SWAT/94-95/57 UM-P-94/122 QMW-PH-94-39 hep-th/9501078 Restricted Quantum Theory of Affine Toda Solitons 5 9 9 1 n a J 8 B. Spence 1 1 v School of Physics 8 7 University of Melbourne 0 1 Parkville 3052 Australia 1 0 E-mail:[email protected] 5 9 / h t - p e h Jonathan Underwood : v i X r Physics Department a University College of Swansea Swansea SA8 2PP UK E-mail:[email protected] Abstract We quantise the reduced theory obtained by substituting the soliton so- lutions of affine Toda theory into its symplectic form. The semi-classical S-matrix is found to involvethe classical Euler dilogarithm function. 1Current Address: Physics Department, Queen Mary & Westfield College, Mile End Road LondonE14NSUK 1 Introduction The affine Toda field theories associated to an affine Kac-Moody algebra gˆ have received a great deal of attention in recent years. The classical equation of motion which describes them is 2µ2 r 22φ+ m H eβαi(φ) =0, (1.1) i i β i=0 X where φ is the dynamical field, taking values in the Cartan subalgebra of ˆg (for whom Chevalley generators are the H and the Coxeter number is h), µ and β are i real parameters, and α are the simple roots of ˆg. Taking β to be real we obtain i a model with a particle spectrum whose properties are algebraic in origin (see, for example,refs. [1,2]),andwhoseS-matrixisexactlyknown[3,2,4,5,6,7,8,9,10]. Ifthecouplingconstantβ ispureimaginarythenthemodelpossessesaseriesof vacuumsolutions. We expect solitonsolutionsto be thoseofminimalenergywhich interpolate these vacua, the difference between the finalandthe initial vacua being known as the topological charge of the soliton. The seminal calculation was made by Hollowood [11] for the Aˆ models using Hirota’s method. This technique was r laterdevelopedtoincludetherestofthetheories[12,13]. Anewmethod,involving theuseofthegeneralsolutiondiscoveredbyLeznovandSaveliev(seee.g. [14]),has recently proved to be very useful in extracting particular properties of the solitons [15]. This method can be motivated by consideration of the form of the canonical energy-momentum tensor r g 2g T =(∂ φ,∂ φ) µν (∂ φ,∂αφ) µν n eβαk(φ) 1 . (1.2) µν µ ν − 2 α − β2 k − Xk=0 (cid:16) (cid:17) Using arguments given in ref. [15] it can be shown that T can be split into a µν sum of two parts, T = C +Θ , where Θ is traceless and C is a total µν µν µν µν µν derivative. Due to the topological nature of solitons, their properties, in particular their energies and momenta, depend only on their behaviour at infinity, and hence we might expect Θ to vanish for soliton solutions. Further work [15, 16] shows µν thatΘ canbewrittenpurelyintermsofthechiralfieldswhicharetheparameters µν of the Leznov-Saveliev solution, so it is plausible that the soliton solutions arise if these fields vanish. InordertofindtheN-solitonsolutionsafurtheransatzfortheformofaconstant of integration appearing in the general solution must be made [17]; the N-soliton specialised solution turns out to be e−βφk = Λ exp W Fˆi1(z ) ...exp W FˆiN(z ) Λ . (1.3) k 1 1 N N k h | | i (cid:16) (cid:17) (cid:16) (cid:17) 1 For a detailed explanation of the quantities appearing in the formula the reader should consult ref. [17]; we will need only the following details. The parameters i j are integers lying between 1 and r which describe the species of the N solitons. z j are complex parameters related to the soliton rapidities ρ in the following way: j q+ z =θ e−ρj| ij|, (1.4) j j q+ ij where θ = 1, and the q are structure constants of gˆ which can be calculated j ± ij from the following formula of ref. [1] q+ γ q(1)=2ıx (1)e−δpBıνπ/h (1.5) p ≡ p· p (ı √ 1). The functions W encode the space-time dependence of the solution, j ≡ − and are given by the formula W =Q exp √2µθ q+ (tsinhρ xcoshρ ) . (1.6) j j j | ij | j − j n o The first major use to which this formalism was put was a calculation of the masses of single solitons of species p. These turned out to be [15]: 4√2h M = µq+ . (1.7) p −β2γ2 | p| p An interesting fact which is not yet fully understood is that these masses are in a certain sense dual to those of the particles. In this letter we will use further features of the solution (1.3) to calculate the Poissonbracketson the N-soliton phase space andwill performa canonicalquanti- sation to extract the S-matrix. 2 Poisson Brackets The classicalphase space of the affine Toda theories has been discussed recently in ref. [18]. Thesymplectic formΩ iscalculatedbyintegratingthe symplectic current over all space at some time, and is simply Ω= dx(δφ,δ∂ φ) (2.1) t Z where ( , ) denotes the usual Killing form on the algebra ˆg. We wish to investigate the phase space of the soliton solutions. To do this, we insert the soliton solutions into the symplectic form Ω. First let us calculate the symplectic form for a one-soliton solution. We will need to use two important propertiesofthissolution,butourmethodwillmeanthatwedonotneedtheexplicit 2 formofthesolutionitself. Thisisatypicalfeatureofcalculationsconcerningsolitons in affine Toda theory. The first feature we need is that the soliton is a relativistic object,andsothe Poincarealgebramustbe realisedonthe phasespace. Whatthis meansforthesolutionisthatthefieldφmustappearonlyasafunctionofu,where u=tsinhρ (x x )coshρ. (2.2) 0 − − Therapidityρandcentreofmassx aretherealparametersofthesolitonsolution. 0 The relationshipofthese parametersto thoseinthe algebraicansatzforthe soliton solution, as well as the explicit dependence of φ upon u, can be determined from the formulæ of [15], [17], but we shall not need these yet. Using the antisymmetry properties of the wedge product we obtain dφ dφ Ω = dx , coshρδx δ(sinhρ), 0 du du ∧ Z (cid:18) (cid:19) δx δ(sinhρ) 0 = dx(∂ φ,∂ φ) ∧ , t x − sinhρ Z δx δ(sinhρ) 0 = T ∧ , (2.3) tx − sinhρ Z where the last step is performed using the expression for the energy-momentum tensor (1.2). Finally, substituting P = dxT =Msinhρ into (2.3), we obtain 01 − R Ω=δξ δρ, (2.4) ∧ whereξ =Mx coshρisthe canonicalvariableconjugatetoρ. Thisisofcoursethe 0 result we expect. In order to evaluate the symplectic form in the more general case of N solitons we consider the form of the solution as t . In the generic situation the → ±∞ solitons will be well separated and since Ω is a local expressionwe can just add up the contributions from each of the solitons in turn. Thus we find N in in Ω= δξout δρout. (2.5) i ∧ i i=1 X We now need to relate these variables to those which parameterise the solution. Let us consider the form of the solution (1.3) as x increases from . The −∞ first significant departure from the vacuum will occur with the soliton of greatest rapidity, ρN. We know from ref. [17] that the greatest non-vanishing power of FˆiN within a representation of level x is x.2 This in turn means that the solution for the component field φ will be the logarithm of a polynomial of degree m in W k k N defined by equation (1.6). As we move through this soliton W becomes much N 2Weareonlyconsideringthetheorieswhereˆgissimply-lacedfromnowon. 3 greater than 1 and so we can ignore all but the highest power in the polynomial as far as calculating the form of the solution for greater x is concerned. This term of coursemultipliesanalgebraicfactorFˆiN(zN). Normalorderingthevertexoperator expression [19], [17] for this yields e−βφk = Λ FiN(z )Λ Wmk Λ exp X (z ,z )W Fˆi1(z ) ... (2.6) h k| N | ki N h k| i1,iN 1 N 1 1 (cid:16) (cid:17) ...exp XN−1,N(zN−1,zN)WN−1FˆiN−1(zN−1) Λk , | i (cid:16) (cid:17) where h−1 z wn(γi)·γj X (z ,z )= 1 e−2πin/h 2 . (2.7) i,j 1 2 − z n=0(cid:18) 1(cid:19) Y The rootsγ arethe simple rootsα up toa sign[17],andw is the Coxeterelement i i of the Weyl group. Thus, aside from a constant shift in the field φ, the only effect is to change each of the Q for j < N by a factor X (z ,z ). Extending this j ij,iN j N argument as we move through each of the solitons in turn leads us to conclude that as t the field φ becomes a sum of one-soliton solutions but with new → −∞ parameters Qin =Q X (z ,z ). (2.8) j j j,p j p p>j Y The rapidities remain unchanged under the transformation to ‘in’ variables. Now all we need to do is relate the variables Q to the ξ . Comparing expressions (1.6) j j and (2.2) we find that up to an irrelevant constant θlnQ x = . (2.9) 0 √2µq+ coshρ | i | Using the mass formula equation (1.7) we obtain 2hθlnQ ξ = , (2.10) β 2 | | and so 2h ξin = ξ + lnX (z ,z ), j j β 2 j,p j p | | p>j X ρin = ρ . (2.11) j j Similar arguments yield the following relationships for the out variables 2h ξout = ξ + lnX (z ,z ), j j β 2 p,j p j | | p<j X ρout = ρ . (2.12) j j 4 Since the symplectic form Ω is trivial to invert in terms of terms of either ‘in’ or ‘out’ variables the appropriate Poisson brackets for the natural variables follow straightforwardly,andcanbeseentobeconsistentifwenotethatX(z ,z )depends j p only on the rapidity difference ρ ρ . j p − 3 Scattering Matrix Havingfoundcanonicalcoordinatesontheclassicalphasespacewecannowquantise the theory in a straightforward manner, simply by replacing the canonical Poisson bracket with the canonical commutators ξin,ρin = ξout,ρout =ı¯h. (3.1) (cid:2) (cid:3) (cid:2) (cid:3) Wecanthenattempttodiscovertheunitarytransformation(analogueofthecanon- icaltransformation)which willbe the S-matrix for the reducedtheory. Fromequa- tions (2.11) and (2.12) we conclude that 2h ξout = ξin+ lnX (z ,z ) lnX (z ,z ) j j β 2  p,j p j − j,p j p  | | p<j p>j X X   ρout = ρin =ρ . (3.2) j j j We can see from these equations that the S matrix S is purely a function of the rapidity differences, and satisfies the equation ∂(logS) 2hı = − lnX (ρ ,ρ ) lnX (ρ ,ρ ) . (3.3) ∂ρ β 2¯h p,j p j − j,p j p  j | | p<j p>j X X   The solution of this is h−1 2hı ıπ S =exp wn(γ ) (γ )Li exp ρ ρ + (δ δ 2n) . ( β 2¯h a · b 2 a− b h ib,B − ia,B − ) | | Xb>anX=0 h (cid:16) (cid:17)i (3.4) We have used the definition of the classical Euler dilogarithm ∞ vs v ln(1 y) Li (v)= = − dy. (3.5) 2 s2 − y s=1 Z0 X Many of the interesting properties of this function were studied by one William Spence in the early nineteenth century [20]. A more recent discussion in the math- ematical literature can be found in ref. [21]. Recent developments relating to conformal field theory are reviewed in ref. [22]. The matrix (3.4) satisfies various conditions, which are essentially related to properties of the time-delay functions X . This function was studied recently in ref. [23], where it was noted that it ij 5 has properties reminiscent of those of S matrices. Indeed, one can think of these properties as being consequences of the fact that the S matrix (3.4) is periodic, symmetric, etc. Let us define h−1 ıπ c(a) c(b) T (ρ)= wn(γ ) (γ )Li exp ρ+ − 2n . (3.6) ab a b 2 · h 2 − n=0 (cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21) X Then the matrix function T (ρ) satisfies the following relations: ab (i) T (ρ+2ıπ)=T (ρ) ab ab (ii) T (ρ)=T (ρ) ab ba (iii) T (ρ+iπ)= T (ρ) ab a¯b − ∗ ∗ (iv) (T (ρ )) =T (ρ) ab ab (v) T (ρ)= T ( ρ)+2T (0) ab ab ab − − (vi) T (ρ+ıη )=0, where l is a free label and i,j,k satisfy a fusing rule; t=i,j,k lt t Pη = 2ξ + c(t)−1, where c(t) is 1( 1) if the root α is black(white), and ξ t − t 2 − t t are the integers in the fusing relation t=i,j,kω−ξtγt =0. P These relations can be shown directly: Property (i) is obvious; properties (ii), (iii) and(iv)followusingargumentsanalogoustothoseusedforthetime-delayfunctions X (ρ) in ref. [23]. Property (vi) similarly follows from an argument analogous to ab that given in ref. [7] when discussing the affine Toda particle S-matrix. Property (v) is most easily proved by noting firstly that the ρ derivative of this equation is true - this follows using eqn. (3.5) and the fact that X (ρ) = X ( ρ) (see ref. ab ab − [23]). Thus the left-hand side of (v) equals the right-hand side up to an additive constant,andputting ρ=0oneseesthatthisconstantmustvanish. Adirectproof of (v) seems more involved - for example, for the A case this property reduces to 1 the equation Li ( eρ) Li (eρ)= Li ( e−ρ)+Li (e−ρ)+2Li ( 1) 2Li (1). (3.7) 2 2 2 2 2 2 − − − − − − The Euler dilogarithm satisfies the following ‘inversion’ relation [22] π2 1 Li ( y)+Li ( 1/y)= (logy)2. (3.8) 2 2 − − − 6 − 2 The function Li (y) is divergent for real y, y > 1; however, one can define a con- 2 tinuous function on the real line by setting [22] π2 1 Li (y)= Li (1/y) (logy)2, for y >1. (3.9) 2 2 3 − − 2 6 Then,usingtheabovetwoequationsandthefactsthatLi (1)=π2/6andLi ( 1)= 2 2 − π2/12, equation (3.7) may be proved. − The constant term in the relation (v) means that the S matrix (3.4) will be invariant under ρ ρ only if it is normalised so that S(ρ = 0) = 1. Physically → − this requirement is obvious - that there be no scattering when the two solitons have the same rapidity. We note that the properties (i)-(vi) above reflect identities satisfied by the Euler dilogarithm. Thus the expression (3.4) satisfies relations expected for an S-matrix. Note, however,thatthe Eulerdilogarithmcanbecontinuedtoamulti-valuedfunctionon the complex plane, minus the segment (1, ) of the real axis [21, 22]. Hence our ∞ proposed S-matrix does not have the expected pole structure. We will comment upon this in the following section. 4 Conclusions and Developments ItisimportanttobeclearaboutwhatthisS-matrixis,andwhatitisnot. Wehave only made a semi-classical approximation to the quantum theory of affine Toda solitons,and so do not expect to see all of the behaviour typicalof a quantum field theory. In particular as noted above, there are no poles in formula (3.4), and we have made no mention of the renormalisationof the soliton masses. The interested reader should consult the recent papers [24] and [25] for a discussion of this latter point. Very little is known about the expected behaviour of the poles on account of difficulties concerning the unitarity of the affine Toda theories in the imaginary coupling r´egime. Another feature expected of the quantum theory which this kind ofapproximationwillnotpossessisthattherewillbenoprocesseswhichchangethe topologicalchargesofthe scattering solitons. This is because suchthese are absent classically. An obvious conjecture worthy of exploration is that a full quantum version of (3.4) involves the quantum dilogarithm of ref. [26]. What we believe we have is the S-matrix of an quantum integrable particle theory, generalising that of Ruijsenaars [27, 28]. In the case of Aˆ-series solitons of equal mass this particle model already reproduces the appropriate scattering shifts. In the more general case however, no such model is known. Starting from the formula for the shifts which follows from Ruijsenaars model, and fixing the rapiditiesinsuchawaythatthesolitonfusingrule[17]issatisfied,wehaveobtained correctexpressions for the shifts when two solitons of different masses scatter (it is reasonablyeasytoshowthatalloftheAˆ-seriessolitonscanbeobtainedbyrepeated fusing from the lightest in this manner). The question is whether this procedure can be implemented at the level of the Ruijsenaars Hamiltonian. This is a non- trivial problem since the fusings correspond to imposing imaginary constraints on 7 the particle rapidities, andit is unclear what effect this will haveon the symplectic structure of the phase space. We remark that the change ρ ρ +2inπ/h is a → symplectictransformation,albeitacomplexone,whichgivesussomehopethatthe above procedure can be made to work. A natural generalisation of this work would seem to be to try and include the breathers. In the sine-Gordon theory at least these breathers are related to the particles, and possess a number of discrete energy levels according to the value of the couplingconstantβ. Itisa long-standingprobleminaffineToda solitontheory to try and characterise the breather spectrum for more general theories, and in particular to see if the particle-soliton correspondence remains. 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