Landaus Last Paper and its Impact on Developments in Mathematics, Physics and Other Disciplines in New Millennium

Landau’s Last P ap er and its Impact on Dev elopmen t s in Mathematics, Ph ysics and Other Discipli n es in New Mi llennium A.L. Kholo denk o 1 375 H . L.Hunter L ab or atories, Clemson Un i versity, Clemson, SC 29634-09 73, USA Abstract. In this pa per we discuss the impact o f the last Landau pap er on ph ysics a nd mathematics to date and ma k e some predictions a bout its lik ely impact on sciences in new millennium. Keywor ds : theor y of strong interactions, a symptotic freedom, standard mo del, uniﬁed ﬁeld theor ies, hypergeo metric functions of m ultiple a rgumen ts, Knizhnik- Zamolo dc hiko v and GKZ equations, mirror symmetry , top ological ﬁeld theories, theory of sing ularities, theory o f knots and links, string theory , theory of ra n- dom fra gmen tatio n and coag ulation pro cesses, chemical kinetics, spin gla sses, po pulation gene tics , economics , computer science, linguistics, fo rensic sc ience P A CS (20 0 8): 12.4 0.-y , 12 .60.-i, 11.25 .Hf , 12.10.j, 02.10 Kn, 11.25.- w, 05.40.- a 1 E-mail address: string@clemson.edu 1 1 In tro duction 1.1 General remarks The brightest r epresen tatives of art, say , Mozart, Mendelsohn, etc., even though they were known as wunderk inds from ear ly age, achiev ed their full maturity only at the time of their death, e.g. recall Requiem (for Moz a rt) or 9-th sym- phony for Beethov en, etc. In the case o f Landa u, his scien tiﬁc abilities ha d bec ome apparent very ear ly and his in ter national acclaim came to him when he was still in his 30 ies. Nevertheless, like with other greates t , the most pro- found of his pap ers were w r itten in a shor t per iod of only 3 years, from 195 6 to 1959 (e.g. s ee the c o mplete lis t of Landau pa pers in this colle ction) pr ior to his tragic fa ta l car accident, whic h happ ened on 7th of January of 196 2. As in the c a se of Mozart or Beethoven, these works sta nd in their own class es since, as if anticipating his own end, with these works he made an attempt to take a lo ok into distant future trying to fores e e cont ours o f the ultimate ” theo ry of everything”. In his la s t pap er ” A b out F undamental P roblems”, written in 1959 on the o ccasion of W olfgang Pauli’s death, he summarized his own eﬀorts as well as those whom he co nsidered as the b e st from his genera t ion o f physicists. In retros pect, very muc h lik e Einstein who wrote his own obituary at the age o f 68, this pa per happ ened to bec o me Landau’s obituar y as well. Abo ut 50 years after this pap er [1] was written, it is impo ssible not to b e a s- tonished by Landau’s vision of the future, esp ecially , in v iew of his own cautious words: ”As is well k no wn, in the case of strong interactions, theoretica l ph ysics at this momen t is to a la rge extent p o werless. Therefore, any attempts at making statements r egarding future directions of this ﬁeld should b e considered as hig hly sp eculativ e (and, hence, prophetic) causing their authors to b e in ev er increasing danger of sho oting s tr aigh t int o the sky .” In spite of these cautious remarks, he go es on in this paper and mak es ”profecies” which happ en to b e higly accur ate. What makes this pap er tr uly remark a ble is the fa c t that in it (in retrosp ect) he actually w as sp eaking on beha lf of the who le generation of physicists o f his time. The famous photograph of Dau sitting next to Geo Ga mo w and Hendr ik Kramer s at the Niels Bohr Institute seminar ca rries m uch more than the ﬂa vor of these exciting tim es. Lo oking back, o ne notices that the peo ple sitting in the front row on this picture: Bohr, Heisen ber g, Pauli, Gamo v, L a ndau and K r amers, although of diﬀerent age at the time when the picture was taken-all died within the same time spa n o f ab out 10- 15 years. Landau’s pap er is a n o bit uary for W. Pauli (died in 1958 ), but then, N.Bohr died in 19 62, H.Kr amers-in 1952, G.Gamov -in 19 6 6, Landau –in 1968 and only W.Heisen b erg died later in 1976 . These circumstances make the conten t of his la st pa per esp ecially signiﬁcant. This is so, b ecause 50 years 2 later up on its reading, corr ectness of Landau prophecies is truly remar k able. In the rest of this pap er we would like to explain why this is indeed so and, based on this, w e also will make a few predictions of what lies ahead. 1.2 Ov erview of t h e last Land au pap er In his obituary , Landau was c o ncerned ab out the future of high energy ph ysics leaving a side predictions for other bra nc hes o f ph y s ics. This is justiﬁable since historically , esp ecially in Landau’s time, the high energ y physics was the ulti- mate source of inspiratio n for the res t o f physics. Landau b egins with discussion of utilit y of metho ds o f renor m alization and renormalizatio n g roup. He notices that in the ca se of strong interactions use o f the renorma lization gr o up meth- o ds leads to pa rado xical r esults: ev en if one b egins with very large coupling s at low energies, at high energies the coupling tends eﬀectiv ely to zero. This phenomenon of ” asymptotic freedom ” was discov ered by Pomeranch uk in 1 9 55. A t the time of this discov ery , to cure such a nuliﬁcation the nonlo calit y of inter- actions was cautiously sugge s ted. This nonlo cality idea was not welcomed by Landau since, in his opinion, all co nsequences of the ma th ematical a pparatus of quan tum ﬁeld theo r y made without an y relia nce on a sp eciﬁc model Hamil- tonian require lo cality and are supp orted by all av ailable experimental da t a, in particular, by sucsessful use of disp ersion r elations. F urthermo re, in supp ort o f his claim, Landa u use d the theory o f random fragmentation developed by F ermi [2] for calculation o f angular distribution of pions pr oduced in (high energy) collisions of tw o nucleons. It req uired the applica tion of concepts of sta t isti- cal thermo dynamics to space-time volumes m uch smaller than those needed for developmen t of the no nlocal theory . These ideas were chec ked exp erimen tally with la r ge degree of sucses s 2 . Next, La ndau notices tha t Heisenberg [3 ] recently expressed the opinion that the existing theor y most likely should undergo sub- stantial c ha nges ca used b y his S- ma trix theory . These c hanges how ever should not b e done at the exp ence of remo ving the lo cality r equiremen t in view of the assumed v alidity of disp ersion relations implying Lorentzian ca usalit y . Landau go es on and makes a cautious r emark ab out usefulnes o f Heisenberg’s nonlin- ear ﬁeld theory also discussed in [3 ] 3 . His reser v ations are based on r esults of his work with R.Peierls (done in 1930 ) whic h questions the existence of wav e functions in rela tivistic qua ntum ﬁeld theories. It sho uld be sa id, how ever, that such negative result he attributes only to the theory of strong interactions. In- deed, he writes : ” Op erators ψ , containing nono bs erv able informatio n, should disapp ear fro m the theo ry and, since the Hamiltonian ca n b e made only fro m ψ ′ s, we necessar ily come to the conclusion that the Hamiltonian metho d for strong in tera ctions o utliv ed itself and should b e burried, of co urse, with all due resp ect”. He goes on by saying that the ” bas is o f new theory sho uld b e ma de out of diag rammatic technique , which uses only diagra ms with ”free” ends, that is 2 Deatails wi ll be discussed b elo w. 3 It should b e nothed thou gh that Landau a vo ids any explicit ment ioning ab out this non- linear ﬁeld theory and only r emar ks that Heisenberg’s assumptions in his opinion are ques- tionable. 3 t ypical for scattering a mplit udes and their analytica l contin uations . The ph ysi- cal foundation o f suc h a forma lism is made of pr inciples of unitarit y a nd locality of in teractions revealing themselves in analytica l prop erties of fundamen ta l ob- jects of this new theor y , for instance, in all kinds of disp ersion relations ..... As r e sult o f s uc h a n approach, the old problem of ”element arity” o f elementary particles lo oses its meaning since it cannot b e formulated without interaction betw een particles.” This p oint of v iew is in acco rd with that made b y Heisen- ber g in [3]. The ab o ve quo t ations taken from Landau pa per comprise to a large extent its conten t. 2 Impact of Landau pap er on high energy physics Since the v a ue of science for so ciety is bas ed on its ability to expla in the pr esen t and to predict the future, we would like to a pply these criteria to Landa u’s last pap er. W e b egin with The theo ry o f random fragmentation [2]. Developmen t of this theory resulted in development of QCD and e xplanation of the pheno m enon of asymptotic free- dom 4 . W e restore some details of this development fo llo wing the recent r eview pap er by T annenbaum [4]. In view of this, for the sake of space, we squee z e the nu mber of key references to the absolute minim um. In the early 60 ies (the time when Landau was inca pacitated but still alive) constr uct ion o f proton accele r- ators with ener gies well ab o ve the thr eshold fo r anti-proton pr oduction, a zo o of new particles was discovered. Gell-Mann and Ne’eman noticed that pa r ticles sharing the same quantum n umbers (spin, parity) follow the symmetry o f the Lie group SU(3). This g roup is built of three genera tors which they asso ciated with three ﬁctitious particles (quarks): u- for ”up”, d-for ”down” and s-for ”strange ” - a ll having spin 1 / 2 and fractio na l c har ge. According to the emerging picture meso ns are ma de of tw o qua rks while the bar ions-of three.This lea d to the discov ery of a new Ω particle in 19 64 based on predictions o f Gell-Mann- Ne’eman theory . This particle had an appa ren t problem with the P auli principle since it was made o f three s-quar k s. T o av o id this proble m, it was sug g ested in the same year that qua rks come in 3 colo ur s. The ma jor bre a kthrough ca me after it w as rea lized that the SU(3) s y mm etry is NOT ass ociated with three quarks but with three colours a nd that co lour-c harged gluo ns ar e the quant a of asymptotically free strong int eractio ns which bind ha drons. QCD a nd the asymptotic fre edom. Although the path to as ymptotic free- dom b egins with Pomeranch uk’s pap er wr it ten in 1955 5 , it had many twists and turns befor e it reached its ﬁnal destination culminating in works by Gross and Wilczek [5] and Politzer [6]. The a bov e mentioned breaktr uth lead to re- newed attention to the Y ang-Mills (Y-M) theory [7] which was un till that time in the shadows b ecause of pro blems with its r enormalizability . The ma jor ad- v ancements in solv ing the renormaliza tion riddle for the Y-M ﬁelds o ccured in 4 Muc h more ab ou t this theory wil l b e said in Section 5. 5 E.g. see Ref.[2] in Landau’s paper [1]. 4 1967 (when Dau was still alive!). In this year F addeev and Popov [8], fo llow- ing ea rlier g eneral idea s by Dirac [9], came up with the cor rect p erturbative scheme for the Y-M ﬁelds which takes into account gauge constraints and gauge redundancy . The famous F addeev-Pop o v auxiliary fermionic ﬁelds (the F addeev-Pop o v ghosts) were incorpo rated into ga uge-in v ariant reno rmalization scheme indep enden tly by Slavnov in 1971 [10] and ’t Ho oft and V e ltma n in 1972 [11]. In the mean-time using re normalization group metho ds Bjor k en ca me up with his famous parton mo del in 19 69 [12] which was sucsessfully tested exper i- men tally and w as instrumen tal in pro ving the a s ymptotic freedom for the Y-M ﬁelds [5 ,6]. In ab out the same p eriod V eneziano , following Regge ideas a bout the likely form o f asymptotics of scattering proces ses at very high energies, and in complete accord with Landau’s predictions (regar ding the cen tral role o f am- plitudes ) came up with his famo us amplitude in 19 68 [13], the year of La ndau’s death. This amplitude, as is well k nown, gave birth to dual r esonance mo dels [14] and str ing theory [15]. Inciden tally , muc h later o n the whole diag rammatic machinery of p erturbativ e calculations in quantum ﬁeld theor ies (of whatever kind, in general, and of QCD in par ticular) was redone in the string-theo retic fashion [16] making practical calculations cons ide r ably ea sier. W e shall discuss this sub ject in some detail in Section 4. In the meantime, we would like to explain wh y it w as nesessar y to come up with string theory in the ﬁrst place. The standar d mo del . The sto ry b egins with t wo names: Goldstone and Higgs. In 1 961 Goldstone published his pap er [17 ] on g lobal symmetry br eaking which is alwa ys b eing accompanied by the emergence o f a mass less particle, the ” Golstone bos on”, while Higgs in 196 4 published a pap er on lo cal symme- try breaking causing emergence of the massive particle s ubsequestly called the Higgs b oson [18]. In b oth cas es, it is rather ea sy to recognize Landau’s in- put since the globa l symmetry br eaking phenomeno n follows directly fro m his work on phenomenolog ical theory of phase transitions [19], while the br e aking of lo cal sy mmetr y ca us ing emer gence of the massive Higgs b oson is contained in his w ork (with V.Ginzburg ) on the phenomenolog ical theory of super c on- ductivit y [20]. Brea king of lo cal symmetry in this ca s e allows to explain the Meissner eﬀect in superco nduct ors: intitially g auge-in v ariant Lagr a ngian of the Ginzburg-Landa u theory of superco nduct ivity (technically known as Lagra ngian for scalar electro dynamics) formally lo oses its manifest gaug e -in v ar iance since the ma s sless electromagnetic ﬁeld acquires mass a s a result of sp on taneous lo cal gauge symmetry bre aking. Emer gence of this mass c a uses the mag net ic ﬁeld to b e exp elled from the bulk of the sup erconductor. This is the microsco pic cause of the mac r oscopically obs erv ed Meissner eﬀect. Since man y con tributions to this centenary volume disc uss this and (related to it) topic(s) in s uﬃ cient detail, w e contin ue with other issues in our revie w. In their Nobe l Prize winning lectures b oth W ein b erg and Salam emphasize d the imp ortance of Goldstone and Higgs (and, consequently , Landau’s) idea s in formulation of the uniﬁed theory of weak and electr omagnetic interactions. In the Lagr angian of the W ein b erg-Salam (W-S) mo del one ca n easily recognize the Ginzburg -Landau (superc onducting) pa rt resp onsible for the emerging Higgs bo son. T o make any ser ious calculations with this mo del requir es use of the 5 renormaliz a tion gr oup methods a pplica ble to the non- Abelian Y-M theor ies in which gaug e inv aria nce is sp on ta ne o usly broken. These methods w ere developed by ’t Ho oft and V eltman [11 ]. With help of these metho ds, it has b ecome p ossi- ble to extend the W-S mo del in o rder to include strong in ter a ctions. The mo del which uniﬁes weak, electromag netic and strong interactions has become known as the standard mo del . Many pr e dictions of this mo del were thoroughly tested exp erimen tally . An excellent summary of both theoretical and exper imen tal results r elated to the s tandard mo del can b e found in a recent monog raph by Bardin and Passarino [2 1]. More relaxe d a nd up t o date exp osition of the satandard mo del ca n b e found in [22]. Higgs b oson and uniﬁed ﬁeld theories . With all the sucses ses of the stan- dard model, it suﬀers from tw o ma jor drawbac ks: a) it needs Higgs (Ginzbur g - Landau) ﬁeld to b e re normalizable, b) it do es not a ccoun t for gravity . Inclusion of gravit y in to existing uniﬁcation scheme is complica t ed by the well kno wn problems of renormalizability of gravit y . Although, as was demonstrated by Utiy ama [23 ], b oth g ra vity and the Y-M theory are obtaina ble as result of im- po sition o f the requirement of lo cal gauge in v a riance with resp ect to: a) external (Lorentz) gro up-for gravit y o r, b) with resp ect to internal (say , isoto pic) sym- metries in the case of Y-M, o n the underly ing ﬁeld-theor etic Lagrangia n, from the point o f view of p erturbative renormaliz a tion group a na lysis, g ra vity es capes treatmets which a r e successful for the Y-M ﬁelds. As result, the string theo ry was prop osed. It is a no nlo cal theory , an o ption which Landau had an ticipated but did not like, and is still in the pro cess o f developmen t which, a t the time of this wr iting, has la sted alr eady for 4 0 years. More on this will b e said in Sections 4 and 5. As we’v e mentioned alr eady , the existing s tring-theoretic for ma lism can b e adapted to the standard mo del [16 ] so that if the string theo r y is sucsessful, then the standard mo del is easily recov erable from it. Having sa id this, w e silently assumed that the sucse s s of string theory dep ends very muc h on its ability to incorp orate and to trea t gravity on equal fo oting w ith the r est of the forces. This leav es the do or totally open for explanation of the exis tence of the Higgs b oson. In the s t andard mo del, even though it is not yet discov er ed, its existence is not questioned 6 bec ause o f the follo w ing : a ) ma n y predictio ns of the standard model not directly in volving this bo son happ en to be corr ect [21], b) this b oson is needed for this mo del to b e r enormalizable, c) only one such b oson is suﬃcient fo r the whole mac hinery to work eﬀectiv ely 7 . The artiﬁciality of the Higgs b oson can be completely removed thoug h if the alr eady published results contained in our latest works [24,25] b ecome commonly a c c epted. In these pap ers we demonstra ted that the auxiliary Ginzburg-Landau (Higg s) functional 6 V ery mu ch the same as the existence of gra vitational wa ves, still to be discov ered. It is int eresting to notice that the Go og le searc h database has 599,000 en tries on H i gg s b oson and 513,000 entries on gravit ational wa ves. In b oth cases one can ﬁnd man y claims that these ob jects will b e disco ve red, if not to d ay , then, at worst, b y tomorro w morning. 7 Theoretically , nothing for b ids us f r om studying the supersymmetric version of the Y-M model (and, hence, of QCD and the standard model). Such a model would r eq uire more than one Higgs boson for its renormalizabilit y . Without disco v ery of the ”standard” Higgs b oson suc h a mo del th us far is only of mathematical interest (e.g. r e ad the next section). 6 of the standard model can be ( without an y appr oximations !) rewritten a s the Hilber t-Einsten functional for gravit y . This means that s uch an equiv alence conv er ts the auxiliary functional for the Higgs ﬁeld in the standar d model int o that for gr avity thus ma king the alr eady exis tin g theory self-c o n tained 8 . Such an a pproach diﬀers substantially from that recently propo sed by Lisi [26] since it do es not r e quire any additional eﬀorts fo r developing the ”theo r y of everything”. It only changes our interpretation of the Higgs b oson (very m uc h like the work by Higg s which r eplaced more narrow treatment of the Ginzburg- Landau functional ta ylored fo r sup ercondactivit y b y ma t hematically bro ader tratment enabling to e xtract the under lying universality of the Higgs-La ndau- Ginzburg phenomenon). 3 Impact of Landau pap er on mathematical ph ysics Exactly in tegrable low dimensio nal qua n tum systems . Although Landau was against the nonlinear ﬁeld theory put forward by Heisen ber g [3] and was a dv o- cating instead his diag rammatic metho ds for amplitudes, his intu ition in this case was incorrect. Subsequent dev elopments revealed that, actually , b oth (his and H eisenberg ’s ) approaches ar e equiv alent . Indeed, in 1979 the ground breaking pap er b y Zamo lodchik ov bro ther s was published in Annals of Ph ysics [27]. In it, in complete accor d with Heisenberg , the ex a ct S- matrices were con- structed for a n umber of manifestly nonlinear relativistic mo de ls in 1+1 dimen- sions. Chronolo gically , this work ca n b e co nsidered as a culmination of eﬀorts of many p eople, b egining with Mc Guir e [28 ] , in 1 964, then Y a ng [29], in 1967 , then B a xter [30 ] in 1971, and many many others . The histor y of discov ery of, now famous , Y ang-Ba xter (Y-B) equations is well do cumen ted in the collection of reprints asse mbled by J im b o [3 1] where o ne can ﬁnd as well many appli- cations o f these equa tio ns to exactly integrable systems. In fa ct, the class ical (quantum) sys tem is exac tly integrable only if the asso ciated cla ssical (quan- tum) Y-B e q uations can b e solved [31,32]. The task of solving these equations was initiated by Bela vin and Drinfel’d [33] and culm unated in Drinfel’ds pap er on Hopf algebras and quan tum Y-B equation [34]. An excelle t r eference on Hopf algebras , qua n tum groups , Y-B equations , Drinfel’d as s ociator s and Kniz hnik - Zamolo dc hiko v (K-Z) equations is the b ook b y Kasse l [3 5] where one can also ﬁnd connections of all these to pics with theor y of knots and links. The sa me topics are discus sed in a somewhat more fo rmal wa y (with emphasis o n sym- plectic asp ects of these issues not pr e sen t in Kassel’s b ook) in the monogra ph by Chari and Pressley [36]. A purely combinatorial, Heisenberg-style exp osition of these topics ca n b e found in our recent work on Heisenber g’s honeycombs [37] in whic h our readers can ﬁnd a v ery elementary intro ductio n to K-Z equa - tions. These equations were disc o vered by Knizhnik and Zamolo dch iko v [38] in connection with their study of the W ess-Zumino-Noviko v- W itten (WZNW) mo del (the nonlinear sigma model with the topo logical WZNW ter m [39]).Such 8 Under such equiv alence the Higgs boson should b e renamed as the Higgs-Y amab e con- formon. 7 a mo del emer ged as a by-pro duct of development s in string theo ry a nd in Y- M theory 9 but is o f independent (from thes e theories) interest and v alue in view of its wide uses in condense d matter [39,42 ] and confor mal ﬁeld theorie s [43] wher e, incidentally , the Ginzburg-Landau description of such mo dels exists [24,32,4 3]. In his lecture notes V ar c henko arg ue d [44] that all hyperg eomet- ric functions of multiple arguments are solutions of the K-Z equa tions. Among these h yp ergeometric fu nctions of multiple ar gumen ts are all Landau a mplitudes as demons trated by Golub ev a [4 5] and will b e discussed in Section 4. Hence, by studying these aplitudes as suggested b y La nda u, one inevitably runs into K-Z equations, quantum gr oups, Y-B equations, knots and links, etc. The Y ang-Mills theor y in v arious dimensions a nd Ginzburg-Landau mo del of sup erconductivity . The r ole of self-duality . Most likely , s ho uld he b e alive, Landau might argue, based o n the knowledge of his time, that the S-matrices obtained b y Zamolo dchik ov br others are of academic interest since they only inv olve nontrivial qua n tum ﬁeld theories in one time and one space dimensions (whic h is r eﬂected in his skepticism of the Heisenberg’s pr o gram). Suc h a co n- clusion, how ever, is not cor rect b ecause of the following. On o ne hand, these quantum models do have s igniﬁcance in r eal life in view of their exp erimen tally suppo rted connections with spin chains [39,42]. On another ha nd, attempts at quantization of higher dimensional nonlinear qua n tum ﬁelds are most likely to be in v ain (in acco rd with Landau!). The co m binator ial ar gumen ts in fav our of this co nclus ion ar e pr esen ted in o ur recent works [37, 46 , 4 7]. They also fol- low from Golub ev a ’s pap er [4 5], and will b e further explained in the following section. In supp ort of our p oin t of view, w e also would like to mention the problem of quantization of pure Y-M ﬁeld. The nontrivialit y of such a quanti- zation alrea dy b egins in 2 dimensions wher e serio us study of pure Y-M living o n Riemannian surfaces w as initiated b y A tiyah and Bott [48]. Their work caused an av a la nc he of subsequent w orks by ma t hematicians. In par ticula r, in tw o dimensions, the complete solutio n of the pure Y-M ﬁeld theor y was found by Witten [49,5 0] whose work was signiﬁca ntly inﬂuenced b y papers o f Atiy a h and Bott [48] a nd Migdal [51]. In dimens io ns higher than tw o the situation for m ally remains muc h less tra ctable since the tw o-dimentional metho ds resist general- ization to higher dimensions. Nevertheless, in some sp ecial cases, e.g. when the Y-M is (anti) self-dual, the theory can b e dev elop ed quite substan tially . W e belie v e, based on the results to b e describ ed immediately be lo w, that only these realizations of the Y-M ﬁeld make physical s ense. W e beg in with very signiﬁcant work by Belavin et al [52 ] done in 1975 . In it, the insta n ton s olution of the Euclidean v er sion of the Y-M ﬁeld theory was found using essen tially ideas of self-duality . In the same pap er the Cher n-Simons (C-S) functional was used for the ﬁrst time in physics literatur e 10 . Shortly thereafter, ’t Ho oft substantially improv ed calculations of Belavin et al (who es sen tially obtained only the lea ding sa ddle po in t result for pure Y-M) b y making these cal- 9 Where it originated f rom the Chern-Si m o ns ﬁeld theory [40,41] which itself is obtainable from the self-dual Y-M theory to b e discussed b e low. 10 The CS theory wa s used later on by Witten [40] in his Fields medal winning paper on th e Jones p olynomial in w hi c h man y ph ysical applications are discussed. 8 culations more r ealistic b y incorp orating fermions in ter a cting via Y-M ﬁelds into these sa ddle p oint-t yp e calculatio ns [53].Atiy ah, Hitchin, Drinfeld and Manin (AHDM instanton) develop ed results b y Belavin e t a l muc h further using meth- o ds of algebra ic geo m etry .A very readable a ccoun t of these and r elated w orks can be found in the review paper by Belavin [54]. Donaldson used essen tially these AHDM results for development of his theory o f top ological classiﬁca tion of four manifolds [55]. In this theor y the anti self-dua l Y-M ﬁelds play prominent r ole since only such ﬁelds can b e asso ciated with complex structures on 4-manifolds. A very rea dable a ccoun t of thes e developmen ts including nice ov erview of the theory o f 4 manifolds is g iv en in a review by Soloviyo v [56 ]. More up to da te information, including that on Seib erg-Witten (S-W) theo ry , which enabled to simplify considerably the Donaldson theory is given in the paper b y T y r in [57]. The mo nograph by Nico laescu provides an unprecede nted amo un t of rea dable material relating the S-W a nd Donaldson theories [58]. At the same time, the S-W theory is just an elab oration o n the phenomenologica l Ginzburg-Landa u (G-L) theory of sup erconductivit y as explained in detail in the monogra ph b y Jost [59 ] 11 . In S-W theo ry the Abelian gauge ﬁeld of G-L theor y is r eplaced by the non-Abelian Y-M gauge ﬁeld, the bos on ﬁeld is repla ced b y the spinor ﬁeld and the G- L co v aria nt deriv ative for scalar ﬁelds is r e pla ced b y that a ppropriate for the Dirac ﬁelds. In co nclusion, we would like to mention that practically all known e x actly int egrable systems can b e obtained from the solutions of the (an ti) se lf- dua l Y-M ﬁeld e q uations of Donaldson’s theory [62]. Similar r esults for the S-W theory a re dis cussed in the mono graph by Marsha k ov [63 ].This means that 4- dimensional physics, including g ra v it y (in this cas e, the self-dual gravity [64 ]) can b e reduced to the study of tw o dimensio nal problems in the s pir it of the progra m outlined b y Heisenberg [3] and in acc ord with re s ult s of our latest works [37,46,4 7] 12 . Hitc hin equations , Langla nds duality , etc . With exception of the S-W theory which is an excellent to ol for clas siﬁcation of 3 and 4 manifolds but, most likely , cannot b e used in its or iginal form for detailed compa rison with ex periment, we hav e not discussed the supersymmetric versions o f Y-M theory . Mathematica lly , to study such v er sions is in ter esting and leads to issues such as mirror symmetry , Langlands duality , Hitc hin’s equations, etc. [65]. How ever, physically , such theories would requir e more than o ne Higgs boson. Since such a bo son r emains to b e discov e red even in the no nsupersymmetric cas e 13 , there is no re a son to go on in this review with descr iption of ramiﬁcatio ns caused by the e ﬀects of sup e rsymmetry (ma y b e, only with the exception of the F addeev-Popov ghos ts which do admit s upersymmetric trea tmen t [66] 14 in the standard Y-M theory). 11 See al so the b ooks by Pismen [60], where emphasis on physical aspects of G-L theory is made, and by Y ang [61], where mathematical asp ec ts of [ 6 0] are discussed in detail. 12 More on this is discussed in Sections 4 and 5. 13 In spite of the very detailed experi m e nt al set ups and theoretical calculations backing these set ups. 14 Kno wn as the BRS-type symmetry named after Becchi, Rouet and Stora. 9 4 Impact of L andau pap er on mathematics 4.1 General remarks In his pap er [1 ], L a ndau empha s ized the most the r esults of his last long and deep pap er on analytical pro perties of the vertex parts (essentially , the scattering amplitudes) in quantum ﬁeld theory [6 7 ]. This work ca n also rightfully to b e considered a s Landau’s las t pap er. In the review by Go lubev a [45] o ur readers can ﬁnd a nice summary of the impact of this pa per on mathematics up to 19 76. Accordingly , to avoid duplications, we shall discuss in this section developments which either happened a f ter this da te or which are not discussed in Golub ev a’s pap er, but ar e of impo rtance. In particular, we would like to men tion some inconsistency in Landau’s r e a - soning. On one hand, he wro te his pap er [67] in order to make some progr ess in the theory of s tr ong interactions. On a nother, the mathematica l metho ds based on ana lysis of F eynman’s diagr ams which he w a s employing do es not dis- tinguish b et ween ”strong ” and ” weak interactions. Because o f this, w e shall discuss ana lytical prop erties of F eynman’s v er tex diag rams for an y quantum ﬁeld theory . Using Go lub ev a’s results we a re for ced to admit that, whatever this diagram might be, there is an equatio n for a hyp e rgeometric-type function of multiple a rgumen ts whos e solution is g iv en b y the vertex part in question. V archenk o demonstra ted [44] that a ll solutions of hypergeo met ric equations of m ultiple arg umen ts are so lutions of the corresp onding Knizhnik-Zamolo dc hiko v equations. These solutions a re a ll also expressa ble in the form the Aomoto- Gelfand type hyperg eometric integrals [6 8 -70] asso ciated with GKZ equations (also o f hype rgeometric type) to b e discussed b elo w. Knowledge of analytical prop erties of the vetex part allows us to connect it via the optical theor em with the exp erimentally observ a ble cro ss se ction [71]. Hence, either such exp erimen- tal da ta ca n b e used to re c onstruct the vertex par t via Kra mers-Kronig-type disp e rsion relatio ns, or the theo retically obtained vertex part (as so me solution of the K-Z or GKZ equation) c an b e used for making predictions ab out the scattering crossection. In both ca ses only 2 dimensional results are actually used in accord with remarks ma de in the preceding section. The ques t ion then emerges: is ther e a wa y to classify a ll so lutions of K-Z equations so that a ll meaningful sca ttering amplitudes can b e obta ine d, r ecognized and classiﬁed? In par t, s uc h a clas siﬁcation might follow from w orks by Belavin and Drinfel’d [33] and Re s hetikhin a nd Wiegmann [72]. And, b ecause of this, all quantum me- chanical a nd ﬁeld-theor etic mo dels can be tr e ated b y the gr o up-theoretic (which are in the essense top o logical [35]) metho ds inv o lv ing theory of knots and liks. Such p oin t of view is develop ed in our rec en t paper on Heisenberg honeycombs [37] and has found its implemetnation in the ﬁeld- t heory case in the works by Connes and Kreimer [73,74 ], C o nnes and Ma r colli [7 5], Kr eimer and collab ora- tors [76], etc. W e would like to reo bt ain some of thes e r e sults having in mind their applications in the next section. Before doing so, it is useful to remind our rea ders a bout developments in 10 mathematics, which were taking place e xactly at the time when Landau’s pa- per [67] was written. Because of these developmen ts, his pap er w as immediately appreciated b y mathematicians and made a big impact on mathematics. In 1958, just less than a y ear b efore Landau’s pap er w as wr itten, Rene Thom w as aw ar ded the highest award in mathematics- t he Fields Medal-for his c o n tribu- tions to the theor y of cob ordism [77]. This theo r y had play ed an imp ortan t r ole in S-W a nd Donaldso n theories , as well as those in volving Jones po lynomial [4 0], etc. Nevertheless, its is the theory of sing ularities and ca t astrophes developed by Thom b et ween 195 4-1962 which is immediately relev ant to La nda u’s work [67]. That this is indeed the case can b e seen fro m the monog raph by Pha m [78] published in 19 67. It is esse ntially the enlarged version of his earlier publication [79], which, in tur n, is part of a still earlier pa per by F otiadi, F roisaa r t, Las- coux and Pha m published in the top mathematical journal ”T op ology” [80]- all devoted to Landau sing ularities. Ma th emaically , all these works are based on results by Thom, Ler a y , Milnor a nd Lefshetz done in the time p erio d aro und publication o f Landau’s pap er. A systematic e x position of these mathema tica l results can be found, in a ddit ion to works alr eady cited, in the monogra ph by Arnol’d, V archenk o and Gusse in- Zade [81 ]. In this b ook, in addition, one can ﬁnd the info r mation ab out the so called mixed Ho dge structures a nd p erio d mapping-results o f contributions of Deligne a nd Griﬃths. An e x cellen t in tro duc- tion to these topics can be found in the mono graph by Carlson, Muller- Stac h and Peters [82]. All these works were employ e d in our analys is of analy tical prop erties o f V eneziano and V eneziano-Like a mplit udes [83-85 ] ev entually cul- minating in [4 7] replacing more traditional string -theoretic mo dels [15] by the tach yon-free mo del ex isting in the usual space- tim e dimens io ns. W e mention this in view of one of the latest papers by Kreimer and B lo ch [76] in which they noticed relev ance of the mixed Hodg e structures to more traditional ﬁeld theories requiring renormalization. Since V eneziano pos tula t ed (guessed) his amplitudes in 19 68 [13 ] many attempts were ma de to derive these amplitudes ﬁeld-theoretically by s t udying high energy asymptotics of scatter ring pro cesses where, as is co mmonly b elieved, the sca tter ing amplitudes should exibit the Regge-type b eha vio ur [86,8 7]. Accordingly , we would like to demonstra te now that, in fact, the V eneziano amplitudes 15 are the backb ones of scatter ing ampli- tudes o f every imaginable ﬁeld theory . T his fac t will be used in the next section in which we are go ing to demons trate muc h wider uses o f V eneziano amplitudes than just those in the scattering pro cesses of hig h energ y physics. 15 W e shall call scatt ering amplitudes as ”V eneziano ampl i tu des” since they all can b e expressed in terms of, may be, linear com binations of Euler’s beta functions of m ultiple ar- gumen ts. The true V eneziano ampli tude has in addition the Regge -type parametrization of these argument s. 11 4.2 V enezian o amplitudes and F eyn man diagrams W e begin with the key iden tity attributed to F eynman [88], page 83, 1 n Q j =1 A λ j j = Γ( λ 0 ) n Q j =1 Γ( λ j ) n Y j =1   1 Z 0 dα j   δ (1 − n P j =1 α j ) n Q j =1 α λ j − 1 j n P j =1 α j A j ! λ 0 (4.1) Here λ 0 = n P j =1 λ j and Γ( λ j ) is Euler g amma function. The factors A j are given, as in Landau’s pap er[67], i.e.as A j = m 2 j − k 2 j . In the present c ase a sligh tly more general identit y is used 16 .Such an identit y is employ ed for wha t ever F eynman int egral of the type I = Z B d d p 1 d d p 2 · ·· A λ 1 1 A λ 2 2 · ·· (4.2) with B , in the case of fermio ns, b eing some p olynomial in k ′ i s with k i being some momenta related to a giv en line on the diagram. Clearly , k i may be a combination of bo th the in ternal momenta, i.e. p i , and the external moment a, e.g. l ′ i , so that a given diag ram is some function o f the external momenta and dimensionality d of spac e (or space- tim e). Use of (4.1) in (4.2) signiﬁcantly simpliﬁes calculations. Details are discussed be lo w. Lo oking a t (4.1) we notice that in the case if A λ j j = 1 ∀ j we o bt ain an impo rtan t identit y Γ( λ 0 ) n Q j =1 Γ( λ j ) n Y j =1   1 Z 0 dα j   δ (1 − n X j =1 α j ) n Y j =1 α λ j − 1 j = 1 . (4.3) This identit y deﬁnes the P oisson-Dirichlet (P-D) probability measure to be dis- cussed in some detail in Section 5. F ur th ermore , without the normalizatio n factor Γ( λ 0 ) n Q j =1 Γ( λ j ) , we rec o gnize in the obta ine d expressio n the V eneziano ampli- tude [8 3-85]. F r om this o bserv a tion the following conclusion ca n be drawn: All F eynman’s diagrams co ntaining lo ops mathematica lly are averages of the sto c has t ic proces ses o f the Poisson-Dirichlet type 17 . Because of this, all such di- agrams can be r ewritten a s linear combinations of V enezia no a mplit udes.W e would like to provide suﬃcien t evidence that this is indeed the case. This task is g reatly simpliﬁed by the signiﬁcant prog ress made to date in a ctual calcu- lations of F eynman’s dia grams. Recent pap ers by Bog ner and W einzier l [89 ] and W eizie r l [90] contain a nice summary of these eﬀo r ts. T o these pap ers, w e 16 In Landau’s paper all λ ′ i s are equal to one. 17 In Section 5 w e shall demonstrate that the same is true for traditional qauan tum mechanics and many other di s c iplines. 12 would lik e to a dd the pap er b y T arasov [91] on h yp ergeometric represent ation of a specia l class of F eynma n’s diag r ams as well a s pap ers b y Smirnov and Smirnov [92] on the use of Gr¨ obner bases for calculation of F eynman integrals. Reading these pap ers and [16] provides a bro ad panora ma o f eﬀorts made in practica l calculations of F eynman diagra ms to date, very muc h in acco rd with and in the spirit of Landau’s predictions. F or the sa k e of space, and without lo oss of generality we would like to con- sider only the b osonic-type diagra ms (t hat is, the diagrams for which the factor B in the numerator of (4.2 ) is identically eq ual to one). This is tota lly justi- ﬁed [89] s inc e the diagra ms which contain nontrivial B -factors can b e reduce d to those in which B = 1.In view of this, we a re left with the calculation o f int egrals of the t yp e I ( α ; s,t,u, ... ) = Z d d p 1 d d p 2 · ·· n P j =1 α j A j ! λ 0 , (4.4) where α = { α 1 , ..., α n } and s, t, u, .. are so me kinematic inv ariants made of exter - nal mo men ta. Calculation of such in teg rals can be found in [89] a nd , therefore, are o f no immediate concer n to us. W e shall co nsider their analy t ical pr operties irrespp ectiv e to a particula r outcome of computation of I ( α ; s,t ,u, ... ) . T o do so, we need to employ a few facts fro m the theory of funct ions of several co m- plex v ariables. This is needed for justiﬁcation of the La uren t-type expansion o f I ( α ; s,t,u, ... ) . If such a justiﬁcation is found, then we can re pr esen t I ( α ; s,t,u, ... ) as I ( α ; s,t,u, ... ) = ∞ X l = −∞ c l ( s, t, u... ) α l (4.5) with l = [ l 1 , l 2 , ..., l n ]; α l = α l 1 1 · ·· α l n n , l 1 + l 2 + .. + l n = ˆ l ; c l ( s, t, u... ) = c l 1 ,l 2 ,...,l n ( s, t, u, ... ) . Clearly , in a ctual co mput ations we do not a n ticipate tha t we s hall use an inﬁnity of co eﬀcien ts c l . But, whatever they are, if suc h ex- pansions do exist, then any F eynman integral is a Poisson-Dir ic hlet average < I ( α ; s ,t,u, ... ) > with the pro babilit y density deﬁned in (4.3). Hence, the task now lies in providing a justiﬁcation that, indeed, s uc h a n expansion do es exist. T o do so, we need to recall a few facts from the theo r y of functions of many complex v a riables. In par ticular, we need to reca ll some mo dels of complex pro- jective space CP n . These are discussed, for instance, in [93] and were emplo yed in our w orks [83-85] on V eneziano amplitudes. Let C n +1 be a complex n + 1 dimensional space and let ω = ( ω 0 , ..., ω n ) be a p oin t in suc h a s pace. If we identify tw o ar bitr ary po in ts ω ′ and ω ′′ in C n +1 via the relation ω ′ = λω ′′ , where λ is some nonzero complex num ber , then the whole space C n +1 can b e sub divided into equiv alence classes with resp ect to pr e viously mentioned rela tion. A quotient C n +1 / ( ω ′ = λω ′′ ) is the complex pro jectiv e space CP n . Each complex line in C n +1 can b e characterized 13 by the unit vector ω 0 = ω | ω | . E v er y line made this wa y in C n +1 is a p o in t in CP n . Because of this, w e notice that p oin ts in CP n are also p oints in C n +1 . This f act can b e used f or construction of another mo del repr esen ting CP n . F or this purp ose, let the line in C n +1 be given para metrically a s z ν = ω 0 ζ , ν = 0 , ..., n with the parameter ζ to be determined as fo llo ws. If a n equa tio n for a complex spher e S n in C n +1 is given by n P i =0 z ν ¯ z ν = 1, then, w e o bt ain: | ζ | 2 n P i =0   ω 0 ν   2 = 1 , implying | ζ | 2 = 1 whic h is just an e q uation for a cir cle S. This means that CP n can be deter mined a s quotient o f S n /S. Because of this, we ca n deﬁne the de fo rmation retrac t of CP n as n P i =0 t i = 1 by identifying t i with   ω 0 i   2 . Thu s obtained equation is an equa t ion for a n − simplex ∆ . W e wen t int o these details only b ecause the δ constaint in the P-D measure (4.3) tells us that the integration in (4.3 ) is done over the simplex ∆ and, hence, such an int egral lies, in fa c t , in the co mplex pro jective spac e (in the case of (4.3), in CP n − 1 ) in acco rd with Golubev a ’s pap er [45 ]. Becaus e o f this, the meaning of auxilia ry α v a r iables in (4.5) c hanges fr o m being r e al to be coming complex. This obser v ation provides the needed justiﬁca tio n for our use o f the theory o f functions of man y complex v a riables. After r ecognition of this fact, we hav e to write down the c o rrect diﬀerential form for CP n . T o do so , w e use r esults of Griﬃths dis cussed b oth in Golubev a’s pap er [4 5] and in our pap er [83 ] on V eneziano amplitudes, e.g. see Section 3.1. of [8 3].Th us, we o bt ain, ω ( w ) = n X i =0 ( − 1) i w i dw [ i ] , (4.6) where dw [ i ] = dw 1 ∧ · · · ∧ dw i − 1 ∧ dw i +1 ∧ · · · ∧ dw n . Ther efore, up to a consta nt, any nontrivial F eynman in tegral inv olving lo ops is expressable as I ( s, t, u, .. ) = Z Γ I ( α ; s,t,u, ... ) n Y j =1 α λ j − 1 j ω ( α ) (4.7) with Γ b eing some homology c y cle in CP n (very muc h like in the more familiar z - plane case o ne consider s an integral o f function f ( z ) a long some contour which do es not cr oss the singularities of f ( z )) t o be determined shor t ly . F or this purp ose, we no tice that if the La ur en t expansion (4.5 ) ca n be us ed in (4.7), then we end up with the sum of terms each of which is the V eneziano amplitude. F rom our w ork [83 ] we know than that each V e ne z iano amplitude represe nts some per iod on the F ermat hypersur face in CP n F ( N ) : x N 0 + · · · + x N n = 0 , N = 1 , 2 , ... (4.8) so that Γ represents one of the p eriods on suc h a surfa c e (v ery muc h like in the case of the complex torus T there are tw o p erio ds). This fact c o nnects the theory 14 of suc h in teg rals with the p erio d mapping theory , Gauss-Manin connec tio ns, hypergeometr ic functions of ma n y v aria bles, etc, e.g. s ee our pap er [8 3 ] and the original works already mentioned, e.g.[68,6 9,81,82]. Suc h a co nclusion depends on the existence of the La uren t-type expansions o f the type given by (4.5) (used in (4.7)). F ortunately , the existence of such a result also follows, alb eit implicitly , from the Lemma and Theorem in Section 1.4 of Golubev a ’s pa per [45]. W e would like now to discuss this to pic in some detail. In pa rticular, fol- lowing Shabat [93], we no tice that up on rescaling , the diﬀerential for m ω ( w ) behaves as follows ω ( w ) = ( − 1 ) ν − 1 w n +1 ν d ( w 0 w ν ) ∧ · · · ∧ d  w ν − 1 w ν  ∧ d  w ν +1 w ν  ∧ · · · ∧ d  w n w ν  , (4.9) implying that a co m bination ω ( w ) / F ( w ) , with F ( w ) being s ome ho mo genous po lynomial of degre e n + 1 , is sca le -in v ar ian t. This is the result cited b oth in our work [83] and in the pap er by Golub ev a [4 5]. It b elongs to Griﬃths [94] who prov ed th at in the pro jective space CP n any closed diﬀeren tial for m should lo ok like Φ = P F ω with degr e e s of homogenous p olynomials P and F r elated to each o th er as deg P + n + 1 = deg F. F eynman dia g rams discussed in Section 1.6 of Golub ev a’s pa per are all sca le-in v a rian t a nd ob ey Cor o llary 2.11 . o f Griﬃths pap er [94].Sca le in v ariance implies renormaliza bilit y and vice versa. In her pap er Golubev a talks ab o ut con vergence of F ey nman in tegr a ls as a function of their dimens ionalit y without paying a tten tion to the renormaliz abilit y issues. Hence, contrary to Landau exp ectations, the diagr ammatic technique he was advocating is allowing us to lo ok only at the renormaliz a ble theor ies. This is not a signiﬁcant drawback though since, as we mentioned alr eady , the La nda u progra m a lso br ings to lig h t the h yp ergeometric f unctions of multiple argumen ts (to be futher discussed below) a nd, with them, the Knizhnik-Zamo lodchik ov equations [38, 44 ] and, hence, the WZNW mo del, etc.The scale-indep enden t int egrals are p erio ds of some v arieties, e.g. of the F er mat-t yp e liv ing in the complex pro jectiv e space. In acco rd with g eneral methods dev elo ped by Griﬃths and nicely summarized in [82], it is of in terest to study these perio ds a s functions of parameters, in the present case - the para meter s are the kinematic inv aria n ts. In addition, as in the theor y of ordina ry in teg rals calculated by the metho d o f residues, in the multidimensional ca se there is analo gous theor y develop ed by Leray [79,80,9 3] and is also discussed in Golub ev a’s pap er. But, a s we know from the simpler, one-dimensional c ase, use of the metho ds of residue theory is essent ially equiv a len t to our ability to obtain the Laurent expans ion fo r a function in question. Hence, we come bac k to the expansion (4.5). In the case of the theo r y of functions of one complex v ar iable the pro cedure for obtaining the L a uren t ex pansion is w ell known. Such a pro cedure is not immediately transfera ble to the case of many c o mplex v ar iables thoug h. T o do this, one nee d to ask a question: what is the a nalog of the Cauch y formula in the cas e of many complex v a riables? Surprisingly , there ar e many analog s of this form ula in the multiv aria ble ca se. Let us r ecall what is involv ed in the one-dimensional ca se. F or a domain D with b oundary ∂ D in z -plane and the 15 well b eha ved function f ( z ) in this domain we hav e the Cauch y form ula f ( z ) = 1 2 π i Z ∂ D f ( t ) dt t − z . (4.10) It can b e used only for z ∈ D . F or z outside D the result is zer o since the fun ction under the integral is holomor phic. With he lp of (4.10) the n − th deriv ative of f ( z ) is obtained as f ( n ) ( z ) = n ! 2 π i Z ∂ D f ( t ) dt ( t − z ) n +1 . (4.11) This result is used for b oth the T aylor and Laur en t expansions of f ( z ) . By complementarit y principle we exp ect that the same holds true in the multidi- mensional case. Hence, we hav e to ﬁnd a multid imensional analo g o f (4.10) ﬁrst. The requirements of s cale in v ariance of integrals of the type given by (4.7) formally leav e us with not to o muc h choice. Our exp erience with V eneziano amplitudes [83 ] tells us, how ever, that the situation is considerably trickier. Indeed, taking into acco un t the ident ity B ( x, y ) B ( x + y , z ) B ( x + y + z , u ) · ·· B ( x + y + ··· + t, l ) = Γ( x )Γ( y ) · · · Γ( l ) Γ( x + y + · · · + l ) , (4.12) e.g. see equation (3.28) in [83], o ne ca n no tice that s tudy of analytica l prop erties of a n y V eneziano a m plitude can b e reduced to that of the elementary Euler ’s beta function B ( x, y ) whic h is just a p eriod of t he simplest F ermat v a riet y ± z N 0 + z N 1 + z N 2 = 0 so that the p erio d in tegral in CP 2 representing this beta function can be wr itten a s I = I z c 1 1 z c 2 2 z c 0 0 ± z N 0 + z N 1 + z N 2 ( dz 1 z 1 ∧ dz 2 z 2 − dz 0 z 0 ∧ dz 2 z 2 + dz 0 z 0 ∧ dz 1 z 1 ) , (4.13) e.g. see (3.10b) of [83]. Here c 1 , c 2 and c 0 are the sa me as x a nd y in B ( x, y ) while an a uxiliary parameter c 0 is used o nly in the pr o jective form o f the b e ta int egral. In a ctual comutations one should even tua lly make a transforma tio n to the a ﬃ ne form where one has to put z 0 = 1 so that the ﬁna l r esult is independent of c 0 . Deatails can b e fo und in [8 3 ]. F o r this (pro jective) perio d to make sence mathematically the following arguments should b e applied. In the integral one should make the fo llowing replacements: z k → z k ξ j , where ξ j = exp( ± i 2 π j N ) (1 ≤ j ≤ N − 1) and k = 0 , 1 , 2 . Substitution of such an ansatz in to (4.13 ) and requiring that the integral I b e indepe nden t of ξ j leads to the (V enez ia no-t y p e) condition j 1 c 1 + j 2 c 2 + j 0 c 0 = N (4.14) with j ′ k s b eing in the range sp eciﬁed a bov e. Suc h a c ondition makes the perio d I nonsingular. Ph ysically , ho wev er , this ca se is unin teresting since the nonsigu- lar expressio ns cannot b e used for scattering amplitudes. The p oles in suc h amplitudes (that is, the resonanc e s) are b eing used in the optical theorem for 16 calculation of expe rimen tally observed cross s e c t ions, e.g.r ead [71 ]. The way out of this appar e n t diﬃculty is explained in o ur work [83], see als o o ur related work [95 ]. In view of this, s o me ca utio n should b e exercised when one wants to use the La uren t expansio n (4.5) in (4.7). This obser v ation shifts a ttention from actual computations of co eﬃcien ts in the Laure nt expansio n (4.5 ) tow ar ds studying of ge ne r al asp ects of such calculations. Our task in this section lies in proving that any scattering amplitude is a linea r combination of the V enezia no- like amplitudes. This pro of is nonconstr uct ive (that is, it do es no t a llo w the eﬀective computation of amplitudes) but it is essential for the discussion pre- sented in the next sec tio n. Aga in, w e would like to remind our readers tha t what we ca ll ”V eneziano amplitudes” are in fact E uler’s b eta functions of m ultiple arguments. The true V eneziano amplitudes inv o lve Regge-type parametr ization of these arguments. Analysis made in [83] indicates that in the ca se when the Laurent e xpansion (4.5) has co untable inﬁnity of terms with nega tiv e p o wers, this counable inﬁnity ca n be s q ueezed in to just o ne (truly V eneziano) amplitude using the appropr iate Regge-type parametriza tion. Such a situation is typical for the meso n and hadro n physics. In the case when such an expans io n has just a few ter ms with negative pow ers, the situation should b e typical for quantum mechanics (see next section), quantum electro dynamics and weak interactions. After these genera l r emarks, ﬁnally , we a re r eady to provide needed (non- constructive) pro of. F or this purp ose we need only to write the m ultidimen- soional analog s of the known Cauch y formulas (4.10), (4.11). In literature there are three types of mult idimensional Cauch y- like form ulas-all reduceable to (4.10), e.g. s e e [9 3]. W e prefer to use the Martinelli-Bo chner (M-B) formula 18 . It is given by f ( z ) = Z ∂ D f ( ζ ) ω M B ( ζ − z ) (4.15) with the M-B form ω M B being deﬁned as ω M B ( ζ − z ) = ( n − 1)! (2 π i ) n n X i =1 ( − 1) i ( ¯ ζ i − ¯ z i ) d ¯ ζ [ i ] ∧ dζ | ζ − z | 2 n (4.16) and, as usual, | z | 2 = n P i =1 | z i | 2 . F or n = 1, beca use of this, we obtain ω M B ( ζ − z ) = 1 2 π i ¯ ζ − ¯ z | ζ − z | 2 dζ = 1 2 π i 1 ζ − z dζ , (4.17) as required. The a nalog of (4.11 ) in the presen t c ase was obtained by Andr e o tti and Norguet (e.g.see [93], page 229) and is given by f ( k ) ( z ) = Z ∂ D f ( ζ ) ω k ( ζ − z ) (4.18) 18 The more general Cauch y-F an tappie formula discov ered by Leray in 1956 i s reducible to that of M- B in the sp ecial case w = ¯ z , e.g. read page 158 (after equation 10) of [93]. W e are int erested just in this sp ecial case in this work. 17 with the form ω k being g iven b y ω k ( ζ − z ) = k ! ( n − 1)! (2 π i ) n n X i =1 ( − 1) i ( ¯ ζ i − ¯ z i ) k i +1 d ¯ ζ α + I [ i ] ∧ dζ | ζ − z | 2 n , (4.19) where k = { k 1 , ..., k n } , α = { α 1 , ..., α n } , I = { 1 , ..., 1 } and k ! = k 1 ! · · · k n ! . Thu s, the Lauren t expansio n (4.5) do es e xists so th at all scatter ing pro cesses in quantum ﬁeld and string theory ar e o btainable a s averages of the sto c ha stic Poisson-Dirichlet type pro cesses. These are go ing to b e disc us sed in some detail in the next s e ction. Before doing so, w e would like to conclude this section with a brief discussio n of hyp e rgeometric equatio ns of multiple arguments asso ciated with F eynman diagrams in order to bring Golub ev a’s pap er [45] up to date. F or the warm up, let us consider a ca lculation of the following g eneric integral I ( z ) = 1 2 π i Z γ e z ζ P ( ζ ) dζ (4.20) where P ( z ) is so me poplyno mial: P ( z ) = z n + a 1 z n − 1 + · · · + a n and the c o n tour γ is chosen in suc h a w ay that all zero s of P ( z ) lie inside. E viden tly , P ( d dz ) I ( z ) = 0 . (4.21) This equation sho uld b e supplemented withthe bo udary conditio ns . They come from consideration of integrals of the type I ( k ) ( z ) | z =0 = 1 2 π i I γ ζ k P ( ζ ) dζ . (4.22) Since k ≤ n, the function ζ k P ( ζ ) has at the p oin t z = ∞ zero of order n − k . F or k ≤ n − 2 w e then obtain res ζ = ∞ ζ k P ( ζ ) = 0 . While for k = n − 1 we obta in, ζ n − 1 P ( ζ ) ∼ 1 ζ for ζ → ∞ . Hence, res ζ = ∞ ζ n − 1 P ( ζ ) = − 1 Ther efore, I ( n − 1) ( z ) | z =0 = 1 and the rest o f deriv a tiv es are being z e r o. This g eneric example can b e broadly generalized. This tas k is a ccomplished in a series of pap ers by Gelfand, Ka pra- nov and Zelevinsky (GKZ) who se r esults ar e summar ized in [96]. In a ddition to this reference, we sha ll follow in part more relaxed exp osition [9 7] desc ribing results of GKZ 19 . F or the sake of spa ce, we a r e not disc us sing the related to GKZ issues of mir ror symmetry . A very access ible exp osition of this topic is given in our w ork [95 ] to whic h we refer our rea ders for details. Reading of this reference sho uld b e suﬃcient for understanding of ﬁner deta ils of mir ror symmetry presented in [97 , 98]. 19 In our exposition we ﬁll in man y gaps in pr e sent ation whic h one m igh t encoun ter while trying to read [97]. 18 Although not mentioned in the review [9 6], the origina l mo tiv ation for sudy- ing the GKZ hyper geometric functions ca m e from obse rv atio n made by GKZ in the earlier pap er [98 ] that the integrals of the type F σ ( α, β ; P ) = Z σ Y i P i ( x 1 , ..., x k ) α i x β 1 1 · · · x β k k dx 1 · · · dx k (4.23) with α = ( α 1 , ..., α m ) ∈ C m , β = ( β 1 , ..., β k ) ∈ C k , P i ( x ) = P ω v ω x ω , where x ω = x ω 1 1 · · · x ω k k so tha t ω = ( ω 1 , ..., ω k ) ∈ Z k and σ is some c a refully chosen k-cycle (whose construction is describ ed in [96, 98])-all are in tegrals of the quan- tum ﬁeld theo ry (qft) alrea dy discussed in La ndau’s pape r s [1, 67] 20 ! Although GKZ promised to write a separ ate pa per devoted to the qft integrals only , to our k nowledge such pap er was never written. Evidently , our equa t ion (4.7 ) falls int o the categ ory of GKZ integrals as required so not muc h else can b e said. T o dea l with in teg rals of the type g iv en in (4.23) consider the following integral very similar to our (4.7). It is given by I ( m ) σ ( u ) = Z σ P u ( x ) m dx 1 x 1 · · · dx k x k (4.24) Here m ∈ Z ,P u ( x ) = P a ∈ A u a x a , a=( a 1 , ..., a k ) , A = { a 1 , ..., a N } is some ﬁnite subset of Z k (usually b eing comprized o f vertices of so me conv e x p olytope P th us leading even tually to the mirror symmetry ar g umen ts as explained in [97 , 98]). F or simplicity σ = σ 1 × · · · σ k is a pro duct of k circles σ i , i = 1 − k , σ i ∈ C , each centered at 0 so tha t P u ( x ) 6 = 0 ∀ ( x 1 , ..., x k ) ∈ σ 1 × · · · σ k . By diﬀerentiation under the integral sign we obtain ∂ I ( m ) σ ( u ) ∂ u i = m Z σ x a P u ( x ) m − 1 dx 1 x 1 · · · dx k x k . (4.25) T o develop this result further some knowledge of solid s tate physics is helpful. W e would like to re mind our rea ders of some useful facts to help the under- standing of what follows. In the case of mo re familiar 3 dimensiona l lattices one can choose (rather a rbitrarily) a cell made of vectors e 1 , e 2 and e 3 so that any vector A o f such a 3 d lattice is decomp osable as A = n 1 e 1 +n 2 e 2 +n 3 e 3 with n i ∈ Z . Suppose now that w e tra nslate this lattice a s a whole by vector b. This will cause us to write: A = n 1 e 1 +n 2 e 2 +n 3 e 3 + b (n 1 +n 2 +n 3 ) . T o mak e such a vector repres en tation well deﬁned we mayrequire that n 1 +n 2 +n 3 = 0 . If, in a ddit ion we ﬁx the or igin, this would res ult in an additional co nstrain t (determining the lo cation o f the orig in) n 1 e 1 +n 2 e 2 +n 3 e 3 = 0 . In the present case we hav e a lattice made of basis se t { a 1 , ..., a N } ≡ A, so that the above tw o conditions are translated in to l 1 + · · · + l N = 0 (4.26a) 20 In mathematics these int egrals ha ve b ec ome known as Aomoto-Gelfand-type i n tegrals as men tioned already . 19 and l 1 a 1 + · · · + l n a N = 0 . (4.26b) Since the vector a in (4.25) b elongs to the set A, the condition (4.2 6b), when applied to the in tegral (4.25), leads to the following GKZ equa tio n [ Y l i < 0  ∂ ∂ u i  − l i − Y l i > 0  ∂ ∂ u i  l i ] I ( m ) σ ( u ) = 0 (4.27) This e q uation is nesessar y but is not suﬃcient for description of the equation for hypergeometr ic function of multiple arg umen ts since it do es no t take into account the homo geneit y of such a function. T o acc oun t for homo geneit y we need to consider w ha t will happ en to the integral I ( m ) σ ( u ) upon r escaling of the externa l parameters u . F or this purp ose some re sults from our pap ers [84, 85] summa r ized in [100] ar e helpful. Omitting technicalities, we would like to present here only the ”b ootom line”. T o do so, we would lik e to consider again the Laurent p olynomials of the t yp e f ( x ) = X a ∈ S σ u a x a . (4.28) Here S σ is the p olyhedral c o ne a s sociated with the Newto n’s p olytope P . In connection with th us deﬁned f ( x ) , consider no w a general deﬁnition o f a quasi- homogenous function of degree d with exp onen ts γ 1 , ...γ n . Such a function is deﬁned by f ( s γ 1 x 1 , ..., s γ n x n ) = s d f ( x 1 , ..., x n ) . (4.29) If w e apply suc h a r e quiremen t to the individual term of the Laurent p olynomial in (4.28), w e obtain n X i =1 γ i a j i = d j j = 1 − N . (4.30) where N is the num b er of terms in the Laurent expa nsion (4.2 8). Equa tion (4 .30) is an equation for the hyperpla ne. Diﬀerent h ype rplanes ma y have diﬀer en t d ′ j s. The polytop e P is made of a co llection of N hyperpla nes. Finally , let us diﬀerentiate (4.29 ) ov er s and le t s = 1 in the end. Thus, we obtain X i γ i x i ∂ ∂ x i f = d f . (4.31) A t this po in t our r eaders already can co rrectly guess that the require d supple- men tal equation(s) to (4.27) are equations describing the po lytope. In view of (4.30), (4.31), these are k equations of the t yp e [ N X i =1 a i u i ∂ ∂ u i − d ] I ( m ) σ ( u ) = 0 . (4.32) 20 W e would like to illus tr ate these g eneral facts by more familiar example of a usual hyper geometric function. F or this, w e need to c ho ose the A-system of basis vectors as follows A =      1 1 1   ,   − 1 0 0   ,   0 1 0   ,   0 0 1      , (4.33) while for the d -vector we c ho ose d =(1 -c,-a,-b). Under s uc h circumsta nces, the GKZ equations can be explicitly written as  ∂ 2 ∂ u 1 ∂ u 2 − ∂ 2 ∂ u 3 ∂ u 4  Φ = 0 (4.34)  u 1 ∂ ∂ u 1 − u 2 ∂ ∂ u 2  Φ = (1 − c )Φ , (4.35)  u 1 ∂ ∂ u 1 + u 3 ∂ ∂ u 3  Φ = − a Φ , (4.36)  u 1 ∂ ∂ u 1 + u 4 ∂ ∂ u 4  Φ = − b Φ . (4.37) F rom the second equation we obta in ∂ 2 ∂ u 1 ∂ u 2 Φ = u − 1 2 ( u 1 ∂ 2 ∂ u 2 1 + c ∂ ∂ u 1 )Φ (4.38) F rom the third and fourth equations we o btain as well ∂ 2 ∂ u 3 ∂ u 4 Φ = u − 1 3 u − 1 4 ( − u 1 ∂ ∂ u 1 − a )( − u 1 ∂ ∂ u 1 − b )Φ . (4.39) Finally , in view of (4.34) we obtain u − 1 3 u − 1 4 ( u 2 1 ∂ 2 ∂ u 2 1 + (1 + a + b ) u 1 ∂ ∂ u 1 + ab )Φ = u − 1 2 ( u 1 ∂ 2 ∂ u 2 1 + c ∂ ∂ u 1 )Φ . (4.40) In this equatio n we hav e to set u 2 = u 3 = u 4 = 1 and u 1 = z in order to get the familiar equation for the Gauss hypergeo metr ic function  z ( z − 1) d 2 dz 2 + [( a + b + 1 ) z − c ] + ab  f ( a, b, c ; z ) = 0 . (4.41) Recall now that the integral repre s en tation for suc h a function is given by [101 ] f ( a, b, c ; z ) = Γ( c ) Γ( b )Γ( c − b ) 1 Z 0 dt t b − 1 (1 − t ) c − b − 1 (1 − tz ) a ≡ < (1 − tz ) − a >, (4.42) 21 where the Poisson-Diriclet av erag e < · · · > is deﬁned by < · · · > = Γ( c ) Γ( b )Γ( c − b ) 1 Z 0 dtt b − 1 (1 − t ) c − b − 1 · ·· (4.43) in a c cord with (4.3). By the principle of complementarit y this mea ns that all F eynman’s diagrams , including those for v er tex parts determining the sca tt ering amplitudes ar e the P- D av erage s inv olving ﬁnite (e.g for a < 0 in (4.42))or inﬁnite (e.g. for a > 0 in (4.42 )) combinations of V eneziano amplitudes in the sence alr eady de s cribed. O ther implications of the results we ha ve just o btained are discussed in the next section. 5 Outlo ok: Impact of Land a u’s last pap er on sciences in n e w millenium 5.1 General commen ts Although V eneziano had guessed (p ostulated) his amplitude [13], and some au- thors hav e criticized suc h an approa c h to scattering pro cesses of hig h energy ph ysics [1 0 2], the results of the previous section demo nstrate that V eneziano amplitudes or their linear combinations are in trinsic ob jects o f high energy scat- tering pro cesses. The questio n arise s : If this is the case, what e ls e can b e sa id ab out these a mplitudes? W e noticed already in (4.3) that all such a mplitudes are P -D av erages . Hence, the task now lies in discussing gene r alities of such t yp es of sto chastic pro cesses. V ery fortunately , without an y reference to high en- ergy ph ysics this tas k was to a large exten t accomplished. References [1 03-105] provide an excellent introductio n into the theo ry of the P -D pro cesses which play the cent ral r ole in the theory of random fr agmen ta tion and coagulation pro cesses. It is suﬃcient to type ”Poisson-Dirichlet” using Go ogle search engine in o rder to ﬁnd ab out 52,00 0 en tr ies. Such a n abundance of entries is caused by the fact that many disciplines fr om the theo r y o f s pin glasses to computer science, from linguistic to fo rensic science, from econo mics to p opulation g e net - ics, from chemical kinetics to random ma t rix theor y , etc.-all inv olve the P-D distributions. Re ma rk ably , none o f these r e f erences men tion applications to the high ener gy physics, quan tum mechanics or qft. The r ole of the coag ulation fragmentation pro cesses in high energ y physics was recog nized so me time ag o by Mekjian, e.g. see [106 ] and references ther e in. His works do not contain, how ever, arguments and results presented abov e, in Section 4, and therefore can be co nsidered as complementary to ours . W e s hall say more a bout this b elo w, in this section. 22 5.2 Random fragmen tation and coagulation pro cesses and the Diric hlet distribution W e b egin with some known facts from proba bilit y theory . F or instance, w e recall that the stationar y Maxwell distribution for v elo cities of particles in a gas is just of Gaussia n-t yp e. It can b e obtained as the stationa r y solution of Boltzmann’s dyna mical equation maximizing Boltzmann’s-type en tr op y 21 . The question arises : Is it p ossible to ﬁnd (discrete or contin uo us) dynamical equa- tions which will provide known probabilit y laws as stable stationary solutions? This task will in volv e ﬁnding of dynamical e quations alo ng with the corre- sp onding Boltzmann-like entropies which will reach their maxima at resp ective equilibria fo r these dynamical equations. W e ar e cer tainly no t in the po sition in this clos ing section o f our pap er to discuss this problem in full g eneralit y . Instead, we fo cus our attention only o n proc esses wihich are descr ibed by the so called Dirichlet distributions. These origina te from the in tegral (equation (2.8) in o ur work [83] on V eneziano amplitudes) attributed to Dirichlet, that is D ( x 1 , ..., x n +1 ) = Z · · · Z u 1 ≥ 0 ,..., u n ≥ 0 u 1 + ··· + u n ≤ 1 u x 1 − 1 1 · · · u x n − 1 n (1 − u 1 − · · · − u n ) x n +1 − 1 du 1 · · · du n . (5.1) A random vector ( X 1 , ..., X n ) ∈ R n such that X i ≥ 0 ∀ i and n P i =1 X i =1 is said to b e Dirichlet distributed with parameters ( x 1 , ..., x n ; x n +1 ) [107 ] if the probability density function for ( X 1 , ..., X n ) is given by P X 1 ,..., X n ( u 1 , ..., u n ) = Γ( x 1 + · · · + x n +1 ) Γ( x 1 ) · · · Γ( x n +1 ) u x 1 − 1 1 · · · u x n − 1 n (1 − n X i =1 u i ) x n +1 − 1 ≡ Γ( x 1 + · · · + x n +1 ) Γ( x 1 ) · · · Γ( x n +1 ) u x 1 − 1 1 · · · u x n − 1 n u x n +1 − 1 n +1 , pro vided that u n +1 = 1 − u 1 − · · · − u n . (5.2) T o get some feeling of such deﬁned distribution, we notice the following p ecu- liar asp ects of this distribution. F o r any discrete dis t ribution, we know that the probability p i m ust b e nor malized, that is P i p i = 1 . Th us, the Dirichlet distribution is dealing with averaging of the probabilities! O r, b etter, it is deal- ing with the pro blem o f eﬀectively selecting the most o ptima l probability . The most primitiv e of these probabilities is the binomial probability giv en b y p m =  n m  p m (1 − p ) n − m , m = 0 , 1 , 2 , ...., n . (5.3) If X is the ra nd om v ariable sub ject to this law of probability then, the e x pec- tation E ( X ) is calculated as E ( X ) = n X m =1 mp m = np ≡ µ. (5.4) 21 As discussed recen tly in our work [25] on the Poinca r e ′ and geometrization conjectures. 23 Consider such a distribution in the limit: n → ∞ . In this limit, if w e write p = µ/n , then the Poisson distribution is obtained as p m = µ m m ! e − µ . (5.5) Next, w e notice that m ! = Γ( m + 1) . F urthermore, w e r eplace m by rea l v alued v ariable α and µ by x . This a llo ws us to introduce the gamma distribution with exp onen t α whose probability densit y is p X ( x ) = 1 Γ( α ) x α − 1 e − x (5.6) for some ga mm a distributed r andom v a riable X . Based on these r esults, we would like to demonstrate now how the Dirichlet distribution can be represented through gamma distributions. Since the gamma dis tr ibut ion orig inates from the Poisson distribution, sometimes in literature the Diric hlet distribution is ca lle d the Poisso n- D irichlet (P-D) distribution [10 8]. T o demonstrate the co nnection betw een the Dirichlet and gamma distributions is relatively e asy . F o llo wing Kingman [108], consider a set of p ositiv e indep enden t gamma distributed ran- dom v aria bles Y 1 , ..., Y n +1 with exp onen ts α 1 , ..., α n +1 . F urthermor e, co nsider the sum Y = Y 1 + · · · + Y n +1 and co ns truct a vector u with compo nen ts: u i = Y i Y . Then, since P n +1 i =1 u i =1 , the co mponents of this vector are Dirichlet distributed and, in fact, indep e ndent of Y . Details a re given in App endix A. Such describ ed Dirichlet distribution is a n equilibr ium measure in v a r ious ﬁelds ranging from spin glasses to computer s cience, from linguistics to g enetics, from forensic science to eco nomics, etc. [103 -105]. F urtheremore, most of fragmentation and coagulatio n pro cesses inv olve the P-D distribution as their equilibrium measure. Some applications of general theory of these pro cesses to nuclear a nd particle ph ysics were initiated in already mentioned series of pap ers b y Mekjian, e.g. se e [106]. T o a void duplications, we would like to rederive some particular results o f Mekjian’s diﬀerently in order to exibit their connections with the previous section. 5.3 The Ew ens sampli ng form ula and V eneziano ampli- tudes This formula w as discussed by Mekjian in [1 09] without any refere nce to the P-D dis t ribution. It is discus s ed in many o t her places, including Ewens own monogra ph [110]. Our expo sition follows work by W atterson [111] where he considers a simple P-D av erage of monomials of the type g enerated by the individual terms in the expansion 22 u n = ( u 1 + · · · + u k ) n = X n =( n 1 ,...,n k ) n ! n 1 ! n 2 ! · · · n k ! u n 1 1 · · · u n k k . (5.7) 22 V ery recently W atterson’s results were successfully applied to some problems in ec onomics [112]. 24 This type of expans io n was used in our work [83] (equations (2.9 ),(2.11)) for calculation of multiparticle V eneziano amplitudes. Not s ur prisingly , W a tter - son’s calculation also results in the mult iparticle V e neziano a mplit ude. Upon m ultiplication b y so me combinatorial factor in a well deﬁned limit such an am- plitude pro duces the Ewens s ampling for m ula playing a ma jor role in genetics. Although in App endix B we r eproduce the Ewens sampling formula (equation (E.6)) without use of the P-D distribution, Kingman [1 13] demonstr ated that ”A sequence of p opulations has the Evens sampling prop ert y if and o nly if it has the P-D limit” That is to s ay , the Ewens sampling fo rm ula implies the P-D distribution and vice versa. In the context o f high energy physics it is the same as to say that the law o f co nserv a tion of ener gy-momen tum which must hold for any scattering amplitude leads to the P-D distribution or, equiv a len tly , to the V eneziano-type for m ula for m ultiparticle amplitudes . Hence, we expect that our readers will consult App endix B prior to reading of what follows. F urther - more, since the vector u is P -D dis t ributed, it is appropriate to men tion at this po in t that e quation (5.7) r epresen ts genetically the Hardy-W einber g law [110] for mating sp ecies 23 . Hence, the Ewens sampling formula provides a reﬁnement of this law accounting for m utatio ns. Considear a sp ecial ca se of (5.2) for which x 1 = x 2 = · · · x K +1 = ε and let ε = θ /K with para meter θ to b e deﬁned later. Then, (5 .2) is co nverted to P X 1 ,..., X K ( u 1 , ..., u K ) ≡ φ k ( u ) = Γ(( K + 1) ε ) [Γ( ε )] K +1 K +1 Y i =1 u ε − 1 i , provided that 1 = X K +1 i =1 u i (5.8) In view o f (5.7), let us consider an av erage P ( n 1 , ..., n K ) ov er the simplex ∆ (deﬁned b y P K +1 i =1 u i = 1) given b y P ( n 1 , ..., n K ) = n ! n 1 ! n 2 ! · · · n K ! Z · · · Z ∆ u n 1 1 · · · u n K K φ k ( u ) du 1 · · · du K . (5.9) A straightforward calculatio n pro duces: P ( n 1 , ..., n K ) = n ! n 1 ! n 2 ! · · · n K ! Γ(( K + 1) ε ) [Γ( ε )] K +1 K Y i =1 Γ( ε + n i ) Γ(( K + 1) ε + n ) . (5.10) Up to a prefacto r , the obtained pr oduct coincides with the multiparticle V enezia no amplitude discussed in our work [83 ]. T o obtain the Ewens sampling formula (equation (B.6)) from (5.1 0) a few additional steps are r equired. These a r e: a) we hav e to le t K → ∞ while allowing many o f n ′ i s in (5.7) to b ecome z e r o (this explains the meaning o f the word ” sampling”), b) we hav e to order remaining n ′ i s in such a way that n (1) ≥ n (2) ≥ · · · ≥ n ( k ) > 0 , 0 , ..., 0 , c) we have to cyclically order the re ma ining n ′ i s in a wa y explained in App endix B by int ro- ducing c ′ i s as n umbers o f remaining n ′ ( i ) s which ar e equal to i . That is w e have 23 E.g. see Wikip e dia where it is known as Hardy-W einberg pr inciple. 25 to make a c hoice b et ween repre s en ting r = P k i =1 n ( i ) or r = P r i =1 ic i under condition that k = P r i =1 c i , d) ﬁnally , just like in the case o f Bos e (F er mi) statistics, we hav e to multip y the r .h.s.of (5.10) by the o b vious ly lo oking combi- natorial factor M = K ! / [( c 1 ! · · · c r !)(( K − k )!]. Under such c o nditions, w e obtain: Γ(( K + 1) ε ) ≃ Γ( θ ) , Γ(( K + 1) ε + r ) = Γ( θ + r ) , Γ( ε + n ( i ) ) n ( i )! = 1 n ( i ) . Less trivial is the result: K ! / [( K − k )! [Γ ( ε ) ] k ] → θ k . Evidently , the factor n ! n 1 ! n 2 ! · · · n K ! in (5.10) no w should b e replaced by r ! n (1) · · · n ( k ) . Finally , a momen t of thought causes us to replace n ′ ( i ) s by i c i 24 in or de r to arrive at the E w ens s ampling formula: P ( k ; n (1) , ..., n ( k ) ) = r ! [ θ ] r r Y i =1 θ c i i c i c i ! (5.11) in ag reemen t with (B.6). This deriv ation was made without any refer ence to genetics and is completely mo del-independent. T o demonstra te connections with high energy physics in general and with V eneziano amplitudes in par ticular, we would like to explain the rationale b ehind this for mula using an abso lute minim um of facts from genetics. Genetic infor mation is s t ored in g enes . These are so me seg men ts ( lo cuses ) of the double str anded DNA molec ule. This fact allows us to think ab out the DNA molecule a s a w o rld line for mesons made o f a pair of quark s . Phe- nomenologica lly , the DNA is essentially the chromosome . Humans and ma n y other sp ecies ar e diplo ids . This means that they need for their repro duction (meiosis) tw o sets of chromosomes-one from ea ch par en t. Hence, we can think of meiosis a s a pro cess analogo us to the meson-meson scattering. W e would like to depict this pro cess g raphically to emphasize the ana logy . Before doing so we need to make a few remar ks. First, the life cy c le for diploids is ra th er bizarre. Each ce ll of a grown organism con tains 2 sets of c hro mosomes. Maiting, how ever, requires this rule to b e changed. The gametes (sex cells) fro m each parent carry only one set of chromosomes (that is, such cells a re haploid !). T he existence of 2 sets of chromosomes makes individual org anism unique b ecause of the following. Consider, for instance, a sp eciﬁc trait, e.g. ”tall” vs ”shor t”. Genetically this prop ert y in enco ded in some gene 25 . A par ticular realiz a tion of the gene (causing the or g anism to b e, say , tall) is called ” allele ”. T ypically , there ar e 2 alleles -one for ea c h o f the c hr omosomes in the t wo chromosome set. F or ins t ance, T and t (for ”tall” and ”short”), or T a nd T or t and t or, ﬁnally , t and T (sometimes o rder matters). Then, if fa t her donates 50% of T cells and 50% of t cells and mo ther doe s same, the oﬀspring is likely go ing to ha ve either TT comp osition with pro ba bilit y 1/4, or tt (with proba bilit y 1/ 4) or tT (with probability 1/4) a nd, ﬁnally , tt with proba bilit y 1/4. But, o ne of the alleles is usually dominant (say , T) s o that we will see 3/4 of tall people in the oﬀspring and 1/4 short. What we just describ ed is the essence o f the Har dy-W ein ber g 24 This is so because the c i n umbers count how many of n ′ ( i ) s are equal to i . 25 Or in many genes, but we talk ab out a gi v en gene f or the sake of argument. 26 law based, o f course , on the or iginal works b y Mendel. Details can b e found in genetics literature [110]. Let us concentrate our attent ion on a par ticular lo cus so that the ge ne tic character(trait) of a particular individual is describ ed by sp ecifying its tw o genes at that lo cus. F or N individua ls in the po pula tion there are 2 N chromosomes containing s uc h a lo cus. F or ea c h allele, o ne is in terested in k nowing the prop or- tion of 2 N chromosiomes at which the g ene is realized as this allele. This gives a probability distribution o ver the set of p ossible alleles w hich describ es a g enetic make-up o f the p opulation (as far as we are only lo oking at some speciﬁc lo cus). The problem now is to mo del the dynamical pro cess by which this distribution changes in time from generation to genera t ion a ccoun ting for m utatio ns a nd selection (caused b y the environment ). Mutation can b e caused just b y c hange of one nucleotide along the DNA strand 26 .Normally , the mutan t alle le is inde- pendent of its pare nt since, once the mutation takes plac e, it is very unlikely that the c orrupt message means a n ything at all. Hence, the m utant can be either ”go od” (ﬁt) o r ”ba d” (unﬁt) for life a nd its contribution can b e igno red. If u is the probability of mutation p er gene p er generation then, the par ameter θ = 4 N u in (6.11). With this infor mation, w e a re ready to resto re the rest of the genetic co nten t of W atters on’s pape r [111]. In par ticula r, random P -D v a riables X 1 , X 2 , ..., X K denote the allele relative frequences in a po pulation consisting of K alleles. Evidently , by construction, they are Dirichlet-distributed. Let K → ∞ and let k b e an experimental sa m ple of repres e n tative frequencies k ≪ K. The comp osition of such a sample will be rando m b oth, bec ause of the nature o f the sampling pro cess, a nd b ecause the p opulation itself is sub ject to random ﬂuctuations. F or this reaso n we av erage d the Har dy-W ein ber g distr ibu- tion (5.7) ov er the P-D distribution in order to arrive a t the ﬁnal result (6.11 ) . This re s ult is an e quilibrium result. Its exp erimen tal veriﬁcation can b e found in [110]. It is of in terest to ar riv e a t it dynamica lly . This is accomplished in the next subsection but in a diﬀerent context. Based on the facts just discussed, it should b e clear that both g enetics and ph ys ics of meson sca tt ering (for whic h V eneziano had prop osed his amplitude) have the s a me combinatorial or igin . All random pro cesses inv o lv ing decomp ositions r = P k i =1 n ( i ) (or r = P r i =1 ic i ) ar e the P-D pro cesses [10 3-105]. T o conclude this subs ection, w e would lik e to illustrate graphica lly why ge- netics and physics o f meson scattering have many things in co mmon. This is done with help of the following 3 ﬁgures. 5.4 Sto c hastic mo dels for second order che mical reaction kinetics in volving V eneziano-lik e amplitudes The role of s t o c ha stic pro cesses in chemical kinetics w a s reco gnized long a g o. A nice summary is co n tained in the pap er by Mc Q uarrie [118]. The purp ose 26 The so called ”Single Nucleotide Polymorphism” (SNP) whic h i s detectable either elec- trophoretically or by DNA m e lting exp erimen ts, etc . 27 Figure 1: The simplest duality diagr am describing meson-meso n scattering [114 ]. The s a me picture descr ib es ”co llis ion” of tw o par en tal DNA’s dur ing meiosis and can b e seen directly under the electro n microscop e. E.g.see Fig.2.3 in [115], page 18. Figure 2: Non- planar lo op Pomeron diagram fo r meso n-meson scatter ing [116]. The same diagra m describe s homolog ous DNA reco m bination, e.g . see Fig.2.2 in [115], page 17. 28 Figure 3: The pla na r lo op meson-baryon sca t tering dua lit y diagram. The same diagram descr ibes the interaction (scattering) b et ween the tr iple a nd do uble stranded DNA helices [117]. of this subsection is to co nnect the results in chemical kine tics with those in genetics in or de r to repr oduce the V eneziano (or V eneziano-like) amplitudes as an equilibr ium measures for the underlying chemical/biologica l pro cesses. F ol- lowing Darvey et al [119 ] we consider a chemical rea ction A + B k 1 ⇄ k − 1 C + D analogo us to the meson-meso n scattering pro cesses whic h trigger ed the discov- ery of the V eneziano amplitudes[13]. Le t the r e s pective concentrations o f the reagents b e a, b, c and d . Then, acc ording to rules of c hemical kinetics, w e obtain the following ”equatio n of motion” da dt = − k 1 ab + k − 1 cd. (5.12) This equation ha s to b e supplemented with the initial condition. It is obta ined by a ccoun ting fo r mass conserv atio n. Speciﬁcally , let the initial concentrations of rea gen ts b e resp ectiv ely: α = A (0) , β = B (0) , γ = C (0) and δ = D (0) . Then, eviden tly , α + β + γ + δ = a + b + c + d, provided tha t for all times a ≥ 0 , b ≥ 0 , c ≥ 0 and d ≥ 0. Account ing for these facts, equation (5.1 2) can be rewritten as da dt = ( k − 1 − k 1 ) a 2 − [ k 1 ( β − α ) + k − 1 (2 α + γ + δ )] a + k − 1 ( α + γ )( α + β ) . (5.13) Thu s far , this is a standard r esult of chemical kinetics. The new element emerges when one cla ims that the v ariables a, b, c and d are r andom but are s till sub ject to mass cons erv atio n. Then, as we know alrea dy fr o m previous subsections, we a re dealing with the P–D-type pr ocess. The new element now lies in the ackno wledging the fact that this pr ocess is dyna mical. F ollowing Kingma n [1 20] we would like to formulate it in precise mathematica l ter ms . F or this purp ose, we intro duce the vector p (t)=(p 1 (t),..., p k (t)) such that it moves ra ndomly on the simplex ∆ deﬁned b y ∆ = { p ( t ); p j ≥ 0 , X k i =1 p i = 1 } (5.14) 29 In the present ca se the p ossible sta tes of the system at time t which co uld lead to a new sta te sp eciﬁed b y a, b, c, d at time t + ∆ t inv o lving not more than one transformatio n in the time in terv al ∆ t are [119]   a + 1 b + 1 c − 1 d − 1 a − 1 b − 1 c + 1 d + 1 a b c d   . (5.15) In writing this matrix, following [119 ], we have assumed that random v ariables a, b, c and d are in teger s. Using (5.15) w e obtain the f ollowing equation of motion P ( a, b, c, d ; t + ∆ t ) − P ( a, b, c, d ; t ) = [ k 1 ( a + 1)( b + 1 ) P ( a + 1 , b + 1 , c − 1 , d − 1; t ) + k − 1 ( c + 1)( d + 1) P ( a − 1 , b − 1 , c + 1 , d + 1; t ) − ( k 1 ab + k − 1 cd ) P ( a, b, c , d ; t )]∆ t + O (∆ t 2 ) . (5.16) In v iew of the fact that t he motion is t aking place on the simplex ∆ , it is suﬃcient to lo ok at the s t o c ha stic dynamics of just one v a riable, say , a (very m uch like in the deter ministic equation (5.13)). This replaces (5.16 ) by the following result: d dt P a ( t ) = k 1 [( a + 1)( a + 1 + β − α ) P a +1 ( t ) + k − 1 [( γ + α − a + 1)( δ + α − a + 1 ) P a − 1 ( t ) − [ k 1 a ( β − α + a ) + k − 1 ( γ + α − a ) ( δ + α − a )] P a ( t ); provided that P α (0) = 1 , α = a and P α (0) = 0 if a 6 = α. (5.17) T o solve this equatio n we intro duce the generating function G ( x, t ) via G ( x, t ) = X a =0 P a ( t ) x a and use this function in (5.17) to obtain the following F okker–Plank-type e qua- tion ∂ ∂ t G ( x, t ) = x (1 − x )( k 1 − xk − 1 ) ∂ 2 ∂ x 2 G + (1 − x )[ k 1 ( β − α + 1) + k − 1 (2 α + γ + δ − 1) x ] ∂ ∂ x G − k − 1 ( α + γ )( α + δ )(1 − x ) G ( x, t ) . (5.18) This equation admits s eparation of v ariables: G ( x, t ) = S ( x ) T ( t ) with solution for T ( t ) in the exp ected form: T ( t ) = exp ( − λ n k 1 t ) , leading to the equation for S ( x ) x (1 − x )(1 − K x ) d 2 dx 2 S ( x )+[ β − α +1+ K (2 α + γ + δ − 1) x ](1 − x ) d dx S − [ K ( α + γ )( α + β ) (1 − s ) − λ n ] S ( x ) = 0 (5.19) This equation is of Lame-type as discussed in [44,4 7] and, therefore , its solution should b e a p olynomial in x o f degree at mos t , where  sho uld be equal to 30 the minim um of ( α + γ , α + δ , β + γ , δ + δ ) . As in quantum mechanics, this implies that the sp ectrum o f eigenv alue s λ n is discr ete, ﬁnite and, a priory nondegenera te. Among thes e eigenv alues there must be λ 0 = 0 since such an eigenv alue corr esponds to the time-indep enden t s olution of (5.1 9) t ypical for the true equilibrium. Hence, for this case we obta in, instead of (5 .1 9), the following ﬁnal result: x (1 − K x ) d 2 dx 2 S ( x ) + [ β − α + 1 + K (2 α + γ + δ − 1) x ] d dx S − [ K ( α + γ )( α + β )] S = 0 , (5.20) where K = k − 1 /k 1 . This cons t ant ca n b e eliminated fro m (5.2 0 ) if we rescale x : x → K x. After this, eq uation (5.20 ) ac q uires the already familiar (e.g. see (4.41)) h yp ergeometric form x (1 − x ) d 2 dx 2 S ( x ) + [ β − α + 1 + (2 α + γ + δ − 1 ) x ] d dx S ( x ) − ( α + γ )( α + β ) S ( x ) = 0 . (5.21) In [1 20] Kingma n obtaine d the F okker-Planck type equatio n ana logous to our (5.18) describing the dyna mical pr o cess whose s t able equilibrium is described by (5.21) (naturally , with diﬀerent co eﬃcien ts) and le a ds to the P-D distribution (5.2) essen tial for obtaining the Ewens sampling form ula. Instead of repr oducing his r esults in this work, w e would like to connect them with results of our Section 4. F or this pur p ose, we b egin with the follo wing observ ation. 5.4.1 Quan tum mec h a nics, h yp ergeometric functions and P-D dis- tribution In our works [37,46] we provided deta iled explana tion of the fact that all exactly solv able 2-b o dy quantum mechanical problems inv olve diﬀerent kinds of s p ecial functions obtainable fro m the Gauss hyper geometric funcftion w ho se int egra l representation is given b y F ( a, b, c ; z ) = Γ( c ) Γ( b )Γ( c − b ) 1 Z 0 t b − 1 (1 − t ) c − b − 1 (1 − z t ) − a dt. (5.22) As is well known from qua n tum mechanics, in the case of a disc t ete sp ectrum all q ua n tum mechanical problems inv o lv e o rthogonal po lynomials. The ques- tion then arises: under what conditions on co eﬃcien ts ( a, b a nd c ) can the inﬁnite h yp ergeometric series who se int egra l re pr esen tatio n is given by (5.2 2) be reduced to a ﬁnite po lynomial? This happens , for instance, if we im- po se the qua n tization condition : − a = 0 , 1 , 2 , .... In such a case we ca n write (1 − z t ) − a = P − a i =1 ( − a i )( − 1) i ( z t ) i and use this ﬁnite expa nsion in (5.22). In view of (5.2 ) w e obtain the co nvergen t gener ating function for the Dirichlet distribution (5.2). Hence, a ll known quantum mechanical pro blems inv olving discrete sp ectrum a re examples o f the P- D sto c hasic pro cesses 27 . F or hyper - geometric functions o f multiple ar gumen ts this was demonstra ted in Section 27 Eviden tly , in the case of con tin uum sp ect rum we are al s o dealing with the P- D processes but the corresp onding h ypergeometric seri es conain now coun table inﬁnit y of terms . 31 4 . Th us , a ll quantum mechanical, quan tum ﬁeld-theo retic a nd s t ring-theo retic pro cesses are the P-D pro cesses . As such they fa ll int o a m uc h larger class of sto c hastic pro cesses kno wn as random coagul a tion and fragmen tation pro cesses . W e would like to conclude this section with the following additional observ atio ns. 5.4.2 Hyp ergeometric functions, Kummer series expansions and V eneziano amplitudes In view of just in tro duced quantization condition, the question arises: is this the only condition reducing the hyperge o metric function to a p o lynomial ? Mo re broadly: what conditions on co eﬃcien ts a, b and c should b e impos ed so that the function F ( a, b, c ; z ) becomes a p olynomial? 28 The answer to this questio n was provided b y Kummer in the ﬁrst half of 19th century [1 01]. W e would like to summar iz e his r e sults and to c onnect them with r esults presented ab ov e. Additional details can b e found in our pap er [47]. By doing so we shall reobtain V eneziano amplitudes for chemical proces s describ ed by equation (5.21). According to gener al theory o f h yp ergeometric eq uations [10 1 ], the inﬁnite series for a hypergeometr ic function degenera tes to a p olynomial if o ne of the nu mbers a, b, c − a or c − b (5.23) is an integer. This co ndition is equiv a len t to the condition that, at lea st one of the eig ht num b e r s ± ( c − 1) ± ( a − b ) ± ( a + b − c ) is an o dd num b er. This observ atio n pro duces 24 solutions for the Gauss h yp ergeometric function (5.2 1) found b y Kummer. Among these he s ingled out 6 (generating all 24) and among these 6 he established that ev ery 3 of them are related to each other. Let us denote these 6 functions (solutions) as u 1 , ..., u 6 . Then, we can represent, say , u 2 and u 6 using u 1 and u 5 as the basis set. W e can do the same with u 1 and u 5 by r epresen ting them thro ugh u 2 and u 6 and, ﬁnally , w e can connect u 3 and u 4 with u 1 and u 5 . Hence, for our purp oses, it is suﬃcient to consider, say , u 2 and u 6 . W e obtain,  u 2 u 6  =  M 1 1 M 1 2 M 2 1 M 2 2   u 1 u 5  , (5.24) with M 1 1 = Γ( a + b − c + 1 )Γ(1 − c ) Γ( a + 1 − c )Γ( b − c + 1) ; M 1 2 = Γ( a + b + 1 − c )Γ( c − 1) Γ( a )Γ( b ) ; M 2 1 = Γ( c + 1 − a − b )Γ(1 − c ) Γ(1 − a )Γ(1 − b ) ; M 2 2 = Γ( c + 1 − a − b )Γ( c − 1) Γ( c − a )Γ( c − b ) . The determinant of this matrix becomes zero if either tw o rows or tw o columns bec o me the same 29 . 28 Inciden tally , i n the case of K -Z-t yp e hypergeometric equations suc h a problem was s o lved only in 2007 [121]! 29 The condition for the determinant to b ecome zero is the resonance condition. It is of cen tr al i mportance i n b o th qun tum m e ch anics and s t ring theory [47]. 32 F or instance, we obtain: Γ( a )Γ( b ) Γ( c − 1) = Γ( a − c + 1)Γ( b − c + 1) Γ(1 − c ) and Γ( c − a )Γ( c − b ) Γ( c − 1) = Γ(1 − a )Γ(1 − b ) Γ(1 − c ) . (5.25) F or c = 1 w e obtain an identit y . F rom [119] we ﬁnd that (5.21) admits 2 independent solutions: S ( x ) = { either F ( − α − γ , − α − δ , β − α + 1 ; K x ) , for β ≥ α or ( K x ) α − β F ( − β − γ , − β − δ , α − β + 1; K x ) , for β ≤ α . (5 .26) Hence, the condition c = 1 in (5.25) causes tw o solutions for S ( x ) to degener ate int o one p olynomial s olution, provided that we make an iden tiﬁcation: β = α in (5.26). Notice that to obtain this re s ult ther e is no need to impo se an extra condition: a = b 30 (whic h, in our case, is the same as γ = δ ) . This makes sence physically b oth in chemistry and in high energy physics. In the case of high ener gy physics, if the V eneziano amplitudes are used for description of, say , π π sc a ttering, in [83], page 54, it is demonstr a ted that pr o- cesses for which ”co ncen tratio ns ” a = b cause this a mplit ude to v anis h. The V eneziano condition: a + b + c = − 1( equa tio n (1 . 5) of [83 ]) has its analog in chemistry wher e it plays the same role, e.g. of mass c o nserv a tion. In the pr esen t case we ha ve α + β + γ + δ = const, and the V eneziano-like amplitude obtaina ble from (5.25),(5.26) is given now by V c ( a, b ) = Γ( − α − γ )Γ( − α − δ ) − c Γ( − c ) | c =1 . (5.27) In view of known symmetry of the hyper geometric function: F ( a, b, c ; x ) = F ( b, a, c ; x ), w e also have: V c ( b, a ) = V c ( a, b ) . This is compatible with the symmetry for V eneziano amplitude. T o mak e the analog y with V eneziano amplitudes complete, we hav e to select the following options in (5.2 7): a) α = 0 , γ = 1 , δ = 1 , 2 , ... or ; b ) α = 1 , γ = 0 , δ = 0 , 1 , 2 , ... These conditions are exa ctly compa tible with those in (1.19 ) of [ 83 ] for V eneziano amplitudes. Finally , in view of (5.22), these ar e the quantization conditions needed for res- onances to exist as required. 6 Conclusions In Einstein’s time to c reate a uniﬁed ﬁeld theory was the ultimate goal of physics. It should b e noted thoug h, that, for instance, Pauli w as not shar ing Einstein’s bele if in the uniﬁed ﬁeld theory . Abdus Salam in his Nob el Lecture given in 1979 31 on pages 5 18-519 writes: ” I must admit I was taken aba ck by Pauli’s ﬁerce pr ejudice against universalism-ag a inst what we w ould to day call uniﬁca- tion o f basic forces-but I did not take it too seriously . I felt this was a leg a cy o f 30 Here a and b ha ve the same meaning as in (5.22) and should not be confused with con- cen tr ations. 31 Readily av ailable at the Google search database. 33 the e x asperatio n whic h Pauli had alwa ys felt a t Einstein’s somewhat formalis- tic attempts at unifying gravit y with electromagnetism- forces which in Pauli’s phrase ”ca nno t b e joined-for Go d ha t h rent them asunder” . Ba sed o n the re- sults discuss ed in this pape r , we sincere ly hop e, that should Pauli b e a live, and b e aw ar e of the enormo us universalit y o f the sto c ha stic theory of random fragmentation-coagulatio n pro cesses, he would change his positio n on univer- salism. By wr iting an o bit uary for Pauli, Landau, in fact, had inagurated this theory . The ide a of a universal theory do es no t ex clusiv ely b elong to physicists though. W e mentioned already in Section 4 that Landau’s pap er [6 7] made a big impa ct on the theory o f s ingularities and cata strophes. Rene Thom-the founding father o f the theory of ca tastrophes-also was lo oking for the theory of everything. In his bo ok [1 22] he made a n attempt to apply the theory of sin- gularities to biology . La ter on Rober t Gilmore using Thom’s r esults developed many applications of catas trophe theory to other disciplines [12 3]. It is fair to say that the ma t hematical theory of catastr ophes is re lated to physical theory of ra ndo m fr a gmen tatio n-coagulation pro cesses as thermo dynamics is re la ted to statistical mec hanics. W e b eliev e, that it is this sto c ha stic theory which rea lly deserves to b e called the universal theory of everything. Hopefully , without causing a g lo bal catastrophe, the millennium into which we are entering is go- ing to b e the millennium of incre dibly versatile succession of hallmarks in the theory of univ ersal catastrophes . App endix A . C onnections b et w een the gamma and Diric hle t dis - tributions Using res ults of our pa per [8 3 ] esp ecially , e quation (3 .27) of this re fer ence, such a connection can be easily established. Indeed, consider n + 1 independently distributed random gamma v aria bles with exponents α 1 , ..., α n +1 . The joint probability density for suc h v a riables is g iv en by p Y 1 , ... Y n +1 ( s 1 , ..., s n +1 ) = 1 Γ( α 1 ) · · · 1 Γ( α n +1 ) s α 1 − 1 1 · · · s α n +1 − 1 n +1 . (A.1) Let now s i = t i t, w he r e t i are chosen in such a wa y that P n +1 n =1 t i = 1 . Then, using such a substitution in (A.1) we obtain at once: p u 1 , ... u n +1 ( t 1 , ..., t n +1 ) = [ ∞ Z 0 t α − 1 e − t ] 1 Γ( α 1 ) · · · 1 Γ( α n +1 ) t α 1 − 1 1 · · · t α n +1 − 1 n +1 , provided tha t 1 = X n +1 n =1 t i . (A..2) Since α = α 1 + · · · + α n +1 , we obtain: ∞ R 0 t α − 1 e − t = Γ( α 1 + · · · + α n +1 ) , so that the density of pro babilit y (A.2) is indeed of Diric hlet-type given b y (5.2). 34 App endix B. Some facts from combinatorics of the symmetric group S n Suppo se w e hav e a ﬁnite set X. ∀ x ∈ X consider a bijection X − → X made of some permutation sequenc e : x, π ( x ) , π 2 ( x ) , ... Because the set is ﬁnite, we must hav e π m ( x ) = x for some m ≥ 1 . A sequence ( x, π ( x ) , π 2 ( x ) , .., π m − 1 ( x )) = C m is called a cycle of le ngth m . The set X can b e sub divided into a disjoint pro duct of cycles so that any p ermutation π is just a pr oduct of these cycle s . Normally such a pr oduct is not uniquely deﬁned. T o make it uniquely deﬁned, we ha ve to assume tha t the set X is ordered a ccording to a certain rule. The, standard cycle r epresen tation can b e constructed by requiring that: a) eac h cycle is written with its largest element ﬁr st, and b) the cycles are written in increasing order o f their resp ectiv e large s t elements. Let N b e some integer and consider a decomp osition of N as N = P K i =1 n i . W e say that n ≡ ( n 0 , ..., n K ) is the partition of N (or n ⊢ N ) . The same re s ult ca n b e achiev ed if, instead we would consider the following decompo sition of N : N = P N i =1 ic i where, a ccording to our conv en tions, we hav e c i ≡ c i ( π ) a s the num b er of cy cles o f length i . The total n umber of cycles then is given b y K = P N i =1 c i . Deﬁne a num b er ˜ S ( N , K ) as the num b er of p erm utations of X with exac t ly K cy cles. Then, the Stir ling nu mber o f the ﬁrst kind can b e deﬁned as S ( N , K ) := ( − 1) N − K ˜ S ( N , K ) . The nu mbers ˜ S ( N , K ) ca n b e obta ine d r ecursiv ely using the following recurrence relation: ˜ S ( N , K ) = ( N − 1 ) ˜ S ( N − 1 , K ) + ˜ S ( N − 1 , K − 1) , N , K ≥ 1; ˜ S (0 , 0) = 1 . (B.1) Use of this recurrence allows us to obtain the following important result: N X K =0 ˜ S ( N , K ) x K = x ( x + 1)( x + 2) · · · ( x + N − 1) . (B.2) Let now x = 1 in ( B . 2) , then we ca n deﬁne the probabilit y p ( K ; N ) = ˜ S ( N , K ) / N ! F urthermor e , one ca n deﬁne yet ano ther probability b y intro ducing a nota tio n [ x ] N = x ( x + 1)( x + 2) · · · ( x + N − 1 ) . Then, we obta in: N X K =0 ˜ S ( N , K ) x K [ x ] N = N X K =0 P K ( N ; x ) = 1 . (B.3) Such deﬁned probabilit y P K ( N ; x ) can be further rewritten in view o f the famous result b y Cauch y . T o obtain his result, we intro duce the gener ating function F N K ( x ) = X K = P N i =1 c i N = P N i =1 ic i N ! 1 c 1 c 1 !2 c 2 c 2 ! · · · N c N c N ! x c 1 1 · · · x c N N (B.4a) 35 and require that ˜ S ( N , K ) x K = F N K ( x ). This can happen o nly if N X K =0 X K = P N i =1 c i N = P N i =1 ic i N ! 1 c 1 c 1 !2 c 2 c 2 ! · · · N c N c N ! = 1 (B.4b) Thu s, we obtain: ˜ S ( N , K ) = K Y i =1 N ! i c i c i ! , provided that K = X N i =1 c i and N = X N i =1 ic. (B.5) In these nota tio ns the E w ens sampling form ula acquires the follo wing canonical form: P K ( N ; x ) ≡ x K [ x ] N K Y i =1 N ! i c i c i ! , provided that K = X N i =1 c i and N = X N i =1 ic. (B.6) References [1] L.Landau, ”On fundamen tal problems”, Collected W orks . V ol.2, p. 421 (Nauk a , Moscow, 196 9) 32 [2] E.F ermi, ”Angular distribution of the pions in high energy nu clear collisions”, Phys.Rev. 81 683 (1951). 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