On the relativistic generalization of Maxwell's velocity distribution

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Notes On Hilbert's 12th Problem

a rX iv:mat h /6511v1[mat h.NT]3M a y261NOTES ON HILBERT’S 12TH PROBLEM SIXIN ZENG Abstract.In this note we will study the Hilbert’s 12th problem for a primitive CM ﬁeld,and the corresponding Stark’s ing the idea of “Mirror Symmetry”,we will show how to generate all the class ﬁelds of a given primitive CM ﬁeld,thus complete the work of Shimura-Taniyama-Weil.Introduction 0.1.Let K be a number ﬁeld,H K be the ideal class group of K ,and let K 0be the Hilbert class ﬁeld of K .The class ﬁeld theory tells us there is a canonical isomorphism Gal (K 0/K )≃H K .In general given any integral ideal d ,let K d be the maximal abelian extension of K unramiﬁed outside d ,and let H d be the generalized ideal class group relative to d ,then a similar isomorphism holds as well:Gal (K d /K )≃H d .On the other hand,Hilbert’s 12th problem asking for an explicit generation of all the abelian extension of K ,more precisely it is asking for ﬁnding a special transcendental function,whose values at some special points would generate all the abelian extension of K .When K is the rational ﬁeld Q ,the transcendental function is the exponen-tial function and the special points are the division points on the unit circle.Indeed by the classical Kronecker-Weber theorem,all the abelian extension of Q can be obtained by adding the roots of unity to Q .When K is an imaginery quadratic ﬁeld,this problem is answered by the theory of complex multiplications.As this theory has very much inﬂuenced the later thinking about this problem,let’s recall some details.So let K be such a ﬁeld,consider the set of elliptic curves E satisfying End (E )⊗Q ≃K ,i.e.,the set of elliptic curves with complex multiplications by K .All such elliptic curves can be constructed by the following way:take the caononical embeddingι:K −→C ,let a be an integral ideal of K ,then a is a rank 2Z -module in C ,i.e.,ι(a )is a lattice in C ,so we can take the quotient C /ι(a )which is an2SIXIN ZENGelliptic curve with complex multiplication by K.From the construction it is clear that the ideal group of K operates on this set of elliptic curves,indeed for any representative b of an ideal class,we have the isogeny:C/ι(a)→C/ι(ab). On the other hand all such elliptic curves are deﬁned over some numberﬁeld, and they are represented by the moduli points on the moduli space of all elliptic curves M1=H/P SL2(Z).In particular the Galois group acts naturally on the moduli points of this set of elliptic curves.So if E=C/ι(a),let℘be a prime of K,E(℘)=C/ι(a℘),s(℘)∈H K≃Gal(K0/K)the Artin symbol,then the main result of the complex multiplication is E s(℘)≃E(℘).In more concrete terms if j is the classical modular function,and let p E be the moduli point of E on M1=H/P SL2(Z),then j(p E)is an algebraic number and generates the Hilbert classﬁeld of K.Similar statements hold for all the ray classﬁelds of K.The idea behind the theory of complex multiplication can be summerized as the following:•Given the numberﬁeld K,in order to solve the Hilbert’s12th problem,ﬁrst we need toﬁnd a suitable class of algebro-geometric objects X,say varieties;•X should be closely related to K such that the ideal classes of K cannaturally act on them;•All such X should be deﬁned over some numberﬁeld,all such X shouldbe living on some natural moduli space so theirﬁeld of moduli are givenby the coordinate function evaluated at the moduli points.The Galoisgroup can naturally acts on them;•Moreover the action of the ideal classes of K and the action of Galoisgroup on the moduli points should be related by the reciprocity law ofclassﬁeld theory and the Kronecker congruence relations;If we canﬁnd such a class of X,then theﬁeld of moduli of X give the answer to Hilbert’s12th problem.For a general numberﬁeld besides imaginery quadratic,this philosophy is diﬃcult to apply.So from now on we will concentrate on a special case.From now on let K be a primitive CMﬁeld of degree2n,n≥1.K is then an imaginery quadratic extension of a totally realﬁeld F,with[F:Q]=n. Further let K∗be the reﬂexiveﬁeld of K,F∗be the maximal totally real subﬁeld of K∗,then K∗is also a primitive CMﬁeld of degree2n,[K∗:F∗]=2. Let{σ1,σ2,···,σn}be the set of archemidean primes of F,lifted to K as the CM types,and letρbe the complex conjugate.When n>1,according to the above philosophy,we naturally consider the set of abelian varieties of dimension n with complex multiplication by K with CM type{σ1,σ2,···,σn},i.e.,those abelian varieties satisfying End(X)⊗Q≃NOTES ON HILBERT’S12TH PROBLEM3 K,and moreover for anyα∈K the action ofαon the space of holomorphic diﬀerentials of X is given as the diagonal action of diag(ασ1,···,ασn).All such X can be constructed in the following way:letι:K→C n be the embedding ι(x)=(xσ1,···,xσn),then for any integral ideal a,ι(a)⊂C n as a Z-lattice is of rank2n,hence the quotient X=C n/ι(a)is an abelian variety,one see that such X satisfying the above conditions.From the construction such set of abelian varieties are naturally associated to K,with the integral ideals of K act on them as isogenies.This is exactly the same as elliptic curves.So next we shall consider the moduli space for these X and their arithmetic properties,it is here some complication arised,let’s be careful.Let O K and O F be the rings of integers of K and F,and let a be an integral ideal of O K.Regarding a as a module of O F,it is of rank2.More precisely we have the following(see for example,Yoshita’s book[19]):Lemma0.1.For any ideal a of O K we have the isomorphism as O F-module: a≃b·ω1⊕O F·ω2,whereω1,ω2∈K,b a fractional ideal of O F.Moreover the ideal class of b is c·N K/F(a)=c·aaρ,where c is a fractional ideal of F, independent of a,and c2=D K/F.In particular O K=c·ω1⊕O F·ω2,and if a satisfying a·aρ=(µ),then a=c·ω1⊕O F·ω2.Now let X=C n/ι(a),by Shimura the polarization of X is given by the Riemann form E(x,y)which is E(x,y)=T r K/Q(ζxyρ),withζ∈K satisfying ζρ=−ζ,Im(ζσi)>0,any i.The“type”of this polarization,by deﬁnition,is the ideal class ofζd K/F aaρ,where d K/F is the relative diﬀerent of K over F. From the above lemma it is clear that b is the type of X.For any moduli problem,in order to have a good moduli space we need to ﬁx a polarization of X.For our class of X,we observe that they all have a large ample cone,in fact the dimension of the ample cone are all of n,hence it makes sense to consider the moduli problem with all the ample classesﬁxed. This is the moduli space of“type polarized abelian varieties”.Concretely for any fractional ideal b of O F,we can constructed a moduli space M n(b)which parametrizes families of abelian varieties with the form C n/v n·ι(b)⊕ι(O F),where v n is a vector in the product of upper half plane H n,and the dot product is deﬁned component-wise.It is well-known that M n(b)=H n/Γ(b)withΓ(b)={α∈SL2(O F)|α≡1mod(b)}.If X is the above abelian variety of type b,the X=C n/ι(a)with a=b·ω1⊕O F·ω2, up to isomorphism we can chooseω1,ω2∈K such thatωX=ω1/ω2satisfyingIm(ωσiX )>0for any i,i.e.(ωσ1X,···,ωσnX)is a vector in H n,hence deﬁnes amoduli point of X in M n(b).4SIXIN ZENGSo in summary given the class of CM abelian varieties with aﬁxed CM type,we can construct a natural moduli space M n(b)for aﬁxed type of X,i.e.,ifX1,X2are of the same type,they live on the same moduli space,but diﬀerenttypes can produce diﬀerent moduli spaces.So although the ideal classes of Kcan act on all of the X,the Galois group can only act on the subclass of Xwith the same type.On the other hand it is easy to show that all X are deﬁned over aﬁniteextension of the reﬂexﬁeld K∗.In fact for an ideal a∗of K∗,we deﬁnea=g(a∗)= τi a∗τi,where{τ1,···,τn}is a CM type of K∗,then g(a∗) acting on X will not change the type of X.It is from this point of view thatin1955,Shimura-Taniyama-Weil(see Shimura’s book[12])established that theﬁeld of moduli of X will generate part of the classﬁeld of K∗,precisely it isthe classﬁeld corresponding to the subgroup of the ideal class group H0K∗= {a∈H K∗|g(a)=(µ),µ∈K}.0.2.In this note we will try to extend the work of Shimura-Taniyama-Weil to cover all the classﬁelds of K.There are actually two problems here.First, the appearance of the reﬂexﬁeld K∗is quite inconvenient,since our abelian varieties X,their moduli spaces M n(a),and the CM points on the moduli are all constructed naturally from the given CMﬁeld K,it is natural for us to look for the invariants from these abelian varieties that directly generate the class ﬁelds of K,instead of the reﬂex K∗.Second and more important problem,is how can we deal with the classﬁelds that corresponding to the isogenies of abelian varieties that live on the diﬀerent type of moduli spaces.Theﬁrst problem can be solved in the following way.Since theﬁelds K and K∗are reﬂex to each other,by Shimura-Taniyama-Weil theory,the classﬁelds of K is generated by the abelian varieties associated to K∗.So the problem is toﬁnd invariants on X that somehow related to Shimura varieties associated to K∗.For this purpose Iﬁnd the following notion of“cone polarized Hodge Structure”quite useful.First we observed the for all the abelian varieties of the CM type the Kahler cone is very large.Indeed as the polarizations are determined by the Riemann form E(x,y)=T r K/Q(ζxyρ),so the polarizations are determined byζ∈K such thatζρ=−ζ,Im(ζσi)>0,any i.Suchζform a cone of dimension n, hence the Kahler cone is of n dimensional.We can also see this by the fact that since End(X)is an order in K,the automorphism group of X is then the unit group of the order,which is of the form Z n−1⊕T orsion.So the Kahler cone of X is necessarily n dimensional.Given such Kahler cone C X of X we can consider the primitive classes of X in the middle dimension relateive to this cone,i.e.,those classes of dimension n such that are annilated by any element in the Kahler cone.NOTES ON HILBERT’S12TH PROBLEM5{α∈H n(X,C) α·x=0,∀x∈C X}This is the generalization of the notion“transcendental lattice”in the theory of K3surface.The Kahler cone and these primitive classes relative to this cone should be considered as the most basic Hodge theoretical invariants of our abelian varieties.So to understand these abelian varieties,we need to construct the appropriate moduli spaces for these invariants.First let’sﬁx the Kahler cone and consider the primitive classes of the middle degree relative to this cone.Such primitive classes carries a natural Hodge structure,indeed for abelian varieties with F-multiplication the primi-tive classes can be characterized as the invariant classes of the automorphism group U F,it carries a Hodge structure of weight n,with the i-th Hodge number to be i n =n!6SIXIN ZENGTheorem0.2.(1)We have a natural isomorphism M P H≃H n/SL2(O F∗);(2)The natural morphisms M n(a)→M C→M P H are allﬁnite.By the theory of Shimura-Taniyama-Weil,the moduli points of X on this moduli space will generate the classﬁeld of K.The natural modular function on the Hilbert modular variety of K∗,when pulled back to M n(a),will become a natural modular function on M n(a).In this way,by directly considering the geometry of X,we can have the classﬁelds of K generated.This answers our ﬁrst question.Note that in our approach we do not emphasis on the use of“ﬁeld of moduli”, indeed since our moduli space of primitive Hodge structures can actually be regarded as the moduli space of abelian varieties with multiplication by F∗,the natural“ﬁeld of moduli”somehow lose its meaning on this moduli space(1). We only need the natural modular functions on the moduli spaces,which give the coordinates of the CM points.In any case our approach is more natural in view of Hodge theory,and serve the purpose of generating classﬁelds of K well.0.3.The second question is more diﬃcult.We need toﬁnd a natural way to interpolate the moduli spaces of diﬀerent types.For this purpose we will use the idea of Mirror Symmetry.Precisely if R1and R2are two ideals suchthat R1R−12is a real ideal,then although the abelian varieties X R1and X R2deﬁned by R1and R2are living on the diﬀerent moduli space,we will showtheir“Mirror partners”X′R1and X′R2are living on a single moduli,and theMirrors’ﬁeld of moduli can be used to generate the classﬁelds.To motivate our idea,let’s consider another approach to the Hilbert’s12th problem,namely the Stark’s conjectures([16]).The point of departure is to consider the Dedekind zeta function and Hecke L-functions for the numberﬁelds.Hilbert’s problem asks for a natural tran-scendental function,indeed for the abelian extension point of view,nothing can be more natural than these L-functions,as they transformed under the Galois group ually the Stark’s conjectures are formulated and studied for a totally realﬁeld,but as we shall see,it is more natural and sim-ple to study it for the CMﬁeld,because all theﬁnite extension of CMﬁelds are necessarily CM,i.e.,any extension of K has no real inﬁnity.The remarkable fact about the Stark’s conjectures is that it can be for-mulated on any classﬁelds of K uniformly,not just those classﬁelds in the Shimura-Taniyama-Weil theory,so in the explicit form it can be used as a guide for us to search for the solution of missing classﬁelds of the old theory.NOTES ON HILBERT’S12TH PROBLEM7 So let K as above,let R be an ideal class of K,recall that the Dedekind zeta function is deﬁned asζK(s)= a1s−1κK+ρK+O(s−1)whereκK=(2π)n R KN(a)s .NatuallyζK(s)= RζK(s,R),with the limitformula:ζK(s,R)=κw Ks n−1(1+δK(R)s)+O(s n+1) thenδK(R)=nγ+nlog2π−log|D K|−w K|D K|1/2ρK(R−1)w Ks n−1(1+δK s)+O(s n+1) withδK= RδK(R)•ifχ=χ0is not trivial,thenL K(s,χ)=ζK(s,R)=−R Kw K0=(−1)m(R K8SIXIN ZENGThe Stark’s conjecture predicts that if we write L K(s,χ)=R K(χ)s n+ O(s n+1),then R K(χ)also has the form of regulators,i.e.it is a determinent of a matrix whose entries are linear combination of logarithm of units of K0. Moreover we should expect R K(χ)σ=R K(χσ)for anyσ∈Aut(C).In our present case K is a primitive CMﬁeld,in this case Stark’s conjec-ture is somehow simple in the following sense:the regulator R K of K is a determinent of(n−1)×(n−1)matrix,and the regulator R Kof K0is a determinent of(mn−1)×(mn−1)matrix.Stark’s conjecture in fact says that the quotient R K/R K is a determinent of(mn−n)×(mn−n)matrix, and this matrix should be diagonalized into m−1blocks,with each block a n×n matrix,further in this case we can use the R K matrix to simplify these blocks,so in this extension there are only m−1essential new units.In other words,U Kas a module of U K is free of rank m−1.This is very much similar to the classical case of imaginery quadraticﬁelds.By some elementry argument we can show that R K=(R K)m·vol(S), where vol(S)is a determinent of(m−1)×(m−1)matrix whose entries are linear combinations of logarithm of units in K0.These are the basis of U Kas the module of U K.On the other hand by the Frobenius determinent formula, χ=χ( Rχ(R)δK(R))=det R1,R2=1(δK(R1)−δK(R2)) Hence we should expect:•δK(R)=log|ηK(R)|;•δK(R1)−δK(R2)=log|ηK(R1)ηK(R2)is a unit in K0.More overthese units should transform under the Galois group according to thereciprocity law.0.4.To prove things like this we need to have a good expression forδK(R), this can be done by using the theory of GL(2)Eisenstein series over the totally realﬁeld F.The Eisenstein series is deﬁned asE(w,s;a)=(c,d)∈(a⊕O F)/U F,(c,d)=(0)ni=1y s i|cσi z i+dσi|−2sThe idea is a classical one,sinceζK(s,R)= a∈R1w2satisfyingIm wσi>0,∀i.Then we haveNOTES ON HILBERT’S 12TH PROBLEM 9ζK (s,R )=N (a 1)s α∈a 1/U F ,α=012n −2πn h F RF [χF (d F )L F (2,χ−1F )i Im(ωi )+πn D −3/2F 0=b ∈d −1F a σ1,χ(bda )|N (b )|−1exp (2πi (n j =1b j ℜ(w j )+i |b j Im(w j )|))]where (1)a,d ∈A ×F such that div (a −1)=a and div (d )=d F ;(2)σs,χ(x )is a function deﬁned asσs,χ(s )= v ∈(finite primes ) 1+χv (w v )q s v +···+(χv (w v )q s v )ord v (x v )if x v ∈d v ,0if x v /∈d v .Here d v denotes the ring of integers of F v ,w v is the prime elementof F v and q v =|d v /w v d v |.So we haveζK (s,R )=−R K10SIXIN ZENGδK(R)=h(ω;a)−log i Im(w i)−logN(a)We reminded that a is the type of R−1,w=(w1,···,w n)∈H n is the CM point deﬁned by R,and h(w,a)is the complicated function deﬁned above.So our basic task is to understand this function.0.5.We notice that hχFsatisfying the following modular properties([19]):for γ∈Γa we have(1)hχF (γω;a)=hχF(ω,a)ifχF is not trivial;(2)h1(γω;a)−log( i Im(γω)i)=h1(ω,a)−log( i Imωi)So in particular we haveχFχF(a)hχF(γω;a)−log( i Im(γω)i)= χFχF(a)hχF(ω,a)−log( i Imωi)which suggests that we may actually have a Hilbert modular formηK(w;a) of parallel weight such that h(w;a)=log|ηK(w;a)|.Classically in case F= Q this is indeed the case asηK is the classical Dedekind eta functionη,it is well-known thatηhas an inﬁnite product expression,which when taking the logarithm translated into a Fourier expansion,which is exactly the above Fourier series.In the higher dimensional case it is not that easy2.Besides the fact that the Fourier expansion is too complicated and diﬃcult to work with,we can not expect by directly exponenciate the above expression of h to get a meaningful function,precisely because of the inﬁnite unit group of F,as shown as the regulator term R F in the leading coeﬃcients of the Fourier expansion.The regulator R F is not a rational number,in fact we expect it to be transcendental, so even if we exponenciate h we can not get anything useful for the arithmetic purpose.In particular we can not expect to get an inﬁnite product expression as the classicalηfunction.0.6.What should I do?It turns out although we can not exponenciate theh function,can not get the explicit formula forηK(w;a),we still can say something quantitatively about it.The idea is to consider a twisted version of Eisentein series:E u,v(w,s;a)=(c,d)∈(a⊕O F)/U F,(c,d)=(0)ni=1e2πi(cσi u i+dσi v i)(Im w i)s|cσi w i+dσi|−2sNOTES ON HILBERT’S12TH PROBLEM11 where(u,v)∈R n⊕R n.This is entirely adopted from the classical Kronecker’s second limit for-mula(see[18]).To explain why we need to develop the twisted Eisenstein series,let’s recall the classical situation,i.e.,when F=Q.In this case the Eisenstein series isE(w,s)=m,n∈Z,(m,n)=0(Im(w))s2[1+(CONST+log Im(w)−4log|η(w)|)s]+O(s2)The twisted Eisenstein series is:E u,v(w,s)=m,n∈Z,(m,n)=0e2πi(mu+nv)(Im(w))s24w∞n=1(1−q n w)and g u,v is the Siegel’s function:g u,v=−q1q zφ(w,z) and we haveg u,v=q112SIXIN ZENG(3)g u,v is also closely related to Dedekindη.In fact as a function of zg u,v has a simple zero at z=0withη(w)as the coeﬃcient,i.e.,|g u,v|=|η(w)|2|q z−1|+O(z2)This shows that the absolute value|η(w)|is not a theta null,but rather a“derivative theta null”.But we note our theta functions arenormalized at z=0to be zero,φ(w,0)=0,if the theta functionnormalized in this way,their derivatives can also be used as the coor-dinates on the moduli space.By abuse of notation we still call|η(w)|a theta null,hence it gives a modular function on the moduli space.From the deﬁnition,E u+1,v=E u,v+1=E u,v,hence g u+1,v=g u,v+1= g u,v,but since|g−v,u|=|q12B2(−v)w g−v,u(w)Then from the periodic condition of g u,v we immediately see that|φ(w,z+1)=φ(w,z);|φ(w,z+w)|=|q−1z|·|φ(w,z)| That is,|φ(w,z)|satisfying the characterization of a theta function,moreover since we know it is an analytic function,it then has to be a theta function itself.This is the idea we would follow in the higher dimensional case,as we observed before,since in the higher dimension we can not expect any explicit inﬁnite product formula for the functionηK(w;a),but we still have all the periodic properties as the1-dimensional case.Now we go back to the higher dimensional case,the Eisenstein series is:E(w,s;a)=(c,d)∈(a⊕O F))/U F,(c,d)=(0)ni=1(Im(w i))s|cσi w i+dσi|−2sWe have the limit formula:E(w,s;a)=−2n−2h F R F s n−1[1+(CONST+logN(a)+log( i Im(w i))−h(w;a))s]+O(s n+1) The twisted Eisenstein series is:NOTES ON HILBERT’S12TH PROBLEM13E u,v(w,s;a)=(c,d)∈(a⊕O F)/U F,(c,d)=(0)ni=1e2πi(cσi u i+dσi v i)(Im w i)s|cσi w i+dσi|−2sand we have the limit formula:E u,v(w,s;a)=−2n−2hF R F log|g−v,u(w;a)|s n+O(s n+1)where log|g−v,u(w;a)|has an explicit Fourier expansion,just like h(w;a).Then we argue as the following:(1)Recall that u=(u1,···,u n)∈R n,v=(v1,···,v n)∈R n,and O F⊂R n as a lattice.From the deﬁnition,for anyα∈O F,we have E u+α,v=E u,v+α=E u,v,i.e.,translate invariant under O F,hence|g−v+α,u|=|g−v,u+α|=|g−v,u|.(2)Now write z=u−vw,i.e.,z=(z1,···,z n),z i=u i−v i w i,∀i.We try to write g−v,u as a function of(w,z),so we deﬁneφ(w,z)=q−12B2(−v)w= i q−112wφ(w,z)|−log(|ηK(w;a)|2 i|z i|)}=0that islim z→0|q1|ηK(w;a)|2 i|z i|=1Also we verify that our theta function is normalized at z=0to be 0.This implies thatηK(w;a)is a theta null.By the classical theory of theta function,theta null naturally gives rise to the modular forms on the moduli space.In fact this should be more or less expected.By Mumford’s theory of algebraic theta function,we may further conclude that these theta nulls in fact deﬁnes the moduli space as an integral scheme over Z,hence have all the expected integral properties.14SIXIN ZENG0.7.In summary we have found the explicit form of the functionδK(R).δK(R)=logN(a)+log i Im(w i)−h(w;a)=logN(a)+log i Im(w i)−log|ηK(w;a)|2=log[N(a) i Im(w i)|ηK(w;a)−2|]where R is an ideal class of O K,a its type,w its CM point on M n(a),and ηk(w;a)is the theta null:∂∂z nφ(w,z)|z=0=ηK(w;a)2By the Stark’s conjecture,we need to understandδK(R1)−δK(R2),i.e., we need to understand the quotientηK(w1;a1)ηK(w2;a)is meaningful,as the modular function evaluating at the CMpoints,and the natural action of Galois group on them is prescribed by the reciprocity law.But when R1and R2are of the diﬀerent type,for example, when R1R−12is a real ideal,thenηK(w;a1)andηK(w;a2)are on the diﬀerent moduli spaces,thus their quotient becomes meaningless.This is precisely the limit of Shimura-Taniyama-Weil’s theory.So what can we do?To go further we need toﬁnd a natural way to inter-polate the diﬀerent moduli spaces,and it is here the idea of Mirror symmetry comes.As we shall see,in this case this quotient will have a meaning similar to the classical one if we consider the complexiﬁed Kahler moduli of the abelian varieties.Why should we interpolate the diﬀerent moduli spaces?The functionδK(R) is deﬁned not on a single moduli space of theﬁxed type,but rather automati-cally been deﬁned on all the moduli spaces,as the type a can vary accordingly. Likewise the modular formηK is a Hilbert modular form on all the type-ﬁxed Hilbert modular varieties,and when the type vary,can be regarded as a mod-ular form on all the moduli spaces.This strongly suggests that we should have a natural way to interpolate all these moduli spaces of diﬀerent types,such that these functions can naturally deﬁned.In other words,when weﬁx the type a,we get the Hilbert modular forms,what then happens if weﬁx the CM pointsωand let the type vary?When we look the explicit form ofδK(R)and h,we note the apparent symmetric roles played by the quantities σIm(ωσi)and N(a).Indeed if we ﬁx the CM pointsωand let a vary,we should have a meaning for the quantity N(a).This can be achieved by considering the Kahler moduli of our abelian varieties.0.8.Mirror Symmetry is usually formulated for the Calabi-Yau varieties, roughly it asserts that Calabi-Yau always come in pairs,X and X′,with theNOTES ON HILBERT’S12TH PROBLEM15“complex moduli”and“Kahler moduli”exchanged.In terms of the Hodgenumber,it means the Hodge diamond of X′is a rotation of X.For abelian varieties,Mirror Symmetry is generally regarded as“trivial”,as the underlying topological type would not change.Nevertheless we canstill talk about it.There are several constructions of Mirror manifold for theabelian varieties,the simplest one I believe,is given by Manin([9]).It goes asthe following:let k be any completeﬁeld,X an abelian variety over k,T bethe algebraic torus of dimension n over k,T≃(k×)n.Then the multiplicativeuniformization is0→P X→T→X→0,where P X is a free abelian groupof rank n,P X is called the period of X.Under this uniformization the Mirrorpartner X′is then0→P X→T∨→X′→0,i.e.,we explicitly indentify theperiods in T and T∨.When k≃C one verify that the complex moduli andKahler moduli of the two are exchanged.In our situation,given the Mirror pair X and X′we will be mainly concernabout the relations of theta functions on them.Since the underlying topologicaltype would not change,we may regard the Mirror transform as a“rotation”of complex structure of X.So to compare the theta functions on X and X′we have toﬁx the underlying real structures.We begin with X=C n/(w·a⊕O F)≃R2n/Z2n,any polarization of X isgiven by an integral skew-symmetric bilinear form on R2n.Given such a formω,we canﬁnd an integral basis{λ1,···,λ2n}of the integral lattice such thatif{x1,···,x2n}is the dual basis,thenω= n i=1δi dx i∧dx n+i,withδ1|δ2|···the elementary divisors.Note in our case the biliear form is given by the trace T r K/Q(ζxyρ)withthe admissibleζ∈K such thatζρ=−ζ,Im(ζσi)>0.Thus we may regard(x1,···,x n)as an integral basis of O F,and(x n+1,···,x2n)as an integralbasis of a.In particular(x n+1,···,x2n)depends on a.In the following we willdenote it as x n+i(a)if we need to use this dependence.Next we introduce the complex structure,so X becomes a complex tori,and we can introduce the complex coordinates.To do this let e i=λi/δi,i=1,2,···,n,and let{z i}be the complex dual of{e i}.Consider the changeof coordinates transform:Ω·(x1,···,x2n)T=(z1,···,z n)TThenΩ=(∆δ,Z)with∆δ=diag(δ1,···,δn)the diagonal matrix,and Zsymmetrical,Im(Z)>0.We recogonize that∆−1δZ is the period matrix of X.Note in our case for abelian varieties with CM by K,the peiod martix Z isnecessarily diagonal∆−1δZ=diag(w1,···,w n),with w=(w1,···,w n)∈H n.In particular we have z i=δi·x i+δi·w i·x n+i(a),we may regard it as the transform from the underlying real coordinates to the complex coordinates.16SIXIN ZENGNow recall the theta function on X is characterized by the periodic condi-tion:θ(z+λi)=θ(z);θ(z+λn+i)=e−2πiz iθ(z)Taking absolute values we have:|θ(z+λi)|=|θ(z)|;|θ(z+λn+i)|=e2πIm(z i)|θ(z)|However from the above coordinates transformation,Im(z i)=δi·Im(w i)·x n+i(a)The Mirror symmetry transform says that we can exchange the complex moduli with the Kahler moduli,while the coordinates w=(w1,···,w n)∈H n can be regarded as the complex moduli of X,where is the Kahler moduli?Our Kahler moduli coordinates are actually in the variables(x n+1(a),···,x2n(a)). Since they are depend on the type a,we want to write the dependency explic-itly,in order to understand the transformation of types.For this end let’s write x n+i=x n+i(O F).The type ideal a as a Z module,is a submodule of O F of full rank,i.e.,if weﬁx an integral basis of O F,then O F/a≃⊕n i=1Z/t i Z,with x n+i(a)=t i x n+i and i t i=N(a)/D F.Thus the positive rational numbers(t1,···,t n)can be conviniently regarded as the coordinates of the ideal a,and under appropriate identiﬁcation,can be regardedas the coordinates of of the Kahler class in the the Kahler moduli.So in particular we haveIm(z i)=δi·Im(w i)·t i·x n+iBut from the above formula,when we exchange the complex moduli(w1,···,w n) and Kahler moduli(t1,···,t n),it’s not going to change the multiplier e2πIm(z i)=e2πδi·t i·Im(w i)·x n+i.Since the absolute value of theta functions can be regardedas a real analytic function on R2n,thus we conclude that for the given Mir-ror pair X and X′,their theta functions’absolute values satisfying the same periodic condition,hence must be only diﬀered by a constant!Recall our previous puzzle,when two ideal classes R1and R2of O K sat-isfying R1R−12is a real ideal,then R1and R2are of the diﬀerent type,sothe corresponding abelian varieties X1and X2living on the diﬀerent moduli spaces,soηK(w,a1)/ηK(w,a2)has no meaning.But we knowηK(w,a)is the theta null of X,thus by the above formula,ηK(w,a)is also the theta null of the Mirror X′.So although X1and X2living on the diﬀerent moduli space,if their Mirror X′1and X′2are in the same moduli space,thenηK(w,a1)/ηK(w,a2) would be meaningful!This is indeed the case,as X′1and X′2are on the sin-gle moduli space,the complexiﬁed Kahler moduli space of X.This is the underlying rationale for us to use the Mirror symmetry.Note from this relation of theta functions we also see that if X is deﬁnes over a numberﬁeld,then the Mirror X′also is deﬁned over that numberﬁeld.。

英语作文伟大的科学家

Great scientists have always been the cornerstone of human progress,contributing significantly to the advancement of knowledge,technology,and societal development. Here are some key points to consider when writing an essay about a great scientist:1.Introduction to the Scientist:Begin by introducing the scientist,mentioning their full name,field of expertise,and the era in which they lived.For example,Albert Einstein was a theoretical physicist who developed the theory of relativity in the early20th century.2.Early Life and Education:Describe the early life and educational background of the scientist.This can include their upbringing,any significant influences,and their journey through formal education.For instance,Isaac Newton was born in Woolsthorpe,England, and his early interest in mathematics and natural philosophy was fostered by his education at the University of Cambridge.3.Major Contributions:Highlight the scientists most significant contributions to their field.This could be a groundbreaking theory,invention,or discovery.For example, Marie Curies research on radioactivity led to the discovery of the elements polonium and radium.4.Impact on Society and Science:Discuss the impact of the scientists work on society and the scientific community.This can include how their discoveries have been applied, the awards they received,and their influence on future generations of scientists.For example,Charles Darwins theory of evolution by natural selection has had profound implications for biology and our understanding of the natural world.5.Personal Life and Challenges:Its also important to touch upon the personal life of the scientist,including any challenges they faced,both professionally and personally.This can humanize the figure and provide context for their work.For example,Albert Einstein faced political persecution and had to flee Nazi Germany.6.Legacy and Current Relevance:Conclude by discussing the lasting legacy of the scientist and the relevance of their work in todays world.This could include ongoing research inspired by their work or how their ideas have been integrated into modern technology and society.For instance,the principles of quantum mechanics,pioneered by scientists like Max Planck and Niels Bohr,are fundamental to the development of modern electronics.7.Citation and References:Ensure that you cite and reference the sources of information you use in your essay.This not only gives credit to the original authors but also addscredibility to your work.nguage and Style:Use clear and concise language,avoiding overly technical jargon unless it is necessary for the context.The essay should be wellstructured,with a logical flow of ideas,and should engage the readers interest.9.Critical Analysis:While its important to praise the achievements of the scientist,its also valuable to provide a balanced view by discussing any controversies or criticisms associated with their work.10.Conclusion:Summarize the main points of your essay,reiterating the significance of the scientists contributions and their enduring influence on the field of science and society.By following these guidelines,you can craft a comprehensive and engaging essay about a great scientist,offering insights into their life,work,and the lasting impact of their discoveries.。

BS EN 12845-2004+A2-2009 固定式消防系统.自动喷水系统设计,安装和维修

批注本地保存成功开通会员云端永久保存去开通
BRITISH STANDARD
BS EN 12845:2004 +A2:2009
Incorporating Corrigendum August 2009
Fixed firefighting systems — Automatic sprinkler systems — Design, installation and maintenance
1 2 3 4 4.1 4.2 4.3 4.4 4.4.1 4.4.2 4.4.3 4.4.4 5 5.1 5.1.1 5.1.2 5.2 5.3 5.4 5.5 6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.3 6.3.1 6.3.2 7 7.1 7.2 7.2.1 7.2.2 7.2.3 7.3 7.3.1 7.3.2
介绍 Introduction ................................................................................................................................................. 10
This British Standard was published under the authority of the Standards Policy and Strategy Committee on 16 November 2004
Amendments/corrigenda issued since publication Date 31 July 2009 Comments Implementation of CEN amendment A2:2009 Correction to National foreword

On the vierbein formalism of general relativity

ˆ
curved spacetime:
ˆ 1 ∂ mc α ˆ µ ˆβ + γ γˆ rα σα − Ψ ( x) = 0 , iγ eα ˆγ ˆ ( x) ˆβ ˆ µ h 4 ∂x
ˆ where the constant matrices γ α
(1-3)
ˆ ˆ ˆ ˆ i ˆ β ˆ ˆβ ˆβ satisfy {γ α = [γ α , γ ] ; the Ricci’s , γ β } = −2η α ;σ α 2
ˆ
equations; and then, discuss some characteristics of the theory. Concretely, taking advantage of this form, we first prove that the "kinetic energy" term, i.e., the quadratic term of time derivative term, in the Lagrangian of the Einstein-Hilbert action is non-positive definitive. And then, we present two groups of coordinate conditions that lead to positive definitive kinetic energy term in the Lagrangian, as well as the corresponding actions with positive definitive kinetic energy term, respectively. Based on the ADM decomposition, the Hamiltonian representation and canonical quantization of general relativity taking advantage of the actions with positive definitive kinetic energy term are discussed; especially, the Hamiltonian constraints with positive definitive kinetic energy term are given, respectively. Finally, we present a group of gauge conditions such that there is not any second time derivative term in the ten Einstein equations.

科技英语翻译1

驱动这些机器的动力装置是一台50马力的感应电动机。
► 2）通顺易懂 ► 译文的语言符合译语语法结构及表达习惯，容易为读者所理解和接受。
► A. When a person sees, smells, hears or touches something, then he is perceiving.
2. Cramped(狭窄的) conditions means that passengers’ legs cannot move around freely.
空间狭窄，旅客的两腿就不能自由活动。
3. All bodies are known to possess weight and occupy space.
忠实、通顺（普遍观点）
► 科技英语文章特点：(well-knit structure;tight logic;various styles)结构严谨，逻辑严密，文体多样
1. 科技翻译的标准：准确规范，通顺易懂，简洁明晰 1）准确规范
所谓准确，就是忠实地，不折不扣地传达原文的全部信息内容。所谓规范，就是译文要符合所涉及的科学技或某个专业领域的专业语言表
实验结果等，而不是介绍这是这些结果，理论或现象是由谁发明或发现的。
► In this section, a process description and a simplified process flowsheet are given for each DR process to illustrate the types of equipment used and to describe the flow of materials through the plant. The discussion does not mention all the variations of the flowsheet which may exist or the current status of particular plants. In the majority of the DR processes described in this section, natural gas is reformed in a catalyst bed with steam or gaseous reduction products from the reduction reactor. Partial oxidation processes which gasify liquid hydrocarbons, heavy residuals and coal are also discussed. The reformer and partial oxidation gasifier are interchangeable for several of the DR processes.

英语介绍科学家

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THANKS !
theories of special relativity and general relativity, and he is also
known for his equation E=mc². Einstein received the Nobel Prize in Physics in 1921 for his work on
quantum theory
第2节
Galileo Galilei
Galileo Galilei
Galileo Galilei was an Italian astronomer, physicist, and mathematician who is credited with making important contributions to the Scientific Revolution. He is known for his work in suppor
of tools and their ability to exhibit empathy. Dr. Goodall's work has led to a greater understanding of chimpanzees and their conservation status. She founded the Jane Goodall Institute in 1977 to promote conservation and animal welfare
第5节
Richard Feynman
Richard Feynman
Richard Feynman was an American theoretical physicist who made numerous contributions to quantum electrodynamics and particle physics. His Nobel Prize-winning work on weak nuclear interactions and quantum chromodynamics helped revolutionize our understanding of these fields. Fe ynman's gift for explaining complex scientific concepts in a clear and concise manner made him a popular lecturer and author. His books "The Feynman Lectures on Physics" and "QED: The Strange Theory of Light and Matter" are considered classics in the field of physics education 以上这些科学家为人类科学事业做出了杰出的贡献，他们的发现和理论不仅改变了我们对世界的认识，也推动了人类社会的发展

自旋轨道角动量耦合

⾃旋轨道⾓动量耦合Chapter 8Spin-Orbit Interaction8.1Origin of spin-orbit interactionSpin-orbit interaction is a well-known phenomenon that manifests itself in lifting the degeneracy of one-electron energy levels in atoms,molecules,and solids.In solid-state physics,the nonrelativistic Schr¨o dinger equation is frequently used as a ?rst approximation,e.g.in electron band-structure calculations.Without relativistic corrections,it leads to doubly-degenerated bands,spin-up and spin-down,which can be split by a spin-dependent term in the Hamiltonian.In this approach,spin-orbit interaction can be included as a relativistic correction to the Schr¨o dinger equation.Let us brie?y review the origins of this correction,following approach of Ref.[80].For this purpose one has to consider the Dirac equation,which is the basic equation for electronic systems,including the electron spin and its rela-tivistic behavior.One obtains the Dirac equation by linearizing the relativistic generalization of the Schr¨o dinger equation.It is Lorentz-invariant and describes the electron spin and spin-orbit coupling from ?rst principles.One naturally arrives at the Dirac equation when starting from the relativistic expression for the kinetic energyH 2=c 2p 2+m 2c 4.(8.1)Inclusion of the electric and magnetic potentials,φand A ,by substituting p ?(ε)2=(c p ?εA )2ing pand H as op erators p =?i80CHAPTER8.SPIN-ORBIT INTERACTION(H2?c2 µp2µ?m2c4)ψ=0.(8.4)pµ=p x,p y,p z are components of the momentum operator,as follows from Eq.8.2.A disadvantage of this di?erential equation is that ones needs the initial values ofψand?t ,similar to the Schr¨o dinger equation;it is referred to as Dirac equation:(H?c µαµpµ?βmc2)ψ=0,(8.6)with the four-component vectorψ.Let us now compare the Dirac equation with the Schr¨o dinger equation.For this we use the non-linearized form[H?εφ?cα·(p?εcA)+βmc2]ψ=0.(8.7)Using the approximation that the kinetic and potential energies are small com-pared to mc2,two components of the spin function can be neglected,and equa-tion8.7takes the form1c A)2+εφ?ε4m2c2E·p?ε2mcσ=ε8.2.SYMMETRIES81B=?1mc(E×p),(8.9)where terms of order(vmc s·B=?εεrdr,(8.11)and the term can be written in the form(s=4m2c2σ·(E×p)=εεrdr×p=1r dV82CHAPTER8.SPIN-ORBIT INTERACTION If the crystal lattice has inversion symmetry(i.e.if the operation r→? r does not change the crystal lattice),one will obtainE(k,↑)=E(?k,↑)and E(k,↓)=E(?k,↓).(8.14) From combination of the equations8.13and8.14it becomes clear that if both time reversal symmetry and inversion symmetry are present,the band structure should satisfy to the conditionE(k,↑)=E(k,↓).(8.15) In other words,the energy cannot depend on the electron spin.Conse-quently,for crystals which have inversion symmetry(like fcc,hcp),spin splitting is not allowed in the bulk,and these solids keep their spin degeneracy.8.3Hydrogen atomConsideration of the electron orbital motion in the Coulomb?eld permits us to visualize spin-orbit interaction.We use the hydrogen atom for which the Schr¨o dinger equation can be solved analytically.In the semiclassical model, moments are treated as vectors with z-components being already quantized. (For spin-orbit interaction,only the case of l≥1is important,since SOC vanishes for l=0.)The energy-level correction due to spin-orbit coupling takes the well-known form[81]:E n,l,s=E n?µs·B l=E n+µ0Ze24π3(v×r)=µ0Ze2[j2?l2?s2]=12[j(j+1)?l(l+1)?s(s+1)],(8.19) where a=µ0Ze222and j=l?1r3should be written as its expectationvalue8.4.SOLIDS838πm2e·ψ?n,l,m1a=?E n·Z2α24π=e2137.Finally,the energy di?erence between two sublevels with opposite spin is△E l,s=?E n·Z2α2n3.(2)Spin-orbit splitting decreases with increas-ing quantum numbers n and l.8.4SolidsBulk crystalsIf a crystal lattice potential has a center of inversion,spin-orbit interaction will not a?ect the band structure,except for certain points of the Brillouin zone (BZ),where the in?uence of spin-orbit interaction can be very important.As an example,we consider the center of a BZ(Γpoint)that has cubic symmetry. In a tight binding model,we can build up the p bands from atomic p wave functions.In a free atom without SOC,there are three degenerate p functions, so three p bands will appear in the model.These bands will all be degenerate at theΓpoint(k=0,see Fig.8.1(a)),transforming one into the other through cubic symmetry transformations at this point.Each of the three bands is also doubly degenerate in the spin,altogether a six-fold degeneracy atΓ.When spin-orbit interaction is included,the bands will split in two sub-bands with fourfold p3/2and twofoldp1/2degeneracy(see Fig.8.1(b)).The bands will now behave di?erently as we move away from theΓpoint depending on whether or not there is inversion symmetry in the lattice.If the crystal has a center of inversion,each band will preserve its spin degeneracy,as illustrated on Fig.8.1(b).The p3/2bands,with j=3/2,will now split into two bands with m j=±3/2and m j=±1/2. (with projections taken e.g.in the direction of the vector k).If a center of inversion symmetry is absent,then the p3/2and p1/2bands will additionally be split by spin-orbit interaction,removing the spin degeneracy. This situation is shown in Fig.8.1(c).The cases considered are really occurring in crystals,e.g.in semiconductors like Ge and InSb.Germanium has a center of inversion symmetry,and we will have the?rst scenario for the top valence band.For InSb,which crystallizes in a zinc-blende lattice,the spin degeneracy will be lifted as we leave the center of84CHAPTER8.SPIN-ORBIT INTERACTIONFigure8.1:In?uence of spin-orbit interaction on the p-levels at the center of the Brillouin zone:(a)Without SOC,six degenerate levels are observed at the Γ-point;(b)SOC leads to a splitting into p3/2and p1/2levels,but leaves the spin degeneracy in lattices with inversion symmetry;(c)in a lattice without inversion symmetry,the spin degeneracy is fully removed,except for theΓ-point.the BZ.This e?ect of symmetry on the electronic band structure of zinc-blende crystals leads to the appearance of a spin-orbit interaction term that is cubic in k(Dresselhaus splitting).SurfacesTermination of the crystal by a surface breaks the3D inversion symmetry.In this way,it permits electron surface states with the same parallel wave vector, but k ,opposite spins to have di?erent energies:E(k ,↑)=E(k ,↓).(8.23) In the absence of external or internal magnetic?elds,spin-orbit splitting is the only possible cause for a splitting of spin degenerate levels at the surface, and this interaction is described by the Rashba Hamiltonian[82].。

量子理论的诞生和发展_从量子论到量子力学_彭桓武

量子理论的诞生和发展———从量子论到量子力学＊彭　桓　武(中国科学院理论物理研究所　北京　100080)摘　要简要叙述,从普朗克1900年首次对电磁波提出量子假设到狄拉克1928年对电子提出相对论性方程这段时间内,量子理论特别是量子力学诞生和发展的演化过程.内容分黑体辐射和量子假设;老量子论的兴与衰;第一条通向量子力学的路———对应原理,包括矩阵力学,狄拉克的q -数;第二条通向量子力学的路———波粒二象性,波动力学;以及量子力学初步成长(指1927年的表象理论、不确定关系、氦原子及氢分子和1928年的狄拉克相对性电子理论)五个部分.关键词量子论,量子力学,矩阵力学,波动力学THE BIRTH A ND GR OWTH OF QUANTUM THEORYFROM QUANTUM HY POTHESIS TO QUANTUM MECHA NICSPE NG Huan -Wu(Ins titut e o f Th eo reti cal Ph ys ics ,C hin es e Aca de my o f Scien ces ,Beijin g 100080,Chi na )Abstract This short histor y c overs the bir th and early gro wth of quantum theory from 1900to 1928,beginning with Planck ′s for mula and the quantum hypothesis for the blac k -body radiation .After a descr iption of the rise and decline of the old quantum theor y in connection with its application in spectrosc opy ,two paths based on the rigorous for mulation of the corr espondenc e pr inciple leading to matr ix mechanics (1925)and Dirac ′s non -c ommuting q -number s (1925)are ex -plained .Another path based on the generalization of the wave -particle aspect of lightquanta is then shown to lead to wave mechanics (1926).Among the wor ks during the early gro wth of quantum mechanics in 1927—1928,repr esentation the -ory ,the uncertainty principle ,two -electron problems ,and Dirac ′s r elativistic theory of electrons are discussed .Key words quantum theor y ,quantum mechanic s ,matr ix mechanics ,wave mechanics＊　2001-01-17收到1　黑体辐射和量子假设量子概念和量子假设首先是在对黑体辐射的谱分析的精细实验基础上作理论研究形成,由普朗克于1900年12月14日提出的.黑体是这样一个物体,它对于从各方向射来的各种频率的辐射都百分之百地吸收.这个理想物体可由壁周围皆处于同一温度下的腔体内部来实现.1859年,基尔霍夫发现,在壁周围皆处于同一温度下的腔体内部建立的热辐射只与温度有关而与壁材料无关.这称为腔辐射或黑体辐射.1884年,玻尔兹曼根据光和辐射都是电磁波,推出辐射压力为辐射能量密度的三分之一,再利用热力学,推出辐射能量密度与绝对温度四次方成正比的斯特藩-玻尔兹曼定律,因为1879年斯特藩已从实验上发现了这条定律.1893年,维恩进一步考虑一个收缩的球形腔,并利用运动界面反射电磁波时的多普勒效应,推导出腔辐射的能量密度对辐射频率的分布函数为频率的立方乘一个只依赖于频率与温度之商的函数.这结果包含了维恩位移定律,也包含了斯特藩-玻尔兹曼定律.但他不能从理论上定出这函数的具体形式,转而从事实验.他与陆末1895年发明了在腔壁上开一小孔,这对腔内辐射状态影响不大,但对不同频率的腔辐射强度可作定量的测量.根据实验结果,维恩于1896年提出半经验公式,将上述函数取为指数衰减型,称为维恩公式.但这只能与实验的高频部分数据弥合得好,到1899年,陆末和普林斯海姆的实验与维恩公式在稍低一些的频率处偏差已很显著.1900年,瑞利根据辐射为电磁振动,将一立方腔电磁场作模分解后,利用经典理论的能量均分定理,每个振动模具有玻尔兹曼常数乘其绝对温度的平均能量,这样得到腔辐射的能量密度对辐射频率的理论分布函数,称为瑞利公式.瑞利公式与当时鲁本斯和库尔鲍姆的实验在低频部分符合得很好,但在高频肯定不能用,因它在对频率积分时发散.普朗克得知这新实验结果验证了瑞利公式在低频的正确,又早知维恩公式在高频的与实验弥合很好,便赶紧做瑞利公式和维恩公式的插值公式.他分别从瑞利公式和维恩公式求出其能量涨落,将二者相加作为插值公式的能量涨落,这样求出插值公式,称为普朗克公式.这插值公式在低频和高频两极限情况下分别退化为瑞利公式和维恩公式,在中间的频率处它与实验数据也符合得很好.1900年10月19日普朗克在德国物理学会上就此作了报告.为给这有牢固的实验基础的普朗克公式一个理论说明,普朗克同瑞利一样用模分解,但不得不放弃能量均分定理而引入一个崭新的能量量子(简称量子)的概念,假设辐射能量在吸收或发射时是以不可分割的整个能量量子进行.量子的能量为该模的振动频率乘以一个固定的常量,现称为普朗克常量.采用这个量子假设,普朗克便能根据统计热力学推导出一个振动模(以后简称振子)的平均能量的量子公式,这样恰好导出他前面凑出的普朗克公式.普朗克于1900年12月14日在德国物理学会上报告了这个理论推导,以及从辐射实验定出的普朗克常量和玻尔兹曼常量的具体数值.这日便成为量子理论(一般为含有普朗克常量的理论的通称)的生日.2　老量子论的兴与衰对黑体辐射这个简单体系的精确而深入的研究,不意却给物理学带来革新.普朗克的量子假设突破了经典理论中能量转移的连续进行方式,先是不怎么引人注意,直到1905年爱因斯坦才将量子假设用来解释光电效应.根据能量守恒,爱因斯坦写下光量子能量等于光电子动能加逸出功,便很好地解释了勒纳在1902年从实验得到的两条结论,即:(1)产生光电子需要起码的频率,而更高频率的光使光电子的动能增加;(2)光强对光电子的动能无影响,只影响光电子的数目.1907年,爱因斯坦又用振子的平均能量的量子公式解释了金钢石、石墨、硼和硅的室温摩尔热容比从经典理论的能量均分定理导出的杜隆-珀蒂(1819)定律所给者偏低,但随温度升高有所改善的老问题.这些工作解释了经典理论不能解释的实验现象,使量子论为大家注意.德拜1910年从辐射振子的能量,据普朗克假设,只能取整数个量子出发,用玻尔兹曼因子作权重直接导出振子的平均能量的量子公式,给普朗克公式添一个简单证明.他在1912年对固体中振动模的分布作简单近似(通称德拜近似)后,用这公式解释了固体的低温热容行为,发展了低温热容的量子理论.但量子论的更系统的运用和发展则是与分子和原子光谱的实验紧密联系的.光谱是复杂的,包含很多条频率不同的谱线.康维1907年想象不同频率的谱线是由大量不同原子或分子产生的,每次吸收或发射只牵涉到某一条一定频率的谱线和一定状态的原子或分子.里兹1908年整理谱线频率,发现其中有如下的组合规则,即某些谱线的频率为另外两条谱线频率之和,因而谱线频率皆可表示为两个光谱项之差.反过来看,并不是任意两个光谱项之差都是谱线,有所谓选择规则需要满足.1912年,Bjerrum将量子概念用到氯化氢和溴化氢气体的红外吸收带时,错误地将分子的转动模———简称转子———的能量量子取为普朗克常量乘转动频率.埃伦菲斯特于1913年对此作了改进,取转子的能量子为普朗克常量乘转动频率的一半.理由是转子只有动能,不像振子还有位能.他发现转子的量子化条件为角动量须为普朗克常量除以圆周弧度的整数倍.弗兰克和赫兹于1913年用电子撞击气体原子,发现能量转移是依不同原子按一定的分立的数量间断地进行,表明原子的确有分立的能级,但不是等间隔的.1913年,玻尔将量子概念用以解释氢原子光谱时,就是利用卢瑟福根据其α粒子散射实验(1911)而建立的核原子模型,用经典力学处理电子绕原子核的圆周轨道,加上角动量等于普朗克常量除以圆周弧度的整数倍的量子化条件,这样定出氢原子的能级,计算结果与氢原子光谱项符合一致.原子从能量高的能级跃迁到低的能级时,从能量守恒得知发射的光量子频率为能级间能量差除以普朗克常量,这称为普朗克-玻尔关系.原子从低能级跃迁到高能级时吸收光量子的频率也由普朗克-玻尔关系决定.量子理论用能级间的跃迁解释光谱,能量转移以整个能量子进行,与按经典电动力学预计的电子应连续地辐射能量而缩小轨道半径的行为迥然不同.玻尔关于氢原子能级的工作,显示了量子理论的巨大威力,使原子的稳定和光谱可以理解,成为后来称为老量子论的典范.1913年,斯塔克发现外加电场时引起氢原子光谱线的分裂,而外加磁场时引起原子光谱线的分裂早在1896年已被塞曼发现;能级分裂这样的小变动可用微扰法处理,但这时玻尔的只考虑圆周轨道便显得过于简单化.1915—1916年,索末菲与威耳逊独立地对多自由度体系的量子化条件给出较为一般的表达,即取每个正则坐标和正则动量的作用量积分分别为普朗克常量的整数倍.对一个可用分离变量法处理的多自由度体系,上述表达从埃伦菲斯特的寝渐不变量原理得到支持.寝渐不变量是指那些在非常缓慢的外界扰动下保持其值不变的量,所以是适宜取作量子化的量.比如一个单摆在往复摆动而绳长非常缓慢地缩短时,容易用经典力学证明单摆的振动总能除以频率是个寝渐不变量,而这正是普朗克假设所选用的.对周期运动而言,用经典力学可以证明每个自由度的作用量积分都是一个寝渐不变量.埃伦菲斯特的关于角动量的量子化条件,如将等式两侧均乘以圆周弧度后,即是转动的作用量积分,所以也符合索末菲或威耳逊的一般表达.索末菲就是用这样的量子化条件认真地考虑了氢原子中电子的三维运动,引入了三个量子数,包含了一些椭圆轨道,得到的能级与玻尔所得的相同,但多数能级是由量子数的不同组合而简并即能级重合在一起.索末菲甚至还考虑到电子的相对论性运动,这样使简并有所分裂,所得的更细致的能级,说来也巧,与氢原子光谱的精细结构符合得很好.不久,史瓦西和依普斯坦1916年在有外加电场的情况下,用类似的方法引入三个量子数,得到能级在电场中分裂,解释了氢原子的斯塔克效应.尽管老量子论的这些发展对原子能级和光谱有重大推进作用,但考虑到碱金属原子光谱的双重结构,索末菲于1920年发现需要引进第四个量子数,以描述后来到1925年乌伦贝克和古德斯密特才正确理解是电子的一个新自由度,名为自旋,它具有半个单位的角动量却具有一整个单位的磁矩,不能用经典力学描述.用四个量子数描述电子的运动,不仅解释了碱金属原子光谱的双重结构和碱土金属原子光谱的三重结构,还按角动量平方的一定修正规则,解释了所谓反常塞曼效应(指原子光谱的双重或三重结构在外加磁场较弱时引起的能级分裂)中从实验数据总结出的朗德分裂因子.并且,泡利1925年据此提出不相容原理,即原子内不可能有两个电子具有完全相同的四个量子数,从而解释了元素周期表的壳层电子结构.这预示量子理论在化学方面的光明前景,同时也指出老量子论的不足之处;量子理论必须改革,才能继续发展.3　第一条通向量子力学的路———对应原理我们注意:先用经典力学求出普遍的运动状态,后加量子化条件以抛弃其绝大部分而只选留少数满足量子化条件的所谓量子态,将得到的公式通过一定的修正后再与物理实际联系,是老量子论的特点.具体与光谱线联系则是按如下的玻尔对应原理(1918)进行.玻尔注意到,当量子数大时按普朗克-玻尔关系所给的跃迁频率与轨道的经典频率的倍频相当;所以,他进一步假设,不受量子数大的限制,量子态间跃迁所发射的光谱线的强度及其极化方向,也与轨道的经典振幅的傅里叶展开中的倍频系数相关.克拉默斯(1919)与科塞尔和索末菲(1919)分别就谱线强度与选择规则(即谱线强度为零或否)给这对应原理许多验证.利用玻尔对应原理,克拉默斯(1924)与克拉默斯和海森伯(1925)将拉登堡(1921)的关于色散电子数的结果修正而给出表达原子的极化率,即原子的感生电偶极矩与外辐射电场之比的量子公式(克拉默斯-海森伯色散公式).1924年,玻恩给出了振动系统受微扰的普遍处理,包括上述色散问题在内.玻恩指出,对于任何物理量,经典的与量子的量有普遍的对应关系,即经典力学中对作用量的微商对应于量子理论中对作用量的差商,亦即对量子数的差商再除以普朗克常量.玻恩并第一次用量子力学作此文标题,这是由于玻恩那时认为严格表述玻尔对应原理即能导致量子力学.果然,一年内量子力学沿这条路以两个数学形式出现.(1)矩阵力学.先说矩阵力学这个形式.首先提出用方阵式的数学来表达物理量的人是24岁的海森伯.他22岁在索末菲那儿拿到博士学位后,去哥本哈根玻尔处,参与克拉默斯-海森伯色散公式的研究.他不满先用经典力学而后加量子化条件的做法,而信念物理学中只应引入可观测量,如光谱线的频率和强度或振幅,振幅的平方给出其强度.因为谱线依赖于高低两个能级,海森伯直接引入振幅方阵,含有频率乘时间的相因子.海森伯考虑恢复力带有振幅的平方项的非简谐振子,为使时间因子得以一致,他建议对振幅方阵的平方用行乘列的乘法,但他那时还不知道这是数学中矩阵的乘法规则.他在度假中开始了这个大胆尝试,将运动方程连同量子化条件一道来求解.量子化条件是用索末菲的作用积分按玻恩的严格表述对应原理用振幅方阵写出.他得到一些初步结果,写成他一人署名的文章(1925).这时海森伯在玻恩教授系里工作.他请玻恩看他的文章并请假去剑桥被邀请作一个月演讲.玻恩学过矩阵的数学课,看出海森伯从里兹组合规则凑出的乘法规则恰是矩阵的乘法规则.玻恩考虑一维的保守系,用正则坐标和动量与哈密顿正则运动方程,但物理量皆用矩阵表示.对保守系有能量守恒,因此哈密顿量必须是对角矩阵.玻恩知道,矩阵的乘法一般是不满足交换律的.从索末菲的作用积分的量子化条件玻恩对应出正则动量与正则坐标的对易矩阵(即颠倒秩序的两个乘积之差)的对角元均相等,为虚数,并含有普朗克常量除以圆周弧度.他先猜想这对易矩阵的非对角元全为零.他请助手约当参加合作,约当几日内便从正则运动方程证明出这个对易矩阵对时间的导数为零,所以是个对角矩阵,如玻恩所猜想.这样,矩阵力学有了自己的量子化条件,简称为对易关系.以玻恩和约当二人署名的这篇文章给出简谐振子的矩阵力学处理和对电磁场的量子化处理,再次证明了关于黑体辐射的普朗克公式.在1925年10月底,玻恩应邀去MIT讲课之前,玻恩与约当紧张合作,并于这时在哥本哈根的海森伯通讯,完成了以玻恩、海森伯和约当三人署名的文章.在这篇文章中,对应经典力学中原则上可用正则变换求解,给出矩阵力学中原则上可用相似变换求哈密顿矩阵的对角化;发展了矩阵力学中逐步近似的微扰理论,包括直接导出克拉默斯-海森伯色散公式;把对易关系推广到多个自由度.这三篇文章奠定了矩阵力学.三人署名的第三篇,印出时间为1926年.注意,由于对易关系的出现,海森伯的矩阵不可能是有限的行或列,这表明体系有无穷多个能级.对无穷行列的矩阵,乘法不一定满足结合律.但如每行只有有限个元不等于零时,则可证明结合律成立.(2)狄拉克的q-数.海森伯1925年夏在剑桥演讲时,比他小一岁的研究生狄拉克在座.狄拉克注意到海森伯方阵的乘法不满足交换律,乃抽象地引入乘法不满足交换律的数,他称为q-数,以与满足乘法交换律的他称为c-数的数相区别.在经典力学中,运动方程和正则变量都可以用泊松符号形式表达,所以他首先寻求两个q-数的泊松符号该如何表达.从泊松符号的代数性质出发,他发现两个q-数的泊松符号与其对易子(即颠倒秩序的两个乘积之差)成正比,比例系数可从经典力学正则坐标和正则动量的泊松符号的特殊值来定.这样,对多个自由度的体系,他得到相应的对易关系,结果与矩阵力学的第三篇文章所得的一样,只是q-数比矩阵更抽象.从数学上讲,人们可以把海森伯、玻恩和约当的矩阵看作是狄拉克的q-数即非对易代数的一种实现,或把后者看作前者的符号化,则这两种表达形式便统一了.运动方程用q-数表达,其形式非常简单:对时间的导数即正比于与哈密顿量的对易子.这对于正则坐标或正则动量或它们的任意组合都适用,还可推广到任何非经典的自由度去.1926年,狄拉克用q-数与泡利用矩阵处理氢原子能级,差不多同时都得到成功.泡利还用矩阵处理了史塔克效应.同年海森伯和约当用矩阵处理带自旋的电子的塞曼效应,直接得到朗德的g-因子公式,而狄拉克用q-数处理康普顿效应(参见下节),得到与实验符合的反冲电子的角分布和散射的X射线.狄拉克还发展了含时间的微扰论,用到光谱跃迁,解决了自发跃迁概率的计算,验证了爱因斯坦关于辐射跃迁的比例系数间的关系式(也参见下节).4　第二条通向量子力学的路———波粒二象性爱因斯坦于1916—1917年从分子与辐射间的动平衡角度给普朗克公式一个新的证明.这证明想象丰富,富于启发.考虑分子高低两个能级和其间的相应跃迁频率的辐射.爱因斯坦引入如下三种跃迁机制:即自发辐射,吸收和诱发辐射,并假设在后两种机制中,其单位时间的跃迁几率与辐射能量密度对频率的分布函数成正比.在热平衡时,注意处于两能级的原子数分别与相应的玻尔兹曼因子成正比,则从能量转移的细致平衡容易导出普朗克公式和上述三种跃迁的比例系数间两个关系式(这关系式在量子力学建立后都得到特别是狄拉克的理论验证).根据狭义相对论,爱因斯坦还想象辐射量子不仅具有能量,而且具有单方向的动量,分子在吸收或发射辐射时,虽然总动量守恒,但辐射与分子间有动量转移.根据上述三种跃迁,利用他在布朗运动中处理随机过程和在电磁理论中运用参考系变换的优势,他计算了(只需准确到分子速度的一次方)包括上述三种跃迁的总平均阻力使分子速度的涨落减少的效应和由于上述三种跃迁都是随机过程而使分子速度的涨落增加的效应,在分子的玻尔兹曼分布与辐射的普朗克分布间的热平衡时,它们恰好抵消,而维持分子速度的麦克斯韦分布不变.这个具有能量和单方向动量的量子的想象,后来被康普顿引用来解释X射线散射实验中的康普顿效应,并赐名为光子.康普顿效应指1922年康普顿所发现的随散射角增大而散射的X射线的波长也稍有增大的现象,康普顿便是用具有能量和单方向动量的光子,与散射体中的自由电子间的弹性碰撞(满足能量守恒与动量守恒)来解释的.玻色1924年认为辐射是全同的光子的集合,又一次从统计热力学推导出普朗克公式,这工作直接启发了玻色-爱因斯坦统计.后来,更直观的实验,博特和盖革(1924—1925)或康普顿和西蒙(1925)用符合计数或用云雾室显示反冲电子和散射的X射线,确凿地证实了电磁波-光子的这种量子的波粒二象性.理论上受电磁波-光子的波粒二象性的启发,德布罗意于1923—1924年反向思维而推广到包括电子在内的任何粒子,都设想具有与其能量和动量相应的波的性质,称为德布罗意波.粒子的速度为德布罗意波的群速,德布罗意波的相速与群速的乘积为光速的平方,光子的德布罗意波即是电磁波.他注意到几何光学中的费马原理与动力学中的最小作用原理相似之处,他指出玻尔对氢原子中电子圆周轨道的量子化条件可以理解为轨道周长须为电子的德布罗意波长的整数倍.但到1926年,他仍只停留在相对论性的自由粒子的情况下,写出克莱因-戈尔登方程.实验证实电子的德布罗意波则是1926年薛定谔发现波动力学以后的事,即戴维孙和革末(1927)与G.P.汤姆孙(1927)分别独立做的电子衍射实验.1926年波动力学的发现,即是沿波粒二象性的另一端达到的.据说德拜在例行的讨论会上建议薛定谔去弄清楚德布罗意的一系列文章,作个报告,特别提到是波就要满足波动方程.薛定谔先也得到和德布罗意一样的相对论性波动方程,觉得与玻尔的氢原子能级不相符后,却进一步作非相对论近似,写出了薛定谔方程.薛定谔认为电子本质是波,应该用波动力学描述电子的运动;经典力学应为波动力学的近似,有如几何光学是波动光学的近似一样,而量子效应即相当于干涉效应.怀着这种信念,对自由电子的德布罗意波,按波动光学惯例用相位的指数函数表示其波函数时,容易发现电子的动量和非相对性能量,可用对空间和时间坐标的偏微分算符乘以普朗克常量除以圆周弧度和适当虚数表示,即用微分算符作用到德布罗意波函数时,将得到相应的物理量的值乘上该波函数.据此薛定谔推广到由普遍的哈密顿量描述的情况,维持微分算符表达形式不变;因为这样,如果这时忽略普朗克常量,波函数的相位便满足熟知的哈密顿-雅可比方程,达到经典力学近似.对于像氢原子中电子的三维运动保守系,哈密顿量不显含时间而有能量守恒.薛定谔方程简化为定态的波动方程,不含时间变量而出现能量参量.定态的波动方程是个齐次方程,由于物理原因波函数必须满足一些条件,如处处单值有限,和(特别对于束缚态而言)波函数在无穷远趋于零等.要求定态的波动方程有不恒等于零的满足上述物理条件的解,这样就定出能量参量的本征值.对于电子的三维运动,由于出现拉普拉斯动能算符,用分离变数法将偏微分方程化为常微分方程时要引入分离常数,这些常数也由波函数满足的条件定出其本征值.一连四篇文章,薛定谔以本征值量子化为题,处理了氢原子能级问题,得到与玻尔和索末菲相同的结果,也发展了微扰理论处理史塔克效应,和含时间的微扰理论得到克拉默斯-海森伯色散公式.在另外一篇文章中,薛定谔出乎他初始意料之外地证明了波动力学与矩阵力学在数学上的等价.在波动力学中,正则动量是用微分算符表示的,正则坐标则用乘法算符表示,其间恰好满足矩阵力学的对易关系.薛定谔指出并证明,某物理量的海森伯矩阵是相应算符夹在两个带能量相位因子定态波函数间的积分,算符左边的波函数连同相位因子须取复数共轭,而哈密顿算符的相应积分即是海森伯能量对角矩阵.反过来,创建矩阵力学的玻恩给波函数以概率解释.当玻恩去美国讲学时,遇到数学家N. Wiener,便与之合作探讨能量的连续谱情况,意欲处理非周期性运动.他们不像狄拉克敢创造不正规函数,感到用矩阵有困难,需要用算符,但没想到像薛定谔的微分算符那么简单.在波动力学问世后,玻恩当即采用波动力学以处理碰撞问题,如用α粒子轰击原子核的卢瑟福散射实验.从薛定谔方程容易导出一个密度和流密度间的连续方程,玻恩解释为概率守恒.薛定谔波函数的绝对值平方为粒子那时在该处出现的概率,或相对概率,如果波函数没有规一化的话.对能量为连续谱中某值的定态方程,玻恩利用他对电磁光学的优势,取波函数为入射波与散射波函数之和这样的解.在两篇文章中玻恩发展了碰撞理论和计算微分散射截面的近似方法,现称为玻恩近似.Wentzel(1926)用玻恩近似公式计算了屏蔽库仑场的散射,结果与经典理论公式一样,与卢瑟福实验一致,支持了玻恩的统计解释.后来狄拉克用他擅长的数学表达方式也给出在动量表象中的玻恩近似公式.。

The analytic structure of an algebraic quantum group

a rXiv:funct-a n /9771v128J u l1997The analytic structure of an algebraic quantum group J.Kustermans 1Institut for Matematik og Datalogi Odense Universitet Campusvej 555230Odense M Denmark July 1997Abstract In [14],Van Daele introduced the notion of an algebraic quantum group.We proved in [5]and [9]that such algebraic quantum groups give rise to C ∗-algebraic quantum groups according to Masuda,Nakagami &Woronowicz.In this paper,we will pull down the analytic structure of these C ∗-algebraic quantum groups to the algebraic quantum group.Introduction Van Daele introduced the notion of an algebraic quantum group in [14].It is essentially a Multiplier Hopf-∗-algebra with a non-zero left invariant functional on it.He proved that these algebaic quantum groups form a well-behaved category :•The left and right invariant functionals are unique up to a scalar.•The left and right invariant functionals are faithful.•Each algebraic quantum group gives rise to a dual algebraic quantum group.•The dual of the dual is isomorphic to the original algebraic quantum group.This category of algebraic quantum groups contains the compact and discrete quantum groups.It is also closed under the double construction of Drinfel’d so it contains also non-compact non-discrete quantum groups.Most of the groups and quantum groups will however not belong to this category.It is nevertheless worth while to study these algebraic quantum groups :•The category contains interesting examples.•The theoretical framework is not technically complicated.In a sense,you have only to worry about essential quantum group problems.Not over C ∗-algebraic complications.At the moment,Masuda,Nakagami&Woronowicz are working on a possible deﬁnition of a C∗-algebraic quantum group.To get an idea of this deﬁnition,we refer to[10].You can also get aﬂavour of it in[5] and[9].This deﬁnition is technically rather involved and C∗-algebraic quantum groups in this scheme have a rich analytical sructure.We proved in[5]and[9]that an algebraic quantum group with a positive left invariant functional gives rise to such C∗-algebraic quantum groups.In this paper,we will pull down this analytic structure to the algebra level.It shows that such algebraic quantum groups are truely algebraic versions of C∗-algebraic quantum groups according to Masuda,Nakagami&Woronowicz.In theﬁrst section,we give an overview of the theory of algebraic quantum groups as can be found in[14]. The second and third section are essentially only intended to assure that an obvious theory of analytic one-parameter groups worksﬁne on this algebra level.The most important results can be found in section4where we prove that the analytic objects of the C∗-algebraic quantum groups are of an algebraic nature.In the last section,we connect the analytic objects of the dual to the analytic objects of the original algebraic quantum group.We end this section with some conventions and notations.Every algebra in this paper is an associative algebra over the complex numbers(not necessarily unital).A homomorphism between algebras is by deﬁnition a linear multiplicative mapping.A∗-homomorphism between∗-algebras is a homomorphism which preserves the∗-operation.If V is a vector space,then L(V)denotes the set of linear mappings from V into V.If V,W are two vector spaces,V⊙W denotes the algebraic tensor product.Theﬂip from V⊙W to W⊙V will be denoted byχ.We will also use the symbol⊙to denote the algebraic tensor product of two linear mappings.We will always use the minimal tensor product between two C∗-algebras and we will use the symbol⊗for it.This symbol will also be used to denote the completed tensor product of two mappings which are suﬃciently continuous.If z is a complex number,then S(z)will denote the following horizontal strip in the complex plane: S(z)={y∈will be denoted by the same symbol as the original mapping.Of course,we have similar deﬁnitions and results for antimultiplicative mappings.If we work in an algebraic setting,we will always use this form of non-degeneracy as opposed to the non degeneracy of∗-homomorphisms between C∗-algebras!For a linear functionalωon a non-degenerate∗-algebra A and any a∈M(A)we deﬁne the linear functionalsωa and aωon A such that(aω)(x)=ω(xa)and(ωa)(x)=ω(ax)for every x∈A.You canﬁnd some more information about non-degenerate algebras in the appendix of[17].Letωbe a linear functional on a∗-algebra A,then:1.ωis called positive if and only ifω(a∗a)is positive for every a∈A.2.Ifωis positive,thenωis called faithful if and only if for every a∈A,we have thatω(a∗a)=0⇒a=0.Consider a positive linear functionalωon a∗-algebra A.Let H be a Hilbert-space andΛa linear mapping from A into H such that•Λ(A)is dense in H.•We have for all a,b∈A thatω(a∗a)= Λ(a),Λ(b) .Then we call(H,Λ)a GNS-pair forω.Such a GNS-pair always exist and it is unique up to a unitary.We have now gathered the necessary information to understand the following deﬁnitionDeﬁnition1.1Consider a non-degenerate∗-algebra A and a non-degenerate∗-homomorphism∆fromA into M(A⊙A)such that1.(∆⊙ι)∆=(ι⊙∆)∆.2.The linear mappings T1,T2from A⊙A into M(A⊙A)such thatT1(a⊗b)=∆(a)(b⊗1)and T2(a⊗b)=∆(a)(1⊗b) for all a,b∈A,are bijections from A⊙A to A⊙A.Then we call(A,∆)a Multiplier Hopf∗-algebra.In[17],A.Van Daele proves the existence of a unique non-zero∗-homomorphismεfrom A to•(ω⊙ι)(∆(a))b=(ω⊙ι)(∆(a)(1⊗b))•b(ω⊙ι)(∆(a))=(ω⊙ι)((1⊗b)∆(a))for every b∈A.In a similar way,the multiplier(ι⊙ω)∆(a)is deﬁned.Letωbe a linear functional on A.We callωleft invariant(with respect to(A,∆)),if and only if (ι⊙ω)∆(a)=ω(a)1for every a∈A.Right invariance is deﬁned in a similar way.Deﬁnition1.2Consider a Multiplier Hopf∗-algebra(A,∆)such that there exists a non-zero positive linear functionalϕon A which is left invariant.Then we call(A,∆)an algebraic quantum group.For the rest of this paper,we willﬁx an algebraic quantum group(A,∆)together with a non-zero left invariant positive linear functionalϕon it.An important feature of such an algebraic quantum group is the faithfulness and uniqueness of left invariant functionals:1.Consider a left invariant linear functionalωon A,then there exists a unique element c∈It is also possible to introduce the modular function of our algebraic quantum group.This is an invertible elementδin M(A)such that(ϕ⊙ι)(∆(a)(1⊗b))=ϕ(a)δbfor every a,b∈A.Concerning the right invariant functional,we have that(ι⊙ψ)(∆(a)(b⊗1))=ψ(a)δ−1bfor every a,b∈A.This modular function is,like in the classical group case,a one dimensional(generally unbounded) corepresentation of our algebraic quantum group:∆(δ)=δ⊙δε(δ)=1S(δ)=δ−1.As in the classical case,we can relate the left invariant functional to our right invariant functional via the modular function:we have for every a∈A thatϕ(S(a))=ϕ(aδ)=µϕ(δa).If we apply this equality two times and use the fact that S(δ)=δ−1,we get thatϕ(S2(a))=ϕ(δ−1aδ) for every a∈A.Not surprisingly,we have also thatρ(δ)=ρ′(δ)=µ−1δ.Another connection betweenρandρ′is given by the equalityρ′(a)=δρ(a)δ−1for all a∈A.We have also a property which says,loosely speaking,that every element of A has compact support(see e.g.[6]for a proof):Consider a1,...,a n∈A.Then there exists an element c in A such that c a i=a i c=a i for every i∈{1,...,n}.In a last part,we are going to say something about duality.We deﬁne the subspaceˆA of A′as follows:ˆA={ϕa|a∈A}={aϕ|a∈A}.Like in the theory of Hopf∗-algebras,we turnˆA into a non-degenerate∗-algebra:1.For everyω1,ω2∈ˆA and a∈A,we have that(ω1ω2)(a)=(ω1⊙ω2)(∆(a)).2.For everyω∈ˆA and a∈A,we have thatω∗(a)=2.We have for everyθ∈M(ˆA)and a∈A thatθ∗(a)=C into the set of homomorphisms from A into A such that the following properties hold:1.We have for every t∈I R thatαt is a∗-automorphism on A.2.We have for every s,t∈I R thatαs+t=αsαt.3.We have for every t∈I R thatαt is relatively invariant underϕ64.Consider a∈A andω∈ˆA.Then the function C:z→ω(αz(a))is analytic.Then we callαan analytic one-parameter group on(A,∆).Except for the last proposition in this section,we willﬁx an analytic one-parameter groupαon(A,∆). We will prove the basic properties ofα.Result2.2Consider z∈z(a∗)for a∈A.Proof:Choose a∈A.Take b∈A.Then we have two analytic functionsC:u→u(a∗)b∗)and C:u→ϕ(bαu(a))These functions are equal on the real axis:we have for all t∈I R that(a∗)b∗)=tϕ(αz(a∗)∗)=ϕ(bαz(a)).So the faithfulness ofϕimplies thatαIn the next result,we extend the group character ofαto the whole complex plane.Result2.3Consider y,z∈C→C→C→C→Corollary2.4Consider z∈We want to useαto deﬁne a positive injective operator in the GNS-space H ofϕ.In order to do so,we will need the following results.Result2.5There exists a unique strictly positive numberλsuch thatϕαt=λtϕfor t∈I R.Proof:By assumption,there exists for every t∈I R a strictly positive elementλt such thatϕαt=λtϕ. It is then clear thatλ0=1and thatλsλt=λs+t for every s,t∈I R.Now there exist a,b∈A such thatϕ(ba)=0.We have that the function I R→.So we see that the function[−t0,t0]→I R+0:t→λt is continuous.ϕ(α−t(b)a)Therefore the function I R→I R+0:t→λt is continuous.We get from this all the existence of a strictly positive numberλsuch thatλt=λt for t∈I R.So ϕαt=λtϕfor t∈I R.C.Proof:Choose a,b∈A.Then we have two analytic functionsC:u→λuϕ(aα−u(b))and C:u→ϕ(αu(a)b)These functions are equal on the real line:we have for all t∈I R thatλtϕ(aα−t(b))=ϕ(αt(aα−t(b)))=ϕ(αt(a)b)So they must be equal on the whole complex plane.We have in particular thatϕ(αz(a)b)=λzϕ(aα−z(b)),henceϕ αz(aα−z(b)) =λzϕ(aα−z(b)). Because A=Aα−z(A),we infer from this thatϕαz=λzϕ.2Λ(αt(a))for a∈A and t∈I R.Proof:We can deﬁne a unitary group representation u from I R on H such that u tΛ(a)=λ−t2 Λ(αt(a)),Λ(b)=λ−tC: u tΛ(a),Λ(b) is continuous.Because u is bounded andΛ(A)is dense in H,we conclude that the function I R→We want to show that every P iz is deﬁned onΛ(A)and that the formula in the above deﬁnition has its obvious generalization to P iz.First we will need a lemma for this.8Lemma2.8Consider a∈A.Then the function C:z→λ−Re zϕ(αz(a)∗αz(a))is bounded on horizontal strips.Proof:Take r∈I R.We will prove that the function above is bounded on the horizontal strip S(ri). We have for every t∈I R thatϕ(αti(a)∗αti(a))=ϕ(α−ti(a∗)αti(a))=λ−itϕ(a∗α2ti(a))which implies that the function I R→I R+:t→ϕ(αti(a)∗αti(a))is continuous.So there exists M∈I R+such thatϕ(αti(a)∗αti(a))≤M for t∈[0,r].Hence we get for every z∈Result2.9Consider z∈2Λ(αz(a))for a∈A.Proof:Take v∈H and deﬁne the function f from C such that f(u)= λ−u C.Deﬁne also for every b∈A the function f b:C such that f b(u)= λ−u C. Then f b(u)=λ−u C which implies that f b is analytic.Choose y∈C around y.By the previous lemma,we get the existence of a positive number M such thatλ−Re uϕ(αu(a)∗αu(a))≤M2for u∈B.This implies easily that λ−u2Λ(αu(a)),Λ(b n) − λ−uC.It is also clear that f(t)= P itΛ(a),v for t∈R.This implies thatΛ(a)∈D(P iz)and that P izΛ(a)=λ−zResult2.10Consider z∈Proof:It is clear that alsoα−i=β−i.Now there exist strictly positive numbersλandµsuch thatϕαt=λtϕandϕβt=µtϕfor t∈I R. Then there exist also positive injective operators P and Q in H such that P itΛ(a)=λ−t2Λ(βt(a))for t∈I R and a∈A.We know thatΛ(A)is a core for both P and Q.We have moreover for every a∈A thatµi2λi2µi2QΛ(a)So we get thatµi2Q.It is clear thatµiλiC thatλ−z2Λ(βz(a))which by the faitfulness ofϕimplies thatλ−z2βz(a).Now we see thatµλα2is multiplicative.Because alsoα2is multiplicative,thisimplies thatµ3Analytic unitary representationsThis section contains the same basic ideas as the previous one.We only apply them in a diﬀerentsituation.We will use the results in this section to introduce complex powers of the modular function of an algebraic quantum group in the next section.We will againﬁx an algebraic quantum group(A,∆)with a positive left Haar functionalϕon it.Let (H,∆)be a GNS-pair forϕ.Let us start of with a deﬁnition.Deﬁnition3.1Consider a function u fromC→zfor every z∈C→ϕ(u−C→ϕ(u−ϕ(u−t a∗)=ϕ(a u∗−t)=ϕ(a u t)for all t∈I R.So they must be equal on the whole complex plane.We have in particular that z a∗)=ϕ(a u z)which implies thatϕ(a u z)=ϕ(a u∗−z=u z.Result3.3Consider y,z∈C→C→C→C→Corollary3.4Consider z∈C: v tΛ(a),Λ(b) is continuous.Because v is bounded andΛ(A)is dense in H,we conclude that the function I R→We want to show that every P iz is deﬁned onΛ(A)and that the formula in the above deﬁnition has its obvious generalization to P iz.First we will need a lemma for this.Lemma3.6Consider a∈A.Then function C:z→ϕ((u z a)∗(u z a))is bounded on horizontal strips.Proof:Take r∈I R.We will prove that the function above is bounded on the horizontal strip S(ri). We have for every t∈I R thatϕ((u ti a)∗(u ti a))=ϕ(a∗u ti u ti a)=ϕ(a∗u2ti a)which implies that the function I R→I R+:t→ϕ((u ti a)∗(u ti a))is continuous.So there exists M∈I R+such thatϕ((u ti a)∗(u ti a))≤M for t∈[0,r].11Hence we get for every z∈Result3.7Consider z∈C into C. Deﬁne also for every b∈A the function f b:C such that f b(c)= Λ(u c a),Λ(b) for c∈C which implies that f b is analytic.Choose y∈C around y.By the previous lemma,we get the existence of a positive number M such thatϕ((u c a)∗(u c a))≤M2 for c∈B.This implies easily that Λ(u c a) ≤M for c∈BNow there exists a sequence(b n)∞n=1in A such that(Λ(b n))∞n=1converges to v.We have for every c∈B and n∈I N that|f(c)−f b n(c)|=| Λ(u c a),Λ(b n) − Λ(u c a),v |≤M Λ(b n)−vThis implies that(f b n)∞n=1converges uniformly to f on B,so f is analytic on B.This implies that f is analytic in y.So we see that f is analytic onResult3.8Consider z∈C thatΛ(u z a)=Λ(v z a)which by the faitfulness ofϕimplies that u z a=v z a.4The analytic structure of an algebraic quantum groupIn[5]and[9],we proved that every algebraic quantum group gives rise to a reduced and universal C∗-algebraic quantum group in the sense of Masuda,Nakagami and Woronowicz.These C∗-algebraic quantum groups have a very rich(complicated?)analytic structure.We show in this section that this analytic structure can be completely pulled down to the algebraic level.Consider an algebraic quantum group(A,∆)and letϕbe a positive left Haar functional of(A,∆).Let (H,Λ)be a GNS-pair forϕ.In[9],we constructed a universal C∗-algebraic quantum group(A u,∆u)out of(A,∆).We denote the canonical embedding of A into A u byπu.Soπu is an injective∗-homomorphism into A u such thatπu(A)is dense in A u.We have also for every a∈A and x∈A⊙A that∆u(πu(a))(πu⊙πu)(x)=(πu⊙πu)(∆(a)x)and(πu⊙πu)(x)∆u(πu(a))=(πu⊙πu)(x∆(a)) As a rule,we will give objects associated with A u a subscript‘u’.So the left Haar weight on(A u,∆u) will be denoted byϕu and its modular group byσu.The scaling group of(A u,∆u)will be denoted byτu,the anti-unitary antipode by R u.The modular group of the right Haar weightϕu R u will be denoted byσ′u.We will also need the co-unit on the universal C∗-algebraic quantum group(A u,∆u).Recall that we have a co-unitεon the algebraic quantum group(A,∆).This gives rise to a non-zero∗-homomorphism εu from A u intoC thatπu(a)belongs to D(δiz u)and that δiz uπu(a)belongs again toπu(A).We will use this fact to deﬁne any complex power ofδon the algebra level.The deﬁnition that we introduce in this paper is much better than the ad hoc deﬁnitions in[5]and [9]because we work here within the framework of the algebraic quantum group(A,∆).Proposition4.1There exist a unique analytic unitary representation w on A such that w−i=δ.We deﬁneδz=w−iz for z∈Proof:We know already thatδiy uπu(A)⊆πu(A)for y∈C into M(A)such thatπu(w y a)=δiy uπu(a)for y∈y for y∈C→A u:y→δiy uπu(a)is analytic.This gives us that the mapping C:y→ϕu(πu(c)∗(δiy uπu(a))πu(b))is analytic. But we have for every y∈C→Notice that we have also proven the following characterization for powers ofδ.Proposition4.2We have for every z∈C thatδy+z=δyδz3.We have for z∈z4.Consider z∈2δt2is self adjoint.We have also the following analyticity property.Result4.5We have for everyω∈ˆA that the function C:z→ω(δz)is analytic.By proposition12.2of[9],we know thatπ(δu)=δr.Hence we get thatπ(δz u)=δz r for z∈Proposition4.6We have for every z∈We want now to prove a generalization of this result for complex parameters.The idea behind this proof is completely the same but we have to be a little bit careful in this case.Recall from[9]that every element ofπu(A)is analytic with respect toσu,σ′u andτu.First we prove a little lemma.15Lemma4.8Consider z∈C into C. We have by the previous proposition for every t∈I R that(σu)t(x)belongs toπu(A),which implies that ω (σu)z((σu)t(x)) =0.Hence f(z+t)=ω((σu)z+t(x))=ω (σu)z((σu)t(x)) =0for t∈I R.This implies that f=0.In particular,ω(x)=f(0)=0.So the density ofπu(A)in A u implies thatω=0.Therefore Hahn-Banach implies that(σu)z(πu(A))is dense in A u.C and a∈A.Then(σ′u)z(πu(a))=(σu)z(πu(δiz aδ−iz)).Proof:We introduce the one-parameter representations L,R,θon A u such thatL t(x)=δit u x R t(x)=xδ−itu θt(x)=δit u xδ−itufor x∈A u and t∈I R.Then L and R commute andθt=L t R t for t∈I R.This implies that L z R z⊆θz(see e.g.proposition3.9 of[8]).We know thatπ(a)∈D(R z)and that R z(πu(a))=πu(aδ−izu).This implies that R z(πu(a))∈D(L z)andthat L z(R z(πu(a)))=πu(δiz u aδ−izu ).So we get thatπu(a)∈D(θz)and thatθz(πu(a))=πu(δiz u aδ−izu).We have moreover thatσu andθcommute and that(σ′u)t=(σu)tθt for all t∈I R.This implies again that(σu)zθz⊆(σ′u)z.Therefore(σ′u)z(πu(a))=(σu)z θz(πu(a)) =(σu)z(πu(δiz aδ−iz)).C.Then(σu)z(πu(A))=(σ′u)z(πu(A)).Now we are in a position to prove a generalization of proposition4.7.Proposition4.11Consider z∈C.Consider elements a1,...,a m,b1,...,b m∈A and p1,...,p n,q1,...,q n∈A such that the equalitym i=1∆(a i)(1⊗b i)= n j=1p j⊗q j holds.Thenmi=1∆u(πu(a i))(1⊗πu(b i))=nj=1πu(p j)⊗πu(q j)We have now the following two analytic functions:andCorollary4.12We have that R u(πu(A))=πu(A).Proof:We know by theorem9.18of[9]that R u((τu)−iThese results imply that al the objects associated to the C∗-algebraic quantum group(A u,∆u)can be pulled down to the algebraic level.Theﬁrst(and typical)result is contained in the next proposition. Proposition4.13There exists a unique analytic one-parameter groupσon A such thatσ−i=ρ.We have moreover thatπu(σz(a))=(σu)z(πu(a))for every a∈A and z∈Proof:By proposition4.11,we know that(σu)z(πu(A))=πu(A)for every z∈C into the set of mappings from A into A such that πu(σz(a))=(σu)z(πu(a))for every a∈A and z∈C thatαz is a homomorphism on A.2.We have for every t∈I R thatαt is a∗-automorphism on A.3.We have for every s,t∈I R thatαs+t=αsαt.Choose z∈A.Thenπu(a)∈D((σu)z)∩Mϕu and(σu)z(πu(a))=πu(σz(a))∈Mϕu.Becauseϕu is invariant underσu,this implies by proposition2.14of[7]thatϕu(πu(a))=ϕu (σu)z(πu(a)) =ϕu πu(σz(a))which implies thatϕ(a)=ϕ(σz(a)).Take a∈A andω∈ˆA.Then there exist b,c∈A such thatω=cϕb∗.We know by proposition8.8of[9]thatπu(a)is analytic with respect toσu.This implies that the functionC→C thatϕu(πu(b)∗(σu)z(πu(a))πu(c))=ϕu(πu(b)∗πu(σz(a))πu(c))=ϕu(πu(b∗σz(a)c))=ϕ(b∗σz(a)c)=ω(σz(a))So the function C:z→ω(σz(a))is analytic.Hence we get by deﬁnition thatσis an analytic one-parameter group on A.Using proposition8.8of[9],we see thatσ−i=ρ.C thatϕσz=ϕ.There exists a canonical GNS-construction(H,Λu,π)for the weightϕu such thatπu(A)is a core forΛu andΛu(πu(a))=Λ(a)for a∈A(see deﬁnition10.2of[9]and theorem10.6of[9]).Denote the modular operator ofϕu by∇and the modular conjugation ofϕu by J(both with respect to this GNS-construction).Then these objects are completely characterized by the following two results.Result4.15Consider z∈Corollary4.16We have for every a∈A that JΛ(a)=Λ(σiBy the remarks after deﬁnition8.4of[9],we know thatπ(σu)t=(σr)tπfor t∈I R.Consequently,π(σu)z⊆(σr)zπfor z∈C and a∈A that(σr)z(πr(a))=πr(σz(a)).Completely similar to proposition4.13,we have the next results.Remember that we have proven in corollary7.3of[5]the existence of a unique strictly positive numberνsuch thatϕr(τr)t=νtϕr for t∈I R. Proposition9.19of[5],proposition8.17of[5]and proposition8.20of[7]imply thatϕr(σ′r)t=ν−tϕr for t∈I R.This gives us also thatϕr(τu)t=νtϕu and thatϕu(σu)z=ν−tϕu for every t∈I R.Proposition4.18There exists a unique analytic one-parameter groupσ′on A such thatσ′−i=ρ′. We have moreover thatπu(σ′z(a))=(σ′u)z(πu(a))for every a∈A and z∈C thatϕσ′z=ν−zϕ.Proposition4.20We have for every a∈A and z∈C.We callτthe scaling group of(A,∆).Result4.22We have for every z∈C thatπr(τz(a))=(τr)z(πr(a)).Now it is the turn of the anti-unitary antipode R u to be pulled down.This is possible thanks to corollary 4.12Deﬁnition4.24We deﬁne the mapping R from A into A such thatπu(R(a))=R u(πu(a))for a∈A. Then R is a∗-anti-automorphism on A such that R2=ι.We call R the anti-unitary antipode of(A,∆).We know by[9]that R rπ=πR u which gives us,as usual,the next result.Proposition4.25We have for every a∈A thatπr(R(a))=R r(πr(a)).19In the rest of this section,we will prove the most basic relations between the objects introduced in this section.In most cases,the proofs consist of pulling down the corresponding relations on the C∗-algebra level.We will make use of the results on the reduced C∗-algebra level because they were proved before the results on the universal level(in fact,most of the latter results depend on the results on the reduced level).First we describe the polar decompositionResult4.26We have for every z∈.2These are immediate consequences of corollary5.4of[5]and theorem5.6of[5].Corollary4.28We have the following commutation relations:1.We have for every z∈C thatτyσz=σzτy.Proof:The remarks before this result imply easily thatσtτs=τsσt for s,t∈I R.Take a,b∈A.Fix t∈I R for the moment.Then we have two analytic functionsC:u→ϕ(σ−t(b)τu(a))and C:u→νuϕ τ−u(b)σt(a)These functions are equal on the real line:we have for every s∈I R thatϕ(σ−t(b)τs(a))=ϕ bσt(τs(a)) =ϕ bτs(σt(a)) =νsϕ(τ−s(b)σt(a))So they must be equal onC→C→Now we prove some relations in connection with the comultiplication.Proposition4.30Consider z∈•(τz⊙τz)∆=∆τz•(τz⊙σz)∆=∆σz•(σ′z⊙τ−z)∆=∆σ′z•(σz⊙σ′−z)∆=∆τzProof:We only prove theﬁrst equality.The others are proven in the same way.Choose a∈A.Take b∈A.By proposition5.7of[5],we have for every t∈I R that(πr⊙πr)(∆(τt(a))(τt(b)⊗1))=∆ πr(τt(a)) (πr(τt(b))⊗1)=∆ (τr)t(πr(a)) ((τr)t(πr(b))⊗1)=((τr)t⊗(τr)t) ∆r(πr(a)) ((τr)t(πr(b))⊗1)=((τr)t⊗(τr)t)(∆r(πr(a))(πr(b)⊗1))=((τr)t⊗(τr)t) (πr⊙πr)(∆(a)(b⊗1)) =(πr⊙πr) (τt⊙τt)(∆(a)(b⊗1))which implies that∆(τt(a))(τt(b)⊗1)=(τt⊙τt)(∆(a)(b⊗1))(*)Choose p,q∈A.Then we have two functionsC:y→(ϕ⊙ϕ)((1⊗p)∆(q)∆(τy(a))(τy(b)⊗1))andC:y→(ϕ⊙ϕ) (1⊗p)∆(q)(τt⊙τt)(∆(a)(b⊗1))We see immediately that the second function is analytic.Because(ϕ⊙ϕ)((1⊗p)∆(q)∆(τy(a))(τy(b)⊗1))=ϕ(pτy(b))ϕ(qτy(a))for y∈A,also theﬁrst function is analytic.We know by(*)that both functions are equal on the real axis,so they must be equal on the whole complex plane.In particular,(ϕ⊙ϕ)((1⊗p)∆(q)∆(τz(a))(τz(b)⊗1))=(ϕ⊙ϕ) (1⊗p)∆(q)(τz⊙τz)(∆(a)(b⊗1)) . Hence the faithfulness ofϕimplies that∆(τz(a))(τz(b)⊗1)=(τz⊙τz)(∆(a)(b⊗1))=(τz⊙τz)(∆(a))(τz(b)⊗1)So we see that∆(τz(a))=(τz⊙τz)(∆(a)).C.Thenετz=ε.Corollary4.33We have thatεR=ε.In the next part,we look at some formulas involving the modular functionδ.Theﬁrst one says that everyδz is a one-dimensional corepresentations of(A,∆).Proposition4.34We have for every z∈Proof:Take a,b∈A.Then we have by proposition8.6of[5]for every t∈I R that(πr⊙πr)(∆(δit)∆(a)(b⊗1))=(πr⊙πr)(∆(δit a)(b⊗1))=∆r(πr(∆(δit a))(πr(b)⊗1) =∆r(δit rπr(a))(πr(b)⊗1)=(δit r⊗δit r)(πr⊙πr)(∆(a)(b⊗1))=(πr⊙πr)((δit⊗δit)∆(a)(b⊗1))which implies that∆(δit)∆(a)(b⊗1)=(δit⊗δit)∆(a)(b⊗1)(∗)We have now two functionsC:u→(ϕ⊙ϕ)(∆(δiu)∆(a)(b⊗1))andC:u→(ϕ⊙ϕ)((δiu⊗δiu)∆(a)(b⊗1))The second function is clearly analytic.Because(ϕ⊙ϕ)(∆(δiu)∆(a)(b⊗1))=ϕ(δiu a)ϕ(b)for u∈A, also the second is analytic.Furthermore,(*)implies that both functions are equal on the real line. So they must be equal on the whole complex plane.In particular,(ϕ⊙ϕ)(∆(δz)∆(a)(b⊗1))= (ϕ⊙ϕ)((δz⊗δz)∆(a)(b⊗1))Hence the faithfulness ofϕimplies that∆(δz)∆(a)(b⊗1)=(δz⊗δz)∆(a)(b⊗1)So we get that∆(δz)=δz⊗δz.C thatε(δz)=1and S(δz)=δ−z.Result4.36Consider y,z∈C→C→Combining the result concerningτwith corollary4.35and proposition4.27,we get the following one.Result4.37Consider z∈C that(σ′r)z=R r(σr)−z R r.So we get the next result.Result4.38Consider z∈C and a∈A thatσ′z(a)=δizσz(a)δ−iz.We end this section with some remarks concerning the right Haar functional on(A,∆).Recall that we have a right Haar functionalϕS on(A,∆)but we do not know(yet)whetherϕS is positive.By the formulaχ(R⊙R)∆=∆R,we have however the following proposition.Theorem4.40The functionalϕR is a positive Haar functional on(A,∆).Because S=τ−i2ϕR.BecauseϕS=δϕ,we have also thatϕ(R(a))=ϕ(δ12)for a∈A.5The analytic structure of the dualIn this section,we will connect the alytic objects associated to the dual of an algebraic quantum groups to the analytic objects of this algebraic quantum groups.So consider an algebraic quantum group(A,∆)with a positive left Haar functionalϕon it.As in the previous section,we will use the notationsσfor the modular group of the left Haar functional,σ′for the modular group of the right Haar functional,τfor the scaling group and R for the anti-unitary antipode. The corresponding objects on the dual quantum group(ˆA,ˆ∆)will get a hat on them,e.g.ˆσwill denote the modular group of the left Haar functional on(ˆA,ˆ∆).First we start with the modular group of the dual quantum group(ˆA,ˆ∆).(Similar results are also considered in[10]).We will introduceﬁrst a temporary notation.Considerω∈M(ˆA).Then we deﬁneωz∈A′such that ωz(a)=ω(τz(a)δ−iz)for a∈A.Lemma5.1Consider z∈where we used result4.36in the second last equality.Consequently,(ωz⊙ι)∆(a)=τ−z [(ω⊙ι)∆(τz(a)δ−iz)]δiz .The other equality is proven in the same way.C.Then we have the following properties.1.We have for everyω∈M(ˆA)thatωz∈M(ˆA).2.Considerω,θ∈M(ˆA).Then(ωθ)z=ωzθz.Lemma5.3Consider z∈Lemma5.4Considerω∈ˆA and a∈A.Then the fuction C:z→ωz(a)is analytic.Proof:There exist b,c,d∈A such thatω=bcϕd∗.We know thatπr(a)is analytic with respect toτr and thatπr(b)is analytic with respect toδr.This implies that the function C→A r:z→πr(b)πr(c))is analytic.ϕr(πr(d)∗(τr)z(πr(a))δ−izrBut we have for every z∈C→Proposition5.5We have for everyω∈ˆA,a∈A and z∈C into L(ˆA)such thatβz(ω)=ωz for z∈2.Choose s,t∈I R.Takeω∈ˆA.Then we have for every a∈A that[βs(βt(ω))](a)=[βt(ω)](τs(a)δ−it)=ω(τt(τs(a)δ−it)δ−is)=ω(τt(τs(a))δ−itδ−is)=ω(τs+t(a)δ−i(s+t)))=[βs+t(ω)](a), implying thatβs(βt(ω))=βs+t(ω).So we have proven thatβsβt=βs+t.3.Lemma5.2implies thatβt is multiplicative for every t∈I R.4.Choose t∈I R.Takeω∈ˆA.Then we have for every a∈A that[βt(ω)]∗(a)=ω(τt(S(a)∗)δ−it)=ω((δit S(τt(a)))∗)=C that b(βz(ω))=βz(ω)(a)=ωz(a))which implies that the function C:z→b(βz(ω))is analytic by lemma5.4So we can conclude from this al thatβz is an analytic one-parameter group onˆA.Choose a∈A.Lemma2.8of[6]implies thatˆσi(ψa)=ψδS2(a).We know by the proof of lemma5.3 that(ψa)i=ψδS2(a).So we see thatˆσi(ψa)=βi(ψa).Henceˆσi=βi which by proposition2.11implies thatˆσ=β.C thatˆσz(ω)(a)=ω(τz(a)δ−iz).The proof of the next result is completely similar(an easier)to the proof of the previous proposition.It is a consequence of the fact thatˆS2(ω)=ωS2forω∈ˆA.Proposition5.7We have for everyω∈M(ˆA),a∈A and z∈,we get easily the following on.2Corollary5.8We have for everyω∈M(ˆA)thatˆR(ω)=ωR.Remembering thatˆσ′z=ˆRˆσ−zˆR,it is now easy to check the next equality.Corollary5.9We have for every a∈A,ω∈M(ˆA)and z∈。

相对论史话英语介绍

相对论史话英语介绍Relativity: A Historical Overview.Relativity is a fundamental concept in modern physics, proposed by Albert Einstein in 1905. It revolutionized our understanding of space, time, and the universe as a whole. There are two main components of relativity: the Special Theory of Relativity and the General Theory of Relativity.The Special Theory of Relativity asserts that space and time are interconnected and relative, rather than absolute and fixed. Einstein introduced the idea that the speed of light is constant in all inertial reference frames, regardless of the observer's state of motion. This led to the famous "principle of relativity," which states that the laws of physics are the same in all inertial reference frames.One of the consequences of special relativity is the relativity of time. This principle states that the passageof time is not absolute; it can vary depending on the observer's reference frame. For example, if two events occur at different points in space, the time interval between these events will be different for observers in different reference frames.Another consequence of special relativity is the relativity of space. This principle suggests that the measurements of space can vary depending on the observer's motion. For instance, two observers moving relative to each other may measure different distances between the same two points in space.The Special Theory of Relativity also gave rise to the famous equation E=mc^2, known as the mass-energy equivalence. This equation states that mass and energy are equivalent and can be converted into each other. This equivalence has profound implications in physics, including the possibility of nuclear energy production and the annihilation of matter.The General Theory of Relativity, published in 1915, isan extension of the Special Theory. It incorporates the idea of gravity as a curvature of spacetime. Einstein proposed that matter and energy curve spacetime, creating a gravitational field that affects the motion of other matter and energy. This theory explained many phenomena that had previously been unexplained, such as the precession of the perihelion of Mercury.The General Theory of Relativity also gave rise to the concept of black holes, regions of spacetime where gravity is so strong that nothing can escape, including light. Black holes are predicted to form when a massive star collapses under its own weight.Relativity has had a profound impact on our understanding of the universe. It has revolutionized our view of space and time, showing that they are not absolute but relative to the observer's reference frame. It has also provided a new understanding of gravity and the curvature of spacetime. The predictions of relativity have been confirmed through numerous experiments and observations, making it one of the most well-tested and accepted theoriesin physics.In conclusion, relativity is a fundamental concept in modern physics that has revolutionized our understanding of space, time, and the universe. The Special Theory of Relativity introduced the idea that space and time are relative and interconnected, while the General Theory of Relativity extended this idea to include the curvature of spacetime and the phenomenon of gravity. The predictions of relativity have been confirmed through numerous experiments and observations, making it one of the most important and well-established theories in physics.。

学术综合英语unit1_5课文翻译

Presenting a speech（做演讲）Of all human creations, language may be the most remarkable. Through在人类所有的创造中，语言也许是影响最为深远的。

我们用语言language we share experience, formulate values, exchange ideas, transmit来分享经验，表达(传递？)价值观，交换想法，传播知识，knowledge, and sustain culture. Indeed, language is vital to think itself.传承文化。

事实上，对语言本身的思考也是至关重要的。

[Contrary to popular belief], language | does not simply mirror reality butalso helps to create our sense of reality [by giving meaning to events].和通常所认为的不同的是，语言并不只是简单地反映现实，语言在具体描述事件的时候也在帮助我们建立对现实的感知。

——语序的调整。

Good speakers have respect for language and know how it works. Words are the tools of a speaker’s craft. They have special uses, just like the tools of any other profession. As a speaker, you should be aware of the meaning of words and know how to use language accurately, clearly,vividly,and appropriately.好的演讲者对语言很重视，也知道如何让它发挥更好的效果。

相对论英文原版

相对论英文原版1简介Relativity Theory,or the Theory of Relativity,is a fundamental concept in modern physics.First proposed by Albert Einstein in1905,it fundamentally transformed our understanding of space,time,and the universe as a whole. The theory has two main components,the Special Theory of Relativity and the General Theory of Relativity.2特殊相对论The Special Theory of Relativity,introduced by Einstein in1905,postulates that the laws of physics are the same for all non-accelerating observers in uniform motion relative to one another.This includes the speed of light,which remains constant regardless of the relative motion of the observer and the source of the light.One of the most significant consequences of Special Relativity is time dilation,which means that time passes more slowly for objects in motion.This effect becomes significant at velocities approaching the speed of light and has been experimentally verified.Another consequence of Special Relativity is length contraction,which means that objects in motion appear shorter in the direction of their motion than they would at rest.This effect is also experimentally verified.3广义相对论The General Theory of Relativity,published by Einstein in1915,builds upon the principles of Special Relativity and introduces the concept of a curved spacetime.According to General Relativity,the curvature of spacetime is determined by the distribution of matter and energy within it.One consequence of General Relativity is the prediction of gravitational waves,which are ripples in spacetime caused by the motion of massive objects.These waves were first detected in2015by the Laser Interferometer Gravitational-Wave Observatory(LIGO).Another consequence of General Relativity is the prediction of black holes,which are regions of spacetime where the curvature is so extreme that nothing,not even light,can escape.Black holes have been observed indirectly through their effects on nearby matter,and directly through the recent imaging of the supermassive black hole at the center of the galaxy M87.4应用The concepts of Relativity Theory have many practical applications in modern technology.For example,GPSsatellites rely on the principles of Special Relativity to make accurate measurements of time and distance.The use of atomic clocks on these satellites allows them to account for the time dilation effect of their high velocities.The development of gravitational wave detectors has the potential to revolutionize our understanding of the universe, allowing us to observe events that are invisible totraditional telescopes.The recent detections ofgravitational waves from merging black holes and neutron stars have already provided unprecedented insights into the nature of these extreme objects.5结论In conclusion,Relativity Theory has profoundly impacted our view of the universe and has led to many technological advancements.While the concepts it introduces can seem strange and counterintuitive,they have been extensively tested and are now widely accepted as fundamental components of our understanding of the physical world.。

Structure, Individuality and Quantum Gravity

a rXiv:g r-qc/5778v 219J ul25Structure,Individuality and Quantum Gravity John Stachel ∗Abstract After reviewing various interpretations of structural realism ,I adopt here a deﬁnition that allows both relations between things that are already individuated (which I call “relations between things”)and relations that individuate previously un-individuated entities (”things between relations”).Since both space-time points in general relativity and elementary particles in quantum theory fall into the latter cate-gory,I propose a principle of maximal permutability as a criterion for the fundamental entities of any future theory of “quantum gravity”;i.e.,a theory yielding both general relativity and quantum ﬁeld theory in appropriate limits.Then I review of a number of current candidates for such a theory.First I look at the eﬀective ﬁeld theory and asymp-totic quantization approaches to general relativity,and then at string theory.Then a discussion of some issues common to all approaches to quantum gravity based on the full general theory of relativity argues that processes,rather than states should be taken as fundamental in any such theory.A brief discussion of the canonical approach is fol-lowed by a survey of causal set theory,and a new approach to the question of which space-time structures should be quantized ends the paper.Contents1What is Structural Realism?3 2Structure and Individuality5 3Eﬀectiveﬁeld theory approach and asymptotic quantization11 4String Theory15 5Quantum general relativity-some preliminary problems175.1States or Processes:Which is primary? (17)5.2Formalism and measurability (20)6Canonical quantization(loop quantum gravity).25 7The causal set(causet)approach29 8What Structures to Quantize?31 9Acknowledgements3321What is Structural Realism?The term“structural realism”can be(and has been)interpreted in a number of diﬀerent ways.1I assume that,in discussions of structuralism,the concept of structure refers to some set of relations between the things or entities that they relate,called the relata.Here I interpret things in the broadest possible sense:they may be material objects,physicalﬁelds,mathematical concepts, social relations,processes,etc.2People have used the term“structural re-alism”to discribe diﬀerent approches to the nature of the relation between things and relations.These diﬀerences all seem to be variants of three basic possibilities:I.There are only relations without relata.3As applied to a particular relation,this assertion seems incoherent.It only makes sense if it is interpreted as the metaphysical claim that ulti-mately there are only relations;that is,in any given relation,all of its relata can in turn be interpreted as relations.Thus,the totality of structural re-lations reduces to relations between relations between relations.As Simon Saunders might put it,it’s relations all the way down.4It is certainly true that,in certain cases,the relata can themselves be interpreted as relations; but I would not want to be bound by the claim that this is always the case. Iﬁnd rather more attractive the following two possibilities:II.There are relations,in which the things are primary and their relation is secondary.III.There are relations,in which the relation is primary while the things are secondary.In order to make sense of either of these possibilities,and hence of the distinction between them,one must assume that there is always a distinctionbetween the essential and non-essential properties of any thing,5For II to hold(i.e.things are primary and their relation is secondary),no essential property of the relata can depend on the particular relation under consider-ation;while for III to hold(i.e.the relation is primary and the relata are secondary),at least one essential property of each of the relata must depend on the relation.Terminology diﬀers,but one widespread usage denotes rela-tions of type II as external,those of type III as internal.One could convert either possibility into a metaphysical doctrine:“All relations are external”or“All relations are internal”;and some philosophers have done so.But,in contradistinction to I,there is no need to do so to make sense of II and III. If one does not,then the two are perfectly compatible.Logically,there is a fourth possible case:IV.There are things,such that any relation between them is only apparent.This is certainly possible in particular situations.One could,for exam-ple,pre-program two mechanical dolls(the things)so that each would move independently of the other,but in such a way that they seemed to be dancing with each other(the apparent relation-I assume that“dancing together”is a real relation between two people).Again,one might convert this possibility into a universal claim:“All relations are only apparent.”.Leibniz monadology,for example,might be interpreted as asserting that all relations between monads are only appar-ent.Since God set up a pre-established harmony among them,they are pre-programmed to behave as if they were related to each other.As a meta-physical doctrine,Iﬁnd IV even less attractive than I.And if adopted,it could hardly qualify as a variant of structural realism,so I shall not mention IV any further.While several eminent philosophers of science(e.g.French and Ladyman) have opted for version I of structural realism,to me versions II and III(in-terpreted non-metaphysically)are the most attractive.They do not require commitment to any metaphysical doctrine,but allow for a decision on the character of the relations constituting a particular structure on a case-by-case basis.6My approach leads to a picture of the world,in which there are entities of many diﬀerent natural kinds,and it is inherent in the nature of each kind to be structured in various ways.These structures themselves are organized into various structural hierarchies,which do not all form a linear se-quence(chain);rather,the result is something like a totally partially-ordered set of structures.This picture is dynamic in two senses:there are changes in the world,and there are changes in our knowledge of the world.As well as a synchronic aspect,the entities and structures making up our current picture of the world have a diachronic aspect:they arise,evolve,and ultimately disappear-in short,they constitute processes.And our current picture is itself subject to change.What particular entities and structures are posited,and whether a given entity is to be regarded as a thing or a relation, are not decisions that are foreverﬁxed and unalterable;they may change with changes in our empirical knowledge and/or our theoretical understanding of the world.So I might best describe this viewpoint as a dynamic structural realism.72Structure and IndividualityA more detailed discussion of many points in this section is presented in[59],[63].It seems that,as deeper and deeper levels of these structural hierarchies are probed,the property of inherent individuality that characterizes more complex,higher-level entities-such as a particular crystal in physics,or a particular cell in biology is ing some old philosophical terminology, I say that a level at has been reached,which the entities characterizing this level possess quiddity but not haecceity.“Quiddity”refers to the essential nature of an entity,its natural kind;and–at least at the deepest level which we have reached so far-entities of diﬀerent natural kinds exist,e.g.,elec-trons,quarks,gluons,photons,etc.8What distinguishes entities of thesame natural kind(quiddity)from each other,their unique individuality or “primitive thisness,”is called their“haecceity.”9Traditionally,it was always assumed that every entity has such a unique individuality:a haecceity as well as a quiddity.However,modern physics has reached a point,at which we are led to postulate entities that have quiddity but no haecceity that is inherent, i.e.,independent of the relational structures in which they may occur.In so far as they have any haecceity(and it appears that degrees of haecceity must be distinguished10),such entities inherit it from the structure of relations in which they are enmeshed.In this sense,they are indeed examples of the case III:“things between relations.”[57]Since Kant,philosophers have often used position in space as a principle of individuation for otherwise indistinguishable entities;more recently,similar attempts have been made to individuate physical events or processes11.A physical process occupies a(generallyﬁnite)region of space-time;a physical event is supposed to occupy a point of space-time.In theories,in which space-time is represented by a continuum,an event can be thought of as the limit of a portion of some physical process as all the dimensions of the region of space-time occupied by this portion are shrunk to zero.Classically, such a limit may be regarded as physically possible,or just as an ideal limit.“An event may be thought of as the smallest part of a process....But do not think of an event as a change happening to an otherwise static object.It is just a change,no more than that”([46],pp.53).See section5.1for further discussion of processes.It is probably better to avoid attributing physical signiﬁcance to point events,and accordingly to mathematically reformulate general relativity in terms of sheaves12Individuation by means of position in space-time works at the level of theories with aﬁxed space-time structure,notably special-relativistic theo-ries of matter and/orﬁelds13but,according to general relativity,becauseof the dynamical nature of all space-time structures,14the points of space-time lack inherent haecceity;thus they cannot be used for individuation of other physical events in a general-relativistic theory of matter and/or non-gravitationalﬁelds.This is the purport of the“hole argument”(see[54]and earlier references therein).The points of space-time have quiddity as such, but only gain haecceity(to the extent that they do)from the properties they inherit from the metrical or other physical relations imposed on them.15 In particular,the points can obtain haecceity from the inertio-gravitational ﬁeld associated with the metric tensor:For example,the four non-vanishing invariants of the Riemann tensor in an empty space-time can be used to individuate these points in the generic case(see ibid.,pp.142-143)16 Indeed,as a consequence of this circumstance,in general relativity the converse attempt has been made:to individuate the points of space-time by means of the individuation of the physical(matter orﬁeld)events or processes occurring at them;i.e.,by the relation between these points and some individuating properties of matter and/or non-gravitationalﬁelds.Such attempts can succeed at the macroscopic,classical level;but,if the analysis of matter andﬁelds is carried down far enough-say to the level of the sub-nuclear particles andﬁeld quanta17-then the particles andﬁeld quanta ofdiﬀering quiddity all lack inherent haecceity.18Like the points of space-time, insofar as they have any individuality,it is inherited from the structure of relations in which these quanta are embedded.For example,in a process involving a beam of electrons,a particular electron may be individuated by the click of a particle counter.19In all three of these cases-space-time points or regions in general relativ-ity,elementary particles in quantum mechanics,andﬁeld quanta in quantum ﬁeld theory-insofar as the fundamental entities have haecceity,they inherit it from the structure of relations in which they are enmeshed.But there is an important distinction here between general relativity one the one hand and quantum mechanics and quantumﬁeld theory on the other:the former is background-independent while the latter are not;but I postpone further discussion of this diﬀerence until Section5b.What has all this to do with the search for a theory of quantum grav-ity?The theory that we are looking for must underlie both classical general relativity and quantum theory,in the sense that each of these two theory should emerge from“quantum gravity”by some appropriate limiting pro-cess.Whatever the ultimate nature(s)(quiddity)of the fundamental entities of a quantum gravity theory turn out to be,it is hard to believe that they will possess an inherent individuality(haecceity)already absent at the levels of both general relativity and quantum theory(see[59]).So I am led to assume that,whatever the nature(s)of the fundamental entities of quantum gravity,they will lack inherent haecceity,and that such individuality as they manifest will be the result of the structure of dynamical relations in which they are enmeshed.Given some physical theory,how can one implement this requirement of no inherent haecceity?Generalizing from the previous examples,I maintain that the way to assure the inherent indistinguishability in of the fundamental entities of the theory is to require the theory to be for-mulated in such a way that physical results are invariant under all possible permutations of the basic entities of the same kind(same quiddity).20I have named this requirement the principle of maximal permutability.(See[63]for a more mathematically detailed discussion.)The exact content of the principle depends on the nature of the funda-mental entities.For theories,such as non-relativistic quantum mechanics, that are based on aﬁnite number of discrete fundamental entities,the per-mutations will also beﬁnite in number,and maximal permutability becomes invariance under the full symmetric group.For theories,such as general rel-ativity,that are based on fundamental entities that are continuously,and even diﬀerentiably related to each other,so that they form a diﬀerentiable manifold,permutations become diﬀeomorphisms.For a diﬀeomorphism of a manifold is nothing but a continuous and diﬀerentiable permutation of the points of that manifold.21So,maximal permutability becomes invariance un-der the full diﬀeomorphism group.Further extensions to an inﬁnite number of discrete entities or mixed cases of discrete-continuous entities,if needed, are obviously possible.In both the case of non-relativistic quantum mechanics and of general relativity,it is only through dynamical considerations that individuation is eﬀected.In theﬁrst case,it is through speciﬁcation of a possible quantum-mechanical process that the otherwise indistinguishable particles are indi-viduated(“The electron that was emitted by this source at11:00a.m.and produced a click of that Geiger counter at11:01a.m.”).In the second case, it is through speciﬁcation of a particular solution to the gravitationalﬁeld equations that the points of the space-time manifold are individuated(“The point at which the four non-vanishing invariants of the Riemann tensor had the following values:...”).So one would expect the principle of maximal per-mutability of the fundamental entities of any theory of quantum gravity to be part of a theory in which these entities are only individuated dynamically.Thomas Thiemann has pointed out that,in the passage from classical to quantum gravity,there is good reason to expect diﬀeomorphism invariance to be replaced by some discrete combinatorial principle:The concept of a smooth space-time should not have any meaningin a quantum theory of the gravitationalﬁeld where probing dis-tances beyond the Planck length must result in black hole creationwhich then evaporate in Planck time,that is,spacetime should befundamentally discrete.But clearly smooth diﬀeomorphisms haveno room in such a discrete spacetime.The fundamental symme-try is probably something else,maybe a combinatorial one,thatlooks like a diﬀeomorphism group at large scales.([67],pp.117)In the next section,I shall look at the eﬀectiveﬁeld theory approach to general relativity and asymptotic quantization,and then,in the following section,at string theory,both in the light of the principle of maximal per-mutability.Section5discusses some issues common to all general-relativity-based approaches to quantum gravity.I had hoped to treat loop quantum gravity in detail in this paper,but the discussion outgrew my allotted spatial bounds;so just a few points about the canonical approach are discussed in Section6,and the fuller discussion relegated to a separate paper,[60].Sec-tion7is devoted to causal set theory,and Section8sketches a possible new approach,suggested by causal set theory,to the question of what space-time structures to quantize.103Eﬀectiveﬁeld theory approach and asymp-totic quantizationThe earliest attempts to quantize theﬁeld equations of general relativity were based on treating it using the methods of special-relativistic quantum ﬁeld theory,perturbatively expanding the gravitationalﬁeld around theﬁxed background Minkowski space metric and quantizing only the perturbations. By the1970s,theﬁrst wave of such attempts petered out with the realiza-tion that the resulting quantum theory is perturbatively non-renormalizable. With the advent of the eﬀectiveﬁeld theory approach to non-renormalizable quantumﬁeld theories,a second,smaller wave arose22with the more modest aim of developing an eﬀectiveﬁeld theory of quantum gravity valid for suﬃ-ciently low energies(for reviews,see[13],[9]).As is the case for all eﬀective ﬁeld theories,this approach is not meant to prejudge the nature of the ulti-mate resolution of“the more fundamental issues of quantum gravity”([9], pp.6),but to establish low-energy results that will be reliable whatever the nature of the ultimate theory.23The standard accounts of the eﬀectiveﬁeld approach to general relativ-ity take the metric tensor as the basicﬁeld,which somewhat obscures the analogy with Yang-Millsﬁelds:Despite the similarity to the construction of theﬁeld strengthtensor of Yang Millsﬁeld theory,there is the important diﬀerencethat the[Riemannian]curvatures involve two derivatives of thebasicﬁeld,R∼∂∂g.([13],pp.4)But much of the recent progress in bringing general relativity closer to other gaugeﬁeld theories,and in developing background-independent quanti-zation techniques,has come from giving equal importance(or even primacy) to the aﬃne connection as compared to the metric(see Sections6,8and [60]).Since the curvature tensor involves only one derivative of the connec-tion,R∼∂Γ,this approach brings the formalism of general relativity much closer to the gauge approach used treat all other interactions.From this point of view,one role of the metric tensor is to act as potentials for theconnectionΓ∼∂g.From this viewpoint,one can reformulate the starting point of general relativity as follows.The equivalence principle demands that inertia and gravitation be treated as intrinsically united,the resulting inertio-gravitationalﬁeld being repre-sented mathematically by a non-ﬂat aﬃne connectionΓ24.If one assumes that this connection is metric,i.e.,that the connection can be derived from a second-rank covariant metricﬁeld g then according to general relativity such a non-ﬂat metricﬁeld represents the chrono-geometry of space-time.But the eﬀectiveﬁeld approach assumes that the true chrono-geometry of space-time remains the Minkowski space-time of special relativity,repre-sented by theﬁxed background metricη25.There is an unique,ﬂat aﬃne connection{}compatible with the Minkowski metricη26,and since the diﬀerence between any two connections is a tensor,Γ−{}-the diﬀerence between the non-ﬂat andﬂat connections-is a tensor that serves to repre-sents a purely gravitationalﬁeld.Thus,the upshot of this approach is to violate the purport of the equiv-alence principle,according to which inertia and gravitation are essentially the same and should remain inseparable.With the help of theﬂat back-ground metric and connection,they have been separated;and a kinematics introduced based on the purely inertial connection,a kinematics that is in-dependent of the dynamics embodied in the purely gravitational tensor.The background metric is assumed to be unobservable,because the eﬀect of the gravitationalﬁeld on all(ideal)rods and clocks is to distort their measure-ments in such a way that they always map out the non-ﬂat chrono-geometry that can be associated with the metric of the g-ﬁeld.If eﬀectiveﬁeld theory did not tell us better,we might be tempted to think of this metric as the true chrono-geometry but then we would be doing general relativity Contrary to the purport of the equivalence principle,inertia and gravita-tion have been separated with the help of the background metric and a kine-matics based on the backgroundﬁelds has been introduced that is indepen-dent of the dynamics of this gravitational tensor.However,the backgroundmetric is unobservable:The eﬀect of this gravitationalﬁeld on all(ideal)rods and clocks is to distort their measurements in such a way that they map out the non-ﬂat chrono-geometry associated with the g-ﬁeld,which if we did not know better,we would be tempted to think of as the true chrono-geometry 27The points of the background metric(ﬂat or non-ﬂat-see note26)are then assumed to be individuated up to the symmetry group of this metric, which at most can be aﬁnite-parameter Lie group(e.g.,the ten parameter Poincar group for the Minkowski background metric)acting on the points of space-time.28Since the full diﬀeomorphism group acting on the base manifold is not a symmetry group of the background metric,29this version of quantum grav-ity does not meet our criterion of maximal permutability.If we choose a background space-time with no symmetry group,each and every point of the background space-time manifold will be individuated by the non-vanishing invariants of the Riemann tensor.But if there is a symmetry group generated by one or more Killing vectors,then points on the orbits of the symmetry group will not be so individuated,but must be individuated by some addi-tional non-dynamical method.30Other diﬀeomorphisms can only be interpreted passively,as coordinate redescriptions of the background space-time and inertialﬁelds.They can be given an active interpretation only as gauge transformations on the gravita-tion potentials h=g−η31Since the eﬀectiveﬁeld approach does not claim to be any more than a low-energy approximation to any ultimate theory of quantum gravity,rather than an obstacle to any theory making such a claim,this approach presents a challenge.Can such a theory demonstrate that,in an appropriate low-energy limit,its predictions match the predictions of the eﬀectiveﬁeld theory for experimental results?32Since these experimental predictions will essentially concern low energy scattering experiments involving gravitons,it will be a long time indeed before any of these predictions can be compared with actual experimental result;and the eﬀectiveﬁeld theory approach has little to oﬀer in the way of predictions for the kind of experimental results that work on phenomenological quantum gravity is actually likely to give us in the near future.In a sense,one quantum gravity program has already met this challenge: Ashtekars(1987)asymptotic quantization,in which only the gravitational in-and out-ﬁelds at null inﬁnity-i.e.,atℑ+(scri-plus)andℑ−(scri-minus)-are quantized.Without the introduction of any background metric ﬁeld,it is shown how non-linear gravitons may be rigorously deﬁned in terms of theseﬁelds as irreducible representations of the symmetry group at null inﬁnity.This group,however,is not the Poincar´e group at null inﬁnity, but the much larger Bondi-Metzner-Sachs group,which includes the super-translations depending on functions of two variables rather than the four paremeters of the translation group.This group deﬁnes a unique kinematics at null inﬁnity that is independent of the dynamical degrees of freedom,and it is this decoupling of kinematics and dynamics that enables the application of more-or-less standard quantization techniques.Just as the quotient of the Poincar group by its translation subgroup deﬁnes the Lorentz group, so does the quotient of the B-M-S group by its super-translation subgroup. Since,in both the eﬀectiveﬁeld and asymptotic quantization techniques,experiments in which the graviton concept could be usefully invoked involve the preparation of in-states and the registration of out states,there must be a close relation between the two approaches;although,as far as I know,this relation has not yet been elucidated in detail.In summary,both the eﬀectiveﬁeld theory and asymptotic quantization approaches avoid the diﬃculties outlined in the previous section by separat-ing out a kinematics that is independent of dynamics.In the former case, this separation is imposed byﬂat everywhere on the space-time manifold by singling out a background space-time metric and corresponding inertial ﬁeld,with the expectation that the results achieved will always be valid to good approximation in the low-energy limit of general relativity.In the lat-ter case,the separation is achieved only for the class of solutions that are asymptoticallyﬂat at null inﬁnity(or more explicitly,the Riemann tensors of which vanish suﬃciently rapidly in all null directions to allow the deﬁni-tion of null inﬁnity).It is then proved that at at null inﬁnity a kinematics can be decoupled from the dynamics at null inﬁnity due to the symmetries of any gravitationalﬁeld there,and that this can be done without violating diﬀeomorphism invariance in the interior region of space-time.Again,this approach presents a challenge to any background-independent quantization program:derive the results of the asymptotic quantization program from the full quantum gravity theory in the appropriate limit.4String TheoryString(or superstring)theory applies the methods of special-relativistic quan-tum theory to two-dimensional time-like world sheets,called strings.33All known(and some unknown)particles and their interactions,including the graviton and the gravitational interaction,are supposed to emerge as certain modes of excitation of and interactions between quantized strings.The fun-damental entities of the original(perturbative)string theory are the strings two-dimensional time-like world sheets-embedded in a given background space-time,the metric of which is needed to formulate the action princi-ple for the strings.For that reason,the theory is said to be“background-dependent.”Quantization of the theory requires the background space-timeto be of ten or more dimensions.34The theory is seen immediately to fail the test of maximal permutabil-ity since the strings are assumed to move around and vibrate in this back-ground,non-dynamical space-time.So the background space-time,one of the fundamental constituents of the theory;is invariant only under aﬁnite-parameter Lie subgroup(the symmetry group of this space-time,usually assumed to have aﬂat metric with Lorentzian signature)of the group of all possible diﬀeomorphisms of its elements.Many string theorists,with a background predominantly in special-relativistic quantumﬁeld theory(atti-tudes are also seen to be background-dependent),initially found it diﬃcult to accept such criticisms;so it is encouraging that this point now seems to be widely acknowledged in the string community.35String theorist Brian Greene,recently presented an appealing vision of what a string theory with-out a background space-time might look like,but emphasized how far string theorists still are from realizing this vision:Since we speak of the“fabric”of spacetime,maybe spacetime is stitched out of strings much as a shirt is stitched out of thread.That is,much as joining numerous threads together in an appropriate pattern produces a shirts fabric,maybe joining numerous strings together in an appropriate pattern produces what we commonly call spacetimes fabric.Matter,like you and me,would then amount to additional agglomerations of vibrating strings like sonorous music played over a muted din,or an elaborate pattern embroidered on a plain piece of material moving within the context stitched together by the strings of spacetime.....[A]s yet no one has turned these words into a precise mathematical statement.As far as I can tell,the obstacles to doing so are far from triﬂing.....We[currently] picture strings as vibrating in space and through time,but without the space-time fabric that the strings are themselves imagined to yield through their orderly union,there is no space or time.In this proposal,the concepts of space ad time fail to have meaning until innumerable strings weave together to produce them.Thus,to make sense of this proposal,we would need a framework for de-scribing strings that does not assume from the get-go that they are vibrating in a preexisting spacetime.We would need a fully spaceless and timeless formulation of string theory,in which spacetime emerges from the collective behavior of strings.Although there has been progress toward this goal,no one has yet come up with such a spaceless and timeless formulation of string theory something that physicists call a background-independent formulation。

On the evolution of universes in quadratic theories of gravity

ON THE EVOLUTION OF UNIVERSES IN QUADRATIC THEORIES OF GRAVITY
JOHN D. BARROW1 AND SIGBJØRN HERVIK2
arXiv:gr-qc/0610013v2 19 Dec 2006
Abstract. We use a dynamical systems approach to investigate Bianchi type I and II universes in quadratic theories of gravity. Due to the complicated nature of the equations of motion we focus on the stability of exact solutions and ﬁnd that there exists an isotropic FRW universe acting as a past attractor. This may indicate that there is an isotropisation mechanism at early times for these kind of theories. We also discuss the Kasner universes, elucidate the associated centre manifold structure, and show that there exists a set of non-zero measure which has the Kasner solutions as a past attractor. Regarding the late-time behaviour, the stability shows a dependence of the parameters of the theory. We give the conditions under which the de Sitter solution is stable and also show that for certain values of the parameters there is a possible late-time behaviour with phantom-like behaviour. New types of anisotropic inﬂationary behaviour are found which do not have counterparts in general relativity.

原子物理讲义 amo_3

Contents3Fine Structure and Lamb Shift23.1Fine Structure (2)3.1.1Kinetic contribution (3)3.1.2Spin-Orbit Interaction (3)3.1.3The Darwin Term (4)3.1.4Evaluation of theﬁne structure interaction (4)3.2The Lamb Shift (4)1Chapter3Fine Structure and Lamb Shift3.1Fine StructureImmediately adjacent to Michelson and Morley’s announcement of their failure toﬁnd the ether in an1887issue of the Philosophical Journal is a paper by the same authors reporting that the Hαline of hydrogen is actually a doublet,with a separation of0.33cm−1.In1915 Bohr suggested that this“ﬁne structure”of hydrogen is a relativistic eﬀect arising from the variation of mass with velocity.Sommerfeld,in1916,solved the relativistic Kepler problem and using the old quantum theory,as it was later christened,accounted precisely for the splitting.Sommerfeld’s theory gave the lie to Einstein’s dictum“The Good Lord is subtle but not malicious”,for it gave the right results for the wrong reason:his theory made no provision for electron spin,an essential feature ofﬁne structure.Today,all that is left from Sommerfeld’s theory is theﬁne structure constantα=e2/¯h c.The theory for theﬁne structure in hydrogen was provided by Dirac whose relativistic electron theory(1926)was applied to hydrogen by Darwin and Gordon in1928.They found the following expression for the energy of an electron bound to a proton of inﬁnite mass:E mc2=⎡⎣1αZn−k+√k2−α2Z2⎤⎦2(3.1)where n is the principal quantum number,k=j+1/2,and j= ±1/2.The Dirac equation is not nearly as illuminating as the Pauli equation,which is the approximation to the Dirac equation to the lowest order in v/c.H=mc2+p22m−e2r+H F S(3.2)Theﬁrst term is the electron’s rest energy;the following two terms are the non relativistic Hamiltonian,and the last term,theﬁne structure interaction,is given byH F S=−p48m3c2+¯h2e22m2c21r3L·S−¯h28m2c2∇2e2r(3.3)The relativistic contributions can be described as the kinetic,spin-orbit,and Darwin terms, H kin,H so,and H Dar,respectively.Each has a straightforward physical interpretation.2M.I.T.Department of Physics,8.421–Spring 200633.1.1Kinetic contribution Relativistically,the total electron energy is E = (mc 2)2+(pc )2.The kinetic energy isT =E −mc 2=(mc 2) 1+p 22−1 =p 2−1p 432+···(3.4)ThusH kin =18p 4m 3c 2(3.5)3.1.2Spin-Orbit InteractionAccording to the Dirac theory the electron has intrinsic angular momentum ¯h S and a magnetic moment µ2=−g e µ0S .The electron g-factor,g e =2.As the electron moves through the electric ﬁeld of the proton it “sees”a motional magnetic ﬁeldB mot =−v c ×E =−v c ×e r 3r =e ¯h mc 1r 3L (3.6)where ¯h L =r ×m v .However,there is another contribution to the eﬀective magnetic ﬁeld arising from the Thomas precession.The relativistic transformation of a vector between two moving co-ordinate systems which are moving with diﬀerent velocities involve not only a dilation,but also a rotation (cf Jackson,Classical Electrodynamics ).The rate of rotation,the Thomas precession,isΩT =12a ×v c 2(3.7)Note that the precession vanishes for co-linear acceleration.However,for a vector ﬁxed in a co-ordinate system moving around a circle,as in the case of the spin vector of the electron as it circles the proton,Thomas precession occurs.From the point of view of an observer ﬁxed to the nucleus,the precession of the electron is identical to the eﬀect of a magnetic ﬁeld.B T =1γeΩT .(3.8)Substituting γe =e/mc ,and a =−e 2r /mr 3into Eq.3.7givesB T =−12e ¯h mc 1r 3L (3.9)Hence the total eﬀective magnetic ﬁeld isB =12e ¯h mc 1r 3L (3.10)This gives rise to a total spin-orbit interactionH so =− µ·B =e 2¯h 22m 2c 21r 3S ·L (3.11)M.I.T.Department of Physics,8.421–Spring 200643.1.3The Darwin TermElectrons exhibit “Zitterbewegung”,ﬂuctuations in position on the order of the Compton wavelength,¯h /mc .As a result,the eﬀective Coulombic potential is not V (r ),but some suit-able average V (r ),where the average is over the characteristic distance ¯h/mc .To evaluate this,expand V (r )about r in terms of a displacement s ,V (r +s )=V (r )+ ∇V ·s +12 ij s xi s xj ∂2V ∂x i ∂x j +···(3.12)Assume that the ﬂuctuations are isotropic.Then the time average of V (r +s )−V (r )is∆V ∼12 13 ¯h mc 2 ∇2V =−16e 2¯h 2m 2c 2∇2 1r(3.13)The precise expression for the Darwin term isH Dar =−18e 2¯h 2m 2c 2∇2 1r (3.14)The coeﬃcient of the Darwin term is 1/8,rather than 1/6.3.1.4Evaluation of the ﬁne structure interactionThe spin orbit-interaction is not diagonal in L or S due to the term L ·S .However,it is diagonal in J =L +S .H so and H Dar are likewise diagonal in J .Hence,ﬁnd-ing the energy level structure due to the ﬁne structure interaction involves evaluating <n, ,S,j,m j |H FS |n,l,S,j,m j >.Note that <H so >vanishes in an S state,and that <H Dar >vanishes in all states but an S state.It is left as an exercise to show thatE FS (n,j )=(α2mc 2) −α22n 4 n j +1/2−34(3.15)Note that states of a given n and j are degenerate.This degeneracy is a crucial feature of the Dirac theory.3.2The Lamb ShiftAccording to the Dirac theory,states of the hydrogen atom with the same values of n and j are degenerate.Hence,in a given term,(2S 1/2,2P 1/2),(2P 3/2,2D 3/2),(2D 5/2,2F 5/2),etc.form degenerate doublets.However,as described in Chapter 1,this is not exactly the case.Because of vacuum interactions,not taken into account,in the Dirac theory,the degeneracy is broken.The largest eﬀect is in the n =2state.The energy splitting between the 2S 1/2and 2P 1/2states is called the Lamb Shift.A simple physical model due to Welton and Weisskopf demonstrate its origin.Because of zero point ﬂuctuation in the vacuum,empty space is not truly empty.The electromagnetic modes of free space behave like harmonic oscillators,each with zero-pointM.I.T.Department of Physics,8.421–Spring20065 energy hν/2.The density of modes per unit frequency interval and per volume is given by the well known expressionρ(ν)dν=8πν2c3dν(3.16)Consequently,the zero-point energy density isWν=12hνρ(ν)=4πhν3c3(3.17)With this energy we can associate a spectral density of radiationWν=1E2ν+B2ν=1E2ν(3.18)The bar denotes a time average and Eνand Bνare theﬁeld amplitudes.Hence,E2ν=32π2hν3c3(3.19)For the moment we shall treat the electron as if it were free.Its motion is given bym¨sν=eEνcos2πνt(3.20)s2ν=e232π4m2ν4E2ν=e2hπ2m2c31ν(3.21)The eﬀect of theﬂuctuation sνis to cause a change∆V in the average potential∆V=ν−V(r)(3.22) V(r+sν)can be found by a Taylor’s expansion:V(r+sν)=V(r)+∆V·sν+12ij∂2V∂sν,i∂sν,jsν,i sν,j+···(3.23)When we average this in time,the second term vanishes because s averages to zero.For the same reason,in theﬁnal term,only contributions with i=j remain.We have,taking the average,V(r+sν)=V(r)+12i∂2V i∂s2ν,i2ν,i(3.24)Since s2ν,i=2ν/3we obtainﬁnallyV(r+sν)=s2ν6∇2V(r)(3.25)Since∇2V(r)=4πZeδ(r),we obtain the following expression for the change in energyδWν=2π2e2s2ν<n , ,m |δ(r)|n, ,m>(3.26)The matrix element gives contributions only for S states,where its value is|Ψn,0,0(0)|2=Z 3πn3a30(3.27)M.I.T.Department of Physics,8.421–Spring20066 Combining Eqs.3.21,3.27into Eq.3.26yieldsδWν=23e2s2νZ4n3a30=23e4Z4m2c3π21n3a20hν(3.28)Integrating over some yet to be speciﬁed frequency limits,we obtainδW=23e4m2c3Z4π2hn3a30lnνmaxνmin(3.29)At this point,atomic units come in handy.Converting by the usual prescription,we obtainδW=43πα3Z4n3lnνmaxνmin(3.30)The question remaining is how to choose the cut-oﬀfrequencies for the integration.It is reasonable thatνmin is approximately the frequency of an orbiting electron,Z2/n3in atomic units.At lower energies,the electron could not respond.For the upper limit,a plausible guess is the rest energy of the electron,mc2.Hence,νmax/νmin∼Z2/(n3α2).For the2S state,this givesδW=16πα3ln8α2=2.46×10−7atomic units=1,600MHz(3.31)The actual value is1,058MHz.Bibliography[1]John M.Blatt,Journal of Comp.Phys.1,382(1967).。

The Theory of Relativity and Its Implications

The Theory of Relativity and ItsImplicationsWhen we talk about the most significant scientific breakthroughs in history, one cannot leave out the theory of relativity. German physicist, Albert Einstein first published his theory of relativity in 1905 and further developed it in 1915, which completely transformed our understanding of space, time, matter, and energy. The theory of relativity is not just a scientific concept; its implications are far-reaching and have altered our perspectives on the universe, technology, and society.Relativity fundamentally demolished the classical understanding of physics. The previous paradigm of physics rested on Newton's laws of motion, which were believed to be absolute, independent of time, space, and the observer. Yet, Einstein argued that these laws are not fundamental and absolute but are only valid in a particular reference frame. He introduced two types of relativity.The first is the special relativity theory, which deals with objects moving at a constant speed relative to each other. It states that all physical laws are the same for all observers moving uniformly relative to one another. The speed of light, which is considered the most fundamental constant in the universe, plays a critical role in this theory. According to Einstein, nothing can travel faster than light, and the speed of light is constant, independent of the speed of the observer. This has profound implications for our understanding of time, space, mass, and energy.The second is the general relativity theory, which deals with objects under acceleration or in a gravitational field. According to this theory, gravity is not a force but is the result of the mass of one object curving the fabric of spacetime, causing other objects to move towards it. General relativity has revolutionized our scientific understanding of the universe. It predicted black holes, gravitational waves, and the big bang itself.The theory of relativity has been tested and verified repeatedly by experiments for over a century, from the bending of light during a solar eclipse to the atomic clocks on GPS satellites continually adjusted for faster time at higher altitude. Our understanding of the universe, from the behavior of galaxies to the fundamental nature of subatomic particles, depends on the theory of relativity.In addition to breakthroughs in the fields of astrophysics and quantum mechanics, the theory of relativity has had a remarkable impact on technology. GPS, for example, relies on the precise measurements of time and position differences for accurate navigation. The clocks on GPS satellites need to be continuously adjusted because time dilation occurs due to differences in the strength of the gravitational field. These adjustments are necessary to keep the GPS system working correctly. This is just one example of how the theory of relativity has transformed our daily lives.But perhaps the most significant implication of the theory of relativity is the transformation of our understanding of space and time. In classical physics, space and time were considered two separate entities, and time was believed to be absolute and universal. Einstein's theory of relativity, however, demonstrates that space and time are intertwined and that the flow of time depends on the observer's relative motion and proximity to gravity. This has profound implications for our social and cultural understanding of time.The theory of relativity went beyond reshaping our physical understanding of the universe; it also has deep implications for philosophy and human thought. The theory of relativity poses the question of the nature of reality, the nature of existence, and the limits of human understanding. The theory suggests that our perceptions of reality are shaped by our sensory experiences, and that the experience of time and space is subjective.In conclusion, the theory of relativity remains one of the most fascinating and profound scientific theories in history, with implications that extend far beyond the realm of physics. It has radically transformed our understanding of space, time, matter, and energy, revolutionized modern physics and technology, and challenged our fundamental beliefs about human existence. Einstein's theory of relativity remains an essentialscientific and philosophical work that will continue to inspire and provoke inquiry for generations to come.。

The Lorentz Force Law (2)

Derivation of the Lorentz Force Law and the Magnetic Field Concept using an Invariant Formulation of the LorentzTransformationJ.H.FieldD´e partement de Physique Nucl´e aire et Corpusculaire Universit´e de Gen`e ve.24,quaiErnest-Ansermet CH-1211Gen`e ve4.e-mail;john.ﬁeld@cern.chAbstractIt is demonstrated how the right hand sides of the Lorentz Transformation equa-tions may be written,in a Lorentz invariant manner,as4–vector scalar products.The formalism is shown to provide a short derivation,in which the4–vector elec-tromagnetic potential plays a crucial role,of the Lorentz force law of classical elec-trodynamics,and the conventional deﬁnition of the magneticﬁeld in terms spatialderivatives of the4–vector potential.The time component of the relativistic gen-eralisation of the Lorentz force law is discussed.An important physical distinctionbetween the space-time and energy-momentum4–vectors is also pointed out.Keywords;Special Relativity,Classical Electrodynamics.PACS03.30+p03.50.De1IntroductionNumerous examples exist in the literature of the derivation of electrodynamical equa-tions from simpler physical hypotheses.In Einstein’s original paper on Special Relativ-ity[1],the Lorentz force law was derived by performing a Lorentz transformation of the electromagneticﬁelds and the space-time coordinates from the rest frame of an electron (where only electrostatic forces act)to the laboratory system where the electron is in motion and so also subjected to magnetic forces.A similar demonstration was given by Schwartz[2]who also showed how the electrodynamical Maxwell equations can be derived from the Gauss laws of electrostatics and magnetostatics by exploiting the4-vector char-acter of the electromagnetic current and the symmetry properties of the electromagnetic ﬁeld tensor.The same type of derivation of electrodynamic Maxwell equations from the electrostatic and magnetostatic ones has recently been performed by the present author on the basis of‘space-time exchange symmetry’[3].Frisch and Wilets[4]discussed the derivation of Maxwell’s equations and the Lorentz force law by application of relativistic transforms to the electrostatic Gauss law.Dyson[5]published a proof,due originally to Feynman,of the Faraday-Lenz law of induction,based on Newton’s Second Law and the quantum commutation relations of position and momentum,that excited considerable interest and aﬂurry of comments and publications[6,7,8,9,10,11]about a decade ndau and Lifshitz[12]presented a derivation of Amp`e re’s Law from the electro-dynamic Lagrangian,using the Principle of Least Action.By relativistic transformation of the Coulomb force from the rest frame of a charge to another inertial system in rela-tive motion,Lorrain,Corson and Lorrain[13]derived both the Biot-Savart law,for the magneticﬁeld generated by a moving charge,and the Lorentz force law.In many text books on classical electrodynamics the question of what are the funda-mental physical hypotheses underlying the subject,as distinct from purely mathematical developments of these hypotheses,used to derive predictions,is not discussed in any de-tail.Indeed,it may even be stated that it is futile to address the question at all.For example,Jackson[14]states:At present it is popular in undergraduate texts and elsewhere to attempt to derive magneticﬁelds and even Maxwell equations from Coulomb’s law of electrostatics and the theory of Special Relativity.It should immediately obvious that,without additional assumptions,this is impossible.’This is,perhaps,a true statement.However,if the additional assumptions are weak ones,the derivation may still be a worthwhile exercise.In fact,in the case of Maxwell’s equations,as shown in References[2,3],the‘additional assumptions’are merely the formal deﬁnitions of the electric and magneticﬁelds in terms of the space–time derivatives of the 4–vector potential[15].In the case of the derivation of the Lorentz force equation given below,not even the latter assumption is required,as the magneticﬁeld deﬁnition appears naturally in the course of the derivation.In the chapter on‘The Electromagnetic Field’in Misner Thorne and Wheeler’s book ‘Gravitation’[16]can be found the following statement:Here and elsewhere in science,as stressed not least by Henri Poincar´e,that view isout of date which used to say,“Deﬁne your terms before you proceed”.All the laws and theories of physics,including the Lorentz force law,have this deep and subtle chracter, that they both deﬁne the concepts they use(here B and E)and make statements about these concepts.Contrariwise,the absence of some body of theory,law and principle deprives one of the means properly to deﬁne or even use concepts.Any forward step in human knowlege is truly creative in this sense:that theory concept,law,and measurement —forever inseperable—are born into the world in union.I do not agree that the electric and magneticﬁelds are the fundamental concepts of electromagnetism,or that the Lorentz force law cannot be derived from simpler and more fundamental concepts,but must be‘swallowed whole’,as this passage suggests. As demonstrated in References[2,3]where the electrodynamic and magnetodynamic Maxwell equations are derived from those of electrostatics and magnetostatics,a more economical description of classical electromagentism is provided by the4–vector potential. Another example of this is provided by the derivation of the Lorentz force law presented in the present paper.The discussion of electrodynamics in Reference[16]is couched entirely in terms of the electromagneticﬁeld tensor,Fµν,and the electric and magnetic ﬁelds which,like the Lorentz force law and Maxwell’s equations,are‘parachuted’into the exposition without any proof or any discussion of their interrelatedness.The4–vector potential is introduced only in the next-but-last exercise at the end of the chapter.After the derivation of the Lorentz force law in Section3below,a comparison will be made with the treatment of the law in References[2,14,16].The present paper introduces,in the following Section,the idea of an‘invariant for-mulation’of the Lorentz Transformation(LT)[17].It will be shown that the RHS of the LT equations of space and time can be written as4-vector scalar products,so that the transformed4-vector components are themselves Lorentz invariant quantities.Consid-eration of particular length and time interval measurements demonstrates that this is a physically meaningful concept.It is pointed out that,whereas space and time intervals are,in general,physically independent physical quantities,this is not the case for the space and time components of the energy-momentum4-vector.In Section3,a derivation of the Lorentz force law,and the associated magneticﬁeld concept,is given,based on the invariant formulation of the LT.The derivation is very short,the only initial hypothesis being the usual deﬁnition of the electricﬁeld in terms of the4-vector potential,which,in fact,is also uniquely speciﬁed by requiring the deﬁnition to be a covariant one.In Section 4the time component of Newton’s Second Law in electrodynamics,obtained by applying space-time exchange symmetry[3]to the Lorentz force law,is discussed.Throughout this paper it is assumed that the electromagneticﬁeld constitutes,to-gether with the moving charge,a conservative system;i.e.eﬀects of radiation,due to the acceleration of the charge,are neglected2Invariant Formulation of the Lorentz Transforma-tionThe space-time LT equations between two inertial frames S and S’,written in a space-time symmetric manner,are:x =γ(x−βx0)(2.1)y =y(2.2)z =z(2.3)x 0=γ(x0−βx)(2.4) The frame S’moves with velocity,v,relative to S,along the common x-axis of S and S’.βandγare the usual relativistic parameters:vβ≡√−(∆x0)2+∆x2=∆x(2.12) since,for the measurement procedure just described,∆x0=0.Notice that∆x is not necessarily deﬁned in terms of such a measurement.If,following Einstein[1],the interval ∆x is associated with the length, ,of a measuring rod at rest in S and lying parallel to the x-axis,measurements of the ends of the rod can be made at arbitarily diﬀerent times in S.The same result =∆x will be found for the length of the rod,but the corresponding invariant interval,S x,as deﬁned by Eqn(2.12)will be diﬀerent in each case.Similarly,∆x0may be identiﬁed with the time-like invariant interval corresponding to successive observations of a clock at aﬁxed position(i.e.∆x=0)in S:S0≡the same value,∆x0,for the time diﬀerence between two events in S,but with diﬀerent values of the invariant interval deﬁned by Eqn(2.13).In virtue of Eqns(2.12)and(2.13)the LT equations(2.8)and(2.11)may be written the following invariant form:S x=−¯U(β)·S(2.14)S 0=U(β)·S(2.15) where the following4–vectors have been introduced:S≡(S0;S x,0,0)=(∆x0;∆x,0,0)(2.16)U(β)≡(γ;γβ,0,0)(2.17)¯U(β)≡(γβ;γ,0,0)(2.18)The time-like4-vector,U,is equal to V/c,where V is the usual4–vector velocity,whereas the space-like4–vector,¯U,is‘orthogonal to U in four dimensions’:U(β)·¯U(β)=0(2.19) Since the RHS of(2.14)and(2.15)are4–vector scalar products,S x and S 0are manifestly Lorentz invariant quantites.These4–vector components may be deﬁned,in terms of speciﬁc space-time measurements,by equations similar to(2.12)and(2.13)in the frame S’.Note that the4–vectors S and S are‘doubly covariant’in the sense that S·S and S ·S are‘doubly invariant’quantities whose spatial and temporal terms are,individually, Lorentz invariant:S·S=S20−S2x=S ·S =(S 0)2−(S x)2(2.20) Every term in Eqn(2.20)remains invariant if the spatial and temporal intervals described above are observed from a third inertial frame S”moving along the x-axis relative to both S and S’.This follows from the manifest Lorentz invariance of the RHS of Eqn(2.14)and (2.15)and their inverses:S x=−¯U(−β)·S (2.21)S0=U(−β)·S (2.22) Since the LT Eqns(2.1)and(2.4)are valid for any4–vector,W,it follows that:W x=−¯U(β)·W(2.23)W 0=U(β)·W(2.24) Again,W x and W 0are manifestly Lorentz invariant.An interesting special case is the energy-momentum4–vector,P,of a physical object of mass,m.Here the‘doubly in-variant’quantity analagous to S·S in Eqn(2.20)is equal to m2c2.Choosing the x-axis parallel to p andβto correspond to the object’s velocity,so that S’is the object’s proper frame,and since P≡mcU(β),Eqns(2.23)and(2.24)yield,for this special case:P x=−mc¯U(β)·U(β)=0(2.25)P 0=mcU(β)·U(β)=mc(2.26)Since the Lorentz transformation is determined by the single parameter,β,then it follows from Eqns(2.25)and(2.26)that,unlike in the case of the space and time intervals in Eqns(2.8)and(2.11),the spatial and temporal components of the energy momentum 4–vector,in an arbitary inertial frame,are not independent.In fact,P0is determined in terms of P x and m by the relation,that follows from the inverse of Eqns(2.25)and(2.26):P0=Thus,from rotational invariance,the general covariant deﬁnition of the electricﬁeld is:E i=∂i A0−∂0A i(3.4) This is the‘additional assumption’,mentioned by Jackson in the passage quoted above, that is necessary,in the present case,to derive the Lorentz force law.However,as written, it concerns only the physical properties of the electricﬁeld:the magneticﬁeld concept has not yet been introduced.A further a posteriori justiﬁcation of Eqn(3.4)will be given after derivation of the Lorentz force law.Here it is simply noted that,if the spatial part of the4–vector potential is time-independent,Eqn(3.4)reduces to the usual electrostatic deﬁnition of the electricﬁeld.The force F on an electric charge q at rest in the frame S’is given by the deﬁnition of the electricﬁeld,and Eqn(3.4)as:F i=q(∂ i A 0−∂ 0A i)(3.5) Equations analagous to(2.24)may be written relating A and∂ to the corresponding quantities in the frame S moving along the x’axis with velocity−v relative to S’:∂ 0=U(β)·∂(3.6)A 0=U(β)·A(3.7) Substituting(3.6)and(3.7)in(3.5)gives:F i=q∂ i(U(β)·A)−(U(β)·∂)A i(3.8)This equation expresses a linear relationship between F i,∂ i and A i.Since the coeﬃcients of the relation are Lorentz invariant,the same formula is valid in any inertial frame,in particular,in the frame S.Hence:F i=q∂i(U(β)·A)−(U(β)·∂)A i(3.9)This equation gives,in4–vector notation,a spatial component of the Lorentz force on the charge q in the frame S,and so completes the derivation.To express the Lorentz force formula in the more familiar3-vector notation,it is convenient to introduce the relativistic generalisation of Newton’s Second Law[19]:dPdτ=γdP iIntroducing now the magneticﬁeld according to the deﬁnition[20]:B k≡− ijk(∂i A j−∂j A i)=( ∇× A)k(3.12) enables Eqn(3.11)to be written in the compact form:dP idγ βt=mc(3.15)∂twhere Eqn(3.12)has been used.Eqn(3.15)is just the Faraday-Lenz induction law,i.e.the magnetodynamic Maxwell equation.This is only apparent,however,once the‘magnetic ﬁeld’concept of Eqn(3.12)has been introduced.Thus the initial hypothesis,Eqn(3.4),is actually a Maxwell equation.This is the a posteriori justiﬁcation,mentioned above,for this covariant deﬁnition of the electricﬁeld.It is common in discussions of electromagnetism to introduce the second rank electro-magneticﬁeld tensor,Fµνaccording to the deﬁnition:Fµν≡∂µAν−∂νAµ(3.16) in terms of which,the electric and magneticﬁelds are deﬁned as:E i≡F i0(3.17)B k≡− ijk F ij(3.18) From the point of view adopted in the present paper both the electromagneticﬁeld tensor and the electric and magneticﬁelds themselves are auxiliary quantities introduced only for mathematical convenience,in order to write the equations of electromagnetism in a compact way.Since all these quantities are completly deﬁned by the4–vector potential, it is the latter quantity that encodes all the relevant physical information on any electro-dynamic problem[21].This position is contrary to that commonly taken in the literature and texbooks where it is often claimed that only the electric and magneticﬁelds have physical signiﬁcance,while the4–vector potential is only a convenient mathematical tool. For example R¨o hrlich[22]makes the statement:These functions(φand A)known as potentialsmanner!In other cases(e.g.Maxwell’s equations)simpler expessions may be written interms of the4–vector potential.The quantum theory,quantum electrodynamics,thatunderlies classical electromagnetism,requires the introduction the4–vector photonﬁeld Aµin order to specify the minimal interaction that provides the dynamical basis of the theory.Similarly,the introduction of Aµis necessary for the Lagrangian formulation ofclassical electromagnetism.It makes no sense,therefore,to argue that a physical conceptof such fundamental importance has‘no physical meaning’.The initial postulate used here to derive the Lorentz force law is Eqn(3.4),whichcontains,explicitly,the electrostatic force law and,implicitly,the Faraday-Lenz inductionlaw.The actual form of the electrostatic force law(Coulomb’s inverse square law)is notinvoked,suggesting that the Lorentz force law may be of greater generality.On theassumption of Eqn(3.4)(which has been demonstrated to be the only possible covariantdeﬁnition of the electricﬁeld),the existence of the‘magneticﬁeld’,the‘electromagneticﬁeld tensor’,andﬁnally the Lorentz force law itself have all been derived,without furtherassumptions,by use of the invariant formulation of the Lorentz transformation.It is instructive to compare the derivation of the Lorentz force law given in the presentpaper with that of Reference[13]based on the relativistic transformation properties of theCoulomb force3–vector.Coulomb’s law is not used in the present paper.On the otherhand,Reference[13]makes no use of the4–vector potential concept,which is essential forthe derivation presented here.This demonstrates an interesting redundancy among thefundamental physical concepts of classical electromagnetism.In Reference[2],Eqns(3.4),(3.12)and(3.16)were all introduced as a priori initialpostulates without further justiﬁcation.In fact,Schwartz gave the following explanationfor his introduction of Eqn(3.16)[23]:So far everything we have done has been entirely deductive,making use only ofCoulomb’s law,conservation of charge under Lorentz transformation and Lorentz in-variance for our physical laws.We have now come to the end of this deductive path.Atthis point when the laws were being written,God had to make a decision.In generalthere are16components of a second-rank tensor in four dimensions.However,in anal-ogy to three dimensions we can make a major simpliﬁcation by choosing the completelyantisymmetric tensor to represent ourﬁeld quantities.Then we would have only6inde-pendent components instead of the possible16.Under Lorentz transformation the tensorwould remain antisymmetric and we would never have need for more than six independentcomponents.Appreciating this,and having a deep aversion to useless complication,Godnaturally chose the antsymmetric tensor as His medium of expression.Actually it is possible that God may have previously invented the4–vector potentialand special relativity,which lead,as shown above,to Eqn(3.4)as the only possible co-variant deﬁnition of the electricﬁeld.As also shown in the present paper,the existence ofthe remaining elements of the antisymmetricﬁeld tensor,containing the magneticﬁeld,then follow from special relativity alone.Schwartz derived the Lorentz force law,as inEinstein’s original Special Relativity paper[1],by Lorentz transformation of the electricﬁeld,from the rest frame of the test charge,to one in which it is in motion.This requiresthat the magneticﬁeld concept has previously been introduced as well as knowledge ofthe Lorentz transformation laws of the electric and magneticﬁelds.In the chapter devoted to special relativity in Jackson’s book[24]the Lorentz forcelaw is simply stated,without any derivation,as are also the deﬁning equations of theelectric and magneticﬁelds and the electromagneticﬁeld tensor just mentioned.Noemphasis is therefore placed on the fundamental importance of the4–vector potential inthe relativistic description of electromagnetism.In order to treat,in a similar manner,the electromagnetic and gravitationalﬁelds,thediscussion in Misner Thorne and Wheeler[16]is largely centered on the properties of thetensor Fµν.Again the Lorentz force equation is introduced,in the spirit of the passagequoted above,without any derivation or discussion of its meaning.The deﬁning equationsof the electric and magneticﬁelds and Fµν,in terms of Aµ,appear only in the eighteenthexercise of the relevant chapter.The main contents of the chapter on the electromagneticﬁeld are an extended discussion of purely mathematical tensor manipulations that obscurethe essential simplicity of electromagnetism when formulated in terms of the4–vectorpotential.In contrast to References[2,24,16],in the derivation of the Lorentz force law andthe magneticﬁeld presented here,the only initial assumption,apart from the validityof special relativity,is the chosen deﬁnition,Eqn(3.4),of the electricﬁeld in terms ofthe4–vector potential Aµ,which is the only covariant one.Thus,a more fundamentaldescription of electromagnetism than that provided by the electric and magneticﬁeldconcepts is indeed possible,contrary to the opinion expressed in the passage from MisnerThorne and Wheeler quoted above.4The time component of Newton’s Second Law in ElectrodynamicsSpace-time exchange symmetry[3]states that physical laws inﬂat space are invariantwith respect to the exchange of the space and time components of4-vectors.For example,the LT of time,Eqn(2.4),is obtained from that for space,Eqn(2.1),by applying the space-time exchange(STE)operations:x0↔x,x 0↔x .In the present case,application of the STE operation to the spatial component of the Lorentz force equation in the secondline of Eqn(3.11)leads to the relation:dP00(4.1)where Eqns(2.5)and(3.4)and the following properties of the STE operation[3]have been used:∂0↔−∂i(4.2)A0↔−A i(4.3)C·D↔−C·D(4.4)Eqn(4.1)yields an expression for the time derivative of the relativistic energy,E=P0:d E(4.5)Integration of Eqn(4.5)gives the equation of energy conservation for a particle moving from an initial position, x I,to aﬁnal position, x F,under the inﬂuence of electromagnetic forces:E F E I d E=qxFx IE·d x(4.6)Thus work is done on the moving charge only by the electricﬁeld.This is also evident from the Lorentz force equation,(3.14),since the magnetic force β× B is perpendicular to the velocity vector,so that no work is performed by the magneticﬁeld.A corollary is that the relativistic energy(and hence the magnitude of the velocity)of a charged particle moving in a constant magneticﬁeld is a constant of the motion.Of course,Eqn(4.5) may also be derived directly from the Lorentz force law,so that the time component of the relativistic generalisation of Newton’s Second Law,Eqn(4.1),contains no physical information not already contained in the spatial components.This is related to the fact that,as demonstrated in Eqns(2.25)and(2.26),the spatial and temporal components of the energy-momentum4–vector are not independent physical quantities.AcknowledgementsI should like to thank O.L.de Lange for asking the question whose answer,presented in Section4,was the original motivation for the writing of this paper,and an anonymous referee of an earlier version of this paper for informing me of related material,in the books of Jackson and Misner,Thorne and Wheeler,which is discussed in some detail in this version.References[1]A.Einstein,17891(1905).[2]M.Schwartz,‘Principles of Electrodynamics’,(McGraw-Hill,New York,1972)Ch3.[3]J.H.Field,Am.J.Phys.69569(2001).[4]D.H.Frisch and L.Wilets,Am.J.Phys.24574(1956).[5]F.J.Dyson,Am.J.Phys.58209(1990).[6]N.Dombey,Am.J.Phys.5985(1991).[7]R.W.Breheme,Am.J.Phys.5985(1991).[8]J.L.Anderson,Am.J.Phys.5986(1991).[9]I.E.Farquhar,Am.J.Phys.5987(1991).[10]S.Tanimura,Ann.Phys.(N.Y.)220229(1992).[11]A.Vaidya and C.Farina,Phys.Lett.153A265(1991).[12]ndau and E.M.Lifshitz,‘The Classical Theory of Fields’,(Pergamon Press,Oxford,1975)Section30,P93.[13]P.Lorrain, D.R.Corson and F.Lorrain,‘Electromagnetic Fields and Waves’,(W.H.Freeman,New York,Third Edition,1988)Section16.5,P291.[14]J.D.Jackson,‘Classical Electrodynamics’,(John Wiley and Sons,New York,1975)Section12.2,P578.[15]Actually,a careful examination of the derivation of Amp`e re’s from the Gauss lawof electrostatics in Reference[3]shows that,although Eqn(3.4)of the present paper is a necessary initial assumption,the deﬁnition of the magneticﬁeld in terms of the spatial derivatives of the4–vector potential occurs naturally in the course of the derivation(see Eqns(5.16)and(5.17)of Reference[3])so it is not necessary to assume, at the outset,the expression for the spatial components of the electromagneticﬁeld tensor as given by Eqn(5.1)of Reference[3].[16]C.W.Misner,K.S.Thorne and J.A.Wheeler,‘Gravitation’,(W.H.Freeman,San Fran-cisco,1973)Ch3,P71.[17]This should not be confused with a manifestly covariant expression for the LT,whereit is written as a linear4-vector relation with Lorentz-invariant coeﬃcients,as in:D.E.Fahnline,Am.J.Phys.50818(1982).[18]A time-like metric is used for4-vector products with the components of a4–vector,W,deﬁned as:W t=W0=W0,W x,y,z=W1,2,3=−W1,2,3and an implied summation over repeated contravariant(upper)and covariant(lower) indices.Repeated Greek indices are summed from0to3,repeated Roman ones from1to3.Also∂µ≡(∂∂x1,−∂∂x3)=(∂0;− ∇)[19]H.Goldstein,‘Classical Mechanics’,(Addison-Wesley,Reading Massachusetts,1959)P200,Eqn(6-30).[20]The alternating tensor, ijk,equals1(−1)for even(odd)permutations of ijk.[21]The explicit form of Aµ,as derived from Coulomb’s law,is given in standard text-books on classical electrodynamics.For example,in Reference[13],it is to be found in Eqns(17-51)and(17-52).Aµis actually proportional to the4-vector velocity,V, of the charged particle that is the source of the electromagneticﬁeld.[22]F.R¨o hrlich,‘Classical Charged Particles’,(Addison-Wesley,Reading,MA,1990)P65.[23]Reference[2]above,Ch3,P127.[24]Reference[14]above,Section11.9,P547.。

Concluding Remarks

a rXiv:as tr o-ph/39269v19Se p23CONCLUDING REMARKS ∗P.J.E.Peebles Joseph Henry Laboratories Princeton University Princeton NJ 08544USA pjep@ Abstract The mood at this conference is summarized in David Hughes’comment,“this decade will be amazing.”We’ve just had a pretty good ten years of advances in cosmology and extragalactic astronomy;why should we expect a repeat,another decade of comparable or even greater progress?The obvious answer is that there still are many more questions than answers in cosmology,and a considerable number of the questions will be addressed by research programs planned and in progress:we certainly are going to learn new things.But beyond that is the fact that there is no practical limit to the hierarchy of interesting topics to explore in this subject.I organize my comments on the state of research,and the prospects for substantial new developments in the coming decade of multi-wavelength cosmology,around the concept of social constructions.Keywords:Cosmology and extragalactic astronomySocial scientists inform us that the alpha members of a community set the social standards and constructions,enforce them by the weight of their authority,and see to it that the young members of the community are taught the standards so they will be remembered and enforced by the next generation of alphas.You have experienced all this in your careers in physical science.There is one minor diﬀerence –we replace the phrase “social construction”with “working hypothesis”–and one big addition –the remarkable fact that some hypotheses become so thoroughly checked as to be convincing approximations to reality.These comments on the ∗In Multi-Wavelength Cosmology,Mykonos,June 20032social constructions of cosmology include elements of their history,thepresent state of the promotions from constructions to established facts, and the prospects for continued additions to our understanding of thereal world.In their book,The Classical Theory of Fields,Landau and Lifshitzoﬀer a very sensible caution about the assumption that the universe isclose to homogeneous and isotropic on the scale of the Hubble length.When this book was published,in the1940s,the evidence for homo-geneity was sparse at best:this was a social construction.Now theobservational tests are tight and believable.Einstein was led to the pic-ture of homogeneity by his reading of Mach’s principle:he felt there hadto be matter everywhere toﬁx inertial motion everywhere.This argu-ment from a philosophical concept led Einstein to an aspect of reality.It is a mystery whether Einstein found the right picture for the rightreason.Landau and Lifshitz assume without discussion that general relativity theory applies on the scales of cosmology,which is fair enough in a surveyof theoretical physics.But at the time–theﬁrst revision of the Russianedition was published in1948–there was just one precision test of thetheory,the precession of the perihelion of Mercury,and hints of twoothers,the gravitational redshift and deﬂection of light.It certainlymade sense to consider the application of the theory to cosmology,butnot to trust it.The searching probes of gravity physics from the tests that com-menced in the1960s give convincing evidence that general relativitytheory is a good approximation on length scales ranging from the lab-oratory to the size of the Solar System,let us say to1013cm.TheHubble length,cH−1o∼1028cm,isﬁfteen orders of magnitude larger. This spectacular extrapolation is a social construction,until checked,which is the purpose of the cosmological tests.The results certainly look promising.An example is the broad con-cordance of evidence that the matter density parameter is in the range0.15<∼Ωm<∼0.3,from analyses of galaxy peculiar velocities,gravita-tional lensing measurements,the SNeIa redshift-magnitude relation,the cluster baryon mass fraction,the galaxy two-point correlation function, the cluster mass function and evolution,and the ratio H o t o of stellar evolution and expansion time scales.A recent addition to the list comes from the wonderfully successful comparison of the theory and measure-ments of the anisotropy of the3K thermal background radiation.This is a demanding test of the gravity theory,which has to describe the prop-agation of irregularities in the radiation distribution through strongly curved spacetime during the expansion factor z dec∼1000since the lastConcluding Remarks3 substantial interaction of matter and radiation,from initial conditions that have to agree with what grew into the structures observed in the distribution of gas at z∼3–in the Lymanαforest–and in the present distribution of galaxies.This has given a new check onΩm,from the apparent detection of a contribution to the temperature anisotropy from the matter distribution at modest redshifts,an eﬀect demanded by the theory ifΩm is signiﬁcantly below unity.Each of these measures of the mean mass density is subject to the haz-ards of interpretation in astronomy.But it is hard to see how systematic errors could aﬀect the many entries in this list all in the same way.Each measure depends on the assumed physics of gravity and the dark sector, which we are supposed to be testing.The test is the consistency:if we were using the wrong physics the broad concordance would be unlikely. The important thing from the point of view of the cosmological tests is not the value ofΩm but rather the convergence of evidence that the estimates of this number are not seriously confused by systematic errors in the observations or byﬂaws in the underlying theory:we have a good approximation to one aspect of the real world.Physical science can’t ex-plain why reality is a meaningful concept,but we can produce examples of approximations to it.This now includes the measurement ofΩm. The evidence that the physics of the standard Friedmann-Lemaˆıtre CDM cosmology is on the right track is a considerable advance,but incomplete.An assignment for this decade is to put the tests of gravity physics applied to cosmology on a systematic basis,in analogy with the program of tests of general relativity on the scale of the Solar System and smaller,though one would of course have to replace the parametrized framework of that program with a framework–maybe parametrized–that is adapted to what is relevant to cosmology.The cosmological principle and general relativity theory are exam-ples of deep advances in physical science that grew out of concepts of philosophy and elegance,which is why we pay attention to such ideas. The lesson is slippery,of course,because our ideas of what is elegant are adaptable.If the cosmological tests had favored the Steady State cosmological model we would be celebrating the perceptive foresight of a diﬀerent group of alphas.And recall the history of opinions of Ein-stein’s cosmological constant,Λ.Einstein came to quite dislike it.Pauli agreed.And Landau and Lifshitz(in the1951English translation by M.Hammermesh)asserted that“it hasﬁnally become clear that there is no basis whatsoever”for the introduction of this term.Others at the time paid no attention to this impressive list of alphas,and they seem to have been on the right track:now there is serious evidence for the detection ofΛ–or a term in the stress-energy tensor that acts like it.4Although most of us would agree that the universe could have done withoutΛ,the dark sector of theΛCDM cosmology is strikingly simple: the dark energy density is close to constant and the dark matter collects in nearly smooth halos by the gravitational growth of small Gaussian de-partures from a homogeneous primeval dark matter distribution.This picture for the dark matter was introduced two decades ago,and for some years was one of a half dozen viable models for galaxy formation. We had useful analytic solutions for explosion models,but serious chal-lenges in an analysis of the physics of a real cosmic explosion.A reliable analysis of the behavior of cosmic strings or monopoles or textures is even more diﬃcult.The CDM model is easy:structure is dominated by particles that move on geodesics,which are readily simulated in nu-merical computations,and there is the added advantage that structure forms later than in isocurvature variants,so an interesting numerical simulation need not deal with a large expansion factor.Simplicity rec-ommended the CDM model.Now we have substantial observational evidence that it is a useful approximation to what actually happened. Is the CDM model complete?One line of thought is that since the dark matter consists of the particles that happen to interact too weakly to be readily observable the dark matter is of course well described as a gas of weakly interacting particles.Another is that the real world seldom is that simple,but that it makes sense to start with the simplest working hypothesis we can get away with,which we will plan to use as a basis for the search for a better approximation,which might in turn lead to a still more complete theory.This is how the physics of the visible sector was discovered.The search for ideas about how the CDM model might be made more complete–if that is required–can be compared to what was happening in the1930s when Fermi,Yukawa,and others were trying out ideas of how elementary particles interact.Ideas then and now may be repre-sented by Lagrangian densities with forms like1L=Concluding Remarks5 Lagrangian the scalarﬁeldφis real,so the Yukawa interaction y a(φ−φa)¯ψaψa just changes the momenta of dark matter particles(represented by the spin-1/2ﬁeld operatorψa for the a th family,where y a andφa are real constants).If the potential V(φ)in the dark sector is close to Yukawa’s form,andµis relatively large,this is a model for the self-interacting cold dark matter picture.If V varies only slowly withφthe a th family behaves as particles with variable mass,m a=y a(φ−φa), and the gradient of the mass is aﬁfth force–a long-range inverse square force of attraction of dark matter particles that adds to the gravitational attraction.This physics traces back to Nordstr¨o m’s(1912)scalarﬁeld model for gravity in Minkowski spacetime,from there to the scalar-tensor gravity theories that were much discussed in the1950s and1960s, and from there to the present-day ideas of dilatonﬁelds that would have observable eﬀects–variable parameters–in the visible sector, and maybe a considerably strongerﬁfth force in the dark sector.A potential energy density V(φ)that varies slowly withφalso appears in a popular model for the dark energy or quintessence.The pedigree is impressive,and it suggests many scenarios for physics in the dark sector even without elaborations comparable to what happened to the model for particle physics after Fermi and Yukawa.To be seen is whether it might lead us to a model that can remedy apparent anomalies–some of which are mentioned below–in the standardΛCDM cosmology.If the present standard cosmology really diﬀers from reality enough to matter it will appear in anomalies.But there is a problem,that we cannot in practice unambiguously connect given physics and initial conditions to the details of cosmic structure that are revealed in the ob-servations.How do we decide whether apparent anomalies are only the result of the diﬃculty of modeling the physics,or whether real failures of the theory have been obscured by the modeling?We ned a new gener-ation of tests of reliability of the hypotheses that are used to model the connection between the theory and observations.The situation is similar to condensed matter physics,where complexity also drives model build-ing,but very diﬀerent in the sense that we have excellent reason to trust the underlying physics of condensed matter.We will gain conﬁdence in the physics of the dark sector by the accumulation of tests,including the examination of alternatives to CDM.This is another assignment for multi-wavelength cosmology.Two apparent anomalies that fascinate me have to do with the cosmic web and the galaxy merger rate.The cosmic web is a striking visual feature of numerical simulations of the CDM model,and the web does predict the observed walls of galaxies.But in maps of halo distributions in the simulations I see chains of dwarfs running into the voids between6the concentrations of the more massive halos,which I don’t see in maps of the real galaxy distribution.Maybe this is a result of the complexity of modeling the connection between theory and observations,exploration of which is part of the research assignment.For now I’m counting the cosmic web of galaxies as a social construction.I hasten to emphasize that I am deeply impressed by the success-ful account the cosmic web of gas oﬀers for the statistical measures of the Lymanαforest.On the other hand,I wish I felt better about the apparent lack of disturbance by whatever added heavy elements to the hydrogen in the Lymanαforest clouds.Another apparent anomaly is the rate of merging of closely bound galaxies.A pair of galaxies separated by a few half-light radii is routinely labeled a merging system,whether at high redshift or low,whether in a group or a rich cluster of galaxies.There is a good reason–simulations and analytic estimates predict the pair will merge in a few crossing times –but is it more than a social construction?The theoretical argument is sound,but only if we have the right physical model for the dark matter, as a nearly collisionless gas,which is not yet something we know.On the observational side,it is often said that the merger remnant of a compact group of spirals is an elliptical,but I also hear that the pattern of element abundances in the progenitors–typically late types–does not look like a promising match to the abundances in a typical early-type galaxy.Also puzzling is the eﬀect of mergers on the shape of the low order galaxy correlation functions.The two-point function is a good approximation to a power law from10Mpc separation down to separations of a few half-light radii.Standard estimates of the cosmic merger rate assume close pairs merge in a few orbit times.If so,what preserves the power law form?Again,I have to qualify these remarks.There is good evidence of mergers at low redshift:we see a clear example in the Centaurus group, where the big elliptical clearly has recently merged with a dusty galaxy, and there are several other classical examples of galaxies that surely are observed in the act of merging.But these spectacular systems do not seem to be all that common:the familiar examples are repeatedly cited. The assignment is to show whether the number of merging galaxies at low redshift really is consistent with the theoretical prediction that galaxies closer than a few half-light radii merge on time scales small compared to the Hubble time.We might consider also that we have to live with quite a few co-incidences within the standardΛCDM Friedmann-Lemaˆıtre cosmology. Heavily advertised nowadays is the coincidence that weﬂourish just as the universe is making the transition from matter-dominated expansionConcluding Remarks7 to an approximation to the de Sitter solution.Maybe related to this is the observation that weﬂourish just as the Milky Way is running out of gas for the formation of new generations of planetary systems like our own,along with the rapid collapse of the global star formation rate density since redshift z=1.Less widely discussed these days is the possible relation to Dirac’s large numbers coincidence:the ratio of the present Hubble length to the classical electron radius is close to another enormous number,the ratio of the electric and gravitational forces be-tween a proton and electron.Another timing coincidence I suppose is unrelated,but also curious,is that in the standard cosmology optically selected galaxies have just now become good mass tracers:the theory seems to predict strong biasing at redshift z=1.In the standard cos-mology the mass of a large galaxy is dominated by dark matter in the outer parts,and by stars near the center.The conspiracy is that the distributions of these two components produce a net mass density run that shows no feature at the transition from high to low mass-to-light ratio.Andﬁnally,theΛCDM cosmology predicts separation-dependent bias of light as a tracer of mass:the ratio of the mass autocorrelation function to the two-point correlation function of optically selected galax-ies is a function of separation.But it is curious that the galaxies seem to give the better approximations to power law forms for the low order correlation functions deep in the nonlinear clustering limit,rather than the mass that is supposed to control the dynamics.It is reasonable to expect that some of these curiosities are nothing more than accidents,and some will be seen not to be curious at all when we have a really good understanding of the theory and its relation to the observations.But it is sensible to be aware of the possibility that some are clues to improvements in the physics.What might come from continued multi-wavelength research on such challenges to cosmology?I expect the paradigms will continue to rest on the relativistic Friedmann-Lemaˆıtre model,or some good approxima-tion to it,because general relativity theory has passed quite demanding checks on the scales of cosmology.But we owe it to our subject to turn these scattered checks into a systematic survey of the constraints on the physics of spacetime and gravity.I do not expect a paradigm shift back to the Einstein-de Sitter model: the lines of evidence for lowΩm are impressively well checked by many independent applications of the theory that depend on quite diﬀerent elements of the astronomy.The evidence of detection of the cosmological constant is serious,too,but not as well checked asΩm.TheΛterm has been debated for more than eight decades;we can wait a few more years8before deciding whether it deserves a place in the list of convincingly established results.In the next ten years multi-wavelength observations,including(in the words of a participant)“millimeter,submillimeter and FIR observations with the imagingﬁdelity currently enjoyed by X-ray,optical,IR and radio astronomers”will produce an enormous increase in our knowledge of cosmic structure,and that is going to drive the development of es-ceedingly detailed models to relate the theory to the observations.The theory of choice will continue to beΛCDM,unless or until the observa-tions drive us to something better.While waiting to see whether that happens a assignment for model builders is to develop a convincing case for how far they have gone beyond curveﬁtting.After a major advance in a physical science,such as we have seen in the past decade in cosmology,there is the tendency to ask whether the subject has now reached completion,requiring only the“addition of decimal places.”You don’t hear this talk among astronomers,and I wouldn’t expect it to be on astronomers’minds in the coming decades, because there is no practical limit to the layers of detail one may study about things like the populations of stars,planetary systems,and civi-lizations that are communicating by radio broadcasts in the Milky Way, in the Magellanic clouds,and on out.We have good reason to expect theﬁrst decade of the21st century will be remembered as a golden time for cosmology,but we can be sure there will be room for productive applications of multi-wavelength cosmology for decades to come.My overall conclusion is that you should pay attention to the alphas–their concepts of simplicity and elegance really have led to deep advances in our understanding of the material world–but then go make the measurements–the alphas have feet of clay like everyone else.I am grateful to Manolis Plionis for inspiration,David Hogg and David Hughes for advice,and the USA National Science Foundation forﬁnan-cial support for this essay.。

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On the relativistic generalization of Maxwell’s velocity distributionJian-Miin Liu*Department of Physics, Nanjing UniversityNanjing, The People's Republic of China*On leave. E-mail address: liu@ABSTRACTSome problems relevant to the relativistic generalization of Maxwell’s velocity distribution are discussed.1Maxwell read his paper on equilibrium velocity distribution before the British Association in Aberdeen on September 21, 1859. In this paper [1], based on two assumptions (1) the velocity distribution function is spherically symmetric and (2) The x-, y- and z- components of velocity are statistically independent, Maxwell derived his equilibrium velocity distribution,M(y1,y2,y3)dy1dy2dy3=N(mK TB2π)3/2 exp[-mK TB2(y)2]dy1dy2dy3,(1a)M(y)dy=4πN(mK TB2π)3/2 (y)2 exp[-mK TB2(y)2]dy,(1b)where y r=dx r/dt, r=1,2,3, is the well-defined Newtonian velocity in the usual inertial coordinate system {x r, t}, y=(y r y r)1/2, N is the number of particles, m their rest mass, T the temperature and K B the Boltzmann constant. Assumptions (1) and (2) reflect structural characteristics of the velocity space, dY2=δrs dy r dy s, r,s=1,2,3,(2)in non-relativistic mechanics. Maxwell’s velocity distribution can be directly deduced from the Boltzmann distribution in the one-particle phase space in non-relativistic statistical mechanics.Maxwell’s velocity distribution is non-relativistic. From the theoretical point of view, as special relativity is a part of the laws of Nature, it is natural to consider the relativistic generalization of Maxwell’s velocity distribution. From the practical point of view, the relativistic equilibrium velocity distribution is necessary when we deal with statistical calculations where most particles crowd in the high-energy region or most particles crowd in the low-energy region but these particles in the low-energy region are not involved in the dealt statistical calculations. Calculations of the nuclear fusion reaction rate are an example of this type.2In our recent work [2,3], we made the relativistic generalization of Maxwell’s velocity distribution through analyzing relativistic velocity space,dY2=H rs(y)dy r dy s, r,s=1,2,3,(3a)H rs(y)=c2δrs/(c2-y2)+c2y r y s/(c2-y2)2, real y r and y<c,(3b)in the usual velocity-coordinates {y r}, r=1,2,3, where c is the speed of light. This velocity space can be represented bydY2=δrs dy’r dy’s, r,s=1,2,3,(4)in the so-called primed velocity-coordinates {y’r}, r=1,2,3, wheredy’r =A r s(y)dy s, r,s=1,2,3,(5a)A r s(y)=γδrs+γ(γ-1)y r y s/y2, (5b) withγ=1/(1-y2/c2)1/2 , (6) because δrs A r p(y)A s q(y)=H pq(y). We call y’r, r=1,2,3, the primed velocity.Using calculation techniques in Riemann geometry, we can findY2=δrs y’r y’s,(7a)Y2=[cy2nc yc y+−]2δrs dy r dy s,(7b)y’r=[cy2nc yc y+−]y r, r=1,2,3,(8a)y’=c2nc yc y+−,(8b)where (y’1, y’2, y’3) and (y1, y2, y3) denote the same point in the relativistic velocity space, y’=(y’r y’r)1/2. The relativistic velocity space is characterized by a finite boundary at y=c and the Einstein addition law in the usual velocity-coordinates and also characterized by unboundedness and the Galilean addition law in the primed velocity-coordinates. Differentiating Eq. (8b) immediately yieldsdy’=dy/(1-y2/c2).(9)The Euclidean structure of the relativistic velocity space in the primed velocity-coordinates convinces us of Maxwell’s equilibrium distribution of primed velocities,P(y’1,y’2,y’3)dy’1dy’2dy’3=N(mK TB2π)3/2 exp[-mK TB2(y’)2]dy’1dy’2dy’3,(10a)P(y’)dy’=4πN(mK TB2π)3/2 (y’)2 exp[-mK TB2(y’)2]dy’.(10b)Inserting Eqs.(5a-5b), (8b) and (9) into Eqs.(10a-10b), we obtain the relativistic equilibrium distribution of Newtonian velocities,P(y1,y2,y3)dy1dy2dy3= N (/)(/)/m K Ty cB2132222π−exp[-mcK TB28(nc yc y+−)2]dy1dy2dy3,(11a)P(y)dy= πc2N (/)(/)/m K Ty cB213222π−(nc yc y+−)2 exp[-mcK TB28(nc yc y+−)2]dy.(11b)This distribution fits to the Maxwellian distribution for low-energy particles (y<<c) but substantially differs from the Maxwellian distribution for high-energy particles. As y goes to c, it falls to zero slower than any exponential decay and faster than any power-law decay [4]. Such a new kind of decay mode has been used to explain the observed non-Maxwellian high-energy tails in velocity distributions of astrophysical plasma particles [4]. The distribution Eqs.(11a-11b) has been also used to calculate the nuclear fusion reaction rate [5]. The nuclear fusion reaction rate based on the relativistic equilibrium distribution, R, has a reduction factor with respect to that based on the Maxwellian velocity distribution, R M,R=tanh QQR M, (12)where Q depends on the temperature, reduced mass and atomic numbers of the studied nuclear fusion reactions. Since 0<Q<∞, 0<tanhQ/Q<1 and0<R<R M. (13) Eqs.(12) and (13) signify much to the solar neutrino problem.3Our paper [2] was submitted to physics-professional journals for publication but refused. From referees, we got a gain to knowledge about other trials of relativistic generalization of Maxwell’s velocity distribution.In a referee’s report (Ref. X/13559/XXX), we read: “The velocity as contrasted to the momentum distribution for classical relativistic particles is well known: N.E.Frankel, K.C.Hines, R.L.Dewar---Energy loss due to binary collisions in a relativistic plasma, Phys.Rev. A20, 2120-2129 (1979)” We went for the mentioned paper but was unsuccessful. For the relativistic generalization of Maxwell’s velocity distribution, there is nothing in the paper except to write down velocity distribution formulaf(w)dw ~ w 2(1122−w c /)5exp{-mc K T w c B 2221−/}dw,(we omitted a proportional coefficient) and to name it the isotropic relativistic Maxwellian velocity distribution. There is no explanation, no derivation and no reference for this formula. In f(w), exponential factor exp{-mc 2/K B T(1-w 2/c 2)1/2} seems to come from substituting the relativistic one-particle energy for the non-relativistic one in Maxwell’s velocity distribution, while factor w 2(1-w 2/c 2)-5/2 is quite out of understanding.In another report (Ref. XXXX/142/2002), a referee wrote: “however, the simple relativistic generalization of Maxwell’s distribution function is just:f(p)=exp{-12K TB [(p 2c 2+m 2c 4)1/2-mc 2]}i.e. the replacement of the non-relativistic kinetic energy by its relativistic equivalent.” Correcting an error in the exponent of f(p) and representing f(p) in terms of velocity, we find referee’s velocity distribution proportional tof(y)dy 1dy 2dy 3 ~ exp{-1K T B [mc y c 2221−/ -mc 2]}dy 1dy 2dy 3.The referee also recognizes substituting the relativistic one-particle kinetic energy for the non-relativistic one in Maxwell’s velocity distribution.Maxwell’s velocity distribution Eq.(1a) can be rewritten as~ exp{-E/K B T}dy 1dy 2dy 3,(14a)E=m(y)2/2.(14b)Two referees, from two different journals, both suggested the replacement of the non-relativistic kinetic energy by its relativistic equivalent, in Maxwell’s velocity distribution, for relativistic generalizing Maxwell’s velocity distribution. They asked~ exp{-E/K B T}dy 1dy 2dy 3,(15a)E=mc y c 2221−/ or [mc y c 2221−/-mc 2] ,(15b)as a relativistic generalization of Maxwell’s velocity distribution, where Eq.(15a) is the same as Eq.(14a).However, Eq.(14a) is linked to the Boltzmann distribution in the one-particle phase space, in non-relativistic statistical mechanics, for a confined many-particle system with weak interactions and zero external forces in volume V,~ exp{-E/K B T}d ω,(16)where d ω=dx 1dx 2dx 3dp 1dp 2dp 3 is a volume element of the phase space, p r =my r , r=1,2,3, while Eq.(15a)does not exist in the framework of special relativity.In non-relativistic statistical mechanics, one can depend upon Boltzmann’s hypothesis on the equality of the probability of equal volume element to derive the Boltzmann distribution. It should be noted that Boltzmann’s hypothesis is appropriate only for uniform phase space structures. The phase space is a direct product of coordinate space and momentum space. In non-relativistic mechanics,coordinate space is dX 2=δrs dx r dx s , r=1,2,3, while momentum space is dP 2=δrs dp r dp s due to velocity space Eq.(2), where p r =my r , r=1,2,3, and P=mY. In the framework of non-relativistic mechanics, the one-particle phase space has a uniform structure in {x r ; p r }, r=1,2,3. On the contrary, using the relativistic relations between momentum and velocity,p r =my r /(1-y 2/c 2)1/2, r=1,2,3,(17a)y=cp/p m c 222+, (17b)we find from Eq.(7b),P 2=[mc p 2n ()()//m c p p m c p p 2221222212+++−]2δrs p r p s , r,s=1,2,3,(18)where p2=δrs p r p s, r,s=1,2,3. The relativistic momentum space can not be of uniform structure in {p r}. The relativistic one-particle phase space can not be of uniform structure in {x r; p r}, either. In the framework of special relativity, we do not have Eq.(16). In the framework of special relativity, we do not have Eq.(15a), either, even for a confined many-particle system with weak interactions and zero external forces in volume V. For the relativistic generalization of Maxwell’s velocity distribution, it is non-sense to put Eq.(15b) in Eq.(15a) or to replace the non-relativistic kinetic energy by its relativistic equivalent in Maxwell’s velocity distribution.ACKNOWLEDGMENTThe author greatly appreciates the teachings of Prof. Wo-Te Shen. The author thanks Dr. P. P. Rucker for helpful suggestions.REFERENCES[1]J. C. Maxwell, Phil. Mag., 19, 19 (1860)[2]Jian-Miin Liu, cond-mat/0108356[3]Jian-Miin Liu, Chaos Solitons&Fractals, 12, 2149 (2001)[4]Jian-Miin Liu, cond-mat/0112084[5]Jian-Miin Liu, nucl-th/0210058。