Unitarity of the SoV Transform for SL(2 , C ) Spin Chains

. We prove the unitarity of the separation of variables transform for SL(2


Introduction
Theory of quantum integrable models is an important part of modern theoretical physics.The solution of such models relies on the Quantum Inverse Scattering Method (QISM) which includes such techniques as the Algebraic Bethe Ansatz (ABA) [1] and Separation of Variables (SoV) [2,3].The ABA allows one to effectively calculate energies and eigenstates of integrable models and to address more complicated problems such as calculating norms [4], scalar products [5] and correlation functions [6,7].Models with infinite dimensional Hilbert spaces, the Toda chain [8] being the most famous example, are, however, beyond ABA's grasp.The solution of such models relies on the SoV method proposed by Sklyanin [2,3].The method consists in constructing a map between the original Hilbert space, H org , in which the model is formulated, and an auxiliary Hilbert space, H SoV .This map is constructed in such a way that a multidimensional spectral problem associated with the original Hamiltonian is reduced to a onedimensional problem on an auxiliary Hilbert space which usually takes the form of the Baxter T − Q relation.Technically constructing the SoV representation is equivalent to finding the eigenfunctions of an element of the monodromy matrix associated with the model.For the Toda chain it was done by Kharchev and Lebedev [9,10].Later, a regular method for obtaining eigenfunctions for models with an R-matrix of the rank one * was developed in [16], and at present the SoV representation is known for a number of models [17][18][19][20][21].
In order to be sure that the spectral problems in the original and auxiliary Hilbert spaces are equivalent it is necessary to show that the corresponding map, H SoV → H org , is unitary (or that the eigenfunctions form a complete set in H org ).If dim H org < ∞ the problem can be solved, at least in principle, by counting the dimensions of the Hilbert spaces.For the models with infinite dimensional Hilbert space, such as the Toda chain, the noncompact SL(2, C) spin chain, etc., the task becomes more difficult.For the Toda chain unitarity was first established by using harmonic analysis of Lie groups techniques [22,23].However this method is quite sophisticated and can hardly be generalized to more complicated cases.The rigorous proof of the unitarity of the SoV transform for the Toda chain based on the use of natural objects for the QISM was given by Kozlowski [24].This technique was later applied to the modular XXZ magnet [25].Later it was realized [26] that there exists a close relation between SL(2, R) symmetric spin chains and the multidimensional Mellin-Barnes integrals studied by Gustafson [27,28] that allowed to greatly simplify the proof of the unitarity of the SoV transform for SL(2, R) symmetric spin chains [29].
In the present paper we apply this technique to the analysis of the noncompact spin chains with the SL(2, C) symmetry group.Such models appear in the studies of the Regge limit of scattering amplitudes in gauge theories, in QCD in particular [30][31][32][33][34], see also [35][36][37] for recent developments.The SoV representation for the SL(2, C) spin chains was constructed in [16].The generalization of Gustafson integrals relevant for the SL(2, C) spin chains was obtained recently in [38].Based on these results, we present below a proof of unitarity of the SoV transform for a generic SL(2, C) spin chain.
The paper is organized as follows: in sect. 2 we recall elements of the QISM relevant for further analysis.The eigenfunctions of the elements of the monodromy matrix are constructed in sect.3.In sect. 4 we calculate several scalar products of the eigenfunctions and discuss their properties.Sect. 5 contains the proof of unitarity of the SoV transform.Sect.6 is reserved for a summary and several appendices contain a discussion of technical details.

SL(2, C) spin chains
Spin chains are quantum mechanical systems whose dynamical variables are spin generators.We consider models with spin generators belonging to the unitary continuous principle series representation, T (s k ,s k ) , of the unimodular group of complex two by two matrices.Namely, each site of the chain is equipped with two sets of generators, holomorphic (S α ) and anti-holomorphic ones ( Sα ), The generators S α k ( Sα k ) satisfy the standard sl(2) commutation relations, while the generators at different sites and holomorphic and anti-holomorphic generators commute, [S α k , Sα k ] = 0.The parameters s k , sk specifying the representation take the form [39] where n k is an integer or half-integer number and ρ k is real, so that 3) The later condition comes from the requirement for the finite group transformations to be well defined while the former one guarantees the unitary character of transformations and anti-hermiticity of the generators, (S α k ) † = − Sα k .The Hilbert space of the model is given by the direct product of the Hilbert spaces at each node.For a chain of length N , In the QISM [1,3,40,41] the dynamics of the model is determined by a family of mutually commuting operators.Namely, one defines the so-called L-operators, which are the basic building blocks in the QISM.The complex variables u, ū are called spectral parameters.The next important object -a monodromy matrix -is given by the product of L operators where ξ k , ξk are the so-called impurity parameters † .The entries of the monodromy matrix, are polynomials in u with the operator valued coefficients, e.g.
where Ξ = N k=1 ξ k and S 0 , S − are the total generators: The entries of the monodromy matrix form commuting operator families [1,42] [ (2.9) In particular, each entry commutes with the corresponding total generator, S α , (2.10) The same equations hold for the anti-holomorphic operators ĀN , BN , CN , DN and, of course, the holomorphic and anti-holomorphic operators commute.Moreover it can be checked that if the impurity parameters satisfy the constraint ξk = ξ * k for all k, the following relations between holomorphic and anti-holomorphic operators hold etc.This ensures that the operators a k and āk in the expansion of A N (u), Eq. (2.7), and ĀN (u), are adjoint to each other a † k = āk (b † k = bk etc.)The commutativity of the operators A N (u), B N (u), C N (u), D N (u) implies that the following families of self-adjoint operators (and similarly for others) are commutative and can be diagonalized simultaneously ‡ .The corresponding eigenfunctions provide a convenient basis -Sklyanin's representation of Separated Variables (SoV) -for the analysis of spin chain models [3].
The operators B N and C N , (A N and D N ) are related to each other by the inversion transformation, see ref. [21] for detail, so it is sufficient to construct eigenfunctions for the operators B N and A N .The eigenfunctions of B N for the homogeneous chain were constructed in ref. [16] and later on for the operator A N , [21].Extending this approach to the inhomogeneous case is rather straightforward.

Eigenfunctions
In this section we present explicit expressions for the eigenfunctions of the operators B N and A N for a generic inhomogeneous spin chain with impurities.We start with the operator B N where the construction follows the lines of ref. [16] with minimal modifications.

B N operator
Let Λ n be an integral (layer) operator which maps functions of n − 1 variables into functions of n variables and depends on the spectral parameters x, x and the complex vectors γ, γ of dimension 2n − 2 (3.1) ‡ The impurity parameters must also satisfy the condition i(ξ k − ξk ) = r k , where r k are (half)integers.
The kernel is given by the following expression where the function D α (z) (propagator) is defined as follows We will assume that the indices α, ᾱ satisfy the condition [α] ≡ α − ᾱ ∈ Z so that the propagator is a single-valued function on the complex plane.It implies that the parameters γ k and x have the form The numbers {m, r 1 , . . ., r 2N −2 } are either integer or half-integer and depending on this we call the corresponding variables integer (half-integer).The continuous parameters σ k and ν are subject to the constraints which guarantee the convergence of the integral (3.1) for a smooth function f with finite support.In the case we are most interested in, γ k + γk = 1, the parameters σ k ∈ R, and the variable ν lies in the strip The operators Λ n possess two important properties: (i) Let ρ be a map which takes M -dimensional vectors to vectors of dimension M − 2 as follows where a ≡ 1 − a.It can be shown that the operators Λ n and Λ n−1 obey the following exchange relation Here γ(γ) is 2n − 2 dimensional vector and the factor ω n is given by the following expression where and Γ is the Gamma function of the complex field C [43] The relation (3.8) is a direct consequence of the exchange relation for the propagators, see (A.4).Its proof is exactly the same as for the homogeneous spin chain.For more details see refs.[16,21].
(ii) Let us choose the vector γ as follows where s k and ξ k are the spins and impurity parameters of the spin chain, respectively.For such a choice of the vector γ the operator B N (x) annihilates Λ N (x|γ) [16,44] Let us define a function where the kernel U x 1 ,...,x N −1 is given by the product of the layer operators, and γ is given by Eq. (3.12).Equation (3.8) guarantees that U x 1 ,...,x N −1 ∼ U x i 1 ,...,x i N −1 for any permutation of x 1 , . . ., x N −1 .The kernel U x becomes totally symmetric for the following choice of the prefactor (x|γ): where Thus the function Ψ (N ) p,x 1 ,...,x N −1 is a symmetric function of the variables x 1 , . . ., x N −1 .Together with Eq. (3.13) it implies that For N = 1 the functions Ψ p (z, z) = π −1/2 e i(pz+pz) form the complete orthonormal system in H 1 = L 2 (C).The aim of this paper is to extend this statement to N > 1. Namely, we will show in sect.5 that if the spins and impurities parameters of the spin chain obey the "unitarity" condition, for all k ( γ k has the form (3.4) with σ k ∈ R ) then the set of functions {Ψ Note that the functions Ψ (N ) p,x are well defined for the complex parameters ν k in the vicinity of the real line.For further analysis, it will be useful to consider regularized functions, Ψ (N ), p,x , by relaxing the last of the conditions (3.22) This can be achieved by shifting the impurity parameter ξ N → ξ N − i ¶ , i.e.

A N operator
Construction of the eigenfunctions of the operator A N follows the scheme described in the previous subsection.We define a layer operator Λ n which maps functions of n − 1 variables into functions of n variables where the kernel is given by the following expression The layer operator Λ n depends on the spectral parameters x(x) and the vector γ(γ) of dimension 2n − 1 which have the form Eq. (3.4).These operators satisfy the exchange relation and the factor ω n is defined in Eq. (3.9).Let Φ (N ) x (z) be the following function where γ is 2N − 1 dimensional vector and the prefactor is given by Eq. (3.16).For such a choice of the function Φ  for the following choice of the vector Taking into account polynomiality of A N (u), see Eq. (2.7), one obtains x (z). (3.30) Again, the variables x k , xk are integers (half-integers) for all k.We will show that these functions, {Φ x (z), x k = x * k , k = 1, . . ., N }, form a complete set in the Hilbert space H N .
The functions constructed in the previous section are given by multi-dimensional integrals.In this section we show that these integrals converge for the parameters ν k in the vicinity of real axis.To this end it will be quite helpful, as was advocated in ref. [16], to visualize the integrals as Feynman diagrams.
The examples for N = 3 are shown in Fig. 1.It will be convenient to convert diagrams (functions) to momentum space In momentum space the function Ψ Let us remark here that the " " regularization is reduced to a multiplication by the factor (p N pN ) 3) The function Ψ (N ), x can be read from the Feynman diagram in Fig. 1 as follows with the integrand J x ({p k }, { ij }) given by the product of the propagators, D α (k).Up to a momentum independent factor where k0 ≡ 0, k−1,k ≡ p and N −1,j = (p 1 + . . .+ p j ).The indices α kj , β kj take the following values where we introduced the notations: In many cases, Feynman diagrams can be evaluated diagrammatically.In particular, the computation of diagrams for the scalar product of Ψ (Φ) functions is based on the successive application of the exchange relation (A.4) to the diagram.Let us consider the scalar product of two functions Ψ (N ), p,x and Ψ (N ), q,y Ψ (N ), q,y , Ψ (N ), p,x = πδ 2 (p − q)(pp) + I , (x, y), where The function I , p (x, y) is given by the Feynman diagram shown in Fig. 2 (left panel), which is a multidimensional integral with the integrand given by the product of the propagators.The diagram can be evaluated in a closed form by successively applying the exchange relation (A.4), that is equivalent to calculating the loop integrals in a certain order.The answer takes the form where 2N −4 − ix, γ 2N −5 − ix, . . ., γ For the sign factor C N (γ) we get : Let us show now that integrations in (4.10) can be done in an arbitrary order.The integrand in (4.10),I x,y (p, { pr }|γ), is given by the product of the propagators D α (k), with each index being of the form α = 1 2 + n 2 +iσ, momentum k being a linear combination of loop momenta, ij , and the external momentum p.Since then for the parameters γ satisfying the unitarity condition (3.22), and x k , y k having the form one obtains for the modulus of the integrand where the underlined variables are: γ = ( 1 /2, . . ., 1 /2), Thus the integral of |I x,y (p, { pr }|γ)| is a particular case of the integral (4.10) which was calculated by performing loop integrations in a certain order.Since all integrals converge under the conditions by Fubini theorem, the integral (4.10) exists and the integrations can be done in an arbitrary order.
The following statements can immediately be deduced from this result: • For any bounded function ϕ(p, x) with a finite support the function where belongs to the Hilbert space H N , Ψ ϕ 2 < ∞, for sufficiently small .
• It follows from the finiteness of the integral I , p (x, y), Eq. (4.9), that the function Ψ (N ), x ( p), Eq. (4.4), exists almost for all p for the separated variables x k close to the real axis: where x ± are defined as follows: The integrals of the functions on the rhs of (4.19) are finite for sufficiently small δ.It follows then from the Lebesgue theorem that the function Ψ The scalar product of the functions Ψ p,y and Φ (N ) x constructed in sect.3.2 can be calculated in a similar way.Note that there is no need to introduce " " regulator here.The corresponding integral is absolutely convergent when Im(ν k + µ j ) > 0 for all k, j, ( x k , y j given by (4.15) ).The scalar product takes the form where Similar to the previous case one can argue that Φ (N ) x is a continuous function of ν k in the vicinity of the real axis.
Finally, the scalar product of the functions Ψ q 2 (z N +1 ) which we need in the proof of Theorem 1, takes the form ‖ Since the integrand is analytic function of is an analytic function of ν k in the vicinity of the real axis. where and The calculation is almost the same as in the previous cases so we omit the details.

SoV representation
In the previous section we constructed the functions Ψ p,x and Φ (N ) x associated with the entries B N and A N of the monodromy matrix (2.5).For a given vector Ψ ∈ H N we define two functions by projecting it on Ψ (N ) p,x and Φ (N ) x : x , Ψ . (5.1) These functions are symmetric functions of the variables x.It was shown by Sklyanin [3] that the transformation Ψ → ϕ (Ψ → χ) reduces the original multi-dimensional spectral problem for the transfer matrix to the set of one-dimensional spectral problems that greatly simplifies the analysis.We want to show that the maps Ψ → ϕ and Ψ → χ can be extended to the isomorphism between the Hilbert spaces, The variables x k , xk take the form , where all n k are either integers or half-integers, and (5.4) The measures are defined as follows dµ (5.6) The weight function µ N (x) is given by the following expression where be the Hilbert spaces of symmetric functions corresponding to the scalar products (5.2): (5.9b) Given that ϕ(p, x) and χ(x) are smooth and compactly supported functions on R 2 × D σ N −1 and D σ N , respectively, we introduce transforms T B N : Note that the function Ψ ϕ depends on the vector γ, Eq. (3.12), which appears in the definition of the function Ψ (N ) p,x .That is T B N ≡ T B N (γ) and the same applies to the operator T A N .In order to not overload the notation we do not display this dependence explicitly.

B system
We begin the proof of the unitarity of the transform T B N with the following lemma: Lemma 1.For any smooth fast decreasing function ϕ on R 2 × D σ N −1 the function T B N ϕ belongs to the Hilbert space H N and it holds with I , (x, x ) given by Eq. (4.11).Let us assume that the function ϕ(ϕ ) has the form ϕ(p, x 1 , . . ., x N −1 ) = κ(p)φ(x 1 , . . ., x N −1 ), (5.13)where φ(x 1 , . . ., x N −1 ) is a symmetric function and the sum goes over all permutations.We also assume that the functions φ ) is an analytic function of ν k in some strip |Imν k | < δ k which vanishes sufficiently fast at ν k → ±∞.Such functions form a dense subspace in the Hilbert space H B,σ N .Since the momentum integral in (5.12) factorizes one has to consider the integrals over (5.15) According to our assumptions only finite number of terms contribute to the sum in (5.15).Let us study behaviour of a particular term in the sum in the limit , → 0. The functions φ, φ are smooth and fast decreasing functions of ν, ν .The function I , (x, x ) contains the factor Γ[ and the product of the Γ-functions where jk ≡ j + k .In the k , j → 0 this function becomes singular at ν k = ν j if n k = n j .Let us shift the contours of integrations over ν k variables to the upper half-plane, Imν k = δ > jk , and pick up the residues at the corresponding poles.After this we can send k , j → 0. Let us consider a generic contribution arising after this rearrangement.It has the form where S(x k ) = x k if k ∈ (i 1 , . . .i M ) and S(x k ) = x p k if k does not belong to this set.The integrand f is given by the product of the functions φ k , φ k , Γ -functions (5.16) and the factor . All these factors are regular on the contours of integration.Moreover, if M ≥ 1 the last factor, A, tends to zero at , → 0. Thus the only non-vanishing contribution comes from the term with M = 0, i.e. when all S N (p, x|q 1 , q 2 , x ) φ(q 1 , q 2 , x ) d 2 q 2 d 2 q 1 dµ B N −1 (x ). (5.28) The kernel S N reads S N (p, x|q 1 , q 2 , x ) = Ψ (N +1) p,x , Ψ (N ) q 1 ,x ⊗ Ψ (1)   q 2 , (5.29) see Eq. (4.23), and x = x 1 + i 1 , . . ., x N −1 + i N −1 .We assume that function φ takes the form φ(q 1 , q 2 , x 1 , . . ., x N −1 ) = κ 1 (q 1 )κ 2 (q 2 ) S N −1 φ 1 (x i 1 ) . . .φ N −1 (x i N −1 ), (5.30)where the sum goes over all permutations and that the functions φ k are local in "n" variable, that is ) and φ n k are compactly supported.The function ϕ(p, y) does not decrease sufficiently fast for large y k in order to justify changing the order of integration after substituting ϕ (p, y) in the form (5.27), (5.28) into (5.25).To overcome this difficulty we, following the lines of ref. [29], consider the integral where the integration contours over ν k are deformed in order to separate the poles due to the Gamma functions, Γ [Z − ω ± iy k ], in the factor Ω. The integral I ω Z (ϕ) is an analytic function of ω.Substituting ϕ(p, y) in (5.34) in the form (5.28) one can show that for Re ω > 1 the integrals over y decay fast enough to allow the change of the order of integration over x, x and y.Thus we obtain 2) (q 1 + q 2 − q 1 − q 2 ) φ(q 1 , q 2 , x) φ(q 1 , q 2 , x ) † q 1 + q 2 q 1 q 2 2 q 1 q 1 q 2 q2 γ2N × R(x, x ) J ( ) ω (Z, ζ, x, x ) d 2 q 1 d 2 q 2 d 2 q 1 d 2 q 2 dµ B N −1 (x) dµ B N −1 (x ) , (5.35) function of the variables x 1 , . . ., x N .It can be shown that the operator A N (x) annihilates the layer operator Λ N (x|γ), A N (x)Λ N (x|γ) = 0, (3.28) ¶ Of course, one also can regularize the function by shifting the parameter γ1 instead of γ2N−2, γ1

Figure 1 :
Figure 1: The diagrammatic representation for the function Ψ (left( and Φ (right) for N = 3.The arrow from z to w with an index α stands for the propagator D α (z − w), Eq. (3.3).

(4. 13 )
Here [a] ≡ a − ā.Details of the calculation can be found in Appendix B.

I
x,y (p, { pr }|γ) = I x,y (p, { pr }|γ) > 0 ,(4.16) p) is a continuous function of ν k in this region.Indeed, let us fix m < N and put u m = Reν m and v m = Imν m , |v m | < δ.One gets the following estimate for the integrand (4.10)

. 5 )
The symbol Dx stands for Dx ≡