QUANTUM RELATIVISTIC TODA CHAIN

Investigated is the quantum relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl’s q we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six-vertex R-matrix. Both independent Baxter’s Qoperators are constructed explicitly as seria over local Weyl generators. The operator-valued Wronskian of Qs is also calculated.


Introduction
Long ago, Baxter in his famous papers [1,2,3,4,5] has introduced the object which is known now as Q-operator.This operator does satisfy the so-called Baxter (or T − Q) equation and besides has many interesting properties.Recently Q-operator was intensively discussed in the series of papers [7,8] in connection with continuous quantum field theory.In [14,21] it was pointed out the relation of Q-operator with quantum Ba ¨klund transformations.In [17], was discovered the relation of Q-operator with Block solutions of quantum linear problem.
Q-operator was used initially for the solution of the eigenvalue problem of XYZ-spin chain, where usual Bethe ansatz fails.The reason is that T − Q equation, together with an appropriate boundary conditions, provides a one-dimensional multiparameter spectral problem which allows one to determine the spectra of both the auxiliary transfer matrix T and the operator Q.In the case of the quantum mechanical integrable chains (e.g., the periodic Toda chain) the appropriate solution of the Baxter equation plays the prominent role in the functional Bethe ansatz and the quantum separation of variables.
Thus the reader should distinguish two approaches to Q-operator: the first one is Baxter's functional equation as the spectral problem, while the second one is the investigation of Q, obeying Baxter's equation, as the operator defined in terms of quantum observables.This paper is devoted to the investigation of Q as an operator.
In quite recent papers explicit constructions of Q-operators were obtained for several models, like the isotropic Heisenberg spin chain, [18], and the periodic Toda chain and other models with the rational R-matrix, [17].In these papers Q-operator was obtained as the trace of monodromies of the appropriate local operators.It is well known that with free boundary conditions for Q, T − Q equation provides a one-parametric family of solutions, so that one may extract two independent solutions with nonzero discrete Wronskian (see [7,8,19]).In [17,18] both independent Q operators were obtained for the models considered.
In this paper, we investigate the exactly integrable model known as "quantum relativistic Toda chain," [13,15,20].Local L operator for the model is constructed with the help of the Weyl algebra generators, commuting on q, and we deal with the case |q| < 1.Here, we do not consider the Jacoby partners to the Weyl algebra, dealing thus with the compact q-dilogarithms (investigation of the modular formulation of the quantum relativistic Toda chain is the subject of the forthcoming paper, [12]).Quantum space of our model is a formal module of an enveloping of the tensor product of several copies of Weyl algebras.The only thing we suggest for the Weyl generators is their invertibility and a q-equidistant spectrum for one of them.Both independent operators Q + and Q − and their Wronskian are calculated locally as the operators acting in the ultra-local Weyl algebra.Actually all our results are to be understood as the well-defined series expansions for functions from the enveloping mentioned.

The model and the results
This section consists of two parts.We formulate the model first, actually just defining the transfer matrix, and then we give the final formulae for Q ± operators and their q-Wronskian.All the sections beyond the introduction are the QUISM-type derivation of these results.

Problem
First of all, we define the relativistic Toda chain L-operator, associated with fth site of a chain, as where x ∈ C is the spectral parameter, λ ∈ C is an extra parameter, common for all sites (i.e., λ is a module), the set of elements u f , v f form the "halfinteger" ultra-local Weyl algebra: and elements with different fs commute.As usual, the whole quantum space is the tensor product of some copies of Weyl modules, and f marks the "number" of given Weyl algebra in this tensor product.Recall, we will always imply |q| < 1.
The correspondence between the relativistic Toda chain and usual Toda chain may be established, for example, in the following parameterization: where in the limit lim The right-hand side of this relation is known as the L-operator for the quantum Toda chain.
In the L-operator as well as in all other objects the spectral parameter x will always couple with u f .So we introduce the useful notation so that for any formal function g(s f ), We define the transfer matrix for the chain with F sites, f = 1,...,F, as (2.8) The matrix T (x 2 ) becomes a polynomial of x 2 with operator-valued coefficients: (2.9) Here it is implied that t F = 1 and (2.10) Later we will argue that {t j } is the commutative set.Equivalently, this means that T x 2 T y 2 = T y 2 T x 2 ∀x, y. (2.11) Note that apart from the trivial t F = 1 all other F coefficients are independent.For a given set {t j } one can define another set {t j } by (2.12) This means simply that (2.13) Baxter's operator Q(x 2 ), by the definition, is an operator commuting with the set of t j , and obeying the Baxter where t 0 is given by (2.10).The model-dependent coefficients of Q(qx 2 ) and Q(q −1 x 2 ) in the right-hand side of (2.15) are the subject of separate calculations, and this particular form will be argued later in this paper.
In what follows, we will see that with this normalization of the coefficients in (2.15) the Baxter equation has a solution entire on x 2 .We will call this solution (2.16) Proposition 2.1.The entire on x 2 solution of (2.15) as a series on λ F is where c −1 ≡ 0, c 0 ≡ 1, and recursively (2.18) Note that J(x 2 ,λ,{t}) is the entire function on all its arguments.The proof of this proposition is a rather simple exercise.
The other solution Q − (x 2 ) must contain a cut with respect to the variable x, and up to this cut we guess Q − (x 2 ) to be entire on x −2 .More exactly, with the s f -notation introduced by (2.6), one may check that also solves (2.15).Obviously, Q + and Q − may be considered as two independent solutions of (2.15).The last definition we need is the q-Wronskian of these two solutions (2.20)

Solution
In this paper, we give explicit expressions for both functions Q ± .The natural question arises: we have got yet the form (2.17) and (2.18), what one may otherwise do.The aim of this paper is to investigate the relativistic Toda chain by QUISM method, to construct local operators M f (x 2 ) such that a trace of their monodromy gives Q ± (x 2 ), to prove the commutativity of the transfer matrices and Q ± and to calculate the Wronskian.Note that in QUISM approach we construct Q ± (x 2 ) not as functions of {t}, but as functions of local u f , v f .This is in some sense a factorization, the simplest analogue of this is the well-known q-exponential formula where conventionally and as the series expansions x n (q; q) n . (2.23) The right-hand side of (2.21) we call the local form of its "global" left-hand side.Now we describe the local form of all solutions.First of all, we introduce a universal function where α and β are complex numbers, and elements q n and q m commute.
Proposition 2.2.The operator Q + (x 2 ), defined by (2.16), (2.17), and (2.18), in the local form is and the operator Q − (x 2 ), defined by (2.17), (2.18), and (2.19), in the local form is Their Wronskian, defined by (2.20), is (2.27) Note that the operators Q + , Q − , and W ((2.25), (2.26), and (2.27)) are the series with respect to u f up to the simple common multiplier λ − f s f .Thus (2.6) is indeed just the useful notation, and actually we do not need the notion of the logarithm of the operator u f .

Integrability
First of all, the integrability of the relativistic Toda chain follows from the commutativity of the transfer matrices (2.8).The origin of it is the famous six-vertex R-matrix.The following relation holds: where , and so forth, the cross product implies the tensor product of the 2×2 matrices, and the six-vertex R-matrix has the form The Yang-Baxter relation (3.1) provides the commutativity of the traces of the monodromies for L 1,f (x) and L 2,y (x), and as the extra multiplier in our definition of the transfer matrix (2.8) is the total shift operator, our modified transfer matrices (2.8) also form the commutative family, (2.11).

Origin of Baxter's equation
The appearance of the six-vertex R-matrix is the criterion of the existence of Baxter's "TQ = Q + Q " relation for our transfer matrix.Here we give a brief description of the method of obtaining Baxter's equation.So, for a given quantum Lax operator, obeying (3.1) with six-vertex R- matrix, let there exists another auxiliary relation where " * " means the 2 × 2 matrix multiplication, 2 × 2 matrix Lh (z) with entries acting in some space h is an auxiliary L-operator, and a scalar with respect to the matrix structure of L f and Lh operator M h,f acts in the tensor product of the spaces f and h.In the case of the usual quantum Toda chain, L is a Sklyanin's Dimer self-trapping L-operator.As usual, the monodromies obey the same relation (3.3), this provides the commutativity of Suppose next that the matrix Lh (z) has the degeneration point, without loss of generality, where ψ h and ψ h are column and row two components vectors with operatorvalued entries.These vectors are known as Baxter's vacuum vectors [6].Analogously, the inverse matrix L−1 (z), being normalized appropriately, has the orthogonal decomposition, Obviously, It is easy to obtain, the following elements are two orthogonal projectors, and because L-operators have the matrix dimension 2, P + P = 1.
Consider (3.3) and its inverse partner in the degeneration point.The projector structure of Lh (1) ±1 means that there exist two scalar (with respect to the 2 × 2 structure of L-operators and ψ and φ-vectors) operators M and M , defined by the following relations: (3.11) 3) for the monodromies may be written as (3.12) Due to (3.11), in the right-hand side there appear the traces of M h,f and M h,f , so if then the fusion relation appears In most particular cases, when h is an infinite-dimensional space, in (3.14) it appears that Q (x 2 ) ∼ Q(qx 2 ) and Q (x 2 ) ∼ Q(q −1 x 2 ).All these are the subject of detailed investigation.

Triangle relations for the quantum relativistic Toda chain
Now we try to guess a form of (3.7) and (3.8) and try to solve (3.11).Taking (3.9) into account, let where a and a + are two essential elements for h-algebra.Writing this decomposition, we do not impose no conditions on a, a + and the unknown factors k 1 and k 2 .The experience of usual Toda chain [17] says that when q → 1, a and a + become usual bosonic annihilation and creation operators, this inspires our notations.In the next derivations we will suggest the invertibility of a and a + , so that the decompositions (4.1) are written without loss of generality.To get something applicable, we introduce an element N, such that a sort of q-oscillator relations hold: first, for any function g let and second let there exists a function [N], The element N is introduced, without loss of generality, as a pair to a. Operators L must form an integrable chain, this provides relations (4.3), and therefore k 1 = k 1 (N) and k 2 = k 2 (N) in (4.1).Thus, the parameterization (4.1) is the general one.The degenerate matrices L± (1) become the orthogonal projectors if one substitutes ).Now we write the explicit form of the triangle relations The pair of equations (4.4) is exactly (3.11), while the pair (4.5) is the dual one.Traces of the monodromies of M and M and of M and M must coincide, they should lead to the same equation (3.14) where extra multiplier appears due to our normalization of the transfer matrix (see (2.8)).But locally M # and M # may be slightly different.
The spectral parameter x in the L-operator (2.1) always stays in the combination x u, therefore the shift of the spectral parameter thus may appear as Due to this property we can put x = 1 for the shortness and omit the spectral parameter in our formulae, the x may be restored subsequently in all equations by the shift u f → xu f .The triangle equations (4.4) and (4.5) are equivalent to two systems up to some operator-valued multipliers.These multipliers are to be integrals of motion in a form of pure product over f.There is only one such integral of motion, it is t 0 , and hence there must exist a monomial function φ(u) ∼ u c such that and therefore (4.10)The same φ(u) is used for M and M because Q must be the same.It is important that in (4.9) the multiplier φ(u) stands from the right of M for M and from the left of M for M .The order of multipliers is governed by Yang-Baxter equation (3.3).
In general, one may put φ(u) to the other sides, this would give another system for M with another solution.We will not investigate such case separately, because there exists an involutive automorphism τ, defined as such that the L-operator is invariant with respect to τ-involution L(1) τ = −σ 3 L(1)σ 3 .(4.12) Also it is important that τ does not change q.Therefore T τ (x 2 ) = (−) F T (x 2 ), and the other case of positions of φ just corresponds to the consideration of M τ .With expressions (4.9) for M and M , system (4.8) is equivalent to −q −1/2 λMa + = vφ(u)M, (4.13b) It is useful to complement system (4.13) by equations with k 1 , k 2 following from (3.3): ) This is the final set of equations that we are going to solve.We will give its solution in two forms.The first one is a formal series solution that admits an interpretation of a and a + as q-oscillator (spectrum of N is the nonnegative integers, and there exists the vacuum vector for a).Another form implies the Weyl algebra parameterization of a, a + , when the permutation between h and f spaces plays the significant role.Actually these two forms differ by the notion of the trace in h-space, and q-oscillator trace will give Q − while the Weyl trace will give Q + .

Series solution
First, we test system (4.13) for the formal operator arguments of M. Just considering the expressions MxM −1 x −1 for several x, one may conclude this trivializes two equivalent relations (4.13e) and (4.13f) of system (4.13).
For further analysis of (4.13) we start from permutation-like (4.13a).Relations like are to be solved as and in the case of (4.13a) this gives Note that G does not depend on av, because any such dependence may be extracted to a n .Now all other relations from (4.13) must give recursion relations for G. Equation (4.13c) is equivalent to Equation (4.13b) gives another permutation-like structure, but with the formal correspondence

.20)
Due to (4.20), M may be rewritten in the form of the other permutation-like structure Moreover, this allows one to write M without negative powers of a or a Apparently, this form is good for q-oscillator representation.Equation ( 4 Here we used φ(u) ∼ u c .Equation (4.23) has only two solutions for φ(u) and [N], corresponding to |q| < 1 and |q| > 1.In our case where α is a complex parameter, [N] is normalized so as [0] = 0.With these φ(u) and [N], equations (4.19) and (4.20) may be solved easily, where u 2 ≡ q s (see (2.6)), and g α,β (n, m) is defined by (2.24).Parameter α is an avoidable scale of u and it is convenient to put it to unity, α ≡ 1.
Note that expressions like (x; q) ∞ in g-function appear as the appropriate solutions of difference relations and the separation between |q| < 1 and |q| > 1 is originated from the unavoidable sign of the quadratic exponent q ±N(N−s) .The other solution of zero curvature condition is With φ(u) defined, the final expressions for M are, in the short form, and in q-oscillator's form where, recall (2.6), u 2 = q s .Substituting φ(u) = −q −1/2 u −1 into the expressions for M and M , (4.9), and using our definition of the transfer matrix (2.8), we obtain the Baxter equation exactly in the form (2.15).
Existence of the form (4.30) allows one to interpret a, a + exactly as q-oscillator generators, such that the spectrum of N is 0,1,2,... (we have normalized [N] so that [0] = 0), and the state |N = 0 is the vacuum, a |N = 0 = 0. Thus one may define the q-oscillator trace of any operator F = F(a, a + , N).If then taking such trace, one has to take a 0 and (a + ) 0 th components and then take the sum over N = 0,1,2,...
In general, one may obtain Q + at once, considering the τ-involution applied to M and to But there are two objections to consider this case: first, τ-involution changes a little the Baxter equation, and second, M τ is the degenerate operator, and hence we will look for another way to obtain Q + operator.

Extraction of a permutation
Solving (4.13), we mentioned the permutation-like relations.In this section, we suppose that the quantum space f and the auxiliary one h are isomorphic.
Our aim is to extract the permutation operator, giving (4.13a) "by hands."As previously, we deal with the case |q| < 1, x = 1, u 2 = q s , and so that we are looking for another realization of the same operator M. We will search for M in the form where Here the first relation is exactly (4.13a), the second one is the consequence of (4.15), and the last one is the definition of the permutation.System (4.13) for operator M can be rewritten as follows: Solution of it is given by M = − λa −1 v −1 u; q ∞ qu 2 ; q ∞ .(4.39) Operator (4.36) with the definitions (4.37) and (4.39) does solve the system of the relations (4.13).Using the series decomposition for the compact quantum dilogarithms, one may obtain the series representation (there is used (v −1 u) n = q n(n+1)/2 u 2n (uv) −n ) for M, (4.36), M = n≥0 q n 2 (q; q) n λ n u 2n q 1+n u 2 ; q ∞ (uv) −n P h,f (uv) n .(4.40) Note that the function g, (2.24), appears in this decomposition

.41)
In this form all the h-space operators a, a + , and N are hidden into the permutation symbol.The permutation operator allows one to calculate the trace in the auxiliary space h in the invariant way via tr inv P h,1 P h,2 where P is the cyclic shift operator for the chain f = 1,2,...,F, F + 1 ∼ 1 The shift is one of the integrals of motion.Now using (4.41) and the definition of the shift operator, one obtains exactly Q + , (2.25), for the trace of M-monodromy up to the shift Now both forms of M-operators have been obtained, (4.22) and (4.41), actually coincide.To show this, we represent P h,f in the following form: where δ(N − s = n) is the projector of N − s into a state with the eigenvalue n With this form of P h,f (4.41) could be written as follows: Now one may take the sum over n using the projectors as the delta symbols, and exactly (4.29) appears

.48)
Such exercises with the projector decomposition of operators are rather formal.One may consider projectors and spectral decompositions of many types, imposing some extra conditions for the spectra of the operators involved.What is actually the difference between both Q operators: the difference is the conjecture about the spectrum of N. Due to the Weyl algebra relations, the spectrum of N must be equidistant, (4.49) In the case when ζ = 0 we get q-oscillator representation.In the case when ζ is the same as for s ∈ Z + ζ, we get the isomorphism between h and f spaces and the permutation extracted representation.In general one may generalize both (5.16) These sums may be calculated with the help of the Rogers-Ramanujan summation formula.Auxiliary relations for this calculations are and the Rogers-Ramanujan celebrated identity is where the series for 1 Ψ 1 is convergent in The results of summations are (5.20) Put now y 2 = q −1 x 2 , then it appeared where y 2 = q −1 x 2 , and

.22)
Here it is used the θ-function notation Θ(x) = (x; q) ∞ qx −1 ; q ∞ (q; q) ∞ = n∈Z (−x) n q n(n−1)/2 , (5.23) such as (5.24) Indeed, due to the equidistant of N 1 and N 2 , F 1,2 depends only on N 1 +N 2 .Now we may calculate the Wronskian.By definition, it is (5.25) Considering the monodromies of Q 1 and Q 2 , standing in the definition of the Wronskian and using (5.7), one may see that the most parts in the subtraction, (5.25), are canceled.Only possible exception is the subspace δ W : a + 1 a 2 = −1.So to calculate the Wronskian, one has to take a trace only over this subspace.In general, let ζ 1 and ζ 2 be the characteristics of N 1 and N 2 , respectively.Then using the definition of F and F, (5.22) equivalence of F and F, one may conclude where Ξ 1,2 is an extra multiplier that may come from the subtraction, δ Wtrace definition and so on.Nevertheless, considering the case s x = ζ 1 modulo Z and ζ 2 = 0, one obtains the following useful form of F 1,2 : F 1,2 (n + s−, s) = (−) n q n(n−1)/2 λ n (q; q) n λ −s Θ q s (q; q) ∞ . (5.27) Extracting now the shift operator as it is described in (4.53), one obtains (2.27).Extra multiplier is equal to unity, this we have checked by a series expansion with respect to λ.

Discussion
The technique and results, given in this paper, are rather formal.We have dealt with the single Weyl pair in each site of the lattice, and q is an arbitrary complex number inside the unit circle.It is well known, this regime is absolutely nonphysical, and thus the results presented are to be considered as just an exercise in the field of q-combinatorial analysis.But, nevertheless, some applications of the results and technique presented may be found.
Talking about the Weyl algebra, people usually keep in mind two aspects: the first one is the Faddeev dualization, when q = ex{iπe iθ } with real θ is the universal unitary regime [10,11,9], and the second one is the finite state q = e 2πi/N .This paper suggests the third aspect, applied in the backward direction yet: several Toda-chain-type models, physical as well, may be obtained from a model with arbitrary q in the limit q → 1 + h, regarded in a special way, such that a rational Weyl algebra mapping is linearized with respect to one of Weyl generators in the first order of h.Our experience in the Weyl algebra exercises says that most our results, especially containing the q-dilogarithms and permutations, may be immediately rewritten in the dualized form.In this way the results may be applied to the physical relativistic Toda chain, [12].It will be done in a separate paper.
The second aspect is also valid, especially in the part of the technique derived.Preliminary considerations show that at the root of unity the model contains the Baxter curve for the Chiral Potts model, the point on Baxter's curve is the spectral parameter of Q-operator, our constant parameter λ is connected with the modulus of Baxter's curve.Remarkable is that in the relativistic Toda chain at the root of unity there appears only one point at Baxter's curve, while in the Chiral Potts model such point lives at each site on the spin chain.This fact makes the relativistic Toda chain much more simple than CPM itself.The investigation of such type models is started in [16]. ) .13d) coincides with (4.19) if one uses the series (4.21).But it is important to note that in general (4.19) and (4.20) are not compatible.Their compatibility condition is the following functional relation for φ(u) and [N]: