS condition provides us a powerful tool, which can be used to deduce this theorem. From (2.115) a linear response can be written as

i Γ()AB;,τ = » A() τ B ÿ , (3.11) Z ⁄

Also the correlation function is expressed as

1 K() AB;τ= A()() τ B + BA τ . (3.12) 2

As it can be seen the response function is a commutator and the correlation is an anticommutator, now if we consider

A(τ) B= f ( τ ) , (3.13)

64 according to the KMS formula (3.9), we will get

BA(τ) = f( τ − iZ β ) . (3.14)

Hence (3.11) and (3.12) will become

i Γ()AB;τ ={} f()() τ − f τ − iZ β , (3.15) Z

and

1 K() AB;τ={} f()() τ + f τ − iZ β . (3.16) 2

$ Suppose f (τ ), is the Fourier transform of a function φ (ω) of frequency ;

dω $ f()τ= e−iωτ φ() ω . (3.17) — 2π

Considering the above Fourier transform the equations (3.15) and (3.16) may be written in the form of

i dω $ Γ()AB;τ = e−iωτ φ() ω {} 1 − e − βZ ω , (3.18) Z — 2π

and

1 dω $ K() AB;τ= e−iωτ φ() ω {} 1 + e − βZ ω . (3.19) 2— 2π

$ Now by eliminating the φ , from the above two equations and keeping in mind that the difference between generalized susceptibility and its conjugate is the response function we will get the fluctuation-dissipation theorem

∗ 2i βZ ω χ()()AB; ω− χ AB ; ω = J() AB ; ω tanh . (3.20) Z 1 2

3.2. Many-body problem

The treatment of the many-particle system is divided into two categories; the first category is related to the study of the macroscopic change of the system’s energy and the number of the particles and the second is considered a microscopic variation of these quantities. In this system since the number of particles is large, it is not possible to distinguish microscopic variation of the energy in neighbouring states and it is assumed as continuos change. A very useful tool that relates these two kinds of multi-particle treatment is the Green’s function [3, 16, 20].

As it was mentioned earlier in this chapter the study of the similarity of the

Gibbs weight factor and the unitary operator of time development of the system, provides us an opportunity to introduce the temperature into the Green’s function, which is equivalent to an imaginary time-ordered Green’s function.

3.2.1. Microscopic property; Green’s function

First we consider the equation of motion in the Heisenberg picture (2.44), where the field operators ψ ( ,tr ) and its hermitian adjoint ψ + ( ,tr )are functions of the time and space

∂ψ1 ∂ ψ + 1 =[]ψ,,,.HH = » ψ + ÿ (3.21) ∂t iZZ ∂ t i ⁄

The Hamiltonian of the system may be given by

Z2 H= −ψ+ ()()()() rt,,,, ∇2 ψ rtdr − ζ ψ+ rt ψ rtdr 2m — — (3.22) +— drdr'ψ+()()()()() rt , ψ + rtvrr ', − ' ψ rt ', ψ rt , ,

the second term which is a number operator has been introduced in the

Hamiltonian, to avoid the collapse of the energy to zero, since we have particles present in the system. Now if we apply the equation of the motion (3.21) to the field operatorψ ( ,tr ), with the Hamiltonian (3.22), we will get

∂ψ (r, t) iZ = ∂t Z2 − ∇2ψ()()()()()()rt, − ζψ rt , + drvrr ' − ' ψ+ rt ', ψ rt ', ψ rt , . 2m —

(3.23)

Let us calculate the equation of the motion of a new physical quantity, instead of the field operatorψ ( ,tr ). This new quantity is

g! 1,2= − iε 1,2» T ψ r , t ψ + r , t ÿ (3.24) 1 ()()()() ( 1 1 2 2 ) ⁄

where T shows that the products of the fields are time-ordered, and ε (12...nn '1'2'... ) is a symbol equal to 1 in the case of bosons and it is equal to ±1, for fermions. To work on the equation of motion for this new quantity we differentiate the g~( 2,1 ) with

respect to t1 . In this differentiation we have aδ (t1 − t2 ) function, because when

t1 → t2 , we will get a discontinuity, otherwise this part will vanish. The contribution of this δ function appears in the differentiation as

t2 +ε ∂ iZδ () t− t ×lim dt g! () 1,2 (3.25) 1 2ε → 0+ — 1∂t 1 t2 −ε 1

Introducing a similar operator for two-particle by

g! 1,2;1',2'= − i2 ε 122'1'» Trtrtrtrt ψ , ψ , ψ+ , ψ + , ÿ 2 ()()()()()()() ( 11 22 2'2' 1'1' )⁄

(3.26)

the equation of motion for this one-particle operator will become

2 ≈∂ Z 2 ’ ∆iZ + ∇1 +ζ ÷ g! 1 ()1,2 = «∂t1 2 m ◊ Z idrdtvr r′′′′ t t g! δ()()()()1,2±— 1 1 1 − 1 δ 1 − 1 + ε 2 1,1;2,1,+

(3.27)

where

ZZδ(1,2) = δ(r1 − r 2) δ ( t 1 − t 2 ) , (3.28)

~ and we have set r2′ = r1′ and t2′ = t1′ + ε , the subscript (+) in the last expression for g 2 indicates the time ordering. In the above equation we can see that, the one-particle operator and two-particle operator are related to each other through this operator equation.

The above differential equation for the corresponding operator can be used also for the Feynman propagator, or the time-ordered Green function. In this case we

~ have to calculate the expectation value of the operator g1 ( 2,1 ), [see equation (2.162)] in a definite temperature /1 kβ , defined by

Tre−β H g! (1,2) G ()1,2 = (3.29) 1 Tre−β H

68 An analytic continuation to a strip (iβZ)in complex plan around the time real axis gives us the generating function defined by

Tre−β H − iH( tf − t o )/ Z g! ()1,2 g1 ()1,2 = (3.30) Tre−β H − iH() tf − t o / Z

~ Since g( 2,1 ) is a time-ordered function we can split the generating function g1 ( 2,1 ),

into two parts as g– ( 2,1 ), when –tt 21 and g ¿ ( 2,1 ), when ¿tt 21 . Of course the order of

~ + the field operator in the g( 2,1 ) in the first case will beψ ( ,tr 11 )ψ ( ,tr 22 ), and

+ corresponding order for the second situation isψ ( ,tr 22 )ψ ( ,tr 11 ). In the case of

g– ( 2,1 ), using the cyclic property of the trace and the time translation equation

iH( t− to)//ZZ − iH( t − t o ) ψ(t) = e ψ ( to ) e (3.31)

it can be shown that

−βHHH−iH( tf − t o )/ Z + − β β −iTreeψ()() rte2,, 2 ψ rte 1 o g– ()1,2 = (3.32) Tre−β H − iH() tf − t o / Z

where we suppose that o ¿ , 21 ¿tttt f , and we set t1 equal to tf. Also by an analytic

continuation of t1 into to + iβZ , we will get a similar expression

−βHHH−iH( tf − t o )/ Z + − β β _iTreeψ()() rte2,, 2 ψ rte 1 o g¿ ()1,2 = (3.33) Tre−β H − iH() tf − t o / Z

therefore

grtrt–( 1,,,;,.f ; 2 2) = ± grt ¿ ( 1 o + iβZ rt 2 2 ) (3.34)

69 Also we obtain a similar condition for the second time variable

grtrt¿( 1,;,,;,. 1 2f) = ± grtrt – ( 1 1 2 o + iβZ) (3.35)

We have reached these conditions by applying the analytic continuation on the complex plane of the time variable, which serves as the KMS condition base on the equivalent of the Gibbs exponential equation and exponential function of the time

evolution. It can be shown that these conditions are satisfied for any order of the g n .

To solve the differential equation (3.27), we assume a system with the interaction between particles equal to zero, and spatially homogeneous, therefore the

spatially points themselves are not of concern, and we will work on r1 − r2 . Also in this case the temperature will be considered as infinity to get β as zero; therefore applying this condition gives us a boundary condition similar to (3.34) and (3.35).

Then after an analytic continuation to the imaginary strip around the real axis of the time (− iβ ) we will reach to the description of the Green’s function for a system of one particle. Since β → 0 , we work on the generating function where the weight factor in numerator and denominator are functions of the time only. First we write the Fourier transformation in a continuous space and discrete time by combining a series and an integral as follows

dk g1,2= eik()() r1− r 2 − inπ t 1 − t 2 / τ g n , k . (3.36) () — 3 ƒ () ()2π n

Theδ ( 2,1 ), is actually in the form of (3.28), hence its spatially Fourier expansion will be in the form of

dk δ1,2 =eik() r1− r 2 δ t − t . (3.37) () — 3 ()1 2 ()2π

70 Employing these Fourier expansion (3.36) and (3.37) in the differential equation of

motion (3.27), when (rv 1 − r1′) → 0 we will have

≈2 2 ’ −inπ() t1 − t 2 / τ nπ ZZ k ƒe∆− +ζ ÷ g()() n, k =Z δ t1 − t 2 . (3.38) n «τ 2m ◊

tim 1 /τπ Now we multiply both sides by e dt1 and integrating we will obtain

≈n p2 ’ inπ t2 / τ π Z imπ t2 / τ ƒeδmn τ∆− + ζ ÷ g() n, k = Z e (3.39) n «τ 2m ◊

Since

iπ t( m− n)/ τ e1 dt = δ (3.40) — 1 mn

hence we have

Z g() n, k = (3.41) ≈nπ Z p2 ’ τ∆− + ζ ÷ «τ 2m ◊

Inserting this expression in the equation (3.36) it gives us

dk Z g1,2 = e−inπ()() t1 − t 2/ τ + ik r 1 − r 2 × (3.42) () — 3 ƒ 2 ()2π n ≈nπ Z p ’ τ∆− + ζ ÷ «τ 2m ◊

In the above expression, we see that it has a simple pole in ([(p 2 2/ m)− ζ ]Z −1 ),

solving this equation using the method of residue technique for t1– t 2 and t1¿ t 2 we get the solution g( 2,1 ) satisfying the boundary condition (3.34) and (3.35) as

exp»−i() t − t» p2 / 2 m −ζ ÿ / Z ÿ dk ik() r− r 1 2 ( ) ⁄ ⁄ t– t: g 1,2 = − i e 1 2 , (3.43) 1 2 () — 3 2π 1_Z exp»−iτ» p2 / 2 m − ζ ÿ / ÿ () () ⁄ ⁄

and for

exp»−i() t − t» p2 / 2 m −ζ ÿ / Z ÿ dk ik() r− r 1 2 ( ) ⁄ ⁄ t¿ t: g 1,2= − i ± 1 e 1 2 1 2 ()() — 3 2π exp»iτ» p2 / 2 m − ζ ÿ /Z_ ÿ 1 () () ⁄ ⁄

(3.44)

The expression for the Green function G( 2,1 ) can be obtained by an analytic continuation of these functions given byτ → −iβZ

exp»−i() t − t» p2 / 2 m −ζ ÿ / Z ÿ dk ik() r− r 1 2 ( ) ⁄ ⁄ t– t: G 1,2 = − i e 1 2 1 2 () — 3 2π 1_ exp»−β»p2 / 2 m − ζ ÿ ÿ () () ⁄ ⁄

(3.45)

exp»−i() t − t» p2 / 2 m −ζ ÿ / Z ÿ dk ik() r− r 1 2 ( ) ⁄ ⁄ t¿ t: G 1,2= − i ± 1 e 1 2 2 2 ()() — 3 2π exp»β»p2 / 2 m − ζ ÿ ÿ _ 1 () () ⁄ ⁄

(3.46)

For a system of many particles the most useful quantity to describe it is the

Green’s function, which was given above for a system of one particle. Now we can extend the above discussion to formulate the equation of the motion for a system of two particles and obtain a very useful relationship for the Green’s function of the different number of particles in the system which is called Hartee-Fock approximation. In this calculation unlike the one-particle system we do not attempt to shift β to zero therefore (3.34) and (3.35) will not be the boundary conditions and we should consider another boundary condition. The procedure to calculate the Green’s function for two-particles will be studied in the following order; first we will obtain the generating function keeping the entire weight factor in the numerator and

denominator, after that we have a shift β → β − (ti f − to )/ Z , we will get to the

72 description of the Green’s function for this system. To work on this system we should

~ include the inter-particle interactions, so the operator applied to g 2 may be written as

∂ Z2 iZ 2 dr n r v r r (3.47) + ∇1 +ζ +— '''()() − ∂t1 2 m

~ After applying this operator, we will get a discontinuity of the g 2 , when t1 passes

through t'1 andt'2 , hence we have a contribution of δ functions, also we get a contribution of three-particle g~ function according to

»∂ Z2 ÿ iZ + ∇2 +ζ + nv g! 12;1'2' …1— Ÿ 2 () ∂t1 2 m ⁄ ZZg!!! g i V d g =δ()()()()()()()11'1 22' ± δ 12' 1 21' ± — 13 3 3 123;1'2'3+

(3.48)

To solve this equation we change the (1) to (3’) and we get

»∂ Z2 ÿ iZ + ∇′2 +ζ + nv g 3'2;1'2' …3— Ÿ 2 () ∂t3' 2 m ⁄ =ZZδ3'1'g 22' ± δ 3'2' g 21' ± i V 3'3 d 3 g 3'23;1'2'3 . ()()()()()()()1 1 — 3 +

(3.49)

Now we multiply this equation by g1 (13')dt'3 dr'3 and integrate from − ∞ to + ∞ , and we get

»∂ Z2 ÿ d3' iZ ′2 nv g 13' g 3'2;1'2' —() …− + ∇3 +ζ + — Ÿ 1()() 2 ∂t3' 2 m ⁄

=ZZg1()()()()11' g 1 22' ± g 1 12' g 21' i V3'3 d 3' d 3 g 13' g 3'23;1'2'3 , ± — ()()()()()1 3 +

(3.50)

73 where we have used

g1 (11') = δ ( 3'1') g ( 13') , (3.51)

and a similar relationship for g1 (12'). In equation (3.50) according to (3.27), we can

substitute for the expression inside the bracket operating to g1 (13') by

»∂ Z2 ÿ iZZ2 nv g i d v g …− + ∇'3 +ζ +— Ÿ 1 ()()()()() 13' = δ 13' ± — 3 3'32 13;3'3.+ ∂t3 ' 2 m ⁄

(3.52)

We obtain an expression for the two-particle Green’s function as

g2(12;1'2') = g 1( 11') g 1( 22') ± g 1( 12') g 1 ( 21') i ± d()()()()()()()3 d 3' V 3'3» g 13' g 3'23;1'2'3− g 13;3'3 g 3'2;1'2'ÿ . Z — 1 3+ 2 + 2 ⁄

(3.53)

When there is no interaction between particles, the above expression can be considered as relating the two-particle Green’s function to the products of two one- particle Green’s functions. This symmetric relationship is called the Hartee-Fock approximation

g2(12;1'2') = g 1( 11') g 1( 22') ± g 1( 12') g 1( 12') g 1 ( 21') (3.54)

Now we will proceed to discuss the Hartee-Fock approximation. We see that the equation of motion in the case of the one-particle Green’s function given by

(3.27) can be written in terms of the two-particle Green’s function, can be manipulated by substituting the Hartee-Fock approximation for this Green’s function

in (3.27). The Fourier transforms of the V (11') and g1 (12) in the interaction term of

(3.27) are given by

74 V(11') = — d( k) eik( x1− x' 1 ) v(ω , k ) , (3.55)

and

g12= d k eik( x1− x 2 ) gω , k . (3.56) 1( ) — ( ) 1 ( )

Hence the interaction term with respect toω and k will become

i d k g,, k eik( x1− x 2 ) V k , (3.57) — ( ) 1 (ω) ( ω )

where

Vω, k= i d k ' g ω + ω ', k + k ' v ω ', k ' . (3.58) ( ) — ( ) 1+ ( ) ( )

Collecting all terms from above, we get for the one-particle Green’s function the equation (3.27)

2 ik() x− x »p ÿ ik() x− x —d() k e1 2 …ZZω− + ζ − V()() ω,. k Ÿ = — d k e 1 2 (3.59) 2m ⁄

The integral in the right hand side is the delta function present in the equation (3.27).

The solution of (3.59) can be obtained by introducing the limit ε → 0 + we will get indeed,

ZA g()ω, k = 1 p2 Zω−i ε − + ζ − V() ω − i ε, k 2m

ZB + p2 Zω+i ε − + ζ − V() ω + i ε, k 2m

(3.60)

To calculate the generating function in terms of the time we take the Fourier transformer of the generating function

75 dω g() t− t,,. k = e−iω() t1 − t 2 g()ω k (3.61) 1 1 2— 2π 1

Now if we substitute the solution for g1 (ω,k) equation (3.60), in the above Fourier transformer and applying the boundary condition (3.34) we will get

dω −iω t − t Z B e ()f 2 — 2π p2 Zω+i ε − + ζ − V() ω + i ε, k 2m (3.62) dω Z A = ± e−iω() to + i β Z − t2 . — 2π p2 Zω−i ε − + ζ − V() ω − i ε, k 2m

As it can be seen in the case of the –tt 21 , we have only the contribution of the term

containing B, whereas in the case of ¿tt 21 , it is a contribution of the term containing A in the equation (3.60). The first integral in the right hand side is taken on a path, which is closed in the lower half-plane and the left hand side integral will be on a path closed in the upper half-plane along the real axis. The conditions which must be satisfied by equations (3.60) and (3.62) are

AB+ =1 (3.63)

and

−iω( tf − t o − i β Z) A= _ Be (3.64)

Again we see an analytic continuation of the time into a strip around the complex plane with a wide equal to βZ . From last two equations for A and B we obtain

_e−iω( tf − t o − i β Z) A = (3.65) 1_ e−iω() tf − t o − i β Z

and

76 1 B = . (3.66) 1_ e−iω() tf − t o − i β Z

Finally the expression for the one-particle generating function may be written as

−i ttgttrr– :,,,− − = dkeik() x1− x 2 Aω k (3.67) 1 2 1()() 1 2 1 2 — () −iω t − t − i β Z 1_ e ()f o

and

−iω( tf − t o − i β Z) ik x− x ±() −i e ttgttrr¿ :,,,− − = dke()1 2 Aω k (3.68) 1 2 1()() 1 2 1 2 — () −iω t − t − i β Z 1_ e ()f o

where

ZΓ(ω,k ) A()ω,, k = 2 2 (3.69) »Zω− ε()() ω,k ⁄ ÿ» + Γ ω , k / 2 ⁄ÿ

we define

1 Γ()ω,k = lim » 2 i δ − V()() ω + i δ , k + V ω − i δ , k ⁄ÿ , (3.70) δ →0+ i

also

p2 1 ε() ω,,,,k= − ζ +» V()() ω + i δ k + V ω − i δ k ÿ (3.71) 2m 2 ⁄

where we have applied an analytic extension of theω , to ω ± iδ above and below the

real axis to close the counter in lower half-plane in the case of –tt 21 for equation

(3.67), and close the counter in upper half-plane in the case of ¿tt 21 for equation

(3.68), and eventually a shift of δ → 0+ .

Chapter IV

4. Field Quantization

To study the Casimir effect, which is actually a quantum-field-theory phenomenon, we need to get familiar with this theory. The quantization of the energy proposed by M. Plank studying the energy spectrum in the black body radiation experiment, leaded physicist to construct a new and powerful mechanics that in fact would not only contradict the classical physics but it was complementary to it. As it was discussed before any quantum system is described by a particular vector space with infinite dimension that is called Hilbert space. In this space the vectors are the states which are described by the functions called vector states. The vector states describe different modes of a system, when the physical quantities in this system have discrete values. In quantum mechanics these physical quantities are replaced by operators (e.g. momentum) acting on vector states to posses a value in that particular state instead of a definite value given in classical mechanics. Thus we say that, the physical quantity is quantized.

78 After the suggestion of quantization of the energy by Plank, Einstein showed that the electromagnetic radiation must be quantized to describe the frequency dependency of the photoelectric phenomenon, on the other hand electromagnetism had a powerful description for this radiation that was a result of the propagation of the field, therefore the idea of quantization of the electromagnetic field and later quantization of any other fields was born [12]. As it was mentioned the quantization of the physical quantities comes from the fact that they have discrete values, but some physical systems like a field are thought to be continuous, so how it is possible to quantize a field. This is actually the idea in the quantum field theory [41].

Associated with each quantized field we consider a particle, for example the quanta of the electromagnetic field is called photon, as the electron is a particle for the quantized Dirac field. On the other hand the quantum mechanics tells us that, there is a particular property for each particle of a system; that is only quantum; it is

1 called the spin of that particle. This vector state is shown by up ↑ with value 2 Z ,

1 or down ↓ with value − 2 Z . Now for each particle as a quantum of the field there must be an associated spin. These spins can have integer or half integer and also zero value. For example for the electromagnetic field, the photon has a spin of one, whereas a scalar field has spin zero and a fermion field describes the particles with spin ½.

In this chapter we are not aiming to work out on the whole discussion of QFT

(Quantum Field Theory), since we do not need all, but instead we will talk about those ideas that would be beneficial in our future discussion. We start with very simple form of the fields that is, the scalar fields.

79 4.1. Canonical commutation relation

From classical mechanics the Poisson bracket of two dynamical variables is defined as

∂f ∂ f ∂ f ∂ f f,. f =1 2 − 1 2 (4.1) []1 2 P ∂qr ∂ p r ∂ p r ∂ q r

One of the important properties of the dynamical variables according to the Poisson bracket is

df∂ f f" ≡ = +[] f,. H (4.2) dt∂ t P

This bracket in quantum mechanics will be replaced by

f,, f→ − i f f , (4.3) [ 1 2]p [ 1 2 ]−

where the right-hand side is the commutation relation. Now for a field the above commutation relation will be

»φx,,,. tΠ y t ÿ = i δ 3 x − y (4.4) ( ) ( ) ⁄− ( )

Π in this relation is the canonical momentum corresponding to the field φ , which is given by

Π( x) = φ"( x). (4.5)

The equation (4.4) is called the canonical commutation relation. As in quantum mechanics this relation indicates that φ and Π must be treated as operators, therefore the quantization of the field is a consequence of this relation.

80 4.2. Scalar field quantization

In this section we will see how a scalar field can be quantized. In the next chapter we will show how a field can be treated as a quantum harmonic oscillator, and hence see how a vacuum actually fluctuates (see 5.2.2). This fluctuation leads us to the concept of zero-point energy, which is associated to the ground state energy of

1 the quantum harmonic oscillator 2 ƒ Zωk that is called vacuum energy. As we will k see since a quantum field and its canonical momentum can be reduced in terms of the annihilation and creation operators given by

d3 p φ ()x= a()() p e−ip.†. x + a p e ip x , (4.6) — 3 () ()2π 2Ep and E Πx =φ" x = dpi3p − ape−ip . x + ape † ip . x , (4.7) ()() — 3 ()()() 2() 2π therefore we can say that the field actually has been quantized. One can show that by using (4.6), (4.7), and (4.4) with an appropriate inversion of Fourier transformation we will obtain

»a p,, a† p′′ ÿ =δ 3 p − p ( ) ( ) ⁄− ( ) »a p, a p′ ÿ = 0, (4.8) ()() ⁄− »a†† p, a p′ ÿ = 0. ()() ⁄−

These commutation relations are analogue to the commutation relations for linear quantum harmonic oscillator.

81 4.3. Complex scalar field quantization

The complex scalar field is a field with a zero spin that dedicate a complex number to each point of space-time, and it may be written in the form

1 φ()x=() φ1()() x + i φ 2 x . (4.9) 2

Here we decompose the field into its real part and imaginary part; associated to each part we can decompose the field by a Fourier decomposition and obtain the

†† corresponding creation and annihilation operators a1, a 2 , a 1 , and a 2 . Similar to the commutation relation (4.8), we can define this relation for these operators as

»a( p),,, a†( p′′′) ÿ»= a( p) a †( p) ÿ =δ 3 ( p − p ) 1 1 ⁄ − 2 2 ⁄ − »a()()()() p, a p′′ ÿ =» a†† p , a p ÿ = 0, (4.10) 1 1 ⁄− 1 1 ⁄− »a p, a p′′ ÿ =» a†† p , a p ÿ = 0. 2()()()() 2 ⁄− 2 2 ⁄

If now we define the following

1 a() p=() a1()() p + ia 2 p , 2 (4.11) 1 aˆ () p=() a1()() p − ia 2 p , 2 the commutation relations will be in the form

»a p,,, a† p′′′ ÿ»= aˆ p a ˆ † p ÿ =δ 3 p − p ( ) ( ) ⁄ −( ) ( ) ⁄ − ( ) »a p, a p′′ ÿ =» a†† p , a p ÿ = 0, (4.12) ()()()() ⁄− ⁄− »aˆ()()()() p, a ˆ p′′ ÿ =» a ˆ†† p , a ˆ p ÿ = 0. ⁄− ⁄−

And the decomposition of the field in terms of the annihilation and creation operator will be

d3 p φ ()x= a()() p e−ip ⋅ x + aˆ† p e ip ⋅ x , — 3 () ()2π 2Ep (4.13) d3 p φ † ()x= aˆ ()() p e−p ⋅ x + a† p e ip ⋅ x . — 3 () ()2π 2Ep

As it can be seen we have quantized φand φ † in terms of aand aˆ .

Now we define the Noether current by

∂L Jµ=δφA − T µν δ x , (4.14) A ν ∂() ∂µφ

This current is conserved if there is symmetry in the system

µ ∂µ J = 0. (4.15)

The above result is a consequence of Noether theorem that says: associated with each symmetry of a system is a conserved quantity. Now if we consider the following symmetry in our system of complex scalar field

φ→ e−iqθ φ, (4.16) φ††→ eiqθ φ , where q is the charge associated with each particle and is a parameter independent of space-time. We will have

δφ= −iq θφ, (4.17) δφ††= iq θφ .

The corresponding Noether current (4.14) will be

∂L ∂ L Jµ = − iqφ + iq φ††† = iq » ∂µ φ φ − ∂ µ φ φ ÿ. () † ()()() ⁄ ∂() ∂µφ ∂() ∂µφ

(4.18)

The second term in (4.14) has been vanished sinceδ xν = 0. The conserved charge will be

Q d3 xJ 0 q N N . (4.19) =— =( a − aˆ )

The first term indicates the total charge of particles created by a† , and the second term gives us the total charge of particles created by aˆ† , which is of opposite sign.

Therefore we call the particles created by the second creation operator as the antiparticles of the particles created by first operator.

4.4. Fermion (Dirac) field quantization

One of the main result of the quantum mechanics as it was mentioned before is an intrinsic angular momentum associated with each particle either fermion or boson. This property has different value for various particles, in this section we will talk about fermions or quanta of the Dirac field. The distinguishing property of this field that separate these particles from bosons is its spin, which is indeed a fractional

1 number. The Dirac field is a field describing spin- 2 particles.

In the quantization of the Dirac field we have to be more conscious, since in this case we are facing not only numbers and functions, but there must be some matrices included in the calculation. We begin with an appropriate form of the

Schrödinger equation for a fermion described by field quanta ψ (x)

∂ iψ()()() x= γ0 − i γ ⋅∇ + m ψ x , (4.20) ∂t where the Hamiltonian for this field has been defined as

84 H=γ0 ( γ ⋅ p + m). (4.21)

In the above equation γand γ 0 are constant matrices that satisfy the following anticommutation properties

»γµ, γ ν ÿ = 2g µν ⁄+ 2 (4.22) ()γ 0 =1. g µν is the inverse metric. Also we define the useful matrices

i σµν= − σ νµ = » γ µ,. γ ν ÿ (4.23) 2 ⁄−

0 If we multiply γ from left on both sides of(4.20), and define for any vector aµ

µ µ a≡γ aµ = γ µ a , (4.24) and we get the Dirac’s equation

(i∂ − m)ψ ( x) = 0. (4.25)

It can be shown that the solution of Dirac equation results in an intrinsic angular momentum, which we call the spin of the corresponding particle. For this we have that for any function f( x) the variation to f′( x) will be related by

µ δf( x) = δ f( x) − ∂µ f( x) δ x . (4.26)

Now if we consider an infinitesimal Lorentz transformation given by

Λµν =g µν +ω µν (4.27) for coordinate xν , where

µ µ ν x′ = Λν x (4.28) the change in ψ ( x) will be given by

ψ′′( x) = S( Λ) ψ ( x). (4.29)

85 The linear transformation S (Λ) takes the form

i S ()Λ =1 − β ω µν . (4.30) 4 µν

For a finite transformation ω µν one obtains

≈i µν ’ ψ′′()()()x= S Λ ψ x =exp∆ − σµν ω ÷ ψ () x , (4.31) «4 ◊ where

βµν= σ µν . (4.32)

Now suppose we have a transformation by an angular transformation, in which we have used (4.30) and (4.29)

≈i µν ’ ψ′()x=∆1 − Jµν ω ÷ ψ () x . (4.33) «2 ◊

Applying (4.26) for (4.27), and (4.28) we will obtain

µν ψ′′′′( x) = ψ( x) + ω xν ∂ µ ψ ( x). (4.34)

Now using (4.31), and (4.33) the angular momentum transformation for infinitesimal

ω µν we will get

1 Jµν= i() x µ ∂ ν − x ν ∂ µ + σ µν . (4.35) 2

From quantum mechanics we recognize the first term in the above equation as the orbital angular momentum operator, but there is a second term which is also related

to the angular momentum J µν , we call this the intrinsic angular momentum or spin.

The Fourier decomposition of the Dirac field in terms of creation and annihilation are given by

3 d p −ip ⋅ xˆ † ip ⋅ x ψ ()x= fpupes()()()() s + fpvpe s s , (4.36) — 3 ƒ () s=1,2 ()2π 2Ep and

d3 p ψ ()x= fpupefpvpe† () ()()ip⋅ x + ˆ () − ip ⋅ x . (4.37) — 3 ƒ ()s s s s s=1,2 ()2π 2Ep

Now we will show how the particle and antiparticle can be deduced from creation and annihilation operators of the Dirac field. For this sake we start with the

Lagrangian of this field as

L =ψ(i ∂ − m) ψ , (4.38) where we define

ψ≡ ψ† γ 0. (4.39)

From classical mechanics we know that the Hamiltonian of a system can be written in terms of its Lagrangian as

Hpq( ,,,,,) = pqr"" r ( pq) − Lqqpq( ( )) (4.40) and

∂ pr = L() q,. q" (4.41) ∂q"r

Therefore the Hamiltonian of our system by employing(4.38), will be

δ L H d3 x" x L d3 x i m . (4.42) =—ψα () − = — ψ() − γ ⋅∇ + ψ δψ"α ()x

Now it can be shown that the solution of the our spinor given by us( p) and v s ( p) satisfy the following relations

−ip ⋅ x − ip ⋅ x (−iγ ⋅∇ + m) us( p) e = γ 0 E p u s ( p) e , (4.43)

87 and similarly

ip⋅ x ip ⋅ x (−iγ ⋅∇ + m) vs( p) e = − γ 0 E p v s ( p) e . (4.44)

† Employing (4.36), (4.37), (4.43), and (4.44) in (4.42) and knowing uγ 0 = u and

† vγ 0 = v , we obtain the Hamiltonian as

H= dpE3» fpfp † − fpfpˆ ˆ † ÿ, (4.45) — pƒ s()()()() s s s ⁄ s=1,2 where due to normalization conditions we put u†† v and v u equal to zero. This

Hamiltonian can not be the correct Hamiltonian for our system of the Dirac field since we will get negative energy eigenvalues. To solve this problem we introduce an anticommutator instead of a commutator given by

†»ˆ ˆ † ÿ 3 »fs()()()()() p,,, f s′′′ p′′′ ÿ = f s p f s p =δ ss δ p − p (4.46) ⁄+ ⁄+ and also we use normal-ordering for fermionic operators, where always we put the creation operator in the left-hand side of the annihilation operator, in this case our normal-ordered Hamiltonian will be in the form

::H= dpE3» fpfp † + fpfpˆ † ˆ ÿ. (4.47) — pƒ s()()()() s s s ⁄ s=1,2

The Noether charge in this case is given by

3» †ˆ † ˆ ÿ » ÿ ::Qqdp= fpfps()()()() s − fpfp s s = qN f − N ˆ . (4.48) — ƒ ⁄ s fs ⁄ s=1,2

As it can be seen again we have an opposite sign for the charge of particles created

† ˆ † by fs compare to the charge of particles created by fs . Therefore we call them antiparticles of the corresponding particles.

The Pauli Exclusion Principle can be expressed according to the property of anticommutation since we have

††††ˆ ˆ fs( p) f s( p) = f s( p) f s ( p) = 0. (4.49)

These equations express that we can not have two particles or antiparticles with the same spin and momentum simultaneously.

4.5. Electromagnetic field quantization

In this section we show how a spin-1 field can be quantized. The very familiar example of these fields can be named as the electromagnetic field, whose quantum is called the photon. This is the quantum of a field that was the beginning of the idea for quantum field theory, and it is the most important component of the quantum electrodynamic. For start we bring some basic formulation and notation from classical electromagnetic theory, for this sake we begin with the concept of

Maxwell’s equations. These equations are given by

∇ ⋅E = ρ, ∂E ∇×B − = j, ∂t (4.50) ∇ ⋅B = 0, ∂B ∇×E + = 0. ∂t

The first two equations are called the inhomogeneous Maxwell’s equations, where and j are the charge and current densities respectively. The next two equations are called the homogenous Maxwell’s equations. Next we introduce the electromagnetic potential (we actually refer to that as the photon field) by four-vector

Aµ ≡ (ϕ,.A) (4.51)

In this notation as we can see the time component of this four-vector is the scalar component and the rest is a 3-vactor. This quantity is related to the electric and the magnetic field through the following relations

89 BA= ∇× , ∂A (4.52) E = −∇ϕ − . ∂t

The field strength tensor Fµν is defined by

»0 EEE1 2 3 ÿ … Ÿ −EBB10 − 3 2 F = … Ÿ. (4.53) µν …−EBB2 30 − 1 Ÿ …3 2 1 Ÿ …−EBB − 0 ⁄Ÿ which is related to the potential according to

FAAµν= ∂ µ ν − ∂ ν µ . (4.54)

The Lagrangian of the electromagnetic field without interaction is given by

1 L = − FF µν . (4.55) 4 µν

The equation of motion according to the inhomogeneous Maxwell’s equation in terms of the field strength tensor will be

µν ν ∂µ F = j , (4.56) where jν is the 4-vector of charge and current density

j µ ≡ (ρ, j) .

Also this equation can be shown by

λµ λ µ λ ( gI−∂ ∂) Aµ = j . (4.57)

The above introduction was as brief as we could bring for the classical electromagnetism theory, now we can proceed in the discussion of quantization of this field. Unlike the scalar and Dirac field the quantization of the electromagnetic field is not straight forward and it has problems, which we will indicate. In the quantization of the scalar and Dirac fields the quanta given by φand ψ for these

90 fields had four components, which were all independent, but in the case of the electromagnetic field, the photon field (the potential) has actually four components, where only two of them are independent. This is a serious problem for our purpose that must be solved in some way.

To surmount this problem we have to modify the classical Lagrangian. In

order to do this modification we must transform the photon field Aµ according to a local transformation given by

AAAµ→ µ′ = µ + ∂ µθ , (4.58) where is any differential function of space-time. It is possible always to perform

this transformation so that the new Aµ satisfy the following condition

µ ∂µ A = 0. (4.59)

The above condition (Lorenz gauge) compensates the introducing an additional part in the Lagrangian that is called the gauge-fixing term given by

2 1 µ LGF = −() ∂µ A . (4.60) 2ξ

Now the equation of motion reads

»λµ≈1 ’ λ µ ÿ λ …gI−∆1 − ÷ ∂ ∂ Ÿ Aµ = j . (4.61) «ξ ◊ ⁄

If we impose an appropriate boundary condition that the condition (4.59) be satisfied after the transformation(4.58), we have in fact no external disturbance on the

Maxwell’s equations.

We have choice for the value of the in equation (4.61), the usual gauge is called the Feynman-‘tHooft gauge, where the value of is considered as unity, therefore the equation of motion (4.61), will takes the form

IAµ= j µ . (4.62)

This equation is similar to equation of motion for massless scalar fields, so we can use the method employed in quantization of this field for the case of electromagnetic field.

We can find a relationship between the photon field and its canonical momentum as

»A x,,,, tΠ y t ÿ = − igδ 3 x − y (4.63) µ( ) ν( ) ⁄− µν ( ) this commutation along other two commutation relations given by

» ÿ Aµ( x, t) , A ν ( y , t) ⁄ = 0, − (4.64) »Πx, t , Π y , t ÿ = 0, µ()() ν ⁄− indicate that our theory (electromagnetic field ) like other fields is quantized, the quantization of this field can be viewed by the Fourier decomposition of the field.

Like zero-spin and half-spin fields discussed in previous sections we Fourier expand the photon field

3 3 µ d k µ−ik ⋅ x ∗ µ † ik ⋅ x A() x=»εr()()()() k a r k e + ε r k a r k e ÿ, (4.65) — 3 ƒ ⁄ r=0 ()2π 2 ωk

µ where ε r is a set of four linearly independent 4-vector, we call this factors polarization vectors. The commutation relations (4.63) and (4.64) after decomposition will implies the following commutation relations

»a k,, a† k′′ ÿ =ζ δ δ 3 k − k r( ) s( ) ⁄− r rs ( ) »a k, a k′ ÿ = 0, (4.66) r()() s ⁄− »a†† k, a k′ ÿ = 0. r()() s ⁄−

In this relations

ζ0= −1, ζ 1 = ζ 2 = ζ 3 = 1. (4.67)

† From above commutation we can see that the commutator of a0 with a0 has the opposite sign compared to the commutator of the scalar fields. This is another problem that must be explained in our frame of quantization. For this purpose we consider the normal order of the Hamiltonian given by

3 H d3 k a † k a k (4.68) ::= — ωkƒ ζ r r()() r r=0

It can be shown that the matrix value of this operator between any two physical states reduces into

2 ′′H d3 k a † k a k (4.69) Ψ::. Ψ =— ωk Ψƒ r()() r Ψ r=1

As it can be seen this value is depending only on two polarization states, and none of them is the scalar polarization state or a0, therefore the wrong sign contribution of the commutators has been corrected, and there will be no problem with our prescription for the quantization. This illustration is applied to any other physical quantities, and therefore the theory works properly.

4.6. Propagators

In this section we discuss the very useful concept of quantum field theory that has a crucial rule in the interaction of the particles. That is the description of the propagator of a particle, which indeed describes how a particle move from one point of space-time to another point. The interaction between particles is a result of their propagation in space-time, so the propagators are the main subject of the dynamic of this theory.

93 To obtain the propagators for each specific field, we will use the method of the Green’s function, and we will show that this quantity actually describes the propagation of the corresponding particle.

4.6.1. Propagator of a scalar field

Consider the following equation

2 4 (Ix +m) G( x − x′′) = −δ ( x − x ), (4.70) where G is the Green’s function associated to the Klein-Gordon equation of motion of the scalar field given by

(I+m2 )φ ( x) = J( x). (4.71) where J( x) is the source. In order to obtain a solution for the equation (4.70), we consider the Fourier transformation of the Green’s function to momentum space given by

d4 p G x− x′ = e−ip ⋅() x − x′ G p . (4.72) () — 4 () ()2π

The operator in the left-hand side of (4.70) can be applied on both sides of (4.72), where the corresponding operator in momentum space will be (−p2 + m 2 ) , and then

d4 p I +m2 G x − x′ = e−ip ⋅() x − x′ − p2 + m 2 G p . (4.73) ()x () — 4 () () ()2π

According to (4.70), the left-hand side of (4.73), will be equal to −δ ( x − x′) , and using the equivalent expression for delta function

d4 p δ 4 x− x′ = e−ip ⋅() x − x′ , (4.74) () — 4 ()2π

94 we will obtain an expression for G( p) ,

1 1 G() p = = . (4.75) p2− m 2 02 2 ()p− Ep

0 The above relation has a pole in p= ± Ep , therefore we have an ambiguity in our description for the propagator of the scalar field. To get over of this problem

Feynman proposed a slightly changed form of the propagator by dragging this quantity into complex plane. This can be done by introducing an infinitesimal small positive imaginary partε (orε′ ), in the denominator of (4.75), and letting that equal to zero in the end of calculation. This gives us the propagator in the new look as

1 1 ∆F ()p = = , (4.76) p2− m 2 + iε′ 0 2 2 ()p−() Ep − iε where

ε′ = 2Ep ε . (4.77)

This is called Feynman propagator. It can be shown that, after an expanding of the denominator of the expression in the most-right hand side of (4.76), and also a

Fourier transformation to main coordinates space, we will get the Feynman propagator as

d3 p i∆ x − x′′′ =» Θ t − t e−ip ⋅() x − x′′ + Θ t − t e ip ⋅() x − x ÿ. (4.78) F () — 3 () () ⁄ ()2π 2Ep

Here the -function is defined as

À1t > 0, Œ 1 Θ()t =Ã 2 t = 0, (4.79) Œ Õ 0t < 0.

The equation (4.78), can be written in the following form

ixx∆F ( −′′′′′) = Θ( tt −) 0φ( xx) φ( ) 0 + Θ( tt − ) 0 φ( xx) φ ( ) 0 , (4.80)

95 or with the notation of time-ordering it will be in the form

′′» ÿ i∆F ( x − x) = 0T φ( x) φ ( x ) ⁄ 0 . (4.81)

In this equation the time-ordering is defined as

ŒÀφ( x) φ ( x′′) t> t , T »φ()()x φ x′ ⁄ÿ ≡ Ã (4.82) ÕŒφ()()x′′ φ x t> t.

In (4.80), the first term in the right-hand side indicates that a particle has been created in time t’ and after propagation in space annihilated in time t, and also the second term explains how a particle is created at time t and annihilated at the later time t’. this can be illustrated by applying the field operator (4.6), on ground state given by

d3 p φ ()x0= eip⋅ x p , (4.83) — 3 ()2π 2Ep and d3 p′ 0φ ()x′′= e−ip′′ ⋅ x p . (4.84) — 3 ()2π 2Ep′

In the above equations we used the creation and annihilation of a particle according to

a† ( p) 0= p , (4.85) and 0a( p′′) = p . (4.86)

Therefore the matrix element of the field operator which is the vacuum expectation value of the correlation of the field will be

d3 p 0φx′ φ x 0= eip⋅() x − x′ . (4.87) ()() — 3 ()2π 2Ep′

This ends our calculation to obtain (4.81), since by substituting the above equation in

(4.78), we will get the expression (4.80).

4.6.2. Propagator of a Dirac field

In this section we follow the same procedure as last section to compute the propagator of a Dirac field. In this case obviously the equation of motion of the field will differ from the case of a scalar field that is

µ (iγ∂µ − m) ψ ( x) = J( x). (4.88)

The Green’s function associated to this differential equation will be

µ x 4 (iγ∂µ − m) S( x − x′′) = δ ( x − x ). (4.89)

Now the Fourier transformation gives us the Green’s function in the momentum space

d4 p S x− x′ = e−ip ⋅() x − x′ S p . (4.90) () — 4 () ()2π

µ x Applying the operator (iγ ∂µ − m) on (4.90), which is obviously a Fourier transformed in momentum space of ( p− m) , for the right-hand side of this equation, and considering (4.74), results to

1 S() p = (4.91) p− m

Multiplying and dividing by p+ m , we will get an expression for the propagator

97 p+ m p + m S() p = = . (4.92) p2− m 2 02 2 ()p− Ep

0 Again we have a pole in p= ± Ep . This problem can be solved by introducing the

Feynman prescription, which is bringing an imaginary part in the denominator, and making the propagator complex. This prescription can be a way to look at the problem as a quantized one, since the complex physical quantity can not be defined in classical mechanics. The corresponding propagator with this prescription will be again called the Feynman propagator for the Dirac field, that is

p+ m S() p = . (4.93) F p2− m 2 + iε

It is easy to show that the Feynman propagator of the Dirac field is related to scalar field by taking a Fourier transformation of the equation (4.93), back to coordinate space. This relationship reads

SFF( x− x′′) =( i ∂ + m) ∆( x − x ). (4.94)

Substituting (4.78), in the above equation leads us to the following relation

′′» ÿ iSFαβ( x− x) = 0T ψ α( x) ψ β ( x ) ⁄ 0 , (4.95) where the time-ordering in this case will be defined as

ŒÀψα( x) ψ β ( x′′) t> t , T »ψ()x ψ () x′ ÿ ≡ Ã (4.96) α β ⁄ ′′ ÕŒ−ψβ()()x ψ α x t > t.

98 4.6.3. Propagator of the electromagnetic field

We start with the equation of motion for an electromagnetic field in terms of the field strength tensor

µν ν ∂µ F = j . (4.97)

But the field strength tensor was related to the potential by the following relation

FAAµν= ∂ µ ν − ∂ ν µ . (4.98)

Therefore the equation of motion for this field will be

λµ λ µ λ ( gI−∂ ∂) Aµ = j . (4.99)

The Green’s function equation for this differential equation after a Fourier transformation in the momentum space will be of the form

λµ2 λ µ λ −( g k − k k) Dµν( k) = g ν . (4.100) the propagator must be in the form

Dµν( k) = ag µν + bk µ k ν . (4.101)

It can be shown that this equation will not give a definite value for the propagator, that is so because the value of b after substituting (4.101) in (4.100) will be cancel out. However this equation can be modified some way. For this reason as we discussed in the section 4.5, we introduce a fixing term with an auxiliary parameter in the Lagrangian given by (4.60), in this situation the equation of motion will be modified as

»λµ≈1 ’ λ µ ÿ λ …gI−∆1 − ÷ ∂ ∂ Ÿ Aµ = j . (4.102) «ξ ◊ ⁄

The corresponding equation for the Green’s function after a Fourier transformation will be

»λµ2 ≈1 ’ λ µ ÿ λ −…g k −∆1 − ÷ k k Ÿ Dµν() k = g ν . (4.103) «ξ ◊ ⁄

Now if we put the expression of the propagator given by (4.101), in the above equation we obtain

λ2 λ»≈1 ’ 1 2 ÿ λ gν= − ak g ν +… a∆1 − ÷ − bk Ÿ k kν . (4.104) «ξ ◊ ξ ⁄

By solving the above equation for a and b we will get

1 a = − , (4.105) k 2 and

1−ξ b = . (4.106) k 4

Using these values and noticing that again we have a situation similar to the scalar and Dirac fields, where there is a pole for the propagator that prevent us for a definite value for this quantity, so we employ the Feynman prescription to make the propagator a complex quantity by adding an infinitesimal small imaginary part in the denominator

1 »kµ k ν ÿ Dµν() k= −2… g µν −()1 −ξ 2 Ÿ . (4.107) k+ iε k ⁄

This is the final form of the propagator for the electromagnetic field, there are some specific cases for this propagator in terms of the particular value of the parameter .

They are

ig iD() k = −µν ,ξ = 1 (4.108) µν k2 + iε and

100

i »kµ k ν ÿ iDµν() k= −2… g µν − 2 Ÿ,ξ = 0. (4.109) k+ iε k ⁄

The first choice is called the Feynman-‘tHooft gauge and the second one is called the

Landau gauge.

4.7. S-matrix

As we have seen in previous chapters the Lagrangian of each specific field, is quadratic in the field operators. These indicate fields with no interaction. We call them free fields. Free fields exist independent of each other that is why if there is no interaction nothing in the world will happen. But the reality is different since we can see the consequence of interaction between fields in the real world. Therefore we can perform experiments that are a measure of interactions, otherwise the apparatus in our experiment with free fields alone have no experimental involvement and hence no experimental result will be achieved.

To include the interaction in our theory we have to manipulate the

Lagrangian for a new form. In this situation we just add the interaction terms into normal free Lagrangian of the field. As it was mentioned earlier, we have the description of the free field Lagrangian for up to a second order of the product of the field operators. The interaction Lagrangian will be given by a third order of the product of the fields and bigger. These terms are coupled with some constants, which are called the coupling constants.

Now we proceed in the theory of the interaction of the fields starting with the evolution equation of a particular state under a total Hamiltonian, consisting of the free part and interaction part given by

101

HHH=0 + I . (4.110)

The free Hamiltonian H0 , is constructed out of free field operators.

Now we define the evolution operators U( t) andU0 ( t) as

ψ 0(t) = U 0 ( t) i , (4.111) and

† ψ(t) = U0( t) U( t) U 0( t) ψ 0 ( t) , (4.112)

where i is the initial state, and ψ 0 (t) is the free state of the fields concerned.

Now if the operators U0 ( t) andU( t) , are unitary the equation (4.112) can be written in the following form

ψ (t) = U0 ( t) U( t) i , (4.113) where ψ (t) , is a state evolving under the total Hamiltonian (i.e. free plus interaction contribution of the Hamiltonian). As it can be seen from (4.111) and (4.113), both kind of states evolve by evolution operators acting on the initial state i .

The Schrödinger’s equation for the free field is

d i U() t i= H U() t i . (4.114) dt 0 0 0

This equation can be solved as

−iH0 t U0 ( t) = e . (4.115)

Now we consider the Schrödinger’s equation in the case of the total Hamiltonian

d iψ() t=()() H + H ψ t . (4.116) dt 0 I

Applying(4.113) on both sides of the above equation we will get

102 dU( t) i= UtHUtUt† ()()()()() = HtUt , (4.117) dt 0II 0 where

† HII( t) ≡ U0( t) H U 0 ( t). (4.118)

HI ( t) is the interaction Hamiltonian H I , in terms of the free fields at time t. The integral solution of (4.117) is

t Ut Ut idtHtUt (4.119) ()()()()=0 − — 1I 1 1 . t0

We assume U( t) →1 as t → −∞ . Puttingt0 = −∞ , in (4.119) we obtain

t U t=1 − i dt H t U t . (4.120) ()()()— 1I 1 1 −∞

In the right-hand side U( t1 ) , can be replaced by same expression of (4.120) resulting in

t ≈t1 ’ Ut=1 − idtHt 1 − idtHtUt . (4.121) ()()()()—1II 1∆ — 2 2 2 ÷ −∞« −∞ ◊

The above procedure can be continued to get the following expression

∞ t t1 tn−1 Ut=1 + − in dtdt?? dtHtHt Ht . (4.122) ()()()()()ƒ —1 — 2 — n I 1 I 2 I n n=1 −∞ −∞ −∞

It can be shown that the above equation is a actually a time-ordered expression given by

n ∞ ()−i t t t Ut=1 + dtdt?? dt» HtHt Ht ÿ. () ƒ —1 — 2 — nT I()()() 1 I 2 I n ⁄ n=1 n! −∞ −∞ −∞

(4.123)

It is convenient to write down the above expression in a compact form

103

»≈t ’ ÿ U t i dt′′ H t ()()=T …exp∆ − — I ÷ Ÿ . (4.124) …«−∞ ◊ ⁄Ÿ

Comparing the results for calculating the evolution operators U0 (t) and U(t), from equations (4.115) and (4.124), one can see the similarity between these formulae as they are both exponential expression i times the Hamiltonian, where in the case of free field the evolution operator contains the free time-independent Hamiltonian, and in the latter case it is the interaction time-dependent Hamiltonian.

Now we define the S-matrix as the evolution operator (4.124), when time goes to infinity

»≈∞ ’ ÿ S=lim U()() t =T … exp∆ − i dtHI t ÷ Ÿ . (4.125) t→∞ — …«−∞ ◊ ⁄Ÿ

To see what is the rule of the S-matrix, consider the time-independent basis vector- states of the free fields given by i , which is an initial state and f , which is another basis state after a time t. In this situation the amplitude of transition from the initial state to the final state is given according to(4.111), and (4.113) by

† limfUtUtUti0( ) 0 ( ) ( ) = lim fUti( ) = fSi . (4.126) t→∞ t→∞

Therefore this amplitude is just the matrix elements of the S-matrix.

4.8. Wick’s theorem

Some times it is convenient and necessary to convert the time-ordered product in terms of the normal ordering, where as it was mentioned before we put all the annihilation operators to the right of all creation operators. The advantage of the normal ordering is that it will be vanished in the vacuum, since the annihilation operators in the most right-hand side annihilate the vacuum.

104 For this sake consider for example a scalar field φ ( x) . This field operator

according to(4.6), can be written in terms of a creation operatorφ− ( x) and an

annihilation operator φ+ ( x) by

φ( x) = φ+( x) + φ − ( x), (4.127) therefore we will have

φ( x) φ( x′′′) = φ( x) φ( x) + φ( x) φ ( x ) + + + − (4.128) +φ−()()()()x φ + x′′ + φ − x φ − x .

The second term in the above equation is the only term that must be manipulated to get a normal-ordered expression out of the field product in the left side of (4.128).

That is because from definition of the normal ordering the annihilation operators must be in the right side of the creation operators, and this is not the case for this term. Now we just change the order of these two field operators in the second term, but before doing that we must subtract this term. So the normal order form of (4.128) will be

::φ( x) φ( x′′′′) = φ( x) φ( x) − φ+( x) φ −( x) + φ −( x) φ + ( x ) (4.129) =φx φ x′′ − » φ x,. φ x ÿ ()()()() + − ⁄−

It can be shown that the commutator on the right-hand side may be written as the vacuum expectation value of the field product

0φx φ x′′ 0= » φ x , φ x ÿ . (4.130) ( ) ( ) +( ) − ( ) ⁄−

Applying the time-ordering on both sides of (4.129), and using (4.130) the final general form of relation between normal-ordering and time-ordering will be as

T »Φ( x) Φ′′′′′′( x) ÿ =::, Φ( x) Φ( x) + Φ( x) Φ ( x ) (4.131) ⁄ 8))))9 where

105

0T »Φ( x) Φ′′′′( x) ÿ 0 ≡ Φ( x) Φ ( x ) . (4.132) ⁄ 8))))9

If the field operator in (4.132), is the same field, since this contraction of the fields describes a vacuum expectation value, it is non-zero when the right-hand side operator creates a particle and the left-hand side operator annihilates that particle.

This in fact defines the propagator, when it describes the movement of a particle created in one point of space-time and annihilated at another point of space-time.

Therefore this quantity can be written in terms of Feynman propagator as following

φ( x) φ ( x) = i ∆( x − x ). (4.133) 8)))91 2F 1 2

The above discussion can be used for the other fields as well such as Dirac’s field.

The conversion of the time-order product to a normal-order product of the fields given by equation (4.131) is called Wick’s theorem, it has many application in quantum field theory especially in the discussion of Feynman amplitude.

4.9. Divergent results in quantum field theory

In quantum field theory sometimes we see that many observables such as mass, charge and others get infinite values. That is so since in the perturbative approximation of quantum field theory following the Feynman diagram technique we confront some closed loops of virtual particles in our calculations. For example in a vacuum polarization in its lowest order, we have a photon that interact with a fermion and its antiparticle counterpart that will be created out of the vacuum and then annihilate, or we can name the fermion self energy in quantum electrodynamic in one-loop order, where a fermion with a specific momentum interact with a photon

106 and then the photon is annihilated and the fermion will obtain its original momentum. In these two examples the perturbations correspond to one-loop corrections to the diagram, but for more accuracy we may introduce higher order loops in the calculation.

Since there is contribution of more than one virtual particle in each loop the momentum and energy of these particles are not constrained by conservation of these quantities outside of the loops, and they can posses any value for these quantities.

Therefore the integration over the energy and momentum for the loop gives us an infinite value or we say it diverges.

To overcome this problem, that is to get a finite result out of infinity we introduce a mathematical and physical ad hoc procedure. The first procedure is called regularization and the second one is renormalization. The regularization will work on the infinite integration or sum, where we introduce a regularization parameter in the theory in order to perform the integral, but often at the end of the calculation we have an expression that depends on the regularization parameter as well as a parameter that is called the scale. To get ride of this situation we perform a procedure that is the renormalization and then we obtain a finite value independent of the regularization parameter. We talk about these procedures in the following subsections.

4.9.1. Regularization

The process of the regularization as it was clarified above makes a loop integral drop faster at high energy and momenta whereas the integral converges.

There are many types of regulator; each regulator corresponds to introducing a specific regularization parameter in the problem and then after solving the integral in

107 the end of the calculation the parameter must be shifted to the real situation, where the original integral was in.

The first regularization process is called the cut-off regularization. In this model instead integrating all loops over infinite momenta we consider an integration over momenta with a certain maximum value of , then the integration will be performed, hence the infinite result for the integration turns into a finite value depending on the cut-off parameter , but by shifting to infinity we will get again an infinite value for the loop integration. This way of regularization is not Lorentz invariant, therefore not appropriate to be compared to other methods.

The other method of regularization is Pauli-Villars regularization. In this procedure we add a fictitious field with a very large mass. These fields are auxiliary and then do not appear in the physical theory. The infinity from the large momenta of the main particles in the loops then will be cancelled out by the infinities arising from these fictitious particles in the loops. The result is dependent on the mass of the auxiliary particles and it will be again infinite when we put these masses equal to infinity.

The third process of regularization is called Lattice regularization. This method is a more geometrical approach for this problem and it is much more complicated. In this scheme we consider the structure of space-time as constructed by a hyper-cubical lattice with fixed grid size. The high momenta of particles will be cut according to the size of the grids. Obviously the result of computing the integral depends on the size of grid, and we will get again infinity when we return back to the real space-time.

The most elegant type of regularization which is used in most calculations of quantum field theory is called the dimensional regularization scheme. This is quite a

108 different technique in a fictitious fractional number of dimensions of space- time(d −ε; ε ∈y) . This relies on the fact that the divergent integral in fact depends on the dimension of space-time and is well-behaved in a nonintegral dimension [45].

Like other techniques in this case the result depends on , and the integral diverges by removing the fractional part of the dimension (i.e. ). The integral in this case can be solved according to following formula

2π m / 2 dm x f() x= dx xm−1 f() x . (4.134) —Γ()m / 2 —

The parts of the integral that depend on the regularization parameter in each regularization method are called the divergent parts of the regularized integral.

4.9.2. Renormalization

The renormalization is a way to get a finite value, for example for energy or the loop amplitude in quantum field theory; this is done by performing some procedure on a physical quantity such mass or a coupling constant in the Lagrangian.

This procedure is called the renormalization of that parameter [25]. In the case of the vacuum expectation value of the energy for a particular field, this quantity is actually the field itself. The renormalization procedure associate with introducing some additional part into Lagrangian is called the counterterm. The counterterms are the product of the field with some coefficient in a form that results in the cancellation of the divergent part of the regularized integral or sum. The concept of renormalization according to the manipulation of the Lagrangian with an additional counterterm is as follows: the Lagrangian with a quantum correction must be in the form

109

(1) 2 ( 2) LL=classical +ZZδ L + δ L + ⋅⋅⋅ , (4.135) where we recover a classical Lagrangian when the quantum effect is neglected with Z → 0 . The superscripts in the corrections are related to the corresponding loop order, for example the second term in the right-hand side of (4.135) is the correction to the Lagrangian at one-loop order, and the third term at the second-loop order and so on. The contribution of the extra term in the Lagrangian manifests itself in our calculation as the counterterm. As it was clarified before the counterterm is a term that counters the dependency of the regularized integral on the regularization parameter. The form of the counterterm is completely arbitrary as long as it gives a proper finite value for the integral independent of regularization parameter [38].

Now we define the renormalized quantities such as mass, charge, and quantum field themselves that in fact do not take into account the contribution of the virtual particle loop effects. These quantities are called the bare quantities and correspondingly we can define a new set of fields and physical constants whose contributions in the Lagrangian gives us a total Lagrangian that is a sum of the original Lagrangian and the counterterm Lagrangian

LBCT= L + L . (4.136)

Hence in order to obtain a finite value for the divergent integral that is related to a physical quantity such as the loop amplitude in quantum field theory, or the vacuum energy for example in the case of the Casimir effect, which is still one quantum field theoretical effect, we have to introduce an appropriate counterterm in the Lagrangian, whose effect is the cancellation of the divergent part in the problem.

110

Chapter V

5. Casimir Effect

One of the most interesting macroscopic effects of quantum field theory, which must be considered as the manifestation of the vacuum fluctuation of a field is called the Casimir force [49, 52]. This is a force between macroscopic bodies either conductors or dielectrics [21]. The Casimir effect is studied in a system with or without boundaries. In the case of a system with boundaries, it is known that this force is highly dependent on the geometry and the topology of such boundaries. It can be worked out that the Casimir force may change its behaviour from an attractive force to a repulsive one according to the shape of the boundary introduced in the system with a certain quantum field. It has been proposed that, the Casimir force is actually a macroscopic version of the Van der Waals force, therefore the Van der

Waals intermolecular interaction can be interpreted as a force originated from zero– point energy. In a more comprehensive point of view, the Casimir energy is described as the distortion of the vacuum due to the presence of either a boundary such as a conductor in the space-time manifold or a background field like a gravitational field whose effect might be thought as curving of the space-time. This

111 energy in the first case was studied by the Dutch physicist Hendrick Casimir, when he proposed and calculated a force between two closed parallel conducting plates, due to a vacuum fluctuation of the electromagnetic field. The investigation in the second case was started by Utiyama and deWitt.

5.1. Casimir effect in various field of physics

The Casimir effect has many contributions in different field of science and technology. In physics it has many important roles in quantum field theory, condensed matter physics, atomic physics, mathematical physics, gravitation and astrophysics [19].

In particular in quantum field theory the contribution of the Casimir effect is noted in quantum chromodynamics, where the Casimir energy plays a vital role in the total nucleon energy. In the Kaluza-Klein field this effect explains effectively the mechanism for the spontaneous compactification of extra spatial dimensions.

Another application of the Casimir effect in quantum field theory can be named as in the bag model of hadrons.

The Casimir effect happens differently for different material boundaries in condensed matter physics. The attractive or repulsive nature of this force strongly depends on the geometrical configuration of the boundaries, the temperature, and the electrical and mechanical properties of the boundaries. This effect is investigated on surface tension and latent heat.

The correction to energy levels of Rydberg states in atomic physics is investigated by considering the Casimir interaction. It has also some applications in quantum electrodynamics.

112 In mathematical physics, the Casimir effect is investigated with a very elegant method involving the zeta function. In this method we relate the zeta function to the heat kernel of a second order differential operator. An appropriate expansion of the heat kernel reveals the behaviour of the Casimir force either in the case of a background field or when there is a material boundary in space-time.

The Casimir effect also arises in a space-time with non-trivial topology, and the polarization of the vacuum resulting from Casimir effect is a very important source of inflation of the universe. The Casimir effect plays a very important role in astrophysics, gravitation and cosmology.

5.2. Vacuum fluctuation

Quantum mechanics has changed our understanding of a vacuum. A vacuum, as it was thought in classical mechanics actually is not an emptied system from all particles, where temperature has been dropped to absolute zero. In the next two subsections I will bring two different ways of interpreting of the vacuum. First I discuss a new and alternative way of understanding this phenomenon using the fluctuation-dissipation theorem. Then I will bring the usual approach to this concept using a quantum field theoretical approach.

5.2.1. An alternative look at the vacuum fluctuations

In this section I will discuss in a new way the concept of vacuum fluctuations, using the fluctuation-dissipation theorem. According to the Heisenberg uncertainty principle, there is uncertainty in the measurement of the observables such as length, energy and momentum. This is not a human error in the measurement of such

113 physical observables, but this in fact comes from an intrinsic character of nature, explained by quantum mechanics. For example an exact measurement of length and momentum simultaneously can not be done since there is an uncertainty in those measurement identified by

Z ∆x. ∆ p ≥ (5.1) 2

that is the product of these two observable can not be less than half of the Z .

We can also see in quantum mechanics there is a similar relationship for time and energy of a system that can be given by

Z ∆E. ∆ t ≥ . (5.2) 2

Therefore an exact measurement of the energy with zero uncertainty needs an infinite time which is not an acceptable value. Hence the energy of a vacuum according to the classical quantum point of view has a deviation and can not be exactly zero. The non-zero energy of the vacuum is called zero-point energy, which was deduced from the Plank’s theories, where according to Plank’s theory zero-point energy is indeed an intrinsic cosmological entity of the universe. The vacuum energy is explained according to the vacuum fluctuation of the quantum fields, where creation and consequently annihilation pairs of virtual particle and anti-particle whose life-times are very short imply this energy variation. This short time is dictated by the uncertainty principle and the virtual particles will be annihilated back to the vacuum after this time. That is so because unlike classical concept of the vacuum, which defines a vacuum as an arena of space with absolutely nothing, the quantum field theory has a different point of view for the vacuum. In this theory the fabric of space consist of fields, or these fields permeate the vacuum. Therefore absolutely

114 nothingness in the classical vacuum is replaced by physical vacuum in modern physics. The physical vacuum is understandable when we assume for example a vacuum in a container with no particle in absolute zero temperature, but in this vacuum there still will be propagation of the electromagnetic waves. This is a wave that does not need material medium for propagation and it is actually ripples in the state of the electromagnetic field. Hence in general it is thought that the structure of the space contains fields.

According to the above discussion, we find dissipation of the energy in the field (as a result of non-zero energy of the vacuum given by Heisenberg uncertainty principle) and from the dissipation-fluctuation theorem in linear response theory discussed in chapter one there must be a fluctuation of the field in that system. Hence we say that the vacuum fields – in particular electromagnetic fields - fluctuate, or there is a vacuum fluctuation for the field.

For example in the case of the electromagnetic field from [39], we have the following relationship

1 ∗ ()()1 2 RR» ÿ ( Ai A2 ) = icoth()()()Zω /2 T Dik ω ;, r1 r 2 − D ki ω ;, r 2 r 1 ⁄ . (5.3) ω 2 { }

In this equation the left-hand side we have a description for the correlation function for the electromagnetic potential A, that indicates the fluctuation of the field and in

R the right-hand side there are electromagnetic retarded Green’s functions Dik . The

Green’s function in this case plays the roles of the generalized susceptibility. As we can see the difference between this quantity and its complex conjugate gives us the imaginary part of the generalized susceptibility that is related to the dissipation of the energy. Hence we have a description for the fluctuation of the field (in this case electromagnetic field) according to the dissipation of the energy in this system.

115 5.2.2. Quantum Field Theory and vacuum fluctuation

From quantum field theory any field such as an electromagnetic field is considered as a set of harmonic oscillators. This is shown by quantization of the field, where we will get a contribution of the creation and annihilation operators.

Comparing the Hamiltonian of the system it gives us an analogy between the quantized field and a harmonic oscillator. We show this analogy by considering the simplest form of the field that is a scalar field [38]. We start by a Fourier decomposing of the field given by

1 φx= d4 p δ p 2 − m 2 A p e−ip . x . (5.4) () 3/ 2 — () () ()2π

Here A(p) is the Fourier component of the field. From other hand the Klein-Gordon equation of field states

(I+m2 )φ ( x) = 0. (5.5)

Now using the above equation the(5.4), will be in the form of

1 I+m2φ x = dppm 4 − 2 + 2 δ pmApe 2 − 2−ip . x = 0 (5.6) () () 3/ 2 — ()() () ()2π

we introduced a -function to insure that (5.5), will be satisfied. Now suppose the value of the field in points of space-time a real number; it is a hermitian operator that is φ( x) = φ † ( x) , this actually implies:

A(− p) = A† ( p). (5.7)

The equation (5.4), now can be written as

1 φx= dppm4 δ 2 − 2 Θ pApe 0−ip . x + Ape † ip . x . (5.8) () 3/ 2 — ()()()()() ()2π

116 is the step function equal to one for values bigger than zero and zero for negative arguments, its value for the argument equal to zero is equal to half. It can be shown that

2 21 » 0 0 ÿ δ()p− m = δ()() p − Ep + δ p + E p , (5.9) 2 p0 ⁄ where

2 2 Ep ≡ p + m . (5.10)

Now (5.8), can be written as

d3 p φ ()x= a()() p e−ip.†. x + a p e ip x , (5.11) — 3 () ()2π 2Ep where

A( p) a() p = . (5.12) 2Ep

Similarly the canonical momentum can also written as

E Πx =φ" x = dpi3p − ape−ip . x + ape † ip . x . (5.13) ()() — 3 ()()() 2() 2π a( p)and a† ( p) in equations(5.11), and (5.13) are indeed the annihilation and creation operator respectively, as in quantum mechanics we discuss for the

Hamiltonian of a quantum harmonic oscillator which can be given by

1 H=Zω () a†† a + aa . (5.14) 2

In quantum field theory the Hamiltonian density is given by

1 2 H = Π2 +() ∇φ + m 2 φ 2 . (5.15) 2 ( )

An integration of the above equation over all space results in the Hamiltonian of the system, the corresponding integrations for each part of equation(5.15), are as follow

117 E —dx3Π 2() x = — dp 3 p » − apape()()()() −−2iEp t + apap† 2 (5.16) ††† 2iEp t ÿ +a()()()() p a p − a p a − p e ⁄ ,

2 2 p iE t 3 3 » −2 p † —dx()∇φ () x = — dp apape()()()() − + apap 2Ep (5.17) ††† 2iEp t ÿ +a()()()() p a p + a p a − p e ⁄, and

2 m iE t 2 3 2 3 » −2 p † mdx—φ () x= — dp apape()()()() − + apap 2Ep (5.18) ††† 2iEp t ÿ a()()()() p a p+ a p a − p e ⁄.

Substituting the above three equations in (5.14), will result in a description of the

Hamiltonian in terms of the creation and annihilation operator

1 H= dpEapap3» †()()()() + apap † ÿ. (5.19) 2 — p ⁄

As it can be seen this equation is completely similar to(5.14), therefore we end up with an analogy between a harmonic oscillator and a quantum field.

It is well known that, the energy of a harmonic oscillator is given by

1 E=( n + 2 ) Zωk (5.20)

Now the energy of this fluctuating field in the ground state for a particular mode k

1 will be 2 Zωk , so the energy of the vacuum, which is called zero-point energy, is identified by

1 2 ƒ Zωk . (5.21) k

118 From the above discussion we conclude that vacuum fluctuates, and this is actually the source of the zero- point energy, and the Casimir effect could be calculated due this energy.

5.3. A simple model

In this section we will calculate the Casimir force, primarily proposed by

Casimir [46]. In this case we consider two parallel, conducting uncharged plates and the field is considered as a massless scalar field. According to the previous section, the zero point energy for the field is

1 E = ƒ Zωk . (5.22) 2 k

There are only some discrete modes in the cavity, with a density of modes given by

nπ k = , (5.23) a

for normal modes, where a is the distance between plates and n is an integer number.

Now the above expression for the energy can be written in the form of

1 ∞ d2 k n 2π 2 E=Z c k 2 + (5.24) ƒ— 2 2 2 n=1 ()2π a

with transverse momentum k. This summation is divergent and to obtain a finite result we have to regularize that. There are many ways to regularize an equation, one of these methods is called dimensional regularization, where we substitute the dimension of the integration equal to a continuous variable d, then the integration will be done and in the end of the calculation the dimension will be changed to the corresponding physical dimension. Also there is a method to treat the square root in

119 the above equation, which is called Schwinger proper-time representation, hence the above expression for the energy may be written as

∞ d ∞ 2 2 2 1 −t k +n π 1d k dt − ( 2 ) 1 E= Z c t2 e a . (5.25) ƒ—d — 1 2 n=1 ()2π 0 t Γ() − 2

Here we have an Euler gamma function present in the equation which is defined as

Γ(n) =( n −1) !, (5.26)

and it can be obtained using

∞ Γ()z = — tz−1 e − t dt . (5.27) 0

The equation (5.25) can be represented in a more convenient form by introducing the

Gaussian integration

∞ 2 2 — ae−()x − b/ c dx= ac π (5.28) −∞

Or in terms of the Euler gamma function it can be written as

∞ b e−ax dx= a −1/ b Γ1 + 1 (5.29) — ()b 0

Applying expression (5.27) and (5.29) in equation (5.25) gives us a simpler expression for the energy

∞ ∞ 1 1 dt 2 2 2 E= − Z c t−1/ 2 −d / 2 e − tnπ / a . (5.30) d / 2 ƒ— 4 π ()4π n=1 0 t

Now we introduce a very useful function that is called Riemann zeta function. This function is defined as

120

∞ ζ ()n= ƒ k −n , (5.31) k=1

this function can be written in terms of the gamma function as

1 ∞ u x−1 ζ x= dx . (5.32) () — u Γ()x0 e −1

Another form of this function may be given in the following form

1∞ 1 ∞ ζ n= e−y y n −1 dy (5.33) () ƒ n — Γ()nk=1 k 0

Applying the zeta function and the gamma function to equation (5.30) gives us

d +1 1 1≈’π ≈d + 1 ’ E= −d / 2 Z c∆÷ Γ ∆ − ÷ζ () − d −1 . (5.34) 4 π ()4π «◊a «2 ◊

Now we use the reflection property for the zeta function to include the odd integer value of d, in the calculation of the energy. The final result for this function will be

1 1 ≈d ’ E= −Z c Γ∆1 + ÷ζ () 2 + d . (5.35) 2d+2π d / 2 + 1a d + 1 « 2 ◊

We have used the dimensional regularization to avoid the shifting the function to infinity, now it is time to get back to the original physical situation for dimensionality in our calculation of the energy, In order to this we put our d equal to

2 in above equation. The values of the gamma function and the zeta function corresponding to this value of d gives us the Casimir energy for a massless scalar field in the case of two parallel conducting plates. The minus differentiation of this energy relative to the separation distance between plates gives us the stress or force per unit area of the plates:

121

π 2 Zc F = − . (5.36) 480 a4

This indicates an attractive force exerted on the plates, as expected from the original calculation of Casimir on parallel plates in an electromagnetic field, obviously the above value must be multiplied by two (corresponding to the two polarization state of this field) to give the correct value of the force on the plates in this field.

5.4. Field energy

The energy of the field as the vacuum expectation value of the energy- momentum tensor is identified as the sum of the zero-point energy over the modes.

This relationship can be easily given starting from the exponential regularization of the zero-point energy, since the sum of this energy over the modes is divergent, which is physically a meaningless value. The oscillating exponential regularization of the zero-point energy is given by

1 −iωa τ 2 ƒ Zωae (5.37) a

To get ride ofωa , we introduce a continuos variable using the Cauchy’s integral formula that is given by

f( z) dz= 2π if z (5.38) î— ()a c z− za

with any complex function f(z). Therefore the expression for the zero-point energy

(5.37), will be

Z ∞ dω2 ω 1 Zωe−iωa τ = e−iωτ ω (5.39) 2 ƒa — ƒ 2 2 a 2−∞ 2πia ωa − ω − i ε

122 where the counter of the integration is closed in the lower half plane. Now suppose we have a massless scalar field. From normalization of this field

dx x∗ x =1 (5.40) — ψa( ) ψ a ( )

for a particular mode a, we can manipulate the right hand side of (5.39), when we get a contribution of a second integration in this equation

∞ ∗ dω ψa( x) ψ a ( x) 1 ZZωe−iωa τ = e−iωτ ω 2 dx (5.41) 2 ƒa — — ƒ 2 2 a −∞ 2πia ωa − ω − i ε

Now since the field satisfies the following equations

2 ≈∂ 2 ’ ∆−2 −ω ÷ ψ a ()x = b (5.42) «∂x ◊

2 ≈∂ 2 ’ ∆−2 −ω ÷G()() x,',' x = δ x x (5.43) «∂x ◊

we get a description for the Green’s function G(x,x’)

∗ ψa( x) ψ a ( x) G() x,' x = ƒ 2 2 (5.44) a ωa − ω

Therefore the equation(5.41) may be written as

−iω τ Z dω 2 −iω() t − t ' 1 Zωea = dx ω G x,;,'. x ω e t → t (5.45) 2 ƒ a — — () a i 2π

using definition of the Green’s function

i G() x,;',' t x t= Tφ()() x , t φ x ',' t (5.46) Z

We rewrite equation (5.45)

123

1 Zωe−iωa τ = dx ∂0 ∂' 0 φ x φ x ' t → t ', (5.47) 2 ƒ a — ( ) ( ) a

where a Fourier transformation has been done. Now consider the energy-momentum tensor for a massless scalar field

µν µ ν1 µν λ T= ∂φ ∂ φ −2 g ∂ φ ∂λ φ (5.48)

but we know that the value of the field equation outside of the source of the field is zero, so the second terms in the energy tensor expression (5.48) will disappear therefore we get

T00( x) = ∂ 0φ( x) ∂ '' 0 φ ( x ) (5.49)

Substituting the above equation in (5.47) results in:

1 Zω = dx T00 x . (5.50) 2 ƒ a — ( ) a

As it can be seen from the above equation the zero-point energy is directly related to the value of the vacuum energy, which is divergent as it was expected [46]. One of the most convenient ways to calculate the Casimir energy is applying the Green function method in the calculation of this effect. This can be observed from the above equation. In fact, in the calculation of the Casimir force we consider the distortion of the zero-point energy by a material body or presence of a background field; otherwise the divergence of the zero-point energy does not indicates any important physical quantity.

The other important task in the question is calculating the force per unit area of a material boundary introduced in the vacuum with a particular field. This quantity can be obtained again by working on the Green function that is in term related to the

124 stress or energy-momentum tensor [47]. In fact the force exerted to the body is obtained by the normal-normal component of the stress tensor

f dxdy T (5.51) = — zz

We need only to consider the Green function corresponding to the field, and then we introduce a reduced Green function g(z,z’) according to a Fourier transformation.

After calculating the reduced Green function, we put this into the following relationship

1 T= ∂ ∂ g() z,' z (5.52) zz2i z z'

Using equations (5.52) for (5.51) after some appropriate transformation we may end up on a description of the force, which comes from this discontinuity of Tzz on the boundaries from inside and outside of the cavity. The same method is valid for the computation on the Casimir energy, except that in this case we have to find out the term T00 from the Green function and then apply equation (5.50) to obtain a distortion of the zero-point energy by the boundary or background field (e.g. gravitational field).

5.5. Green’s function method for calculating the Casimir

Force

The general Green function method for calculating the Casimir force for different geometrical boundaries was explained in the last section. In this section we use this method to show how to compute this force for various boundaries with different geometrical configurations. A demonstration of the simple case of a massless scalar field for parallel plates, spherical shell, and cylindrical shell will be

125 given. We apply the electromagnetic field in the case of a spherical shell, where a disappointing result was obtained in terms of a repulsive force. From now on for simplicity we assume Z =c =1.

5.5.1. Casimir force on conducting parallel-plates

The most important part of the calculation is the evaluation of the Green’s function, followed by applying certain manipulations to evaluate the vacuum expectation value of the normal-normal components of the energy-momentum tensor.

We start by definition of the Poisson’s equation for this scalar field with a source K

−∂2φ = k (5.53)

Obviously the Green function expression is given by substituting the fieldφ , by the

Green function G and the source k by a delta function

−∂2G( x,'' x) =δ ( x − x ) (5.54)

The above equation may be written in a more transparent way, where a reduced

Green function g(z,z’) depending on k and satisfies a similar equation

2 ≈∂ 2 ’ 2 2 2 ∆−2 −λ ÷ g()() z,'', z = δ z − z λ = ω − k (5.55) «∂z ◊

The reduced Green function is in fact a Fourier transformation of the main Green function, where the variable z indicates the normal component on the plates

dd k dω G x,',' x= eik.''() x− x e − iω() t − t g z z (5.56) () —d — () ()2π 2π

126 As it can be seen from the above transformation, we have considered a general dimensionality d in our calculation, which can be treated as the real dimension of the problem that is equal to 2 at the end of the calculation. Applying the Dirichlet boundary condition to the reduced Green function, equation (5.55) can be easily solved and we get

1 g() z, z '= − sinλ z sin λ () z − a (5.57) λsin λa < >

where we have considered the greater or lesser value of z and z’, shown by > and <.

Now according to equation (5.52) we can obtain the normal-normal component of the vacuum expectation value of stress tensor as

ZZi T= ∂ ∂ g() z, z ' =λ cot λ a , z → z ' = 0, a (5.58) zz2i z z ' 2

Now to calculate the force per unit area of the boundaries, we should consider the discontinuity of Tzz on each plate, we mentioned earlier that, the Casimir energy is actually a distortion in the vacuum energy due to presence of a geometrical boundary. Therefore we consider for example the right boundary with an infinitesimal thickness. The Green function in this case will vanish on the boundary according to Dirichlet condition and its value in far away from that surface will be eikz . A similar equation to (5.57) for the reduced Green function in this region will be obtained as

1 g() z, z '= sin λ () z − a eiλ() z> − a (5.59) λ <

The corresponding stress tensor can be written as

ZZi T= ∂ ∂ g() z,','. z =λ z = z = a (5.60) zz2i z z ' 2

127

We employ a complex frequency rotation

ω→i ζ,. λ → i ρ (5.61)

Now an integration of the discontinuity of the stress tensor on the boundary relative to the transverse momentum and frequency results in the force per unit area of the surface

Zc dd k dζ F= −ρcoth ρ a − 1 (5.62) —d — () 2()2π 2π

Again we have used the general dimensionality for the integration, and we reduce it to the real physical dimension of two at the end of the calculation. In a polar coordinates system the above integration get the simpler form

A ∞ ρ F= −Z cd +1 ρd d ρ (5.63) d +1 — 2ρa ()2π 0 e −1

The Ad+1 can be evaluated as

≈d +1 ’ ∆ ÷ 2π «2 ◊ A = , (5.64) d +1 d +1 Γ()2

where we have used the Gaussian integration. Applying the zeta function (5.32), and introducing the gamma function identity

−1/ 2 2z− 1/ 2 1 Γ()()()2z = 2π 2 Γ z Γ( z + 2 ) (5.65)

we get to the expression for the force as

Γ(1 +d )ζ (d + 2) F= −Z c() d +1 2−d −2π − d / 2 − 1 2 (5.66) ad +2

128 Substituting the value of two for the dimension d we recover our result for the

Casimir force exerted per unit area of the surface of the parallel plates in a scalar field (5.36).

5.5.2. Casimir force on a conducting spherical shell

In this section we show how the Green’s function method can be employed to calculate the Casimir stress on a perfectly conducting uncharged sphere [48]. Like parallel plates this stress was suggested by Casimir as an attractive force, which could play the role of Poincare stabilizing stress for a model of the electron. This would explained many problems in physics, but unfortunately as it was worked out by Boyer the Casimir stress on a sphere is not attractive and indeed it is repulsive, a disappointing result. Boyer in his calculations has made many assumptions which do not have a physical basis, for example to proceed in the calculation of the stress, he suggested that the main sphere is surrounded by a larger sphere, whose radius could approach to infinity.

A most convenient method for computing the Casimir stress on a spherical shell in the electromagnetic field by using the Green’s function was developed by K.

A. Milton, L. DeRaad, and J. Schwinger. This method actually confirms the result of

Boyer for a repulsive stress on the sphere with a more significant figure of accuracy in compare with Boyer’s result which was of only one significant figure.

Indeed if we start with relations analogue to Maxwell’s equations, this functions relates the curl and divergence of the Green’s dyadic for the electric field and magnetic field as follows ed ∇⋅Φ = 0 , (5.67)

129 ed d d ∇ ⋅Γ = −∇δ (r − r′). (5.68)

The second coupled of first-order equation reads as egd ed f d d ∇×Γ′′ −iω Φ = ∇×1 δ ( r − r ) , (5.69) ed egd −∇× Φ −iω Γ′ = 0 , (5.70) also the second-order equation can be expressed by f f 2 2 »d d ÿ (∇ +ω) Γ′′ = −∇× ∇×1 δ ()r − r ⁄ , (5.71)

f f d d (∇2 +ω 2 ) Φ =i ω ∇×1 δ ( r − r′) . (5.72)

In these equations we have defined f f f d d Γ′′ = Γ +1δ (r − r ) , (5.73) f where the tensor Green’s functionΓ( x, x′) is called the Green’s dyadic for the electric field. This quantity in fact relates the electric field and polarization. For a dielectric media the polarization is defined as d d PE= ε0 χe , (5.74) where e is called the electric susceptibility. Now we express a relation between the electric field and the polarization, which gives a definition for the Green’s dyadic for the electric field as df d E( x) =— dx′′′ Γ( x,. x) ⋅ P( x ) (5.75)

Introducing the vector spherical harmonics, which is the most convenient way to express the Green’s function for a spherical shell in terms of the angular d momentum L

d1 d XLYlm()θ,, φ= 1/ 2 lm () θ φ , (5.76) »l() l +1 ⁄ÿ

130 where l is the orbital quantum number, and Ylm (θ, φ ) is the spherical harmonic, which from quantum mechanics it is defined by separating the spherical wave- function by the product

ψ(r,, θ φ) = R( r) Θ( θ) Φ ( φ ) . (5.77)

The product of the last two terms on the right-hand side of (5.77) is called the

spherical harmonicYlm (θ, φ ) . Two very important properties of the vector spherical harmonics are the orthonormality and the sum rule given by d d dΩ X∗ ⋅ X = δ δ , (5.78) — l'''' m lm ll mm and

l d 2 2l + 1 ƒ X lm ()θ, φ = , (5.79) m=− l 4π respectively. Now we are in a position to expand the Green’s dyadic functions in terms of the vector spherical harmonics

d »di d ÿ ′ Γ =ƒ …fl X lm + ∇× g l X lm Ÿ, (5.80) lm ω ⁄ and

d »di d ÿ ! ! Φ =ƒ …gl X lm − ∇× f l X lm Ÿ. (5.81) lm ω ⁄

Now we substitute (5.80) and (5.81) in (5.70), we will immediately get the following relations

g!l= g l , (5.82) and

! 2 Dl f l= −ω f l . (5.83)

d d Employing the following identity for an arbitrary vector function V( r )

131

1 »d ÿ ddd dd d rll++1 dXVr Ω⋅∗ =Ω∇× d» X ∗ ÿ ⋅∇× Vr , (5.84) 2 …()()() Ÿ lm lm r dr ⁄ — — ⁄ in (5.69), it gives the second-order differential equation

d f d d D2 g! i d′′ X∗ ′′» ′′ r ′′ r ′ ÿ (5.85) ( l+ω) l = ω— Ω lm ()() Ω ⋅ ∇ ×1 δ − ⁄ , and the relation

1 d f! = f −δ ()() r − r′′ X ∗ Ω , (5.86) l lr 2 lm where we have used (5.82), and

1 d 2 l( l +1) D= r − . (5.87) l r dr2 r 2

Another very important result which is a second-order differential equation can be obtained by employing the following identity

ddd d d d D dΩ X∗ ⋅ V r = − d Ω X ∗ ⋅» ∇× ∇× V r ÿ, (5.88) l— lm()() — lm ( ) ⁄ which is

d f d d D2 f d′′ X∗ ′′ ′′» ′′ r ′′ r ′ ÿ (5.89) ( l+ω) l =−Ω— lm ()() Ω⋅∇×∇× 1 δ − ⁄ .

In this equation we used(5.88), and (5.84) in the second-order equations (5.71) and

(5.72). Now (5.85) and (5.89) are second-order differential equation which must be

solved to find out the expressions for g!l and f l . One of the most convenient way to solve equations is using the Green’s function method, where those equations will be replaced by a similar differential equation applied to an auxiliary function, which is

called Green’s function ∆l , and on the right-hand side we will have instead a

δ (r− r′) function.

2 2 (Dl+ω) ∆ l ( r, r′′) = −( 1/ r) δ ( r − r ) . (5.90)

132 After solving this equation the value of the main function will be calculated as the convolution of the Green’s function and the source is the function in the right-hand side of the main equation. We apply the above procedure to solve the differential equations (5.85) and (5.89), and then using (5.82) and (5.86) we will get the solutions for these differential equations d ! ′′′»∗ ÿ gl()()()() r= g l r = − iω ∇ × G l r,, r X lm Ω ⁄ (5.91) d d 2∗ 2 ∗ fl( r) =ω F l( r, r′′′′) X lm ( Ω) +( 1/ r) δ ( r − r) X lm ( Ω ) , (5.92) d ! 2 ∗ fl( r) =ω F l( r,, r′′) X lm ( Ω ) (5.93)

where FGl and l are the scalar Green’s functions corresponding to fl and g l

respectively that play the rule of ∆l in (5.90).

The next step in our calculations is the variation of the vacuum energy due to the presence of the conducting spherical shell. For this purpose we need to manipulate the Green’s dyadic function in terms of the Green’s functions Fl and Gl.

In order to do that we use relations (5.91), (5.92), and (5.93) in equations (5.80) and

(5.81). For example Green the dyadic function for electric field will be f d d 2 ∗ Γ(r,,, r′′′ω) =ƒ { ω Fl( r r) X lm( Ω) X lm ( Ω ) lm d dd e »G r r′′ X X ∗ ÿ −∇× l()()(), lm Ω lm Ω ⁄ ×∇} (5.94) 1 d d f ′′′∗ +2 δ()()()()r − rƒ Xlm Ω X lm Ω −1 δ r − r . r lm

We do not know what the solutions of Fl and Gl are yet. To find out this we present a general form of the solution for the Green’s function (5.90), which is in the form of a Bessel’s differential equation:

d2 y dy x2+ x + x 2 −α 2 y = 0. (5.95) dx2 dx ()

133 In this equation , is a real or complex number, and it is referred to as the order of the Bessel function if it is an integer. We will not talk about the solution for this equation, and rather we will consider the more appropriate form of the Bessel’s equation for our discussion that is worked out in the spherical coordinate system. The general form of this differential equation in the spherical coordinate system is

d2 y dy x2+2 x +» x 2 − n() n + 1 ÿ y = 0 . (5.96) dx2 dx ⁄

The two linearly independent solutions to this equation are called the spherical

Bessel functions jn and yn. The following are also important functions for this equation

(1) hn( x) = j n( x) + iy n ( x), (5.97) and (2) hn( x) = j n( x) − iy n ( x), (5.98) theses are called Hankel functions [ for more detail in Bessel’s functions see

Appendix A]. Now the solution of (5.90) for Green’s functions Fl and Gl for a sphere of radius a is given by the Bessel’s and Hankel functions for r and r’ inside and outside of the sphere according to

À ! ŒGl= − A G ikj l( kr) j l ( kr′), inside: Ã (5.99) ! ′ ÕŒFl= − A F ikj l()() kr j l kr and

À ! ()1() 1 ŒGl= − B G ikh l() kr h l () kr′ , outside: Ã (5.100) ! ()1() 1 ′ ÕŒFl= − B F ikh l() kr h l () kr , where k = ω . Each of the above equations is actually a part of the main Green’s function of the sphere, the other part of these Green’s functions is related to the vacuum part of them, which is

134

0 (1) Gl( rr,.′) = ikjkr l( <) h l ( kr > ) (5.101)

In this equation < and > indicate the smaller and bigger values of r and r’, therefore

0 ! ÀŒGGGl= l + l , Ã (5.102) 0 ! ÕŒFGFl= l + l .

It can be shown that according to the boundary condition for vanishing tangential components of the electric fields and normal components of the magnetic field, the values of AF,G and BF,G will be given by

−1 (1) AF= B F = h l( ka) /, j l ( ka) (5.103) and

»()1 ÿ d( kahl () ka) / d() ka AB=−1 = ⁄ . (5.104) GG » ÿ d() kajl () ka/ d() ka ⁄

Then the two angular integrations in the spherical coordinate systems that are important for the rest of our calculation are

d∗ d d» frX′′ ÿ» grX ÿ frgr∗ , (5.105) — Ω ()()()()l′′ m ⁄ ⋅ lm ⁄ = δ ll′′ δ mm

d∗ d d» f r′ X ÿ» g r X ÿ — Ω ∇×()()l′′ m ⁄ ⋅ ∇× lm ⁄ (5.106) 1 » d∗ d ∗ ÿ =()rfr′′′() () rgr() + ll()()() +1 frgr δll′′ δ mm , rr′′ … dr dr ⁄Ÿ where f(r ) and g(r ) are the spherical Bessel’s functions. If we consider r = r’ the right-hand side of (5.106) can be written in the form

1 »d d ÿ rf rg+ l l +1 fg δ δ 2 …() ()() Ÿ ll′′ mm r dr dr ⁄ (5.107) À1 d» d ÿ ¤ =k2 fg + rf rg δ δ . Ã2 2 …() Ÿ ‹ ll′′ mm Õk r dr dr ⁄ ›

In this stage we can calculate the total Casimir energy of the spherical shell.

The general form of the energy density for the electromagnetic field is written as

135 1 dd d d u() x=» E()()()() x ⋅ E x′′ + H x ⋅ H x ÿ . (5.108) 2 ⁄ x→ x′

But the product of the electric fields in different points of space-time indicates the electric Green’s function, whose Fourier transform is

d d ∞ dω f iE()() x E x′′=— e−iω() t − t′ Γ() r,,, r ω (5.109) −∞ 2π since the magnetic field and electric field is interrelated through d ∂H d = −∇× E. (5.110) ∂t

Combining (5.109) and (5.110) we will have

dd∞ dω 1 df e iH x H x′′′= − e−iω() t − t′ ∇×Γ r,,. r ω ×∇ (5.111) ()() — 2 () −∞ 2π ω

Now employing (5.94), which gives the value of the electric Green’s function in

(5.109) and (5.111), and also integrating over whole space we will get the value of the energy

1 ∞ dω d d E dr e−iω() t − t′ kFrrGrrX2 »! ′′′! ÿ X ∗ =—() — ƒ{ l()(),, + l ⁄ lm()() Ω ⋅ lm Ω 2i −∞ 2π lm dd de − ∇×»X Ω ⋅» F! r,,. r′′′′ + G! r r ÿ ⋅ X ∗ Ω ÿ ×∇ lm()()()()} l l ⁄ lm ⁄ r= r′

(5.112)

Here according to (5.94) the second term in the curly bracket is the result of the direct substitution of first term of (5.94) into (5.111) and the second term in (5.109), the first term in the curly bracket comes from direct substitutions of the first term of

(5.94) into (5.109), and also using the second term of this function into (5.111). In this situation we have used the identity (5.72), where the right-hand side of this equation will be zero when r→ r′. In this calculation the Fl and Gl are substituted

! ! by FGl and l , which are the value of these Green’s functions after subtracting the

136 vacuum counterpart from the main Green’s function. Also the terms involving delta function in (5.94) will be vanished as r→ r′.

Now observing the form of the energy expression in (5.112), we can see that the first term in the curly bracket has the form of (5.105) and the second term in this bracket is similar to (5.106). Therefore employing these two equations in the above expression for the energy we reduce the form of the expression for the energy into

1 ∞dω ∞ E=()2 l + 1 e−iωτ rdrkFGrr2 2 2 »! + ! ÿ () , 2i — 2π — ( l l ⁄ −∞ 0 (5.113) 1 dÀ d ¤ ’ +r r′′′» F! r,,. r + G! r r ÿ 2 Ã l()() l ⁄ ‹ ÷ r drÕ dr′ ›r= r′ ◊

Now we can separate the second integration inside and outside of the sphere, from zero to a; the radius of the sphere and from a to infinity. Also since r = r’ (5.99) and

(5.100) will be

À ! 2 ŒGl= − A G ik » j l () kr ÿ⁄ , inside: (5.114) Ã 2 Œ ! » ÿ ÕFl= − A F ik j l () kr ⁄ and

2 ÀG! = − B ik» h()1 kr ÿ , Œl G l () ⁄ outside: (5.115) Ã 2 ŒF! = − B ik» h()1 kr ÿ . Õ l F l () ⁄

In the integral (5.113) the second term in the second integration will be vanished and we have only the contribution of the first term that will be written as

∞ a dω ŒÀ 2 E l ekrdrAAjkr−iωτ 3 2 » ÿ = −ƒ()2 + 1 —Ã — ()()G + F l ⁄ l −∞ 2π ÕŒ 0 (5.116) ∞ 2 Œ¤ +r2 dr B + B» h()1 kr ÿ . — ()GF 1 () ⁄ ‹ a ›Œ

137 Now multiplying and dividing the above equation by a; the radius of the sphere, and after a complex transformation τ/ a→ i ε into a Euclidean space we will get the above equation in the form

1∞ 1 ∞ Às′ s ′′ e ′ e ′′ ¤ E= −ƒ()2 l + 1— dyeiε y y Ãl + l + l + l ‹ , (5.117) 2π al=1 2 −∞ Õ sl s l′′ e l e l › where sl and el are the modified spherical Bessel’s functions.

A discussion on equation (5.117) is needed to complete the calculation. In order to this let us suppose

′ λl= ()s l e l . (5.118)

Equation (5.117) now can be written in a new but convenient form 1∞ 1 ∞ d E l dyeiε y x 2 (5.119) = −ƒ()2 + 1— log() 1 − λl , 2π al=1 2 −∞ dx x= y . An approximation for the above equation for ε = 0 , for a given l when l → ∞ is

1 ∞ d J l dx x2 x (),0=() 2π + 1— log() 1 − λl () π 0 dx (5.120) 1∞ dz 3 ≈3 = . 2π — 2 32 0 ()1+ z

Therefore the value of the Casimir energy for a conducting spherical shell will be given by

El ≈ −(1/ 2 a)( 3/ 32) = −( 1/ 2 a)( 0.09375) , l → ∞ . (5.121)

This is a negative energy, therefore the Casimir effect on this geometrical boundary unlike the Parallel conducting plates is a repulsive force.

It can be shown that the stress on this shell can be obtained by

1 ∂E F = − . (5.122) 4π a2 ∂ a

138

5.5.3. Casimir force on a conducting cylindrical shell

As it was mentioned in the previous section after calculating the force between two conducting uncharged parallel plates, Casimir assumed that this must be a common effect for any kind of geometrical boundaries. Since in the case of parallel plates the value of this stress was obtained as an attractive force, he suggested the same for the a spherical shell, but as it was shown by Boyer and confirmed by

Milton, DeRaad, and Schwinger that is in fact a repulsive force. It was tempting to do an approximate calculation considering an intermediate configuration medium such as a cylindrical shell that will have a zero value for this force. More precise attempts were made to compute this stress [31]. Among these efforts is the calculation made by DeRaad and Milton, where again they used the Green’s dyadic formulation [26]. Although the calculations for theses geometrical boundaries increased our knowledge on the behaviour of this force, but the prediction of this behaviour was impossible or at least a false result. This is actually the most bizarre behaviour of the quantum fluctuation of the field, which has not been explained yet.

In this subsection we will give a brief discussion for the calculation of the stress, putting the emphasis on the results obtained by this method. A comparison between results obtained for various geometrical bodies will be given at the end.

We start with the electric and magnetic Green’s dyadic functions. Like in the f f spherical case, we call the functions Γ′′(r,; r ω) and Φ(r,; r′ ω) respectively. Again we introduce some relation similar to the Maxwell’s equations for these functions.

The divergence of the electric Green’s function is f ∇⋅Γ′ = 0, (5.123) and this divergence for the magnetic Green’s function is also zero

139 f ∇ ⋅Φ = 0. (5.124)

The first-order equations are f f f ∇×Γ′ −iω Φ = ∇×1, (5.125) and f −∇×Φ −iω Γ′ = 0 , (5.126) where we define f f f Γ′′′′(r, r ;ω) = Γ( r , r ; ω) + 1 δ ( r − r ) . (5.127)

Similar to (5.80) and (5.81), we can expand theses tensor Green’s functions in terms

! of four auxiliary functions fm, g m , f m , and g! m that are functions of r and h. In this

case instead of the vector spherical harmonics we will have χmh ( θ, z) , where

1 χ = eimθ e ihz . (5.128) mh 2π

Theses expansions are f ∞ ∞ dh Γ′′()r,;;,, rω =ƒ {() ∇× zˆ fm()() r h ω χ mh θ z — 2π m=−∞ −∞ (5.129) i + ∇×()()()} ∇× zˆ g r;,,, hω χ θ z ω m mh and f ∞ ∞ dh Φ()r,;;,, r′ ω =ƒ {() ∇× zˆ g!m()() r h ω χ mh θ z — 2π m=−∞ −∞ (5.130) i − ∇×()()()} ∇× zˆ fˆ r;,,. hω χ θ z ω m mh

Now we define the reduced Green’s functions Fm and Gm by d d f! r,,,, r= χ∗ θ′′′′ zM ∗ F r r (5.131) m ( ) mh ( ) m ( )

d ∗d ∗ gm( r,,,, r) = − iχ mh( θ′′′′ z)N G m ( r r ) (5.132) where d M ≡ ∇× zˆ, (5.133)

140 d N ≡ ∇×( ∇× zˆ). (5.134)

The boundary conditions on a conducting cylinder for dyadic Green’s functions are f θˆ⋅Γ′ = 0, (5.135) r= a f zˆ ⋅Γ′ = 0, (5.136) r= a f rˆ⋅Φ = 0, (5.137) r= a where a is the radius of the cylinder. These boundary conditions for the corresponding reduced green’s functions Fm and Gm imply

Gm ( a, r′) = 0, m ≠ 0, (5.138)

∂ Fm () r, r′ = 0, (5.139) ∂r r= a

h2 G()() a, r′′+G G a , r = 0, (5.140) ω 2 0 0

where we define

1 G F ()r,,, r′′= D F() r r (5.141) m mω 2 m and

1 G G ()r,,. r′′= D G() r r (5.142) m mω m

The differential operator Dm is defined as

2 2 Dm= d m − h +ω , (5.143) and

1 ∂ ∂ m2 d= r − . (5.144) m r∂ r ∂ r r 2

141 It can be shown that the reduced Green’s functions for the interior and exterior of the cylinder are given by, for r,, r′ < a

1 iπ » H(λ a) ÿ Grr(), ′ =… JrHr()()λ λ − m JrJr()() λ λ ′ Ÿ ωm2 λ2 m< m > J λ a m m … m () ⁄Ÿ 1 + G G ()r,, r′ λ 2 m

(5.145) and 1 iπ » H′ (λ a) ÿ Frr(), ′ =… JrHr()()λ λ − m JrJr()() λ λ ′ Ÿ ω2m2 λ 2 m< m > J′ λ a m m … m () ⁄Ÿ 1 + G F ()r,. r′ λ 2 m

(5.146)

Also for r,, r′ > a

1 iπ » J(λ a) ÿ Grr(), ′ =… JrHr()()λ λ − m HrHr()() λ λ ′ Ÿ ωm2 λ2 m< m > H λ a m m … m () ⁄Ÿ 1 + G G ()r,, r′ λ 2 m

(5.147)

1 iπ » J′ (λ a) ÿ Frr(), ′ =… JrHr()()λ λ − m HrHr()() λ λ ′ Ÿ ω2m2 λ 2 m< m > H′ λ a m m … m () ⁄Ÿ 1 + G F ()r,. r′ λ 2 m

(5.148)

Here Jm is the Bessel’s function of first kind and Hm is the Hankel’s function, with

λ2= ω 2 − h 2. (5.149)

GF, For m ≠ 0 , Gm (r, r′) the interior and exterior of the cylinder can be determined precisely.

142 Now we can work out to determine the Casimir stress on a conducting cylindrical shell. For this purpose we must subtract the stress tensor outside the cylinder from the inside one. This tensor is

1 2 2 THErr=2 ( ⊥ − r ), (5.150)

The directions of the electric and magnetic field specified in this equation is a result of the boundary conditions for these fields.

Now we introduce the following integration by

∞dω 1 ∞ ∞ dh L ≡ —ψ() ω ƒ — , (5.151) −∞2π 2 πm=−∞ −∞ 2 π where ψ( ω) is a high-frequency cutoff. The second integration comes from

∞ dω f iE x E x′′′= e−iω() t − t′ rˆ ⋅Γ r,;, rω ⋅ r ˆ (5.152) r()() r′ eff — () −∞ 2π

∞ dω 1 f iH x H x′′= − e−iω() t − t′ zˆ ⋅Φ ⋅∇ × z ˆ, (5.153) z()() z eff — −∞ 2πi ω and ∞ dω 1 f iH x H x′′= e−iω() t − t′ θˆ ⋅Φ ⋅∇ × θ ˆ. (5.154) θ()() θ eff — −∞ 2πi ω

A detailed calculation using (5.129), (5.130), (5.131), and (5.132) in above equations results in the following relations

»1 ≈H′ () zm2ω 2 H() z ’ ÿ iE2() a+ =L …∆ h 2 m − m ÷ Ÿ, (5.155) r z∆ H z z2 H′ z ÷ …«m() m () ◊ ⁄Ÿ

»1 ≈J′ () zm2ω 2 J() z ’ ÿ iE2() a− =L … −∆ h 2 m − m ÷ Ÿ, (5.156) r z∆ J z z2 J′ z ÷ …«m() m () ◊ ⁄Ÿ

»λ 2 H( z) ÿ iH2 () a + =L … − m Ÿ, (5.157) z ′ …z Hm () z ⁄Ÿ

143

»λ 2 J( z) ÿ iH2 () a − = L …m Ÿ, (5.158) z ′ …z Jm () z ⁄Ÿ and

»1 ≈H′ () zm2 h 2 H() z ’ ÿ iH2() a + =L …∆ω 2 m − m ÷ Ÿ, (5.159) θ z∆ H z z2 H′ z ÷ …«m() m () ◊ ⁄Ÿ

»1 ≈J′ () zm2 h 2 J() z ’ ÿ iH2() a − =L … −∆ω 2 m − m ÷ Ÿ. (5.160) θ z∆ J z z2 J′ z ÷ …«m() m () ◊ ⁄Ÿ

Now employing the above equations in the stress tensor equation (5.150) we obtain an expression for the force per unit area of the cylinder

»2 ÿ 1λ ≈HHJJm′ m′′ m′m ′′ 2 ’ F=Trr()() a − − T rr a + = − L …∆ + + + + ÷ Ÿ. ′′ 2i … z« Hm H m J m J m z ◊ ⁄Ÿ

(5.161)

In the case of the spherical shell similar to (5.119), this equation can be treated in a more convenient form as

1» 1 d ≈π 2 »′ 2 ÿ ’ ÿ F= − L …2 zln∆ 1 − z()() Hm J m z ÷ Ÿ . (5.162) 2i a dz « 4 … ⁄Ÿ ◊ ⁄

We are not going through the detail of the evaluation of this integral, but rather we indicate the main point of this evaluation. Before that we should emphasis that unlike the spherical shell, the role of a high-frequency cutoff in this calculation is very important, since it leads us to determine a finite value for the corresponding integral.

In fact the shell must be transparent to the high frequency modes and including this term (high-frequency cutoff) does not affect the calculation, when it will be finally independent of this term.

It can be shown that the force on the unit area of the shell can be reduced in the following form

144 1 1 F = −2 ()SRR + + 0 . (5.163) ()2π 2a4

Where S is

1 5 S = −2ln 2πε + 8 , (5.164) and R0 is

11 2 1 RRR0=2lnε + 0 + 0 − 4 , (5.165) with

1 R0 = 0.7511, (5.166) and

2 R0 = 0.1774. (5.167)

Also we have

R = −0.0437. (5.168)

The logarithm part will be eliminated between S and R0 and the final result for this force will be

1 1 F = − 2 ()0.3409 . (5.169) ()2π 2a4

As it can be seen this is a non-zero force, which is actually similar to the case of two parallel conducting plates which gives us an attractive force.

145 5.5.4. Comparison of the Casimir force for different geometrical boundaries

A comparison between the values of the Casimir forces for different geometrical boundaries that so far we have considered may be of interest. Theses forces are given for the unit of area of the corresponding geometrical body. We obtained before

0.0691 F Cylinder = − , (5.170) ()2a 4

0.0411 F = − , (5.171) plate a4 where this force in the case of a sphere is

0.05879 F sphere = 4 . (5.172) ()2a

As we can see the magnitude of theses forces do not differ much, and they are almost of the same magnitude except for a different sign in the case of the sphere. Another calculation for the cylinder of square cross section with side L obtained by Lukosz

[40] shows that this force is actually about half the magnitude of the round cross section cylinder

0.0382 F = − . (5.173) L,cylinder L4

But as it is discussed in [26], Lukosz has not include the exterior modes in the calculations, which as it was considered in the previous sub-sections it is actually a necessary part in the computing the stress on the bodies.

146 5.6. Casimir force as a limiting case of Van der Waals interaction

One of the most important thermodynamic quantities of a body is the free energy. This quantity as it was introduced in section 2.2.5, is the Hamiltonian of the system, and it will be shown that any change in this quantity due to variation of permittivity of the medium gives us an expression for the interatomic interaction of the Van der Waals force, whose effective characteristic range is much larger in comparison to the interatomic distances.

The Van der Waals force can be considered as an interatomic interaction at relatively large distances. This force is brought through the long-wavelength electromagnetic field, which includes the thermal fluctuation as well as the zero- point oscillation. As it was mentioned this force has a definite influence on the macroscopic thermodynamical quantities of the system.

Since the Van der Waals force is an interaction acting at large distances between the particles of a medium, the important wavelengths are in the scale of the inhomogeneity of that body. The inhomogeneity can be regarded for example as a gap between two very narrow parts of that body, therefore in a limiting case when the medium is conducting the Van der Waals interaction, it manifests itself as the

Casimir force [30].

In this section we start with the calculation of this force, by computing the contribution of the long-wavelength electromagnetic field to the free energy. We suppose this contribution comes from a small variation of the Hamiltonian of the system according to a small change in the permittivity of the medium, this change in the Hamiltonian related to a small variation of the free energy is given by

147

δFH= δ ˆ , (5.174)

Now we introduce the Matsubara operators. Similar to the discussion in the chapter

0, the Matsubara operator is defined by replacing the real variable t by the imaginary variable −iτ , then the it can be defined by

ˆ ˆ ′ À ˆ M τHH0ˆ − τ 0 ŒΨα()()τ,,r = e ψ α r e (5.175) Ã ˆ ˆ ˆ M τHH0ˆ † − τ 0 ÕŒΨα()()τ,.r = e ψ α r e

The H0, is the free part of the Hamiltonian. The Green’s function in this case will be of the form

1 ˆMM ˆ Gαβ()τ1,, τ 2= −T τ Ψ 0 α()() τ 1 Ψ 0 β τ 2 σˆ (5.176) σˆ 0 0 where ? , is the averaging with respect to the states of a system of non-interacting 0

ˆ particles, or Gibbs distribution with Hamiltonian H0 , and σ is the S matrix discussed in section 4.7. This design of the field operators is very beneficial for the calculating the thermodynamic quantities of a macroscopic system.

In this case we obtain the free energy (5.174) with respect to Matsubara operators as

À 1 ˆ M δFTH= τ δ σˆ , Œ σˆ 0 Œ 0 (5.177) Ã 1/T Œσˆ =TexpˆjMM ⋅ A ˆ d3 xd τ , Œ τ — — Õ 0 where

Vˆ ˆj A ˆ d3 x, (5.178) lm = −— ⋅ is the interaction part of the Hamiltonian that describes the interaction of the particles with the long-wave electromagnetic field. The variation of the free energy can be shown to be

148

∞ 1 δFF= δ − 1 T ζ;,;,, rr δ ζ rrdxdx3 3 (5.179) 02 ƒ —Dik() s 1 2 P ki() s 2 1 1 2 s=−∞ 4π

whereDik is the temperature Green’s function andPki is the polarization operator,

and ζ s are the values of the frequencies satisfying Zζs = 2 π sT . The change in the polarization operator is related to the variation of the permittivity of the medium, for the above equation we can write that

T ∞ F F2 r r i r d3 x (5.180) δ= δ0 − ƒ— ζsD ll() ζ s;,,. δε() ζ s 4π s=0

And from the equation

()()1 22 2 ()() 1 2 Ei E k= ω /, c A i A k (5.181) ( )ω( )( ) ω we deduce the following relation

E 2 Dik(ζ s;,;,.r r′′) = − ζ s D ik( ζ s r r ) (5.182)

Taking into account equation (5.182), equation (5.180) can be written as

T ∞ F FE r r i r d3 x (5.183) δ= δ0 + ƒ—Dll() ζ s;,,. δε() ζ s 4π s=0

An isothermal small deformation of the medium by displacement u, and force density f, results in a change in the free energy of the system given by

δ F= −—f ⋅ u d3 x. (5.184)

If P0 is the pressure on the body we know that

f0 =-, ∇P0 (5.185) and also

∂ε δε= − ∇⋅() ρu , (5.186) ∂ρ therefore using(5.184), (5.185), and (5.186) in (5.183) we obtain a very useful expression for the force density acting on the body

149

∞ T »E ∂ε ÿ f = −∇P0 −ƒ ρ ∇ …Dll() ζ s ;,. r r Ÿ (5.187) 4πs=0 ∂ ρ ⁄

In order to write the above equation in terms of the stress tensor exerted on the corresponding medium we will have

À≈ ’ ¤ ∂P0 T ∂ Œ∂ε(i ζ s , r) E Œ fi = − +ƒ Ã∆ε() i ζs,;, r − ρ ÷D ll() ζ s r r ‹ ∂xi4π ∂ x i ÕŒ« ∂ ρ ◊ ›Œ

T ∂ E − ƒε()i ζs,;,. rD ll() ζ s r r (5.188) 4π ∂xi

To get a description similar to

∂σ ik fi = , (5.189) ∂xk for the force acting on the medium, we have to manipulate only the last term in

(5.188), in order to do this we separate the Green’s function in terms of r and r’ and put r = r’ at the end of the calculation. After this prescription the last term will be

T À∂ ∂ ¤ E −ƒ Ãε()r′′ + ε () r ‹Dll () r,. r (5.190) 4π Õ∂xi ∂ x i′ ›

Each term in the bracket in the above expression can be written as

∂ ∂ ∂ EEHH» ÿ εDll=2 ε DD ik + ik ⁄ − D ll . (5.191) ∂xi ∂ x k ∂ x i

Hence the stress tensor will be in the form

∞ À T Œ 1 »∂ε(i ζ s , r) ÿ σik= −P0 δ ik −Ã − δ ik… ε() i ζ s , r − ρ Ÿ 2πƒ 2 ∂ ρ s=0 ÕŒ ⁄ 1 ¤ ×DEEHH()()()ζ;,,;,;,;,.r r + ε i ζ r D ζ r r − δ DD()() ζ r r + ζ r r ‹ lls s iks2 iklls iks ›

(5.192)

150 To show that the Van der Waals interaction can be the limiting case for the Casimir force, we consider the most simple geometrical configuration in the development of the Casimir effect, which is a body consist of two parts that are separated by a very narrow gap equal to l. Suppose the x-direction of the coordinate system is perpendicular to the gap in this case we can work out on the stress tensor by the intermolecular interactions as the Van der Waals force between these two separated parts of the medium. In this situation the pressure in the absence of the Van der

Waals stress is regarded as zero. Hence the stress applied, let say on the right-hand side body is calculated as the momentum flux into that body through this surface from the gap between those bodies. Therefore the value of the permittivity is equal to unity and from (5.192), the stress tensor will be given by

∞ T EEE F=σxx() l =ƒ{D yy()()() ζ n;,;,;, l l + D zz ζ n l l − D xx ζ n l l 4π n=0 HHH +Dyy()()()ζn ;,;,;,.l l + D zz ζ n l l − D xx ζ n l l }

(5.193)

On the other hand the temperature Green’s function must satisfies the following differential equation

»∂2 ζ 2 ÿ n ′′ …−δil ∆ +2 ε()i ζ n, r δ il ŸD lk()() ζ n ; r , r = − 4 πZ δ ik δ r − r . ∂xi ∂ x l c ⁄

(5.194)

Applying the above differential equation on each component of the temperature

Green’s function and solving them, we reach to the solutions of these equations as

≈d 2 ’ 2 ′′ ∆w−2 ÷Dzz ()() x, x = − 4πδ x − x , (5.195) «dx ◊

≈d2 ’ 4π w 2 2 ′′ ∆w−2 ÷Dyy () x,, x = − 2 δ () x − x (5.196) «dx ◊ εζ n

151 iq d D()x,,, x′′= − D () x x (5.197) xyw2 dx yy

iq d 4π D()x,,. x′′′= − D () x x −δ () x − x (5.198) xxw2 dx yx w 2

Where q is a parameter in the Fourier transformation of the Green’s function

Dik(ζ n ;,r r) intoDik(ζ n ,;,q x x) , and w is

2 1/ 2 ≈εζ n 2 ’ w=∆2 + q ÷ . (5.199) «c ◊

Now we can solve the equations (5.195)-(5.198), the solution to theses equations are given by

4π Dzz =coshw3 () x − x′ . (5.200) w3∆

This is a solution when w1=w2=w3, where different subscription for w indicates the corresponding value of this quantity for the left part, gap, and right part of the body respectively. Also

(w1+ w 3)( w 2 + w 3 ) ∆ =1 − e2w3 l . (5.201) ()()w1− w 3 w 2 − w 3

Also for the rest of the equations we get the solutions as follow

4π w 3 ′ Dyy =2 coshw3 () x − x , (5.202) ζ n ∆1 where

εw+ w ε w + w 2w3 l ( 1 3 1)( 2 3 2 ) ∆1 =1 − e . (5.203) ()()ε1w 3− w 1 ε 2 w 3 − w 2

And

152 4πiq ′ Dxy= D yx = −2 sinhw3 () x − x , (5.204) ζ n ∆1

4π q2 ′ Dxx = −2 coshw3 () x − x . (5.205) ζ n w3∆ 1

EH It can be worked out from the above solutions for the Dikand D ik and by a Fourier

EH transformation for Dik(ζ n,q ; x , x) and D ik( ζ n , q ; x , x) and finally with a change to a

2 new integration variable q=ζ n p −1 / c , that the corresponding force per unit area of the body will be

−1 ∞ À T ∞ Œ»()()s+ p s + p ≈2 pζ l ’ ÿ F l= ζ 3 p 2 1 2 expn − 1 () 3 ƒ n — Ã…∆ ÷ Ÿ πcn=0 s+ p s − p« c ◊ 1 ÕŒ …()()1 2 ⁄Ÿ (5.206) −1 » ÿ ¤ ()()s1+ pε 1 s 2 + p ε 2 ≈2 pζ n l ’ Œ +…exp∆ ÷ − 1 Ÿ ‹dp , s− pε s − p ε « c ◊ …()()1 1 2 2 ⁄Ÿ ›Œ where

2 s1=ε 1 −1 + p , (5.207)

2 s2=ε 2 −1 + p . (5.208)

In order to remove the temperature dependency of the force per unit area given by

(5.206) we are only considering lT/ cZ 0 1. The sum in this formula can be written in the form of an integral over dn= Z dζ/ 2 π T . Therefore the final form of the force per unit area of the body exerted by Van der Waals interaction when the medium is an inhomogeneous medium will be given by

−1 ∞ ∞ À» ÿ Z 2 3 Œ ()()s1+ p s 2 + p ≈2 pζ l ’ F() l= p ζ Ã…exp∆ ÷ − 1 Ÿ 2π 2c 3 —— s− p s − p« c ◊ 0 1 ÕŒ …()()1 2 ⁄Ÿ (5.209) −1 ¤ »()()s1+ pε 1 s 2 + p ε 2 ≈2 pζ l ’ ÿ Œ +…exp∆ ÷ − 1 Ÿ ‹dpdζ . s− pε s − p ε « c ◊ …()()1 1 2 2 ⁄Ÿ ›Œ

153 A limiting case is an electrostatic field when (ζ → 0), the permittivities will be

ε10and ε 20 , in this situation the equation for the force will be given in the form

−1 ∞ ∞ 3 À Zc x Œ»()()s+ p s + p ÿ F=Ã…10 20 ex −1 Ÿ 32π 2l 4—— p 2 s− p s − p 0 1 ÕŒ …()()10 20 ⁄Ÿ (5.210) −1 ¤ »()()s+ pε s + p ε ÿ Œ +…10 10 20 20 ex −1 Ÿ ‹ dpdx , s− pε s − p ε …()()10 10 20 20 ⁄Ÿ ›Œ where we have used a change in the integration variable x= 2 plζ / c . If both bodies are metal we have a reduced form of the above equation, since for metal

ε→ ∞as ζ → 0 , therefore the second term in the above integration under the curly bracket will vanish and the first term for large values of according to definition

(5.207) and (5.208) will be equal to the unity, so the reduced form of the integration reads

ZZc∞ ∞ x3 dpdxπ 2 c F = = . (5.211) 2l 4—— 2 x l 4 16π 0 1 p() e −1 240

As it can be seen this is the actual value resulted by calculation of the Casimir force in the case of two uncharged conducting parallel plates. Hence as we worked on this problem we deduce that the Casimir effect is actually the macroscopic limiting case of the Van der Waals intermolecular interaction when the wavelength in this situation is in the scale of the gap introduced in the medium.

154 5.7. Zeta-function method for calculation of the Casimir force

So far we have worked out on the problem of the Casimir force, using a classical approach to compute the Casimir energy and corresponding force using the classical Green’s function. In this method we tried to find the Green’s function related to each particular case of geometrical boundaries for a specific field, then by using the expression for the energy according to these Green’s functions we ended up on a relation for the Casimir force.

Parallel to this technique there is a most elegant approach to the problem. In this model we start with the definition of vacuum energy. As this energy is divergent, we know that introducing any kind of distortion in the vacuum will regularize the energy. The distortion of the vacuum could be done by the presence of a background field like gravity or geometrical material boundaries [14]. This defines a finite value for the energy which is the Casimir energy.

The regularization of the energy by for example a material body is mathematically equivalent to regularizing the sum of the ground state energy

(vacuum energy) by a technique that is called the zeta-function regularization. In this technique we simply introduce an external parameter s, as the power of the ground state energies and a dimensional scale , to keep the dimension of the energy independent of s. After applying the above procedure, the regulator s must be zero in the end of the calculation, or in the other word we remove the regulator at the end.

There are other methods for regularizing the energy such as the frequency- cutoff regularization, where the convergence is achieved by an exponential damping function [19]. In this section and subsequent sections we concentrate on the zeta- function regularization methods. To begin we will talk about a most simple and general form of the zeta-function that is called Riemann zeta- function, and then we

155 generalized the concept in a spectral zeta-function associated to a differential operator [24, 37, 56, 10]. We will talk about the zeta-function regularization model using the latter zeta-function, where we will show how this function can be written in terms of the heat kernel. In this model the expansion of the heat-kernel with corresponding coefficients shows us the form of divergent part of the energy.

5.7.1. Riemann zeta-function

We briefly review the definition and some basic properties of the Riemann zeta- function. This function is very useful in the calculation of the Casimir energy for a massless scalar field, where a sum over discrete modes of the field in some direction reduces to a Dirichlet series.

The Riemann zeta-function was first considered by Euler as a Dirichlet series given by

∞ 1 ζ ()s = ƒ s . (5.212) n=1 n

This is a function of a complex variable s. This function converges for Re(s) > 1, and it is defined in this complex half-plane. With a simple pole at s =1, with residue equal to 1, Riemann zeta-function can be extended to a meromorphic function on { , by the following relationship

−s / 2 s−(1 −s) / 2 1− s πΓ( 2) ζ(s) = π Γ( 2 ) ζ (1 − s) . (5.213)

As it was mentioned this function has a simple pole at s =1, and otherwise it is regular. One of the most interesting properties of this function according to Riemann hypothesis is that it has non-trivial zeros, which all lie at Re(s) = 1, the trivial zeros are located at -2, -4, … .

156 This zeta-function was first studied in the number theory in relation to the prime numbers. The prime decomposition of a natural number using the Riemann zeta- function is in fact equal to the Euler product

−1 ζ ()s=∏(1 − p−s ) , (5.214) p where p is a prime number. The Riemann zeta-function may be represented by a

Mellin integral

1∞ 1 ζ s= ts−1 dt. (5.215) () — t Γ()s0 e −1

This form of representing the zeta-function is very important since as we will see later the general form of a spectral zeta-function corresponding to a second-order differential operator is also written in the form of a Mellin like integral, which is related to the heat-kernel (diffusion operator). This is a very powerful statement that relates statistical mechanics to number theory that has proved to be a fertile ground for condensed matter physics.

5.7.2. Generalized zeta-function

In the last subsection we defined the Riemann zeta-function and gave some of its important properties. Similarly we define a more general form of the zeta-function constructed from the eigenvalues of a differential operator. To proceed in this discussion consider a second-order self-adjoint elliptic operator D defined on a compact manifold Ωwith boundary ∂Ω , suppose the eigenvalue of this operator

is λn , then we define the zeta-function associated with operator D as

157

−s − s ζs=tr µ−2 D = µ − 2 λ , (5.216) () {() } ƒ()n where we will not include the zero eigenvalue of D in the sum. is a scale that makes the zeta-function dimensionless [14]. Like the Riemann zeta-function this function has definite value – it converges - in the complex half-plane when

Re(s) > d / 2, where d is the dimension of the manifold [22]. Also this function can be extended as a meromorphic function, which is regular in the whole complex plane except that it has a poles only at s=2 and s = 1 [32]. The zeta-function can be expressed as an integral by a Mellin-like transform given by

∞ 1 s−1 − 2 ζ()s=ƒ dt texp() − λn µ t Γ()s — 0 (5.217) 1 ∞ =— dt ts−1tr{} exp() − Dµ − 2 t , Γ()s 0

−2 the exponential expression in the first integral (e−λn µ t ), is called the heat-kernel

(diffusion operator), associated with the differential operator D. We will talk about this function and its asymptotic expansion in next section.

5.7.3. The heat equation

As we have seen in the previous section the generalized zeta-function is constructed from a Mellin-like transformation of the exponential function of eigenvalues of a second-order differential operator. The zeta-function can still be obtained if one does not know the eigenvalues of the operator. This can be achieved considering the heat-equation of the corresponding operator.

The Helmholtz equation (Heat equation) corresponding to an operator H reads

158

∂t K( x, y t) + HK( x , y t) = 0, (5.218) where K( x, y t) is called the local heat kernel. The local heat kernel can be expressed in terms of the eigenvalues and eigenfunctions of operator H

K( x, y t) =ƒ exp( −λn t) φ n( x) φ n ( y) . (5.219) n

This is defined for operators with a discrete or continuos spectrum. The local heat kernel has the following property

K( x, y t) →δ ( x − y) , t → 0. (5.220)

The global heat kernel is defined as

Kt» Ht ÿ Kxxtgdxd t (5.221) ( ) =trexp- ( ) ⁄ =— Tr( ,) =ƒ exp( −λn ) . n

Therefore the equation (5.217), may be written as

1 ∞ ζ ()s= — ts−1 K() t dt, (5.222) Γ()s 0 for an operator H in the form

H= ∆ + m2 , (5.223) which is the Hamiltonian operator for a field that is free of any potential and boundary conditions. In this case the solution of (5.218), will be

≈2 ’ 0 1 ()x− y K() x, y t= exp∆ − − tm2 ÷ . (5.224) () d / 2 ∆ ÷ ()4πt «4t ◊ d is the dimension of the manifold. The asymptotic solution for t → 0 , can be written in the form

∞ ()0 n Kxyt(),,,.= K() xytƒ axytn () (5.225) n=0

The coefficients an are called the heat kernel coefficients of the expansion. Theses coefficients possess different values according to situations in which problems are

159 considered, for example when we have a manifold with a background potential the coefficients will get values proportional to the potential [35]. For a manifold in general when there is no boundary, these coefficients also can be expressed in terms of geometrical properties of the manifold such as the Riemann curvature tensor, its contractions, and covariant derivative. In this way they do not depend on the dimension of the manifold [19]. These coefficients of expansion also are called

Seeley-deWitt coefficients.

By introducing boundaries in the manifold, there will be additional coefficients with half integer number that are related to the boundary condition. In this situation the coefficients have different values according to the nature of the condition imposed on these boundaries such as Dirichlet or Neumann conditions

[17]. We will talk about the determination of the heat kernel coefficients in Appendix

B: Heat Kernel Expansion.

The off-diagonal coefficients by means of recurrence relations are given by

» ÿ n+( x − y) ∇x ⁄ a n( x,,, y) = Ha n−1 ( x y) (5.226)

where a0 ( x, y) = 1 [55]. This relation is not applicable for a manifold with boundaries.

For the diagonal contribution of the heat kernel the asymptotic expansion will be in the form

∞ 1 n K() x,. x t= d / 2 ƒ an () x t (5.227) ()4πt n=0

In our case the diagonal (coincidence limit) coefficients up to 3 will be given by [22]

160

a0 ( x) =1,

a1 ()() x= − V x ,

12 1 a() x= V() x − V′′() x , (5.228) 2 2 6

13 1» 12 1 ÿ ′′ ′ ′′′′ ax3 () = − Vx() +… VxVx()() + Vx() + Vx() Ÿ. 6 6 2 10 ⁄

The general form of Seeley-deWitt coefficients in terms of the geometrical properties of the manifold for first few coefficients are tabulated in the literature, the first three coefficients in this case are [14]

a0 ( x) =1,

a1 () x= kR, (5.229) a x= AWeyl2 + B» Ricci 2 −1 R2 ÿ + C ∇ 2 R + DR 2 . 2 ()()() 3 ⁄

However there is another part contributing to the asymptotic expansion. This corresponds to the diagonal part of the heat kernel and it contains an exponential part(e−k() x/ t ) . The corresponding coefficients of expansion run over half-integers and they are functions of the geometrical properties of the boundary; such as the second fundamental form and intrinsic curvature of the boundary. This is actually the boundary contribution in the heat kernel expansion. The calculation for determining of heat kernel expansion for manifolds with a boundary can be found in a beautifully extended De Witt expansion by D. M. McAvity and H Osborn in [43, 44, 54] and also in [13, 23]. According to (5.221), in this situation (5.227) will be in the form

∞ 1 d n d−1 n Kt() = dxaxgtn() + dxbxgt n () . (5.230) d / 2 ƒ(—Ω — ∂Ω ) ()4πt n=0

It must be noted that the boundary dependent coefficients with half integer number are the same in two sides of the boundary whereas these coefficients for integer numbers are equal but opposite in sign.

161

5.7.4. Zeta function expansion

Now we are in a position to meromorphicaly extend the zeta function in terms of the heat kernel coefficients of expansion. If we define

A a x g dd x n= — n ( ) , Ω (5.231) d −1 Bn= b n () x g d x, —∂Ω

and

CABn= n + n. (5.232)

Then, the zeta function defined by (5.222) will be in the form

1 ŒÀ∞ C Œ¤ ζ s=n + f s . (5.233) () d / 2 Ãƒ 1 () ‹ Γ()()s 4π ÕŒ0 s−()2 d − n ›Œ

This is a meromorphic function of the complex variable s. The function f(s) is an analytic function of s. As it was discussed in section 0, the zeta function has simple poles for s ≤ 2, where its residues are determined by Cn [14, 22].

5.7.5. Zeta functional approach to the vacuum energy

The vacuum or zero-point energy as the sum of the ground states energy is given by the expression (5.21)

1 E = ƒ ZωJ . (5.234) 2 J

In this equationωJ is the eigen-frequency value of a second-ordered differential operator H, which is the Hamiltonian of the system. Consider a pure Schrödinger operator P as

162

∂2 P= − + V() x . (5.235) ∂x2

Here V( x) is the potential, the eigenvalue corresponding to this operator is λ j . Now the Hamiltonian can be written as

H= P + m2 , (5.236) so that the vacuum energy in this situation takes the form

1/ 2 1 2 E=ƒ()λJ + m , (5.237) 2 J

2 where we have put Z =c =1, and ωj= λ j + m . This energy is divergent and it must be regularized using one of the elegant and modern ways of the regularization of this energy that has been used in many areas of the physics, the so called zeta function regularization method. In this model basically we form one kind of the summation that can be considered as zeta function. This can be achieved by adding to the exponent of the expression under sum a parameter, which is called the regulator. Also a normalization scale must be included in the expression to make the dimension of the energy independent of the regulator. We call this a regularized energy

2s 1 −s µ 2 2 E0 () s=ƒ()λJ + m . (5.238) 2 J

We brought the abovementioned scale as µ and the regulator as s. In this procedure at the end of the calculation the regulator must be shifted to zero. That means the regulator can be removed since that is not a physical quantity. Hence according to

(5.216) the sum in the above equation is actually a zeta function corresponding to the

Hamiltonian operator. This equation may be written by the Mellin-like transform

(5.217) as

163

µ 2s 1 ∞ E s= dt ts−3/ 2exp» − tλ + m 2 ÿ . 0 () ƒ 1 — ()J ⁄ 2 J Γ()s − 2 0 (5.239)

The sum runs over the parameter J that is corresponding to the eigenvalue λJ , therefore by separating this factor in the exponent from m2 , which is e−tλJ and considering the definition of heat kernel from (5.221), also applying the heat-kernel expansion (5.227), the above equation takes the form

2s ∞ µ a Γ(s + n − 2) 2 2−s − n E s= n m (). (5.240) 0 () ƒ 3/ 2 1 2 n=0 ()4π Γ()s − 2

This is the final form of the regularized vacuum energy in zeta function regularization prescription. As it was mentioned before after any regularization in any kind, the regulator must be removed from calculation to give a consistent result according to the physical situation in the problem. This means in our case the value of s must be put equal to zero. But as it can be seen for some value of integer n for s → 0, the gamma function in the numerator of our vacuum regularized energy diverges due to the definition of the gamma function, where it is infinite for zero and negative integer values. This corresponds to n ≤ 2 . Therefore we will get parts of this equation that are associated to the divergent section of the energy equation. This divergent part is given by [18, 19]

−m4≈1 4µ 2 1 ’ m 3 E0(div ) =2∆ +ln 2 − ÷ a0 − 3/ 2 a 1/ 2 64π«s m 2 ◊ 24 π m2≈1 4µ 2 ’ m +2∆ +ln 2 − 1 ÷ a1 + 3/ 2 a 3/ 2 (5.241) 32π«s m ◊ 16 π 1≈ 1 4µ 2 ’ −2∆ +ln 2 − 2 ÷ a2 . 32π «s m ◊

So far we have been employing the procedure that leads us to determine the energy of the vacuum, starting from the concept of vacuum fluctuation from previous

164 chapters. But as we have seen in this way we had to deal with some inappropriate results that give us the infinities in the problem, to get over this we introduce the regularization operations. In fact all approaches of the regularization model must result in the same expression for the corresponding physical quantity, but unfortunately as the regulator in each model is removed at the end of the calculation these procedures in some parts have irregularities that are the divergent part. Also sometimes the regularized quantities still depend on the unphysical arbitrary normalization scale introduced in the calculation. We talked about this crucial situation in the last chapter, where we introduced the concept of the renormalization.

This physical trick, is discussed in QFT to deal with irregularities in the computation of physical quantities, is also used in our case for determining the vacuum energy.

The quantity which will be renormalized is in fact the classical background field. In this procedure we remove the divergent part of the vacuum energy and it will be absorbed by the classical energy associated with the background field [18, 19]

EEEEE= + + − (5.242) %((&(('class0( div ) %(&(' 0 0( div )

The first curly bracket contains the classical energy associated with renormalization of the parameters of the classical system, and the second part is the renormalized vacuum energy of the quantum field.

5.7.6. An advanced look at the Casimir energy

The Casimir energy has been discussed in its original approach by the

Green’s function models. These methods are very powerful. The modern way of dealing with this problem is the zeta function regularization.

165 To proceed with the discussion let us separate the temporal and spatial part of the differential operator D describing the field equation as

2 DD= −∂0 + s . (5.243)

The eigen-frequency of Ds, are

2 ωn= λ n . (5.244)

Therefore using (5.238) for the regularized vacuum energy we have

1 2s 1 E0 () s=µ ζ s () − + s , (5.245) 2 2

with ζ s being the zeta function associated with spatial operator Ds. At s = 0 , we have obviously a pole that contributes to the divergence of the non-regularized energy, keeping this in mind and recalling the equation (5.233), the residue corresponding to this pole will be

1 CD( ) − 2 s (5.246) 2 ()4π 2

As it was discussed before, C2 is a geometrical coefficient of the heat kernel expansion equal to A2 + B2, where these coefficient are given by (5.231). The pole in

(5.245) has to be absorbed by the classical background field [14, 36]. The Casimir energy has an ambiguity proportional to C2, because there are many ways to remove the pole from the regularized energy, then we can use different renormalization schemes to define a definite Casimir energy. One of these renormalization schemes is called the minimal subtraction scheme, which is simply removing the divergent part or the pole in the problem. Adopting this method the Casimir energy will be defined in a new form by (to not to get confuse of s for the spatial part of the differential operator with regulator, from now on we use ε instead of s, to show the regulator)

166 1 1 2ε »1 1 ÿ ECasimir=lim µ ζ s() −2 + ε + ζ s () − 2 − ε ⁄ 2ε →0 2 (5.247) »CD ÿ 1 1 2 ()s 2 =…PPζs () −2 + ε − 2 ln µ Ÿ . 2 …()4π ⁄Ÿ

Here we actually consider the principal part (PP) of the zeta function, in which we just consider the finite part of this meromorphic function. In this equation the following property has been used

» ÿ »ÿ′ »ÿ PP f(ε) ζs( ε) ⁄= f( ε) PP ⁄ ζ s( ε) + f ( ε) Res ⁄ ζ s ( ε ) , (5.248) where f (ε) = µ 2ε . (5.249)

The derivative of the above function is

dµ 2ε f ′()ε= = µ2ε ln µ 2 . (5.250) dε

And the residue is given by (5.246).

5.7.7. Various behaviours of the Casimir effect with different geometrical boundaries

Rescaling the metric and mass results into a rescaling of the eigenvalue of the

Ds, that is

ÀŒg→ κ 2 g s s Ω λ→ κ−2 λ . (5.251) Ã −1 n n ÕŒm→ κ m

Then rescaling of the eigenvalue applied on (5.238) gives

2− 1 2ε ζs( κg s;;;;. κ m ε) = κ ζ s( g s m ε ) (5.252)

(5.248) helps us to determine the rescaled Casimir energy in terms of the scale and non-scaled Casimir energy as

167

2− 1 ECasimir( g s;; m) C2 ( g s m) lnκ ECasimir()κ g s ;. κ m = − 2 (5.253) κ()4π κ

2 Now since the radius of the manifold gsand κ g s are connected by

2 R(κ gs) = κ R( g s ) (5.254)

2 the equation (5.253) may be written in terms of the radius of the manifold κ gs in the form

ε− εln ( µR) ER() = 0 1 , (5.255) Casimir R where R is the radius of this manifold and

À C( g ) ε = 2 s Œ 1 2 Ã ()4π . (5.256) Œ E g R g R g ÕŒε0 =Casimir()()() s s + ε1 ln () µ s

Now from the discussion in [14], we will see that the sign of C2(gs) actually is the determining factor in the behaviour of the Casimir effect for different geometrical boundaries. That is so since for a positive value of this geometrical term (C2 (gs)>0), we have a negative energy with minimum value

ERmin= − ε 1/ min , (5.257) whereas for a negative C2, (C2(gs)<0), the energy has a maximum positive value

ERmax= + ε 1/. max (5.258)

The maximum and minimum radiuses of the manifold, where the corresponding

−1 extremum energies occur are µexp( 1+ ε0 / ε 1 ) . If C2(gs) = 0 then ε1 will be vanished and the energy is zero E = 0, this in fact corresponds to the situation when the space-time manifold is flat and there are thin boundaries in the problem that the second fundamental form in both sides of the boundaries have the same value with opposite sign therefore they cancel out each other. Also by inspection of (5.251), if

168 κ → ∞ Ω m → 0 therefore from (5.253) E → 0. Hence a massless field in a space- time with trivial topology produces a zero Casimir energy. Also from (5.257) and

(5.258) we conclude that the repulsive or attractive behaviour of the Casimir force is related to the sign of the geometrical coefficient of heat kernel expansion C2, and it is strongly dependent on the shape and geometry of the boundary introduced into field.

5.7.8. One-loop effective action

In the context of the quantum field theory, the corrections to the exact propagator or Green’s function in an interaction process or even in a vacuum polarization or self energy diagram comes as the closed loops of virtual particles in the Feynman diagram technique. This correction also comes to the picture when we are working on propagation amplitude problems. The lowest order of this correction, which contains one loop, is called the one-loop correction. The one-loop approximation first arose in quantum electrodynamic; it is very important function and has also many applications in other areas of physics.

In this section we will show how the action of one-loop theory can be related to the zeta function of a differential operator, which can be the Hamiltonian of the system. And also we will investigate that how the corresponding energy is approximated in terms of the Casimir energy.

Let start with the definition of one-loop effective actionΓ , where broadly is used instead of the ground state energy in quantum field theory. We will see that this action is related to the Euclidean one-loop partition function or vacuum-vacuum transition amplitude that is given by following functional integral

169

Z= eS()φc — D Φ eS()Φ , (5.259)

where S (φc )is a classical action and S (Φ) is an action of a free field – quadratic in the fields – which may be written in the form

1 S()Φ = − dV Φ L Φ. (5.260) 2 —

In this equation L, is a self-adjoint, non-negative, second-ordered differential operator of the form

L= −∆ + V + m2 , (5.261)

V is the background potential. Observing(5.260), we see that (5.259) is an integral of a Gaussian functionexp(−x2 ) , whose solution is

−1/ 2 −S()φ 2 −S()φ À1 2 ¤ Z= ec »det/() Lµ ÿ = ec expÃ − lndet/() L µ ‹ . (5.262) ⁄ Õ2 ›

µ is a normalization scale for dimensional reason. Now the one-loop approximation action is defined as

1 Γ = −lnZSL = + ln det() /µ 2 . (5.263) 2

For the second term in the right-hand side we have

∞ L 2 ln det= − dt t−1 Tr e −tL / µ . (5.264) 2 — µ 0

For the above relation we have used the following definition [14]

−2 ≈d ’ det()µL= exp∆ − ζ () s ÷ . (5.265) «ds s=0 ◊

170 This integral is divergent and we must regularize it. To do so, we employ the zeta function regularization method by introducing a regularizing function

tε . (5.266) Γ()1 + ε

By removing the regulator , the above expression goes to unity. Therefore equation(5.263) leads

∞ ε −1 1t −tL / µ 2 1 Γ =S − dtTr e = S − ζ2 ε . (5.267) — L / µ () 20 Γ() 1 + ε 2 ε

As it can be observed the one-loop effective action is expressed in terms of the zeta function of the corresponding differential operator. This indicates that there must be a close relationship between the one-loop effective energy and the Casimir energy, since as it was mentioned before the Casimir energy is now described by the principal part of the corresponding zeta function. This relationship is

»1 ÿ 1 µ ψ(1) − ψ ( − 2 ) ⁄C2 EEeff= Casimir + , (5.268) 2 ()4π 2 where the digamma function is defined as follow

dln Γ( s) ψ ()s = . (5.269) ds

The Casimir energy and effective energy agree within the ambiguity of the renormalization of the Casimir energy, which was discussed before as the term proportional to the heat-kernel expansion coefficient C2. In the case when C2 = 0, the

Casimir effect coincides with the one-loop quantum effect [14]. This is a very interesting point as in the non-trivial topology of the space-time manifold, the ambiguous part related to renormalization of the Casimir energy will not vanished, this indicates that for example in the curved space-time the one-loop effect that is a quantum field theory effect will differ with the Casimir effect, which is a

171 consequence of the quantum field fluctuation. And the difference is proportional to

C2.

5.8. Casimir effect in the Kaluza-Klein model

There have been many attempts to unify the fundamental forces. This effect was started by Einstein. He spent many years of his life to achieve a unified frame- work that explains different interactions in nature such as gravitation and electromagnetism. Although his effort was not successful but the investigation in this area has continued, and some spectacular outcomes were achieved. The electroweak theory was the first result of these series of research. In this theory the electromagnetism was put in the same frame of formulation as the weak interaction.

Later the nuclear strong interaction found its way to be accompanied with two other forces through the mechanism of the supersymmetry. The big problem in this way was to entangle the gravitation, and putting this force in the same formulation as other interactions. The first model for unifying the gravitation with other forces was introduced by the mathematician Kaluza, which was confirmed later by Klein. This model called Kaluza-Klein or KK theory in fact aims to unify the gravitation with electromagnetism and it provides the basis for further investigation in the unified description of all fundamental interactions, such as superstring theory.

The Kaluza-Klein theory is in fact a multi-dimensional prescription of the space-time. In this model the true dimensionality of space-time is considered as d=4 + N , N is the number of extra dimensions of the space that are compactified.

This means that the additional dimensions form a compact space with geometrical

172 size of order of Plank length (10-32mm). In the original Kaluza-Klein theory the number of compact dimensions is equal to one; therefore the dimensionality of space-time is regarded as five.

The most advanced development in the theory of everything that tries to formulate all kind of fundamental interactions in a united way is called superstring theory. Similar to Kaluza-Klein theory, the superstring theory takes advantage of the extra compact dimension of the space-time to formulate the interactions, which seemed to be impossible in the frame work of a four dimensional space-time. The explanation of the mechanism of the dimensional compactification in this theory is the main importance as the model. The dynamics of hidden dimensions can be associated with the Casimir effect, where a non-zero energy associated with this effect for non-trivial geometry is calculated. Therefore the zero-point energy stabilizes the geometry of the extra dimensions [46].

To see how the Casimir energy participates in compactified space we start with a general case, where our manifold is in the form of the Minkowski four

4 N dimensional space-time plus a N-sphere with radius a and volume VN (MS× ). in this situation according to the equation (5.52)

1 T00= ∂ 0 ∂′′′ 0 G() x,;,, y x y (5.270) 2i where x are the coordinates in M4, and y the coordinates for SN . The Casimir energy for a massless scalar field will be calculated

iV uaVT=00 = −N dkd 3ωω 2 gyyk,;.′ 2 − ω 2 (5.271) () N 4 — — () 2() 2π c

µ Here we have used the reduced Green’s function g( y,; y′ k kµ ) as the Fourier transform ofG( x,;, y x′′ y ) . And c is the counter of integration with respect to ,

173 encircling the poles on real axis in both directions. Since our reduced Green’s function is a function of the coordinates in SN it is much appropriate to use the N- dimensional spherical harmonics for this Green’s function. Now applying the

m following operator on the spherical harmonicsYl ( y) , results in:

2 2 2 2m ≈M l 2 2 ’ m ()∇N +k −ω Y l () y =∆ −2 + k − ω ÷ Yl () y (5.272) «a ◊

where

2 Ml = l( l + N −1) , (5.273)

2 N is the eigenvalue of the Laplacian ∇N over the S , and corresponding degeneracy is

(2l+ N − 1)( l + N − 2) ! D = . (5.274) l ()N−1 ! l !

Here we must keep in mind that the reduced Green’s function satisfies the following

2 2 2 2 2 (∇N +k −ω) g( y,;. y′′ k − ω) = − δ ( y − y ) (5.275)

Applying the (5.272) on the above relation gives us

1 Ym y = . (5.276) l () 2 2 2 2 ()Ml / a+ k −ω

And considering the following sum

m m∗ Dl ƒYl()() y Y l y = , (5.277) m VN we substitute (5.276) in the above equation, therefore the vacuum energy density corresponding to the compact space SN will be in the form

i ∞ D u a= − d3 k dω ω 2 l . (5.278) () 4 — — ƒ 2 2 2 2 c+ ()2π l=0 ()Ml / a+ k −ω

Now c+ is the counter that encircles the poles on the positive real axis in a clockwise sense. As it can be seen the vacuum energy of a massless scalar field in MS4 × N is

174 reduced as the energy of a massive scalar field in the Minkowski space, since as we know we had a Fourier transform for the reduced Green’s function that transform from coordinates in M4 into the momentum space k, with an integral over this

quantity, with a mass description given by M l .

The simplest case of the above vacuum energy for the non-trivial geometry described above occurs when the number of dimensionality of the compact space is equal to one (N = 1). Here the N-sphere actually reduces to a circle and our model will return back to its original form that is described by Kaluza-Klein theory. With N

=1 the mass Ml is equal to l, therefore the sum in the right-hand side of (5.278) may be written in terms of a coth function

ex+ e− x 2 cothx = = 1 + . (5.279) ex− e− x e2 x −1

This sum will be

2 ∞ 1/ 2 a Dl aπ »2 2 ÿ ƒ =1/ 2 coth aπ() k − ω 2 2 2 2 2 2 … ⁄Ÿ l=0 l+ a() k −ω ()k −ω (5.280) aπ ≈2 ’ ∆ ÷ =1/ 2 1 + 1/ 2 . 2 2 ∆2πa() k 2− ω 2 ÷ ()k −ω «e −1 ◊

Notice the degeneracy for N =1 has values of DDl=0=1 and l ≥ 1 = 2.

The above expression for the vacuum energy density contains two parts the first part corresponding to the cosmological term and it is actually divergent, which after an appropriate cutoff regularization ofω - b−1 , this energy will be

2π a u - , (5.281) cosmo b5 b5 is at Plank scale.

175 The Casimir energy associated with the compact space in the Kaluza-Klein model is determined from the second part of(5.280). That is [46]

5.0558077× 10−5 u = − . (5.282) Casimir a4

This energy is responsible for the stabilizing of the compact manifold.

Similarly for higher additional dimensions N >1, we have contribution of the cosmological energy as

V u ∝ N . (5.283) cosmo bN +4

This energy is the same for both odd and even values of N but the Casimir energy for odd N is computed as

1∞ 2 2π u() a= −Re dy» y2 − i() N − 1 y ÿ D . (5.284) Casimir 64π 2a 2 —0 ⁄ iy e2π y −1

For examples these energies for N = 3 and N = 5 are calculated as

7.5687046× 10−5 u() a = 3 a4 (5.285) 4.2830381× 10−4 u() a = . 5 a4

The situation is different when the N is an even integer number, since the Casimir energy also diverges, but it can be shown that by introducing a cutoff regularization

[46], this energy can be written as

1 u=»αln() a / b + γ ÿ , (5.286) Casimira4 N N ⁄ where

1 ∞ dt 2 α =ImD»() N − 1 it − t 2 ÿ , (5.287) N16π 2—0 e 2πt − 1 it ⁄

176 and

1∞ dt ≈ 1 2 γ = −ImD» N − 1 it − t 2 ÿ N 2 2πt ∆ it () ⁄ 32π —0 e − 1« 2 { } »()N−1 2 t2 + t 4 ÿ ×…ln + 1 Ÿ (5.288) …16 ⁄Ÿ 2 N −1’ »2 ÿ +ReDit () N − 1 it − t ⁄ arctan÷ . {} t ◊

Also by starting from the classical Einstein equation

1 8πG R− R g + Λ g = − N T ren , (5.289) AB2 N AB N ABc4 AB

where A,B= 0,1,…,N-1, and GNN and Λ are gravitational and cosmological constants in N dimensions. We put for the expression of the energy in the right-hand side the corresponding Casimir energy related to the compact manifold, therefore after a few steps, we find the self-consistent value of the radius of the N-sphere [19]

1/ 2 »8π ()N + 4 ÿ a= … CN Ÿ l Pl , (5.290) …NN()−1 ⁄Ÿ here CN is a constant whose values depend on the dimensionality of the compact manifold.

There have been many attempts for different spontaneous compactified manifolds. In the above discussion we just consider the simple case of an N-sphere, for more complicated geometries the expression of the energy and compactification radius will be different. For example we consider the extra manifold with geometry of L two-dimensional noncommutative tori. Here following the [34], we find the

Casimir energy as

177

−1 1» 1≈d + 1 ’≈ d + 1 ’ 1 u=d / 2 … L+ d +1 Γ∆ L + ÷∆ v2L L + ÷ d +1 4 π ()4π π «2 ◊« 2 ◊ a (5.291) 2 d+1≈ d + 1 ’ 2()L− 1 ÿ −λθ Γ∆ − ÷v2L () L −1 a Ÿ , 2« 2 ◊ ⁄ and the compactification radius as

1 d+1 d + 1 2L+ d − 1 »−Γ()L + v2L () L + ÿ a = 2 2 . (5.292) …2L+ d + 1 d +1 Ÿ …λθ π ()L−1 Γ() − 2 v2L ⁄Ÿ

In the above equations

−s ∞ »1 ÿ vN () s = ƒ …2 2 Ÿ , (5.293) n+... + n n1 ,..., nN = 1 1 N ⁄ and

λ 2Γ(L + d −3 ) λ 2 = 2 , (5.294) θ L+ d +1 2()L+ d −3 64π2 θ 2

where is regarded as the noncommutativity of the coordinates in compact manifold

2LL− 1 2 »y,. y ⁄ÿ = iθ (5.295)

178

Chapter VI

6. A New Approach to the Casimir Energy for a Massive Scalar Field

Apart from the methods we already discussed, in this thesis a simpler and more powerful algorithm is proposed. So far we have seen in the previous chapter how to calculate the Casimir energy in a cavity in the form of two parallel planes using the Green’s method and also the zeta function approach. For the case of a massless scalar field the corresponding energy can be determined by employing these models, where in the end of the calculations we will get a contribution from Riemann zeta function that can be treated in a very simple manner resulting in a finite value of the energy for this field. The situation is much complicated when we have instead a massive scalar field. In that case the sum in the expression of the energy can not be regularized by the simple Riemann zeta function.

In this chapter I introduce a new method to regularize the massive scalar field that is basically obtaining a finite value for the sum expression in the calculation. The model in which I will work on is based on the dimensional regularization scheme. As it was clarified in subsection 4.9.1, this is a technique for extracting a finite value out

179 of the infinity for a divergent integral by implicating the integral in a fictitious fractional dimension of space, since the integration is well-behaved in the fractional dimension. The regularization parameter must be removed at the end of the calculation but with no surprise we shall get the divergent part of the problem in terms of the regularization parameter. To get over this problem we use an appropriate renormalization by introducing counterterms in the calculations which results in a physically acceptable value for the energy. In order to employ the dimensional regularization in this model we use an expansion formula for the sum that is called

Euler-Maclaurin summation formula. This expansion gives us the opportunity to convert the sum into integration, where the dimensional regularization can be used.

For this purpose we start with recalculating the Casimir energy for a massless scalar field for the case of the two parallel hyper-planes cavity with the Dirichlet boundary condition and extend our calculation for the massive scalar field with our new technique explained above.

6.1. Casimir energy for a scalar field

The vacuum energy of a field is given by a sum over the modes of the ground state energies as

1 E = ƒ 2 Zωn. (6.1) n

Hereωn , is the eigen-frequency value of the Hamiltonian operator. Since the eigen- frequency value correspond to infinite modes, the above sum for high frequencies is divergent. To obtain a finite value for this energy it must be regularized. The zeta function regularization method for the vacuum energy was discussed in subsection

180 5.7.5. As it was mentioned before introducing a material boundary or a background field that implies a periodicity condition for the quantum field regularizes the vacuum energy in the form of the Casimir energy. The boundary conditions applied by a material body in the vacuum constraint in the medium some specific direction that eliminates some modes in the calculation of the energy. Associated with this constraint we get some district modes which appear in the calculation under a summation over those modes.

Now the eigen-frequency value of the Hamiltonian operator appeared in (6.1) is given by

2 2 2 ωk=c() n π /, a +K T + m (6.2)

where a is the separation between planes and KT are the eave numbers in the transverse directions. Suppose the discrete modes obey the Dirichlet boundary condition

φ ( z) = 0. (6.3)

In this equation z are the points on the boundaries (i.e. z = 0, a), andφ ( x) is a scalar field that satisfy the following Klein-Gordon equation of motion of the field

(∂2 +m 2 )φ ( x) = 0. (6.4)

Next since in the transverse directions we have continuos modes the contribution of these free modes in the energy (6.1) is a d-1 dimensional integration, where d is the dimension of the space concerned in the problem. We consider a particular length L in the transverse directions and L>> a . The vacuum energy with the constraint of hyper-planes will be actually the Casimir energy for this particular geometrical boundary for the scalar fields and it is given by

181

d −1 1 ≈L ’ 2 E dd −1KK n a 2 m 2 (6.5) =∆ ÷ ƒ— TT()π /. + + 2« 2π ◊ n

In the above equation we put Z =c =1 for convenience. We will write this equation in the following form

d −1 d −1 / 2 (d −3) / 2 () ∞ ∞ 1≈L ’ 2π 2 2 1 2 2 2 E= kTTT d k() nπ /. a+ k + m ∆ ÷ ƒ—0 () () 2« 2π ◊ Γ()()d −1 / 2 n=1 2

(6.6)

To solve the integral in (6.5) we have used spherical coordinates, where a multidimensional integral will be reduced to a one-dimension as

2π d / 2 f() k dd k= k d −1 f() k dk. (6.7) —Γ()d / 2 —

The reason we act in this way is that we are constructing an integral which can be solved by employing the Beta function defined as

r ∞ t Γ(r +1) Γ( s + 1) dt= B r +1, s + 1 = . (6.8) — r+ s +2 () 0 ()1+ t Γ()r + s + 2

Therefore the equation (6.6) will take the following form by using the beta function

d −1 ∞ d / 2 1 Γ() −d/ 2d +1 / 2 () L / 2 2 E= π() » am/. π + n2 ÿ (6.9) d ƒ () ⁄ 2Γ() − 1/ 2 a n=1

The sum in the above equation for the case of a massless scalar field (i.e. m = 0) is indeed a Riemann zeta function formally given by

∞ ƒ nd =ζ () − d . (6.10) n=1

By substituting the Riemann zeta function in (6.9), we reach the general formula for the Casimir energy of a massless scalar field [6] for parallel hyper-plane as

182

d −1 L −()d +1 / 2 E= − Γ()d +1 ()4π ζ () d + 1 . (6.11) ad 2

This energy for a real physical situation with dimension of the space equal to three

(i.e. d = 3) will be

π 2L 2 E = − . (6.12) 1440 a3

This is the standard value of the Casimir energy for the case of the massless scalar field, which was obtained in the section 5.3.

6.2. Casimir energy for a massive scalar field

In the formulation of the Casimir energy for a scalar field given by equation

(6.9) the last expression in the right-hand side is a sum over infinite discrete modes of the field, which is identified by the integer value n, this term will be divergent for high frequencies of the field fluctuations. As it was discussed in the last section, this sum in the case of a massless scalar field reduces to a Riemann zeta function regularization, where a finite value for the sum will be achieved. Now in the case of a massive scalar field it is not that simple and an alternative regularization procedure is provided by us.

The original way of treating this problem can be found in [6, 51]. I will not go through the detail of this calculation and I just show the important part of that. I start by consider the following sum from (6.9)

−s / 2 ∞ »2 2 ÿ −s / 2 ≈’s ≈’≈’ m n S =π Γ∆÷ƒ … ∆÷∆÷ + Ÿ , Res > 1, (6.13) «◊2 n=−∞ … «◊«◊π a ⁄Ÿ

Considering the Jacobi ϑ function

183

∞ 2 ϑ ()z;, x= ƒ e−πn x e2 π nz (6.14) n=−∞ we will get an expression for the S as

1−s »∞ ÿ am K1−s ()2 m an S =… Γs−1 + 2 Ÿ , (6.15) ()1−s / 2 ()2 ƒ ()1−s / 2 π …n=−∞ Ÿ ()m an ⁄ where, the sum runs over negative and positive integer values except zero, and K is a modified Bessel function that is discussed in Appendix A: Bessel’s functions. In this situation the energy may be written as

d −1 ∞ L −d +1 / 2 d + 1 Kd +1 / 2 (2 amn) E= − 4π () ma () () (6.16) d () () ƒ ()d +1 / 2 a n=1 ()amn

As it can be observed there is a sum in the formulation of the energy that must be controlled in a proper manner to give the right expression for the energy. The above equation can also be written in the following integral form

d −1 (d +1) / 2 L −()d +1 / 2 π ∞ ()()d−1 / 2 − am2 / nt E= − d ()4π dt t e () ϑ () 0;1− 1 a 2 —0 (6.17) d −1 ∞ 2 2 L 1 −d / 2 d −1 ≈−2 t +() ma ’ = −()2π dt t log∆ 1 − e ÷ . ad 2 —0 « ◊

This equation determines a formulation for the Casimir energy of a Massive scalar field, but still the presence of the integration indicates that more work is needed on this calculation. Some approximate results are available in literature especially in [6,

46, 51]. These values are given by

7LL2π 2 1 1 2 E≈ − m2 , when ma 0 1, (6.18) 8 1440a3 192 a

184 and

2 3/ 2 L≈ m ’ −2ma E≈ ∆ ÷ e, when ma 2 1. (6.19) 16 «π a ◊

As we can see from (6.18) we have 7/8 of the standard result of the a massless scalar field given in (6.12), minus the correction for the contribution of the massive field.

6.3. A new technique

The regularization of the sum in (6.9) can be done by the using usual zeta function regularization scheme. Suppose we are working for the real dimension of the space equal to three (i.e. d = 3). In this model the sum

2 d / 2 S=» am/,π + n2 ÿ (6.20) ƒ () ⁄ n is written in the following form

2s s µ 2 3/ 2− ζ=»am/. π + n2 ÿ (6.21) p ƒ () ⁄ 2 n

This is a generalized zeta function that can be expressed in terms of the heat-kernel associated with a Laplace type operator that is the Hamiltonian of the system. As we know the heat-kernel can be expanded [6, 9], so we will get the expression for our corresponding sum in terms of the heat-kernel expansion. The divergent part is related to a number of the heat-kernel coefficients of the expansion when s = 0 [19].

In this situation we see the energy depends on the regularization parameter and mass scale , which is not desirable. To overcome this problem we have to introduce a renormalization scheme, but as it was mentioned above the divergent part of the problem after regularization correspond to some coefficient of the expansion that

185 makes the renormalization difficult. To deal with this situation we have to introduce counterterms in the energy expression.

As we saw in 4.9.1 there are some very well-known methods for regularizing divergent integrals. The most elegant of these schemes is the dimensional regularization, where a fictitious fractional dimension for the integration is employed in place of the positive integer number of the dimension originally appearing in the expression of the integral. For this reason we subtract (or add) a small real number from the true dimensionality of the integration. This is called the regularization parameter and then we proceed to calculate the integral in the normal way. In the end we remove the regulator, but by taking out the regulator we will get the divergent part, where we have an integral depending on the regularization parameter. The next prescription will be the renormalization of the physical quantities eliminating this dependency and obtaining a finite value.

In this new model of regularization I will employ one of the most useful mathematical expansions that relates the sum of an expression to the lower and upper value of the function itself and a number of the function’s derivatives. This expansion is called Euler-Maclaurin sum formula

Λ−1 Λ 1 ƒ fn = f() n dn − » f()()0 + f Λ ⁄ÿ —0 2 n=1 (6.22) ∞ B 2r »()2r− 1() 2 r − 1 ÿ +ƒ f() Λ − f ()0 ⁄ . r=1 ()2r !

In this expansion B2r are the Bernoulli numbers, which are a sequence of signed rational numbers with the following identity

∞ n x Bn x x ≡ ƒ . (6.23) e−1n=0 n !

This number can be defined by the contour integral

186 n! z dz B = . (6.24) n 2πiî— ez− 1 z n+1

Now by inspection according to(6.20), for d = 3 we have

3/ 2 f am/.2 n2 (6.25) n =(()π + )

Next since in the summation (6.20), the upper limit of the sum is infinity ( Λ = ∞ ), the first integral in right-hand side of (6.22) will be

∞ ∞ 3/ 2 f()() n dn= am/.π 2 + n2 dn (6.26) —0 — 0 ( )

This is a divergent integral, since for the upper limit of the integration we have an infinite value that makes the integration infinite. We must regularize this integration.

The most convenient form of the regularization is the dimensional regularization, where we change the dimension of problem to a fractional that is 1-, then the integration will be in the following form

2 3/ 2 µε— (()am/. π + n2) d 1− ε n (6.27)

The mass scale in this integral expression plays an important role, while it avoids the dependency of the energy on the arbitrary regularization parameter . Now we try to evaluate this integral with the new fractional dimension, for this reason we solve the integral in spherical coordinates according to (6.7), then we get

()1−ε / 2 ε 2 3/ 2 2π µ 2 3/ 2 µε()am/ π + ndn2 1− ε = nam − ε »()/π + ndn2 ÿ —() Γ()1−ε — ⁄ 2 (6.28) ()1−ε / 2 ε 3/ 2 π µ 3−ε / 2 2 =()am/π n2 » 1 + () n π / am ÿ d n2 . 1−ε — () ⁄ () Γ()2

187 The above integration can be written as

(1−ε ) / 2 ε 2 3/ 2 π µ 5−ε 3/ 2 µεam/ π+ n2 d 1− ε n = ma / π α−ε / 2 1 + α d α , —()() 1−ε ()() — Γ()2

(6.29) where we have defined

α= ()n π /. ma 2 (6.30)

Now using the definition of the Beta function given by (6.8), for the integration in the right-hand side of (6.29) we have

−ε3 − 7 + ε r=, r + s + 2 = − Ω s = , (6.31) 2 2 2 and then

≈2−ε − 5 + ε ’ ΓΓ( 2−ε) ( − 5 + ε ) B r+1, s + 1 = B , = 2 2 . (6.32) () ∆ ÷ −3 «2 2 ◊ Γ()2

Therefore

5 −ε 3/ 2 ()1−ε / 2 2−ε − 5 + ε 2 π ≈ma ’ ΓΓ( ) ( ) ≈ ma ’ µεam/ π + n2 d 1− ε n = 2 2 . — ()() 1−ε ∆ ÷ − 3 ∆ ÷ ΓΓ()2«π ◊ () 2 « πµ ◊ (6.33)

After shifting the regulator to zero we have a finite value for this integral and it is given by

3/ 2 1/ 2− 5 5 ∞ 2 π ΓΓ(1) ( ) ≈ma ’ ()am/π + n2 dn = 2 . (6.34) —0 () 1 −3 ∆ ÷ ΓΓ()2() 2 «π ◊

The next term to be evaluated is

f()()0= am /π 3 . (6.35)

Now we write f (Λ) in terms of a divergent integral [11] in the form

188

3/ 2 Λ f()()Λ = am/π2 + Λ2 = 3 nam() / π2 + ndnam2 + () / π 3 . (6.36) ( ) —0

This is a divergent integral since when Λ → ∞ , the integral diverges. A similar procedure given for the integral (6.26) may be employed for regularizing purposes using dimensional regularization. In this case by using (6.7) the integral will be written in the following form

()1−ε / 2 2 π 2 3µεnam() / π+ ndn2 1− ε = 3 µ ε n 1 − ε () am / π + ndn 2 —Γ()1−ε — 2 (6.37) ()1−ε / 2 1 π am ()1−ε / 2 2 =µε n2 n π / am + 1 d n2 . 1−ε —() () () 2 Γ()2 π

Using Beta functions and following the procedure from (6.29) to(6.32), we obtain for the above integral

3µε— n() am / π 2 + n2 d 1− ε n 1−ε / 2 4 −ε (6.38) π () ≈ma ’ ΓΓ()3−ε() − 4 + ε ≈ ma ’ = 32 2 . 1−ε ∆ ÷ −1 ∆ ÷ ΓΓ()2«π ◊ () 2 « πµ ◊

In this equation as we shift the regulator to zero, we will get a divergent term corresponding toΓ( −2) . Now let us consider the Gamma function expansion [53]

n n ()−1 »1 −1 ÿ Γ() −n +ε =… − γ +ƒ k + O() ε Ÿ, (6.39) n! ε k=1 ⁄ where n is a nonnegative integer and = 0.5772…, is the Euler-Mascheroni constant, and also the exponent expansion

AAO−ε =1 −ε ln + ( ε 2 ) . (6.40)

With these two expansion equations we can write the divergent part of (6.38) as

−ε ≈ε ’≈’am1 ≈’ am 1 3 Γ∆ −2 + ÷∆÷ = − ln ∆÷ −γ + . (6.41) «2 ◊«◊πµ ε «◊ πµ 2 4

189 TheOO(ε) and ( ε 2 ) for =0 will vanish since they are multiples of . The integral

(6.36) corresponds to the divergent part of the energy regularization and it may be written in the following form

3µε— n() am / π 2 + n2 d 1− ε n 1−ε / 2 3−ε 4 (6.42) 3π () Γ() ≈am ’ » 1≈ am ’ 1 3 ÿ =2 −ln −γ + . 1−ε 1 ∆ ÷ …∆ ÷ Ÿ 2Γ()2 Γ() − 2 «π ◊ ε« πµ ◊ 2 4 ⁄

This gives the regularized expression for this integral, but as we can see it depends on the regularization parameter and the mass scale . This is not physically desirable and we must get ride of this situation by an appropriate renormalization scheme. We will discuss about this later to obtain a finite value for the energy independent of the regulator and mass scale. Next we work on the rest in the expression of the sum (6.22). We see that ∀r ∈ N; f (2r− 1) ( 0) = 0, and

1/ 2 r=1 → f()1 ()() Λ = 3 Λ( ma /π 2 + Λ2 ) −1/ 2− 3/ 2 r=→2 f()3 ()() Λ=Λ 9() ma /π2 +Λ2 −Λ 3 3()() ma / π 2 +Λ 2 −3/ 2 −5/ 2 r=→3 f()5 ()() Λ=−Λ 45() ma /π2 +Λ2 +Λ 90 3()() ma / π 2 +Λ 2 −7 / 2 −45 Λ5()()ma /π 2 + Λ 2 @

(6.43)

It is noted that for high frequency values, gets very large therefore f(2r− 1) =0 for r ≥ 3. Hence the only derivative parts remaining in our sum expression correspond to r = 1, 2. The first equation of (6.43) is written in its integral form as

190

1/ 2 f()1 ()()Λ =3 Λ( ma /π 2 + Λ2 ) (6.44) Λ » 2−1/ 2 2 1/ 2 ÿ =3n2() am /π + n 2 +() am / π + n 2 dn . —0 … () () ⁄Ÿ

For Λ → ∞ the above integral diverges and we treat this again by dimensional regularization. The integral will become

−1/ 2 1/ 2 » 22 2 2 2ÿ 1− 3µε n() am / π+ n +() am / π + n dε n . (6.45) — … ( ) ( ) ⁄Ÿ

Again following the above procedure to evaluate the integral, we obtain

∞ » 2−1/ 2 2 1/ 2 ÿ 3n2() am /π+ n 2 +() am / π + n 2 dn —0 … ( ) ( ) ⁄Ÿ 3 (6.46) π 1/ 2≈am ’ »ΓΓΓΓ()2()− 3() 1 () − 3 ÿ =32 + 2 , 1∆ ÷ … 1− 1 Ÿ ΓΓΓ()2«π ◊ …() 2() 2 ⁄Ÿ

(3) whose coefficient from (6.22) will be B2 / 2!. Now the f (Λ) =9 − 3 = 6, since the denominators in this equation for large value of , will be and 3 for

2 −1/ 22 − 3/ 2 (()am/π+ Λ2 ) and(() am / π + Λ2 ) respectively. The coefficient for this part

3/ 2 2 2 is B4 / 4!. Therefore the divergent sum ƒ(()am/π + n ) will be regularized as n

5 4 3 −2≈’≈’ma − 3 am» 1≈ am ’ 1 3 ÿ 7 ≈’ am 1 ∆÷∆÷+… −ln∆ ÷ −γ + Ÿ + ∆÷ − . (6.47) 5«◊«◊π 8 π ε« πµ ◊ 2 4 ⁄ 6 «◊ π 120

Now by collecting all contributing terms in the above regularization for the sum expression of the energy (6.9), we obtain the energy in the following form

2 2 2 2 2 L a5 L a 4 »1≈ am ’ 1 3 ÿ L π 1 E=3 m + 2 m … −ln∆ ÷ −γ + Ÿ + 3 . (6.48) 30π 32 π ε« πµ ◊ 2 4 ⁄ 1440 a

The third term of (6.47) has been discarded since it is independent of a [6]. In this technique as we can see the result consists of a contribution of massless scalar field part plus some correction for the case of massive scalar field. The massless part

191 completely agrees with the standard value given by other author and the rest is as much as we can go in order to regularize the energy.

6.3.1. Casimir energy and renormalization

As it can be seen the result for the regularization with our new technique has a very simple and consistent form but with no surprise we observe that the regularizing of this energy ends up with a divergent part depending on a regulator and scale . These are completely arbitrary unphysical quantities in the sense that we have introduced them to obtain a finite value for the energy, but our mission is not accomplished and we need some physical procedure to avoid this dependency.

As we know from perturbative quantum field theory the loop contribution in interaction theory results in divergent quantity such as the amplitude, in that case as it was discussed in 4.9.2 after regularization we borrow a procedure that makes the situation more reliable. This was actually the renormalization scheme, where we have to renormalize some physical quantities such as mass, charge or other coupling constants appearing in the Lagrangian. The bare quantities correspond to introducing counterterms into the Lagrangian that makes the condition free of divergent parts associated with the loop effect. In the case of the vacuum energy or vacuum expectation value of the physical quantities, the corresponding renormalizing quantity is considered as the background field. The eliminated divergent part after prescribing a proper counterterm for the problem is actually transferred to the background field [19]. The form of the counterterm can be completely arbitrary as long as it removes the regulator depending divergent part of the energy [38].

192

Conclusion

Now I present the conclusion of my work in this thesis. Particular emphasis will be on some conclusions, which by my knowledge have been considered for the first time. The sequence and materials discussed in the different chapters have been organised in such a way that lead the reader to the main subject of this project that is the Casimir effect.

The main topics are: the linear response theory based in quantum mechanics and thermodynamics. The KMS state as an alternative way of understanding the fluctuation-dissipation theorem and an effective method for formulating the many- body problem using the Green’s function. These discussions have been used in my new interpretation of the concept of vacuum field fluctuations.

The Casimir effect is actually an interaction between the field with a background external field or material boundaries, where some particular modes of the field fluctuations are restricted due to boundary conditions. These restrictions can be obtained by introducing the material body in the vacuum or due to periodic conditions in a space-time manifold with non-trivial topology.

We also studied the behaviour of the Casimir effect for different geometrical boundaries. We saw that actually a specific kind of field can have completely different effects for different shapes of the material bodies. For example we demonstrated that the Casimir effect for a cylinder is an attractive stress, whereas this field on a sphere has a repulsive stress effect. Therefore the dependency of the

Casimir effect on the geometry of the boundary was investigated and it was shown that this dependency is related to coefficient of heat-kernel expansion in a zeta

193 function regularization approach. But on the other hand the coefficients of the heat- kernel expansion are functions of geometrical properties such as curvature. Hence the strong dependency of the Casimir force on the geometrical configuration of the boundaries is understandable.

The Casimir effect is also a topological dependent phenomenon; this means that the vacuum fluctuation of the field in a space-time with a topology that is accompanied with a periodic condition for some modes of the field, like a circle in a one-dimensional space, may result in a definite value for the Casimir energy. This is actually the mechanism of the spontaneous compactification of the extra dimension in the Kaluza-Klein theory, which is the key point in the superstring theory for its additional dimensionality.

Summarizing the main results obtained in this thesis are: a new interpretation of vacuum fluctuation and the Casimir energy regularization.

In this research I realized that the vacuum fluctuation can indeed be interpreted or it is better to say; it can be deduced using a different point of view taken from linear-response theory. In this new perspective I tried to use the fluctuation-dissipation theorem for a field together with the uncertainty principle.

According to Heisenberg principle the energy of a system can not possess any definite value; like zero, in the case of the vacuum this non-zero energy of the system is called the zero-point energy that is interpreted by quantum mechanics. According to Plank’s second theory the zero-point energy is actually an intrinsic cosmological entity of the universe. Now from an advanced point of view in physics the structure of the space consists of field, therefore for any physical vacuum there exist an intrinsic natural energy (zero-point energy) dissipated in the field. This energy of the vacuum according to the fluctuation-dissipation theorem implies a fluctuation of the

194 field in the vacuum. This is a new - as long as I know - and completely equivalent perception of the vacuum field fluctuation with a classic prescription of the quantum field theoretical approach to this phenomenon.

My second contribution to the Casimir effect is actually introducing a new method for regularizing the Casimir energy for a massive scalar field in the case of two parallel planes with Dirichlet boundary conditions. In this approach I have used a very useful expansion of a sum in terms of the corresponding integrals, the value of the function itself and its consequent derivatives. This is called the Euler-Maclaurin summation formula. The integral in the expansion and its equivalent integral form of the function and its derivatives later are regularized by dimensional regularization scheme.

The main purpose in the development of this new model for the regularization can be explained as following: the usual calculation of the Casimir energy for a scalar field starts by a sum of the zero-point energy over different modes of the vacuum field fluctuations. Obviously this sum is divergent, that means the zero-point energy of the vacuum has an infinite value which is an expected result. By introducing two parallel plates (in general any kind of material bodies with different geometrical configuration) we disturb the vacuum and this disturbance will result in a Casimir energy that is a consequence of the limitation of the field modes in some directions. Now those distinct modes present a sum in the calculation which is again divergent. In the case of a massless scalar field we regularize this sum by a Riemann zeta function, but for a massive scalar field this divergent energy needs more extra condition for its regularization. In the discussion of the vacuum energy we have a similar situation of infinite energy. We will get rid off infinity by a zeta function regularization method, where an arbitrary regulator in the power of the eigen-

195 frequency is introduced, and the calculations will proceed by writing the zeta function in terms of the heat-kernel by a Mellin-like transform. Later the heat-kernel can be expanded and finally the vacuum energy is regularized in terms of the heat- kernel expansion.

By following a similar procedure for the massive scalar field we can regularize the Casimir energy for this field, but as in any regularization model we will end up in an expression whose divergent part depends on some coefficients of expansion. This is actually a complicated result for the rest of the calculation since we have to borrow an appropriate renormalization scheme to obtain a finite value for the energy and the renormalization in this case requires introducing some counterterms associated with the coefficient of expansions involved in divergent parts of the regularization.

In this new model instead of working on the corresponding sum appearing in the calculation of the energy and regularizing that by zeta function regularization, we employ the dimensional regularization that is a well-known procedure for regularizing the divergent integrals. The advantage of this model unlike the cutoff regularization method is that it is actually Lorentz invariant and the result of regularization has a very simple form of divergent part that makes the renormalization much easier.

The above approach regularizes the Casimir energy for the massive scalar field. A similar procedure may be used for other kind of fields and even also for other geometrical boundaries. By the way a final renormalization scheme is needed to obtain a finite value for those energies. But as it is clear there is no such a definite recipe for any situation where a renormalization operation is needed. This final step

196 must be worked out consciously by renormalizing physical quantities with an appropriate counterterm in the calculation.

204

Appendices

Appendix A: Bessel’s functions

The Bessel’s differential equation given by

d2 y dy x2+ x + x 2 −α 2 y = 0, (A.1) dx2 dx () is obtained when a solution to the Laplace’s equation

∆ϕ = 0, (A.2) in cylindrical or spherical coordinates is investigated. In this equation is called the order of the Bessel function, which can be a real or complex number. The order of the Bessel’s function when we are working in cylindrical coordinate system is an integer number, and this order for spherical coordinate is half-integer numbers.

The solution of the Bessel’s equation may be given by the Bessel’s function of the first kind or second kind or a complex combination of these solutions. The

Bessel’s function of first kind denoted by J(x) for integer order is defined by a

Taylor series expansion around x = 0 as

m 2m+α ∞ ()−1 ≈x ’ Jα () x = ƒ ∆ ÷ . (A.3) m=0 m!Γ() m +α + 1« 2 ◊

For integer order Bessel’s equation the solutions of first kind has the following property

n J−n()()() x= −1 J n x . (A.4)

Therefore these functions are not linearly independent and the other solution of this equation linearly independent to the Bessel’s function of first kind is called Bessel’s

205 function of second kind denoted by Y(x). For non-integer order Bessel’s equation this function of second kind is given by

J( x)cos(απ ) − J( x) Y() x = α− α . (A.5) α sin ()απ

This function for integer value of is obtained as a limit from non-integer to integer order

Yα( x) = lim Y α ( x) . (A.6) α →}

The Bessel’s function of first kind and second kind are defined by an integral form, where these definitions help us to derive various properties of these functions.

The integral forms of functions of the first kind and the second kind are

1 π −i() nτ − xsin τ Jn () x= e dτ , (A.7) 2π —−π and

1π 1 ∞ α Y() x=sin() x sinθ − αθ d θ −» eαt +() − 1 e− α t ÿ e − xsinh t dt , (A.8) α π—0 π —0 ⁄ respectively.

As it was mentioned before a complex combination of the Bessel’s first and second functions can be named as another solution to the equation. Two linearly independent solutions in this case which is Hankel functions are given by

H()1 ( x) = J( x) + iY( x) α α α (A.9) ()2 Hα()()() x= J α x − iY α x .

Two linearly independent Hankel’s function can be written as following

−απi 1 J( x) − e J( x) H() () x = −α α , α isin ()απ (A.10) απi 2 J()() x− e J x H() () x = −α α . α −isin ()απ

206 For a complex argument of the Bessel’s functions we define the modified Bessel’s function of the first and second kind denoted by I(x) and K(x) respectively given by

−α Iα( x) = i J α ( ix) π I()() x− I x (A.11) K() x = −α α . α 2 sin ()απ

The above functions are linearly independent solution of the Bessel’s equation in the following form

d2 y dy x2+ x − x 2 +α 2 y = 0. (A.12) dx2 dx ()

Another form of the Bessel’s equation may be written by

d2 y dy x2+2 x +» x 2 − n() n + 1 ÿ y = 0. (A.13) dx2 dx ⁄

In this case the two linearly independent solutions of this equation are called the spherical Bessel’s function denoted by jn and yn. These two linearly independent functions are related to the ordinary Bessel’s function by the following relationship.

π j() x= J() x n2x n+1/ 2 (A.14) π y() x= Y() x . n2x n+1/ 2

There are similarly spherical Hankel functions defined by

h()1 ( x) = j( x) + i y( x) n n n (A.15) ()2 hn= j n()() x − i y n x .

207

Appendix B: Heat Kernel Expansion

In this appendix we study the heat kernel associated with an elliptic second- order partial differential operator of the Laplace type acting on a smooth section of a vector bundle over a Riemannian manifold. We also investigate the diagonal values of the heat kernel coefficients; these are called HMDS (Hadamard–

Minackshisundaram–De Witt–Seeley) coefficients.

We start by definition of a Laplace type differential operator F as

FQ= −I + . (B.1)

In this equation is the generalized Laplacian defined by

TMVV∗ ⊗ I=trg ∇ ∇ , (B.2) where the derivative is a connection, or covariant derivative. The first is the Levi-

Civita connection, which is a covariant derivative on the tensor product of a cotangent bundleTM∗ of the manifold and a vector bundle V, and the second connection is a covariant derivative on the vector bundle V over manifold M. Also Q in the equation (B.1) is a smooth Hermitian section of the endomorphism bundle.

Now consider the well defined operator U( t) =exp( − t F ) for t > 0, which is a bounded operator on the Hilbert space of the vector bundle V. The kernel of this operator is defined as following

U( t x, x′′) = exp( − t F)δ ( x , x ) , (B.3) where δ ( x, x′) is the Dirac function along the diagonal of MM× , it can be regarded as an endomorphism from the fiber of V over x’ to the fiber V over x [9].

The heat kernel is indeed the solution of the heat equation given by

208

(∂t +F) U( t x, x′) = 0, (B.4) with initial condition

U(0+ x , x′′) =δ ( x , x ) . (B.5)

For the purpose of computing of the asymptotic expansion of the heat kernel, let consider the heat kernel diagonally in the product of the manifold by itself MM× . This makes a concentration on the heat kernel locally, which is the aim of the asymptotic expansion. We define some useful mathematical quantities that will be used later in our computation. First we define the world function (x,x’) as the geodetic interval that is the half the square of the length of the geodesic connecting the point x and x’ given by

1 σ ()x,,. x′′= r2 () x x (B.6) 2

Next the Van Vleck-Morette determinant is defined by determinant of the mixed second derivatives of the world function

−1/ 2 −1/ 2 ∆( x, x′′′′) = g( x) det( −∇µ ∇ ν ′σ ( x , x)) g( x ) . (B.7)

It satisfies

−1/ 2 1/ 2 1 ∆D ∆ =2 ( d −Iσ ), (B.8) where

µ D ≡σ ∇µ (B.9)

With the boundary condition

[∆] =1. (B.10)

The square bracket means the coincidence limit x→ x′ is taken [8]. And the operator P(x,x’) is defined as a parallel transport operator along the geodesic from x to the point x’. This is again an endomorphism from fiber V over x’ to the fiber of V

209 over x. Now consider the following equation which consists of all the above functions

−d / 2 ≈1 ’ Utxx0 (),′′′′=()() 4π t ∆ xx , exp∆ − σ ()() xxPxx , ÷ , . (B.11) «2t ◊

It can be shown that this function satisfies the equation (B.5). The above equation is the heat kernel when our manifold is flat and the bundle has a curvature of zero with a vanishing endomorphism. This is called the free case of the heat equation. To work out in a generalized situation we have to introduce an auxiliary function that is called transport functionΩ(t x, x′) . The transport function satisfies a transport equation given by [8, 9]

≈∂ 1 −1 − 1/ 2 1/ 2 ’ ∆+D + P ∆ F ∆ P ÷ Ω() t = 0, (B.12) «∂t t ◊

with the initial condition

Ω0x , x′ = I , (B.13) ( ) x= x′ where I is an identity endomorphism. Now the generalized form of the equation

(B.11) will be

−d / 2 ≈1 ’ Utxx(),′′′′′=()() 4π t ∆ xx , exp∆ − σ ()() xxPxx , ÷ , Ω() txx , . (B.14) «2t ◊

Now let consider a Mellin transformation of Ω(t) ,

∞ 1 −q −1 bq = dt t Ω() t . (B.15) Γ() −q —0

The inversion of the above transformation provides us an analytic integral expressing the transport function as

210

c i 1 + ∞ q Ω()t = dqt Γ() − q bq (B.16) 2πi —c− i ∞ where c is a negative constant. By deforming the integral contour to the right and considering the simple poles in non-negative integer values of the q due to presence of the gamma function we arrive at

k N −1 ()−t Ω()t =ƒ bk + R N () t , (B.17) k =0 k! where

1 cN + i ∞ R t dqtq q b (B.18) N () =— Γ() − q 2πi cN − i ∞

N with N− 1 < cN < N . At t → 0 , RN () t is of order O( t ) that is smaller than last term of the sum in this limit. Therefore the asymptotic expansion of Ω(t) is obtained as

k ∞ ()−t Ω()t x,. x′ - ƒ bk (B.19) k =0 k!

Next we give the formula for the trace of heat kernel by

Tr2 exp−tF = dvolx tr Utxx , . (B.20) L ( ) — ( ) V ( ) M

Using the above formula and (B.14) plus formula (B.19) and defining the following

B=Tr2 b = d vol x tr b x , x , (B.21) qL q— ( ) V q ( ) M we will arrive at Minackshisundaram-Pleijel or Schwinger-De Witt asymptotic expansion as t → 0

k ∞ −d / 2 ()−1 Tr2 exp−tF- 4π t B . (B.22) L ()() ƒ k k =0 k!

The Bk are called Hadamard–Minackshisundaram–De Witt–Seeley (HMDS) coefficients. On the other hand the HMDS coefficients bk satisfies the functional equation,

211

≈1 ’ −1 − 1/ 2 1/ 2 ∆1+D ÷ bq = P ∆ F ∆ Pb q−1 . (B.23) «q ◊

This is a result of substituting (B.16) into (B.12), with the limit q → 0

Db0 = 0, (B.24) in which satisfies the initial condition

b x,. x′ = I (B.25) q ( ) x= x′, q = 0

According to (B.17) and (B.18) the equation (B.23) for non-negative values of q will reduces to De Witt recursion relation [27, 28]. This equation is very helpful since it determines all the HMDS coefficients bk.

In order to calculate the HMDS coefficients from functional equation (B.23), in coincidence limit; x close to x’, the solution for this equation at integer values of q = k = 0, 1, 2, … can be written as

−1 − 1 ≈1 ’≈ 1 ’ −1 bk = P∆1 + D ÷∆ M 1 + D ÷ M ...() 1 + D M , (B.26) «k ◊« k −1 ◊

The zero order of this coefficient is given by

b0 ( x,,. x′′) = P( x x ) (B.27)

In this equation M is the operator in the right-hand side of (B.23)

MPFP=−1 ∆ − 1/ 2 ∆ 1/ 2 . (B.28)

The Taylor series of the coefficients may be written as

bk= Pƒ n n b k . (B.29) n≥0

−1 1 Similarly for the inverse operators such as (1+ k D) this expansion is given by

−1 − 1 ≈1 ’ ≈’n ∆1+D ÷ =ƒ ∆÷ 1 + n n , (B.30) «k ◊n≥0 «◊ k where we have the contribution of eigenvalue for this operator in the right-hand side.

212 Now from the above expansion for the corresponding inverse operators; the

scalar product n bk of (B.29) and using operator solution (B.26) we will get

−1 −1 ≈n ’ ≈nk −1 ’ −1 n bk =ƒ ∆1 + ÷ ∆ 1 + ÷ ...() 1 + n1 n1,..., nk− 1≥ 0 «k ◊ « k −1 ◊ (B.31)

× n M nk−1 n k − 1 M n k − 2... n 1 M 0 .

On the other hand the vectors of the vector space and its dual space are given by

n ()−1 ′ ′ n =µ′′... µ = σµ1 ... σ µn . (B.32) 1 n n!

m=µ′′′... µ = D D δ x , x . (B.33) 1m (µ1′′ ... µm ) ( )

Also the matrix elements of M can be written as

»n ÿ ()−1 ′ ′ m M n=µ... µ M ν ... ν =… ∇ ... ∇ M σν1 ... σ ν n Ÿ . (B.34) 1m 1 n (µ1 µm ) …n! ⁄Ÿ

Therefore we obtain an expression for the computing the matrix element of the operator M

≈m ’ ν1... ν n µ1... µmMZ ν 1 ... ν n= ∆ ÷ δ (µ ... µ m ... µ ) «n ◊ 1n n+ 1 m (B.35) ≈’m ≈’ m (ν1 ... νn− 1 ν n )ˆ ( ν1 ... νn− 2 ν n − 1 ν n ) +∆÷δ(...µ µYX µ ...) µ − ∆÷1 δ (... µ µ µ ...) µ , «◊n−11n− 1 n m «◊ n − 2 1 n− 2 n − 1 m where XXµν=» ∇... ∇ µ′′ ν ÿ , µ1... µn ( µ 1 µ n ) ⁄ YYν=» ∇... ∇ ν ′ ÿ , (B.36) µ1...() µn µ 1 µ n ⁄ ZZ=» ∇... ∇ ÿ . µ1... µn ( µ 1 µ n ) ⁄

In the above equations we have

213

µ′′′′ ν µ ν α X =ηα η µ′′′′ˆ ν µ′ µ ν YDXXA=1ν′′ + 2 ν (B.37) Z= Xµ′′ ν 1ˆζ ζ − A A − D Xµ′′ ν 1 ˆ ζ + A + P−1 QP ()()µ′′′′′′′ ν µ ν ν{} µ µ −1/ 2 1/ 2 ζµ′′′=DD µ ζ = ∆m ∆ .

For the purpose of completing the task, we have to introduce some definitions for particular terms present in the above equations. These parameters are

µ′′ µ ην= ∇ ν σ , (B.38) and ν −1 APPµ′′=γ µ ∇ ν , (B.39)

and also

À ()−1 n Œζ= ƒ ζ , Œ n≥2 n! ()n Ã . (B.40) 1n ! ≈ ’ Œζ= ƒ ƒ tr∆ γ ... γ ÷ Œ n n n () 1≤k ≤[] n / 22k n1 ... nk ≥ 2 n1 !... n 2 ! «()1 ()k ◊ Õ n1 +... + nk = n

Thus the values of the terms in (B.37) can be determined, which helps us to evaluate the matrix elements given in (B.35) that gives the scalar product (B.31), where this product results to evaluation of the coefficients of heat kernel expansion (B.29).

The values of the first three coefficients of the heat-kernel expansion by using above procedure are tabulate in many references such as [8, 14]. They are

[a1 ] = P 2 1 []a2 = P − 3 Z()2 (B.41) a P3 1 P, Z 1 P 1 Jµ P 1 J µ 1 Z . []3 = −2{}()2 − 2() ∇µ − 3 µ () ∇ + 3 + 10 ()4

These are value of coefficients in coincidence limit; where x is close to x’. Finally in these equations the terms are given as following:

214

α J µ≡ ∇ αR µ 1 ˆ PZQR≡[] = − 6 1 ZZQRRRRR»µ ÿ 1 µνˆ 1 µν 1 µναβ 1 ()2 ≡ µ ⁄ =I −2R µν R +1{} 30 µν − 30 µναβ − 5I ZZQQJQ≡»µ1 µ 2 ÿ =I2 −1»R,,,» R µν ÿ» ÿ − 2 µ ∇ ÿ ()4 µ1 µ 2 ⁄ 2 µν ⁄ ⁄ 3 µ ⁄ +2RQRQJJJµν ∇ ∇ + 1 ∇ µ ∇ −2R , ∇ µ ν − 8 µ 3µ ν 3 µ{} µν9 µ −4 ∇RR ∇µ αβ −6 RRR ν γµ −10 R αβ RR µ + Rµναβ R R 3 µαβ µν γ3 α µβ µν αβ ˆ 3 2 1µν 2 µν 4 α β µν +1{ −14IIRRRRRRR − 7 ∇µ ∇ ν + 21 µν − 7 µ ν ∇ α ∇ β 4µ 1 µαβ 1 αβµ3 µαβγδ −∇∇+∇∇36µRRRRRRRR 42 µαβ +∇∇ 21 µαβ −∇ 28 µαβγδ ∇ 2αβγ 2 µναβ 2 α βµνλ −189RRRRRRRRRβγα + 63 αβ µν − 9 αβµνλ

16µν σρ αβ 88 αβ µν νρ +189RRRRRRαβ µν σρ + 189 µνσρα β }.

(B.42)

197

References

[1] lecture 1: Linear response theory. In: http://ocw.mit.edu/NR/rdonlyres/Physics/8-512Spring2004/27759392-B28D- 4CD9-BF15-D093AE8CEB8E/0/lecture_1_rev2.pdf.,

[2] Linear Response Theory In: http://webusers.physics.uiuc.edu/~efradkin/phys582/LRT.pdf.,

[3] Abrikosov A A, Gorkov L P and Dzyaloshinski I E 1963 Methods of quantum field theory in statistical physics (Englewood Cliffs, N.J. New York: Prentice- Hall Dover Publications)

[4] Ahmadi N and Nouri-Zonoz M 2006 Topology, mass and Casimir energy Nuclear Physics B 738 269-82

[5] Allen P B 1995 Linear Response Theory and Kubo Formulas. In: Transport in Solids, (Lausanne

[6] Ambjorn j and Wolfram S 1983 Properties of the vacuum. I. Mechanical and thermodynamic Annals of physics 147 1-32

[7] Ashby N 1961 Martin-Schwinger methods in the many-body problem. (Cambridge: Harvard University)

[8] Avramidi I G 1991 A covariant technique for the calculation of the one-loop effective action Nuclear physics B 355 712-54

[9] Avramidi I G 1999 Covariant techniques for computation of the heat kernel Rev.Math.Phys. 11 947-80

[10] Avramidi I G and Esposito G 1999 Heat kernel asymptotics of Gilkey-Smith boundary-value problem AMS/IP Stud.Adv.Math. 13 15-34

[11] Basagoiti M A V and Moreta J G A new approach to non-renormalizable perturbation theory. In: arxiv.org/pdf/math/0402259v3,

[12] Berestetskii V B, Lifshitz E M and Pitaevskii L P 1982 Quantum electrodynamics (New York: Pergamon Press)

198 [13] Bernasconi F, Graf G M and Hasler D 2003 The heat kernel expansion for the electromagnetic field in a cavity. In: arXiv:math-ph/0302035v1,

[14] Blau S K, Visser M and Wipf A 1988 Zeta functions and the Casimir energy Nuclear physics B 310 163 - 80

[15] Bogdanoff I 2002 The KMS state of spacetime at the Plank scale Chinese journal of physics 40 149-58

[16] Bogolubov N N and N.N. Bogolubov J 1982 Introduction to quantum statistical mechanics (Singapore: World Scientific)

[17] Bordag M, Elizalde E and Kirsten K 1996 Heat kernel coefficients of the Laplace operator on the D-dimensional ball J.Math.Phys. 37 895-916

[18] Bordag M, Elizalde E, Kirsten K and Leseduarte S 1997 Casimir energies for massive fields in the bag Phys.Rev.D 56 4896-904

[19] Bordag M, Mohideen U and Mostepanenko V M 2006 New developments in the Casimir effect. In: Elsevier preprint,

[20] Brouwer P 2005 Theory of Many-Particle Systems. ed Lecture notes for P654: Cornell University)

[21] Casimir H B G 1948 On the attraction between two perfectly conducting plates Proc. Kon. Nederland. Akad. Wetensch B51

[22] Cognola G, Elizalde E and Zerbini S 2006 Heat-kernel expansion on non compact domains and a generalized zeta-function regularization procedure Journal of Mathematical Physics 47

[23] Cognola G, Vanzo L and Zerbini S 1990 A new algorithm for asymptotic heat kernel expansion for manifolds with boundary Physics Letters B 241 381-6

[24] Deitmar A 2003 Panorama of zeta functions Mathematical Works.

[25] Delamotte B 2004 A hint of renormalization Am.J.Phys. 72 170 -84

[26] DeRaad L L and Milton K A 1981 Casimir self-stress on a perfectly conducting cylindrical shell Annals of physics 136 229-42

199 [27] DeWitt B S 1966 Dynamical Theory of Groups and Fields American Journal of Physics 34 1209

[28] DeWitt B S and Stora R 1984 Infinite-dimensional Lie Groups in ”Relativity, Groups and Topology II” Les Houches

[29] Fermions I, Rocca F, Sirugue M and Testard D 1969 On a class of equilibrium states under the Kubo-Martin-Schwinger boundary condition Communications in Mathematical Physics 13 317-34

[30] Genet C, Intravaia F, Lambrecht A and Reynaud S 2004 Electromagnetic vacuum fluctuations, Casimir and Van der Waals forces Annales de la Fondation Louis de Broglie 29 311-28

[31] Gosdzinsky P and Romeo A 1998 Energy of the vacuum with a perfectly conducting and infinite cylindrical surface Physics Letters B 441 265-74

[32] Hawking S W 1977 Zeta function regularization of path integrals in curved spacetime Communications in mathematical physics 55 133 - 48

[33] Hertel P 2001 Lecture on theoretical physics - Linear response theory. University of Osnabruck)

[34] Huang W-H 2001 Casimir effect on the radius stabilization of the noncommutative torus Physics Letters B 497 317-22

[35] Khusnutdinov N R 2006 zeta-function approach to Casimir energy with singular potentials Phys.Rev. D 73

[36] Kirsten K and Elizalde E 1996 Casimir energy of a massive field un a genus-1 surface Physics Letters B 72

[37] Lagarias J C 1999 Number theory Zeta functions and dynamical Zeta functions Contemporary Math. 237 45-86

[38] Lahiri A and Pal P B 2001 A first book of quantum field theory (Boca Raton: CRC Press/Narosa Pub. House)

[39] Landau L D and Lifshitz E M 1980 Statistical physics vol 2 (New York: Pergamon Press)

200 [40] Lukosz W 1971 Electromagnetic Zero-Point Energy and Radiation Pressure for a Rectangular Cavity Physica 56 109-20

[41] Mandl F 1959 Introduction to quantum field theory (New York: Interscience Publishers)

[42] Martin P C and Schwinger J 1959 Theory of Many-particle system. I Physical review 115 1342-73

[43] McAvity D M and Osborn H 1991 Asymptotic expansion of the heat kernel for generalized boundary conditions Class. Quantum Grav 8 1445-54

[44] McAvity D M and Osborn H 1991 A DeWitt expansion of the heat kernel for manifold with a boundary Class. Quantum Grav 8 603-38

[45] McComb W D 2004 Renormalization methods : a guide for beginners (Oxford: Oxford University Press)

[46] Milton K A 2001 The Casimir effect : physical manifestations of zero-point energy (New Jersey: World Scientific)

[47] Milton K A, Cavero-Pelaez I and Kirsten K 2006 Local and global Casimir energies in a Green's function approach. In: arXiv:hep-th/0612299v1,

[48] Milton K A, DeRaad L L and Schwinger J 1978 Casimir self-stress on a perfectly conducting spherical shell Annals of physics 115 388-403

[49] Nguyen T T 2003 Casimir effect and vacuum fluctuations. In: www.hep.caltech.edu/~phys199/lectures/lect5_6_cas.pdf Department of Physics and Astronomy, Ohio University)

[50] Ozcan M 2006 Casimir energy density for spherical universes in n-dimensional spacetime Classical and Quantum Gravity 23 5531-46

[51] Pinto A C A, Britto T M, Bunchaft R, Pascoal F and Rosa F S S d 2003 Casimir effect for a massive scalar field under mixed boundary conditions Brazilian journal of physics 33 860-6

[52] Plunien G, Muller B and Greiner W 1986 The Casimir Effect Physics reports 134 87—193

201 [53] Srednicki M 2007 Quantum field theory (Cambridge: Cambridge University Press)

[54] Vassilevich D V 2003 Heat kernel expansion: User's manual Physics Reports 388 279-360

[55] Wilk S F J, Fujiwara Y and Osborn T A 1981 N-body Green's functions and their semiclassical expansion Physical Review A 24 2187 - 202

[56] Zywina D 2005 The zeta function of a graph. In: http://math.berkeley.edu/~zywina/zeta.pdf.,

202

Bibliography

[57] Kaluza-Klein theory. In: http://en.wikipedia.org/wiki/Kaluza-Klein_theory: Wikipedia)

[58] Arnold V I 1978 Mathematical methods of classical mechanics (New York: Springer-Verlag)

[59] Avramidi I G 2004 Heat kernel asymptotics of Zaremba boundary value problem Mathematical Physics, Analysis and Geometry 7 9-46

[60] Bar C 1998 Heat operator and zeta function estimates for surfaces Arch. Math. (Basel) 71 63-70

[61] Birrell N D and Davies P C W 1982 Quantum fields in curved space (Cambridge: Cambridge University Press)

[62] Branson T P, Gilkey P B, Kirsten K and Vassilevich D V 1999 Heat kernel asymptotics with mixed boundary conditions Nuclear physics B 563 603-26

[63] Goldstein H, Poole C and Safko J 2002 Classical mechanics (San Francisco: Addison Wesley)

[64] Graham N, Jaffe R L and Weigel H 2002 Casimir effects in renormalizable quantum field theories International Journal of Modern Physics A 17 846-69

[65] Greiner W 1993-1994 Quantum mechanics (Berlin New York: Springer- Verlag)

[66] Isham C J 1989 Modern differential geometry for physicists (Singapore Teaneck, N.J: World Scientific)

[67] Kastler D and Takesaki M 1979 Group duality and the Kubo-Martin-Schwinger condition Communications in mathematical physics 70 193-212

[68] Kremp D, Schlanges M and Bornath T 1985 Nonequilibrium real time Green's functions and the condition of weakening of initial correlation Journal of Statistical Physics 41 661-70

[69] Landau L D and Lifshitz E M 1977 Quantum mechanics : non relativistic theory (Oxford Pergamon Press)

[70] Landau L D and Lifshitz E M 1980 Statistical physics vol 1 (New York: Pergamon Press)

[71] Lang S 1999 Complex analysis (New York: Springer)

203 [72] Mendes V S and Prado T G 2005 Casimir effect for diferential forms in real compact hyperbolic spaces Brazilian journal of physics 35 1104-5

[73] Osborne A D 1999 Complex variables and their applications (England: Addison Wesley Longman)

[74] Parisi G 1988 Statistical field theory (Redwood City,: Addison-Wesley )

[75] Salcedo L L 2004 Covariant derivative expansion of the heat kernel The European Physical Journal C - Particles and Fields 37 511-23

[76] Schiff L I 1968 Quantum mechanics (New York: McGraw-Hill)

[77] Schwinger J 1992 Casimir energy for dielectrics: Spherical geometry Proc Natl Acad Sci U S A. 89 11118–20

[78] Struik D J 1961 Lectures on classical differential geometry (Reading: Addison- Wesley)

[79] Thirring W 1997 Classical mathematical physics : dynamical systems and field theories (New York: Springer)

[80] Tierz M and Elizalde E 2001 Heat kernel-zeta function relationship coming from the classical moment problem. In: arXiv:math-ph/0112050v1,

[81] Wesson P S Zero-point fields, gravitation and new physics. (Waterloo: Department of physics, University of Waterloo) p 21

[82] Young A 2003 Eigenvalues and the heat kernel.