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Generation of matter from stimulated vacuum fluctuations Desrochers, Michael J. 2017

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Generation of Matter from StimulatedVacuum FluctuationsbyMichael J. DesrochersB.Sc., Simon Fraser University, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Michael J. Desrochers 2017AbstractThrough the connection between quantum mechanical vacuum effects and atime dependent deformation of space-time, an avenue to excite the quantumfields that lay on top of space-time is possible. It is through this approach,energy can be given to a spatially compact region of space-time, throughthe deformation of its boundary, to generate matter that is screened by saidboundary. While this particle genesis is described generally, the focus will bewithin 1+1-dimensions for the photon field and a natural photon-screeningboundary, a conductor. This work explores the simplest components of thisproceedure, starting from the dynamics of an isolated piece of a conductordrifting in the vacuum, which itself does not exhibit the Casimir effect. Thenit will be shown that by stirring the vacuum through a time-dependent de-formation of a conducting boundary around a spatially compact space-time,photons will be excited. This deformation will be via directly moving one ofthe boundaries or by changing the reflectivity as a function of time.iiLay SummaryThis manuscript examines the way the vacuum can interact with conductors,which are objects that can block electromagnetic radiation. The idea isthat one can stir the vacuum with a conductor and pop electromagneticradiation, which are photons, into existence. In other words, make nothingglow. The number of photons, as well as the energy density is determined inthe case of two parallel plate conductors. In addition, the idea of stirringthe vacuum without moving the mirror is introduced. This is accomplishedby changing the amount the electromagnetic sector of the vacuum that isallowed into the surrounding conductors, which is called the skin depth, asa function of time. This plays the same role as moving the conductor, but ismore attainable in a laboratory environment. This phenomena has not beenobserved in an experiment, so the theoretical ground work presented hereshould guide future experiments.iiiPrefaceThis thesis is a new look at the field of vacuum fluctations and the motionalCasimir effect. The author, M.J. Desrochers, has done independent calcu-lations and analysis that build on previous authors. Any credit to previouswork is described and cited in detail throughout the manuscript.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 The Casimir Effect: an Overview . . . . . . . . . . . . . . . . 11.1 The Static Casimir Effect . . . . . . . . . . . . . . . . . . . . 21.2 Stress-Energy Tensor . . . . . . . . . . . . . . . . . . . . . . 51.3 The Nature of the Divergent aspect of the Vacuum Expecta-tion Value of the Stress-Energy Tensor . . . . . . . . . . . . 61.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Single Mirror in 1+1 Dimensions . . . . . . . . . . . . . . . . 82.1 Quantum Langevin Equation for a Single Mirror UndergoingVacuum Fluctuation . . . . . . . . . . . . . . . . . . . . . . . 92.2 Motional Forces Enacted on a Single Conductor . . . . . . . 102.2.1 Derivation of the S-Matrix for Scattering Photons . . 112.2.2 Linear Response . . . . . . . . . . . . . . . . . . . . . 132.3 An analogy to the Lorentz-Abraham Problem in ClassicalElectrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Scattering Properties . . . . . . . . . . . . . . . . . . . . . . 152.5 A free Conductor undergoing Vacuum Fluctuations . . . . . 182.5.1 A Realistic Conductor Undergoing Vacuum Fluctua-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 Precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22v2.6.1 Resolution of Acausal Behavior . . . . . . . . . . . . . 222.6.2 Acausal effects on a realistic conductor . . . . . . . . 222.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 The Motional Casimir Effect in 1+1 Dimensions . . . . . . 263.1 Introduction - The Definition of a Particle . . . . . . . . . . 273.2 Motional Casimir Effect and the Energy Density . . . . . . . 273.2.1 The Fulling-Davies-Moore Approach . . . . . . . . . . 283.2.2 Dynamics of the Vacuum due to a Sinusoidally MovingConductor . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Determination of the Photon Number . . . . . . . . . . . . . 333.3.1 The Number Density . . . . . . . . . . . . . . . . . . 343.3.2 Photon Mode Functions . . . . . . . . . . . . . . . . . 343.3.3 The Intermediate Mode Function . . . . . . . . . . . 353.4 Method 1: Krylov-Mitropolski-Bogoliubov . . . . . . . . . . 363.4.1 The Modification to the KBM Method . . . . . . . . 373.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.3 Result: Number of Photons Generated . . . . . . . . 393.4.4 Problems with the MKBM Approach . . . . . . . . . 413.5 Method 2: WKB . . . . . . . . . . . . . . . . . . . . . . . . . 413.5.1 Number Operator . . . . . . . . . . . . . . . . . . . . 453.5.2 Photon Number Results . . . . . . . . . . . . . . . . . 463.5.3 Results: Energy Density . . . . . . . . . . . . . . . . 513.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6.1 The Meaning of Spatial In-homogeneity in the EnergyDensity . . . . . . . . . . . . . . . . . . . . . . . . . 533.6.2 Detection of Particles - The Number generated andwhere they will be . . . . . . . . . . . . . . . . . . . . 554 Imperfect Conductors, Scattering and Time-Dependent Re-flectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.1 Statically Translucent Casimir Effect . . . . . . . . . . . . . 564.1.1 Scattering Formalism . . . . . . . . . . . . . . . . . . 574.1.2 External Pressure Calculation . . . . . . . . . . . . . 584.1.3 Cavity Pressure and The Static Casimir Effect . . . . 594.1.4 General Reflectivity . . . . . . . . . . . . . . . . . . . 614.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 624.2 Time Dependent Reflectivity . . . . . . . . . . . . . . . . . . 634.3 Specific Form of the Time-Dependence . . . . . . . . . . . . 684.3.1 Modelling Time-Dependent Reflectivity . . . . . . . . 68vi4.3.2 Choosing a Form for the Reflectivity . . . . . . . . . . 694.4 Casimir Effect with One-Mirror Time-Dependent Reflectivity 714.4.1 Form of the Casimir Force . . . . . . . . . . . . . . . 714.4.2 Adiabatic Result . . . . . . . . . . . . . . . . . . . . . 724.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 734.5 Casimir Effect with Two-Mirror Time-Dependent Reflectivity 744.5.1 Form of the Two-Mirror . . . . . . . . . . . . . . . . . 744.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1 Discussion of the Results . . . . . . . . . . . . . . . . . . . . 775.2 Experimental Observation of the Motional Casimir Effect . . 785.2.1 Fabry-Perot Measurements . . . . . . . . . . . . . . . 785.2.2 Measurement of the DCE through Time-DependentReflectivity . . . . . . . . . . . . . . . . . . . . . . . . 785.3 Higher-Dimensional Considerations . . . . . . . . . . . . . . 795.3.1 Path-Integral Method . . . . . . . . . . . . . . . . . . 795.3.2 Time-Dependent Surfaces . . . . . . . . . . . . . . . . 795.4 Space-time Horizons, Hawking Radiation, Cosmological Con-stant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.5 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Appendix A Calculation of the Stress Energy Tensor in 1+1Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.1 Determination of 〈0 |Tµν | 0〉 in Minkowski Space . . . . . . . 86A.2 Determination of 〈0 |Tµν | 0〉 inside the Cavity . . . . . . . . . 88Appendix B Linear Response for the Force Acting on a SingleMirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Appendix C Solutions to Moore's Equation . . . . . . . . . . 91viiAppendix D The Modified KMB Method . . . . . . . . . . . . 99D.1 Averaged Coefficients . . . . . . . . . . . . . . . . . . . . . . 100D.2 Other useful Terms . . . . . . . . . . . . . . . . . . . . . . . 102D.3 Calculation of the Differential Equation for the Time Depen-dent Bogoliubov Coefficients . . . . . . . . . . . . . . . . . . 102Appendix E Numerical Laplace Transform Inversion . . . . . 116E.1 Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . 122E.1.1 Definitions and Useful Integrals . . . . . . . . . . . . 122E.1.2 Further Simplifications . . . . . . . . . . . . . . . . . 123E.2 Numerical Laplace Transform Inversion . . . . . . . . . . . . 124E.3 The Rest of the Bogoliubov Coefficients . . . . . . . . . . . . 124Appendix F Solutions to the WKB method of solving the mov-ing mirror problem . . . . . . . . . . . . . . . . . . . . . . . . . 125F.1 First Order Term . . . . . . . . . . . . . . . . . . . . . . . . 125F.2 Derivation of the Mode Function . . . . . . . . . . . . . . . . 129F.2.1 m = p Case . . . . . . . . . . . . . . . . . . . . . . . . 129F.2.2 m 6= p Case . . . . . . . . . . . . . . . . . . . . . . . . 130Appendix G Reflective Two-Mirror System . . . . . . . . . . . 131G.1 Energy Density on the Left and Right sides of the Two-MirrorPartially Reflective System . . . . . . . . . . . . . . . . . . . 131G.2 Energy Density in the Cavity of the Two-Mirror Partially Re-flective System . . . . . . . . . . . . . . . . . . . . . . . . . . 132Appendix H Path-Integral Methods in Higher Dimensions . 133H.1 Formulation of the Path Integral Method . . . . . . . . . . . 133H.1.1 Generating Functional and the Path Integral Measure 134H.1.2 Bounding Surfaces . . . . . . . . . . . . . . . . . . . . 135H.2 Generating functional for two conductors . . . . . . . . . . . 136viiiList of Tables3.1 The first few γk that solve equation (3.5) and encode the rightmost mirror's world-line into the definition of 〈0 |Tµν | 0〉. . . . 303.2 Expressions which appear in the equation of motion for Cmn (t),(3.14), are written in an expansion of the small parameter  . 423.3 The particular solution that satisfies the equation of motion(3.31). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 The Full Solution that solves the Cauchy Problem (3.33) withthe differential equation (3.31) . . . . . . . . . . . . . . . . . . 443.5 The required Bogoliubov coefficients for the t > T solution tothe Klein Gordon equation (3.12), evaluated precisely at thepoint when the mirror motion stops at t = T . Note that in theexpression for the number operator (3.9), the ξmn coefficientsare not required. . . . . . . . . . . . . . . . . . . . . . . . . . 46D.1 Terms that will appear in the calculation of the Bogoliubov co-efficients used in the Modified Krylov-Mitropolski-Bogoliubovmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99D.2 The components of the decoupled equation of motion for the Bo-goliubov Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . 113ixList of Figures1.1 A schematic of what is known as the Static Casimir effect.The Casimir force fcasimir, expressed in equation (1.2), is de-pendent on the parallel-plate conductor separation L0. . . . 32.1 This is a single conductor undergoing scattering from bothsides by the vacuum modes of a photon field, which is repre-sented as a scalar field for simplicity. . . . . . . . . . . . . . . 112.2 In order for there to be causal response from some output ef-fect by some input stimulation, it is required that the responsefunction be entirely analytic in the upper half imaginary plane. 172.3 The difference between the first order (in O (Ω−n)) and thezeroth order solutions, z1 (t) and z1 (t) respectively. There isnot much feature to note other than the size of the correctionbeing smaller than the Planck scale. This is for a silicon con-ductor of mass m = 2.492× 10−6kg, undergoing an initial ve-locity of v0 = 0.2µms and initial acceleration of a0 = 0.01µms2 ,which mirrors conditions found in experiments such as thatdone in [1]. What is breaking the right/left symmetry is theinitial velocity v0 and initial acceleration a0, which is movingforward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 The difference between the second order (in O (Ω−n)) and thezeroth order solutions, z2 (t) and z0 (t) respectively. There issome oscillation structure visible at length scales deep in thePlanck scale, but would be unlikely or impossible to detect.This is for a silicon conductor of mass m = 2.492 × 10−6kg,undergoing an initial velocity of v0 = 0.2µms and initial ac-celeration of a0 = 0.01µms2 , which mirrors conditions found inexperiments such as that done in [1]. . . . . . . . . . . . . . . 21x2.5 The acausal behavior of a conductor undergoing motionalforces from the vacuum. The mass of the conductor in ques-tion is m0 = 2.49 × 10−6kg, which is the mass of the siliconobjects used to measure the Casimir effect in a table top ex-periment [1] and the magnitude of the force is f0 = 10−7N. . . 243.1 The conformal map introduced by Moore and used by Fulling,Davies and Moore to describe the dynamic Casimir effect. Theconformal symmetry of the wave equation in 1+1-dimensionsessentially permits us to map the dynamic Casimir effect tothe static case. . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 〈0 |Tµν | 0〉 inside the cavity with the right-most wall oscillatingat 1 Hz with an amplitude 10% that of the cavity length. Theaverage separation length is a micron. (a) The energy densityand the adiabatic limit is recovered as the driving frequencyis so small that the speed of light has enough time to reach allpoints within the cavity before the next motion of the mirroris achieved. (b) The momentum density in the cavity favorsthe moving mirror side and vanishes on the side that is notmoving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 〈0 |Tµν | 0〉 inside the cavity with the right-most wall oscillat-ing at 2 times the lowest photon mode frequency, which isω = 2ω0  1Hz, where ω0 = picL . The amplitude of the right-most mirror's motion is 10% that of the cavity length, whichis itself a micron. The plot is evaluated for 3 periords of os-cillation. (a) The energy density and it fluctuates wildly witha variety of peaks and valleys. Later in this chapter, this spa-tial inhomogeneity will be associated with particle production.(b) The momentum density in the cavity and now no longerfavors the moving mirror side. . . . . . . . . . . . . . . . . . . 32xi3.4 Two different views in the Total Photon number as a functionof the dimensionless time τ = 12ω1t where ω1 =piL0and whereL0 is the initial position of the right hand conductor in the twoconductor setup. Notice that the amount of photon generationis significantly more than what is predicted by Dodonov[2].This is due to the slight alteration of the first (m = 1) ofthe hierarchy of equations of motion for the mode functionwhile the mirror is moving (3.20), which imply a great dealof sensitivity to the method used to solve this problem. Dueto numerical limitations, there are higher order terms in theNumber operator sum (3.9) that could not be included at thistime. The prediction is that the number of photons generatedshould be even greater than what is shown in this Figure,and will be corroborated with the perturbation theory methoddepicted in Figure 3.5 . . . . . . . . . . . . . . . . . . . . . . 403.5 A graph of the photon number growth of the various methodsemployed in this manuscript, with respect to the dimensionlesstime τ = 2L0pi . It is noted that the prediction from this work'sModified MKBM calculation is in agreement with the newWKB calculation of this section. The difference between thesetwo plots should be from the fact that higher order modecontributions in the MKBM method could not be includeddue to numerical limitations. . . . . . . . . . . . . . . . . . . 473.6 The amount of photons generated for the first few driving fre-quencies of the mirror, which are proportional to ω1 =cpiL by1, 1.5 and 2. Note that the p = 1.5 case is so small comparedto the resonant values that it isn't visible on the first graph,so a log scale plot is also given. . . . . . . . . . . . . . . . . . 483.7 A demonstration that the dominant photon generation is thatwhich has a frequency that is equal to the driving frequencyof the mirror. In particular, if the driving frequency is a half-integer multiple of ω1 =cpiL0, then none of the modes are favored. 493.8 A demonstration that the dominant photon generation is thatwhich has a frequency that is equal to the driving frequencyof the mirror. In particular, if the driving frequency is a half-integer multiple of ω1 =cpiL0, then none of the modes are favored. 503.9 A Depiction of the energy density contained between the twomirrors as a function of space and time for p = 2. . . . . . . 513.10 A Depiction of the energy density contained between the twomirrors as a function of space and time for p = 3. . . . . . . . 52xii3.11 Information propagation of the point-mirror's trajectory, de-picted by worldline z (t). In this depiction, spacetime event 0will not realize that the mirror has begun moving at all. Point1 will be able to witness that the mirror has begun movingand how it moved for some time after, but will not be able todetermine the trajectory after the time where the photon rayγ1 intersects the world-line. In the adiabatic limit the world-line z (t) is moving so slow that it is almost exactly moving inthe time direction only, which means that all observer will ba-sically see the same location for the mirror. A mirror movingthis slowly will not create photons or produce inhomogeneusenergy densities. . . . . . . . . . . . . . . . . . . . . . . . . . 544.1 The decomposition of left and right going modes in the CasimirSetup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 The case of the time-dependently reflective mirror, where anexternal current is applied that introduces a time dependenceto the reflectivity for each mode of the photons on either sideof the mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Time depdendent reflectivity for the right most mirror, whilethe left mirror is left at perfect reflectivity (r1 [ω] = −1). Thisshows that no matter how fast the reflectivity is changed, theresult will only be akin to the adiabatic motional Casimir ef-fect, and therefore will not produce photons. This is truebecause while there is time-dependence, the result is spatiallyhomogeneus and is a similar result to what was shown in Fig-ure 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73xiiiChapter 1The Casimir Effect: anOverviewThe concept of nothing has been a fascination of human beings throughoutrecorded history. One of the earliest records of these thoughts came fromParmenides of the 5th century BC who would argue that nothing cannotexist as to speak of a thing, one has to speak of a thing that exists.[3].During this time, philosophical axioms that emerged to produce differentconclusions regarding the concept of nothing. It is understandable that thissort of discourse occurred so early, it is quite curious that we are born into aworld of structure and laws when it is seemingly equally likely that there benothing at all. This early interest with the void, or the absence of somethingwas not restricted to musings on nature, but also to practical purposes. Asearly as 1770 BC, the need for the concept of nothing was fulfilled by ahieroglyph found in accounting balance sheets to keep track of money[4].However the concept of a unit of nothing remained unclear and subject toever changing interpretations and choices of axioms about what is correct.Ultimately, without access to sufficient empirical methods, these argumentswould remain insubstantial.With the advent of quantum theory and its list of experimental valida-tions, one can now finally take a meaningful step forward towards under-standing the profound nature and the irreducible elements of nothing, orwhat is more precisely known as the vacuum, in a way that is not just ra-tional but empirical and falsifiable. In particular, it can and will be shownthat one can stir the vacuum to produce excitations in the form of tangibleparticles. Either the vacuum is not truly nothing or Parmenides' claim iswrong and one can speak about nothing as well as interact with it.The presence of vacuum amplitudes in quantum field theory reveal thatmatter, for a brief moment of time, can have a probability of being gen-erated from the vacuum before promptly being reabsorbed at some latertime. These vacuum processes, which mostly manifest in the form of di-vergent quantities, complicate the discussion in quantum field theories with1unbounded and nonsensical predictions1. One singular example of this is theprediction that the energy of the ground state of a quantum field is infinite.In spite of this, there is a sector of the discussion on vacuum effects that pro-duces finite predictions for various dynamical quantities that can and havebeen confirmed in experiment. This sector, which includes the static anddynamic Casimir effect, is the main point of interest of this study.In this Chapter, a terse review is given to illuminate what the StaticCasimir effect is and provide a foundation to move onto a more complexdiscussion. For a thorough2review on the static Casimir effect, the readeris invited to read a review by Bordag [6] or a discussion by DeWitt [6].1.1 The Static Casimir EffectThe static Casimir effect, or just Casimir Effect has been observed exper-imentally by Lamoreaux [7] in 1997 and later by Bressi [1] in 2002,but was predicted as early as 1948 by Casimir and Polder[8]. Under specificgeometric conditions imposed within space-time, namely that of a boundedor compact spatial region, there is, in addition to the divergent quantities, afinite contribution in some dynamical quantities. This finite aspect appearsstrictly due to the geometry of the space-time of the system in question.The archetypal example of this is with the energy density of the vacuummodes of a photon field in between two perfect, parallel-plate conductors.Since the conductors permit no photon modes at all in their interiors, theseplates serve to provide a compact spatial region between their enclosure.1The resolution to these problems appear in the form of Renormalization and of sub-tracting divergent measurements to produce a finite result.2For a more light-hearted introduction the reader is refered to an article byA.Lambrecht [5].2Figure 1.1: A schematic of what is known as the Static Casimir effect.The Casimir force fcasimir, expressed in equation (1.2), is dependent on theparallel-plate conductor separation L0.Suppose that these two parallel plate conductors are separated by a dis-tance L0 in the z-axis, then the vacuum expectation value (VEV) of theStress-Energy tensor Tµν (x, t) can be determined as3,〈0 |Tµν | 0〉 = Tdiv1 0 0 00 13 0 00 0 13 00 0 0 13+ pi2~c720L40−1 0 0 00 1 0 00 0 1 00 0 0 −3 (1.1)where Tdiv is a divergent quantity that is unbounded. The technicalreason for the finite term comes from the geometry of the space-time. Acompact spatial region results in, instead of an uncountable Fock-space likethat in free space, a countable4Fock-space, which itself results in a finitecontribution to the divergent term in Equation (1.1). In particular, thepressure appears as a force on both of the parallel plate conductors, and inorder to calculate this force5one subtracts the contributions from both sides,3Note that the divergent quantity will be discussed in more detail at the end of thischapter.4Countable is the mathematical term to describe a discrete set. It is defined asforming an equivalence class with the set of natural, or counting numbers.5Using the Lorentz Force Law.3leaving the finite force,fcasimir= −pi2~c240AL40(1.2)where A is the area of the conducting plate.The main mathematical feature that generates these finite force contribu-tions is the countable, or discrete, Fock-space that appears from restrictingthe system to a spatially bounded region. With the intention to pursuethe most irreducibly complex components of these vacuum effects, it will beuseful to model the photons in 1+1-dimensional, with a scalar field insteadof a vector field. This serves as the least complicated physical system thatcontains all the relevant features one needs for a more general approach6.To obtain the VEV of the stress-energy tensor 〈0 |T00| 0〉 of this scalarfield, for both outside the two-conductor system where the space-time isMinkowskian and inside where the spatial region is compact, one will get theresults,〈0 |T11| 0〉Minkowski= − limτ→0(~2cpiτ2)withφω (x, t) = e−i(ωt−kx) , ω ∈ R〈0 |T11| 0〉Interior= − limτ→0(~2cpiτ2)− ~pic24L20withφn (x, t) = e−iωt sin(npiL0x), n ∈ N(1.3)where the divergent term was calculated with point-splitting regulariza-tion, and φω (x, t) is what is known as a mode function of the quantumfield,φˆ (x, t) =∫dωφω (t, x) aˆω+h.c. or φˆ (x, t) =∑nφn (t, x) aˆn+h.c. (1.4)For a more detailed calculation in 1+1-dimensions, for the full stressenergy tensor, please refer to Appendix A. In this case, since the discussionis in 1 + 1 dimensions, the pressures are forces and the conductors are point-like in geometry. The Lorentz force law then determines the force,fcasimir= 〈0 |T11| 0〉Interior− 〈0 |T11| 0〉Minkowski=~pic24L20(1.5)6A minor detail, but the lack of polarization states in the scalar field corresponds toTµν having exactly half the value determined for a fully vector photon.4where it is noted that due to this subtraction knowledge of the divergentquantity is never truly required.It is important to know that all the geometric and conductive propertiesof the conductor appear in the calculation through the mode functions ofthe quantum field in question. In second quantization of a field subjectto Dirichlet and Cauchy boundary conditions, the information about theCauchy conditions is not used and the terms that represent that informationis used to represent the creation and annihilation operators. However theDirichlet conditions remain in these mode functions to preserve these physicalboundary conditions in the quantum limit. This fact is how the geometry ofspace-time is introduced and will be demonstrated in Chapter 2.1.2 Stress-Energy TensorThe dynamical quantities relevant to the discussion of finite vacuum effectscan be determined through the stress energy tensor, a conserved tensor cur-rent of the Poincaré group7, which can be determined for a type of matterwith Lagrangian Lmatteron a space-time background with metric gµν ,Tµν (x, t) = − 2√−det (g) δ(√−det (g)Lmatter)δgµν(1.6)where for the most part, the calculation will be done in a flat space-timebackground and so in this case gµν (x, t)→ ηµν .For a scalar field, this quantity corresponds toTµν (x, t) =12[(∂tφ)2 + (∂xφ)2 (∂xφ) (∂tφ) + (∂tφ) (∂xφ)(∂tφ) (∂xφ) + (∂xφ) (∂tφ) (∂tφ)2 + (∂xφ)2](1.7)where φ is either a classical or quantum field representing the matterin question. In the situations explored in this manuscript, the VEV of thestress energy of a quantum field is considered. Inserting the quantum fieldin equation 1.4 into the VEV of Tµν will reduce the expression to[9] the7The Poincaré group is the Lorentz group along with the operation of space-time trans-lations. Translations in space and time, rotations and boosts are all transformations con-tained in the group.5following,〈0 |Tµν (x, t)| 0〉 =∞∑n=1Tµν (φn (x, t) , φ∗n (x, t)) (1.8)=12∞∑n=1[φn,tφ∗n,t + φn,xφ∗n,x φn,xφ∗n,t + φn,tφ∗n,xφn,tφ∗n,x + φn,xφ∗n,t φn,xφ∗n,t + φn,xφ∗n,x]which is the result most calculations that follow will start off with. InAppendixH, the path integral formalism will be briefly introduced to discusshigher dimensional models and the stress-energy tensor can be determinedfrom the corresponding generating functional for the theory,〈0 |Tµν | 0〉 =∫ DφTµν (φ) e− i~ (S[φ]+〈J |φ〉)∣∣∣J=0∫ Dφe− i~S[φ]= Tµν(δδJ (x))Z [J ]Z [0]∣∣∣∣0(1.9)where Z [J ] is the generating functional, and J (x) is an external currentcoupling to the photon fields, where we note that Tµν has all instances of φreplaced with functional derivatives of the external current J (x).1.3 The Nature of the Divergent aspect of theVacuum Expectation Value of theStress-Energy TensorIn equation (1.1), the form of the divergence has been left out as a properdiscussion must be given in order to make sense of it. This sector of vacuumeffects, while not the focus of this work, is of some interest and will be brieflydescribed here. There are two main ways of discussing this divergence andthe physical implications of these approaches differ a bit8. In one way, theway that the author finds most appropriate, is through what is known asdimensional regularization.This calculation is similar to calculating the two-point correlation func-tion between the scalar fields at the same space-time point. Point-splitting8This choice of wording is a bit whimsical as the difference is between positive infinityand negative infinity for the divergence, but is apparently inconsequential, outside ofgravity , as energy is always measured as a difference which subtracts out this divergentquantity.6regularization introduces an inaccuracy in time by evaluating Tµν (x, t) attwo times t and t′ = t+ τ , where τ is this inaccuracy. The result, within thecavity enclosed between the two mirrors in 1+1-dimensions, then becomeswhat we see in equation (1.4),〈0 |Tµν (x, t)| 0〉 = limτ→0∞∑n=1Tµν (φn (x, t) , φ∗n (x, t+ τ))={− limτ→0(~2cpiτ2)− ~pic24L20}δµνThis, in the author's opinion, is an expression of the uncertainty principlefor energy and time∆t∆E ≥ ~ (1.10)where if one did evaluate the energy of the quantum field at a specifictime t, one would have absolutely no idea of what the actual energy is. Inthis manner, it makes sense that the energy density T00 would be divergentif we make the uncertainty in time, τ , to be null.Another way to view this divergence is to assume that there must existsome high energy cutoff to the calculation, which would correspond to apositive but very big VEV of the energy density.9This approach has ledto some progress in the discussion on how one can relate the cosmologicalconstant to the high-frequency cutoff of a quantum field theory [10]. Onecrucial difference between these approaches is that they are as far apart aspossible from one another! One produces a positive and huge energy densitywhile the time-splitting regularization approach produces a negative andhuge10value.1.4 DiscussionA brief review of the static Casimir effect has been demonstrated, which isnot truly anything new, but is an important compontent of understandingthis work. The eventual point that will be reached in Chapter 3 is that itis possible to stir the energy density inbetween the mirrors by moving thebounding conductors, and this will excite the vacuum. The next step thatwill complete this analysis is to rule out any effects of the vacuum that cancome from a single conductor.9This is similar to the difference between∑∞n=1 n = − 112 but∑Nn=1 n = ”big number"and comes from how one calculates the divergent part of the energy density. Look inAppendix Afor more.10This is assuming that the time-splitting  is not zero but very small7Chapter 2Single Mirror in 1+1DimensionsIt is prudent to begin this program by studying the elementary componentsinvolved, namely a single object undergoing vacuum fluctuations, and thedynamics of the vacuum fluctuations themselves. In the case of the Casimireffect and nearly every other related topic concerning vacuum fluctuations,the object experiencing fluctuations is a conductor that can block electro-magnetic radiation, and couple to the Quantum Electrodynamics (QED)vacuum. An analogy one can use to understand this relationship is that ofthe spoon and a pot of cooking broth. The spoon interacts with the fluctu-ating broth by blocking the soup from entering its interior, and through thisinteraction we can both stir the soup and also witness small fluctuations inthe spoon's motion as well as a dampening to any driven motion. All theseeffects will be present in the vacuum-conductor case, albeit incredibly smallwith respect to any macroscopic measurement. This behavior becomes lesscomplex and more easily measurable when two mirrors are involved, but isstill important to review to voice a complete discussion.In this section the Quantum Langevin equation for a single mirror cou-pled to the QED vacuum will be derived and from this the dynamics of thevacuum fluctuations can be determined. These dynamics, for both a per-fect conductor and a conductor that is translucent, have been explored tocompletion by Marc-Thierry Jaekel and Serge Reynaud[11]11, but form thefoundation of later ideas.A partially translucent conductor allows both transmission and reflectionof vacuum modes, or in our soup analogy, a slotted spoon that lets somesoup pass through it while reflecting the rest. This will, as shown in thisChapter, result in a motional force acting on the mirror that will dependon the conductive properties of the mirror. At the end of this Chapter, afinal discussion on the importance of imposing translucent properties to theconductor and how this will affect the causal structure of the system will be11There are many more references that will be introduced later in the Chapter.8reviewed.2.1 Quantum Langevin Equation for a SingleMirror Undergoing Vacuum FluctuationIt is important to understand that the conductors used in the Casimir effectare classical objects. In 1+1 dimensions, a point conductor is represented bya world-line z (t) and the photon field is a quantum mechanical object, whichis the Klein-Gordon mass-less vector, the photon Aµ (x, t). For the purposeof pedagogy the photon wil be approximated as a scalar-field φ (x, t) as thepolarization states do not greatly affect the calculation. For our discussion,there will not be any loop-order effects such as some fermion self-energy orhigher energy processes of this nature. For a discussion on the calculationof these effects, read Bordag[12].The Lagrangian for such a theory can be inferred,L =m2(dzdt)2+ Ua (z) + Λ (φ (z (t) , t) , z (t)) +∫12∂µφ∂νφgµνdx (2.1)where Λ (φ, z) is a yet to be understood coupling between the classicalconductor and the quantum photon field and Ua (z) is the potential for someapplied and external force acting on the conductor. In the most general case,this coupling buried in the definition Λ, is non-local in time, but must becausal. The equations of motion for the conductor is then,md2z (t)dt2= Fa (t) +∂Λ∂z (t)︸ ︷︷ ︸Fm(t)(2.2)In order to fully decompose this coupling a Volterra expansion of Λ isperformed which gives,Fm (t) =∂Λ∂z (t)(t, z (t)) =∞∑n=1n∏j=1(∫ ∞tj0dτj)hn (τ1, . . . , τn)x (t− τj)=∫ ∞t10dτ1h1 (τ1) z (t− τ1)+∫ ∞t10∫ ∞t20h2 (τ1, τ2) z (t− τ1) z (t− τ2) dτ1dτ2 +O(z3)9where the causal properties of this coupling are represented through theanalytic properties of the kernels hn, in particular that their Fourier trans-forms have no poles in the upper half complex plane. This condition on thepoles results in the kernels hi (τ) to vanish for negative times,τ < 0 =⇒ hn (τ) = 0 (2.3)which makes the convolutions in the integrals above only contribute from thestate of the conductor's position z (t) at a time in the past and not from anyfuture states. In this case, we have a description where the mirror's positionat varying different past times t − τ1 act as an input to the force acting onthe mirror.The higher order terms in the Volterra series correspond to increasinglymore non-linear effects due to the classical mirror's position where we needto include different types of correlations between the state of the systemnow and in the past. While, such an expansion does not have any methodto determine the size of the error of truncating at a specific n, we will workwith the assumption that the linear response, i.e., the first term of this series,is appropriate for our purposes and assume the form of the motional forceFm (t) to bemd2z (t)dt2= Fa (t) + Fm (t) (2.4)Fm (t) =∫ +∞−∞χ (τ − t) z (t) dτ (2.5)where this linear response function can be calculated by studying thevacuum expectation value of the photons. The properties of χ as well as it'scausal structure will be described in detail within later sections.2.2 Motional Forces Enacted on a SingleConductorA complete discussion of the motional forces exerted on a conductor mov-ing through the vacuum needs to contain information about the conductiveproperties of the mirror in question. A simple analogy that highlights thisneed is that of a slotted spoon stirring a broth. The bigger the slots onthe spoon, the less soup will pass through and not be reflected back. Thedifference, in the case of a mirror and a conductor, is that the frequency ofthe vacuum photon modes of the vacuum soup become unbounded in thedescription of a quantum field and can lead to photons with an unbounded10momentum p = ~ωc being stopped by the mirror. As such, it is importantto build a description that is robust enough to incorporate scattering witha high-frequency cut-off, which, in principle will allow these modes to passthrough the conductor without difficulty.To this effect, one way to obtain the linear response function χ (t) of themotional force described in equation (2.5) in a manner that is suited to dis-cuss the conductive properties of the mirror itself, is through the a scatteringformalism. If one imposes a sole conductor to empty, Minkowskian space,a separation of the vacuum modes of a scalar field occurs, where there arenow left and a right scalar field modes that both respect the sole boundarycondition that they cannot exist at the conductor. Decomposing these quan-tum fields into left-going and right-going modes, a scattering process can beunderstood, and which is represented in Figure (2.1) below,Figure 2.1: This is a single conductor undergoing scattering from both sidesby the vacuum modes of a photon field, which is represented as a scalar fieldfor simplicity.2.2.1 Derivation of the S-Matrix for Scattering PhotonsOne derives the S-Matrix for this system by evaluating Dirichlet boundaryconditions at the point mirror it's self, where simply the photons vanish inthe perfect mirror limit, ie., φ (x0, t) = 0 ∀t. The solution of the classical11scalar field is,ΦRk (x, t) = Ck e−i(ωt+kx)︸ ︷︷ ︸Left\in moving mode− e−2ikx0e−i(ωt−kx)︸ ︷︷ ︸Right\out moving modeΦLk (x, t) = Ck e−i(ωt−kx)︸ ︷︷ ︸Right\in moving mode− e2ikx0e−i(ωt+kx)︸ ︷︷ ︸Left\out moving modewhere the undetermined constants Ck that are usually defined by solvingthe Cauchy problem will be turned into creation and annihilation operatorsin second quantization and x = x0 is the position of the boundary condition.It is clear that the in and out-going modes satisfy the relationship,ϕRout[ω] = −e−2ikx0ψRin[ω]ψLout[ω] = −e2ikx0ψLin[ω]and this can be tabulated in the S-matrix below,[ϕRout [ω]ψLout [ω]]= S (ω)[ϕLin [ω]ψRin [ω]](2.6)whereS (ω) =[0 (−1) e−2ikx0(−1) e+2ikx0 0](2.7)and where the column vector is defined asΦin/out(x, t) =[ϕin/out(ωt− kx)φin/out(ωt+ kx)]=∫ ∞−∞dk2piΦin/out[ω;x] e−iωt (2.8)This column vectors Fourier transform is defined as,Φin/out[ω;x] =[e−2ikxϕin/out[ω]e+2ikxψin/out[ω]]= exp (−2iηx)[ϕin/out[ω]ψin/out[ω]](2.9)where η =[ −1 00 1](2.10)This notation reduces the scattering to a simple linear algebra problemwhile increasing some of the notational complexity. However, the benefit of12this method shows in how it covers imperfect conductive boundaries. A gen-eralization of the S-Matrix in equation (2.6) with some reflection amplitudeR (ω) = |r (ω)|2 and transmission amplitude T (ω) = |s (ω)|2 for mode ω, is,S (ω) =[s (ω) r (ω) e−2ikx0r (ω) e+2ikx0 s (ω)](2.11)where r (ω) ∈ [−1, 0] and s (ω) ∈ [0, 1] are the reflection and transmissioncoefficients, respectively. The scattering now looks like,[ϕout [ω]ψout [ω]]=[s (ω) r (ω) e−2ikx0r (ω) e+2ikx0 s (ω)]e−iωηx[ϕin [ω]ψin [ω]](2.12)This generalization of the perfect reflection limit corresponds to a scalarfield of the form,Φk (x, t) = θ (x− x0)Ck[e−i(ωt−kx) + r (ω) e−2ikx0e−i(ωt+kx) + s (ω) e−i(ωt−kx)]+ θ (x0 − x)Ck[e−i(ωt+kx) + r (ω) e2ikx0e−i(ωt−kx) + s (ω) e−i(ωt+kx)]which is the analogue to the perfect mirror case seen above. This scalarfield can be determined by imposing the Dirichlet boundary condition atx = x0,Φk (x0, t) =[1 + r (ω) + s (ω)]2e−iωt cos (kx0) (2.13)which, in the limit that r → −1 and s → 0, corresponds to the perfectmirror limit.In general the transmission and reflection functions r (ω) and s (ω) re-spectively, are not related to one another. However, as will be shown in thebelow sections, they will be related if unitary scattering is imposed.2.2.2 Linear ResponseWith the assumption that the fields are stationary around the mirror, theresult of the linear response becomes,χ [ω] = 3imτ∫ ω0(ω − ω′)ω′ [1− s [ω] s [ω′]+ r [ω] r [ω′]] (2.14)this result is determined by Jaekel[13] and its derivation is reviewedbriefly in the Appendix B. This stationary limit that the calculation is13done physically means that the speed of light is considered infinite and theautocorrelation between any two points of the vacuum around the mirror isconstant in time. So this can be interpreted as an adiabatic limit. As such,the calculation would have to be redone if the intention is to find the forceson a mirror in the two parallel plate capacitor system where one mirror ismoving quickly. Since this is a singular mirror sitting in the vacuum, thiscalculation of the linear response is sufficient.2.3 An analogy to the Lorentz-Abraham Problemin Classical Electrodynamics.From the Quantum Langevin equation (2.4), in the perfect conductor limitthe equation of motion for the mirror's world-line will be shown to be givenby equation (2.15)below,m0d2zdt2(t) = m0τd3zdt3(t) + Fa (t) , τ =~6pim0c2(2.15)where m0 is the mass of the conductor. This is an interesting result as itis exactly, up to a difference in constants, the the Lorentz-Abraham equationfor an electron moving through an electromagnetic field, which is given bymed2zdt2(t) = meτL-Ad3zdt3(t) + Fa (t) , τL-A =µ06pimec(2.16)where µ0 is the vacuum permeability, me is the mass of the electron andand c is the speed of light. The solution to this is famous for containingeither pre-accelerations or unbounded velocities when an applied force, Fais exerted on the electron, even if this force is applied only for a brief mo-ment [14, pg.489-492]. This problem occurs from the fact that we are talkingabout a semi-classical reduction of a more general theory of Quantum Elec-trodynamics, where one is mixing a totally classical object, namely the ideaof a point particle with well-defined trajectories, with a classical field that isa reduction from a quantum field.In the vacuum fluctuation induced case we are introducing a classicalpoint mirror and examining the vacuum expectation value of forces actingon it by the fluctuating vacuum, but unlike the Lorentz-Abraham case, thereis actually a physical condition that one can impose to resolve these acausalproblems. This resolution will be discussed in subsection 2.6.Equation (2.15) displays the motional force derived for a conductor un-dergoing vacuum fluctuations in the perfect reflectivity case, which is a con-sequence of equation (2.5). In the above section, and in the work of Jaekel14and Reynaud[15], the linear response function was determined from the vac-uum modes in Equation (2.14) where s [ω] , r [ω] are the transmission andreflection coefficients and are components of the S-matrix,S (ω) =(s (ω) r (ω) e−2iωz0r (ω) e2iωz0 s (ω))(2.17)and where z0 is the location of the mirror in space. It should be notedthat there are some fairly severe restrictions on this S-matrix that must beadhered to in order to avoid nonsensical results. These are discussed in thefollowing section.2.4 Scattering PropertiesThe scattering process mentioned above requires a detailed discussion. Inorder for this scattering process to avoid complications, such as complexforces, acausal scattering12, mode absorption into the conductor, etc, theremust be conditions imposed on the S-matrix (2.17). These are detailed below.i): Reality Condition and Real ForcesIn order for this theory to make sense forces that have no imaginary com-ponent must result from a calculation. In order for this to be the case, thereflectivity and transmission coefficients, r [ω] and s [ω] respectively, musthave real inverse Fourier transforms. The reason for this is that the Casimirforce is proportional to r (t).This shown in equation (4.14) of Section 4. Infrequency space, where the scattering formalism is defined, this reality con-dition is represented as,r∗ [ω] = r [−ω] , s∗ [ω] = s [−ω] (2.18)Proof.r (t) = r∗ (t) ; Reality Condition∫ +∞−∞dω2pi r [ω] e−iωt =∫ +∞−∞dω2pi r∗ [ω] e+iωt= − ∫ −∞+∞ dω2pi r∗ [ω] e+iωt ; Order of Integration change=∫ +∞−∞dω˜2pi r∗ [−ω˜] e−iω˜t ; ω →-ω˜(2.19)12It should be noted that the scattering can be causal, that is, in-going modes in thefuture do not produce affects on outgoing modes in the past, while the accelerationscalculated in equation (2.16) are still a-causal. Imposing causality on scattering and onaccelerations of the conductor must be done separately. This is seen later in this Chapter.15ii): Causality Condition and the Absence of Pre-accelerationsIn order for the scattering of photon modes off this mirror to not exhibit anyacausal behavior, such as input modes causing changes in the output modesbefore the input modes actually reach the mirror, the S-matrix must satisfythe condition that in the frequency domain there are no poles in the upperimaginary plane. This is true for any linear response theory [16]. It shouldbe noted that a causal S-matrix permits casual scattering but in general willnot correspond to the production of equations of motion for the mirror thatare causal. This distinction is made in the next subsections.Proof. To produce a response theory where there are no effects that precedethe cause of the same effect, as one wishes the convolution below,Youtput(t) =∫ +∞−∞dτχ (t− τ)Xinput(τ) (2.20)which represents the response of the output Youtput(t) to an input Xinput(t),to vanish when τ < t. This requires that χ (t), the linear response function,must vanish for negative values. In the frequency domain, this means thatyou can use the Cauchy Residue theorem to represent this expression,χ (t) =∫ +∞−∞dω2piχ [ω] e−iωt = Resz∈I±(χ [z])− limR→∞∫C±Rdθ2piχ[Re−iθ]e−iRe−iθt(2.21)where I± is either the upper or lower imaginary plane, C±R is the an upperor lower semi-circular path of radius R and θ is the parameterization alongC±R .16Figure 2.2: In order for there to be causal response from some output ef-fect by some input stimulation, it is required that the response function beentirely analytic in the upper half imaginary plane.One has the choice of a contour that is in the upper-half imaginary planeor the lower-half imaginary plane. To restrict χ (t) to vanish for negativearguments, it is therefore required that there be no poles in I+.iii): Unitary Condition and the conservation of modesIn a physical conductor, the conductivity is never perfect and the conduc-tor does absorb some of the modes in a scattering event where they thenattenuate. Assuming unitary scattering is the analog of assuming inelasticscattering and will simplify the calculation of the forces, as will be seen inthe next sections. In order to make scattering unitary, one simply requires,S (ω)S† (ω) = I (2.22)or in terms of the reflectivity and transmission,r (ω) r∗ (ω) + s (ω) s∗ (ω) = 1 , s (ω) r∗ (ω) + r (ω) s∗ (ω) = 0 (2.23)iv): Realistic Conductor Conditions, Ultraviolet frequency cutoffsand Causal SystemsThe unphysical condition one imposes in the problem of the motional vac-uum forces on a point-conductor is that that the mirror is perfectly reflecting17all modes of the vacuum. In a physical situation, the mirror will eventuallybecome translucent for higher frequency modes of the photon field, allow-ing energetic modes to escape. These modes have increasingly large andunbounded momenta which must be perfectly reflected by the mirror in theperfect conductivity limit. The solution, which is determined by Jaekel[17], is to impose a high-frequency cut-off on the mirrors reflectivity so thatthese large frequency modes can pass through the mirror unrestricted. Thenatural frequency cut-off to choose for a conductor is the plasma frequency,which is the minimum frequency electromagnetic waves can travel throughthe conductor without attenuation.2.5 A free Conductor undergoing VacuumFluctuationsThe objective here is to understand what the Quantum Langevin equationsof motion, and their solutions, looks like for a conductor undergoing vacuumfluctuations. If we demand a form of the S-matrix where we also have thats [ω] = 1 + r [ω], then a possible form for the reflectivity that satisfies all theproperties listed above is,r [ω] = − 11− iωΩ, s [ω] = 1 + r [ω] (2.24)where Ω is the plasma frequency or high frequency cut-off. Using thedefinition of the response function χ [ω] and this form of the reflectivityallows one to write the motional force Fm (t) in a power series of the inverseof the plasma frequency Ω, and expanding in the inverse plasma frequency.The motional force is then,Fm (t) = mτd3zdt3+mτ2Ωd4zdt4+3mτ10Ω2d5zdt5+O (Ω−3) (2.25)The equation of motion for this is now[d2dt2+ τd3dt3+τ2Ωd4dt4+3τ10Ω2d5dt5+O (Ω−3)] z (t) = 0 (2.26)where the solutions are trivial to find for the initial data13 (x (0) , v (0) , a (0) , j (0)) =13j is the jerk of the particle[18].18(0, v0, a0, j0). These solutions are, for O (Ω−n),n=0 : z0 (t) = v0tn=1 : z1 (t) = a0τ2e−tτ + (a0τ + v0) t− a0τ2n=2 : z2 (t) = −C1 + C2t+ C3e−Ωτ eiω0 + C4e−Ωτ e−iω0(2.27)whereω0 =√Ωτ− Ω2 (2.28)is an oscillation frequency of the second order correction for a particle andCi are coefficients that depend on the initial data.As one can plainly see, the critically dominant term in all of these is theclassical, uncharged, free particle solution x (t) = v0t, as τ is a time-scalethat is typically τ ∼ 10−47s for a conductor in used in experiments[1]. Thatis the corrections to the position are on the same order, ∆x ∼ c10−47m whichis deeply within de Brolige wavelength of the conductor, λDB' 10−28m andfar below the Planck length scale , `planck∼ 1.6×10−35m. This is appropriateas one does not witness vacuum effects on the motion of single conductors,even in experiments where the Casimir effect is measured.The only way to push these corrections to be comparable to the de Broligewavelength of the conductor is, perhaps, near a space-time singularity wherethe curvature becomes unbound, which is outside the scope of this study.An illustration of these effects are provided below,2.5.1 A Realistic Conductor Undergoing VacuumFluctuationsIn what follows these vacuum corrections are determined for a system thatcan be examined in a table-top experiment. The first and second ordercorrections are in Figure (2.3) and (2.4) respectively. The mass used is2.49µg, which is the same mass used to measure the two-conductor Casimireffect by G.Bressi [1], where this point conductor undergoes an initialvelocity of v0 = 0.2µms and initial acceleration of a0 = 0.01µms2 .19Figure 2.3: The difference between the first order (inO (Ω−n)) and the zerothorder solutions, z1 (t) and z1 (t) respectively. There is not much feature tonote other than the size of the correction being smaller than the Planck scale.This is for a silicon conductor of mass m = 2.492 × 10−6kg, undergoingan initial velocity of v0 = 0.2µms and initial acceleration of a0 = 0.01µms2 ,which mirrors conditions found in experiments such as that done in [1].What is breaking the right/left symmetry is the initial velocity v0 and initialacceleration a0, which is moving forward.20Figure 2.4: The difference between the second order (in O (Ω−n)) and thezeroth order solutions, z2 (t) and z0 (t) respectively. There is some oscillationstructure visible at length scales deep in the Planck scale, but would beunlikely or impossible to detect. This is for a silicon conductor of massm = 2.492× 10−6kg, undergoing an initial velocity of v0 = 0.2µms and initialacceleration of a0 = 0.01µms2 , which mirrors conditions found in experimentssuch as that done in [1].212.6 PrecursorsAs foreshadowed in Section 2.3, while the scattering on the conductor canbe causal, the overall equations of motion can present acausal solutions, orprecursor effects. This can be seen by the introduction of an applied force tothe particle undergoing these motional forces from the vacuum. The mostgeneral way to resolve these issues is through a high-frequency cutoff on themodes being reflected [17, 19] as will be shown below.2.6.1 Resolution of Acausal BehaviorA way to determine the acausal behavior, which occurs when an applied andexternal force is exerted on the conductor, is to cast the problem in termsof a linear response and look at its analytic properties in the frequencydomain. Instead ensuring the casual structure of the S-matrix, which wasaccomplished by guaranteeing that it is analytic in the upper imaginaryplane, (refer to Section 2.4), the mechanical admittance Y (τ) is studied.The mechanical admittance can be determined from the equation of motionfor the conductor, which for our purposes is equation (2.5), and representsthe linear response of the velocity of the conductor for a given applied forceFa(t),Fa[ω] = Y [ω] v [ω] (2.29)Jaekel [17, 19] have done extensive and complete analysis on thisand the resolution comes through high-frequency cut-off on the contributionsfrom scalar field modes to the motional force. This will shift any poles in theupper imaginary plane of the domain of Y [ω]down to the lower half. Dueto the imposition of a finite plasma frequency, a realistic conductor will notsuffer the precursors that plague the Lorentz-Abraham case.As will be shown in the very next section, this analysis is not requiredas all the acausal behavior occurs well below the Planck scale and is notmeasurable through the particle's motion.2.6.2 Acausal effects on a realistic conductorWhile there is much discussion on maintaining the causal properties of theconductor's motion, seen above and in the work of Jaekel [17] since themotional force produces changes that can only be seen deep in the Planckscale, it is not required to maintain consistency, at least in the point-mirrorcase. In Figure 2.5, both the precursors and the triviality of their effects are22shown for a conductor a mass and initial value data commonly found in atable top experiment.From equation (2.5), the equation of motion for a perfect conductor un-dergoing vacuum fluctuations, one finds the classic solutions to the Lorentz-Abraham problem of an electron. Consider some brief, constant force actingon the conductor,Fa (t) = F00 ; t < 01 ; t ∈ [0, T ]0 ; t > 0(2.30)If the initial velocity and position are zero, that is, the particle is initiallydoing nothing, while choosing an acceleration at t = T that eliminates theunbounded acceleration for t > 0, then the solutions for the velocity, positionand acceleration become,a (t) =F0m(1− e−Tτ)etτ ; t < 0(t− τe (t−T )τ); t ∈ [0, T ]0 ; t > T(2.31)This clearly demonstrates that before applied force even touches the con-ductor, the point conductor begins to respond in a way that depends on themagnitude of the force, F0. However, as shown in Figure 2.5, the resultingpre-acceleration occurs well below the Planck-scale.23Figure 2.5: The acausal behavior of a conductor undergoing motional forcesfrom the vacuum. The mass of the conductor in question is m0 = 2.49 ×10−6kg, which is the mass of the silicon objects used to measure the Casimireffect in a table top experiment [1] and the magnitude of the force is f0 =10−7N.24It should also be noted that in Equation (2.31) after t = 0 kinematicsare, for all intents and purposes, identical to that of a free, uncharged, pointparticle that has been given an initial velocity. This is different from theusual Lorentz-Abraham case and is entirely due to the small magnitude ofτ in equation (2.5), with respect to the time-scales of the setup, as seenthrough its definition,τ =~6pim0c2' 2.5× 10−45s (2.32)While these effects are small in the 1+1 dimensional case, where the conduc-tor is a point particle, it is possible to consider the effects in higher dimen-sions where the conductor has internal geometry and finite size. However,this is outside the scope of this current program.2.7 DiscussionThe simplest elements of the discussion of Vacuum fluctuations are the QEDvacuum itself, conductors embedded in this vacuum, and the interaction ofmultiple conductors. The focus of this chapter was on the discussion of asingle conductor and all the vacuum effects it undergoes, and main point isthat the vacuum exerts negligible effects on the single conductor. This is tobe expected, however a future point of interest would be to study the effectsa violently moving conductor has on the vacuum, its dynamics14. This is,however, outside of the scope of this work.14The dynamics of the vacuum are commonly studied through the Vacuum expectationvalue of the stress energy tensor 〈0 |Tµν | 0〉.25Chapter 3The Motional Casimir Effect in1+1 DimensionsTwo elementary components of the discussion on vacuum effects have beendiscussed now. A brief discussion on the dynamics of a single conductorundergoing vacuum fluctuations has been discussed in Chapter 2and alsoa discussion on the vacuum effects of two stationary conductors has beendiscussed in 1. All the cards are in play to discuss the motional Casimireffect, which is what happens to the dynamics between the two parallelplate conductors if one conductor moves. The motional Casimir effect is atheoretical prediction that is well known and a review on the current progresson it can be found by Dodonov[20].A stark difference between what has been discussed thus far is the possi-bility of exciting photons from the QED vacuum. By driving the conductorsinto motion one can effectively stir the vacuum violently enough to ex-cite matter, which is the overall theme of this project. While one could, inprinciple, excite the vacuum with a single conductor, the internal reflectionof the photon modes inbetween two mirrors greatly amplifies the effect. Inwhat follows, a classical example of the motional Casimir effect is studied,then a discussion on the different types of observables one can calculate torepresent the amount of matter generated is given.The motional casimir effect, and the particles that are generated is a wellknown theoretical result and has been discussed before in journals such asNature[21]. The main challenge yet to be overcome is the lack of experimen-tal observation of photons that are shaken from the void. This chapter'sobjective is to produce a more solid estimate of the number of photons gen-erated, where they will excite out of the vacuum and a clearer understandingof what a particle actually is.263.1 Introduction - The Definition of a ParticleSince this Chapter will be discussing particle generation, it is appropriate totalk about what a particle actually is first, especially as there will be twodifferent observables used to probe particle15generation.In classical mechanics and in gravitation, the concept of a particle is fairlystraight forward. It is in principle an object sufficiently small that it's tra-jectory throughout space and time can be represented by a one-dimensionalworld-line. In quantum mechanics, this definition blurs as one now has a sumover all world-lines the particle takes simultaneously, and assigns a proba-bilistic weight of e−i~Sto each path. However, even in this limit, the conceptof a particle through its world-line is intact. In quantum field theory, the par-ticle states of a quantum field are only defined in the non-interacting, linearlimit of the theory, which is all encoded in the Fock-space of the theory.Once one includes interactions, the exact nature of the particle is lost asone cannot define a linear function space like the Fock space for this theory.Of course there is perturbation theory, which retains the idea of a particlethrough the Feynman diagrams, radiative corrections to propagators andwhat not. In effect, this is simply the representation of a non-linear theoryin terms of the linear concept of a particle and is inexact and merely relieson the familiar definition of a particle to describe something more complex.To make progress, one must rely on sensible observables such as theenergy density and number density operator, defined below, that work withinthe limits of our understanding and sort of avoid our ignorance of true non-linear dynamics.By studying the motional Casimir effect, this manuscript will shed somelight on these seemingly unconnected observables and show how one can useboth to detect a particle.3.2 Motional Casimir Effect and the EnergyDensityTo introduce motion to the conductors that enclose some volume of space-time will excite the vacuum contained within and introduce particle content.Of course, the concept of a particle is actually an idealization of a more real-15In this manuscript, the particles will be photons. But in general one can generate anyquanta of a quantum matter-field so long as you can screen said matter. Note that this caninclude quarks or other bosonic intermediary particles such as those in the electro-weakcase, but cannot include the graviton since one cannot screen gravity.27istic description of nature, which is discussed above. However the concept ofparticle content between these two conductors can be described by havingsome finite energy density16within the cavity.3.2.1 The Fulling-Davies-Moore ApproachA discussion of the energy density between two moving conductors tradi-tionally, and with good reason, cannot begin without referring to the workof Fulling and Davies[22] and of Moore [23] who used conformal transfor-mations in the usual 1+1-dimensional system to obtain the following closedform expressions for the vacuum expectation value of the stress energy tensorTµν (x, t) within the two-mirror,〈0 |T00| 0〉 = 〈0 |T11| 0〉 = lim→0(− 12pi2)− 124pi[F (v) + F (u)]〈0 |T10| 0〉 = 〈0 |T01| 0〉 = − 124pi[F (v)− F (u)]F (u) =[pi22(R′ (u))2+32(R′′ (u)R′ (u))2− R′′′ (u)R′ (u)](3.1)u = t− x , v = t+ x (3.2)where the term R (u) is the inverse conformal mapping which takes ad-vantage of the conformal symmetry of the 1+1-dimensional wave equationin mapping the moving mirror problem into the stationary mirror problem.This is visualized in figure 3.1 below,16In fact, by the end of this Chapter, it will be shown that it is not just any finite energydensity that is required, but one that is spatially in-homogeneous.28Figure 3.1: The conformal map introduced by Moore and used by Fulling,Davies and Moore to describe the dynamic Casimir effect. The conformalsymmetry of the wave equation in 1+1-dimensions essentially permits us tomap the dynamic Casimir effect to the static case.where the transformation R maps the sum and difference of the spaceand time coordinates to the new conformal coordinates w and s,R (t− x) = w − sR (t+ x) = w + s(3.3)The proceedure is to simply solve the VEV of Tµν for the static Casimircase, in the w, s coordinates, which was done in Section 1 and in moredetail in Appendix A, then invert the conformal transformation back to(t, x) coordinates using the inverse conformal mapping R (t± x).The explicit form of R (u) can be determined for a given world-line z (t)by solving Moore's equation which is described extensively in [23]. The resultis below and a complete derivation of it can be found in Appendix (C),R (t∓ x) =∞∑k=0∫ t∓xτγk (s) ds (3.4)where γ0 =cz(t) and the functionsγk (s) are given by, solving equation(3.5) below, ∑`i=01(2i+ 1)!(qc)2i d2idt2i(γ`−i) = 0 ∀` ∈ N (3.5)where the first few terms are listed in Table 3.1.29γ1 = − 13!z2cd2dt2(1z (t))γ2 =(13!)2 1c4z2 (t)d2dt2(z2d2dt2(1z (t)))− 15!c4z4 (t)d4dt4(1z (t))Table 3.1: The first few γk that solve equation (3.5) and encode the rightmost mirror's world-line into the definition of 〈0 |Tµν | 0〉.3.2.2 Dynamics of the Vacuum due to a SinusoidallyMoving ConductorUsing the Fulling-Davies-Moore method, the VEV of the stress energy ten-sor is visualized in Figure 3.1 using various numerical methods. The caseconsidered for the entirety of this effort is that of the right most conductorbeing driven to move sinusoidally. The world-line of such a system is givenby,z (t) = L0[1 +  sin(ppicL0t)](3.6)where L0 is the initial mirror position,  is the unit-less amplitude of themirror's motion, where it is noted in particular that the energy density ischanging in between the plates with the same period as the mirror's oscilla-tions.30(a)(b)Figure 3.2: 〈0 |Tµν | 0〉 inside the cavity with the right-most wall oscillatingat 1 Hz with an amplitude 10% that of the cavity length. The averageseparation length is a micron. (a) The energy density and the adiabaticlimit is recovered as the driving frequency is so small that the speed of lighthas enough time to reach all points within the cavity before the next motionof the mirror is achieved. (b) The momentum density in the cavity favorsthe moving mirror side and vanishes on the side that is not moving.31(1b)(b)Figure 3.3: 〈0 |Tµν | 0〉 inside the cavity with the right-most wall oscillating at2 times the lowest photon mode frequency, which is ω = 2ω0  1Hz, whereω0 =picL . The amplitude of the right-most mirror's motion is 10% that of thecavity length, which is itself a micron. The plot is evaluated for 3 periordsof oscillation. (a) The energy density and it fluctuates wildly with a varietyof peaks and valleys. Later in this chapter, this spatial inhomogeneity willbe associated with particle production. (b) The momentum density in thecavity and now no longer favors the moving mirror side.32The energy density is changing with respect to the ground-state energydensity, which is indicative of particle content occurring between the mirrorsin the classical field theory sense. However, in order to fully understand thenature of this particle content, at least another observable, such as the Pho-ton Number Operator must be considered. It will be learned that a changingenergy density is not sufficient to argue that there is particle content. Theenergy density must have spatial inhomogeneity.3.3 Determination of the Photon NumberIn the spirit of the preceding discussion, the approach of the rest of thissection is to find the VEV of the photon number operator17, as the enclosedregion of space-time between the mirrors changes. Later sections will displaydifferent methods used to solve this problemWhy the photon number won't be zero, which one would usually ex-pect from a state that represents the vacuum, is due to the time-dependentchanges in the vacuum state itself. The vacuum state,|0〉, changes as theenclosed region of space-time is altered through moving the mirror, whicheffectively changes the available volume contained18, and any change to thevacuum state will permit the possibility of photon production.The relationship between the vacuum states for these different regionsof space-time is described by the Bogoliubov transformation[24]. The Bo-goliubov transformation relates two different vacuum states, lets call them|0〉1 and |0〉2 ,through the annihilation operators (and through a complexconjugation, the creation operator as well) that annihilates them,aˆn |0〉1 = 0 |0〉1 , bˆn |0〉1 = 0 |0〉2 (3.7)where the transformation itself is defined as,bˆm =∑n√mn(ξnmaˆn + η∗nmaˆ†n), bˆ†m =∑n√mn(ξ∗nmaˆ†n + ηnmaˆn)(3.8)where ξnm and ηnm are what are known as the Bogoliubov coefficients.The majority of the work is done by finding out what these Bogoliubov17It operates on an element of the Fock space, which is a super-position of free particlestates.18As will be shown, this change must be done beyond the adiabatic limit, or in otherwords, fast enough with respect to the propagation of the information about the changingboundary.33coefficients are through solving a Cauchy problem between when when theconductor starts moving and when it stops moving. Through this, chain ofinitial value problems, the number of particles in between the mirror aftermoving it, as well as the energy density between the mirror, can be easilyfound.3.3.1 The Number DensityThe vacuum expectation value of the number density operator for the al-tered space-time can be easily determined,Nm =〈0in∣∣∣b†mbm∣∣∣ 0in〉=〈0in∣∣∣∣∣[∑n√mn(ξ∗nmaˆ†n + ηnmaˆn)][∑l√ml(ξlmaˆl + η∗lmaˆ†l)]∣∣∣∣∣ 0in〉= m∑n1n|ηnm|2 (3.9)where Nm are the number of photons created in the mode m with energyEm = ~ mpiLfinaland where Lfinalis the final position of the right-most mirrorafter it stops moving.3.3.2 Photon Mode FunctionsThrough this Bogoliubov transformation the form of the scalar field's modefunctions can be determined in all regions. The quantum scalar field beforethe mirror moves (t < 0), after the mirror is done moving (t > T ), as well asduring motion t ∈ [0, T ] are, defined asφˆ (t, x) =∞∑n=1φn (t, x) aˆn + φ∗n (x, t) aˆ†n (3.10)where the mode function over all times is defined as,φn (t, x) =1√mpi∞∑m=1Cmn (t)√z0z (t)sin(mpiz (t)x)(3.11)where z (t) is the equation for the world-line trajectory for the right mostmirror, about the the position z0 > 0, and the left most mirror is assumedto be at z = 0.34The coefficient Cmn (t), which is defined in equation (3.12) is the time-dependent part of the mode function that guarantees that φn (t, x) satisfiesthe Klein-Gordon equation of motion.Cmn (t) =δmne−iωmt ; t < 0C˜mn (t) ; t ∈ [0, T ]ξnme−iωmt + ηnme+iωmt ; t > T(3.12)where the intermediate mode function C˜mn (t) must be determined from theright hand mirror trajectory z (t) and the equation of motion. The t > Texpression for Cmn would enforce a generic Bogoliubov transformation onthe creation and annihilation operators, as seen below,φ (t, x)=∞∑n=1[φn (t, x) aˆn + φ∗n (t, x) aˆ†n]=∞∑n=11√npi∞∑m=1([αmne−iωmt + βmne+iωmt]aˆn)+∞∑n=11√npi∞∑m=1([α∗mne+iωmt + β∗mne−iωmt]aˆ†n); Insert Mode Function=∞∑m=11√mpi( ∞∑n=1√mn[αmne−iωmt + βmne+iωmt]aˆn)+∞∑m=11√mpi(√mn[α∗mne+iωmt + β∗mne−iωmt]aˆ†n); Rearrange=∞∑m=11√mpi( ∞∑n=1√mn[αmne−iωmtaˆn + β∗mne−iωmtaˆ†n])+∞∑m=11√mpi( ∞∑n=1√mn[βmne+iωmtaˆn + α∗mne+iωmtaˆ†n])=∞∑m=11√mpi(e−iωmtbˆm + e+iωmtbˆ†m); New Mode Functions(3.13)In order to relate these regions properly the Cauchy problem betweenwhen the mirror is stationary and when it starts moving, as well as when itstops moving to now remain stationary at t > T , must be solved19.3.3.3 The Intermediate Mode FunctionIn order to calculate C˜mn (t) for a relatively simple right mirror trajectoryz (t), the following differential equation must be solved, which was calculated19This method is inspired by Dodonov [2], and will be used to produce some originalwork.35by Dodonov[2, 25],C¨nm (t) + ω2m (t)Cnm (t) (3.14)=∑j 6=m2v (t)hmjC˙nj (t) + v˙ (t)hmjCnj (t) + v2 (t)∑`6=jhjmhj`Cn`hmj = (−1)m+j 2mjj2 −m2 , v (t) =z˙ (t)z (t)where it is noted that this hmj numerical factor never diverges due to thelack of diagonal terms for hmj .20Equation (3.14) is found by substitutingthe mode function (3.12) for t ∈ [0, T ] into the wave equation, and has theeffect of ensuring that φˆ (x, t) keeps all it's causal properties. To furthersimplify the task of solving this problem, the simple sinusoidal trajectoryintroduced in equation (3.6) is used,z (t) = L0 (1 +  sin (pω0t)) , ω0 =picL0(3.15)where  is a small parameter that reflects the small oscillations of theworld line about z = L0.In what follows, two methods are used and contrasted to complete theanalysis and determine some numbers for the particle content within themirrors,3.4 Method 1: Krylov-Mitropolski-BogoliubovThe general idea behind this method is to infer the form of the intermediateregion t ∈ [0, T ] by something that is the most convenient to work with. Inpractice this will prove to not be very convenient and computationally heavy.The ansatz for the form of the mode function will be,∀t ∈ [0, T ] : φn (t, x) = 1√mpi∞∑m=1Cmn (t)√z0z (t)sin(mpiz (t)x)Cmn (t) = αmn (t) e−iωmt + βmn (t) e+iωmtwhere the new Bogoliubov-Like coefficients, αmn (t) and βmn (t), aretime-dependent. This will turn the differential equation (3.14) into a messy,20Inserting solution found in Equation (3.12) will give you these hmj through the useof inner products like∫sin(mpiLx)sin(npiLx)dx, etc.36mixed differential equation for these new functions, which is seen below,I1mn (t)− I2mn (t) = 0 (3.16)I1mn (t) = 4ω0∑jhmj[{α˙nj − ijω0αnj}cos (2ω0t) e−ijω0t]−4ω0∑jhmj[ω0αnj (t) sin (2ω0t) e−ijω0t]+4ω0∑jhmj[{β˙nj + ijω0βnj (t)}cos (2ω0t) e+ijω0t]−4ω0∑jhmj[ω0βnj (t) sin (2ω0t) e+ijω0t]I2mn (t) = α¨mne−iωmt + β¨mne+iωmt − ω2m[αnme−iωmt + βnme+iωmt]− 2iωm[α˙nme−iωmt − β˙nme+iωmt]+ ω2m[αnm (t) e−iωmt + βnm (t) e+iωmt]− 2ω2m sin (2ω0t)[αnm (t) e−iωmt + βnm (t) e+iωmt]In order to turn this into a more tractable problem, it is prudent to find amethod that, if possible, will split them into two separate differential equa-tions for βmn and αmn , and this can be done with a modification to theKrylov-Mitropolski-Bogoliubov (KBM) method.3.4.1 The Modification to the KBM MethodThe KBM method[26] is first described below. For an equation of motionsuch as in equation (3.17),dx (t)dt= f (t, x (t)) (3.17)where  is a small positive parameter, a time-averaged approximation can befound in the form,dx (t)dt∼  1T∫ T0f (t, x) dt (3.18)when f (t, x) is periodic in time with period T . Note that x (t) treated asa constant with respect to this integral, for this time interval. What this issaying is that the change in x (t) is small enough with respect to the changein f (t, x) that it can be treated as a constant in the time-averaging.This is not sufficient enough for the equations of motion (3.16) as it willnot neatly isolate the equation into two equations of motion for both αnj (t)37and βnj (t). The modified KBM method can be predicted by noting that allterms with αnj (t)'s have a factor of e−ijω0tand terms with βnj (t)'s have afactor of e+ijω0t. Exploiting this, (3.16) can be then multiplied by e±imω0tbefore averaging to decouple two separate equations of motion for α and β21.The rather extensive details of this procedure are demonstrated in AppendixD.The resulting equations of motion, once averaging over the period T =2piω0, will be second order, but this can be avoided by introducing a slow-timevariableτ =ω02t , ω0 =picL0(3.19)and keep the equations of motion up to the order . Choosing to do thisrestricts the resulting analysis of the photon number to short times related tothe internal periods of oscillation for the lowest photon modes, while cuttingout details that will occur within a larger time interval.3.4.2 ResultsFollowing the derivation described above and demonstrated in Appendix D,the following uncoupled differential equations for α and β are demonstratedbelow along with the results of Dodonov[2] ,Desrochers Calculation (this work)dαn1dτ = 3αn3 (t)− 2βn1 (t) (m = 1)dαnmdτ = (m+ 2)αn(m+2) − (m− 2)αn(m−2) (m ≥ 1)dβn1dτ = 3βn3 (t)− 2αn1 (t) (m = 1)dβnmdτ = (m+ 2)βn(m+2) (t) + (m− 2)βn(m−2) (t) (m ≥ 1)Dodonovs Calculation [2]dαn1dτ = 3αn3 (t)− βn1 (t) (m = 1)dαnmdτ = (m+ 2)αn(m+2) − (m− 2)αn(m−2) (m ≥ 1)dβn1dτ = 3βn3 (t)− αn1 (t) (m = 1)dβnmdτ = (m+ 2)βn(m+2) (t) + (m− 2)βn(m−2) (t) (m ≥ 1)(3.20)21In Dodonov's [2] derivation, which is referenced in many later articles, the results aresimply shown without explanation. This is the author's original take on the calculation,and will result in a slightly different result.38While deceptively trivial, the mixing term in the m = 1 equations ofmotion controls the rate of photons generated,dNtotaldτ= −∞∑n=11nηn1 (τ) ξn1 (τ) (3.21)where this would be zero if the mixing terms are zero, for example. Asseen in Appendix D, the proceedure is quite involved and prone to smallmistakes that could drastically change the outcome of this m = 1 term, orindeed, any aspect of the calculation.3.4.3 Result: Number of Photons GeneratedAfter some numerical work, which is described in Appendix E, the differencein the number of photons generated between the Desrochers calculation andthe Dodonov calculation is fairly substantial[2, 20]. For a realistic situation,the average distance between the two mirrors would be about a micron andif the mirror is driven to oscillate by 10% of this distance, one unit of slowtime will correspond tot1 =2L0pi= 6.4× 10−6s (3.22)and during one step of this slow time there will be 700 times more photonsgenerated than what is predicted by Dodonov. This is seen in figure 3.4below,39(3.23)Figure 3.4: Two different views in the Total Photon number as a function ofthe dimensionless time τ = 12ω1t where ω1 =piL0and where L0 is the initialposition of the right hand conductor in the two conductor setup. Notice thatthe amount of photon generation is significantly more than what is predictedby Dodonov[2]. This is due to the slight alteration of the first (m = 1) ofthe hierarchy of equations of motion for the mode function while the mirroris moving (3.20), which imply a great deal of sensitivity to the method usedto solve this problem. Due to numerical limitations, there are higher orderterms in the Number operator sum (3.9) that could not be included at thistime. The prediction is that the number of photons generated should be evengreater than what is shown in this Figure, and will be corroborated with theperturbation theory method depicted in Figure 3.5403.4.4 Problems with the MKBM ApproachThere are currently no experiments that have been performed which measurethe generation of photons from the excited vacuum, via the motional Casimireffect. This is due to a variety of possible factors, one of which is the difficultyin oscillating small conductive objects at the length scales required to see anyphotons. Another possibility is the lack of a strong theoretical prediction forthe number of photons expected in such an experiment.The sole reliance on Dodonovs Calculation, which is as far as the authorknows, the only one in the literature, could be the cause of a lack of exper-imental motivation. A point of contention with this calculation is the levelof approximation used to gain the result, as well as the likely possibility ofmaking a mistake in the expansive list of steps required. Since the photongeneration rate is entirely dependent on the m = 1 equation in equation(3.20), it is possible that this result is model dependent as this term couldchange if any of the approximations or steps taken were chosen differently.For these reasons it is important to find another method that can give amore reliable result.3.5 Method 2: WKBAs seen above, the modified KBM method used in the previous subsectionprovides equations of motion that are very sensitive to the terms in the firstin the hierarchy of equations of motion. This, along with the assumption ofthe slow time-scale and of the form of the mode functions when the mirror ismoving casts some doubt on the calculation. Another approach is to use theWKB method where instead of assuming the form of the time-dependence ofthe mode function, Cmn (t), in equation (3.12), we suppose the more generalform,Cmn (t) =∞∑k=0kCkmn (t) = C0mn (t) + C1mn (t) + · · · (3.24)where  is the fraction of the mirror's oscillation compared to it's spatiallocation as seen in (3.15), rewritten below,L (t) = L0 [1 +  sin (Ωt)] (3.25)Such an expansion permits the problem to be generally solved, order-by-order in this small parameter , which, in effect, allows one to focus the workon the relevant parts of the calculation.41v (t) = L˙(t)L(t)= Ω cos (Ωt)∑∞k=0 k+1 sink (Ωt)v˙ (t) = L˙(t)L(t)−L¨(t)L˙(t)L2(t)= Ω2∑∞k=0 k+1 (−1)k+1 v˙k (t)v (t)2 = Ω2 cos2 (Ωt)∑∞k=0 (k + 1) 2+k sink (Ωt)ωm (t) =npiL0[1+ sin(Ωt)]= npiL0∑∞k=0 k sink (Ωt)ω2m (t) =n2pi2L20∑∞k=0 (k + 1) k sink (Ωt)hmj = (−1)m+j 2mjj2−m2(3.26)Table 3.2: Expressions which appear in the equation of motion for Cmn (t),(3.14), are written in an expansion of the small parameter  .Consulting Table (3.2) for the relevant expressions in a power series of ,the equation of motion for Cmn, at all orders is,iC¨imn (t) +∑k,in2pi2L20(k + 1) k+i sink (Ωt)Cimn (t)=∞∑k=0∑jj 6= mk+i+1[2Ω cos (Ωt) sink (Ωt)hmjC˙inj (t)]+∞∑k=0∑jj 6= mk+i+1[Ω2 (−1)k+i+1 v˙k (t)hmjCinj (t)](3.27)+∞∑k=0∑jj 6= m∑`` 6= jk+i+2[Ω2 cos2 (Ωt)∞∑k=0(k + 1) sink (Ωt)hjmhj`Cin`]where from now on it is assumed that the diagonal components of habare vanishing for a = b, which allows the summation to be written simply.Zeroth Order TermTheO (0) differential equation is found below by keeping terms proportionalonly to 0 in equation 3.27, and is just the usual wave equation which is found42in the adiabatic limit to the Casimir problem,C¨0mn (t) +c2n2pi2L20C0mn (t) = 0where it's general solution,C0mn = η0mne−iωnt + ξ0mne+iωnt(3.28)The Cauchy problem is done at where the mirror just begins to movewhich means that the solution is,C0mn (0) = δmne−iωnt =⇒ η0mn = δmn , ξ0mn = 0 (3.29)C0mn = δmne−iωnt(3.30)First Order TermThe O (1) equation is a bit more involved, but is as below,d2C1mndt2+ ω2n (0)C1mn= −e−inω0t [2ω2n (0) sin (Ωt) δmn + 2Ω cos (Ωt)hmniωn (0) + Ω2 sin (Ωt)hmn]︸ ︷︷ ︸Vmn(t)(3.31)The general and particular solution of this is of the form,C1mn (t) = η1mne−iωmt + η1mne+iωmt + C1,pmn (t) (3.32)where the particular solution and the general solution coefficients are foundusing basic methods which are detailed in Appendix F .In this case, the particular function needs to be evaluated for p = m andp 6= m separately, as well as for n = m and n 6= m. The former classificationis defined first and is tabulated in Table 3.343(m 6= p) C1,pmn (t) =[(2n2p2 −m2)δmn −((−1)m−n 4p2n2(p2 −m2) (n2 −m2))δm 6=n]sin (pω0t)+ i[(0) δmn +((−1)m−n 2np3(p2 −m2) (n2 −m2))δm 6=n]cos (pω0t)(m = p) C1,pmn (t) =n2p2[2(pω0t cos2(p2ω0t)− pω0t2)− 2 cos(p2ω0t)sin(p2ω0t)]δpn+12p[2p(pω0t cos2(p2ω0t)− pω0t2)]hpn+12p[(i4npω0t− 2p) cos(p2ω0t)sin(p2ω0t)]hpnTable 3.3: The particular solution that satisfies the equation of motion (3.31).As the initial value has no terms proportional to , any Cauchy problemfor the higher order differential equations must vanish when the mirror beginsto move at t = 0. Therefore the initial value problem requires that the fullsolution and its derivative to vanish there,C1mn (0) = 0 , C˙1mn (0) = 0 (3.33)In the Appendix F this is done for all the cases and the results aretabulated in Table 3.4.(p = m) C1mn (t) = C1,pmn (t)=n2p2[2(pω0t cos2(p2ω0t)− pω0t2)− 2 cos(p2ω0t)sin(p2ω0t)]δpn+12p[2p(pω0t cos2(p2ω0t)− pω0t2)]hpn+12p[(i4npω0t− 2p) cos(p2ω0t)sin(p2ω0t)]hpn(p 6= m) C1mn (t) =12[−i pmBmn −Amnp2 −m2]e−iω0mt +12[−Amn + i pmBmnp2 −m2]e+iω0mt+[Amnp2 −m2]cos (pω0t) +[Bmnp2 −m2]sin (pω0t)Table 3.4: The Full Solution that solves the Cauchy Problem (3.33) with thedifferential equation (3.31)44This result shows that there is a peak in the scalar field's magnitudewhen the driving frequency is the same for any photon mode ω0n. Considertable 3.4, in the p = m solution, the presence of the factors ofω0 makes thisexpression much larger than those found in the m 6= p case. This meansthat by driving the mirror at a frequency nω0 one can enhance how manyphotons are generated with energy En = ~ωn.3.5.1 Number OperatorThe number density operator has been derived in equation (3.9), for thet > T solution to the Klein-Gordon equation (3.12). Equation (3.9)uses onlythe ηmn Bogoliubov coefficient, which can be found by solving yet anotherCauchy problem at t = T . This is done by noting that only the time-dependent function Cmn (t) in Equation(3.12), rewritten below, needs to beequated at t = T .φn (t, x) =1√mpi∞∑m=1Cmn (t)√z0z (t)sin(mpiz (t)x)(3.34)This, in principle, is how one actually must define a general Bogoliubovtransformation. To get the ηmn for both the p = m and p 6= m case, one sim-ply adds the Cmn (t ∈ [0, T ] ;T ) = Cmn (t > T ; T ) and C˙mn (t ∈ [0, T ] ;T ) =C˙mn (t > T ; T ) equations, then solves for ηmn for the t > T region. Theresults of this simple procedure are tabulated (3.5) below,45(p 6= m)ηmn =2[−Amn + i pmBmnp2 −m2]+2[Amnp2 −m2]e−imω0T(cos (pω0T )− i pmsin (pω0T ))+2[Bmnp2 −m2]e−imω0T(sin (pω0T )− i pmcos (pω0T ))(p = m)ηpn = n2p2[[pω0t− i](2 cos2(p2ω0T)− 1)+ [i2pω0 − 2] t cos(p2ω0T)sin(p2ω0T)]δpn−n2p2[sin2(p2ω0T)− cos2(p2ω0T)]δpn+(−1)m−n 2pn2p (n2 − p2)[[p2ω0T − ip] (2 cos2(p2ω0T)− 1)]δp 6=n+(−1)m−n 2pn2p (n2 − p2)[[(i4npω0T − 2p) + i2p2T + 4n]cos(p2ω0T)sin(p2ω0T)]δp 6=n+(−1)m−n 2pn2p (n2 − p2)[p (2nω0T + i)[cos2(p2ω0T)− sin2(p2ω0T)]]δp 6=nTable 3.5: The required Bogoliubov coefficients for the t > T solution tothe Klein Gordon equation (3.12), evaluated precisely at the point when themirror motion stops at t = T . Note that in the expression for the numberoperator (3.9), the ξmn coefficients are not required.3.5.2 Photon Number ResultsUsing the analysis in the preceding sections, the number of photons generatedcan be determined numerically. In Figure (3.5)we find a verification of theresults presented using the Modified KBM method, showing that the numberof photons generated in a short time stepτ ' 6.4× 10−6s (3.35)is about 700 times greater than what Dodonov predicted [2].Note that inthis perturbation theory methodology, one can see small oscillations in thenumber of photons, which were averaged out in the modified KBM method.46Figure 3.5: A graph of the photon number growth of the various methodsemployed in this manuscript, with respect to the dimensionless time τ = 2L0pi .It is noted that the prediction from this work's Modified MKBM calculationis in agreement with the new WKB calculation of this section. The differencebetween these two plots should be from the fact that higher order modecontributions in the MKBM method could not be included due to numericallimitations.What is more is that since there is no time-averaging of the equationsof motion, nor truncations of terms that are beyond the small time scale(3.19) , which if the reader recalls, truncates all second order terms in thedifferential equation, the number density can be evaluated for longer times.This means one can find how many photons you can realistically generate byoscillating a point conductor, a micron away from another, at an amplitudeof 10% of the distance, for common time scales, which is visualized in Figure3.647Driving Frequency Mode pPhotons Generated for DifferentComparison of the Total Numberp=1p=1.5p=2PhotonNumber (Log Scale) [Unitless]10−610−311000Dimensionless Time [Unitless]0 0.25 0.5 0.75 1 1.25Figure 3.6: The amount of photons generated for the first few driving fre-quencies of the mirror, which are proportional to ω1 =cpiL by 1, 1.5 and 2.Note that the p = 1.5 case is so small compared to the resonant values thatit isn't visible on the first graph, so a log scale plot is also given.What Figure (3.6) shows is that if the driving frequency of the right-handed mirror (or indeed, the left-handed mirror if you preferred) is an48integer multiple of the lowest photon mode ω1, then resonance will occurthat greatly amplifies the number of photons excited from the vacuum. If anexperimental apparatus could oscillate a conductor separated by a micron inthis way, the void between them should then glow violet.Figures 3.8 and 3.7 show how many photons are generated various modesof frequency ωn , for various driving frequencies. As seen in these figures, theamount of photons generated in modes that are not at the driving frequencyis trivial in comparison to those generated at the driving frequency.Figure 3.7: A demonstration that the dominant photon generation is thatwhich has a frequency that is equal to the driving frequency of the mirror.In particular, if the driving frequency is a half-integer multiple of ω1 =cpiL0,then none of the modes are favored.49Figure 3.8: A demonstration that the dominant photon generation is thatwhich has a frequency that is equal to the driving frequency of the mirror.In particular, if the driving frequency is a half-integer multiple of ω1 =cpiL0,then none of the modes are favored.503.5.3 Results: Energy DensityFurthermore, it is possible to calculate the VEV of the stress energy tensordirectly from the mode functions (3.12)by the definition (1.3). A depictionof this for a driving force that is double and triple the natural frequency ofthe system, ω0 =picL0is displayed in Figures 3.9and 3.10 respectively.Figure 3.9: A Depiction of the energy density contained between the twomirrors as a function of space and time for p = 2.51Figure 3.10: A Depiction of the energy density contained between the twomirrors as a function of space and time for p = 3.A comparison of these energy densities from those calculated in theFulling-Davies-Moore method[22, 23], depicted in Figure (3.3), is that theprofiles are quite a bit different. While both of these calculations are donefor a small amplitude driving of the right-most mirror, the Fulling-Davies-More method includes many more orders in perturbation theory, where thecalculation in this subsection keeps it only to first order. Computationally,the O (2) terms are much more difficult to calculate and so have not beenconsidered in this preliminary analysis.523.6 Discussion3.6.1 The Meaning of Spatial In-homogeneity in theEnergy DensityIt can be shown that in the motion of the conductor in the adiabatic limit,where  = 0 exactly, there will be no generation of photons as well as afully homogeneous energy density in the cavity between both conductors.The latter can be easily proved by looking at the mode function in the timeinterval t ∈ [0, T ] in the adiabatic limit,φn (t, x) =1√mpi∞∑m=1Cmn (t)√z0z (t)sin(mpiz (t)x)(3.36)where Cmn (t) for all regions will be the same, due to the Cauchy problemdefined at t = 0 and t = T ,∀t ∈ R : Cmn (t) = δmne−iωmt (3.37)Since in the t > T time interval, the Bogoliubov coefficients are ξmn =δmne−iωmtand ηmn = 0, the number of generated photons (3.9) will alwaysbe zero.The interpretation of the above solution is that one is moving the mirrorso slowly that the propagation of the information that the right mirror hasmoved will be instantaneous throughout the cavity. In other words, theenergy density is homogeneous in space and every observer in the cavity willinstantly know the exact position of the moving mirror boundary.The effect of going beyond the adiabatic limit by treating  as a small,yet finite, parameter is to enable relativistic causality where now the motionof the mirror is fast enough that there will be delay due to informationnow traveling at the speed of light22. This will cause the vacuum to reactdifferently at each spatial point and therefore have spatial in-homogeneityin the energy density. Both of these cases are depicted in Figure (3.11)22By cutting out the time dependence of the length of the cavity,which is equivalent tosetting  = 0, the mode function (3.36)no longer satisfies the wave equation. In truth theonly thing that guarantees that the information travels at the speed of the light is thatthe solutions to the mode functions satisfy the wave equation, which is Poincaré invariant.53Figure 3.11: Information propagation of the point-mirror's trajectory, de-picted by worldline z (t). In this depiction, spacetime event 0 will not realizethat the mirror has begun moving at all. Point 1 will be able to witnessthat the mirror has begun moving and how it moved for some time after,but will not be able to determine the trajectory after the time where thephoton ray γ1 intersects the world-line. In the adiabatic limit the world-linez (t) is moving so slow that it is almost exactly moving in the time directiononly, which means that all observer will basically see the same location forthe mirror. A mirror moving this slowly will not create photons or produceinhomogeneus energy densities.Therefore one cannot generate matter from the vacuum if there is notsufficient spatial in-homogeneity in the energy density in the cavity.An obvious contradiction to this statement are the various types of matterthat are modelled using a homogeneous energy density, such as a gas in abox or homogeneous and isotropic matter found in the Friedman-Walker-Robertson-Lamaître spacetime used in Cosmology. A possible resolutionto this contradiction is that of the energy scale. The discussion in thiswork details microscopic genesis of matter from vacuum fluctuations,whilethe models of matter just described are macroscopic matter that appears in54much larger quantities has been assumed to have existed for a much longerthan the relatively new matter generated in the motional Casimir effect. Inthis picture, new matter must spring forth from spatially inhomogeneousenergy densities, but can form into homogeneous energy densities later on.3.6.2 Detection of Particles - The Number generated andwhere they will beThese two means of describing the nature of the matter content in theCasimir system, the number of particles and the energy density, while dif-ferent, have an important connection as shown above. The number operatorwill reveal how many photons will be generated for a given mode and theenergy density will reveal the likely location of the space-time event of thisgenesis through the location of its peaks. This conclusion about the energydensity is admittedly derived from some liberal use of Ockhams razor, yetstands to be the most likely interpretation given that if the energy densityis homogeneous, there will be no photons detected.On the practical side, the length scales in the experiments attemptedthus far are of the order of a micron, so the location of the a detector mustbe determined with a great deal of accuracy, if it can be reduced to such asize at all. Figure 3.10 shows that while the energy density is smaller forhigher modes, the location of their generation from the vacuum will be morespread out. It should be noted that using the Fulling-Davies-Moore method,as depicted in Figure 3.3, the energy density is far more spread out than thecases shown in Figure 3.9 and related. The only way to get a clearer pictureof this would be to calculate higher order terms in both theories.55Chapter 4Imperfect Conductors,Scattering andTime-Dependent ReflectivityIn Chapter 3, it was shown that matter does indeed spring out of the vacuumif you stir it enough with some object that couples or interacts with thevacuum modes of the respective quantum field. In the cases discussed, aconductor is what is doing the stirring as it can restrict photons from enteringit's interior depending on its conductivity. In principle, one could do thesame for the vacuum modes of any other kind of matter, provided therewas a similar analogy to the conductor. In this Chapter, the Casimir effect,both static and motional is demonstrated for a fixed two conductor systemundergoing changes to its reflectivity. What this will show, is that it is notneeded to actually move the conductor to stir the vacuum as one only needsto control the conductive properties as a function of time.The analogy to this in the soup-spoon example, is having a slotted spoonthat, as a function of time, goes from a solid spoon to one that is slotted sonumerously that it is more like a loop instead. This sort of time-dependentreflectivity is more attainable in a laboratory setting, and so this new cal-culation can introduce a new method of detecting something akin to themotional Casimir effect.4.1 Statically Translucent Casimir EffectThis work has explored moving the conductors that enclose some regionof space-time in the above Chapter and we have discussed the effects onallowing a single mirror to be translucent in the Chapter before it. With theseelementary pieces of information one is now competent enough to inquireabout the Casimir effect with translucent bounding conductors. This ideahas been explored in the static reflectivity limit by Jaekel [11]and willbe briefly described in the following section.564.1.1 Scattering FormalismAs described in Chapter 2, specifically equation (2.13), the imposition ofa boundry value problem, via some conductor, at the boundaries, can bebest described as a reflection and transmission of modes from both sides ofthe boundary. Since this is the natural language of this problem, scatteringformalism is the best way to approach the discussion when there are twoconductors separating three different regions of space-time.In each region there is a scalar field that represents a photon field. Thisscalar-field will be separated into it's left and right moving modes which,given their location with respect to the mirrors, will be either within thecavity, going out from it or going into it. This will be denoted in the subscriptas seen in Equations (4.1), while it's left-going or right-going nature will bedenoted by writing the field as ψ or ϕ respectively. In the definition below,the superscript will denote the direction but afterwards it will be assumedthat the field symbol used will determine the field's direction. This setup isdepicted in Figure (4.1).ΦL(x, t) = ϕin (x− t) + ψout (x+ t)ΦCav(x, t) = ϕcav (x− t) + ψcav (x+ t)ΦL(x, t) = ϕout (x− t) + ψin (x+ t)(4.1)Figure 4.1: The decomposition of left and right going modes in the CasimirSetup.Similar to the analysis for one mirror in Chapter 2, there will be, for eachmirror, an S-matrix that related the boundary conditions that represent the57reflective properties of the mirror. Let the left mirror be denoted by mirror1 and the right mirror be denoted by mirror 2 , and then their S-matrixequations can be written down as,(ϕcav(ω)ψout(ω))= S1 (ω)(ϕin(ω)ψcav(ω))(4.2)(ϕcav(ω)ψout(ω))= S2 (ω)(ϕcav(ω)ψin(ω))(4.3)Si (ω) =(si (ω) ri (ω) e−2iωqiri (ω) e2iωqi si (ω))(4.4)where si [ω] and ri [ω] are the transmission and reflection coefficients ofmirror i for the photon-field with mode ω, and where qi is the spatiallocation of mirror i with respect to the origin. Later, their difference will besimply defined asq = q2 − q1 (4.5)It is recalled here that in this notation, perfect reflection is denoted by ri =−1 and s = 0.To properly determine the Casimir effect in the cavity, the cavity modes,as well as the out-going modes, must be written in terms of a linear responsewith respect to the in-going field. The linear response matricies for both ofthese descriptions is the Global Scattering Matrix S (ω) and the ResonanceMatrix R (ω)[11] respectively,Φout(ω) = S (ω) Φin(ω)Φcav(ω) = R (ω) Φin(ω) (4.6)where the matrix elements are given below,R11 =s1(ω)d(ω) R12 =s2(ω)r1(ω)d(ω) e−2iωq1R21 =s1(ω)r2(ω)d(ω) e−2iωq2 R22 =s2(ω)d(ω)(4.7)d (ω) = 1− r1 (ω) r2 (ω) e−2iω(q2−q1) (4.8)4.1.2 External Pressure CalculationTo obtain the vacuum expectation value of the pressures acting on eachmirror, the definition (1.8)is employed for each region depicted in Figure(4.1). In this case, the pressures acting from the Vacuum modes outside themirror is considered.58Using the global and resonant scattering matricies defined in equation(4.6) to represent all the scalar functions in terms of in-going modes, providesthe following results.The pressures on the left, right and inside the cavity of the mirror systemare written below, and directly calculated in Appendix G. They are,〈0∣∣TL11 (x, t)∣∣ 0〉 = limτ→0∫ ∞0dω T00({φLω (x, t) , φL∗ω (x, t+ τ)})=∫ ∞−∞dω|ω|2pie−iωτ=~4pilimτ→0lim+→0[1(+ iτ)2+1(− iτ)2]= limτ→0[− ~2cpiτ2]〈0∣∣TL11 (x, t)∣∣ 0〉 = limτ→0∫ ∞0dω T00({φLω (x, t) , φL∗ω (x, t+ τ)})=〈0∣∣TR00 (x, t)∣∣ 0〉where point-splitting regularization has been used to get the divergent, neg-atively infinite vacuum energy23and the space-time input (x, t) is put in as areminder that this quanity can in general not be a constant. This is the sameresult as we get in full Minkowski, even though the left and right sides of themirror are actually half of the Minkowski space24. Within flat space-time,one only sees a difference in this value if there is a compact spatial region,where there will be a finite contribution. This was shown in the first sec-tion of this manuscript in equation , and we reproduce it with this notationbelow.4.1.3 Cavity Pressure and The Static Casimir EffectIn the case between the two mirrors the pressure is below in equation (4.10),wherethe resonance matrix is used to write the cavity scalar field in terms of the23In the end of Chapter 1 it was noted that this diverent sector of Vacuum energy canbe treated with point-splitting regularization, which is akin to a sort of Energy Uncer-tainty Principle interpretation of the divergence, while one can impose a high-frequencycutoff. The former gives a negatively unbounded energy while the latter gives a postivelyunbounded energy. This sector of vacuum energies is not the focus of this work as it willalways be subtracted in the Lorentz force law on the conducting surface.24This is due to the fact that they both have uncountable Fock-spaces and therefore nofinite contribution to Tµν .59input scalar field,〈0 |T cav11 | 0〉 =12limτ→0∫ ∞−∞dω2pi|ω| g (ω) e−iωτ (4.9)g (ω) =1− |r1 (ω) r2 (ω)|2∣∣1− r1 (ω) r2 (ω) e2iω(q2−q1)∣∣2 (4.10)where further details are in Appendix G. With the pressures acting on themirror from both sides, which in 1+1-dimensions are forces, the total forceacting on each mirror can be given, directly from the Lorentz force law, by,fL =〈0∣∣TL00 − TCav00 ∣∣ 0〉 = f , fR = 〈0 ∣∣TCav00 − TR00∣∣ 0〉 = −f (4.11)f =12limτ→0∫ ∞−∞dω2pi~ |ω| [g (ω)− 1] e−iωτIt is simple to show that in the perfect reflection limit, where r1 (ω) r2 (ω) =1, the static Casimir force is recovered for both mirrors,f=12limτ→0∫ ∞−∞dω2pi~ |ω| [1− g (ω)] e−iωτ ;Definition=12limτ→0∫ ∞−∞dω2pi~ |ω|[1− piq∞∑n=0δ(ω − npiq)]e−iωτ ; Cauchy ResidueTheorem=12limτ→0[− ~cpiτ2− ~cpi2q2∞∑n=0ne−inpiqτ]= limτ→0− ~2cpiτ2 − ~cpi2q2 e−i(τcpiq)(e−i(τcpiq)− 1)2= limτ→0− ~2cpiτ2 + ~2picτ2︸ ︷︷ ︸divergences subtracted+~cpi24q2+(~pi3c3480q4)τ2 +O (τ4) ; Taylor Expansion= +~cpi24q2(4.12)where the usual cancellation of the divergences is noted.604.1.4 General ReflectivityA direct calculation for a static reflectivity follows. The definition C [ω] =~ |ω|will be used where the square brackets are for Fourier space functions,f =12limτ→0∫ ∞−∞dω2pi~ |ω| [g (ω)− 1] e−iωτ=∫ ∞0dω2pi~ |ω|[1− |r (ω)|2|1− r (ω) e2iωq|2 − 1]=∫ ∞0dω2pi~ |ω|[r∗ (ω) e−2iωq + r (ω) e2iω1 − 2 |r (ω)|2|1− r (ω) e2iωq|2]=∫∞0dω2pi ~ |ω|[r(ω)e2iωq−|r(ω)|2|1−r(ω)e2iωq|2 +r∗(ω)e−2iωq−|r(ω)|2|1−r(ω)e2iωq|2]=∫∞0dω2pi ~ |ω|[r(ω)e2iωq−|r(ω)|2(1−r(ω)e2iωq)[1 + r∗ (ω) e−2iωq +O (r∗2)]]+∫∞0dω2pi ~ |ω|[r∗(ω)e−2iωq−|r(ω)|2(1−r∗(ω)e−2iωq)[1 + r (ω) e2iωq +O (r∗2)]] ; Taylor Exp.(In the above, at each order any |r (ω)|2 cancels leavingwhat is below by mathematical induction)=∫∞0dω2pi ~ |ω|[r(ω)e2iωq(1−r(ω)e2iωq) +r∗(ω)e−2iωq(1−r∗(ω)e−2iωq)]=∫ +∞−∞dω2pir(ω)e2iωq(1−r(ω)e2iωq)~ |ω|=∫ +∞−∞dω2pi( ∞∑`=1r` (ω) e2iω`q)~ |ω|=∞∑`=1∫ +∞−∞dω2pir` (ω) ~ |ω| e−iω(−2`q) ; Fubini Thm.=∞∑`=1∫Rdt1r (t1)∫Rdt2r (t2) · · ·∫Rdt`r (t`)︸ ︷︷ ︸`timesC(−2`q −∑`i=1ti); Convolution=∞∑`=1~ |−2`q| [∫R dt r (t) eiωt∣∣ω=0]` ; 2`q  cti=[− 14piq2] ∞∑`=1[r1 [0] r2 [0]]``2;Only zeromodes contriubte!= −[ζr[0] (2)4piq2](4.13)Therefore the result for the force, in a general reflectivity case, isf = −[ζr[0] (2)4piq2](4.14)61where the ζyx is defined asζx (y) =∞∑`=1(x)``y(4.15)In the perfect reflectivity limit, which is r1 [0] = r2 [0] = −1 the usual staticCasimir force is recovered,f = −~cζ−1 (2)4piq2= − 14piq2∞∑`=1[−1]``2= − 14piq2(pi26)= − ~pic24q2− 4.138× 10−15N for q = 1µm (4.16)4.1.5 DiscussionAs seen in Equation (4.14),what Jaekel's work [11] calculates is that theCasimir force will depend only on the zero frequency modes of the reflectivity.What this means is that all the higher energy photons, and how they reflector pass through the conductor, will not change the Casimir effect. This isthe analog of the static Casimir effect, but in terms of a changing reflectivityinstead of a changing parallel plate distance L0.While one can tweak the reflectivities slowly, there won't be particleproduction as there is no inhomogeneity in the energy density within thecavity25. A fully non-adiabatic approach is required to see this productionand is done by the author in the next section.25The details on why inhomogeneity in the energy density is crucial for particle produc-tion is argued in the end of Chapter 3624.2 Time Dependent ReflectivityIt is well known that in elementary electrodynamics[14] if the mirror has acertain thickness to it with some conductivity σ in that thickness, then themirror will dissipate any initial amplitude A of the photons to the value Ae1within a distanced = ω√µ2(√1 +( σω)2+ 1)− 12(4.17)This is what is known as the skin-depth and it in general depends on thereflection and transmission coefficients of the conductor. The main idea ofthis work will be to extend the work done by Marc-Thierry Jaekel and SergeReynaud[11] by changing this skin depth (4.17) as a function of time, effec-tively creating an analog to the motional Casimir effect, which was discussedin full detail in Chapter 3.To achieve this, an external field will be introduced, that will play therole of controlling the reflectivity of the mirror as a function of time. In thiscase the reflectivities of both mirrors, r1 (ω, t) and r2 (ω, t) will be endowedwith time dependence as depicted in Figure 4.2.This will demands a specific form for the transmission coefficients too asthe scattering must meet a variety of realistic conditions such as unitarity,causality and other requirements. A review on what properties the S-matrixneeds to uphold is found in Chapter 2.63Figure 4.2: The case of the time-dependently reflective mirror, where an ex-ternal current is applied that introduces a time dependence to the reflectivityfor each mode of the photons on either side of the mirror.To proceed with this calculation we will insert the cavity mode functionsinto the definition of the vacuum expectation value of the pressure acting onthe mirror, then through the use of the now time-dependent resonance matrixR (ω, t)(4.6), the cavity mode functions are transformed into the incoming,Minkowskian, scalar field modes. Below is direct calculation of this, whereu = t− x and v = t+ x64〈0∣∣∣TˆCav11 (x, t)∣∣∣ 0〉= limτ→0∫ ∞0dω T11 ({(φcav (u) + ψcav (u)) , (φcav (v + τ) + ψcav (v + τ))})= limτ→0∫ ∞0dω T11({(R11ϕin(u) +R12ψin(v) +R21ϕin(v) +R22ψin(u)), · · ·})= limτ→0∫ ∞0dω T11({RA (t)ϕin(u) +RB (t)ψin(v) , RA (t)ϕin(v − x) +RB (t)ψin(u+ τ)})where RA = R11 +R21 , RB = R12 +R22=12limτ→0∫ ∞012dω ∂0(RA (t)ϕin(u) +RB (t)ψin(v))∂0(R∗A (t)ϕ∗in(u− x) +R∗B (t)ψ∗in(v + x))+12limτ→0∫ ∞012dω ∂0(RA (t)ϕin(v + τ) +RB (t)ψin(u))∂0(R∗A (t)ϕ∗in(v) +R∗B (t)ψ∗in(u))+12limτ→0∫ ∞012dω ∂1(RAϕin(v) +RBψin(u))∂1(R∗Aϕ∗in(u+ τ) +R∗Bψ∗in(v + τ))+12limτ→0∫ ∞012dω ∂1(RAϕin(u+ τ) +RBψin(v + τ))∂1(R∗Aϕ∗in(u) +R∗Bψ∗in(v))+12limτ→0∫ ∞012dω((+iω)RAϕin(u) + (−iω)RBψin(v)) ((−iω)R∗Aϕ∗in(u+ τ) + (+iω)R∗Bψ∗in(v + τ))+12limτ→0∫ ∞012dω((iω)RAϕin(u+ τ) + (−iω)RBψin(v + τ)) ((−iω)R∗Aϕ∗in(u) + (+iω)R∗Bψ∗in(v))=12limτ→0∫ ∞0dωω28pi |ω|(RAe−iω(t−x) +RBe−iω(t+x))(R∗Ae+iω(t+τ−x) +R∗Be+iω(t+τ+x))+12limτ→0∫ ∞0dω|ω|8pi(RAe−iω(t+τ−x) +RBe−iω(t+τ+x))(R∗Ae+iω(t−x) +R∗Be+iω(t+x))+12limτ→0∫ ∞0dω|ω|8pi(RAe−iω(t−x) −RBe−iω(t+x))(R∗Ae+iω(t+τ−x) −R∗Be+iω(t+τ+x))+12limτ→0∫ ∞0dω|ω|8pi(RAe−iω(t+τ−x) −RBe−iω(t+τ+x))(R∗Ae+iω(t−x) −R∗Be+iω(t+x))( above are all the terms that from the time-independent case)+12limτ→0∫ ∞0dω8pi |ω|(|RA,0 (t)|2 e+iωτ +RB,0 (t)R∗A,0 (t) e−iω(2x−τ))+12limτ→0∫ ∞0dω8pi |ω|(−iωRA (t)R∗A,0 (t) e+iωτ − iωRB (t)R∗A,0 (t) e−iω(2x−τ))+12limτ→0∫ ∞0dω8pi |ω|(RA,0 (t)R∗B,0 (t) e−iω(−τ−2x) + |RB,0 (t)|2 R∗B,0 (t) e+iωτ)+12limτ→0∫ ∞0dω8pi |ω|(−iωRA (t)R∗B,0 (t) e−iω(−τ−2x) − iωRB (t)R∗B,0 (t) e−iωτ)+12limτ→0∫ ∞0dω8pi |ω|(iωRA,0 (t)R∗A (t) eiωτ +RB,0 (t)ψin(t+ x) (iω)R∗A (t)ϕ∗in(t+ τ − x))+12limτ→0∫ ∞0dω8pi |ω|(iωRA,0 (t)R∗B (t) e−iω(−τ−2x) + iωRB,0 (t)R∗B (t) eiωτ)65The time-dependent aspect of this calculation now follows,〈0∣∣∣TˆCav11 (x, t)∣∣∣ 0〉Time-Dependent+12limτ→0∫ ∞0dω8pi |ω|(|RA,0 (t)|2 e+iωτ +RB,0 (t)R∗A,0 (t) e−iω(2x−τ))+12limτ→0∫ ∞0dω8pi |ω|(−iωRA (t)R∗A,0 (t) e+iωτ − iωRB (t)R∗A,0 (t) e−iω(2x−τ))+12limτ→0∫ ∞0dω8pi |ω|(RA,0 (t)R∗B,0 (t) e−iω(−τ−2x) + |RB,0 (t)|2 R∗B,0 (t) e+iωτ)+12limτ→0∫ ∞0dω8pi |ω|(−iωRA (t)R∗B,0 (t) e−iω(−τ−2x) − iωRB (t)R∗B,0 (t) e−iωτ)+12limτ→0∫ ∞0dω8pi |ω|(iωRA,0 (t)R∗A (t) eiωτ +RB,0 (t)ψin(t+ x) (iω)R∗A (t)ϕ∗in(t+ τ − x))+12limτ→0∫ ∞0dω8pi |ω|(iωRA,0 (t)R∗B (t) e−iω(−τ−2x) + iωRB,0 (t)R∗B (t) eiωτ)= limτ→0∫ ∞−∞dω2pi{ |ω|4(|RA (ω)|2 + |RB (ω)|2)} e−iωτ+ limτ→0∫ ∞−∞dω2pi{18 |ω|(|RA,0 (t)|2 − iωRA (t)R∗A,0 (t) + iωRA,0 (t)R∗A (t))} e−iωτ+ limτ→0∫ ∞−∞dω2pi{18 |ω|(−iωRB (t)R∗B,0 (t) + iωRB,0 (t)R∗B (t) + |RB,0 (t)|2)} e−iωτ+ limτ→0∫ ∞−∞dω2pi{18 |ω|(+iωRB (t)R∗A,0 (t) + iωRB,0 (t)R∗A (t) +RB,0 (t)R∗A,0 (t))}e−iω(τ+2x)+ limτ→0∫ ∞−∞dω2pi{18 |ω|(−iωRA (t)R∗B,0 (t) + iωRA,0 (t)R∗B (t) +RA,0 (t)R∗B,0 (t))} e−iω(τ−2x)= limτ→0∫ ∞−∞dω2pi{g1 (ω, t)} e−iωτ +∫ ∞−∞dω2pi{g2 (ω, t)} e−iω2x+∫ ∞−∞dω2pi{g3 (ω, t)} e−iω(−2x)66The result of the force acting, or up to a factor c, the energy density26,on any space-time location in the cavity is then given below in equation(4.18)where there is now a spatial dependence that comes from independentlychanging the conductive properties of each the mirror independently.f (x, t) =12limτ→0∫ ∞−∞dω2pic (ω) [1− g0 (ω, t)− g1 (ω, t)] e−iωτ (4.18)+−∫ ∞−∞dω2pi{g2 (ω, t)} e−iω2x −∫ ∞−∞dω2pi{g3 (ω, t)} e+iω2xg0 (ω, t) =12|ω|1−(1− |r (ω, t)|2)∣∣1− r (ω, t) e2iω(q2−q1)∣∣2g1 (ω, t) =18 |ω|(|RA,0 (ω, t)|2 + |RB,0 (ω, t)|2)+18 |ω|(iω[RA,0 (ω, t)R∗A (ω, t)−RA (ω, t)R∗A,0 (ω, t)])+18 |ω|(iω[RB,0 (ω, t)R∗B (ω, t)−RB (ω, t)R∗B,0 (ω, t)])(4.19)g2 (ω, t) =18 |ω|(RB,0 (t)R∗A,0 (t) + iω[RB,0 (t)R∗A (t) +RB (t)R∗A,0 (t)])g3 (ω, t) =18 |ω|(RA,0 (t)R∗B,0 (t) + iω[RA,0 (t)R∗B (t)−RA (t)R∗B,0 (t)])Where the RA (ω, t) and RB (ω, t) are a sum of the time-dependent res-onant matrix components, defined in Equation (4.6), defined below,RA (ω, t) = R11 (ω, t) +R21 (ω, t) = 0 (4.20)RB (ω, t) = R12 (ω, t) +R22 (ω, t) (4.21)R11 =s1(ω,t)d(ω,t) R12 =s2(ω,t)r1(ω,t)d(ω)R21 =s1(ω,t)r2(ω,t)d(ω,t) e−2iωq2 R22 =s2(ω,t)d(ω,t)(4.22)Note that the first term g0 (ω, t) is the adiabatic limit27calculated byJaekel[11], and the other expressions are exact expressions for beyondthe adiabatic limit that include the dynamics due to the changing conduc-tivity.26Recall that in 1+1-dimensions, the pressures are equal to forces, and the pressure areequal to the energy density in Tµν .27Adiabatic in the change in the reflectivity of the mirror.674.3 Specific Form of the Time-DependenceThe intention of this subsection is to model a conductor that has a sinusoidalreflectivity, which will be a common way to change the reflectivity in anexperimental setup.First, it is imporant to know that it is not sufficient to simply make thereflective properties of the conductor time-dependent in any way you want asthe scattering off the mirror, described through the S-matrix, must be causaland produce a Casimir force that is real. One can permit a non-unitary scat-tering, where the mirror absorbs vacuum modes that come into contact withit, but for our purposes it is assumed the conductor's properties are unitary.Non-unitary scattering off of a single mirror has been explored briefly byJaekel[15] for one mirror, but such considerations would only be usefulin a high-precision calculation needed to compare to, as of now, non-existantexperimental observation. This calculation is, for now, a demonstration thatone can in principle get the motional Casimir effect through the deformationof the boundarys conductivity.4.3.1 Modelling Time-Dependent ReflectivityFor our purposes, an appropriate form of the reflectivity and transmissioncoefficients are required and they must satisfy the unitarity conditions dis-played in equations (2.23). A convenient choice that satisfies all of theseproperties in the time-independent limit is the following,ri (ω) = − 11− iωΩ, si (ω) = 1 + ri (ω) (4.23)where i is the mirror number as depicted in Figure 4.2, where the leftmirror is 1 and the right mirror is 2. One can introduce time-dependenceby multiplying the reflectivity by a complex number A ∈ C that depends ontime,ri (ω, t) = −A0 (t) (AR + iAI) 11− iωΩ, si (ω, t) = 1 + ri (ω, t) (4.24)then demand that it provides unitary S-matrix scattering is encoded inthe equation below,s2 (ω, t) s∗2 (ω, t) + r2 (ω, t) r∗2 (ω, t) = 1 (4.25)In order for this condition to be satisfied, a form for A that satisfiesEquation (4.25) is required and can be found by substituting Equation (4.24)68into the unitarity condition,s2 (ω, t) s∗2 (ω, t)= (1 + r2 (ω, t)) (1 + r∗2 (ω, t))= 1 + r2 (ω, t) + r∗2 (ω, t) + |r2 (ω, t)|2= 1−A0 (Ar + iAi) 11− iωΩ−A0 (Ar − iAi) 11 + iωΩ+A20(A2r +A2i) 11− ω2Ω2= 1 +[−A0 (Ar + iAi)(1 + iωΩ)−A0 (Ar − iAi)(1− i ωΩ)+A20A2r +A20A2i] 11− ω2Ω2= 1 +(A20A2r +A20A2i − 2A0Ar + 2A0Ai ωΩ)+ iA0 (−Ar ωΩ −Ai +Ai +Ar ωΩ)︸ ︷︷ ︸=0 11− ω2Ω2= 1 +[A20A2r +A20A2i − 2A0Ar + 2A0Ai ωΩ]︸ ︷︷ ︸must be equal to −A20A2r − A20A2i11− ω2Ω2This provides an algebraic equation for A, where any assumed form of eitherthe imaginary or real parts of the number A can get a form that will maintainunitarity,A20(A2r +A2i)−A0 (Ar −Ai ωΩ)= 0 (4.26)The other unitary condition also gives us this exact same equation,s∗ (ω, t) r (ω, t) + s (ω, t) r∗ (ω, t) = 0=⇒ A2r +A2i −Ar +AiωΩ= 0 (4.27)4.3.2 Choosing a Form for the ReflectivityThe reflectivity desired is one where static reflectivity is at t = 0, at whichtime-dependence occurs for t > 0. As such, the time dependent functionA0 (t) must be unity for t = 0. The choice of which A0 (t) happens for latertimes must be chosen in a way where the S-matrix properties outlined inChapter 2 and is considered below.There will be problems if Ai , time imaginary part of A (t) , is allowed tobe non-zero, must be chosen carefully as to not make A diverge for ultravioletand infrared modes respectively. This problem occurs due to theωΩ term inEquation (4.26).This can be resolved by carefully choosing what AI will be and solvingfor AR. A possible choice that satisfies the Cauchy Problem outlined aboveis to choose Ai = −12 sin(ωΩ)e−ωΩ, then find the Ar that will satisfy (4.26),69the unitarity condition. This gives us the form for the real part,Ar =12± 12√1− sin2 (t) sin2(ωΩ)e−2ωΩ + 2ωΩe−ωΩ sin (t) (4.28)This choice is one that: a): is regular for ω = 0, for all times, b): isregular for ω →∞for all times, c): keeps Ar a real number and d): Satisfiesthe Cauchy problem thatr (ω, 0) = − 11− iωΩ(4.29)The reflectivity of the mirror is therefore,r2 (ω, t) = − (Ar + iAi) 11− iωΩ, s2 (ω, t) = 1 + r2 (ω, t) (4.30)Ai = −12cos(ωΩ)e−ωΩ , Ar =12±12√1− cos2(ωΩ)e−2ωΩ + 2ωΩe−ωΩ cos(ωΩ)(4.31)From this expression one can find the Reflection amplitude byR (ω, t) = r (ω, t) r∗ (ω, t)=1211− ω2Ω2cos2 (t)(1 +√1− cos2(ωΩ)e−2ωΩ + 2ωΩe−ωΩ cos(ωΩ))+1211− ω2Ω2cos2 (t)(12ωΩe−ωΩ cos(ωΩ))= cos2 (t)[1 +74(ωΩ)2 − 12(ωΩ)3+O((ωΩ)4)]However, making Ai zero will give the result AR ∈ {0, 1} which corre-sponds to A = A0 (t)AR, which is agreeable with the intentions of wantingsimple oscilitory reflectivity.r2 (ω, t) = − cos (t) 11− iωΩ, s2 (ω, t) = 1 + r2 (ω, t) (4.32)R (ω, t) = r (ω, t) r∗ (ω, t)=11− ω2Ω2cos2 (t)= cos2 (t)[1 +(ωΩ)2+(ωΩ)3+O((ωΩ)4)]704.4 Casimir Effect with One-MirrorTime-Dependent Reflectivity4.4.1 Form of the Casimir ForceTo gain insight into this calculation the left mirror located at q1 = 0 will beleft stationary and will be a perfect conductor so that r1 = −1 and s1 =0, while the right hand mirror (mirror 2 in this case), has time-dependentconductive properties determined by equations(4.31) and (4.24).With thiswe can calculate the force, re-written below,f (x, t) =12limτ→0∫ ∞−∞dω2pic (ω) [1− g0 (ω, t)− g1 (ω, t)] e−iωτ (4.33)−∫ ∞−∞dω2pi{g2 (ω, t)} e−iω2x −∫ ∞−∞dω2pi{g3 (ω, t)} e+iω2xg0 (ω, t) =12|ω|(1− |r2 (ω, t)|2)∣∣1 + r2 (ω, t) e2iω(q2−q1)∣∣2g1 (ω, t) =18 |ω|(|RA,0 (ω, t)|2 + |RB,0 (ω, t)|2)+iω8 |ω|[RA,0 (ω, t)R∗A (ω, t)−RA (ω, t)R∗A,0 (ω, t)]+iω8 |ω|[RB,0 (ω, t)R∗B (ω, t)−RB (ω, t)R∗B,0 (ω, t)]g2 (ω, t) =18 |ω|(RB,0 (t)R∗A,0 (t) + iω[RB,0 (t)R∗A (t) +RB (t)R∗A,0 (t)])g3 (ω, t) =18 |ω|(RA,0 (t)R∗B,0 (t) + iω[RA,0 (t)R∗B (t)−RA (t)R∗B,0 (t)])where the RA (ω, t) and RB (ω, t) are defined below, turn out to vanishin this case,RA (ω, t) = R11 (ω, t) +R21 (ω, t) = 0 (4.34)RB (ω, t) = R12 (ω, t) +R22 (ω, t) =s2 (ω, t) [1− 1]1 + r2 (ω, t) e−2iωL0= 0 (4.35)These terms vanish as R11 (ω, t) and R21 (ω, t) are proportional to s1 (ω, t),which vanishes as mirror 1 is a perfect conductor and the RB (ω, t) vanishes71as the limit in equation (4.35) also vanishes.R11 =s1(ω,t)d(ω,t) R12 =s2(ω,t)r1(ω,t)d(ω)R21 =s1(ω,t)r2(ω,t)d(ω,t) e−2iωq2 R22 =s2(ω,t)d(ω,t)(4.36)d (ω) = 1− r1 (ω, t) r2 (ω, t) e−2iω(q2−q1) (4.37)In this limit, the only term that survives is the now time-dependentg0 (ω, t) term, which, as a consequence, eliminates any space-dependence ofthe force as the last two terms in the force expression (4.33) vanish. Thefinal expression for the force is,f (t) =12limτ→0∫ ∞−∞dω2pic (ω)1− |ω|(1− |r2 (ω, t)|2)∣∣1 + r2 (ω, t) e2iω(q2−q1)∣∣2 e−iωτ(4.38)4.4.2 Adiabatic ResultUsing the result (4.14) obtained in the time-independent, imperfect conduc-tivity case, the time-dependent case where one mirror is a perfect conductoris,f (t) = − ~c4piq2∞∑`=1[−r2 [0, t]]``2Where the reflectivity isr1 [ω] = −1 , r2 [ω, t]|ω=0 = − cos (t)11− iωΩ∣∣∣∣ω=072Figure 4.3: Time depdendent reflectivity for the right most mirror, whilethe left mirror is left at perfect reflectivity (r1 [ω] = −1). This shows that nomatter how fast the reflectivity is changed, the result will only be akin to theadiabatic motional Casimir effect, and therefore will not produce photons.This is true because while there is time-dependence, the result is spatiallyhomogeneus and is a similar result to what was shown in Figure DiscussionAs shown in the above analysis, it is impossible to generate photons if oneof the mirrors has perfect reflectivity. The main issue that causes this isthat the terms g2 (ω, t) and g3 (ω, t) in the Casimir force expression foundin Equation (4.33) vanish in this case. The resolution to this problem willbe to either allow the static reflectivity mirror to have a realistic reflectivitybound by some plasma frequency Ω, while the other side reflects in a time-dependent way, or to let both sides reflect in a time-dependent way, as willbe shown in the next section.734.5 Casimir Effect with Two-MirrorTime-Dependent ReflectivityThe result above does not exhibit spatial dependence in the energy densitybetween the two conductors and thus is very similar to an adiabaticallymoving mirror, which has been shown to not generate photons [2]. This isbecause in the adiabatic limit the motion of the boundary conditions for themirror does not introduce any new terms to the wave-equation, as they allexist at higher orders in the non-adiabatic limit. For more on this, refer tothe end of Chapter (3) for a full discussion.The intention of this section is to simply demonstrate that in order toachieve the creation of photons, one must either reflect the both the mirrorsin a time-dependent fashion or to allow one to have static, but imperfectreflectivity while the other one changes as a function of time. The analysisof the number of photons generated through this process or the exact formof the energy density is a work in progress and will appear in a completework at a later time.4.5.1 Form of the Two-MirrorIn this case the two mirrors will reflect the same external current, as depictedin 4.2. Their reflectivities will both have the form derived in equation (4.26).This means that the terms RA and RB in equations (4.35) are as below,r1 (ω, t) = r2 (ω, t) = r (ω, t) (4.39)RA (ω, t) = R11 (ω, t) +R21 (ω, t)=s (ω, t)(1 + e−2iωL0r (ω, t))d (ω, t)=(1 + r (ω, t))(1 + e−2iωL0r (ω, t))|1 + r (ω, t) e2iωL0 |2=1 + r (ω, t)|1 + r (ω, t) e2iωL0 |2 +r (ω, t) e−2iωL0 + r2 (ω, t) e−2iωL0|1 + r (ω, t) e2iωL0 |2RB (ω, t) = R12 (ω, t) +R22 (ω, t)=s (ω, t) (1 + r (ω, t))d (ω, t)=1 + r (ω, t)|1 + r (ω, t) e2iωL0 |2 +r (ω, t) + r2 (ω, t)|1 + r (ω, t) e2iωL0 |274and their temporal derivatives are,RA,0 (ω, t) =ddt(1 + r (ω, t)|1 + r (ω, t) e2iωL0 |2 +r (ω, t) e−2iωL0 + r2 (ω, t) e−2iωL0|1 + r (ω, t) e2iωL0 |2)=r,0 (ω, t)|1 + r (ω, t) e2iωL0 |2 −2 (1 + r (ω, t)) r,0 (ω, t) e2iωL0|1 + r (ω, t) e2iωL0 |3+r,0 (ω, t) e−2iωL0 + 2r (ω, t) r,0 (ω, t) e−2iωL0|1 + r (ω, t) e2iωL0 |2−2r (ω, t) r,0 (ω, t) + r2 (ω, t) r,0 (ω, t)|1 + r (ω, t) e2iωL0 |3RB,0 (ω, t) =ddt(1 + r (ω, t)|1 + r (ω, t) e2iωL0 |2 +r (ω, t) + r2 (ω, t)|1 + r (ω, t) e2iωL0 |2)=r,0 (ω, t)|1 + r (ω, t) e2iωL0 |2 − 2(1 + r (ω, t)) r,0 (ω, t) e2iωL0|1 + r (ω, t) e2iωL0 |2+r,0 (ω, t) + 2r (ω, t) r,0 (ω, t)|1 + r (ω, t) e2iωL0 |2+r (ω, t) r,0 (ω, t) e2iωL0 + r2 (ω, t) r,0 (ω, t) e2iωL0|1 + r (ω, t) e2iωL0 |2What is important to notice is that neither of these turn out to be zerounless they are in the perfect mirror limit, and so the spatial dependence ismaintained in the expression of the force,(4.33), namely the last two termsthat have g1 (ω, t) and g2 (ω, t) as integrands. This provides a form for theforce,f (x, t) =12limτ→0∫ ∞−∞dω2pic (ω) [1− g0 (ω, t)− g1 (ω, t)] e−iωτ−∫ ∞−∞dω2pi{g2 (ω, t)} e−iω2x −∫ ∞−∞dω2pi{g3 (ω, t)} e+iω2x (4.40)4.6 DiscussionIn this Chapter a review of the static reflectivity case of the Casimir ef-fect with partially translucent conducting boundaries was derived and dis-cussed28. It was shown that this provides an analogy to adiabatic limit of the28This is a review of the work of M.T. Jaekel[11]75motional Casimir effect limit, where the propagation of information withinthe cavity is treated to be infinite and no particles are excited from the pho-ton vacuum. An introduction to the time-dependent reflectivity case wasderived by the author and it was demonstrated that this case is an analogto the motional Casimir limit discussed in depth in Chapter (3).However it was shown that to perturb the reflectivity of one side of theCasimir apparatus while leaving the other side at static reflectivity, onecannot generate photons unless both the mirrors are imperfect conductors.A plausible reason for this is that the imposition of perfect conductivityis unphysical and destroys the non-adiabatic dynamics derived in Equation(4.18). As seen in Chapter 2 a perfect conductor with an infinite plasmafrequency results in strange pheomonea such as acausal behaviour. What thisresult suggests is that lowering the reflectivity on the right side of the mirrorwill allow modes that were previously reflecting back and forth perfectlywithin the Casimir setup, to begin to leak out from the right side. Strangely,they will leak out in a way that is spatially uniform. A deeper analysis ofthis is required to fully understand this result and will be presented in futurework.The crescendo of this Chapter was the demonstration that changing thereflectivity of the mirrors can produce an inhomogeneous energy densitywithin the parallel plate capacitor system, and as argued at the end of Chap-ter 3, will require the production of photons. An analysis of the exact amountof photons or the behaviour of the energy density is a work in progress andwill appear in later work.76Chapter 5Discussion5.1 Discussion of the ResultsThrough a deeper understanding of the 1+1-Dimensional case, of both themathematical mechanisms of the finite part of the divergent stress energytensor, and the physical objects that inhibit quantum fields (ie. the con-ductor), a path way to explore higher dimensional phenomena is opened. In1+1-dimensions the only boundaries one can establish is that of the pointboundary, and this simplified the calculation while maintaining much of thephysics.It was shown in Chapter 1 that the resonance occuring within an enclosed,spatially compact conducting surface will produce a measurable force on thebounding conductors, which is a well known result. Exploring the effectsof a single point conductor was examined in Chapter 2 and it was shownthat vacuum fluctuations can affect a single piece of free conductor, but theeffect is neither measurable or significant to any deeper physics so long asthis conductor does not contain a spatially compact region of space-time.While one can likely excite the vacuum with an isolated free conductor,a much better way is studied in Chapter 2, where it was shown that for themotional Casimir effect, which is caused by moving the conductors, photonscan be excited. In particular, new results show that at integer multiplesof the resonant frequency of the conductor,picL0, one can greatly amplify theamount of photons excited from the vacuum state of the photon field. It isstill possible to obtain some yeild of photon generation at frequencies that area non-integer multiple to this frequency, but the effect is greatly diminished.The very nature of a particle was also explored and it was determined thatphotons will be excited from the vacuum, so long as the resulting time-dependent energy density of the region is spatially inhomogeneous. Thismeans that there is a certain threshold one must stir the vacuum so thatit doesn't have enough time to react to the changes of the boundary.Also, an alternative way to produce the motional Casimir effect has beenintroduced in Chapter 4, where instead of moving condutors at small scalesone can instead change the reflectivity of the conductor as a function of77time. This produces a time-dependent boundary in a similar way to simplymoving it. This is more attainable in a laboratory setup, and should resultin an easier way to make experimental verification of the motional Casimireffect. It was shown that under specific conditions it is possible to producea spatially inhomogeneous vaccum expectation value of the energy density,which is characteristic of photon generation. Further details, such as theyeild of photons generated through this process and the manner of time-dependence enacted on the conductor will be studied in future work.5.2 Experimental Observation of the MotionalCasimir Effect5.2.1 Fabry-Perot MeasurementsIn Chapter 4, the final expression for the Casimir force acting on a conductorin the static reflectivity limit was derived29and expressed in Equation (4.14).This expression is interpreted as being a sum over the number of internalreflections ` of a photon mode trapped in the Casimir setup. It was alsonoted by Jaekel[11] that this is in direct analogy to the Fabry-Perotexperiment.A unique idea would be to attempt a Fabry-Perot experiment that candetect the static or dynamic Casimir effect by simply looking at the trans-mitted, out-going light from the Casimir setup on some detector. While theinternal reflections of a lazer can be viewed as a pattern of rings and dots on ascreen or camera, it would be interesting to see what sorts of patterns wouldbe seen from strictly the plane-waves generated from the vacuum fluctuationsalone. A calculation that could predict what one would see by changing thereflectivity of the conductors or by vibrating them would be a new way ofdetecting the yet to be seen motional Casimir effect.5.2.2 Measurement of the DCE through Time-DependentReflectivityAs of the time of this writing, there has been no experimental evidence ofparticle production through the dynamic Casimir effect by virtue of a fluc-tuating conductor. The limitations in experiment are such that it wouldbe difficult to vibrate nano-scale structures such as the chromium coated29This was derivation of M.T. Jaekel[11]78silicon pieces used by Bressi[1] to detect the static Casimir effect. Fur-thermore, as predicted by the author of this manuscript, in order for thereto be an appreciable amount of photon excitation, the frequencies of thesevibrations must be exactly the frequencies that are integer multiples of30ω1 =picL0.A way to circumvent these difficulties is to apply a time-depdendentcurrent to the conductors that changes the reflectivities very quickly as afunction of time. As seen in Chapter4, this will produce photon excitationif done correctly. An experimental setup that can do this has been devisedalready by a third-party and some stronger theoretical analysis than whatwas done in Chapter 4 is currently underway.5.3 Higher-Dimensional Considerations5.3.1 Path-Integral MethodConsiderable effort has been put towards moving the Casimir effect to higherdimensions by those such as Golestanian[27], Li [28][29] and[12]. In Appendix H,a discussion on the Casimir effect in general space-time dimensions through the use of path integrals is given31. A a uniquederivation of the path-integral measure required to encode the conductor's lo-cation and reflective properties is derived and the way to higher-dimensionaltreatments are established. For more on this, refer this final Appendix.5.3.2 Time-Dependent SurfacesAs path integral methods are perfect for decribing Dirichilet and Neumannboundaries, which is effectively done by integrating over a specific set ofquantum field configurations like that done for Gauge theories through theFadeev-Popov method, a generalization of what was done in Chapter 3 andChapter 4 done for higher-dimensional geometry.In particular, a conducting bounding surface that changes location as afunction of time would impose a time-dependent path-integral measure in thegenerating functional of the theory. This change in the path-integral measureis seen in Appendix H.7, and would provide a new generating functional for30L0 is the conductor place separation.31This was a chapter that was meant to be included in this manuscript, but was notready in time.79the scalar field within the cavity,ZB [J ] =1N∫Dφ∏x∈Stδ (φ (x)) e−i~Smatter[φ]− i~∫dDxJ(x)φ(x)(5.1)where St is a surface that depends on time , N a normalization factor andJ (x) is an external source that one can probe the scalar field φ (x) with.In the same sense, since the transmission and reflection coefficients are inactuality given by a Dirichlet boundary condtion as seen in Equation (2.13),the time-dependence of the reflectivity can be encoded in this generatingfunctional as well (5.1) as well.5.4 Space-time Horizons, Hawking Radiation,Cosmological ConstantWhile one cannot screen gravitions in the manner that photons are screenedby conductors, it is possible to explore the Casimir effect in gravitationalsystems through the presence of spatially compact Horizons. A space-timehorizon can serve as a natural boundary condition for a region of space-timeand this, as shown in Chapter 3, should produce static or dynamic Casimireffects depending on the motion of the horizon. It was also shown in Chapter4 that this boundary need only reflect a few modes to create this effect.In the case of an evaporating Black hole, if the motion of the horizonis non-adiabatic, or if the horizon accelerates, then the production of parti-cles from the vacuum state of the quantum fields restricted should occur inaddition to the space-time dependent energy density within the black hole.Another consideration in this vein of thought would be that of the particlehorizon and its potential to produce a very small, but finite vacuum energydensity through the Casimir effect, which could be interpreted as the sourceof the Cosmological constant in Einstein's Field equations. Through thisidea, if the calculation is tractable, it could be possible to predict the radiusof the particle horizon just from the Cosmological constant's value.Both of these possible horizon-induced vacuum effects would appearthrough the mathematical mechanism of a discrete or countable Fock spacefor the bounded quantum fields. Physically, an argument against this ideawould be that these effects would come as a result of some accumulatedreflection of the vacuum modes due to these boundaries, but as they arehorizons, these modes would never actually touch them. However, sincequantum fields can be interpreted as covering all of space-time and are eventied to an innate feature of space-time, the fiber-bundle, it is possible that80by some correlations the horizon can be felt inside just as the interior of aCasimir setup feels the conductors through the change in energy densitywithin.A thorough calculation of these effects has a reasonable potential to eithertie the Casimir effect to these gravitational phenomena or to rule them outas considerations and is being considered.5.5 Final ThoughtsThe Casimir effect is a phenomena that's been studied for almost a century,but what allows it to remain relevant and singular with respect to other areasof study is its direct relation quantum field theory and time-dependent space-times. Future work will use what was learned here to expand the discussionand hopefully discover new physics.81Bibliography[1] G Bressi, G Carugno, R Onofrio, and G Ruoso. 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The intention is toprovide a complete and pedagological description of all the results providedthroughout the chapters and discussion.85Appendix ACalculation of the StressEnergy Tensor in 1+1DimensionsA.1 Determination of 〈0 |Tµν| 0〉 in MinkowskiSpaceIn order to know exactly what divergence to subtract from the VEV of theStress energy tensor〈0 |Tµν | 0〉inside the cavity, one must know the divergenceoutside of the cavity. This section will calculate the divergent 〈0 |Tµν | 0〉 onthe outside of the Casimir system, which is the same as that of the MinkowskisystemEnergy Density and PressureIn 1+1-dimensions, the pressure and the energy density are the same as seenin the following calculation.86〈0 |T00| 0〉 = 〈0 |T11| 0〉=∫dωT00 (φn (x, t) , φ∗n (x, t)) ; Definition= limτ→0∫dωT (φω (x, t) , φ∗ω (x, t+ )) ; Point Splitting= ~2c limτ→0∫ ∞−∞dω ∂0 [ϕω (t− x)] ∂0 [ϕω (t+ τ − x)] ; Definition of Tµν+ ~2c limτ→0∫∞−∞ dω ∂1 [ϕω (t− x)] ∂1 [ϕω (t+ τ − x)]=~2climτ→0∫ ∞−∞dω (iω) [ϕω (t− x)] (+iω) [ϕω (t+ τ − x)]+ ~2c limτ→0∫∞−∞ dω (iω) [ϕω (t− x)] (−iω) [ϕω (t+ τ − x)]= ~c limτ→0∫ ∞−∞dωω22pi |ω|[e−iω(t−x)e+iω(t+τ−x)]= ~c limτ→0∫ ∞−∞dω2pi|ω| eiωτ= ~4cpi limτ→0lim+→0[1(+ iτ)2+1(− iτ)2]: Fourier Transform= limτ→0(− ~2cpi1τ2); Result(A.1)〈0 |T01| 0〉 = 〈0 |T10| 0〉=∫dωT01 (φn (x, t) , φ∗n (x, t)) ; Definition= limτ→0∫dωT01 (φω (x, t) , φ∗ω (x, t+ )) ; Point Splitting= ~2c limτ→0∫ ∞−∞dω ∂0 [ϕω (t− x)] ∂1 [ϕω (t+ τ − x)] ; Definition of Tµν+ ~2c limτ→0∫∞−∞ dω ∂1 [ϕω (t− x)] ∂0 [ϕω (t+ τ − x)]= ~c limτ→0∫ ∞−∞dω|ω|ω2pi |ω|[e−iω(t−x)e+iω(t+τ−x)]= ~c limτ→0∫ ∞−∞dω2piω eiωτ= 0 : Odd function(A.2)Therefore the full result in Minkowski space, or indeed, Half-Minkowskispace, is,〈0 |Tµν | 0〉Minkoski= − limτ→0(~2cpiτ2[1 00 1])(A.3)87A.2 Determination of 〈0 |Tµν| 0〉 inside the CavityInside of the cavity, the Fock-space is countable and therefore the Lebesqueintegral to use is the sum, which is displayed below for〈0∣∣T 00∣∣ 0〉=∞∑n=1T (φn (x, t) , φ∗n (x, t)) ; Definition= lim→0∞∑n=1T (φn (x, t) , φ∗n (x, t+ )) ; Point Splitting= lim→0∞∑n=112(ωnL0e+it sin2 (knx) +(ωnL0)e+it cos2 (knx)); Substitution= ~2L0 lim→0∞∑n=1e+i cnpiL0ωn ; Trig Identity= ~pi2L20lim→0∞∑n=1e+i cnpiL0n ;= ~pi2L20lim→0e+i cpiL0(e+i cpiL0 − 1)2 ; ∞∑n=1neing(u) =eig(u)(eig(u) − 1)2= ~pi2L20[lim→0(− L20pi22)− 112− lim→01240c2pi2L202 +O (4)] ; Taylor Expansion=[−lim→0~2cpi2− ~pic24L20]; Result(A.4)The momentum density will still vanish however,〈0∣∣T 01∣∣ 0〉 = 〈0 ∣∣T 10∣∣ 0〉 = 0 (A.5)88Appendix BLinear Response for the ForceActing on a Single MirrorIn the process of deriving the fluctuation-dissipation theorem, which linksthe linear response that defines the motional force, to the auto-correlationfunction of the QED vacuum, Jaekel [13] provides a derivation of the linearresponse function directly from the definition of the stress energy tensor in1+1-dimensions displayed in Equation (1.8). The form of the force which,is proportional to the mirror's motion δz (t), is given through the Lorentzforce law and is expressed below,〈0 |Fm| 0〉 = 〈0 |δF (t)| 0〉 − δz (t) ∂t 〈0 |G (t)| 0〉 − dδz (t)dt〈0 |G (t)| 0〉 (B.1)〈0 |δF (t)| 0〉 =∫Rdω′2pi∫Rdω2pie−i(ωt+ω′t′)iωiω′Tr{[1 00 −1]δCout[ω, ω′]}〈0 |G (t)| 0〉 =∫Rdω′2pi∫Rdω2pie−i(ωt+ω′t′)iωiω′Tr{[1 00 −1] [I − S (ω′)S (ω)]}δCout[ω, ω′]=∫dω′′2pi[δS(ω, ω′′)Cin(ω′′, ω′)ST(ω′)+ S (ω)Cin(ω, ω′′)δST(ω′, ω′′)]δS(ω, ω′)= iω′δz[ω − ω′] [S (ω) [ 1 00 −1]−[1 00 −1]S(ω′)]where δS (ω, ω′) is the first perturbation to the S-matrix in the movingmirror limit in the laboratory frame. Cin(ω, ω′) is the fourier transform of thetwo-point correlation function of the scalar field, represented in the scatteringformalism applied by Jaekel The expressions that are a function of〈0 |G (t)| 0〉 correspond to the energy flux seeping into the conductor andcorrespond to the part of the Lorentz force in the second part of Equation(B.2)〈Fz〉 =∫∂VT zαnαdA− 1c2ddt∫VSzdV (B.2)89Note that in unitary scattering, ST (ω)S (ω) = I and so this expressionvanishes, as expected.From this, in the unitary scattering case, the Static Vacuum limit, andafter some reorganization, the force can then be represented asFm (t) = 〈0 |δF (t)| 0〉 =∫dω2piχ [ω] δz [ω] e−iωt (B.3)χ [ω] = χ [ω] = 3imτ∫ ω0(ω − ω′)ω′ [1− s [ω] s [ω′]+ r [ω] r [ω′]] (B.4)90Appendix CSolutions to Moore's EquationThe equation in the method used is referred to as Moore's Equation andit is detailed in Moore's paper [23].This work simply expands his analysis toinclude all the details of the calculation that were left out.Reduction 1):Express Moores EquationThe conformal mapping, which is depicted in Figure 3.1, is expressed belowt− z (t) = f (w − 1) , t+ z (t) = f (w + 1) (C.1)f−1 (t− z (t)) = w − 1 , f−1 (t+ z (t)) = w + 1 (C.2)=⇒ R (t+ z (t)) = R (t− z (t))− 2 R (ρ) ≡ f−1 (ρ) (C.3)We note that if we take the derivative of this, we get the differentialequation,∂R (t± x)∂t=∂R∂ρ∂ρ∂t=∂R∂ρ= ±∂R (t± x)∂x(C.4)The solution to this will grant the inverse conformal transformation,which will provide the real space-time solution to 〈0 |Tµν | 0〉.Reduction 2): Change Coordinates to ξ = xz(t)The change to a unitless distanceξ = xz(t) maps the cavity interval to theunit intervalx ∈ [0, z (t)] ⇐⇒ ξ ∈ [0, 1] (C.5)This means that we are going from (x, t)→ (ξ, t) and hence,∂∂x=∂ξ∂x∂∂ξ+∂t∂x∂∂t=1q∂∂ξ(C.6)∂∂t=∂ξ∂t∂∂ξ+∂t∂t∂∂t= − xz (t)2∂z∂t∂∂ξ+∂∂t(C.7)91Hence we have0 =∂R (t± x)∂t∓ ∂R (t± x)∂x=[(1∓ ξz˙ (t)) ∂∂ξ± z (t) ∂∂t]R (t, x)= 〈(1∓ ξz˙ (t)) ,±z (t)〉 ·〈∂∂ξ,∂∂t〉R (t, x) = 0 (C.8)This final form is written so that the characteristic of the PDE is clearlyvisible.Reduction 3): Find a useful form of R (x, t)The form for R (t, ξ) that will be chosen isR (t, ξ) = g (ξ, t) +∫1z (t)dt (C.9)where,g =∞∑n=0∂ng (ξ, s)∂n∣∣∣∣=0n (C.10)andg(k−1) (ξ, s) =k∑j=0αkj (s) (±ξ)j (C.11)The reason for choosing this form is due to the fact that the partialdifferential equation (C.8) is not easy to solve exactly. Each term in Equation(C.9) is explained below,1. Find the main time dependence of the function.The second term, which is the integral of the inverse of the mirror'sworldline is justified in this part. This form is due to an approximationto the characteristic equation for the above differential equation.Wehave that∂ξ∂τ = (1∓ ξz˙ (t)) and ∂t∂τ = ±z (t) so the characteristic equa-tion is,∂ξ∂t=1∓ ξ ∂z∂t±z (t) = ±1z (t)− ξ z˙ (t)z (t)(C.12)To proceed here we make an adiabatic approximation so that z˙  1(much less than the speed of light) and that the characteristic curve is∂ξ∂t=1∓ ξ ∂z∂t±z (t) = ±1z (t)− ξ z˙z' ± 1z (t)(C.13)92=⇒∫dξ + C = ±∫1z (t)dt =⇒ C = ∓ξ +∫1z (t)dt (C.14)which means that R (t, ξ) ' h(∓ξ + ∫ 1z(t)dt) for some general h thatsolves the IVP and BC32. The form of this f is easily found. At x = 0,we recall the boundary conditionsf (w − s)|s=0 = t− x|x=0 =⇒ f (w) = t (C.15)g (w + s)|s=0 = t+ x|x=0 =⇒ g (w) = t (C.16)Since the wall is not moving, there is no reasonable conformal trans-formation other than the identity that it can take here, sog (w) = f (w) = w = t ⇐⇒ t = w (C.17)This then means thatR (t, 0) ' h(∫1z (t)dt)= f−1(∫1z (t)dt)=∫1z (t)dt (C.18)2. Find the slow time-dependenceThis refers to the first part of Equation (C.9). We can now supposethat the solution is of the formR (t, ξ) = ∓ξ +∫1z (t)dt (C.19)where ξ can be considered a parameter with respect to time (as in itdoesn't change). However since we are dealing with an approximationwe say it is a slowly varying function of time (this is the method offast and slow variables), which then has the formR (t, ξ) = g (∓ξ, t) +∫1z (t)dt (C.20)where we introduce a slow time variable s = t, with 0 <   1 suchthat∂g∂s has a value that cannot be ignored in the slow time limit.32Initial value problem and boundary condition.93Reduction 4): Express Moore's Differential Equation in Slowtime VariablesEquation (C.9) wil now be expressed using the results of the previous reduc-tions. Set s = t where  is a small parameter and then we get that[(1∓ ξz˙ (t)) ∂∂ξ ± z (t) ∂∂t]R (t, x) = 0 ; Starting0 =[(1∓ ξ∂z∂s)∂∂ξ ± z (t)  ∂∂s]R (t, x)=[(1∓ ξ∂z∂s)∂∂ξ ± z (t)  ∂∂s] (g (ξ, t) +∫1z(t)dt);Reduction 3=[(1∓ ξ∂z∂s) ∂g∂ξ ± z (t) ∂g∂s ± z (t) ∂∂t(∫1z(t)dt)]=[(1∓ ξ∂z∂s) ∂g∂ξ ± z (t) ∂g∂s ± 1]; Result(C.21)Reduction 5): Assume a form for gIn this reduction, a form for g (±ξ, t) is derived. The idea is to express g asan expansion in the small parameter .g =∞∑n=0∂ng (ξ, s)∂n∣∣∣∣=0n (C.22)The first term of this is, recalling that s = t , should be the below due to theapproximation made for the characteristic of the PDE, found in Equation(C.19),g0 (ξ) = g (ξ) 0 = ∓ξ (C.23)Plugging this into Moores equation shows that it will be symmetric in ξ,so the ambiguity of the ±ξ can be ruled out,R (t+ z (t)) = R (t− z (t))− 2 ;Moores Equation.R (t+ (ξ = 1)) = R (t− (ξ = −1))− 2 ; Evaluated at mirror.∫1zdt+ 1 +∞∑n=1gn (1, s)=∫1zdt− 1 +∞∑n=1gn (−1, s)− 2 ; Algebra∞∑n=1gn (1, s) =∞∑n=1gn (−1, s) ; It is symmetric(C.24)94Reduction 6): Obtain a Hierarchy of equations for gnWe plug this the result above into[(1∓ ξ∂z∂s) ∂g∂ξ ± z (t) ∂g∂s ± 1]= 0 to getthat0 =[(1∓ ξ∂z∂s) ∂g∂ξ ± z (t) ∂g∂s ± 1]; Differential Eq.=[∂∂ξ (∓ξ + u (s))± 1]0 ; Order in powers of .+∞∑n=1[∂gn∂ξ∓ ξ ∂z∂s∂g(n−1)∂ξ± z (t) ∂g(n−1)∂s]n = 0=⇒ ∀n ≥ 1 : ∂gn∂ξ ∓ ξ ∂z∂s ∂g(n−1)∂ξ ± z (t) ∂g(n−1)∂s = 0 ; Find Heirarchy of Eq.(C.25)Reduction 7): Express each gn (ξ, s) as another small parameterexpansionThe objective here is to express gn (ξ, s) in the formg(k−1) (ξ, s) =k∑j=0αkj (s) (±ξ)j (C.26)so thatg0 (ξ, s) =∞∑j=0α1j (s) (±ξ)j (C.27)1. This means that α10 (s) = 0Since we want to recover g0 (ξ, s) = ∓ξ, we getg0 (ξ, s) = α10 (s) + α11 (s) (±ξ) +O(ξ2)= 0∓ ξ2. We have that α00 (s) =∫1z(s)ds , the anti-derivative of1z(s) , sinceg(k−1) (ξ, s) =∑kj=0 αkj (s) (±ξ)j for k = 0 gives,g−1 (ξ, s) =0∑j=0α0j (s) (±ξ)j = α00 (s) (C.28)We demand this as a way to absorb∫z−1 (t) dt in the expansionR (t, ξ) = g (∓ξ, t) + ∫ 1z(t)dt.95Reduction 8): Get a form for the αjk (s)The objective here is to plug the result in the previous reduction into[∂gn∂ξ∓ ξ ∂z∂s∂g(n−1)∂ξ± z (t) ∂g(n−1)∂s]= 0 (C.29)to get a simple result. This analysis is below,[∂gn∂ξ ∓ ξ ∂z∂s ∂g(n−1)∂ξ ± z (t) ∂g(n−1)∂s]= 0 ; Differential Eq.=⇒ ∑nj=0 (j + 1)αj+1k (s)− j ∂z∂sαjk−1 + z (t) ∂αjk−1∂s = 0 ; Plugged in=⇒ αjk (s) = −1j[− (j − 1) ∂z∂s + z (s) ∂∂s]αj−1k−1 (s) ; Solve for αjk (s)= − zj(s)j[−z−j (s) (j − 1) ∂z∂s + z−j+1 (s) ∂∂s]αj−1k−1 (s) ; Multiply by zj (s) z−j (s)= − zj(s)j ∂∂s[z(−j−1) (s)αj−1k−1 (s)]; Chain Rule= − zj(s)j ∂∂s[z(−j−1) (s)(− zj−1(s)j−1 ∂∂s[z(−j−2) (s)αj−2k−2 (s)])]; Recursive Iteration= (−1)2 zj(s)j(j−1) ∂2∂s2[z(−j−2) (s)αj−2k−2 (s)]; Reorganize= (−z(s))jj!djdsj[α0k−j (s)];Principle ofMathematical Induction(C.30)Reduction 9): Enforce the symmetry Dirichlet ConditionThe boundary condition is simply the symmetry condition derived in equa-tion (C.24). This turns out to be important as it gives a means to solve forαjk in general. This is done below,1. Apply Boundary ConditionsAt each order k we have this Symmetry Condition,gk (−1, s) = gk (+1, s) ; Symmetry condition∑kj=0 αjk (s) (−1)j =∑kj=0 αjk (s) ; Impose ±ξexpansionα0k − α1k + α2k + · · · = α0k + α1k + α2k + · · ·∑kj=0 α2jk (s) =∑kj=0 α2jk (s) + 2∑kj=1 α2j+1k ; Isolate Even\Odd0 =∑kj=1,3,5 αjk (s) ; Odd solutions vanish.0 =k∑j=1,3,5(−z (s))jj!djdsj[α0k−j (s)](C.31)96k∑j=1,3,5(−z (s))jj!djdsj[α0k−j (s)]= 0 (C.32)2. Even k solutions vanish.In the above, we have this for all k ∈ {2, 3, 4, . . . }, but only the odd kare non-vanishing. This is true asα01 (s) = 0 (C.33)This carries forward recursively for all even k, so only odd k survive.You can show this by taking the equation (C.32) and solving for eachk = 2`.3. Get tidy formula for odd k.From (C.32), we get for odd k=2`+1∑j=1,3,5(−z (s))jj!djdsj[α02`+1−j (s)]; Only odd kare non vanishing=2∑`j=0,2,...(−z (s))j+1(j + 1)!dj+1dsj+1[α02`−j (s)]; j → j + 1= −z (s)2∑`j=0,2,...(−z (s))j(j + 1)!djdsj[dα02`−j (s)ds]; Factor −z (s)= −z (s)∑`j=0(−z (s))2i(2i+ 1)!d2ids2i[dα02`−2j (s)ds]; j → 2j= −z (s)∑`j=0(−z (s))2i(2i+ 1)!d2ids2i[γ`−j (s)] ; γn = dds(α02n (s))(C.34)This gives the result,∑`i=0(z (s))2i(2i+ 1)!d2ids2i[γ`−j (s)] = 0 (C.35)97Reduction 10): Write R (ξ, t) in terms of γ`.With the main result (C.32), the inverse conformal mapping R (u) can beexpressed quite neatly,R = g (ξ, t) +∫1z(t)dt ; Definition=∞∑k=0∂kg (ξ, s)∂k∣∣∣∣=0k +∫1z (t)dt︸ ︷︷ ︸α00(s); Absorb∫1z(t)dt=∞∑k=0k∑j=0αkj (s) (±ξ)j k ; Expand gk=∑∞k=0∑kj=01j! (∓z (t))j djdtj[αk−j,0 (s)] kξj ; Recursive Def of αkj (s)=∑kj=0∑∞k=01j! (∓z (t))j djdtj[αk−j,0 (s)] kξj ; Tonelli's Theorem=∑kj=01j! (∓z (t))j djdtj[∑∞k=0 αk−j,0 (s)] kξj ;=∑kj=01j!djdsj[∑∞k=0 αk−j,0 (s)] k (∓ξz (t))j ;=∑kj=01j!djdtj[∑∞k=0 αk−j,0 (t)] (∓x)j ; Use xz(t) = ξ=∑∞k=0 αk,0 (t∓ x) ; Taylor's Theorem=∑∞k=0∫ t∓xτ γk (s) ds ; Define in terms of γk(C.36)98Appendix DThe Modified KMB MethodHere the explicit details of the Modified Krylov-Mitropolski-Bogoliubov (KMB)method are expressed. As far as the author is concerned, this is the onlypublicly available calculation using this method. The terms that are usedin the determination of the number operator density are as follows in TableD.1A+±mj =12 {δp,n±j + δp,−n∓j} B+±nj = i2 {δp,n±j − δp,−n∓j}A−±nj =12 {δp,n∓j + δp,−n±j} B−±nj = − i2 {δp,n∓j − δp,−n±j}hm(m+2) =m2(m+2)(m+1) hm(2−m) =m2(m−2)(m−1)hm(m−2) = −m2 (m−2)(m−1) hm(−m−2) = −m2 (m+2)(m+1)Table D.1: Terms that will appear in the calculation of the Bogoliubovcoefficients used in the Modified Krylov-Mitropolski-Bogoliubov method.where these will be used in decoupling the equations of motion in termsof α and β, then solving for them. Their explicit calculation is below,99D.1 Averaged CoefficientsThe time-averaging mentioned requires the calculation of these termsA+±nj=1T∫ T0cos (ωpt) ei(ωn±ωj)tdt=1T(ω2p − ω20 (n± j)2) [ωpeiω0(n±j)T sin (ωpT ) + iω0 (n± j) eiω0(n±j)t cos (ωpT )− iω0 (n± j)]=1Tω20(p2 − (n± j)2)ω0[peiω0(n±j)T sin (pω0T ) + i (n± j) eiω0(n±j)t cos (pω0T )− i (n± j)]=ω02piω20(p2 − (n± j)2)ω0[p sin ei2pi(n±j) + i (n± j) eiω0(n±j)t cos (2pip)− i (n± j)]=i (n± j)2pi(p2 − (n± j)2)[ei2pi(n±j) − 1]=0 ; |n± j| 6= plimx→±p ix[ei2pix−1]2pi(p2−x2) = limx→±pi([ei2pix−1]+i2pixei2pix)−4pix =−2pi−4pi(±p±p); n± j = ±p=0 ; |n± j| 6= p12; n± j = p12; n± j = −p=12{δp,n±j + δp,−n∓j}B+±nj =1T∫ T0sin (ωt) ei(ωn±ωj)tdt=1T(ω2 − ω20 (n± j)2) [iω0 (n± j) ei(ωn±ωj)t sin (ωt)− ωei(ωn±ωj)t cos (ωt)]T0=ω02piω20(p2 − (n± j)2)[iω0 (n± j) ei2pi(n±j) sin (2pip)− ωei2pi(n±j) cos (2pip) + ω]= − p2pi(p2 − (n± j)2)[ei2pi(n±j) − 1]= −0 ; |n± j| 6= plimx→±p p(e+2ipix−1)2pi(p2−x2) = limx→±pp2ipie+2ipix−4pix = limx→±p2ipi−2pi(p±p); n± j = ±p=0 ; |n± j| 6= p+ i2; n± j = p− i2; n± j = −p=i2{δp,n±j − δp,−n∓j}100A−±nj =1T∫ T0cos (ωt) ei(−ωn±ωj)tdt =1T∫ T0cos (ωt) e−i(ωn∓ωj)tdt=1T(ω2 − ω20 (−n± j)2) [ωeiω0(−n±j)t sin (ωt) + iω0 (−n± j) eiω0(−n±j)t cos (ωt)]T0=1T(ω2 − ω20 (n∓ j)2) [ωe−iω0(n∓j)T sin (ωT )− iω0 (n∓ j) e−iω0(n∓j)T cos (ωT ) + iω0 (n∓ j)]=1Tω20(p2 − (n∓ j)2)ω0 [pe−iω0(n∓j)T sin (pω0T )− i (n∓ j) e−iω0(n∓j)T cos (pω0T ) + i (n∓ j)]=ω02piω20(p2 − (n∓ j)2)ω0 [−i (n∓ j) e−i2pi(n∓j) cos (2pip) + i (n∓ j)]= − i (n∓ j)2pi(p2 − (n∓ j)2) [e−i2pi(n∓j) − 1]=0 ; |n∓ j| 6= p− limx→±p ix[e−i2pix−1]2pi(p2−x2) = − limx→±p i([e−i2pix−1]−2piixe−i2pix)−4pix = − i2(2pi)4pi (±p±p) ; n∓ j = ±p=0 ; |n∓ j| 6= p12; n∓ j = +p12; n∓ j = −p=12{δp,n∓j +12δp,−n±j}B−±nj =1T∫ T0sin (ωt) ei(−ωn±ωj)tdt =1T∫ T0sin (ωt) e−iω0(n∓j)tdt=1T(ω2 − ω20 (−n± j)2) [−ωe−iω0(n∓j)t cos (ωt)− iω0 (n∓ j) e−iω0(n∓j)t sin (ωt)]T0=−1T(ω2 − ω20 |n∓ j|2) [ωe−iω0(n∓j)T cos (ωT ) + iω0 (n∓ j) e−iω0(n∓j)T sin (ωT )− ω]=−ω02piω20(p2 − |n∓ j|2) [ω0pe−i2pi(n∓j) cos (2pip)− iω0 (n∓ j) e−i2pi(n∓j) sin (2pip)− ω0p]= − p2pi(p2 − |n∓ j|2) [e−i2pi(n∓j) − 1]=0 ; n∓ j 6= plimx→±p −p(e−i2pix−1)2pi(p2−x2) = +i2pi−4pix ( p±) = ; n∓ j = ±p=0 ; |n∓ j| 6= p− i2; n∓ j = +p+ i2; n∓ j = −p= − i2{δp,n∓j − δp,−n±j}The other terms that will be used are,1011T∫ T0dt =1T(T ) = 1 (D.1)1T∫ T0e−i2ωmtdt = B−−mm (D.2)D.2 Other useful TermsIn the equation of motion (3.14) for Cmn (t), the time-dependence of themode function for the scalar-photons in Chapter 3, the term hmn appears,hmn = (−1)m−n 2mnn2 −m2 (D.3)For convenience we express the relevant values for these factors below,hm(p−m) = (−1)m−(p−m)2m (p−m)(p−m)2 −m2 = (−1)2m+p 2pm− 2m2p2 − 2pm+m2 −m2=2m (2−m)p (p− 2m)hm(m−2) = (−1)2m2m (m− 2)(m− 2)2 −m2 =2m (m− 2)4− 4m+m2 −m2 = −m2(m− 2)(m− 1)hm(2−m) = (−1)m+2−m2m (2−m)(2−m)2 −m2 = −2m (m− 2)4− 4m+m2 −m2 =m2(m− 2)(m− 1)hm(m+2) = (−1)m−(m+2)2m (m+ 2)(2 +m)2 −m2 =m2(m+ 2)(m+ 1)hm(−m−2) = (−1)m+2+m2m (−m− 2)(m+ 2)2 −m2 = −2m (m+ 2)4 + 4m+m2 −m2 = −m2(m+ 2)(m+ 1)D.3 Calculation of the Differential Equation forthe Time Dependent Bogoliubov CoefficientsAs this calculation is sensitive to the slightest error, it is important to gothrough it in excessive detail to get the equation of motion for αmn (t) andβmn (t) for the mode function of the scalar field while the mirror is moving(D.8).102Reduction 1): Small Oscillation ApproximationThe small oscillation form that we shall take here is L (t) = L0 [1 +  sin (ωt)],then we have L˙ = ω cos (ωt), hence the time dependent frequency and otherrelevant terms take on the forms,ωm (t) =mpiL (t)=mpiL0− mpiL0sin (cωt) +O (2)=mpiL0[1−  sin (ωt)]ω2m (t) =mpiL (t)=m2pi2L20− 2m2pi2L20sin (ωt) +O (2)=m2pi2L20[1− 2 sin (ωt)] +O (2)v (T ) =L˙ (t)L (t)=1L0[1−  sin (ωt)]L0ω cos (ωt) = ω cos (ωt) +O(2)(D.4)v˙ (t) = −ω2 sin (ωt) +O (2) (D.5)Reduction 2): Assume the form found for t > TThe αmn (t) and βmn (t) are promoted to being time dependent in thismethod, as described in Part 3. However we are dealing with the case wherethey are slowly varying with respect to the time-scale of this problem. Thismeans that α˙mn ∼ β˙mn ∼ . This slowly varying nature permits us to cancelout a lot of terms when we put Cmn (t) back into the wave equation. First,we calculate the derivatives of the time-dependent part of the mode functionwith this new time-dependence on α, β,Cnm (t) = αmn (t) e−iωmt + βmn (t) e+iωmt (D.6)C˙nm (t) = α˙mne−iωmt + β˙mne+iωmt − iωmαmn (t) e−iωmt + iωmβmn (t) e+iωmt(D.7)C¨nm (t) = α¨mne−iωmt + β¨mne+iωmt − iωmα˙mne−iωmt + iωmβ˙mne+iωmt− iωmα˙mne−iωmt + iωmβ˙mne+iωmt − ω2mαmne−iωmt − ω2mβmne+iωmt= α¨mne−iωmt + β¨mne+iωmt − ω2m[αmne−iωmt + βmne+iωmt]− 2iωm[α˙mne−iωmt − β˙mne+iωmt]Now we can calculate new equations of motion for αnm and βnm byplugging Cnm (t) into the wave equation.103Reduction 3a): Solve for the First part of the Equation of MotionThe differential equation for these Cmn (t) can be decomposed into two parts,0 =∑j[2v (t)hmjC˙nj (t) + v˙ (t)hmjCnj (t)]︸ ︷︷ ︸I1mn−[C¨nm (t) + ω2m (t)Cnm (t)]︸ ︷︷ ︸I2mn(D.8)In what follows I1mn will be determined first, then using the modified KBMmethod. Before this is done, we will introduce a dimensionless slow timevariableτ =12ω0t ,∂∂t=dτdt∂∂τ=12ω0∂∂τ(D.9)where this represents a slow time scale.104Now we put (D.6) into this part of the wave equation.I1mn=∑j[2v (t)hmjC˙nj (t) + v˙ (t)hmjCnj (t)]; Definition= ∑j[4ω0 cos (2ω0t)hmj(α˙nj e−iωjt + β˙nj e+iωjt)]+∑j[4ω0 cos (2ω0t)hmj(+iωjβnj (t) e+iωjt − iωjαnj (t) e−iωjt)]; Substitution−∑j[4ω20 sin (2ω0t)hmj(αnj (t) e−iωjt + βnj (t) e+iωjt)]= ∑j[4ω0 cos (2ω0t)hmj(α˙nj e−ijω0t + β˙nj e+ijω0t)]+∑j[4ω0 cos (2ω0t)hmj(−iωjαnj (t) e−ijω0t + iωjβnj (t) e+ijω0t)] ; ωj = jω0−∑j[4ω20 sin (2ω0t)hmj(αnj (t) e−ijω0t + βnj (t) e+ijω0t)]= 4ω0∑jhmj[{α˙nj − ijω0αnj}cos (2ω0t) e−ijω0t] ; Rearrange−4ω0∑jhmj[ω0αnj (t) sin (2ω0t) e−ijω0t]+4ω0∑jhmj[{β˙nj + ijω0βnj (t)}cos (2ω0t) e+ijω0t]−4ω0∑jhmj[ω0βnj (t) sin (2ω0t) e+ijω0t]=∑jhmj2 2ω20︸︷︷︸smalldαnjdτ− i4jω20αnj cos (2ω0t) e−ijω0t ; Slow Time−∑jhmj[4ω20αnj (t) sin (2ω0t) e−ijω0t]+∑jhmj2 2ω20︸︷︷︸smalldβnjdτ+ i4jω20βnj (t) cos (2ω0t) e+ijω0t−∑jhmj[4ω20βnj (t) sin (2ω0t) e+ijω0t]= −4ω20∑jhmj[ijαnj cos (2ω0t) e−ijω0t + αnj (t) sin (2ω0t) e−ijω0t] ; Result+4ω20∑jhmj[ijβnj (t) cos (2ω0t) e+ijω0t − βnj (t) sin (2ω0t) e+ijω0t](D.10)where we take any term of O (2) ∼ 0.105Reduction 3b): Time-Averaged Equations of motion for the firstpartTo get simple equations of motion, the method chosen is inspired by theKrylov-Bogolyubov-Mitropolsky (KBM) method. A summary of this is asfollows. If we have an equation of motion such asdx (t)dt= f (t, x (t)) (D.11)where  is a small positive parameter, then an approximation that is validwhen  is small is, the time averaged version,dx (t)dt∼  1T∫ T0f (t, x) dt (D.12)where we average over some time interval T while leaving x (t) to be constantin this time interval.In our case, the method will be tweaked a bit since we want to isolatean equation of motion for αnj (t) and isolate a separate equation of motionfor βnj (t). This can be done by noting that all terms with αnj (t)'s have afactor of e−ijω0t and βnj (t)'s have a factor of e+ijω0t. If we multiply theequation of motion (D.8) by e±imω0t before averaging, then it turns out thatthe equations of motion for α, β decouple in a simple way.To get equations of motion for αnm (t), we multiply the whole equation bye+imω0t and average it over the period of the lowest mode, which is T = 2piω0where ωn = nω0,106I1,αmn= −4ω0∑jhmj[ijω0αnj1T∫ T0cos (2ω0t) e−i(j−m)ω0tdt]; Averaged E.O.M−4ω0∑jhmj[ω0αnj (t)1T∫ T0sin (2ω0t) e−i(j−m)ω0tdt]+4ω0∑jhmj[ijωjβnj (t)1T∫ T0cos (2ω0t) e+i(j+m)ω0tdt]−4ω0∑jhmj[−ω0βnj (t) 1T∫ T0sin (2ω0t) e+i(j+m)ω0tdt]= −4ω0∑jhmj[ijω0αnjA−+jm + ω0αnj (t)B−+jm]; Appendix D+4ω0∑jhmj[ijω0βnj (t)A++jm − ω0βnj (t)B++jm]= −4ω0∑jhmj[ijω0αnj12{δ2,(j−m) + δ2,(m−j)}]+4ω0∑jhmj[ω0αnji2{δ2,(j−m) − δ2,(m−j)}]+4ω0∑jhmj[ijω0βnj12{δ2,(j+m) + δ2,(−j−m)}]−4ω0∑jhmj[ω0βnji2{δ2,(j+m) − δ2,(−j−m)}]= −2iω20[hm(m+2) (m+ 2)αn(m+2) + hm(m−2) (m− 2)αn(m−2)]; Sums collapsed+2iω20[hm(m+2)αn(m+2) − hm(m−2)αn(m−2)]+2iω20[hm(2−m) (2−m)βn(2−m) + hm(−2−m) (−2−m)βn(−2−m)]−2iω20[hm(2−m)βn(2−m) − hm(−2−m)βn(−2−m)]= −2iω20[hm(m+2) (m+ 1)αn(m+2) + hm(m−2) (m− 1)αn(m−2)]; Terms Collected−2iω20[hm(2−m) (m− 1)βn(2−m) + hm(−2−m) (m+ 1)βn(−2−m)]= −2iω20[m2(m+ 2) (m+ 1)(m+ 1)αn(m+2) − m2(m− 2) (m− 1)(m− 1) αn(m−2)]; hmn terms used.−2iω20[−m2(m− 2) (m− 1)(m− 1) βn(2−m) −(m2(m+ 2) (m+ 1)(m+ 1))βn(−2−m)]= −imω20[(m+ 2)αn(m+2) − (m− 2)αn(m−2)]; Result−imω20[(m− 2)βn(2−m) − (m+ 2)βn(−2−m)](D.13)therefore,I1,αmn = −imω20[(m+ 2)αn(m+2) − (m− 2)αn(m−2)](D.14)− imω20[(m− 2)βn(2−m) − (m+ 2)βn(−2−m)]107wherehjm = (−1)j−m 2jm(m2 − j2) = (−1)j−m 2jm(m− j) (m+ j) (D.15)To get equations of motion for βnm (t), we multiply the whole equation bye−imω0t and average it over the period of the lowest mode, which is T = 2piω0where ωn = nω0. This is displayed on the next page.108I1,βmn= −4ω0∑jhmj[ijω0αnj1T∫ T0cos (2ω0t) e−i(j+m)ω0tdt]; Averaging−4ω0∑jhmj[ω0αnj (t)1T∫ T0sin (2ω0t) e−i(j+m)ω0tdt]+4ω0∑jhmj[ijωjβnj (t)1T∫ T0cos (2ω0t) e+i(j−m)ω0tdt]−4ω0∑jhmj[ω0βnj (t)1T∫ T0sin (2ω0t) e+i(j−m)ω0tdt]= −4ω0∑jhmj[ijω0αnjA−−jm + ω0αnj (t)B−−jm]; Appendix A4+4ω0∑jhmj[ijω0βnj (t)A+−jm − ω0βnj (t)B+−jm]= −4ω0∑jhmj[12{δ2,(j+m) + δ2,(−j−m)}ijω0αnj]+4ω0∑jhmj[i2{δ2,(j+m) − δ2,(−j−m)}ω0αnj (t)]+4ω0∑jhmj[12{δ2,(j−m) + δ2,(−j+m)}ijω0βnj (t)]−4ω0∑jhmj[i2{δ2,(j−m) − δ2,(−j+m)}ω0βnj (t)]= −2iω20{− (m− 2)hm(2−m)αn(2−m) − (m+ 2)hm(−2−m)αn(−2−m)} ;Sums Collapsed+2iω20{hm(2−m)αn(2−m) (t)− hm(−2−m)αn(−2−m) (t)}+2iω20{hm(m+2) (m+ 2)βn(m+2) (t) + hm(m−2) (m− 2)βn(m−2) (t)}−2iω20{hm(m+2)βn(m+2) (t)− hm(m−2)βn(m−2) (t)}= 2iω20{(m− 1)hm(2−m)αn(2−m) + (m+ 1)hm(−2−m)αn(−2−m)}; Terms collected+2iω20{hm(m+2) (m+ 1)βn(m+2) (t) + hm(m−2) (m− 1)βn(m−2) (t)}= 2iω20{{m2(m− 2)(m− 1) (m− 1)}αn(2−m)}; hmn terms.+2iω20{{−m (m+ 2)2 (m+ 1)(m+ 1)}αn(−2−m)}+2iω20{{m2(m+ 2)(m+ 1)(m+ 1)}βn(m+2) (t)}+2iω20{{−m2(m− 2)(m− 1) (m− 1)}βn(m−2) (t)}= −imω20{(m+ 2)αn(−2−m) − (m− 2)αn(2−m)}; Result+imω20{(m+ 2)βn(m+2) (t)− (m− 2)βn(m−2) (t)}(D.16)where,I1,βmn = −imω20{(m+ 2)αn(−2−m) − (m− 2)αn(2−m)}(D.17)+ imω20{(m+ 2)βn(m+2) (t)− (m− 2)βn(m−2) (t)}109Reduction 4a): Solve for the Second part of the Equation ofMotionNow the second term, I2mn = C¨mn (t)+ω2m (t)Cnm (t) = C¨mn (t)+ω2mCnm (t)−2ω2m sin (2ω0t)Cnm (t), is calculated by substituting the mode functions intothe wave equation,I2mn= C¨mn (t) + ω2m (t)Cnm (t) = C¨mn (t) + ω2mCnm (t)− 2ω2m sin (2ω0t)Cnm (t)= α¨mne−iωmt + β¨mne+iωmt︸ ︷︷ ︸O(2)−ω2m[αnme−iωmt + βnme+iωmt]− 2iωm[α˙nme−iωmt − β˙nme+iωmt]+ ω2m[αnm (t) e−iωmt + βnm (t) e+iωmt]− 2ω2m sin (2ω0t)[αnm (t) e−iωmt + βnm (t) e+iωmt]= −imω20[dαnmdτe−imω0t − dβnmdτe+imω0t]− 2m2ω20[αnm (t) sin (2ω0t) e−imω0t + βnm (t) sin (2ω0t) e+imω0t]110Reduction 4b): Time-Averaged Equations of motion for theSecond partAs mentioned before, if we multiply the equations of motion above by e±imω0t,then we will be able to get equations of motion for α\β's respectively.To get equations of motion for αnm (t), we multiply the whole equation bye+imω0t and average it over the period of the lowest mode, which is T = 2piω0where ωn = nω0I2,αmn = −imω20dαnmdτ 1T∫ T0dt︸ ︷︷ ︸1−dβnmdτ1T∫ T0e+2imω0tdt︸ ︷︷ ︸=δm0− 2m2ω20αnm (t) 1T∫ T0sin (2ω0t) dt︸ ︷︷ ︸B−+mm+βnm (t)1T∫ T0sin (2ω0t) e+2imω0tdt︸ ︷︷ ︸B++mm= −imω20dαnmdτ − dβn0dτ︸︷︷︸=0− 2m2ω20− i2 {δ2,(m−m) − δ2,(m−m)}αnm (t)︸ ︷︷ ︸=0+βnm (t)i2δ2,2m − δ2,−2m︸ ︷︷ ︸=0= −imω20[dαnmdτ]− im2ω20βn1 (t)whereI2,αmn = −imω20[dαnmdτ]− im2ω20βn1 (t) (D.18)111To get equations of motion for βnm (t), we multiply the whole equation bye−imω0t and average it over the period of the lowest mode, which is T = 2piω0where ωn = nω0I2,βmn = −imω20dαnmdτ 1T∫ T0e−2imω0tdt︸ ︷︷ ︸=δm0−dβnmdτ1T∫ T0dt︸ ︷︷ ︸=1− 2m2ω20αnm (t) 1T∫ T0sin (2ω0t) e−2imω0tdt︸ ︷︷ ︸B−−mm+βnm (t)1T∫ T0sin (2ω0t) dt︸ ︷︷ ︸B+−mm= −imω20[dαnmdτδm0 − dβnmdτ]− 2m2ω20− i2δ2,(2m)︸ ︷︷ ︸δm,1− 6 δ2,(−2m)︸ ︷︷ ︸δm,−1αnm (t) + i2 {δ2,0 − δ2,0}︸ ︷︷ ︸=0βnm (t)= −iω20[0 · dαn0dτ−mdβnmdτ]+ im2ω20δm,1αnm (t)= imω20dβnmdτ+ iω20αn1 (t)thereforeI2,βmn = imω20dβnmdτ+ iω20αn1 (t) (D.19)112Reduction 5): Solve for decoupled, hierarchy of equations ofmotion.The result for the above is tabulated below,I1,αmn = −imω20[(m+ 2)αn(m+2) − (m− 2)αn(m−2)]+imω20[(m+ 2)βn(−2−m) − (m− 2)βn(2−m)]I2,αmn = −imω20 dαnmdτ − iω20m2βn1 (t) δm1I1,βmn = −imω20[(m+ 2)αn(−2−m) − (m− 2)αn(2−m)]+imω20{(m+ 2)βn(m+2) − (m− 2)βn(m−2)}I2,βmn = imω20dβnmdτ + iω20m2αn1 (t) δm1Table D.2: The components of the decoupled equation of motion for the Bogoli-ubov Coefficients.For the alphas-favoring terms, we find that,I2,αmn = I1,αmn−imω20[dαnmdτ]− im2ω20βn1 δm1 = −imω20[(m+ 2)αn(m+2) − (m− 2)αn(m−2)]+ imω20[(m+ 2)βn(−2−m) − (m− 2)βn(2−m)]=⇒dαnmdτ=[(m+ 2)αn(m+2) − (m− 2)αn(m−2)]+[−mβnmδm1 + (m− 2)βn(2−m) + (m+ 2)βn(−2−m)]113The first few terms are listed below,m = 1dαn1dτ=[3αn3 + αn(−1)]+[−βn1 − βn1 + 3βn(−2−1)]= 3αn3 (τ)− 2βn1 (τ)m > 2dαnmdτ=[(m+ 2)αn(m+2) − (m− 2)αn(m−2)]+[−mβnmδm1 + (m− 2)βn(2−m) + (m+ 2)βn(−2−m)]︸ ︷︷ ︸=0= (m+ 2)αn(m+2) − (m− 2)αn(m−2)The same thing is done for the betas,I2,βmn = I1,βmnimω20dβnmdτ+ iω20αn1 (t) = −imω20{(m+ 2)αn(−2−m) − (m− 2)αn(2−m)}+ imω20{(m+ 2)βn(m+2) (t)− (m− 2)βn(m−2) (t)}dβnmdτ= (m− 2)αn(2−m) − (m+ 2)αn(−2−m) − αnm (t) δ1m+ (m+ 2)βn(m+2) (t)− (m− 2)βn(m−2) (t)the first few terms are,m = 1dβn1dτ= −3 αn−3︸︷︷︸=0+ (1− 2)αn1 − αn1 (t) δ11+ 3βn3 (t) + βn−1 (t)︸ ︷︷ ︸=0= 3βn3 (τ)− 2αn1 (τ)114m > 2dβnmdτ= (m− 2)αn(2−m)︸ ︷︷ ︸=0 ∀m>1− (m+ 2)αn(−2−m)︸ ︷︷ ︸=0−αnm (t) δ1m︸ ︷︷ ︸=0+ (m+ 2)βn(m+2) (t)− (m− 2)βn(m−2) (t)= (m+ 2)βn(m+2) (t) + (m− 2)βn(m−2) (t)Reduction 6): Differences from LiteratureIn the literature, Dodonov calculates [20, 2] that the equations of motionare as below. They differ for the m = 1 term and this cannot be repli-cated by the author. For my result there are terms from the integrals,1T∫ T0sin (2ω0t) e−2imω0tdt︸ ︷︷ ︸B−−mmand1T∫ T0sin (2ω0t) e+2imω0tdt︸ ︷︷ ︸B++mm, that end up can-celing out the βn1 and αn1 terms in his equations below.Dodonovs Calculationdαn1dτ = 3αn3 (t)− βn1 (t)dαnmdτ = (m+ 2)αn(m+2) − (m− 2)αn(m−2) (m ≥ 1)dβn1dτ = 3βn3 (t)− αn1 (t)dβnmdτ = (m+ 2)βn(m+2) (t) + (m− 2)βn(m−2) (t) (m ≥ 1)My Calculationdαn1dτ = 3αn3 (t)− 2βn1 (t)dαnmdτ = (m+ 2)αn(m+2) − (m− 2)αn(m−2)dβn1dτ = 3βn3 (t)− 2αn1 (t)dβnmdτ = (m+ 2)βn(m+2) (t) + (m− 2)βn(m−2) (t)115Appendix ENumerical Laplace TransformInversionThis appendix provides the way to get numerical results for the equationsof motion derived in Appendix D, which are corrections to the equationsderived in the work of Dodonov [2].Reduction 1: Symmetric CoordinatesFirst we write the Bogoliubov functions in a new coordinate system thatis more symmetricank = ξnk + ηnk , bnk = ξnk − ηnk (E.1)a˙n1 = 3an3 − 2an1 , b˙n1 = 3bn3 + 2bn1 (E.2)a˙nk = (k + 2) ank+1− (k − 2) ank−1 , b˙nk = (k + 2) bnk+2 + (k − 2) bnk−2 (E.3)Reduction 2: Solve using Generating Function MethodTo solve we must recast the equations of motion in a simpler form. The toolthat will be used is the generating function method, where the equations ofmotion of the said generating function for one of the ξnk (τ) , ηnk (τ) are recastin terms of the below,An (τ, z) =∞∑k=1ank (τ) zk , Bn (τ, z) =∞∑k=1bnk (τ) zk(E.4)116First we do the ank series. The time and z derivatives can be related asseen through the application of the third step below,∂an∂τ (τ, z)=∑∞k=1∂ak(τ)∂τ zk ; Definition= [3an3 (τ)− 2an1 ] z ; Apply Equation of Motion+∑∞k=2[(k + 2) ank+2 (τ)− (k − 2) ank−2 (τ)]zk= [3an3 (τ)− 2an1 ] z ; mzk = zk−m+1 ∂zm∂z+∑∞k=2[z−1 dzk+2dz ank+2 (τ)− z3 dzk−2dz ank−2 (τ)]= [3an3 (τ)− 2an1 ] z+(∑∞k=0[z−1 dzk+2dz ank+2 (τ)− z3 dzk−2dz ank−2 (τ)])− 2z−1dz2dzan2 (τ) + 2z3dz−2dzan−2 (τ)︸ ︷︷ ︸k=0,=0− z−1dz3dzan3 (τ) + z3dz−1dz=0︷ ︸︸ ︷an−1 (τ)︸ ︷︷ ︸k=1= −2zan1 + z−1∑∞k=2dzkdz ank (τ) ;Cancellation−z3∑∞k=0dzkdz ank (τ) + 0︸︷︷︸-2,-1 terms =0= −an1 (τ)(z−1 + 2z)+(z−1 − z3) dAdz ; Use reduction A(E.5)117Now we do the same for the bnk series,∂Bn∂τ (τ, z)=∑∞k=1∂bnk (τ)∂τ zk ; Definition= [3bn3 (τ) + 2bn1 ] z ; Apply Equation of Motion+∑∞k=2[(k + 2) bnk+2 (τ)− (k − 2) bnk−2 (τ)]zk= [3bn3 (τ) + 2bn1 ] z ; mzk = zk−m+1 ∂zm∂z+(∑∞k=0[z−1 dzk+2dz bnk+2 (τ)− z3 dzk−2dz bnk−2 (τ)])− 2z−1dz2dzbn2 (τ) + 2z3dz−2dzbn−2 (τ)︸ ︷︷ ︸k=0,=0− z−1dz3dzbn3 (τ) + z3dz−1dz=0︷ ︸︸ ︷hn−1 (τ)︸ ︷︷ ︸k=1= +2zbn1 + z−1∑∞k=0dzk+2dz bnk+2 (τ) ; Cancellation−z3∑∞k=0 dzk−2dz bnk−2 (τ)= +2zbn1 + z−1∑∞k=2dzkdz bnk (τ)−z3∑∞k=0dzkdz bnk (τ) + 0︸︷︷︸-2,-1 terms =0= −bn1 (τ)(z−1 − 2z)+ (z−1 − z3) dBdz ; Use reduction A(E.6)Reduction A. We note thatdYdz=∞∑k=1ynkdzndz= yn1 +∞∑k=2ynkdzndz(E.7)This gives the equations of motiondAndτ= −an1 (τ)(2z + z−1)+(z−1 − z3) dAdz(E.8)dBndτ= −bn1 (τ)(z−1 − 2z)+ (z−1 − z3) dBdz(E.9)where it will be noted that these two differential equations depend onlyon the +2 or −2 multiplied by z. In the following reduction, both of thesedifferential equations will be solved simultaneously, leaving these numbersas a constant C ∈ {+2,−2}. Note that to recover the result of Dodonov[2,pg 53] C = 1 and the z−1 term that is summed to it is replaced with ±1.118Reduction 3: Solve Cauchy ProblemThe initial value problem comes from the initial value problem of the Bo-goliubov coefficients,ξnk (0) = δkn ηnk (0) = 0 =⇒ ank (0) = δkn bnk (0) = 0 (E.10)The result of this isA (τ, 0) = 0 , B (0, z) =∞∑k=1δnk zk = znA (0, z) =∑∞k=1 δnk zk = zn B (0, z) =∑∞k=1 δnk zk = znA (τ, 0) = 0 B (τ, 0) = 0(E.11)In general, the solution to such a differential function is of the form aboveis,Reduction 4: Derivation of the SolutionTo solve this, the method of characteristics is applied to the following CauchyProblem∂u∂τ+(z3 − z−1) ∂u∂z= −fC (τ)[Cz +1z](E.12)u (0, z) = zn (E.13)where fC (τ) is either αn1 (τ) or βn1 (τ) corresponding to whether C isequat to −2 or 2 respectively. The integral curves are tangent to〈1, z3 − z−1,−f (τ)[Az +1z]〉(E.14)so the following is true,d−→γdλ= τ (λ)z (λ)w (λ) = 1(z3 − z−1)−f (τ) [Cz + 1z ]−→γ (0) = −→γ s (0) = τ0 (s)z0 (s)w0 (s) = 0ssn119where the solution is equal to u (τ, z) = w (τ, z). The solution of the ODE33above is then provided below,τ (s, λ) = τ0 (s) + λ (E.15)s = ±√√√√√1− e−4λ(1−z21+z2)1 + e−4λ(1−z21+z2) , z = ±√√√√√1− e+4λ(1−s21+s2)1 + e+4λ(1−s21+s2)(E.16)τ =14ln(z2 − 1z2 + 1)− 14ln(s2 − 1s2 + 1)(E.17)where the intention is to find w (λ),w (λ) = w0 (s)−∫ λ0 fC (τ)[Cz + z−1]dλ ; Definition= w0 (s)−∫ λ0 fC (τ)[Cz2+1z2]dλ=(1−e−4τ(1−z21+z2)1+e−4τ(1−z21+z2))n2; Substitute z− ∫ λ0 fC (τ)[C√1−e+4λ(1−s21+s2)1+e+4λ(1−s21+s2) −√1+e+4λ(1−s21+s2)1−e+4λ(1−s21+s2)]dλ(E.18)33Ordinary Differential Equation.120Reduction 5: Apply the Initial Value ProblemHere we apply the boundary value problem at the initial Cauchy slice att = 0 to get, for each Bogoliubov generating functional.H (0, z) = zn= C1(z2 − 1z2 + 1)−∫ 00√√√√√1 +(1−z21+z2)e−4σ1−(1−z21+z2)e−4σ yn1 (τ − σ) dσ︸ ︷︷ ︸=0= C1(z2 − 1z2 + 1)⇐⇒C1 (z) =(1 + e0 z2−1z2+11− e0 z2−1z2+1)m=((z2 + 1)+ e0(z2 − 1)(z2 + 1)− e0 (z2 − 1))m=(2z22)m= (z)2m= zn2=⇒C1 (τ, z) =1 + e−4τ(z2−1z2+1)1− e−4τ(z2−1z2+1)n2Reduction 6: Express the SolutionNow since z is just a dummy variable, it can be set to zero and the resultwill then be, ∫ τ03− e−4τ√1 + e−8ταn1 (τ − σ) dσ = −f (τ)−n2(E.19)∫ τ0−(1− 3e−4τ√1 + e−8τ)βn1 (τ − σ) dσ = −f (τ)−n2f (τ) ≡ 1 + e−4τ1− e−4τ (E.20)121E.1 Inverse Laplace TransformIn order to obtain βn1 (τ) or αn1 (τ), an inverse transformation must be em-ployed to Equation (E.19) to isolate the laplace transforms of α and βthrough the convolution theorem,αn1 {s} =L{−f (τ)−n2}L{3−e−4τ√1+e−8τ} , βn1 {s} = −L{−f (τ)−n2}L{1−3e−4τ√1+e−8τ}(E.21)Then to obtain βn1 (τ) or αn1 (τ) an inverse laplace transform is performed.To proceed, a few integral results and definitions must be introduced,E.1.1 Definitions and Useful IntegralsIn what follows, a list of definitions, theorems and integrals will be intro-duced. They will be used in the next subsection.Integral 1:∫ 10xλ−1 (1− x)µ−1 (1− βx)−ν dx = B (λ, µ)F1 (ν, λ, λ+ µ;β)(E.22)which is equation 3.197.3 in Ref. [30].Definition: Beta Function :B (x, y) =∫ 10tx−1 (1− t)y−1 dt = Γ (x) Γ (y)Γ (x+ y)(E.23)Definition: Hypergeometric Function:2F1 (a, b, c, d;x, y) =∞∑m=0∞∑n=0(a)m+n (b)m (c)n(d)m+n n!m!xmyn (E.24)Kummers Identity 12F1 (a, b, a− b+ 1;−1) (E.25)=√pi2aΓ (1 + a− b){1Γ(1 + 12a− b)Γ(12 +12a)}which is discussed in the following Reference [31, Pg.557] in Equation15.1.21.122Kummers Identity 22F1 (a, b, a− b+ 2;−1)=√pi2a (b− 1)Γ (a− b+ 2){1Γ(a2)Γ(32 +a2 − b) − 1Γ(12 +a2)Γ(1 + a2 − b)}(E.26), which is discussed in the following Reference [31, Pg.557] in Equation15.1.22.E.1.2 Further SimplificationsReduction 1: Simplify f(t)This reduction deals in using Integral 1 (defined above) to write the Laplacetransform of f (τ),L{f (τ)−n2}=∫∞0(1+e−4τ1−e−4τ)−n2e−τsdτ ;Laplace Transform=∫ 01 y−1 (1− y)n2 (1− (−1) y)−n2 (y) s4 dy ; y = e−4τdτ = −14 dyy= − ∫ 10 y s4−1 (1− y)n2 (1− (−1) y)−n2 dy= B(s4 ,n2 + 1)F(n2 ,s4 ,s4 +n2 + 1;−1); Integral 1=Γ( s4)Γ(n2+1)Γ( s4 +n2+1)F(n2 ,s4 ,s4 +n2 + 1;−1)(E.27)Reduction 2: Simplify The restThis step uses Integral 1 (above) to do the same for the denominator. Theresult is given by using Kummers Identities 1 and 2 listed in Equations (E.25)and (??) respectively. The result is,L{αn1 (τ)}=Γ(s4+ n2+ 1)sΓ(s4)Γ(n2+ 1) (12s2F1(12, s8, 1− s8,−1)2F1(n2, s4, s4+ n2+ 1,−1) − 4(s+ 4) 2F1(12, s8+ 12, 32+ s8,−1)2F1(n2, s4, s4+ n2+ 1,−1))L{βn1 (τ)} (E.28)=Γ(s4+ n2+ 1)sΓ(s4)Γ(n2+ 1) (4s2F1(12, s8+ 12, s8+ 32,−1)2F1(n2, s4, s4+ n2+ 1,−1) − 12(s+ 4) 2F1(12, s8, 1 + s8,−1)2F1(n2, s4, s4+ n2+ 1,−1))123Reduction 3: Generalized Kummer's IdentityThere are still a lot of Hypergeometric functions in the results listed above inEquation (E.28) that can be reduced using a new theorem for a generalizedKummer Identity developed by R.Vidunas[32]. This procedure, however,is quite laborious and not exactly required as Numerical Laplace inversiontechniques can digest these hypergeometric functions with enough compu-tational time. This is the approach this manuscript will take as an originaltake on the problem will be done in later sections using perturbation theory.E.2 Numerical Laplace Transform InversionTo obtain αn1 (τ) and βn1 (τ), a numerical inverse Laplace transform is donewith a code done by the author, where the method used is the Stehfestalgorithim[33]. The result is tabulated in a list of data files used to producethe photon number.E.3 The Rest of the Bogoliubov CoefficientsIt is noted that only the m = 1 coefficients have been determined thus farwith the methods above. In order to obtain the m > 1 coefficients, for anyn , the Heirarchy of equations of motion derived in Chapter 3 and found inEquation 3.20 are applied.124Appendix FSolutions to the WKB methodof solving the moving mirrorproblemF.1 First Order TermThe particular solution to the differential equation at first order in  is asbelow,C1,pmn (t) = −e−iωnt∫e+iωmtVmn (t)Wmndt+ e+iωnt∫e−iωmtVmn (t)Wmndt (F.1)Wmn (t) = e−iωmt (iωmeiωmt)− eiωmt (−iωme−iωmt) = 2iωm (F.2)These integrals contained in the particular solution are different for whetheror not p = m or p 6= m, and need to be treated differentlyReduction 1): p 6= m CaseIn the case where the driving frequency ω0p is different than any photon modefrequency ω0m, the particular solution for the O () terms in the Scalar Fieldmodes will as below. The following integrals will be useful34∫e±imω0t cos (pω0t) dt =1ω0 (p2 −m2)e±imω0t [±i n cos (pω0t) + p sin (pω0t)]∫e±imω0t sin (pω0t) dt =1ω0 (p2 −m2)e±imω0t [±im sin (pω0t)− p cos (pω0t)]34Once again, the notation δm 6=p ={0 ;m = p1 ;m 6= p is used to be concise.125hmn =(−1)m−n 2mnn2 −m2 δm6=p (F.3)The first term is,∫e+iωmtVmn (t)Wmndt=−12iωm[2ω2n∫e+iωmt sin (ωpt) δmndt+ 2iωpωnhmn∫e+iωmt cos (ωpt) dt]−12iωm[hmnω2p∫e+iωmt sin (ωpt) dt]=−12iωm[2n2ω0δmn(p2 −m2) e+imω0t [im sin (pω0t)− p cos (pω0t)]]+ 2ihmn(pnω0(p2 −m2)eimω0t [p sin (pω0t) + im cos (pω0t)])− 12iωm[hmnω2p(1ω0 (p2 −m2)eimω0t [im sin (pω0t)− p cos (pω0t)])]= −[n2δmn(p2 −m2)e+imω0t sin (pω0t) + ipn2δmnm (p2 −m2)e+imω0t cos (pω0t)]− hmn(eimω0t[p2nm (p2 −m2) sin (pω0t) + ipnmm (p2 −m2) cos (pω0t)])− hmn(eimω0t[p22m (p2 −m2) sin (pω0t) + ip32m (p2 −m2) cos (pω0t)])= eimω0t{[−(n2p2 −m2)δmn + p2(2n+m2m (p2 −m2))(−1)m−n 2mn(n2 −m2) δm 6=n]}sin (pω0t)+eimω0t{−i[+(n2pm (p2 −m2))δmn + p(2nm+ p22m (p2 −m2))(−1)m−n 2mn(n2 −m2) δm 6=n]}cos (pω0t)126and the second term is,∫e−iωmtVmn (t)Wmndt=−12iωm[2ω2nδmn∫e−iωmt sin (ωpt) dt+ 2iωpωnhmn∫e−iωmt cos (ωpt) dt]−12iωm[hmnω2p∫e−iωmt sin (ωpt) dt]=−12iωm[2ω2nδmn(1ω0 (p2 −m2)e−imω0t [−im sin (pω0t)− p cos (pω0t)])]− 12iωm[2iωpωnhmn(1ω0 (p2 −m2)e−imω0t [p sin (pω0t)− im cos (pω0t)])]− 12iωm[hmnω2p(1ω0 (p2 −m2)e−imω0t [−im sin (pω0t)− p cos (pω0t)])]= −e−imω0t[(− n2p2 −m2)δmn]sin (pω0t)− e−imω0t[(2p2n2m (p2 −m2) −p2m2m (p2 −m2))(−1)m−n 2mn(n2 −m2) δm6=n]sin (pω0t)= −e−imω0t[(− p2m2m (p2 −m2))(−1)m−n 2mn(n2 −m2) δm 6=n]sin (pω0t)− e−imω0t{i[(n2pm (p2 −m2))δmn]}cos (pω0t)− e−imω0t{i[(− 2pnm2m (p2 −m2))(−1)m−n 2mn(n2 −m2) δm6=n]}cos (pω0t)− e−imω0t{i[(p32m (p2 −m2))(−1)m−n 2mn(n2 −m2) δm6=n]}cos (pω0t)The final result for the p 6= m particular funciton is,C1,pmn (t) =[(2n2p2 −m2)δmn −((−1)m−n 2np2m(p2 −m2) (n2 −m2))δm 6=n]sin (pω0t)i[(0) δmn +(p3m (p2 −m2))(−1)m−n 2mn(n2 −m2) δm 6=n]cos (pω0t)127Reduction 2): p = m CaseThe case where the driving frequency of the mirror is the same as a photonmode is done here. The following integrals will be useful∫e±ipω0t cos (pω0t) dt= −e±ipω0tpω0[(pω0t cos2(p2ω0t)− pω0t2)± (ipω0t∓ 1) cos(p2ω0t)sin(p2ω0t)]∫e±ipω0t sin (pω0t) dt=e±ipω0tpω0[±i(pω0t cos2(p2ω0t)− pω0t2)+ (pω0t∓ i) cos(p2ω0t)sin(p2ω0t)]hmn =(−1)m−n 2mnn2 −m2 δm6=p (F.4)The first term is calculated,∫e+iωmtVmn (t)Wmndt=−12iωm[2ω2n∫e+iωmt sin (ωpt) δmndt+ 2iωpωnhmn∫e+iωmt cos (ωpt) dt]− 12iωm[hmnω2p∫e+iωmt sin (ωpt) dt]= iω0n2m∫e+iωmt sin (ωpt) δmndt+ ω0pnmhmn∫e+iωmt cos (ωpt) dt+ ihmnω0p22mω2p∫e+iωmt sin (ωpt) dt=n2mpe+ipω0t[−(pω0t cos2(p2ω0t)− pω0t2)+ (ipω0t+ 1) cos(p2ω0t)sin(p2ω0t)]δmn− 12me+ipω0t[[2n+ p](pω0t cos2(p2ω0t)− pω0t2)]hmn+12me+ipω0t[([p− 2n] ipω0t+ [p+ 2n]) cos(p2ω0t)sin(p2ω0t)]hmn128and the second term is,∫e−iωmtVmn (t)Wmndt=−12iωm[2ω2nδmn∫e−iωmt sin (ωpt) dt+ 2iωpωnhmn∫e−iωmt cos (ωpt) dt]− 12iωm[hmnω2p∫e−iωmt sin (ωpt) dt]=n2pme−ipω0t[(pω0t cos2(p2ω0t)− pω0t2)+ (ipω0t− 1) cos(p2ω0t)sin(p2ω0t)]δmn− nme−ipω0t[pω0t cos2(p2ω0t)− pω0t2− (ipω0t+ 1) cos(p2ω0t)sin(p2ω0t)]hmn+p2me−ipω0t[(pω0t cos2(p2ω0t)− pω0t2)+ (ipω0t− 1) cos(p2ω0t)sin(p2ω0t)]hmn=n2pme−ipω0t[(pω0t cos2(p2ω0t)− pω0t2)+ (ipω0t− 1) cos(p2ω0t)sin(p2ω0t)]δmn+12me−ipω0t[[p− 2n](pω0t cos2(p2ω0t)− pω0t2)]hmn+12me−ipω0t[([p+ 2n] ipω0t− [p− 2n]) cos(p2ω0t)sin(p2ω0t)]hmnThe final result for the p = m term is then,C1,pmn (t) =n2p2[pω0t(2 cos2(p2ω0t)− 1)− 2 cos(p2ω0t)sin(p2ω0t)]δpn+(−1)m−n 2pn2p (n2 − p2)[p2ω0t(2 cos2(p2ω0t)− 1)]δp 6=n+(−1)m−n 2pn2p (n2 − p2)[(i4npω0t− 2p) cos(p2ω0t)sin(p2ω0t)]δp 6=nF.2 Derivation of the Mode FunctionF.2.1 m = p CaseThis case is simple as C1,particularp,n (t) already satisfies the Cauchy problem,which is that the mode function and its time derivative must vanish at t = 0.Therefore,C1pn (t) = ξ1pne−ipω0t + η1pne+ipω0t + C1,particularpn (t) (F.5)129C1,particularmn (t) =[n2p ω0t(2 cos2(p2ω0t)− 1)− 2n2p2cos(p2ω0t)sin(p2ω0t)]δpn+ (−1)m−n2pn2p(n2−p2)[p2ω0t(2 cos2(p2ω0t)− 1)] δp 6=n+ (−1)m−n2pn2p(n2−p2)[(i4npω0t− 2p) cos(p2ω0t)sin(p2ω0t)]δp 6=nη1mn = 0ξ1mn = 0(F.6)F.2.2 m 6= p CaseIn this case, the cauchy problem requires the following relations for thecoefficients on the general solution,ξ1mn + η1mn = −iAmn1p2 −m2 (F.7)− ξ1mn + η1mn = +ipmBmn1(p2 −m2) (F.8)which gives,η1mn =12[−Amn + i pmBmnp2 −m2], ξ1mn =12[−i pmBmn −Amnp2 −m2](F.9)The full solution to the Cauchy problem is then,C1mn (t) =12[−i pmBmn −Amnp2 −m2]e−iω0mt +12[−Amn + i pmBmnp2 −m2]e+iω0mt+[Amnp2 −m2]cos (pω0t) +[Bmnp2 −m2]sin (pω0t)130Appendix GReflective Two-Mirror SystemG.1 Energy Density on the Left and Right sides ofthe Two-Mirror Partially Reflective System〈0∣∣∣TL00 (x, t)∣∣∣ 0〉= limτ→0∫ ∞0dω T00({φLω (x, t) , φL∗ω (x, t+ τ)})= limτ→0∫ ∞0dω12[T00(φLω (x, t) , φL∗ω (x, t+ τ))+ T00(φLω (x, t+ τ) , φL∗ω (x, t))]= limτ→0∫ ∞0dω12T00([ϕinω (t− x) + ψoutω (t+ x)],[ϕin∗ω (t+ τ − x) + ψout∗ω (t+ τ + x)])+ limτ→0∫ ∞0dω12T00([ϕinω (t+ τ − x) + ψoutω (t+ τ + x)],[ϕin∗ω (t− x) + ψout∗ω (t+ x)])=12limτ→0∫ ∞0dω12(∂0[ϕinω (t− x) + ψoutω (t+ x)]∂0[ϕin∗ω (t+ τ − x) + ψout∗ω (t+ τ + x)])+12limτ→0∫ ∞0dω12(∂1[ϕinω (t− x) + ψoutω (t+ x)]∂1[ϕin∗ω (t+ τ − x) + ψout∗ω (t+ τ + x)])+12limτ→0∫ ∞0dω12(∂0[ϕinω (t+ τ − x) + ψoutω (t+ τ + x)]∂0[ϕin∗ω (t− x) + ψout∗ω (t+ x)])+12limτ→0∫ ∞0dω12(∂1[ϕinω (t+ τ − x) + ψoutω (t+ τ + x)]∂1[ϕin∗ω (t− x) + ψout∗ω (t+ x)])=12limτ→0∫ ∞0dωω2214pi |ω|[e−iω(t−x) + e−iω(t+x)] [eiω(t+τ−x) + eiω(t+τ+x)]+12limτ→0∫ ∞0dω|ω|8pi[e−iω(t−x) − e−iω(t+x)] [eiω(t+τ−x) − eiω(t+τ+x)]+12limτ→0∫ ∞0dω|ω|8pi[e−iω(t+τ−x) + e−iω(t+τ+x)] [eiω(t−x) + eiω(t+x)]+12limτ→0∫ ∞0dω|ω|8pi[e−iω(t+τ−x) − e−iω(t+τ+x)] [eiω(t−x) − eiω(t+x)]=12limτ→0∫ ∞0dω|ω|2pi[eiωτ + e−iωτ]=∫ ∞−∞dω|ω|2pie−iωτ=~4pilimτ→0lim+→0[1(+ iτ)2+1(− iτ)2]= limτ→0[− ~2cpiτ2]131G.2 Energy Density in the Cavity of theTwo-Mirror Partially Reflective SystemThe energy density, or pressure as they are the same, is calculated for withinthe cavity, below,〈0∣∣∣TˆCav11 (x, t)∣∣∣ 0〉= limτ→0∫ ∞0dω T11 ({(φcav (t− x) + ψcav (t+ x)) , (φcav (t+ τ − x) + ψcav (t+ τ + x))})= limτ→0∫ ∞0dω T11({(R11ϕin(x− t) +R12ψin(x+ t) +R21ϕin(x− t) +R22ψin(x+ t)), · · ·})= limτ→0∫ ∞0dω T11({RAϕin(t− x) +RBψin(t+ x) , RAϕin(t+ τ − x) +RBψin(x+ t+ τ)})where RA = R11 +R21 , RB = R12 +R22=12limτ→0∫ ∞012dω ∂0(RAϕin(t− x) +RBψin(t+ x))∂0(R∗Aϕ∗in(t+ τ − x) +R∗Bψ∗in(t+ τ + x))+12limτ→0∫ ∞012dω ∂0(RAϕin(t+ τ − x) +RBψin(t+ τ + x))∂0(R∗Aϕ∗in(t− x) +R∗Bψ∗in(t+ x))+12limτ→0∫ ∞012dω ∂1(RAϕin(t− x) +RBψin(t+ x))∂1(R∗Aϕ∗in(t+ τ − x) +R∗Bψ∗in(t+ τ + x))+12limτ→0∫ ∞012dω ∂1(RAϕin(t+ τ − x) +RBψin(t+ τ + x))∂1(R∗Aϕ∗in(t− x) +R∗Bψ∗in(t+ x))=12limτ→0∫ ∞0dωω28pi |ω|(RAe−iω(t−x) +RBe−iω(t+x))(R∗Ae+iω(t+τ−x) +R∗Be+iω(t+τ+x))+12limτ→0∫ ∞0dω|ω|8pi(RAe−iω(t+τ−x) +RBe−iω(t+τ+x))(R∗Ae+iω(t−x) +R∗Be+iω(t+x))+12limτ→0∫ ∞0dω|ω|8pi(RAe−iω(t−x) −RBe−iω(t+x))(R∗Ae+iω(t+τ−x) −R∗Be+iω(t+τ+x))+12limτ→0∫ ∞0dω|ω|8pi(RAe−iω(t+τ−x) −RBe−iω(t+τ+x))(R∗Ae+iω(t−x) −R∗Be+iω(t+x))=12limτ→0∫ ∞0dω|ω|8pi(|RA|2 e+iωτ +RAR∗Be+iω(τ+2x) +RBR∗Ae+iω(τ−2x) + |RB |2R∗Be+iωτ)+12limτ→0∫ ∞0dω|ω|8pi(|RA|2 e−iωτ +RBR∗Ae−iω(τ+2x) +RAR∗Be−iω(τ−2x) + |RB |2 e−iωτ)+12limτ→0∫ ∞0dω|ω|8pi(R∗ARAe+iωτ −R∗BRAe+iω(τ+2x) −R∗ARBe+iω(τ−2x) + |RB |2 e+iωτ)+12limτ→0∫ ∞0dω|ω|8pi(|RA|2 e−iωτ −RBR∗Ae−iω(τ+2x) −RAR∗Be−iω(τ−2x) + |RB |2 e−iωτ)=12limτ→0∫ ∞0dω|ω|4pi(|RA|2(e+iωτ + e−iωτ)+ |RB |2(e+iωτ + e−iωτ))=12limτ→0∫ ∞−∞dω2pi( |ω|2(|RA (ω)|2 + |RB (ω)|2)e−iωτ)=12limτ→0∫ ∞−∞dω2pi(|ω|)(1− |r (ω)|2∣∣1− r (ω) e2iω(q2−q1)∣∣2)e−iωτ=12limτ→0∫ ∞−∞dω2pi|ω| g (ω) e−iωτ132Appendix HPath-Integral Methods inHigher DimensionsIn this next section a entry point to a higher dimensional discussion is intro-duced using the path integral formulation of quantum field theory, followedby some calculations of the static and dynamic Casimir effect for a simpleexample.H.1 Formulation of the Path Integral MethodThe main objective is to calculate the time-ordered vacuum expectation valueof the stress energy tensor, so that the pressures acting on the surfaces, andhence the Casimir effect can be determined. In the path-integral formulation,this is represented as,〈0 |Tµν | 0〉 =∫ DφT 00 (φ) e− i~ (S[φ]+〈J |φ〉)∣∣∣J=0∫ Dφe− i~S[φ] = Tµν(δδJ (x))Z [J ]Z [0]∣∣∣∣0(H.1)where J (x) is an external source that is used to stimulate the quantumfield φ, which itself will be a scalar field representing photons.The basic prerequisite of using path integrals to discuss the Casimir effectand the related phenomena found in previous chapters, is that of imposingboundary conditions for equation (H.1), which will appear around a spatiallycompact region of space-time for the quantum fields that are to exhibit theaforementioned effects. The method of encoding this geometrical data intothe path-integral takes inspiration from the gauge fixing method for bosonicquantum fields, which was introduced by Faddeev [34] and later gener-alized as functional integration over geometries later on [35].The path integral in equation (H.1) is performed over the entire set of fieldconfigurations of φ (x). This means that the quantum fluctuations of φ (x),which do not respect the conductor's boundary conditions, will be includedin the calculation. These fluctuations of φ (x) will violate the properties133of the conductor, which, in fact, totally ignores the conductor's existence.To avoid this we must restrict the path integral measure to integrate overonly those field configurations, both gauge and physical, that respect thefollowing perfect conductor condition35,∀x ∈ S : φ (x) = 0 (H.2)In the case for a translucent, realistic conductor, the boundary conditionsfor the left and right going modes of the photon field will instead be morecomplex but easily represented by imposing another Dirichlet condition inequation (H.2), but that will be beyond the scope of this section.H.1.1 Generating Functional and the Path IntegralMeasureTo make sense of how this will be accomplished, the path integral measure,and the changes it will undergo, must be considered in detail. Consider afunction φ (x), and let us represent it in a orthonormal function space withbasis un (x), such thatφ (x) = limN→∞N∑n=1anun (x) ,∫un (x)um (x) dx = δnm (H.3)where an are constants that specify exactly which function is desired out ofall that exist in L2, the set of square integrable functions. The path integralof some functional F [φ (x)] is now a function of the coefficients ai,F [φ (x)] = F ({an}∞1 ) (H.4)and the path integral of this functional can then be defined as,∫Dφ (x) F [φ (x)] = limN→∞N∏n=1(∫dan√2pi)F (a1, . . . , aN ) (H.5)where all the quantum fluctuations of φ (x) are represented as integra-tions over these coefficients an, over their entire range. To impose the bound-ary condition φ (x) = 0, at a point x0 ∈ R1,3 a delta distribution of the formbelow would be used to modify the integral measure,φ (x0) 'N∑n=1anun (x0) = 0 =⇒ δ (φ (x0)) ' δ(a1u1 (x0) +N∑n=2anun (x0))(H.6)35As seen in Section 4, for the imperfect conductor consideration the boundary conditionchanges to one that oscillates instead of vanishing, as seen in equation 2.13134This is in fact a restriction on the possible range of ai as seen below,∫Dφ (x) δ (φ (x0)) F [φ (x)]= limN→∞N∏n=2(∫dan√2pi)[∫da1√2piδ(a1u1 (x0) +N∑n=2anun (x0))F (a1, . . . , aN )]where we note that the delta distribution above does not collapse the pathintegral, but instead reconfigures it to respect the boundary conditions. Thisprocedure must be done for all x ∈ S and so the path integral measure picksup an uncountable product of delta functions evaluated at all points,∫Dφ∏x∈Sδ (φ (x))F [φ (x)] (H.7)This modification of the integration measure is similar to what one woulddo for a general Lebesgue integral in measure theory [36, pg.19], and theoverall effect is a restriction to a subset of the entire configuration space forφ (x). While on the topic of the path integral measure, it should be notedthat since there is no change in the symmetries that the field φ (x) exhibits,there should be no additional anomalies [37] in the path-integral measureeither.H.1.2 Bounding SurfacesUsing the path integral Fourier transform of this we can then write∏u∈Rdδ (φ (y (u))) =∫Dω exp{i∫ddu√det |hab (u)|ω (u)φ (y (u))}(H.8)hab (u) = `a`b + gab (H.9)where `a is the covariant form of the surface normal of S.In the above analysis we have introduced a new field ω (u) defined onthe space of parameterizations for the manifold u ∈ S. This new field,similar to how a ghost works in a gauge theory, can be interpreted as a fieldthat destroys field configurations φ (x) that do not respect the boundaryconditions, where the ghost did the same but for field configurations thatdidn't respect the particular gauge fixing choice.The main idea here is to treat the conducting surfaces Si in n dimensionsas a manifold embedded in R1,3, at which point the parametric definition of135the embedding of the manifold is defined as,xµ = yµ (ua) (H.10)where xµ ∈ R1,3 and ua ∈ Rd where d is the spatial dimension of the mani-fold.36H.2 Generating functional for two conductorsThe next step in this analysis is to produce the static Casimir effect byintroducing two such alterations to the path integral measure described inEquation H.8 so that the generating functional isZ [J ] =1Z0∫Dφ∏u1∈S1δ (φ (y1 (u1)))∏u2∈S1δ (φ (y2 (u2))) e− i2~∫d4xLφ[x]+J(x)Φ(x)(H.11)then calculate the vacuum expectation value of the stress energy tensor,defined in Equation 1.9. This has been done already by [27], in his appendi-cies. For other geometries work has also been done by Li [28][29] andby Bordag[38].36d = 1 for a line conductor, d = 2 for a surface, as in the usual Casimir case, d = 3 fora realistic conductor with volume, etc136


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