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The topological Casimir effect on a torus van Caspel, Moos 2013

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The Topological Casimir Effect on a TorusbyMoos van CaspelA 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 2013? Moos van Caspel, 2013AbstractThe conventional Casimir effect manifests itself as a quantum me-chanical force between two plates, that arises from the quantization ofthe electromagnetic field in the enclosed vacuum. In this thesis the ex-istence is discussed of an extra, topological term in the Casimir energyat finite temperatures. This topological Casimir effect emerges due tothe nontrivial topological features of the gauge theory: the extra energyis the result of tunneling transitions between states that are physicallythe same but topologically distinct. It becomes apparent when exam-ining, for instance, periodic boundary conditions. I explicitly calculatethe new term for the simplest example of such a system, a Euclidean4-torus. By dimensional reduction, this system is closely related to twodimensional electromagnetism on a torus, which is well understood. Itturns out that the topological term is extremely small compared tothe conventional Casimir energy, but that the effect is very sensitiveto an external magnetic field. The external field plays the role of atopological theta parameter, analogous to the ? vacuum in Yang-Millstheory. The topological Casimir pressure and the induced magneticfield show a distinctive oscillation as a function of the external fieldstrength, something that can hopefully be observed experimentally.iiPrefaceMost of the content in this thesis has been previously published in a pa-per [1], co-written by my collaborator ChunJun (Charles) Cao, my super-visor Ariel Zhitnitsky and myself. This being theoretical physics, I think itis fair to say that the three of us were involved in all parts of the research,through discussions and thorough editing cycles of the paper. Parts that Idid not contribute to, like the appendix, have not been reproduced here butare only cited. Bits and pieces from the rest of the publication have foundtheir way into this work, but the vast majority has been rewritten in orderto cater to a wider audience. I believe that this thesis will be readable to anygraduate student with a basic knowledge of quantum field theory, not justto experts in the field. Hopefully this will make it easier for a new studentto continue our research in the future, helping them answer the questionsthat we have asked ourselves over the last year. Aside from the computa-tions that can be found in the published paper, I have added a significantamount of material to put the research in a broader perspective of physics? including a discourse on related topological systems and two appendices.Charles Cao created the numerical plots in section 5, which have beenreproduced here with his explicit permission. I want to thank him for ourmany discussions, and for his pleasant company on the travels to presentour work at several conferences. It was a joy working with him. I am alsovery grateful to professor Ariel Zhitnitsky, for proposing and supporting thisresearch during the two years of my MSc program. It was inspiring to workon something so new and exciting, and to get rewarded for our effort witha publication.Moos van Caspel, 2013iiiContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Topological Sectors and Winding Numbers . . . . . . . . . . 52.1 Homotopy 101 . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Aharonov-Bohm Effect: Theta States . . . . . . . . . . . 62.3 The Yang-Mills Vacuum: Instantons . . . . . . . . . . . . . . 83 Electromagnetism in Two Dimensions . . . . . . . . . . . . . 103.1 Hamiltonian Approach . . . . . . . . . . . . . . . . . . . . . . 113.2 Path Integral Approach . . . . . . . . . . . . . . . . . . . . . 123.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Topological Casimir Effect in Four Dimensions . . . . . . . 164.1 Decoupling of the Topological and Conventional Parts . . . . 174.2 Computing the Topological Pressure . . . . . . . . . . . . . . 185 The External Magnetic Field as a Theta Parameter . . . . 235.1 Instantons in an External Field . . . . . . . . . . . . . . . . . 235.2 Pressure and the Magnetic Response Functions . . . . . . . . 256 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . 31Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Appendix A: The Conventional Casimir Effect . . . . . . . . . 36A.1 2d Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.2 The EM Casimir Effect on a Torus . . . . . . . . . . . . . . . 38A.3 Thermal Corrections . . . . . . . . . . . . . . . . . . . . . . . 39Appendix B: Mathematical Tools . . . . . . . . . . . . . . . . . . 41B.1 Wick Rotation and Thermal Field Theory . . . . . . . . . . . 41B.2 Poisson Resummation . . . . . . . . . . . . . . . . . . . . . . 42ivList of Figures1 Loops with various winding numbers . . . . . . . . . . . . . . 62 Instantons as tunneling between degenerate vacua . . . . . . 93 The topological pressure as a function of ? . . . . . . . . . . . 214 The topological pressure as a function of the external mag-netic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 The induced magnetic field as a function of the external mag-netic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 The magnetic susceptibility as a function of the external mag-netic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30v1 IntroductionThe idea of a mysterious quantum force, arising purely from the vacuum,was a controversial issue for several decades. When Hendrik Casimir firstpredicted his eponymous effect [2] in 1948, the existence of vacuum fluctua-tions was not yet well established. Quantum field theory was still in its earlydevelopment. Later theoretical advances brought a better understanding ofthe quantum vacuum and the Casimir effect became a well-studied part ofphysics. Regardless, it took nearly 50 years before the effect was quantita-tively measured in an experiment [3], although qualitative confirmation ofthe Casimir force was achieved in the ?70s.The Casimir effect arises from the quantization of fields under boundaryconditions. Within the boundary only certain field modes are allowed, pro-viding a vacuum energy that is different than in free Minkowski space. Thisenergy difference is dependent on the size of the system, such that a realforce is generated on the boundary. The famous example, as first studiedby Casimir, is that of two parallel, perfectly conducting plates. Even in aperfect vacuum, these neutral plates feel an attractive force that is purelyof quantum mechanical origin. The Casimir energy for this system is givenbyEC ? (EBC ? EMinkowski) = ?h? c pi2 L2720 a3 , (1)where a is the separation distance between the plates of length L. Since thesystem can lower its energy by decreasing a, this results in the well-knownCasimir pressureP = ? 1L2??EC?a= ? h? c pi2240 a4 . (2)The pressure rapidly falls off with distance and becomes typically relevanton micrometer scales or smaller. That is why there has been an increasedinterest in the Casimir effect with the recent advent of nanotechnology.Since its original prediction, the Casimir effect has been studied for manydifferent configurations and many different fields [4]. In addition to physicalboundaries like metal plates, the effect can also occur on nontrivial topolog-ical spaces. For example, the periodicity of space can produce a similar fieldquantization as the conducting plates and will also produce a vacuum pres-sure. Typically these calculations are done for simple scalar fields. Whilethe electromagnetic field is more relevant for experiments, it can often be1reduced to a scalar field computation by treating the two polarization direc-tions separately.There are however situations in which the electromagnetic field can pro-duce a unique kind of vacuum effect, that cannot be described in terms ofscalar fields. One of these is the topological Casimir effect , in which an addi-tional vacuum pressure arises due to the gauge invariance of the field, whendefined on a topologically nontrivial manifold. Whereas the conventionalCasimir effect results from the vacuum fluctuations of physical photons, thetopological effect is solely due to a new type of excitations, known as instan-tons.1In the topological Casimir effect, mappings between the gauge groupand the spacetime topology create an infinite number of degenerate vacuumstates. These so-called topological sectors correspond to the same physicalstate, even though they are topologically distinct. The instanton excitations,which contribute to the system?s vacuum energy, can be interpreted as tun-neling events between the vacuum states. Because the topological Casimireffect has an entirely different source than the conventional photon fluctua-tions, it shows some fascinating behavior that is worth studying. Examplesinclude a sensitivity to external magnetic fields and possible applications incosmology.It should be noted that this is not the first time that the vacuum energyis computed for the electromagnetic field on nontrivial topological spaces.Earlier this year, Gerard Kelnhofer formulated the theory for a general com-pact manifold, in a very formal mathematical manner [6]. However, he doesnot discuss any of the physical consequences of the topology. Older paperssuggest that it is necessary to sum over topological sectors, but did notperform the calculations [7]. In this work, I will focus on an intuitive under-standing of the problem, discussing the physical context and the possibilityof experimental observation. Recreating the topological Casimir effect ina lab would be a unique opportunity to probe the topological propertiesof the quantum vacuum in a controlled environment, but there are manychallenges involved that need to be considered.This thesis follows largely the structure of our recent paper [1], with1A remark must be made here on the terminology. In some older literature, the regularCasimir effect on a topological space is also called the ?topological Casimir effect?. However,in this work the term is used in a much stricter sense. Here the topological Casimir effectarises purely from topological fluctuations, instead of real propagating degrees of freedomlike photons. For the latter, I will use the term conventional Casimir effect, to distinguishit from the newly discussed topological contributions. This naming convention was firstused in [5].2the addition of some extra background material. This material serves toput the topological Casimir effect in a broader light and to show some ofthe motivation for this work. As a matter of fact, the topological Casimireffect is very closely related to other systems in physics, sharing the sameunderlying mathematics. In section 2, I will start with a brief explanationof some of this math and discuss two important examples of topology inphysics: the Aharonov-Bohm effect and the Yang-Mills vacuum. Both aredirectly analogous to the systems that are examined later in the text, and Iwill regularly refer back to these examples.In section 3 I review a toy model for the topological Casimir effect,namely 2d electrodynamics on a circle ? a well-studied system with a lot ofliterature. In two spacetime dimensions, photons cannot exist. This meansthat there is no conventional Casimir effect, which would be caused by thezero-point energy of the photon modes. Nevertheless, we will see that thereare topological excitations that give the theory its vacuum energy. These2d instantons can be easily generalized to higher dimensions in order todescribe the 4d topological Casimir effect.By generalizing the 2d electromagnetic theory, the topological Casimireffect is formulated on a Euclidean 4-torus in section 4. In this case, it canbe shown that the topological and conventional part of the vacuum energyare completely separated. Since the conventional Casimir effect can be com-puted through known methods, I will focus on the topological part. Thispart can be related to the 2d theory from section 3 by using the techniqueof dimensional reduction, allowing us to directly apply the previous resultsto this new system. Unfortunately, the topological pressure turns out to bevery small, compared the conventional Casimir force. Without some kind ofcharacteristic behavior, it is unlikely that the topological Casimir effect canbe measured.We are in luck, because the topological Casimir effect interacts with anexternal magnetic field in a very specific manner, as discussed in section 5.While the conventional Casimir effect is unchanged by the application of anexternal field, the topological pressure shows an interesting oscillatory be-havior as a function of the field strength Bext. In fact, the external magneticfield plays the role of a topological theta parameter, in close analogy withthe examples from section 2. Under the influence of this theta parameter,the energy of the instanton configurations is shifted and they will create aninduced magnetic field. The induced field and the corresponding magneticsusceptibility also show an unusual variation as Bext is changed. Hopefullythis can be use to the distinguish the topological Casimir effect from theconventional one.3Section 6 concludes this work with a short discussion on experimentalconsiderations and future directions for the topological Casimir effect. Whilethis is a theoretical paper, I think it is important to keep in mind our relianceon observational data to confirm these new concepts. I cannot complete thisthesis without exploring some of these practical aspects of the effect. Finallyin the Appendix, I review the conventional Casimir effect and discuss a fewmathematical tools that are used throughout the text.42 Topological Sectors and Winding NumbersTopology has been a booming subject within physics in the last decade. Itis an abstract field of mathematics, and as such its applications are oftennot obvious. But in recent years it has become clear that in a wide range ofphysics, from condensed matter to high energy, the consequences of topologycannot be ignored; it can, in fact, lead to some very interesting phenomena.The greatest example is the rapid development in the theory of topologicalinsulators, which started in 2005 and still sees many new publications eachweek.Although topological effects can be found in many areas of physics, theyall have certain features in common. This universality is a result of theunderlying mathematical principles, which I will discuss briefly in this sec-tion. This will be, when possible, from a physicist?s point of view, choosingintuitive understanding over mathematical rigor. For a more mathematicalapproach, see [8].2.1 Homotopy 101Two spaces are topologically equivalent (or homotopic) if you can contin-uously deform one into the other, without cutting them or gluing partstogether. The canonical example is the donut and the coffee mug, whichhave the same topology: the donut?s hole turns into the handle of the mug.However, the donut is inequivalent to a ball, since one cannot remove thehole in a continuous deformation.It is clear that the number of holes in a space is important for the topol-ogy. This is however not a very well-defined quantity, especially in higherdimensions. Therefore we introduce homotopy groups. These groups, la-beled by pin, consist of the mappings between the topological space and then-sphere Sn. The first homotopy group, also known as the fundamentalgroup pi1, has a fairly intuitive definition in terms of loops: its elements areunique loops that cannot be continuous deformed into one another.For example, a space with one hole through it (like the circle S1) cancontain loops that wrap around this hole an integer number of times, seefigure 1. Loops with different winding numbers are topologically distinctand correspond to different elements of the fundamental group. As a result,we can conclude that pi1(S1) ' Z. A torus T2 ? S1?S1 has two independentwinding numbers, one for each circle, yielding pi1(T2) ' Z2. Contrarily, ona sphere S2 any loop can be contracted to a point, such that pi1(S2) ' 0. Ifall homotopy groups of a space are empty, as is the case with Euclidean or5Figure 1: Illustration of loops with different winding numbers. The blackdot represents a hole in the space, such that a loop cannot pass it when beingcontinuously deformed. a: n = 0. The loop can be continuously contractedto a single point. b: n = 1. The loop wraps once around the hole, it cannotbe contracted to a point. c: n = 2. The loop wraps twice around the hole.Minkowski space, the space can be considered topologically trivial.In general it can be very hard to compute the homotopy groups of agiven space, but in the context of physics this is typically not necessary.We will see that the previous example, where the fundamental group isequivalent to the integers, is in fact of the most interest to us. The integerscorrespond to the so-called topological sectors, which play an important partin the topological Casimir effect. In the rest of this section I will discuss twofamous instances of topological sectors in physics that will demonstrate tworelated concepts: theta states and instantons.2.2 The Aharonov-Bohm Effect: Theta StatesWhile often covered in basic quantum mechanics classes, it might be a sur-prise to some that the Aharonov-Bohm effect is topological in nature. Anelectron moving around a solenoidal magnet will acquire a complex phase,even though it does not pass through any electromagnetic field. To show therole of topology, let us confine the electron to a circle around this magnet,so that its movement can be described by the angular coordinate ?. Theclassical action is given byS[?] =?dt(me2 ??2 + ?2pi ??), ? ?eh?c? (3)with ? the flux through the solenoid. It is immediately clear that the secondterm is a full derivate and thus depends only the beginning and end positionof the electron. This is characteristic of a topological term in the action:it does not depend on the path taken. With every full circle clockwise, theterm increases by ?.6In classical mechanics, the topological term would not contribute to theequations of motion. However, after canonical quantization we find theenergy eigenvalues to beEn =12me(n??2pi)2, n ? Z, (4)where we work in natural units h? = c = 1 from here on. The spectrumis shifted, depending on ?, but note that there is a 2pi periodicity. Thecorresponding eigenfunctions are ?n = ein?, showing the familiar phasechange as the electron moves around.The problem becomes more interesting when we add a finite temperatureand view it as a quantum statistical system. In this case we need to computethe partition function, defined by the path integralZ =?D? eiS[?] =?ED? e?SE [?]. (5)In the second expression we have switched to Euclidean space by means ofa Wick rotation (see appendix B.1). The rotated time ? ? ?it is periodicwith period ?, which means that all paths must satisfy ?(?)? ?(0) = 2pik,for some integer k. This integer is known as the winding number, as itcounts the number of loops around the circle during one time period. In thepartition function one must sum over all winding numbers.With this, the Euclidean action becomes:SE =? ?0d?(me2 ??2 ? i?2pi ??)= ?i?k +? ?0d? me2 ??2 (6)where we have made use of the topological term as a total derivative. In-serting this into the path integral, we can writeZ =?kei?k??(?)??(0)=2pikD? e?12? ?0 d? me??2. (7)This final form shows very clearly the physics of the system: the partitionfunction is a sum over path integrals in different topological sectors, whichare represented by the winding number k. The winding numbers originatefrom the mapping of the paths onto the circle, or pi1(S1) ' Z.The topological ?-term emerges as a complex weight of the topologicalsectors. Although ? has a specific physical meaning in this case, i.e. the mag-netic flux through the system, its role in topological effects is very universal;we shall see several other examples. An important remark is that even at? = 0, the topological sectors still contribute to the partition function.72.3 The Yang-Mills Vacuum: InstantonsYang-Mills theory is a non-Abelian gauge theory that forms the basis for thedescription of the weak and strong forces in the Standard Model. Here wewill work with the SU(2) gauge group, since it is generated by the familiarPauli spin matrices. In the vacuum state, the potential must be a pure gaugeconfiguration as given by:A? =igU(x) ??U ?(x) , U(x) ? SU(2) (8)where g is the coupling constant and we have fixed our gauge such that A0 =0. This defines the gauge U on R3. However, by assuming that U is constantat infinity, the boundary is shrunk to a point and the space is compactifiedinto a 3-sphere S3. That leaves us with the mapping U : S3 ? SU(2), whichcan conveniently be characterized by the homotopy group pi3(SU(2)) ' Z.Physically, that means that every gauge U can be assigned an integerwinding number, that signifies how many times U(x) wraps around thespatial 3-sphere. Gauges with different winding numbers cannot be contin-uously deformed into each other. The consequences of this are very deep:configurations of pure gauges with different winding numbers must be sep-arate, local minima of the Hamiltonian. After all, when deforming one intothe other we must pass through configurations that do not correspond to apure gauge, and thus have a nonzero energy. See [9] for details.These local minima provide an infinite number of topologically distinctvacuum states |n?, known as the topological sectors of the theory. Theyare connected by gauge transformations with a nonzero winding number,also known as large gauge transformations (LGTs). The result is sometimescalled a degeneracy of the vacuum, but one must take care not to confuse itwith the regular definition of degeneracy in quantum mechanics - these localminima all correspond to the same physical state, despite being topologicallydifferent. In fact, the generic ground state of the system can be written as|?? =?ne?i?n|n?. (9)This is known as the theta vacuum, and the similarities to eq. (7) are striking:again ? serves as the complex weight of the topological sectors. Indeed,the ? parameter would appear in the Yang-Mills Lagrangian density as thetopological termLtop =ig2?16pi2 F???F?? (10)8Figure 2: As a result of mappings between the gauge group and the space-time topology, there are an infinite number of winding states |n? correspond-ing to local minima of the Hamiltonian. These degenerate vacuum states aretopologically different, even though they resemble the same physical state.An instanton configuration A?k of topological charge k represents tunnelingbetween these topological sectors.with F??? = 12????F?? being the dual field strength. Just as in the Ahoronov-Bohm effect, this term is a total derivative, such that the action depends onlyon the boundary conditions. It does not contain the metric tensor, anothercharacteristic property of topological terms. In quantum chromodynamics,the apparent lack of such a term is known as the strong CP problem.2If all the winding states |n? correspond to the same physical state, thenwhat are the consequences of this topological vacuum structure? The mostimportant feature is a new type of excitations, called instantons. Instantonsrepresent tunneling through the energy barriers between vacuum states -they are non-perturbative, classical solutions to the Euclidean equations ofmotion. When solving a double-well problem in the path integral approach,instantons will emerge. In the case of the Yang-Mills vacuum, there canbe tunneling between any two vacua |n? and |n??, giving the correspondinginstanton a so-called topological charge k = n? n?. See figure 2.Instantons play a crucial role in the rest of this thesis. They are real,topological excitations that need to be taken into account when computingpartition sums and expectation values. In the next section we will showthat instantons can arise in electrodynamics, we will explicitly constructthem and see how they will lead to the topological Casimir effect.2A nonzero theta term in the QCD Lagrangian would be a large source of CP-violationin the strong sector. That makes it possible to measure ? with great precision and thecurrent experimental upper limit is |?|< 2 ? 10?10. Because there are no symmetryarguments why the term should vanish, this poses a fine-tuning problem.93 Electromagnetism in Two DimensionsBefore arriving at the topological Casimir effect, it is instructive to study amuch simpler system: Maxwell electrodynamics on a two-dimensional, com-pact spacetime. This is a well-understood problem, but it will prove to bean exceptionally good toy model for topological effects in higher dimensions.Even better, we will be able to generalize our computations from this sectionto four-dimensional spacetime and use them to describe the TCE.With two spacetime dimensions, the theory of electromagnetism is quitedifferent than what we are used to. The most interesting property is thatelectromagnetic waves cannot exist in just one spatial dimension. Afterall, the Maxwell equations require that the polarization of an EM-wavebe perpendicular to its direction of propagation. As a result, the waveequation has no solution other than the zero mode. In terms of quantumfield theory, this means that the model has no photons and is completelydevoid of physical propagating degrees of freedom. In that sense the theorycan be considered ?empty?. However, we will see that when formulatingthe theory on a circle, topological excitations will emerge ? much like theinstantons in section 2.3.In the Yang-Mills vacuum, the existence of topological sectors and in-stantons is due to the mapping between the compactified Euclidean spaceand the gauge group, characterized by pi3(SU(2)) ' Z. On the other hand,in the 2d Maxwell theory we have a mapping between the one dimensionalcompactified space S1 and the Abelian gauge group U(1). Here too we finda nontrivial vacuum structure, given by pi1(U(1)) ' Z. The tunneling be-tween these topological sectors is what provides the theory with its energyspectrum and its interesting features.A more intuitive way to view the topological sectors is to consider peri-odic boundary conditions, up to a gauge.A?(L, t) = A?(0, t) + ??? (11)where L is the size of the space and ? is some scalar function. However,an electron field would pick up a phase eie?(x) ? U(1) under this gaugetransformation. If we require the matter field to be single valued on thecircle, this puts certain restrictions on the allowed gauges, for example? = 2pikeLx , k ? Z. (12)In other words, the periodicity of the matter field quantizes the allowedgauges ?. For k 6= 0, ? represents a large gauge transformation of winding10number k, as mentioned in section 2.3. We shall see below that the boundarycondition (11) is satisfied by a classical instanton configuration of topologicalcharge k. Again one has to sum over all allowed k in the partition sum, toinclude all excitations in the system. The topological features of the theoryarise very naturally in this way.Two-dimensional electromagnetism on a circle is also known as the SchwingerModel without fermions, which has been solved in a number of differentways [10]. The presence of instanton configurations in this model is wellestablished and in fact has been shown to be required for a consistent the-ory [1]. Without the contribution of the topological sectors, the Schwingermodel would violate the Ward Identities, one of the most fundamental state-ments about symmetries in quantum field theory.Because the model is so simple, it is possible to do the canonical quan-tization and compute all quantities using the Hamiltonian approach. It ishowever more useful for us to take the path integral approach, which is mucheasier to generalize to higher dimensions. We will discuss both methods be-low.3.1 Hamiltonian ApproachThe Hamiltonian approach is exactly like the analysis in section 2.2 of theelectron on a circle. The single zero mode of the electromagnetic field playsthe role of the particle. To see this, we fix the gauge as followsA0 = 0, ?1A1 = 0. (13)This means the scalar electric field is given by E = A?1. There can be nomagnetic field in one spatial dimension, since the vector cross product hasno meaning. Classically, Gauss? Law implies that ?xE = 0, which showsthat we are dealing only with one x-independent zero mode. As mentionedearlier, no physical propagation degrees of freedom can exist in this system,as there can be no polarization perpendicular to the momentum.We can then quantize the theory, taking A1 and E as canonical conju-gates:[A1(x), E(x)] = ih? ?(x? y). (14)The periodic boundary conditions are due to the identification of the gaugeequivalent configurationsA1 ? A1 +2pineL, n ? Z (15)11as discussed above. With that, the rescaled variable eA1L takes the placeof the coordinate ? in the Aharonov-Bohm effect of section 2.2. By perfectanalogy, the conjugate momentum is quantized E = en and the Hamiltonianwith its energy eigenvalues becomesH = ? 12L ?d2dA21, En =12n2e2L. (16)In studies of the Schwinger model it is also customary to add a thetaparameter, although the interpretation is not as clear as in the Aharonov-Bohm effect, where it represents magnetic flux. The parameter enters thecomputations in the same way, by shifting the energy levels:En =12e2L(n+ ?2pi)2. (17)Including the theta term, the thermal partition sum takes the following formZ(?) =?n?Ze??En =?n?Ze?12 e2L? (n+ ?2pi )2 , (18)where ? is the inverse temperature. We will now derive this same result usingEuclidean path integral calculations, in which the role of the instantons ismuch more obvious.3.2 Path Integral ApproachBy performing a Wick rotation, we can define the same theory on the Eu-clidean torus of size L??. See appendix B.1. In this formulation the appear-ance of topological sectors becomes immediately clear. In order to satisfy theboundary conditions (11, 12), we can introduce the classical instanton-likepotentialA(k)0 = 0 , A(k)1 =2pikeVx0, (19)where V = L? is the volume of the Euclidean space. The integers k classifythe instantons, which represent transitions between the topological sectors.The field strength of the classical configuration is given byE(k) = ?0A(k)1 =2pikeV. (20)Please note that this E-field should not be confused with the real electricfield that was derived in Minkowski space with the Hamiltonian approach.12Instead, (20) is an unphysical, complex configuration in Euclidean spacewith no clear interpretation. It is proportional to the so-called topologicalcharge density Q = e2piE(k) such that the integral?d2x Q(x) = e2pi?d2x E(k)(x) = k (21)is the topological charge of the instanton.For the path integral, we split the potential into the classical config-uration plus the quantum fluctuations around it: A? = A(k)? + ?A?. Thefluctuating field ?A? must satisfy periodic boundary conditions, such thatthe action S = 12?d2xE2 can also be split into a classical and a quantumpart - the cross terms vanish when integrating the full derivative. As aresult, the partition function can be written asZ =?k?Z?DA e?12?d2xE2 = Zq?k?Ze?2pi2k2e2V , (22)where the exponentials represent the classical instanton configurations andthe prefactorZq =?D?A1 e?L2?d2x ?A?21 (23)is the path integral over the fluctuating field. By the same reasoning as in theHamiltonian approach, the A0 = 0 gauge implies that ?A? is x-independent.,which makes the computation of Zq a simple quantum mechanical problem.Zq =?D?A1 e?L2? ?0 d?(?A?1)2 =?D?a1 e?L2 ( 2pieL)2? ?0 d?(?a?1)2 (24)where in the second step we have changed integration to the dimensionlessvariable ?a1 = eL2pi ?A1, which fluctuates between 1 and 0 according to (15).This makes the partition sum equivalent to that of a free particle with massm ? L(2pieL)2 and for this Gaussian path integral the result is known to beZq =?m2pi? =?2pie2V. (25)Inserted into eq. (22), this gives the full partition sum.In Euclidean space, the topological ?-term enters the partition functionas the complex weight of the topological sectors, just like we saw in sec-tion 2.2. The final result isZ(?) =?2pie2V?k?Ze?2pi2k2e2V+ik?. (26)13This expression is exactly the same as the result obtained by the Hamiltionapproach in (18), as can be seen by applying the Poisson resummation for-mula (see appendix B.2). The two forms are said to be dual. See [11] fora more detailed comparison between the Hamiltonian and the path integralapproach of the Schwinger model.One thing of note about the partition sum (26) is that it converges.Unlike the conventional Casimir effect, here there is no need for a regular-ization scheme. While there is no closed form for the expressions (18, 26),they correspond to a special function that is well studied: the Jacobi Thetafunction ?. The two dual forms are very usual for studying the asymptoticbehaviour of the system, for example in the low temperature limit.3.3 InterpretationFrom the partition function it is easy to calculate the free vacuum energyF = ? 1? lnZ and any other thermodynamic quantity. Expressions (18, 26)show clearly that the free energy depends on the size of the system L, justlike in the conventional Casimir effect. This is crucial for our work, since itholds the promise of a measurable effect: the vacuum pressure on the system.The system could lower its energy by moving the boundaries, providing areal force.In the conventional Casimir effect, the vacuum pressure arises from thezero-point energy of the electromagnetic field modes, i.e. the photons. How-ever, in our two-dimensional system these photonic modes do not exist andas such cannot contribute to the zero-point energy. What then causes thispressure that our computations suggest? This energy comes from topologi-cal excitations at a nonzero temperature, rather than the regular propagat-ing degrees of freedom. These excitations, or instanton configurations (19),can be interpreted as the tunneling transitions between degenerate vacuumstates. The vacuum states all correspond to the same physical state, butare topologically distinct and related to each other by large gauge transfor-mations like (12).The energy associated with the tunneling processes is dependent on thetopological charge (21), or the separation between the topological sectors.It is also the energy of this tunneling that depends on the system?s size,providing the theory with what we call the topological Casimir effect - avacuum pressure purely caused by topological degrees of freedom. The un-derlying cause of this effect is the interaction between the gauge field anda nontrivial compact spacetime manifold, characterized by the fundamentalgroup pi1(U(1)) ' Z. This means that it can never be reduced to a problem14in terms of scalar fields, a technique that is often used in the conventionalCasimir effect (see appendix 6).The question is whether this effect would be really measurable, or whetherit is just a mathematical glitch that can be removed by renormalization orthe redefinition of variables. After all, if the degenerate vacua correspond tothe same physical state, then why should tunneling between them result in aphysical effect? This is a question that has plagued physicist ever since theYang-Mills vacuum was first studied. However, as mentioned before thereis clear evidence from the Schwinger model that topological sectors are realand must be taken into account for a consistent theory [1].Another concern is whether the topological effects in this section arejust an interesting quirk of two-dimensional spacetime. Since we live infour spacetime dimensions, this would mean that the effect is irrelevantfor any experimental measurement, no matter how interesting the theoryis. Fortunately, the topological Casimir effect is much more general thanthat and the next section deals with its description in 4d. Many of thecomputations from this section will carry over, yielding a very simple modelthat may allow the measurement of topological vacuum effects in the lab.154 Topological Casimir Effect in Four DimensionsIn two-dimensional electromagnetism, the nontrivial structure of the vac-uum arises due to the mapping of the gauge group U(1) onto the manifoldS1. A naive generalization to 4d spacetime would follow the reasoning ofsection 2.3, by compactifying the Euclidean space to S3. Unfortunately thisdoes not work: while S3 is not topologically trivial, its mappings to U(1)are. In the language of homotopy groups, we can write pi3(U(1)) ' 0, mean-ing that on this manifold all the gauges can be continuously transformedinto one another. There will be no degenerate vacua, no tunneling and notopological Casimir effect. In order to recover the topological vacuum effectsin higher dimensions, we need to look at different manifolds ? like thosewith a toroidal topology.Consider a four-dimensional Euclidean box, of size ? ? L1 ? L2 ? L3.When we apply periodic boundary conditions to the opposing sides of thebox, this creates a space with the topology of a 4-torus S1 ? S1 ? S1 ? S1.This manifold does provide topologically nontrivial mappings to the gaugegroup U(1), and in fact there are several independent winding numbers thatcount the wrapping around the different loops on the torus. The calculationsfor this system are difficult and have been done in [6]. However, we canmake some assumptions that will greatly simplify the theory, allowing us todescribe the 4d topological effects in terms of the 2d computations.If we assume that L3 is much smaller than L1 and L2, this suppressesthe contribution of all but one of the winding numbers, for reasons thatwill become clear below. Taking a slice of this system in the xy-plane nowprecisely recovers the topological features of the 2d Schwinger model, atechnique known as dimensional reduction. It also simplifies the calculationof the propagating degrees of freedom, analogous to the simple example oftwo large parallel plates in the conventional Casimir effect. In fact it israther tempting to interpret this 4-torus simply as two large parallel plateswith periodic boundary conditions, at finite temperature.These two portions of the vacuum energy, resulting from the topologicalsectors and the physical photons respectively, are completely decoupled inthis system. This means that the two contributions can be computed sepa-rately. Convenient, since the conventional Casimir effect for these boundaryconditions has been studied many times. I have nothing new to add thereand the details are discussed in appendix A.2. Most of this section will con-sequently focus on the topological part, such that in the end we can compareits effects to the conventional vacuum pressure.164.1 Decoupling of the Topological and Conventional PartsIt is not difficult to generalize the 2d instanton potential (19) to higherdimensions. The boundary conditions (11, 12) are more or less unchanged,but can be applied in any of the directions. One way we can define theclassical instanton configurations isA?top =(0, ? pikeL1L2y,pikeL1L2x, 0), (27)where k is is the integer-valued topological charge, and L1, L2 are the di-mensions of the plates in the x and y-directions respectively, which areassumed to be much larger than the distance between the plates L3. Whengoing around a loop in the xy-plane, this potentional picks up a large gaugetransformation that corresponds to tunneling from one topological sector toanother, separated by winding number ?n = k. The configuration providesa topological magnetic flux in the z-direction:Btop = ~?? ~Atop =(0, 0, 2pikeL1L2), (28)in close analogy with the 2d case in eq. (20). Technically, the periodicboundary conditions would also allow different sets of instantons with theirown winding numbers, related to loops in different directions. However,the magnetic flux for these configurations would be proportional to 2pieL1L3or 2pieL2L3 , which correspond to a much larger energy because L3  L1, L2.Since we are only considering low temperatures, the instantons of eq. (27)would dominate the partition sum such that we can safely neglect all others.With this, the Euclidean action of the system becomes12?d4x{~E2 + (Bq + Btop)2}, (29)where the integration is over the Euclidean torus L1 ? L2 ? L3 ? ? and ~Eand B are the photonic quantum fluctuations of the electromagnetic field.These terms were not present in the 2d model, but must here be taken intoaccount due to the presence of real propagating physical photons in fourdimensions. We find that the action can be easily split into the sum of atopological and a quantum part, because of the vanishing cross term?d4x Bq ?Btop =2pikeL1L2?d4x Bz = 0 (30)17Here the fact is used that Bq is periodic over the domain of integration. Asa result, there is no coupling between the conventional quantum fluctuationsand the classical instanton potential (27).As a result, the partition function of the system can be written as Z =Z0 ? Ztop and both parts can be computed separately. The conventionalpart Z0 is nothing but the electromagnetic Casimir on the torus, whichcan be reduced to a simple scalar field calculation. After all, it does notdepend on the topological sector k of the theory, so that we can only considerthe trivial sector k = 0. The resulting expression requires zeta-functionregularization, but is not much more difficult to obtain than that for thecanonical example of the parralel conducting plates. The details are workedout in appendix A.2. As usual, the conventional Casimir effect manifestsitself as a pressure on the plates in the xy-plane due to the zero-point energyof the photon modes, found to be:P0 = ?2pi215L43(31)at zero temperature. A finite temperature will add a small correction, ascomputed in appendix A.3.With the conventional part accounted for, we can now focus purely on thecontribution from the topological sectors. For this we will invoke argumentsof dimensional reduction to relate our problem to that of section 3. It isthen possible to find the topological pressure and compare it to eq. (31), tosee whether the topological features of the vacuum result in a measurableeffect.4.2 Computing the Topological PressureThe instanton potential (27) assumes periodic boundary conditions up toa large gauge transformation, but only in the x and y directions. In otherwords, all the relevant topological features of the theory live in the xy-plane,which we can consider as a 2-torus. This makes things extraordinarily easy,because the partition sum of topological excitations on a 2-torus is knownfrom section 3. Some minor adjustments have to be made due to the extradimensions, but most of the work has been done.Taking 2d slices in the xy-plane recovers the Schwinger model withoutfermions. Summing over all these slices is done simply by the integral in the4d action (29). Comparing the instanton configurations (28) and (20), it is18clear that the classical action of the 4d instantons will be slightly different:Stop =12?d4x( 2pikeL1L2)2= 2pi2k2?L3e2L1L2(32)as apposed to the 2d action S2d = 2pi2k2e2?L . This difference is partly explainedby the obvious substitution L, ? ? L1, L2 that follows from the definition ofthe 2-torus slices. The rest of the disparity can be interpreted as a rescalingof the coupling constant, i.e. the electric chargee22d =e24d?L3,e24pi ? ?. (33)This is a common trick of dimensional reduction, and such a redefinitionis in fact necessary in order for the units of the actions to be consistent.In 4d, e2 ? ? is the dimensionless fine-structure constant, whereas in 1+1dimensions the QED coupling constant has units of (length)?2. With theredefinition (33), the two actions are equal. We now apply the same sub-stitutions to the resulting 2d partition function (18, 26) to acquire the 4dexpressions:Ztop =?2pi?L3e2L1L2?k?Ze? 2pi2k2?L3e2L1L2 =?n?Ze? e2n2L1L22?L3 . (34)Again, the two expressions are dual, related by the Poisson resummationformula. We have chosen to take ? = 0, for now, because there is no physicalargument for such a term in the Lagrangian. In section 5 we will furtherinvestigate how a theta term might emerge in the topological Casimir effect,under the influence of an external magnetic field.For the purpose of notation is is convenient to introduce the dimension-less parameter? ?2?L3e2L1L2(35)such that the topological partition sum takes the extremely simple formZtop(?) =?pi??k?Ze?pi2?k2 =?n?Ze?n2? . (36)While this series again has no closed form, there are still some interestingremarks we can make.19Consider the case where the size of the plates becomes infinite, or L1L2 ??. Here the parameter ? approaches zero. Using the second expression ofeq. (36), we see that all terms vanish, except for the one with n = 0. Accord-ingly, Ztop = 1 and all topological features of the theory disappear. Only theconventional Casimir effect Z0 remains. Also of interest is the asymptoticlimit where ?  1. In this case we can include the first nonzero n in thesecond sum of (36), to findZtop ? 1 + 2 e?1/? , ?  1 (37)The pressure on the plates is defined as the derivative of the free energyF = 1? lnZ towards the separation distance L3, divided by the area L1L2.Ptop =1?L1L2???L3lnZtop (38)In the asymptotic limit, the pressure can be calculated analytically fromeq. (37). We use the approximation ln(1 + 2e?1/? ) ? 2e?1/? to find:Ptop ?e2?2L23e?1/? , ?  1. (39)As expected, the topological pressure is exponentially suppressed in thislimit. Comparing this with the conventional Casimir pressure (31), it isclear that the topological effect cannot be measured in this case. Measuringthe conventional pressure in the lab is no easy feat, and this contribution isfar smaller.While ?  1 can be examined analytically, it is more interesting tostudy a system where ? ' 1. Is this a reasonable regime, considering theassumption that L3  L1, L2? At this point it is useful to look at somenumbers, just to get a sense of the orders of magnitude involved. Let us takeL1 = L2 = 1mm, L3 = 0.01mm and a temperature of 1K. Inserting thesenumbers into eq. (35) gives us ? = 0.5. Such values are not implausible foran experimental measurement and satisfy all our assumptions necessary forusing dimensional reduction as well as the low temperature approximationfor the conventional Casimir effect. The result is a value for ? that is welloutside the asymptotic limit of eq. (39).Unfortunately there is no analytical expression for Ptop in the regimewhere ? ' 1 and we must resort to using numerical calculations in orderto study its behavior. The series is slow to converge and saturates aroundk ? 1000. Figure 3 shows the numerical result in the range 0 ? ? ? 3. Thetopological pressure on the vertical axis is measured in units of 2e2L21L22.20Figure 3: A numerical plot of the topological pressure in the z-direction ona 4-torus, as a function of ? ? 2?L3/e2L1L2. Pressure is measured in unitsof 2L21L22e2 .The numerical graph shows several interesting features. First of all, theexponential behavior of eq. (39) when ? ? 0 is clearly reproduced. Moreimportant however is the large peak around ? ? 0.4, where the topologypressure is of order 1 in its given units. It seems that fine-tuning our pa-rameters to reach this maximum is our best bet at producing a measurableeffect. The largest topological pressure possible in this system is thus ap-proximatelyP maxtop ?2e2L21L22. (40)To put this in perspective, let us compare this result with the conventionalCasimir effect of eq. (31). It is enlightening to look at the dimensionlessratio of these two pressures:Rmax =|P maxtop ||P0|?15L43e2pi2L21L22= 154pi3? ?L43L21L22. (41)Even at this maximum, the ratio is tiny. Using the example values for theparameters that were mentioned earlier, we find Rmax ? 10?7, a negligi-21ble contribution. It appears that any topological effect will be completelydrowned out by the regular propagating degrees of freedom.Before concluding this section, there are a number of remarks that canbe made about these computations. The presence of the topological sectorsproduces a vacuum pressure with a positive sign, i.e. the plates will berepelled from each other. This is in contrast with the conventional Casimireffect, which manifests itself in this system as an attractive force. Also ofnote is the appearance of the coupling constant in the final expression. Theinverse coupling constant in the exponent exp(?1/e2) is a characteristicfeature of quantum tunneling processes [12], which is consistent with ourinterpretation of the instantons as tunneling events between the degeneratevacua.While eq. (41) may suggest that the topological Casimir effect can neverbe measured, it is still way too soon to give up hope. We have found thatthere is a real, physical force associated with the existence of topologicalsectors, even though it is small. On a compact manifold like a 4-torus,the gauge freedom of the electromagnetic theory causes a degeneracy ofthe vacuum state, that allows for a new type of tunneling excitations calledinstantons. The instantons are completely decoupled from the photons in thesystem and provide an extra contribution to the Casimir pressure, as givenin eqs. (36, 38). This underlying topological vacuum structure is analogousto that of the non-Abelian Yang-Mills field of section 2.3, but is special inthat it could be directly measured, at least in theory, through the topologicalCasimir effect.Due to its unusual origin, the topological Casimir effect has some uniquefeatures that separate it from its conventional counterpart. It is these qual-ities that will hopefully make it possible to measure the effect in a lab,despite it being relatively tiny. Most significant is the sensitivity towards anexternal magnetic field, which interacts with the instantons and creates atype of theta vacuum. The conventional Casimir effect is not affected by anexternal field, making it a promising tool to probe the topological featuresof the theory. The next section will focus on the response of our system tosuch a magnetic field.225 The External Magnetic Field as a Theta Param-eterBecause the Topological Casimir effect is so much smaller than the conven-tional Casimir pressure, some sort of probe is needed to detect it: somethingthat is sensitive to the topological sectors but not to the fluctuating pho-tons. Fortunately such a thing exists, in the form of an external magneticfield. In this section I will show that the external field interacts with theinstantons by shifting the energy levels of the excitations. Apart from af-fecting the topological pressure, this also induces a nonzero magnetic field inthe system due to the presence of the topological sectors. Interestingly, theexternal magnetic field shows all the behavior of a topological theta term,as discussed in sections 2 and 3.Classically, an external magnetic field should have no influence on theconventional Casimir effect, simply due to the linearity of the Maxwell equa-tions. Electromagnetic modes do not interact with each other. On thelevel of QED it is possible to have photon-photon interactions, but theyare greatly suppressed. The lowest order Feynman diagrams have four ver-tices. This suggests that any such interaction is proportional to the factor?2B2ext/m4e, which is incredibly small. Even in an external field as largeas 1 Tesla this amounts to an order of 10?20, much smaller still than thetopological contribution. The exact correction to the Casimir effect due tophoton-photon interaction is calculated in [13], confirming this quick esti-mation.Because of its negligible effect on the conventional Casimir pressure, theexternal magnetic field is an ideal tool to probe the topological properties ofour system. In the rest of this section we will assume a constant field Bextin the z-direction. First we will calculate how this modifies the partitionsum, before looking at the various response functions of the system, suchthe induced magnetic field and the magnetic susceptibility.5.1 Instantons in an External FieldIncluding a constant Bext in our Euclidean action (29) is fairly straight-forward3. The total magnetic field is now comprised of three parts: B =Bq+Btop+Bext. The external field will add an uninteresting constant termB2ext to the action, but more important are the cross terms. As mentioned3Note that a magnetic field is invariant under a Wick rotation, unlike an electric field,so we do not have to worry about the physical interpretation of Bext in the Euclideanaction.23above the interaction term Bq ?Bext should vanish, which is clearly the casedue to the periodicity of Bq over the domain of integration ? just like ineq. (30). The conventional and topological parts are still decoupled, nothinghas changed there. Explicitly we find(Bextz +2pikL1L2e) ??d4xBqz = 0. (42)The conventional Casimir effect is unchanged and Z0 can be computed asnormal. It is the last cross term that will be making the difference:2 Btop ?Bext =4pikL1L2eBext. (43)This new term is nonzero and depends on the topological sector k. It canbe considered as a source term in the action, which will be useful later on.As a result, the energy levels of the instantons get shifted and the classicalEuclidean action becomesStop = ?L1L2L3( 2pikL1L2e+Bext)2= pi2?(k + ?eff2pi)2, (44)where, in close analogy with eq. (17) from 2d electrodynamics, we havedefined the effective theta parameter?eff = Bextz L1L2e. (45)The rest of the partition sum computations is the same as before. Theprefactor ?pi? , that arises from the fluctuations around the instanton con-figuration and was derived via dimensional reduction, is unaffected by theexternal field. Inserting the shifted energy of eq. (44) into the partition sum(36) gives us the final expressionZtop(?, ?eff ) =?pi??k?Zexp[?pi2?(k + ?eff2pi)2](46)Before studying the physical response functions of the system, a few re-marks should be made about the interpretation of the effective theta param-eter. At first sight it appears that ?eff enters the partition sum in exactlythe same way as in eq. (18), but there are some subtle differences. After all,eq. (18) was derived using the Hamiltonian approach in Minkowski space,whereas our analysis of the 4d system is all done in the Euclidean metric.24The description of instantons as tunneling configurations between topolog-ical sectors is not possible in Minkowski space and conversely, it is unclearhow to solve the 4d system using canonical quantization like in section 3.1.This is problematic for our interpretation that the external magnetic fieldcreates a theta vacuum. In the Euclidean path integral approach, a thetavacuum appears in the action as a complex weight of the topological sector,as seen in eqs. (7) and (26), rather than the shift of the energy levels thatwe find in (46).To resolve this issue, consider the dual representation of (46) after Pois-son resummation:Ztop(?, ?eff ) =?n?Zexp[?n2?+ i ?eff n]. (47)In this form, the role of the magnetic field as a theta parameter becomesmore clear. The partition function is obviously 2pi-periodic in ?eff , whichsatisfies all the criteria of a topological theta term. As is necessary in Eu-clidean space, the theta term enters the expression as a complex phase andcouples to the topological integer n. However, it should be emphasized thatn does not label the original instantons that we constructed in section 4.Instead the terms in this series correspond to some kind of dual configu-rations, with a classical action proportional to ??1. This means that the?-state created by the magnetic field cannot easily be described in terms ofwinding numbers and topological sectors.From eq. (47) it follows that the asymptotic behavior is largely the sameas in section 4.2. In the limit ? ? 0, once again all topological featuresvanish as only the n = 0 term remains, leaving Ztop = 1. The asymptoticlimit ?  1 also includes the terms n = ?1 to yield:Ztop ? 1 + 2 cos(?eff ) e?1/? , ?  1, (48)similar to (37). In the general case, we again need numerical tools to inves-tigate the pressure and the magnetic behavior of the system.5.2 Pressure and the Magnetic Response FunctionsNow that the partition function is resolved, finding the topological pressure(38) is easy. To get a first impression of the external field?s influence, letus look at the small-? limit, where there is an analytical solution. Eq. (48)yieldsPtop ?e2?2L23cos(?eff ) e?1/? , ?  1. (49)25Figure 4: A numerical plot of the topological Casimir pressure as a functionof ?eff ? eL1L2Bext, given in units of 2L21L22e2for three different values of ? .A clear 2pi periodicity is seen and local extrema are between odd and eveninteger multiples of pi.While the pressure is still exponentially suppressed, what is striking is thefactor cos(?eff ). This suggest that the force on the system will switch be-tween being attractive and repulsive, depending on the strength of the ex-ternal magnetic field Bext ? ?eff . The same thing is seen in Figure 4, anumerical plot of the topological pressure as a function of ?eff , for variousvalues of ? ' 1.Figure 4 also shows clearly the oscillatory behavior with respect to ?eff .The extrema of the function are at integer multiples of pi, with the odd mul-tiples corresponding to a negative Casimir pressure. We see that, by playingaround with the extra parameter ?eff , it is possible to further amplify thetopological Casimir effect. Unfortunately the increase is not enough to makea significant difference in comparison with the conventional pressure. A 3dplot shows a maximum pressure only few times larger than what we foundin eq. (40).However, the oscillating variation of the topological Casimir pressure26in an external magnetic field is a unique feature that might be possibleto measure experimentally. It serves mainly to distinguish the topologicalcontribution from the conventional Casimir effect, which is not sensitiveto an external field. Bext takes the role of a theta parameter by causinginterference between the topological sectors. This shifts the energy spectrumof the instantons to produce the variation shown in Fig. 4.In addition to changes in the topological pressure, it is interesting tostudy how Bext affects the magnetic properties of the system. After all, theinstanton excitations carry a magnetic field (28). In the case where Bext = 0,the magnetic field configurations of instantons with opposite topologicalcharge (but the same energy) will cancel each other, resulting in a vanishingnet magnetic field. This can also be seen from symmetry arguments: withoutan external magnetic field the system is P and CP invariant, which meansthere can be no induced field. However, a nonzero theta parameter ?eff isknown to break these symmetries. The external magnetic field shifts thespectrum, such that instantons of opposite k have a different action andtheir corresponding magnetic fields no longer cancel in the partition sum.The result is an induced B-field that counteracts the applied magnetic field.The expectation value of the total magnetic field of the system can becomputed from the partition function as follows:?B? = ? 1?V???BextlnZtop = ?e?L3????efflnZtop (50)The reasoning for this expression is similar to that for the pressure. If thefree energy is lowered by changing the strength of the external magneticfield, then the system can emulate this by inducing its own magnetic fieldproportionally. This is also analogous to the calculation of expectation val-ues in QFT, as the functional derivative of the partition function towardsa source term4. If this argument does not convince you, we can insert thepartition sum (46) into eq. (50) to explicitly find:?B? =??piZtop?k?Z(Bext +2pikL1L2e)exp[??pi2(k + ?eff2pi )2], (51)which is intuitively clear as a thermodynamic expectation value. Bext + 2pikL1L2eis the total field strength of the configuration labeled by winding numberk, while the exponential is its Boltzmann factor e??E . Dividing by Ztop4Except that in this case, the source is physical and is thus not set to zero after takingthe derivative.27ensures the normalization and cancels the prefactor ??pi. As mentionedabove, when Bext = 0 the series is antisymmetric in k and all terms willcancel in pairs.In the limit L1L2 ??, all the topological effects vanish and we recover?B? = Bext as expected. To see this, it is possible to rewrite eq. (51) as?B? = Bext ?e?L3????effln?k?Zexp[??pi2k2 ? ?pik?eff]. (52)The second term vanishes when ? ? 0. Unfortunately this cannot be di-rectly seen from the asymptotic expression (48), because of the way ?effdepends on L1L2.Just as with the pressure, it is more enlightening to study a numericalplot of the induced field, for values of ? ' 1. Such a plot is shown in figure 5.Again the oscillatory behavior is immediately obvious, which is consistentwith the interpretation of the external magnetic field as a theta parameter.Figure 5: A numerical plot of the induced magnetic field in units of 1L1L2eas a function of ?eff . The oscillatory behaviour becomes more pronouncedfor large ? .28Furthermore, ?B? vanishes when ?eff reaches an integer multiple of pi. Thisis particularly interesting because it means that, for a nonzero multiple of pi,the external magnetic field is completely canceled by the contribution fromthe instantons.The expression (50) for the expectation value of the magnetic field is veryreminiscent of the way magnetization is computed in statistical physics. Upto a sign, ?B? is equivalent to the magnetization per unit volume:?m? = 1?V???HlnZ, (53)where H is the magnetic field strength in the medium. This leads us todefine the magnetic susceptibility of the system:?mag = ?1?V??2?B2extlnZtop = ?2???2??2efflnZtop. (54)The susceptibility represents the magnetic response of the free energy to-wards an external source, in this case the external magnetic field Bext ? ?eff .It is a dimensionless quantity (in natural units), as the right-hand side ofeq. (54) shows. Because of this, we can easily use the asymptotic expres-sion (48) to conclude that the susceptibility is again exponentially suppressed? e?1/? when ?  1. For the more general regime we must again resort toa numerical plot, seen in figure 6.The plot of the magnetic susceptibility shows some very unique features.Most importantly, ?mag flips sign such that the system will switch betweenbeing diamagnetic and paramagnetic, depending on the strength of the ex-ternal field. This kind of behavior is not often seen in condensed mattersystems. Furthermore, the susceptibility does not vanish when ?eff = 0.This makes sense intuitively, since ?mag represents the sensitivity to an ex-ternal magnetic field.Not only are the induced magnetic field and the susceptibility of use forpossible experiments, they also hold some theoretical interest. Because ofthe universality of topological theta terms, these quantities are very closelyrelated to the ones of other topological systems. In the 2d electrodynamicsfrom section 3, as well as 4d QCD mentioned in section 2.3, one can definea so-called topological density that is completely analogous to our inducedmagnetic field (50). Likewise, our magnetic susceptibility corresponds to thetopological susceptibility of other systems with a theta term. The suscepti-bility plays a role even when ? = 0, like in QCD where it is connected to themass of certain mesons through the Witten-Veneziano relation. As such the29Figure 6: A numerical plot of the magnetic susceptibility ?mag as a functionof ?eff for different values of ? . It oscillates with ?eff as it should and doesnot vanish at ?eff = 0. The magnetic susceptibility is dimensionless innatural units.topological Casimir effect allows us to indirectly investigate theories thatare otherwise difficult to access.To conclude this section, we have seen that an external magnetic field in-teracts with the topological Casimir effect. It takes the role of a topologicaltheta parameter and the system shows all the universal behavior associatedwith this. The addition of the theta parameter modifies the topologicalpressure and the magnetic field of the system, both of which are periodicas a function of the external field strength. This very specific variation issomething that could possibly be measured in a lab. The external magneticfield is an ideal tool to probe the topological features of our setup, sinceit leaves the conventional Casimir effect entirely unaffected. As such, thesensitivity towards an external field is a unique and integral part of the topo-logical Casimir effect that sets it apart from other features of the quantumvacuum.306 Discussion and ConclusionThe Casimir effect is an exciting new manifestation of topology in the quan-tum vacuum. It arises from tunneling configurations between the topologicalsectors, the degenerate vacuum states that are topologically different eventhough they correspond to the same physical state. However, all this theorymeans very little until the effect has been experimentally confirmed. In thiswork I have discussed several times the possibility of measuring the topo-logical Casimir effect, but this should not be mistaken for an experimentproposal. A collaboration with an experimental physicist, somebody whoknows what is and isn?t possible in a lab, would be helpful for suggesting arealistic setup. Regardless, this thesis would not be complete without someshort discussion on this matter.Although I am generally optimistic that the topological Casimir effectcan be observed, some major challenges would have to be overcome. The firstquestion that readers are likely to ask themselves, is how one would createthe Euclidean 4-torus that our theory is formulated on. It is important tokeep in mind here that the system I discuss in the text is not intended tobe the most general description of the topological Casimir effect. Instead,the 4-torus was chosen as the simplest configuration that demonstrates theessential features of the theory. For practical applications in the lab a slightlydifferent setup may be useful. It is however probable that some sort ofperiodic boundary conditions are needed to realize the required nontrivialtopology of the spacetime.A Euclidean 4-torus corresponds to a 3-torus at finite temperature, asshown in appendix B.1. To create such a system, one must engineer a boxwith periodic boundary conditions on opposing sides, in all three directions.Please note though that the periodicity in the z-direction is not used in thederivation of section 4, so the same topological effects are possible without it.Creating periodicity in x and y-direction might be as simple as connectingthese sides with a superconducting wire. Having the plates in the xz-planeand the yz-plane be metallic would still allow the magnetic field from theinstantons in the z-direction. This setup would need a closer investigationbefore making any conclusions, though. Another option would be to createactual curved boundaries, like a ring or a 2-torus. Formulating the theoryfor such a configuration would be much more complicated. All things con-sidered, I am convinced that with the current state of material science, itshould be possible to create a system that exhibits the topological Casimireffect.Of course, creating the system is only half the battle. Since the topo-31logical effects are so small, the actual detection will also be very difficult.Aside from the practical challenges of observing such a tiny effect, thereare many corrections that need to be accounted for ? even more so whendealing with real materials instead of idealized boundary conditions. Theaddition of an external magnetic field will add a distinctive variation to theCasimir pressure, but it will still be orders of magnitude smaller than theconventional Casimir effect. It would be interesting to see if one could gener-ate physical photons from the induced magnetic field, for example by addingtime-dependence to the external field, similar to the dynamical Casimir effect[14]. However, this subject needs further research to make any quantitativepredictions. The experimental observation of the topological Casimir effectis still a long way off, but as of yet there is no reason to believe that it isimpossible.Assuming that the topological Casimir effect can be recreated in the lab,what would be the implications? Due to the universality of the topologi-cal behavior (see section 2), observation of the effect could tell us thingsabout completely different systems. Topological Casimir experiments wouldprovide a means to answer very fundamental questions about the quantumvacuum, not only in QED but also in QCD. An important example is thepossible application in cosmology, where the topological Casimir effect hasbeen suggested as a dark energy candidate [5, 16]. In this case the QCDvacuum, as described in section 2.3, yields a nonzero Casimir energy whendefined on an expanding Hubble universe. An order of magnitude estima-tion of this energy is in very good agreement with the observed dark energydensity. If true, this would be an amazing opportunity to test cosmologicalproperties in a controlled environment, something that is quite rare.In conclusion, there is interesting new physics that emerges when con-sidering the electromagnetic field on nontrivial topological spaces. Thiseffect is well established in 1 + 1 dimensions: correspondence between thegauge group and the spacetime topology leads to the existence of topologicalsectors, labeled by an integer winding number. Instanton excitations pro-vide tunneling between these sectors and their thermal fluctuations yield anonzero vacuum energy. For a consistent theory it is necessary to sum overall winding numbers ? all possible instantons ? in the partition function.In 2d QED, the topological excitations are the only degrees of freedom inthe theory, since physical photons cannot exist. On the other hand, in fourspacetime dimensions the topological vacuum energy must compete with theconventional Casimir energy, that originates from the zero-point energy ofthe photon modes. On the toroidal system that is discussed in this text,the topological Casimir effect is several orders of magnitude smaller than32its conventional counterpart. However, it may still be studied through someof its unusual properties, like the sensitivity towards an external magneticfield.Under the influence of an external magnetic field, the topological Casimireffect shows a unique oscillating behavior as a function of the field strength.By varying the external field, one can switch between an attractive and re-pulsive Casimir force, or between a paramagnetic and diamagnetic system.The role of the external magnetic field here is directly analogous to the topo-logical theta parameter in the QCD vacuum, or ? = pi in strong topologicalinsulators.All in all, I think that this is a meaningful discovery that will havesignificant implications for our understanding of the quantum vacuum. Thetopological Casimir effect provides a direct means to experiment with thephysics of topological sectors and theta states, opening the way to fascinatingnew research.?33References[1] C. Cao, M. van Caspel and A.R. Zhitnitsky, Phys. Rev. D 87, 105012(2013).[2] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap 51, 793 (1948).[3] S.K. Lamoreaux, Phys. Rev. Lett. 78 5 (1997);U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998);H.B. Chan et al., Science 291, 1941 (2001);G. Bressi et al., Phys. Rev. Lett. 88, 041804 (2002);S.K. Lamoreaux, Rept. Prog. Phys. 68, 201 (2005);G.L. Klimchitskaya et al., Rev. Mod. Phys. 81, 1827 (2009).[4] G. Plunien, B. Muller and W. Greiner, Phys. Rept. 134, 87 (1986);M. Bordag, U. Mohideen and V.M. Mostepanenko, Phys. Rep. 353 1(2001);K.A. Milton, J. Phys. A 37 R209 (2004);M. Bordag, G.L. Klimchitskaya, U. Mohideen and V.M. Mostepanenko,Advances in the Casimir Effect, Oxford University Press (2009).[5] A.R. Zhitnitsky, Phys. Rev. D 86, 045026 (2012)[6] G. Kelnhofer, Nucl. Phys. B 867, 110 (2013).[7] B.S. DeWitt et al., Physica 96A, 197 (1979);C.J. Isham, Proc. Roy. Soc. Lond. A 362, 383 (1978);C.J. Isham, Proc. Roy. Soc. Lond. A 364, 591 (1978);R. Banach and J.S. Dowker, J. Phys. A 12, 2527 (1979).[8] M. Nakahara, Geometry, Topology and Physics, Institute of PhysicsPublishing (2003).[9] M. Srednicki, Quantum Field Theory, Cambridge University Press(2007).[10] I. Sachs and A. Wipf, Helv. Phys. Acta 65, 652 (1992),I. Sachs and A. Wipf, Annals Phys. 249, 380 (1996),S. Azakov, H. Joos and A. Wipf, Phys. Lett. B 479, 245 (2000).[11] S. Azakov, Int. J. Mod. Phys. A 21, 6593 (2006) [hep-th/0511116].[12] H. Suzuki and H. Yasuta, Phys. Lett. B 400, 341 (1997).34[13] D. Robaschik et al., Ann. Phys. 174 401 (1987).[14] G. T. Moore, J. Math. Phys. 11, 2679 (1970),S. A. Fulling and P. C. W. Davies, Proc. R. Soc. London, Ser. A 348,393 (1976),C. M. Wilson et al., Nature 479, 376 (2011).[15] D. F. Walls and G. J. Milburn, Quantum Optics, Springer (2008).[16] A.R. Zhitnitsky, Phys. Rev. D84, 124008 (2011).[17] M. Le Bellac, Thermal Field Theory, Cambridge University Press(2000).35Appendix A: The Conventional Casimir EffectIn this appendix I discuss some relevant points of the conventional Casimireffect, which is frequently compared to the topological contributions in themain text. None of the physics in this section is new, and in fact there havebeen many books written about the subject. The goal of this appendix ismainly to derive the Casimir pressure on a 4-torus, as given in eq. (31). Fi-nite temperature corrections will also be discussed. For most other things,such as the Green?s Function formalism or the subtle interpretation of reg-ularization, the reader will be referred to several standard works [4].A.1 2d Scalar FieldThe Casimir effect arises from the quantization of a field within certainboundary conditions, which is different than in free Minkowski space. Asan illustration, let us start with a simple massless scalar field ?(x, t) in 1+1dimensions. Canonical quantization defines the field in terms of creationand annihilation operators, such that the operator a??k creates a scalar par-ticle of wave vector k. The number operator n?k = a??ka?k counts the numberof particles of wave vector k in the system. The Hamiltonian of the sys-tem is given by H =?h??k(n?k + 1/2), just as for a collection of quantumharmonic oscillators. We see that even in the vacuum where nk = 0, eachmode contributes a zero-point energy of h??k/2. In free Minkowski space,all wavelengths are allowed and k forms a continuous spectrum. The totalvacuum energy on an interval of length L becomes:EMink(L) =h?L2? ???dk2pi?k. (A.55)Now we impose boundary conditions on the field, confining it to a boxsuch that?(0, t) = ?(a, t) = 0. (A.56)As a result the allowed wavelengths are quantized, yielding kn = pin/awhere n is a positive integer. The vacuum energy of the system within theseboundaries now becomes:Ebound =h?2??n=1?kn =pih?c2a??n=1n (A.57)This energy is divergent, but a finite quantity can be acquired by subtractingthe Minkowski vacuum energy EMink(a). This difference is known as the36Casimir energy Ecas ? Ebound ? EMink. To compute the Casimir energy,we introduce a regulator e?c?k into the integral and sum, and then take thelimit ? ? 0 at the end. This gives usEMink = lim??0ch?a2pi? ?0dk e?c?kk = lim??0h?a2pic?2 (A.58)whereas the energy with boundaries yieldsEbound = lim??0ch?pi2a??n=1e?cpi?n/a n = lim??0h?a2pic?2 ?pih?c24a . (A.59)Subtracting EMink cancels the divergent term in the expansion and offersthe Casimir energyEcas = ?pih?c24a . (A.60)Since the energy depends on the size of the box a, it produces a vacuumpressure. This is known as the Casimir effect. The method used here is calledthe mode summation technique and is intuitively the most clear. However,for more complicated systems this method might not work and one mustresort to other tools, such as path integrals or the energy momentum tensor.The result (A.60) can be obtained more easily using zeta function reg-ularization. Using the analytical continuation of a zeta function, we canrewrite (A.57) asE = lims?0pih?c2a??n=1ns?1 = lims?0pih?c2a ?R(s? 1) (A.61)where the Riemann zeta function ?R(z) is defined as?(z) =??n=11nz. (A.62)Of course, in general the zeta function is only properly defined for z ? 2,otherwise the series diverges. It is however possible to create an analyticcontinuation onto the complex plane, a technique that is very well studied.This continuation assigns a value of ?R(z) for any z ? C. It suggests that?R(?1) = ? 112 , which yields the same Casimir energy as in eq. (A.60). Toprove this in a rigorous manner requires a lot of math and for that I willrefer to the literature [4]. With some difficulty it can be shown that these37zeta function procedures are mathematically equivalent to the subtractionof the Minkowski vacuum energy and that it is more than just a useful trick.Other regularization schemes, like point-splitting, are also possible.Aside from imposing fixed boundary conditions like eq. (A.56), one canalso consider the Casimir effect on compact topological spaces. For example,when the scalar field ?(x, t) is defined on a ring of circumference a, theperiodic boundary conditions cause quantization of the allowed momenta.The result is a Casimir energy similar to (A.60), up to a numerical factor. Itis these periodic boundary conditions that cause the conventional Casimireffect on the 4-torus.A.2 The EM Casimir Effect on a TorusIn section 4 the electromagnetic Casimir effect on a Euclidean 4-torus isdiscussed. The compactified time dimension, corresponding to a finite tem-perature, will not be considered for now. In the low temperature regime, thethermal correction is very small as we will derive at the end of this appendix.That leaves a 3-torus with size L1 ? L2 ? L3, where L3  L1, L2.Switching from a scalar field to the electromagnetic field does not changemuch, for the conventional Casimir effect with periodic boundary conditions.The two polarization directions act as independent massless scalar fields andonly contribute a factor 2. This reduction to a scalar field is often possible,although for some boundary conditions the process is less trivial. Workingfrom now on in natural units, the Casimir energy becomes:ECas =12?k,??(?)k =?k?k, (A.63)where ? denotes the polarization. Due to the periodic BCs, the momentumk is quantized. For the z-direction, this implies kz = 2pin/L3. Because L1and L2 are much larger than L3, we can use the approximation that kx andky are continuous. This simplifies the computations a lot.?n,k|| =?k2|| +(2pinL3)2, k|| ??k2x + k2y (A.64)To find the Casimir energy, we must integrate over kx and ky and sum overn. This expression can be transformed into another zeta function by usingthe parametric integral??s =? ?0dtt?ts/2?(s/2) e?t?2 . (A.65)38When inserted into the formula for the Casimir energy, this form allows usto integrate out k|| and perform the zeta function regularization:ECas = lims??1L1L22pi? ?0dk|| k||?n??sn,k||= lims??1L1L22pi?n? ?0dtt?ts/2?(s/2)? ?0dk|| k||e?t(k2||+(2pinL3)2)= lims??1L1L22pi?n? ?0dtt?ts/2?1?(s/2) e?t(2pinL3)2= lims??1L1L22pi ??(s/2? 1)?(s/2)?n(2pinL3)2?s= lims??18pi2L1L2L33?22? s ?R(s? 2) = ?2pi245 ?L1L2L33. (A.66)With this bit of unpleasant algebra, we obtain the regularized vacuum energyof the torus. The final step is to compute the pressure in the z-direction,given byPCas = ?1L1L2??ECasdL3= ? 2pi215L43. (A.67)This is the expression used in section 4.2 to compare against the topologicalCasimir pressure. It is about an order of magnitude larger than the Casimireffect for parallel conducting plates (2), but the power law behavior is exactlythe same.In the above derivation we have neglected the periodic boundary con-ditions in the x and y-direction, by assuming that the momentum in thosedirections is continuous. Without that approximation, several correctionswould be added to eq. (A.67) of the form ? 1L21L23and ? 1L22L23. Compared tothe leading order term, these are highly suppressed because L1 and L2 aremuch larger than L3. We do not consider them further.A.3 Thermal CorrectionsUp until here, the Casimir effect computations of this appendix have beenat zero temperature. Consider now the system at a finite temperature T .This causes thermal fluctuations which means that the modes of the elec-tromagnetic field are no longer all in the ground state nk = 0. In order to39define the free energy and pressure at finite temperature, it is necessary tocompute the partition sumZ =?k,?(?nexp[???k(n+12)])=?k,?exp[?12??k]1? exp[???k], (A.68)which yields the free energyF =?k(?k +2?ln [1? exp (???k)]). (A.69)The first term clearly corresponds to the Casimir energy (A.67) that hasalready been calculated, while the second term is known as the thermalcorrection. Interestingly, the correction term is convergent and does notneed to be regularized. It is however still important to subtract the thermalvacuum energy of the space without the periodic conditions.FMink =2V?? d3k(2pi)3 ln [1? exp (???k)] = ?pi2V45?4 (A.70)This is simply the free energy of black-body radiation as given by basicstatistical mechanics.Computing the thermal correction for the torus is not trivial, but can bedone in the low-temperature limit.F therm = L1L2pi?? ?0dk|| k||?nln??1? exp(? ??k2|| +(2pinL3)2 )?? .(A.71)Assuming ?  L3, we can use the approximation ln[1? e???]? ?e??? aslong as n 6= 0. Keeping only the terms with n = 0,?1 the integral can beperformed to findF therm ?L1L2pi?[?R(3)?2?4pi?L3exp(?2pi?L3)](A.72)Subtracting the background energy FMink and calculating the correspondingpressure finally yields the small thermal correction for low temperatures:P thermCas ? ?pi245?4 +8pi?L33exp(?2pi?L3)(A.73)40Appendix B: Mathematical ToolsB.1 Wick Rotation and Thermal Field TheoryThe Wick rotation is a crucial part of the path integral approach for de-scribing quantum field theory at finite temperature. It transforms a problemin 3+1 dimensional Minkowski space into one formulated in 4d Euclideanspace, by switching to an imaginary time coordinate [17]. The technique ismotivated by the similarities between the partition function as defined inQFT, and the statistical partition function:ZQFT =?D? ei?dt L[?] ?? Zstat =?Dm e??H[m] (B.74)where ? is a quantum field and m some kind of local order parameter.Consider a field in 4d Minkowski space. The metric of the coordinatesis x?x? = t2 ? |x|2. Now we apply the transformationt ?? ? ? ?it (B.75)which produces the Euclidean 4-vector product: |xE |2= ?2 + |x|2. In otherwords, the transformation switches between a Minkowski and a Euclideanmetric. How would such a transformation affect the action of the system?For example, for a free scalar field we find:?d4x(12 (???)2 ?12m2?)?? i?d? d3x(12 (???)2E +12m2?)(B.76)The transformed integrand is known as the Euclidean Lagrangian LE . Thefactor i cancels with the one in the partition sum. Note that the terms ofLE are typically real, since the Lagrangian only contains even powers of ?t.The exception are topological theta terms, as seen in eq. (7). The Wickrotated partition sum becomesZQFT =?D? e??d? LE [?]. (B.77)To introduce the finite temperature into this expression, there is onlystep left: we identify the imaginary time with the inverse temperature ?.Z(?) =?D? e?? ?0 d? LE [?]. (B.78)Here it is necessary that the fields ? that one integrates over satisfy theperiodic boundary conditions ?(? = 0) = ?(? = ?). That why we consid-ered a Euclidean 2-torus and 4-torus in sections 3 and 4 respectively, the41imaginary time component is also periodic in nature. For a more rigorousderivation, one can analyze the partition sum Z = Tr[exp(??H)] to arriveat eq. (B.78).B.2 Poisson ResummationThe Poisson Sum formula, in the form that was used many times in thiswork, is a result from Fourier analysis that states??n=??exp[?a (n+ x)2]=?pia??k=??exp[?pi2k2a+ 2pi i x k]. (B.79)for positive a. Here we will show a quick proof.Consider the sum?n f(n + x), for some function f(x). This sum isperiodic in x, with period 1. As a result it can be written in terms ofdiscrete Fourier coefficients:??n=??f(n+ x) =??k=??ck e2pi i k x (B.80)where the coefficients ck are defined asck ?? 10dx??n=??f(n+ x) e?2pi i k x=??n=??? n+1ndx f(x) e?2pi i k x=? ???dx f(x) e?2pi i k x= f?(k) (B.81)with f?(k) the Fourier transform of f(x). Substituting this into eq. (B.80)yields??n=??f(n+ x) =??k=??f?(k) e2pi i k x. (B.82)Now we can choose for f(x) the Gaussianf(x) = e?ax2 ?? f?(k) =?piae?pi2k2a . (B.83)Inserting this into eq. (B.82) recovers precisely the formula (B.79) that wewanted to prove.42

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