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a rXiv:h ep-ph/984378v123Apr1998Vorton FormationC.J.A.P.Martins ∗and E.P.S.Shellard †Department of Applied Mathematics and Theoretical Physics University of Cambridge Silver Street,Cambridge CB39EW,U.K.In this paper we present the first analytic model for vorton forma-tion.We start by deriving the microscopic string equations of motion in Witten’s superconducting model,and show that in the relevant chiral limit these coincide with the ones obtained from the supersonic elastic models of Carter and Peter.We then numerically study a number of solutions of these equations of motion and thereby suggest criteria for deciding whether a given superconducting loop configuration can form a vorton.Finally,using a recently developed model for the evolution of currents in superconducting strings we conjecture,by comparison with these criteria,that string networks formed at the GUT phase transition should produce no vortons.On the other hand,a network formed at the electroweak scale can produce vortons accounting for up to 6%of the critical density.Some consequences of our results are discussed.I.INTRODUCTION As first pointed out by Witten [1],cosmic strings can in some circumstances (typically when the electromagnetic gauge invariance is broken inside the string)behave as ‘superconducting wires’carrying large currents and charges—up to the order of the string mass scale in appropriate units.The charge carriers can be either bosons or fermions (see [2]for a review).The former type occurs when it becomes energetically favourable for a charged Higgs field to have a non-zero vacuum expectation value in the string core;the latter happens when fermions couple to the string fields creating fermion zero modes.It is well known that arbitrarily large currents are not allowed—there is a criticalvalue beyond which the current saturates.In other words,for large enough wind-ing number per unit length,the superconducting condensate is quenched down,suppressing the current flow.Also,the current can decay by magnetic flux-line tun-nelling;this can be used to impose constraints on allowed particle physics models.If superconducting strings carry currents,they must also carry charges of similar magnitude.This includes not only charges trapped at formation by the Kibble mechanism but also the ones due to string inter-commuting between regions of the string network with different currents.Just like with currents,charge densities cannot have arbitrarily large magnitude—there is a limit beyond which there will no longer be an energy barrier preventing the charge carriers from leaving the string.A rather important point is that the presence of charges on the string tends to counteract the current quenching effect discussed above.In fact,numerical simula-tions of contracting string loops at fixed charge and winding number have shown [3]that a‘chiral’state with equal charge and current densities is approached as the loop contracts.In this limiting chiral case,quenching is in fact eliminated completely. This has several important consequences.Strings that have trapped charges as a consequence of a phase transition can become superconducting even if the formation of a condensate was otherwise energetically unfavoured.More importantly,a string with both a charge and a current density will have a non-zero angular momentum. In the cosmological context,these strings would of course interact with the cosmic plasma,originating a number of interesting consequences.The most remarkable of these,however,has to do with the evolution of string loops.If a superconducting string loop has an angular momentum,it is semi-classically conserved,and it tries to resist the loop’s tension.This will at least increase the loop’s lifetime.If the current is too large,charge carriers will leave the string accompanied by a burst of electromagnetic radiation,but otherwise it is possible that dynamically stable loops form.These are called vortons[4]—they are stationary rings that do not ra-diate classically,and at large distances they look like point particles with quantised charge and angular momentum.Their cosmological significance comes from the fact that they provide very strong constraints on allowed particle physics models, since they behave like non-relativistic particles.According to current belief[4,5], if they are formed at high enough energy scales they are as dangerous as magnetic monopoles,producing an over-density of matter in disagreement with observations. On the other hand,low-mass vortons could be a very interesting dark matter can-didate.Understanding the mechanisms behind formation and evolution is therefore an essential cosmological task.The overwhelming majority of the work done on cosmic strings so far was con-cerned with the structureless Goto-Nambu strings(but see[6]and references therein for some exceptions).In the case of work on vortons,this means that somewhat ad-hoc estimates had to be made for some properties of the cosmic string network—notably for microscopic quantities such as current and charge densities.This is de-spite the fact it has been recognised a long time ago that,even though they might be computationally very useful[7–9],Goto-Nambu models cannot realistically be expected to account for a number of cosmologically relevant phenomena,due to the very limited number of degrees of freedom available.Two such phenomena are the build-up of small-scale structure and charge and current densities.In this paper wefill this important gap by discussing the problem of vorton formation in the context of the superconducting string models of Witten[1]and of Carter and Peter[10](sections II and III).Strangely enough,the issue of the conditions for vorton formation has been so far neglected with respect to those of their stability and cosmological consequences.We will start by introducing these models and determining the microscopic string equations of motion in each case.It will be shown that in the relevant chiral limit these equations coincide—this also provides thefirst conclusive evidence of the validity of the supersonic elastic models of Carter and Peter[10].We then proceed to study the evolution of a number of loop solutions of these equations numerically(sections IV and V),and from the results of this analysis parameters will be introduced which characterise the loop’s ability to evolve into a vorton state(section VI).Finally,we discuss a very simple phenomenological model for the evolution of the superconducting currents on the long cosmic string network [11],based on the dynamics of a‘superconducting correlation length’(sections VII–VIII).Using this model we can therefore estimate the currents carried by string loops formed at all relevant times,and thus(in principle)decide if these can become vortons(section IX)and calculate the corresponding density(section X).Based on our results,we don’t expect any GUT vortons to form at all.This is essentially because the friction-dominated epoch is very short for GUT-scale strings [7],so their currents and charges are never large enough to prevent them frombecoming relativistic—and therefore liable to losses.Even if they did form,they wouldn’t be in conflict with the standard cosmological scenario if they decayed soon after the end of the friction-domination epoch.Hence we conclude that,in contrast with previously existing estimates[4,5], one cannot at the moment rule out GUT superconducting string models.We should point out at the outset that there are essentially three improvements in the present work which justify the different end result for GUT-scale strings.Firstly,by analysing simple(but physically relevant)loop solutions of the microscopic string equations of motion for the Witten model,we can get a much improved idea of how superconducting loops evolve and of how(and under which conditions)they reach a vorton state.Secondly,by using a simple model for the evolution of the currents on the long strings[11]we can accurately determine the typical currents on each string loop at the epoch of its formation.Finally,the use of the analytic formalism previously introduced by the present authors[7,9]allows us to use a quantitative description throughout the paper,and in particular to determine the loop sizes at formation.As will become clear below,when taken together these allow a detailed analysis of the process of vorton formation to be carried out,either in the Witten model(as is done in this paper)or any other that one considers relevant.In contrast,note that Davis&Shellard[4]restrict themselves to the particular case of the initial Brownian Vachaspati-Vilenkin loops with Kibble currents,and do not consider the subsequent evolution of the network.On the other hand,Brandenberger et al.[5] make rather optimistic order-of-magnitude estimates about the process of relaxation into a vorton state.As it turns out,for high energy GUT scales,all these loops be-come relativistic before reaching a vorton state.Finally,neither of these treatments has the benefit of a quantitative model for the evolution of the long-string network [7]which allows one to accurately describe the process of loop production.On the other hand,as we lower the string-forming energy scale we expect more and more efficient vorton production,and the’old’scenario still holds.Therefore intermediate-scale superconducting strings are still ruled out,since they would lead to a universe becoming matter-dominated earlier than observationally allowed.Fi-nally,at low enough energy scales,vortons will be a dark matter candidate.For example,for a string network formed around T∼102GeV(typical of the elec-troweak phase transition)they can provide up to6%of the critical density.A more detailed discussion of these issues is left to a forthcoming publication[12]. Throughout this paper we will use fundamental units in which¯h=c=k B= Gm2P l=1.II.WITTEN’S MICROSCOPIC MODELAsfirst pointed out by Witten[1],a low-energy effective action for a supercon-ducting string can be derived in a way that is fairly similar to what is done in the Goto-Nambu case(see for example[2]).One has to adopt the additional as-sumptions that the current is much smaller than the critical current and that the electromagnetic vector potential Aµis slowly varying on the scale of the condensate thickness.The derivation then proceeds as in the neutral case,except for the use of the well-known fact that in two dimensions a conserved current can be written as the derivative of a scalarfield.One obtainsS= √2γabφ,aφ,b−qAµxµ,aǫab−γφ,b d2σ(2.1)−1−gFµνFµν;(2.2) the four terms are respectively the usual Goto-Nambu term,the inertia of the charge carriers,the current coupling to the electromagnetic potential and the external electromagneticfield(ǫab is the alternating tensor);note that this applies to both the bosonic and the fermionic case[2].Recalling the usual definitionsA a=xµ,a Aµ,(2.3)F ab=Fµνxµ,a xν,b=A b,a−A a,b,(2.4) and definingΥab to be the stress-energy tensor of the scalarfieldφΥab=φ,aφ,b−1√−γγabφ,b +1−γ˜ǫab F ab=0,(2.8) and∂a √µ0Υab xα,b +√µ0Υab Γασρxσ,a xρ,b=(2.9)√=gαλ(gαµ,ν+gνα,µ−gµν,α),(2.11)2and Fαis the Lorentz force1Fα=ℓf uα−xα,a xσ,a uσ ,(2.13) using the same procedure as described in[13,7].As shown in[11],plasma effects are subdominant,except possibly in the presence of background magneticfields—either of‘primordial’origin or generated(typically by a dynamo mechanism)onceproto-galaxies have formed.Hence one expectsAharonov-Bohm scattering [14]to be the dominant effect,and consequently we have [7]ℓf =µµ0γ00,(2.17)and choosing the standard gauge conditionsσ0=τ,˙x ·x ′=0,(2.18)(with dots and primes respectively denoting derivatives with respect to the time-like and space-like coordinates on the worldsheet as usual)the string equations of motion in an FRW background with the line element ds 2=a 2 dτ2−dx 2 (2.19)(which implies that γ00=a 2(1−˙x 2))have the form[ǫ(1+Φ)]˙+ǫa ǫΦ,(2.20)andǫ(1+Φ)¨x +ǫǫ ′+ ˙Φ+2˙a ℓd =a 2H +1ǫ.(2.24)Note that the Witten action is ‘microscopic’in the sense of being built using only the properties of the underlying particle physics model [1].In the next section we will analyse the equations of motion obtained form the action for the elastic supersonic models of Carter and Peter [10],which is is this sense ‘macroscopic’.III.SUPERSONIC ELASTIC MODELSIn order to account for phenomena such as the build-up of charge and current densities on cosmic strings,one must introduce additional degrees of freedom on the string worldsheet.One such class of models,originally introduced by Carter and co-workers is usually referred to as elastic models(see[6]and references therein,on which the following two subsections are based).A.Basics of elastic modelsIn general,elastic string models can be described by a Lagrangian density de-pending on the spacetime metric gµν,backgroundfields such as a Maxwellian-type gauge potential Aµor a Kalb-Ramond gaugefield Bµν(but not their gradients) and any relevant internalfields(that will be discussed in detail below).Note that the Goto-Nambu model has a constant Lagrangian density,namelyL GN=−µ0.(3.1) Upon infinitesimal variations in the backgroundfields,and provided that indepen-dent internalfields are keptfixed(or alternatively that their dynamic equations of motion are satisfied),the action will change byδS=−1−γd2σ,(3.2)whereTµν=2δLδAµ(3.4) is the worldsheet electromagnetic current density,andWµν=2δLNote that U and T are simply constants for a Goto-Nambu string,U=T=µ0,(3.8) but they are variable in general—hence the name‘elastic strings’.In particular, one should expect that the string tension in an elastic model will be reduced with respect to the Goto-Nambu case due to the mechanical effect of the current. Since elastic string models necessarily possess conserved currents,it is convenient to define a‘stream function’ψon the worldsheet that will be constant along the current’sflow lines.The part of the Lagrangian density L containing the internal fields is usually called the‘master function’,and can be defined as a function of the magnitude of the gradient of this stream function,Λ=Λ(χ),such thatχ=γabψ;aψ;b,(3.9) where the gauge covariant derivative is defined asψ;a=∂aψ−eAµxµ,a.(3.10) Note that the definition ofχdiffers by a minus sign from that of Carter[6];the reason for this will become clear below.This‘dynamic’term contains charge cou-plings,whose relevance will be further discussed below.Nevertheless,whether or not these or other background gaugefields are present,it is always the form of the master function which determines the equation—or equations—of state.There is also a dual[15]potential˜ψ,whose gradient is orthogonal to that ofψ, and the corresponding dual master function˜Λ=˜Λ(˜χ)such that˜χ=γab˜ψ;a˜ψ;b,(3.11) with the obvious definition for˜ψ;a.The duality between these descriptions means that thefield equations for the stream functionψobtained with the master function Λare the same as those for the dual potential˜ψobtained with the dual master function˜Λ.However,there will in general be two different equation of state relating the energy density U and the tension T;these correspond to what is known as the ‘magnetic’and‘electric’regimes,respectively corresponding to the cases˜χmg<0<χmg(3.12) andχel<0<˜χel,(3.13) that are respectively characterised by space-like and time-like currents.In the degenerate null state limit,however,there will be a single equation,U=T=µ0.(3.14) Note that the distinction between a given model and its dual disappears in the absence of charge couplings;such models are then called‘self-dual’for obvious reasons.In each case the equation of state provides the expressionsc2E=TdU =νdν,(3.16)for the extrinsic(that is transverse,or‘wiggle’)and for the sound-type(longitudinal or‘woggle’)perturbations of the worldsheet.Both of these must obey c2≥0(a requirement for local stability)and c2≤1(a requirement for local causality).These two speeds can be used to characterise the elastic model in question;in particular there is a straightforward but quite meaningful division of the models into supersonic (that is,those obeying c E>c L),transonic(c E=c L;only in the null limit is this common speed unity)and subsonic(c E<c L).B.Supersonic(superconducting)modelsCarter and Peter[10]have recently proposed two supersonic elastic models to describe the behaviour of current-carrying cosmic strings.The Lagrangian density in the magnetic regime is˜Λmg =−m2+˜χ2k0m2σ −1,(3.17)mσbeing the current carrier mass(which is at most of the order of the relevant Higgs mass);this is valid in the range−1k0m2σ<1−k0m2σm2=1+k0m2σk02mσm2−1+k0m2σ2ln 1−˜χk0m2σ<1−e−2m2/k0m2σ,(3.21)and the corresponding equation of state isUm2+k0m2σk0m2σ/m2 −1.(3.22)These models are supersonic for all space-like,and weak time-like currents,with the exception that in the null limit˜χ=0one has c L=c E=1.C.Equations of motionWe now derive the microscopic equations of motion for elastic cosmic string mod-els.It is convenient to start by defining the quantityΘab≡˜Λγab−2∂˜Λthen recalling the definition of ˜χ,(3.11),onecanfind the free string equations of motion in the usual (variational)way,obtaining(√−γΘab Γαµνx µ,a x ν,b =0.(3.24)Also in a similar way to what was done in section II,the effect of the frictional forces is accounted for by introducing a term√βT 3b (3.26)(note that ˜Λis negative).Of course we now have a further equation for the scalar field ˜ψ,namely ∂a √∂˜χγab ˜ψ;b =0.(3.27)Furthermore,the spacetime energy-momentum tensor and electromagnetic cur-rent will be given by√−γΘab x µ,a x ν,b δ(x −x (σ,τ))d 2σ(3.28)and√−γγab ∂˜Λ−(3)gT 00=a−˜Λ+2∂˜Λ−(3)gJ 0=−2ea∂˜Λ∂˜χ√∂˜χǫ˜ψ;0,(3.33)j≡−j1=−2e ∂˜Λǫ.(3.34)Again,for the reasons explained above,a particularly relevant situation will be that of a chiral current,that is one in whichγab j a j b=0.(3.35) This is equivalent to˜ψ′2=ǫ2˙˜ψ2,(3.36) and therefore it implies that∂ρ∂σ(3.37)and that the total(spacetime)charge and current are also equal.Note that in the chiral case one also has˜χ=0,2∂˜Λ−γ∂˜Λ−γ∂˜ΛD.The chiral limitWe now consider the(common)chiral limit of the two supersonic elastic models of Carter and Peter[10],defined by the Lagrangian densities(3.17)and(3.20), respectively for the magnetic and electric regimes.Also,as we did in section II for the Witten model,we will interpret the charge coupling and the scalarfield as being renormalised and neglect the coupling to external electromagneticfields.Then,with our usual gauge choices and definitions of the damping and friction length-scales,the microscopic string equations of motion(3.24)simplify to[ǫ(1+Ψ)]˙+ǫaǫΨ,(3.44)andǫ(1+Ψ)¨x+ǫǫ ′+˙Ψ+2˙aµ0γ00.(3.46)That is,these are exactly the same equations of motion as those of Witten’s model(2.20–2.21)if one identifies the corresponding scalarfields,φ≡˜ψ.(3.47) Then,the worldsheet charge and current densities also coincide,ρw=qǫ˙˜ψ,(3.48)j w=q˜ψ′x(τ)=r(τ)(sinθ,cosθ,0);(4.1)we also need an ansatz for the scalarfield˜ψ(orφ),which we will take to be˜ψ=√r2 =1,(4.4)¨r+ 1−n4t4c2.(4.6)Note that opposite signs of n correspond to left and right moving currents;naturally it always appears as n2in any relevant equation,and we will therefore be taking n to be positive.Infigure1we plotted some relevant evolutionary properties of chiral supercon-ducting loops with different n’s inflat spacetime.Note that these loops never collapse to zero size,and that their microscopic velocity is always less than unity (unlike in the Goto-Nambu case).Furthermore,there is a static solution withn=1t c˙r=0;(4.7)in this case the energy is equally divided between the string and the current.It should also be noted that energy is transferred back and forth between the string and the current as the loop oscillates.We can easily determine the following quantities(the averages are over one oscillation period)r22−n2,(4.8)t2c n2,(4.9)˙r2 =1E total =1−n,(4.11)12E2string2n;(4.12) note that the energy of these loops is E total/t c=2πµ.Finally,two other points that will have further relevance below.Firstly,a loops with a given conserved number n will reach a maximum microscopic velocity(and corresponding Lorentz factor)given by1˙r2max=1−4n2,γmax=µ0(F(τ)+nσ)t c.(5.1) The winding number per unitσand the function F are also constrained as before. In terms of these quantities the total energy of the loop can be written asE total=µ0ℓtotal=µ0a 1+n2t2c˙r a2r2 1a2r2 ¨r+(1−˙r2)4πnt cn=n is a variable parameter obeying0≤n=0corresponds to the Goto-Nambu case,while thev t2,ℓstring√n,namelyℓtotal1−n goes from zeroto unity we go from the Goto-Nambu case to the static case where the energy is split equally between the string and the current;the positive sign corresponds to the current branch,where the ratio of the energies in the string and in the current decreases until it vanishes whenn=2ℓstringℓstring−11/2.(5.9)In practice,it is not easily conceivable that in cosmological contexts loops can be formed with more energy in the current than in the string itself.Therefore, although for the sake of completeness we will be discussing the current branch in the remainder of this section,we will neglect it afterwards.Thus from(5.3)one obtains the evolution equation forℓ(ℓi,t i,n i,t) and other relevant quantities.As we will see below,a crucial quantity will be the the maximum velocity reached by each loop configuration during it evolution, v max(ℓi,t i,n i).Infigures3–5we plot the cosmological evolution of some relevant GUT-scale chiral circular loops.We should mention that in order to save space,only one out of every forty points resulting from the numerical integrations is plotted,and this is the reason why some plots show irregularities.Figure3shows some relevant properties of the evolution of chiral circular GUT-scale loops formed at t=t c;all have an initial total energy E total/2πµt c=10,but the distribution of the energy between the string and the current varies. Obviously,loops with higher currents will have smaller physical radii,and hence they will be less stretched by expansion and enter the horizon earlier,at which point they start oscillating—as can be confirmed in3(a-b).Regarding the velocities,note the significant differences between loops in the‘string branch’(which still reach fairly high microscopic velocities,but never v=1)and in the‘current branch’(which quickly become non-relativistic).Therefore the latter ones should definitely become vortons,and so it is perhaps fortunate that,as we pointed out above,we do not expect loops with such high currents to be produced in the early universe (at least,for GUT-scale networks).Note that in one of the cases shown the initial current is so high that the loop‘overshoots’and acquires a fairly large velocity,but friction quickly slows it down again.On the other hand,in the string branch the velocity is reduced with respect to the Goto-Nambu case,and a more detailed investigation will be needed to set14up some criterion defining which velocities will allow vorton formation—recall that relativistic velocities will imply charge losses and it will therefore be unrealistic to make any definite claims or predictions about such cases.The evolution of the fraction of the loop’s energy in the current is particularly il-luminating(see3(c)).This will obviously decrease while the loop is being stretched, and it will start oscillating when the loop falls in side the horizon.The oscillations are around the state with equipartition of the energy between the string and the current,which as we saw corresponds to a static solution inflat spacetime.Note that the effect of the friction force is to reduce the amplitude of these oscillations, so one can see that friction is in fact crucial for vorton formation.Naturally,loops with smaller velocities will undergo oscillations with smaller amplitudes,so again we confirm that these are the strongest vorton candidates.Finally,we have plotted the parametern once the loop is‘free’—that is,much smaller than the damping length defined in(2.22). On the other hand,radiative backreaction also tends to damp these energy os-cillations,and consequently increasen becomes a constant in this limit—hence its usefulness)˙r2 =1n2),(5.10)E string2E2total=1−3n;(5.12) the variance of the fraction of the energy in string is therefore∆E string4n).(5.13)Infigure4we show chiral loops with the same initial conditions as3,but starting to evolve at the epoch t⋆when when friction becomes negligible[7].The differences are self-evident.Now,after afirst period of growth of the total radius due to expansion,there is no mechanism forcing the loops to return this extra energy back to the medium when they fall inside the horizon.Consequently there is also no velocity damping(all loops will have microscopic velocities larger than0.5) and the energy oscillations between the string and the current always have a large amplitude—so thatthe region of the space of initial conditions that will originate them—because as we said the effect of friction is to increasen∼1,in which case velocity is so small that friction does not significantly affect the loop.Note that asn and v needs to be looked at in more detail,and we shall do that in the next section.VI.CRITERIA FOR VORTON FORMATIONIn the previous section we saw that the evolution of chiral superconducting cosmic string loops depends sensitively on the conditions at formation.In particular,one would need to know in which cases one ends up with a vorton.Clearly,since we are not including radiative mechanisms at this stage,our crite-rion should be that loops whose velocity is always small(in a sense that will need to be made more precise)will become vortons,while those who are relativistic at some stage will suffer significant charge losses,so that their fate cannot be clearly asserted until a rigourous quantum-mechanical treatment of these processes is available. Thus we will explore in more detail the phase space of possible initial condi-tions in order to determine relevant properties of these loops.Figure6shows the maximum microscopic velocity v max(ℓi,t i,n and to the base-ten logarithm of the initial string radius relative to the horizon;recall that we only consider loops having initially most of their energy in the string(in other words,loops in the string branch).Note that the friction length-scale corresponds to about−1.5in the vertical axis on thefirst plot,and to0on the last(where it is equal to the horizon, by definition).It can be seen that any loop initially larger than the horizon will inevitably become relativistic.This is essentially because expansion will(temporarily,at least) decrease the fraction of the loop’s energy in the current(and hencen(neglecting radiation),so we will need fairly high initial currents in order to get non-relativistic velocities.Finally,for the case of loops being produced with sizes between the friction length-scale and the horizon,which is of course the cosmologically relevant case during the16。

相对论原文（网摘）论动体的电动力学大家知道，麦克斯韦电动力学 -- 像现在通常为人们所理解的那样 -- 应用到运动的物体上时，就要引起一些不对称，而这种不对称似乎不是现象所固有的。

比如设想一个磁体同一个导体之间的电动力的相互作用。

在这里，可观察到的现象只同导体和磁体的相对运动有关，可是按照通常的看法，这两个物体之中，究竟是这个在运动，还是那个在运动，却是截然不同的两回事。

如果是磁体在运动，导体静止着，那么在磁体附近就会出现一个具有一定能量的电场，它在导体各部分所在的地方产生一股电流。

但是如果磁体是静止的，而导体在运动，那么磁体附近就没有电场，可是在导体中却有一电动势，这种电动势本身虽然并不相当于能量，但是它 -- 假定这里所考虑的两种情况中的相对运动是相等的 -- 却会引起电流，这种电流的大小和路线都同前一情况中由电力所产生的一样。

诸如此类的例子，以及企图证实地球相对于“光媒质”运动的实验的失败，引起了这样一种猜想：绝对静止这概念，不仅在力学中，而且在电动力学中也不符合现象的特性，倒是应当认为，凡是对力学方程适用的一切坐标系，对于上述电动力学和光学的定律也一样适用，对于第一级微量来说，这时已经证明了的。

我们要把这个猜想（它的内容以后就称之为“相对性原理”①）提升为公设，并且还要引进另一条在表上看来同它不相容的公设：光在空虚空间里总是以一确定的速度V传播着，这速度同发射体的运动状态无关。

由这两条公设，根据静体的麦克斯韦理论，就足以得到一个简单而又不自相矛盾的动体电动力学。

“光以太”的引入将被证明是多余的，因为按照这里所要阐明的见解，既不需要引进一个具有特殊性质的“绝对静止的空间”，也不需要给发生电磁过程的空虚空间中的每个点规定一个速度矢量。

这里所要阐明的理论 -- 像其他各种电动力学一样 -- 是以刚提的运动学为根据的，因为任何这种理论所讲的，都是关于刚体（坐标系）、时钟和电磁过程之间的关系。

对这种情况考虑不足，就是动体电动力学目前所必须克服的那些困难的根源。

a-Galactosylceramide as a Therapeutic Agent for Pulmonary Mycobacterium tuberculosis InfectionIsabel Sada-Ovalle1,Markus Sko¨ld1,Tian Tian1,Gurdyal S.Besra2,and Samuel M.Behar11Division of Rheumatology,Immunology,and Allergy,Brigham and Women’s Hospital and Harvard Medical School,Boston,Massachusetts; and2School of Biosciences,University of Birmingham,Edgbaston,Birmingham,United KingdomRationale:Invariant natural killer T(iNKT)cells are a unique subset of Tcells that recognize lipid antigens presented by CD1d molecules.Recentstudies have shown that iNKT cells can protect mice against Mycobac-terium tuberculosis(Mtb)infection.We sought to determine whetherpharmacological activation of iNKT cells by a-galactosylceramide(a-GalCer)could be used to treat tuberculosis(TB).Objectives:We hypothesized that a-GalCer,either alone or incombination with isoniazid,could be used to treat pulmonary TB.Methods:The ability of a-GalCer–activated iNKT cells to suppressMtb replication was evaluated using an in vitro coculture system.Totest its potency in vivo,mice infected with virulent Mtb were treatedwith a-GalCer alone or in combination with isoniazid.Measurements and Main Results:Quantitative colony-forming unitcounts were compared for both experimental systems.Our resultsshow that a-GalCer plus isoniazid controls bacterial growth betterthan a-GalCer or INH alone,and single or multiple a-GalCer admin-istrations prolong the survival of the mice infected via the aerosolroute.Conclusions:Our results demonstrate that a-GalCer administrationcan improve the outcome of Mtb infection,even when transmitted by the aerosol route.However,a combination of isoniazid and a-GalCer treatment has a synergistic effect on infection control. We conclude that more efficient treatment of TB will be achieved through a combination of classic chemotherapy and modulation of the host immune response.Keywords:CD1d;antibiotics;natural killer T cells;tuberculosis;lipids Tuberculosis(TB)is caused by Mycobacterium tuberculosis(Mtb) and represents an important global threat to health,especially in developing countries where new and efficient therapies are urgently needed.Although current antibiotic regimens against Mtb are effective against drug-susceptible strains,patients require long-term follow-up and can suffer severe side effects.In addition, the prevalence of multidrug-and extensively drug-resistant Mtb strains is on the rise(1).Treatment of drug-resistant bacterial strains is more complicated,requiring hospital supervision(2). Recent efforts to develop new therapeutic regimens are promis-ing,but their efficiency remains to be tested in larger populations. In the current study,we investigated whether the specific activa-tion of iNKT cells,which we have previously shown is capable of activating macrophages(M w)to suppress intracellular Mtb rep-lication,can act therapeutically against established pulmonary TB(3).Invariant natural killer T(iNKT)cells are a sublineage of T lymphocytes that are defined by their expression of a canonical V a14-J a18T-cell receptor(TCR)a-chain.The ab-TCR expressed by iNKT cells recognizes self and foreign lipid antigens presented by the antigen-presenting molecule CD1d. iNKT cells influence host immunity during infectious diseases through production of soluble mediators,such as IL-4and IFN-g,by reciprocal NKT–dendritic cell(DC)activation and by elaboration of chemokines that recruit inflammatory cells(4–6).The discovery that CD1d presentation of a-galactosylcer-amide(a-GalCer)rapidly activates murine and human iNKT cells has greatly increased our understanding of the potential role that iNKT cells play in host immunity(7,8).The synthetic glycosphingolipid a-GalCer is a potent iNKT cell agonist,and even though it is not a natural ligand for iNKT cells,a-GalCer and its analogs are commonly used experimentally(9).iNKT cell activation by a-GalCer in vivo induces downstream activa-tion of several lymphoid and myeloid cell populations,including B and T cells,NK cells,M w,and DCs,which in turn can modify multiple immune responses(10–13).For example,a-GalCer has been used to analyze the immunomodulatory properties of activated iNKT cells in mice and patients with malignancies (14–16).Administration of a-GalCer activates both the innate and adaptive immune system and has raised the possibility of using a-GalCer for therapeutic purposes(8,17).The number of iNKT cells in the lung is increased after infection of mice with virulent Mtb(18).CD1d,which presents antigen to iNKT cells,is up-regulated by IFN-g and is also increased in the lungs of infected mice(19).We havepreviously(Received in original form December23,2009;accepted infinal form May17,2010) Supported by the National Institutes of Health grant R01HL80312(S.M.B)and a Parker B.Francis Foundation postdoctoral fellowship(I.S.O).G.S.B.was supported by a Personal Research Chair from Mr.James Badrick,Royal Society Wolfson Research Merit Award,as a former Lister Institute-Jenner Research Fellow,the Medical Council,and The Wellcome Trust(084,923/B/08/7).M.S. was supported by the Swedish Heart Lung Foundation.Present address for I.S-O.is Immunochemistry Department,National Institute of Respiratory Diseases,Me´xico.Present address for M.S.is Department of Microbiology,Tumor and Cell Biology, Karolinska Institutet,Stockholm,Sweden.Present address for T.T.is Department of Dermatology,Brigham and Women’s Hospital and Harvard Medical School,Boston,Massachusetts. Correspondence and requests for reprints should be addressed to Samuel M. Behar,M.D.,Ph.D.,Division of Rheumatology,Immunology,and Allergy, Brigham and Women’s Hospital,Smith Building Room516C,One Jimmy Fund Way,Boston,MA02115.E-mail:sbehar@Am J Respir Crit Care Med Vol182.pp841–847,2010Originally Published in Press as DOI:10.1164/rccm.200912-1921OC on May27,2010 Internet address:shown that innate iNKT cells recognize Mtb-infected M w, produce IFN-g,and kill intracellular bacteria both in vitro and in vivo(3).Although these data suggest that iNKT cells can mediate protective immunity against TB,several laboratories have found that iNKT cells are not required for optimal immunity to Mtb(20,21).One explanation is that iNKT cells make only a small and possibly redundant contribution to antimycobacterial immunity.Another possibility is that after infection,iNKT cells become anergic,as has been shown after BCG and viral infection(22).Given the potential ability of activated iNKT cells to suppress bacterial growth,we wished to determine whether administration of a-GalCer could enhance the antibacterial activity of iNKT cells and whether a-GalCer could serve a role in the treatment of established disease. METHODSMiceSix-to8-week-old female C3H/HeJ(C3H)(highly susceptible to Mtb) and C57BL/6(B6)(relatively resistant to Mtb)mice were obtained from Jackson Laboratories(Bar Harbor,ME).All mice were housed in a BSL3facility under specific pathogen–free conditions in the Animal Biohazard Containment Suite(Dana Farber Cancer Institute,Boston, MA)and used in a protocol approved by the institution.Bacteria and In Vitro Infection of Peritoneal Macrophages Macrophages were harvested by peritoneal lavage4days after intraperitoneal injection of sterile3%thioglycolate medium.M w were purified by positive selection using CD11b-microbeads accord-ing to manufacturer’s directions(Miltenyi Biotec,Auburn,CA).The purified cells were95%F4/801CD11b1,as determined byflow cytom-etry.Purified M w(0.53106/well)were cultured in24-well plates in RPMI1640medium(Invitrogen Life Technologies,Carlsbad,CA) supplemented with10%fetal calf serum(HyClone,Rockford,IL), penicillin/streptomycin,L-glutamine,sodium-pyruvate,2-ME,nonessen-tial amino acids,essential amino acids,and N-2-hydroxyethylpiperazine-N9-ethane sulfonic acid(all from Gibco,Carlsbad,CA).Virulent Mtb (H37Rv)was grown to mid-log phase in Middlebrook7H9medium supplemented with10%albumin/dextrose/catalase(BD Biosciences, San Jose,CA).Bacteria were opsonized for5minutes using RPMI 1640with2%human serum(Gemini Bio-Products,West Sacramento, CA),10%fetal bovine serum,and0.05%Tween80and then washed twice with complete medium without antibiotics.Bacteria were passed through a5-m m syringefilter(Millipore,Billerica,MA),counted in a Petroff-Hausser chamber,and added to enriched M w for2hours before extensive washing to remove extracellular bacteria.The length of infection was2hours for all experiments.Recombinant mouse IFN-g(US Biological,Swampscott,MA),a-GalCer,and a-C-GalCer were used at the concentrations indicated.Colony-Forming Unit Determination(In Vitro Model) Bacterial growth was quantified72hours after coculture with naive splenocytes with or without IFN-g and/or a-GalCer(Day4).The cells were lysed after removing the supernatant by adding distilled water. Serial dilutions were plated on Middlebrook7H11agar plates.The number of bacteria contained in the supernatant was less than10%of the cell-associated colony-forming units(CFU),independently of whether splenocytes were added.The colonies were enumerated after 3weeks at378C.Bacteria and InfectionsVirulent M.tuberculosis(Erdman strain)were passed through mice and subsequently grown once in Middlebrook7H9supplemented with the oleic acid–albumin–dextrose complex and stored at808C.For each experiment an aliquot was thawed,sonicated twice for10seconds with a cup horn sonicator,and then diluted in0.9%NaCl20.02%Tween80. Mice were infected via the aerosol route as described previously(23). For intravenous infections,mice were injected via the lateral tail vein with0.253106to4.03106live mycobacteria.The inoculum was confirmed by plating.Treatment of Mice with a-GalCera-GalCer was prepared in a vehicle of0.5%polysorbate20.The mice were injected with a low(100m gÁkg21)or high(500m gÁkg21)dose of a-GalCer,or injected with control vehicle.The following treatment regi-mens were tested:(1)intraperitoneal injection of100m gÁkg21a-GalCer on Days5,3,and1before aerosol infection;(2)intraperitoneal injection of100m gÁkg21a-GalCer on Days1,5,and9after aerosol infection;(3) intratracheal injection of100m gÁkg21a-GalCer on Day1after aerosol infection;(4)intraperitoneal injection of500m gÁkg21a-GalCer on Day1 after aerosol infection;(5)intraperitoneal injection of500m gÁkg21 a-GalCer on Day1after aerosol infection and then again every30th day;(6)intraperitoneal injection of500m gÁkg21a-GalCer on Day30 post aerosol infection and then again every30th day.a-GalCer and Antibiotic TreatmentGroups of C3H/HeJ mice were infected via the intravenous route and injected with a low dose of a-GalCer or vehicle on Day1post infection (p.i.).On Day14p.i.,isoniazid(100mg/L)was added to the drinking water.The drinking water was changed twice weekly during the treatment period and the mice had access to the water ad libitum. The isoniazid treatment was stopped on Day28p.i.and the CFU was determined in lung and spleen on Day36p.i.Measurement of IFN-g Lung ProductionBefore removal,lungs were perfused by injecting5to10ml of sterile PBS into the right ventricle of the heart after severing the inferior vena cava.The lungs were homogenized in0.9%NaCl20.02%Tween80 with a Minibeadbeater-8and centrifugation for20minutes at3,000 rpm,then supernatants were recovered and IFN-g was analyzed by a standard ELISA sandwich.CFU Determination(In Vivo Model)Three,5,or12weeks after infection,mice were killed and their left lung and spleen were aseptically removed and individually homoge-nized in0.9%NaCl–0.02%Tween80.Viable bacteria were enumer-ated by plating organ homogenates onto7H11agar plates.Colonies were counted after3weeks.StatisticsData were expressed as parisons of data among experimental groups were performed using analysis of variance with Bonferroni post test.A survival analysis was performed using Kaplan-Meier method and comparisons were done with log-rank test.Values of P,0.05were considered to be statistically significant. RESULTSa-GalCer–Activated Splenocytes Reduce Intracellular Replication of Mtb In VitroWe have previously shown that nonimmune splenocytes have innate antimycobacterial activity that is mediated by iNKT cells,which recognize Mtb infected M w and restrict intracellular bacterial growth(3).Although CD1d expression by the infected M w is required,which implies cognate recognition(3),it is uncertain whether CD1presents a microbial antigen,or alter-natively,whether cytokine production costimulates weak rec-ognition of an endogenous(self)antigen(24).Because only1to 2%of splenocytes are iNKT cells,we wondered whether pharmacological activation of iNKT cells with a-GalCer would enhance the growth suppression of Mtb mediated by iNKT cells in vitro.Coculture of naive splenocytes with Mtb-infected M w reduced the number of viable CFU as reported previously (Figure1A)(3).Addition of a-GalCer led to a further re-duction in CFU,as did the addition of IFN-g,and adding both a-GalCer and IFN-g led to an additive,and possibly coopera-842AMERICAN JOURNAL OF RESPIRATORY AND CRITICAL CARE MEDICINE VOL1822010tive,effect on CFU control.Treatment with a-GalCer was associated with a dose-dependent increase in IFN-g production (Figure1B).Thus,although iNKT cells have the physiological potential to recognize infected M w,their pharmacological activation by the synthetic lipid a-GalCer leads to a more potent mycobactericidal effect.Systemic Therapy with a-GalCer Prolongs the Survivalof Susceptible Mice after Low-Dose Aerosol Infectionwith M.tuberculosisWe next wished to determine the potential of a-GalCer to treat mice with pulmonary TB.To observe protection,we selected C3H mice,which are a susceptible mouse strain that develops extensive lung pathology and succumbs to infection earlier than B6mice(25–27).Previously,we demonstrated that a single dose of a-GalCer(100m gÁkg21)administered intraperitoneally within5days of Mtb infection by the intravenous route was effective in prolonging the survival of susceptible C3H mice (28).However,a-GalCer administered by the intraperitoneal route did not protect mice against low-dose aerosol(LDA) infection with virulent Mtb(Figure2A).There are several important differences between the systemic and pulmonary routes of infection that may affect the efficacy of a-GalCer. We hypothesized that a-GalCer–activated iNKT cells acted locally.Therefore,a-GalCer(100m gÁkg21)was administered intratracheally after LDA infection with Mtb.a-GalCer-treated mice had prolonged survival in two separate experiments (Figure2A).These data indicate that locally activated iNKT cells can provide protection against LDA Mtb infection.Ishii and colleagues reported that intraperitoneally adminis-tered a-GalCer(10m g/100m l vehicle)has a protective effect on the lung after bleomycin treatment(29).Based on these results, we analyzed whether a high dose of a-GalCer(500m gÁkg21) administered systemically protected mice challenged with LDA pared with mice treated with vehicle alone,a-GalCer administered intraperitoneally on Day1post infection pro-longed the survival of susceptible C3H/HeJ mice(Figure2B). The mean survival time(MST)of vehicle-treated mice was60 days.In contrast,the mice that received a single injection of a-GalCer(500m gÁkg21)had a MST of106days.These data show that systemic administration of high-dose a-GalCer pro-tects mice against pulmonary TB.Having established that systemic a-GalCer could protect mice against LDA Mtb infection,we next determined whether repeated administration of a-GalCer would provide longer-term protection.Mice were given a-GalCer on Day1after Mtb infection and then administered a-GalCer every30days, a treatment regimen that significantly prolonged their survival compared with vehicle-treated control mice(MST5124d vs. 31d;P50.0009).However,repeated monthly injections with a-GalCer had only an incremental effect on the survival compared with animals injected with a single dose of a-GalCer on Day1after LDA infection(MST5124d vs.106d;P5not significant)(Figure2C).Thus,similar to the low-dose a-GalCer treatment regimen after intravenous MtbinfectionFigure2.Treatment with a-galactosylceramide(a-GalCer)(100m gÁkg21or500m gÁkg21)significantly prolongs the survival of mice infected bythe aerosol route.(A)Survival of mice after the administration ofa-GalCer(100m gÁkg21)by the intraperitoneal or intratracheal route(n510).Mice were divided in four groups.One group(solid diamonds)received a-GalCer group by intraperitoneal administration on Days25,23,and21.A second group(solid squares)received a-GalCer groupby intraperitoneal administration on Days1,5,and9.A third group(closed circles)received a-GalCer group by the intratracheal route onDay1.Mice were infected with Mycobacterium tuberculosis(Mtb)bylow-dose aerosol(LDA)on Day0.(B)Vehicle(open circles)or a-GalCergroup(500m gÁkg21)(solid circles)was administered intraperitoneally1day after LDA infection with Mtb.(C)Survival of mice after theadministration of a-GalCer(500m gÁkg21)by the intraperitoneal route(n510).No animals died in the uninfected group treated witha-GalCer.Mice received vehicle(open circles),a-GalCer on Day1byintraperitoneal administration(solid circle),or multiple doses ofa-GalCer(solid squares)starting on Day1and then every30days.(D)Mice(solid diamonds)received multiple intraperitoneal administra-tions starting on Day30and then every30days.C and D are from thesame experiment and the data are separated for clarity.Survivalanalysis was performed using Kaplan-Meier method and comparisonswere done with log-rank test.*P,0.05,***P,0.0005.Sada-Ovalle,Sko¨ld,Tian,et al.:iNKT Cells,INH,and Tuberculosis843(28),a single dose of a-GalCer injected on Day1after LDA Mtb infection significantly prolonged the survival of the infected mice and repeated administration did not provide any additional benefit.Importantly,treatment with a-GalCer did not diminish the survival of uninfected mice and there were no deaths in mice repeatedly treated with a-GalCer,even after200days of observation(data not shown).This is consistent with Kimura and colleagues,who found that although a-GalCer treatment attenuated the development of bleomycin-induced pulmonary fibrosis,it did not appear to have any adverse effects on the pulmonary architecture(29).Three weeks after a-GalCer treatment of infected mice,no histopathological changes in the lung were observed compared with vehicle-treated control mice(data not shown).However,this does not rule out other subtle changes in the number or volume of granuloma or changes at later time points(28).These results demonstrate that systemic activation of iNKT cells can provide protection to susceptible mouse strains against LDA Mtb infection.a-GalCer Treatment of Susceptible Mice with EstablishedMtb Infection Prolongs Their SurvivalThe ability of a-GalCer to protect mice against LDA Mtb infection suggests that it could be useful as an adjunct treatment against TB.A critical issue is whether a-GalCer had efficacy against established TB disease.To evaluate this question,treatment of mice infected with a low dose of aero-solized Mtb was postponed until Day30post infection.At this time point,the bacterial counts in the lung are approximately 10,000-fold greater than the initial inoculum,and the immune response has controlled bacterial replication leading to the establishment of the plateau phase of the infection.Starting on Day30,mice were treated with a-GalCer(500m gÁkg21) every30days.Delayed administration resulted in a significant increase in survival compared with vehicle control mice(MST5 56d vs.31d;P50.02)(Figure2D).This indicates that act-ivated iNKT cells can alter the course of established TB infection.a-GalCer Administration Increases Lung IFN-g Production and Controls Bacterial ReplicationInjection of a-GalCer into naive mice leads to rapid but transient activation of iNKT cells(22).Still,we observed a long-term beneficial effect after a single injection of a-GalCer in Mtb aerosol-infected mice.A dose-dependent stimulation of IFN-g (Figure1B)but not IL-4(data not shown)was observed when a-GalCer is added to a coculture of splenocytes and Mtb-infected M w.We hypothesized that the beneficial effect of iNKT cells is mediated in part by IFN-g production.To determine whether iNKT cell activation induces long-term IFN-g production in the lung,a-GalCer(500m gÁkg21)or vehicle was injected intraperi-toneally into resistant B6or susceptible C3H mice24hours after LDA Mtb infection.Lung CFU and IFN-g production were measured in total lung homogenates3weeks post infection. Although the data did not reach statistical significance,the IFN-g concentration was elevated in lung homogenates3weeks after a-GalCer treatment compared with vehicle-treated control mice (Figure3A).Importantly,the bacterial burden in the lungs of mice treated with a-GalCer was significantly reduced3weeks after aerosol infection compared with vehicle-treated control mice(B6and C3H;Figures3B and3C).In addition,treatment of B6mice with a-C-GalCer(30),an analog of a-GalCer that has greater Th1-inducing capability,led to a greater CFU reduction (Figure3B).a-GalCer Has an Additive Effect When Administeredwith IsoniazidWe have shown that a-GalCer treatment ameliorates pulmo-nary TB and prolongs survival of susceptible mice,even in those with established pulmonary TB.However,a-GalCer treatment did not eradicate the bacterium or fully cure infected mice, indicating that the use of a-GalCer as a single therapeutic agent is not feasible.Because immune modulation using a-GalCer and conventional antibiotics reduces bacterial growth by dis-tinct mechanisms,we considered whether a-GalCer could have a role as an adjunct therapy.Therefore,we determined the efficacy of a-GalCer when administrated with or without the first-line antibiotic isoniazid(INH).We were particularly in-terested in testing this combination in a different disease model, one in which there was significant systemic disease.Therefore, we infected mice by the intravenous route to rapidly dissemi-nate the bacteria and establish a systemic infection with a high peripheral bacterial burden.Five weeks after Mtb infection,the combination of a-GalCer and INH significantly reduced bacte-rial burden in spleen and lung tissue.Importantly,even under conditions of disseminated disease with a high bacterial burden, the combination of a-GalCer plus INH had an additive effect in controlling bacterial growth in the lungs and in the spleen (Figure4).Our results suggest that a combination of traditional chemotherapy and modulation of the immune response during pulmonary TB can be used for more efficient control of Mtb replication.DISCUSSIONThe promise of immunotherapy,an approach that may have important implications for drug-resistant disease(31,32),has yet to show itself as an important part of the arsenal against TB. Although IFN-g,IL-2,IL-12,and thalidomide have been tried as immune modulators in patients with pulmonary TB,the results of experimental studies and clinical trials have been only moderately encouraging(33–37).Although many clinical stud-ies using IFN-g,IL-12,and thalidomide have demonstrated the potential of these agents to enhance immune responses to infection(38,39),a clinical response has beenfleeting.Obtain-ing a therapeutic effect will require unraveling the mechanism of action in addition to optimizing the timing,dose,and route of administration.The complex and myriad effects ofcytokinesmost likely explain the difficulty in implementing the clinical use of these different agents.Immune therapy with a-GalCer has also been suggested as a therapeutic option for human diseases,including multiple sclerosis,type1diabetes,rheumatoid arthritis,and lung cancer (16,40,41).Given the low frequency of these iNKT cells,it is remarkable that their activation could change the outcome of disease.Furthermore,given the complex interactions between Mtb and host immunity within the microenvironment of infected foci,simple activation of iNKT cells with a-GalCer in vivo is unlikely to be sufficient to elicit a curative response.A more realistic strategy would be to combine a-GalCer with conventional antibiotics.Pharmacological activation of iNKT cells by a-GalCer modulates adaptive immunity to several different microorgan-isms,including Mtb,Pseudomonas aeruginosa,Cryptococcus neoformans,and several different viruses(3,42,43).However, it has also been shown that specific iNKT cell activation with a-GalCer can exacerbate disease caused by Leishmania dono-vani infection(44).Therefore,the use of synthetic glycolipids to potentiate the function of iNKT cells must be carefully studied. Wefind that a-GalCer and INH have an additive effect in reducing the bacterial burden in Mtb-infected mice,and that a-GalCer prolongs the survival of infected mice with estab-lished pulmonary TB after Mtb infection via the respiratory route.How does a-GalCer modify the natural course of pulmonary TB?We have previously shown that the ability of a-GalCer to prolong the survival of Mtb-infected mice and to ameliorate disease requires CD1d(28)and is therefore dependent on the action of iNKT cells.We have recently shown,using an in vitro model of infection,that iNKT cells can kill intracellular Mtb and induce host protection after being adoptively transferred into mice infected with Mtb via the aerosol route(3).However, other actions of a-GalCer may be important.For example,Fujii and colleagues have shown that intravenous administration of a-GalCer into mice induces DC maturation.This effect is mainly mediated by TNF-a and IFN-g,and mature DCs can induce greater T-cell responses because of more efficient antigen presentation than exposure to mycobacterial antigens alone(45).Recently,a-GalCer incorporation into BCG has been shown to enhance the conventional CD81T-cell response;thus,it does appear that iNKT cell activation is modulating T-cell priming(46).The ability of iNKT cells to produce IFN-g after a-GalCer stimulation is one of their most important features to mediate host protection and may have therapeutic implications(47,48). Given the importance of IFN-g in controlling Mtb growth and the fact that a-GalCer administration induces IFN-g production by iNKT cells and by NK cells after downstream activation(49), one explanation for how a-GalCer ameliorates pulmonary TB may simply be via the induction of IFN-g.However,our in vitro data show that a combination a-GalCer and IFN-g is more efficient in controlling bacterial growth than a-GalCer or IFN-g treatment alone(Figure1A).This suggests that a-GalCer–activated iNKT cells reduce Mtb replication via an IFN-g independent mechanism,perhaps via direct effector functions.In this study we wanted to determine whether a-GalCer could ameliorate TB after infection via the respiratory route and if a-GalCer could ameliorate established pulmonary TB. Therefore,we infected mice via the aerosol route to mimic the natural route of Mtb infection and we used a high dose of a-GalCer(500m gÁkg21),which has been shown by others to influence the immune response in lung tissue(29).We found that a single dose of a-GalCer systemically or intratracheally significantly prolonged the survival of LDA Mtb-infected mice. We also observed that delayed administration of a-GalCer dramatically improved the outcome of established infection as determined by survival.We have previously shown that repeated a-GalCer admin-istration does not have a greater beneficial effect on survival after intravenous Mtb infection than a single a-GalCer in-jection.In the present study wefind similar results using a model of LDA Mtb infection.The LDA Mtb-infected group that received multiple a-GalCer administrations30days apart lived only18.5more days compared with the infected group that received a single a-GalCer dose.Chiba and colleagues have previously shown using an in vivo model of Mycobacterium bovis bacille Calmette-Gue´rin(BCG)infection that the re-sponsiveness of NKT cells changes during the natural course of the infection,including a transient period of anergy(22).If a-GalCer activation of NKT cells also leads to a period of anergy,the timing of repeated a-GalCer administration could be a critical variable that affects the NKT cell response.At present,one of the most important approaches against TB infection is the design of new therapeutic options,including adjuvants and vaccines.Although current treatment is effective against susceptible Mtb strains,the emergence of multidrug-and extensively drug-resistant strains requires the use of new strategies that can enhance the immune response.We show that the use of a-GalCer as adjuvant therapy influences the immune response by enhancing iNKT cell production of IFN-g,which presumably activates lung M w to kill Mtb and induces DC maturation to improve antigen presentation to T cells.Similar effects have been observed in other infection models(50–52). Using a clever strategy to exploit the adjuvant properties of a-GalCer clinically,Venkataswamy and colleagues incorpo-rated a-GalCer into the cell wall of BCG.A better immune response was elicited by the modified BCG(46).New and efficient antibiotics are likely to emerge on the market to help fight pulmonary TB.Unfortunately,there will be an ongoing battle between any new anti-TB drug candidate and the de-velopment of drug resistance.The inclusion of immunomodu-latory agents as adjuvant therapy may prove to be beneficial in the treatment of TB.Because a-GalCer treatment in people with cancer has been proven to be safe(53),we believe that further studies with a-GalCer are warranted todetermine。

第5章 相对论基础5-1 相对性原理1. 伽利略相对性原理● 伽利略相对性原理：一切彼此作匀速直线运动的惯性系，对于描写机械运动的力学规律来说是完全等价的，并不存在任何一个比其它惯性系更为优越的惯性系，与之相应，一个惯性系的内部所作的任何力学的实验都不能够确定这一惯性系本身是在静止状态，还是在作匀速直线运动。

● 伽利略相对性原理解释：在一个惯性参照系K 中，质点的质量、位矢、速度、加速度和质点所受的力分别为：Fa v r m ,,,,，在另一个相对于参照系K 以速度R v 作匀速直线运动的惯性参照系K '中，该质点的质量、位矢、速度、加速度和质点所受的力分别为：F a v r m ''''' ,,,,。

伽利略相对性原理指出，无论在参照系K 中，还在在参照系K '中，描写机械运动的力学规律的牛顿定律应该具有相同的形式：在参照系K 中：a m F =在参照系K '中：a m F ''='● 伽利略相对性原理来源：在经典力学的时空观是绝对时空观，绝对时空观得到的坐标变换为伽利略坐标变换，由伽利略坐标变换得到，在参照系K 和参照系K '中的加速度相等，经典力学认为，在参照系K 和K '中，质点的质量和所受的力都相等，所以在参照系K 和K '中描写机械运动的力学规律的牛顿定律具有相同的形式，所以经典力学的概念满足伽利略相对性原理。

伽利略坐标变换：t v r r R -='，t t ='得加速度变换为：a a=' 经典力学认为：m m ='，F F ='所以由参照系K 中的牛顿定律：a m F =可以推出参照系K '中的牛顿定律：am F ''=' 两个参照系中的牛顿定律形式相同2. 洛伦兹坐标变换● 洛伦兹坐标变换的来由：根据伽利略坐标变换，电磁学方程在参照系K 和K '中具有不同的形式，电磁学方程不满足相对性原理，为了使电磁学方程满足相对性原理，洛伦兹提出了洛伦兹坐标变换。

Game Theoretic and Machine Learning Techniques for Balancing GamesA Thesis Submitted to theCollege of Graduate Studies and Researchin Partial Fulfillment of the Requirementsfor the degree of Master of Sciencein the Department of Computer ScienceUniversity of SaskatchewanSaskatoonByJeffLongc JeffLong,August2006.All rights reserved.Permission to UseIn presenting this thesis in partial fulfilment of the requirements for a Postgrad-uate degree from the University of Saskatchewan,I agree that the Libraries of this University may make it freely available for inspection.I further agree that permission for copying of this thesis in any manner,in whole or in part,for scholarly purposes may be granted by the professor or professors who supervised my thesis work or,in their absence,by the Head of the Department or the Dean of the College in which my thesis work was done.It is understood that any copying or publication or use of this thesis or parts thereof forfinancial gain shall not be allowed without my written permission.It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis.Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to:Head of the Department of Computer Science176Thorvaldson Building110Science PlaceUniversity of SaskatchewanSaskatoon,SaskatchewanCanadaS7N5C9AbstractGame balance is the problem of determining the fairness of actions or sets of actions in competitive,multiplayer games.This problem primarily arises in the context of designing board and video games.Traditionally,balance has been achieved through large amounts of play-testing and trial-and-error on the part of the designers.In this thesis,it is our intent to lay down the beginnings of a framework for a formal and analytical solution to this problem,combining techniques from game theory and machine learning.Wefirst develop a set of game-theoretic definitions for different forms of balance,and then introduce the concept of a strategic abstraction.We show how machine classification techniques can be used to identify high-level player strategy in games,using the two principal methods of sequence alignment and Naive Bayes classification.Bioinformatics sequence alignment,when combined with a3-nearest neighbor classification approach,can,with only3exemplars of each strategy, correctly identify the strategy used in55%of cases using all data,and77%of cases on data that experts indicated actually had a strategic class.Naive Bayes classification achieves similar results,with65%accuracy on all data and75%accuracy on data rated to have an actual class.We then show how these game theoretic and machine learning techniques can be combined to automatically build matrices that can be used to analyze game balance properties.AcknowledgementsMany thanks to my supervisor,Michael C.Horsch,for his invaluable support, advice and for reading this so many times he must surely be bored of it by now.Thanks also to all participants in the study,and the valuable advice of my thesis committee.Thanks also to NSERC for funding the years of study that went into this work.ContentsPermission to Use i Abstract ii Acknowledgements iii Contents iv List of Tables vi List of Figures viii 1Problem Description and Motivation1 2Game Theory62.1External Balance (8)2.2Internal Balance (10)2.3Game Theory in Games (11)3Machine Learning for Constructing Game Matrices from Data133.1Machine Learning in Games (17)3.2Experimental Methods and Terminology (18)3.2.1Experimental Testbed (19)3.2.2Sequence Alignment (20)3.2.3Nearest Neighbors Methods (22)3.2.4Naive Bayes Classification (23)3.2.5Boosting and the ADABoost Algorithm (25)4An Empirical Study of Abstract Strategy Identification from Data284.1Pre-Data Survey (28)4.2Data Collection (29)4.3Data Classification (32)4.3.1Structure of Data (33)4.3.2Needleman-Wunsch Alignment Methodology (34)4.3.33-nearest Neighbor Classification (39)4.3.4Naive Bayes Classification Methodology (40)4.3.5Naive Bayes with Boosting Methodology (46)4.4Constructing Game-Balance Matrices (48)4.5Results and Analysis (50)5Conclusion and Future Work565.1Conclusion (56)5.2Future Work (57)A The Warcraft III Ladder65B Pre-Data Survey66C Questionnaire72D Sequence Alignment Scoring Matrices78List of Tables2.1A4x4matrix game that illustrates internal imbalance,where thefirstrow and column represent play frequencies for each player’s optimalmixed strategy.Note that no row or column is dominated by any other.11 4.1Results of the pre-data survey issued to participance,showing thenumber of participants who agreed that the strategies proposed bythe study’s author were common and viable Warcraft III strategies..30 4.2Results of game data classification,showing the number of times eachstrategy appeared according to human experts.Note that the totalstrategies listed will be higher than the total number of games,asparticipants were permitted to list more than one strategy per game.32 4.3Results of100test runs using Needleman-Wunsch sequence alignmentfor classification and selecting exemplars at random (37)4.4Breakdown of Needleman-Wunsch alignment results by game strategyusing the edit-distance scoring matrix (37)4.5Breakdown of Needleman-Wunsch alignment results by game strategyusing the customized scoring matrix (38)4.6Results a test run using Needleman-Wunsch sequence alignment forclassification and selecting a set of‘good’exemplars according to asimple heuristic (39)4.7Results of100test runs using Needleman-Wunsch sequence alignmentwith a3-nearest neighbor approach for classification (40)4.8Breakdown of Needleman-Wunsch alignment results with3-nearestneighbor approach by game strategy (41)4.9Results of100test runs using Naive Bayes classification,randomlyselecting half of the data as a training set for each run (43)4.10Breakdown of Naive Bayes classifier results by strategy (44)4.11Results of100test runs using Naive Bayes classification,with the en-hancement of10additional nodes representing the presence or absenceof individual unit types in the build (44)4.12Breakdown of Naive Bayes classifier with enhancement of10additionalnodes results by strategy (45)4.13Results of100test runs using Naive Bayes classification,using theADABoost boosting algorithm with M equal to5 (47)4.14Breakdown of the boosted Naive Bayes classifier M equal to5for theADABoost boosting algorithm (47)4.15A game balance matrix,with rows and columns constructed usingstrictly the labelling provided by human experts over14games (50)4.16A game balance matrix,with rows and columns constructed using thelabelling provided by Needleman-Wunsch sequence alignment over14games (50)4.17A game balance matrix,with rows and columns constructed usingNaive Bayes classification over14games (50)4.18A game balance matrix,with rows and columns constructed using thekNN Needleman-Wunsch alignment classifier,run on100unlabelledOrc vs.Night Elf games (51)4.19A measurement of the difference in alignment score and posterior prob-ability between correctly and incorrectly classified samples,over100random test runs (55)D.1The character encoding key for Orc and Human military units (79)D.2The character encoding key for Undead and Night Elf military units.80 D.3The scoring matrix used for Orc strategy classification (81)D.4The scoring matrix used for Human strategy classification (82)D.5The scoring matrix used for Undead strategy classification (83)D.6The scoring matrix used for Night Elf strategy classification (84)List of Figures4.1The structure of the Naive Bayesian classifier for strategy prediction.Each UnitX node represents the character at position X in the gamedescription (42)4.2The structure of the enhanced Naive Bayesian classifier for strategyprediction.Each UnitX node represents the character at position Xin the game description,and each TypeX node is a boolean valuerepresenting the presence or absence of a specific unit type (46)Chapter1Problem Description and MotivationPeople who play any sort of competitive game on a regular basis have an intuitive sense of what‘game balance’means.Indeed,many a losing player will,sometimes with justification and sometimes not,blame his impending loss not on his own lack of skill,but on imbalances present in the game.Take,for example,a game of chess.The player who is assigned to play black may feel that he has been forced to play the weaker side in the game,and thus a loss may not be due to his skill, but rather because the odds were stacked against him.Available online statistics indicate that such a complaint could be well-founded,and that black is at a significant disadvantage over white in chess[Chessgames,2005].Similarly,players will often come to identify individual game elements with the monikers‘overpowered’or‘underpowered.’This is especially prevalent in online competitive video games.Certain elements of the game—such as a particular military unit or weapon,for example—may seem overly effective,thus marginalizing other elements that they overpower.In the same manner,elements that are too weak will simply go unused by experienced players.Problems of this sort have long plagued online multiplayer games;the mutalisk unit,from the popular strategy game Starcraft [Blizzard Entertainment,1998],has been cited by designers as a prime example of a game element that was overpowered and difficult to balance[Cadwell,2002].However,while players are intuitively familiar with the concepts brieftly sketched above,intuition is generally where the familiarity ends.While some classical games have been modeled from a complexity theory standpoint(chess,for instance,is EXPTIME-complete[Fraenkel and Lichtenstein,1981]),to my knowledge there has been no formal definition of these particular problems in an academic setting,espe-cially coming from the perspective of a design standpoint.Some expertise and advice on‘rules of thumb’exist in industry,of which Cadwell[Cadwell,2002]is a prime ex-ample.Carpenter[Carpenter,2003]uses risk analysis and spread-sheeting techniques to balance individual elements of multiplayer role-playing games,and Kennerly[Ken-nerly,2003]suggests the use of data mining for improving game design.A few other tips and tricks are available[Adams,1998,2002,Rouse III,2005],and usually con-sists mainly of the author’s personal advice and experiences.Furthermore,none of these articles demonstrate rigorous proof or testing of the suggested techniques,nor do they suggest a formal and general-purpose framework for discussing game-balance and its most important properties.For the purposes of this thesis,the term game is used to refer primarily to so-called parlor games-often manifested by board games such as chess,and card games such as Blackjack-and multiplayer computer games.However,many of the concepts and issues raised are applicable to an even wider variety of competitive domains,such as professional sports.We also restrict our attention to games of the two-player variety,though in general the principles discussed are intended to apply to games with numerous players.To further examine and codify the problem of balance,which at this point has only been discussed as an informal concept,we divide it into two distinct categories: external balance and internal balance.We will formalize both of these concepts in Chapter2.External balance refers to the balance between two opposing players in an asymmetric game who are assigned different roles,or action sets,at the game’s outset.Many games exhibit this asymmetric property;the example of chess,in which white movesfirst,has already been mentioned.In the domain of online games,it is extremely common for players to choose a role or faction at the game’s outset that dictates their available action set for the remainder of the game.The game is said to be externally balanced if neither player has an advantage over the other due simply to the role initially assigned or chosen.Internal balance refers to the balance between different available actions or ele-ments of an individual player.If a given action is too weak,then the optimal strategymay be to never use it.Similarly,an action that is too strong may be overused,and could in fact become the game’s only viable course of action.Unlike problems of ex-ternal balance,internal balance problems can exist even in symmetric games where both players have identical action sets available.For instance,online real-time strat-egy games are symmetric so long as both players select the same initial faction,but the players may have military units available to them that should never be built under optimal play.This is an internal balance problem.Cadwell[Cadwell,2002]contends that all forms of imbalance boil down to an elimination of choices.We see that the definitions given here agree with that claim. In the case of internal balance,it is evident that if the effectiveness of available actions is too disparate,several player choices have been eliminated.In the case of external balance,if one role in the game is stronger than the other,then that role should always be selectedfirst,if a player is permitted to choose her own role.Again, assuming both players are trying their utmost to win the game,it seems reasonable to say that a player choice exists only if the two sides of the game,although different, are equally likely to win the game.On the point of external balance,it should be noted that in many games a handicap is sometimes desirable so as to even the playingfield between two players of disparate skill.Therefore,it might sometimes be argued that factions of different strengths are desirable so as to provide such a handicap,so that one player can volunteer to play the‘harder’side.However,it is our opinion that a handicap should be an explicit element of a game,configurable by the players,rather than hidden away in the game’s opposing roles.A handicap is of little use,after all,if players mustfirst be experts at the game to even realize that the handicap exists.Both forms of imbalance are highly undesirable in a competitive game.Cadwell [Cadwell,2002]describes poor game balance as the factor that often stands in be-tween a good design and a good game;Carpenter goes so far as to call it the“holy grail”of game design[Carpenter,2003].External imbalance can create discontent and frustration among players before the game has even begun,particularly for the player who is forced into the weaker role.Internal imbalance means that the gamecontains extraneous elements,that exist only to frustrate and confuse inexperienced players.From a designer’s point of view,it makes little sense to devote resources to creating a game element that expert players will quickly come to ignore.The most formal of publicly available industry methods of which we are aware for dealing with these problems is the Risk Analysis method by Carpenter[Carpenter, 2003].This is essentially a probabilistic sampling method which simulates the pitting of game elements against one another and generates a probabilistic model of the outcome using multiple iterations.This method requires an extensive user-defined description of the process being simulated,as well as a knowledge of’average’player statistics.The method provides a pair-wise comparison of game elements,but for truly complex systems,the number of iterations required to obtain a meaningful output will be significant.Kennerly advocates the use of data mining techniques [Kennerly,2003],involving simple comparisons of statistics and rates of change. Such techniques can inform a designer that a given aggregated rate(such as the rate at which a certain character class improves in an online role-playing game,for instance)needs to be changed,although how to change the rate or by how much is left to the discretion of the designer.Beyond the above methods,it appears for the most part that game balance is achieved through extensive amounts of play-testing and trial-and-error on the part of panies such as Blizzard Entertainment run extensive beta tests that last for months and involve thousands of players,in part to obtain data on game balance issues Blizzard Entertainment[2005].Balance patches for online strategy games continue to be released for months or even years after a game goes public. Board games and card games generally do not have this option,and so it is even more imperative they be properly balanced before a commercial release.An analytical method for extracting balance problems and principles from game data,or better yet identifying and correcting game balance issues without the need for generating large amounts of data,would be a significant contribution in both domains.The contributions of this thesis are twofold.First,we reframe and formalize the problems of game balance using the language of economic game theory.We showhow game theoretic concepts and tools can be used to analyze games in extensional form and extract properties relating to game balance.Secondly,we propose building formal game models of manageable size through the use of strategic abstractions,i.e., a high-level strategy implemented by the player.We provide and evaluate several different machine learning techniques for automatically classifying these high-level strategies from real game data,and then show how the results can be used to build game theoretic matrices.The remainder of this thesis is organized as follows.Chapter2gives a brief overview of the game theory used in this work and presents game theoretic definitions of balance,as well as a brief overview of the general application of game theory to games.Chapter3presents a general overview of the machine learning approach used in this thesis,as well as describing all of the individual technologies used for experiments.Chapter4describes in detail the experimental work of this thesis in automatically classifying strategies from game data,and presents an analysis of the results.Finally,Chapter5suggests directions of future work and concludes the thesis.Chapter2Game TheoryIn this chapter,wefirst present the game theoretic notation used in this thesis and then go on to formally define the concepts of external and internal balance and show how they can be identified once a game has been reduced to a normal form matrix.As discussed in Chapter1,game balance for competitive,multi-player games is essentially ensuring that the following two broad properties hold.Thefirst property we call external balance.For a2-person game to be externally balanced,we require that both players have an equal chance at winning the game regardless of the game’s potential starting conditions.This means we do not want the starting conditions of the game to give an inherent advantage to one player.A symmetric game is trivially externally balanced,because the starting conditions are identical for both players. The second property we call internal balance.For a2-person game to be internally balanced,we require that for each available action or game element available to a player,it is useful in at least one circumstance.In other words,we do not want to confuse players with options that should never sensibly be taken.For the subsequent work of this thesis,it is necessary to formalize these concepts. To do so,we turn to the language of game theory,as commonly used in the economic sciences.In spite of its name,game theory has rarely been applied to actual games, especially in the context of the game balance problem studied here.We will use game theory to describe the formalisms that will be our objects of study and analysis.A game,G,consists of an n-dimensional array of payoffelements,where n is the number of players in the game.This matrix representation is said to be the normal form of the game.The rows and columns of the matrix represent actions or strategiesthat are available to the players.Each payoffelement is an n-tuple,which specifies the payoffs to each of the n players.A payoffis the reward given to a player for the outcome that a particular array element ually,these payoffs may be any element of the set of real numbers;sometimes they can even be qualitative in nature.It is assumed that each player’s preference is to maximize this payoff.In this thesis,we consider only2-player games,in which case the array is a 2-dimensional matrix M,in which the rows represent possible actions of player1 (termed the row player)and the columns represent the actions of player2(termed the column player).The element M(i,j)is a pair of payoffvalues,indicating the payofffor each player when player1selects action i and player2selects action j.We also restrict our attention to zero-sum games;that is to say,games that require that the sum of the payoffs in each particular matrix cell is zero.This model is an accurate reflection of competitive games,since in such games one player winning requires that the other has lost.In fact,two-player,zero-sum games are often termed “strictly competitive”in game theoretic language.In this thesis,we represent the payoffvalue of a win as1,the value of a loss as-1,and a draw(in games where such is permitted)as0.Payofftables throughout this thesis show only the payofffor the row player;the column player’s payoffis of course the negation of these payoffelements.The solution concept used in this thesis is that of mixed strategy Nash equilib-rium.A set of strategies is said to be in Nash equilibrium if no player can improve her expected payoffby changing strategies,provided that the other players also do not change strategies.Each row in the payoffmatrix is said to represent a pure strat-egy for the row player(and similarly with columns for the column player).A mixed strategy for a player is a probability distribution over the set of his pure strategies, that defines the probabilistic frequency that each pure-strategy will be selected by the player.While many games will not contain a pure-strategy Nash equilibrium,at least one mixed strategy Nash equilibrium is guaranteed to exist.This mixed strat-egy will often be referred to as a player’s optimal strategy.The standard method of finding the optimal mixed strategy for a matrix game is the Simplex algorithm,agood description of which can be found in Owen[Owen,1995]or Papadimitriou and Steiglitz[Papadimitriou and Steiglitz,1998].The value of a game,v,is the expected payoffthat player1will receive upon playing her optimal strategy;player2will receive its negation.In the case of two-player,zero-sum games,this value will be unique.Calculating the value of a game is straightforward once the optimal strategies of the game are known;usually it can be obtained simply as a byproduct of calculating these strategies via the Simplex method.It should be noted that while probably the most common,Nash equilibrium is not the only solution concept in game theory.There are others that exist as well; Azhar,McLennan and Reif[Azhar et al.,1992]and Papadimitriou[Papadimitriou, 2005]discuss some of them,as well as some of the shortcomings of Nash equilibrium. Perhaps the most significant short-coming is that in the general case,multiple Nash equilibria may exist,and there is no known efficient means tofind all equilibria points. However,two convenient properties of Nash equilibria are obtained by restricting the proposed model to two-person,zero-sum games.Thefirst is that while it is possible for multiple equilibrium mixed strategies to exist,they are all guaranteed to result in the same expected payoff.Secondly,all equilibrium strategies are interchangeable; that is to say,if(σ1,σ2)is an equilibrium pair,that is to sayσ1is an optimal mixed strategy for player1andσ2is an optimal mixed strategy for player2,and(τ1,τ2)is also an equilibrium pair,then so are(σ1,τ2)and(τ1,σ2)[Owen,1995].2.1External BalanceTo define external balance in game theoretic terms,we relate it to the concept of the value of a two-person,zero-sum game.Definition2.1A two-player competitive game is externally balanced if neither player, given perfect skill,has a higher probability of winning the game than the other.We will now show how this definition relates to the game-theoretic value of the game through the following theorem.Theorem2.1A two-player competitive game G is externally balanced if and only if the corresponding zero-sum matrix game has a value of0.Proof2.1The following equivalences can be used to obtain both directions of the proof.Let G be an n×m zero-sum matrix game where n and m are the number of pure strategies available to players1and2respectively,and where element g ij is1if player 1wins the game when her action is i and player2’s action is j,and-1if player1 loses.A payoffof0indicates a draw.Let x and y be the optimal mixed strategies for players1and2respectively,where x i and y i represent the players’frequency of play for pure strategy i.G has a value of0if and only if the following sum holds:n i=1mj=1x i y j g ij=0(2.1)As previously defined,g ij∈{−1,0,1},with1representing a win for player1,−1 representing a loss,or0representing a draw.We can partition this sum according to the values of g ij.i,j:g ij=1x i y j g ij+i,j:g ij=−1x i y j g ij+i,j:g ij=0x i y j g ij=0(2.2)Clearly,the terms where g ij=0can all be dropped,as they do not affect the equality.Furthermore,since the g ij are constant in both the other sums(1for the first sum,and-1for the second),they can be factored from the summation,yielding the following:i,j:g ij=1x i y j−i,j:g ij=−1x i y j=0(2.3)i,j:g ij=1x i y j=i,j:g ij=−1x i y j(2.4)Now,let p1be the probability that player1wins the game,and p2be the probability that player2wins the game,given perfect play by both sides.Since by the definition ofG,player1wins the game when g ij=1,and the x i y i define a probability distribution over these possible outcomes,we see that:x i y j(2.5)p1=i,j:g ij=1x i y j(2.6)p2=i,j:g ij=−1By equation2.4,this is true if and only if:p1=p2(2.7) 2.2Internal BalanceNext we consider the issue of internal balance in game theoretic terms.Definition2.2A two-player competitive game is said to be internally balanced if for every available action i,there exists some optimal strategy x for which x i has a non-zero frequency of play.One obvious method of determining this is simply tofind a player’s optimal mixed strategy,x.If x i>0for all i,then the definition of internal balance is immediately satisfied.Since optimal strategies are interchangeable in a two-player, zero-sum game,we need not concern ourselves withfinding all possible optimal strategies;finding a single one for which this property holds is sufficient.However, whilefinding a single strategy could prove that the property does hold,we would have tofind all optimal strategies to prove that it does not,and in general there is no known method for doing this[Owen,1995].Therefore,it would be convenient if an even simpler criteria could be found for determining internal imbalance.One such game-theoretic concept that seems intuitively plausible is that of domination.We say that the ith row dominates the kth row if,in a matrix game G,g ij≥g kj for all j and g ij>g kj for at least one j. The situation is similar for columns.A dominated row or column can be eliminated from the game without affecting optimal strategies for either player.Table2.1:A4x4matrix game that illustrates internal imbalance,where thefirst row and column represent play frequencies for each player’s optimal mixed strategy. Note that no row or column is dominated by any other.//33%33%-100%-11100%01However,while it is evident that a game is not internally balanced if it contains a dominated row or column,the converse does not hold.The proof is through the following simple counter-example in Table2.1.It can be found through standard means that the optimal strategy profile x for player1consists of the vector(0,0,1.0,0),whereas the profile y for player2consists of(0,0.33,0.33,0.33).Both players have one or more options available that should never be played under their optimal strategies.However,an examination of the matrix reveals that no row or column is dominated by any other.This means that a lack of domination is not sufficient to demonstrate internal balance in a game.Therefore,for this thesis,we must resort to calculating a strategy profile for each player,and ensuring that no action has a zero-frequency of play.Should wefind an optimal strategy such that this property holds,we call the game internally balanced. Should wefind a dominated action,we say the game is internally imbalanced.Should we fail tofind either a strategy where no action has a zero-frequency of play or a dominated action,we cannot say whether or not the game is internally balanced. Therefore,the problem of determining internal balance is a semi-decidable problem.2.3Game Theory in GamesIn spite of its name,as a modelling tool game theory has more often been used to describe economic,social and political situations than actual games.However, the most basic and famous game-playing algorithm,minimax search,is based upon。