p,q-Duality and Hamiltonian Flows in the Space of Integrable Systems or Integrable Systems as Canonical Transforms of the Free Ones

Variation of coupling constants of integrable system can be considered as canonical transformation or, infinitesimally, a Hamiltonian flow in the space of such systems. Any function $T(\vec p, \vec q)$ generates a one-parametric family of integrable systems in vicinity of a single system: this gives an idea of how many integrable systems there are in the space of coupling constants. Inverse flow is generated by a dual"Hamiltonian", $\widetilde T(\vec p, \vec q)$ associated with the dual integrable system. In vicinity of a self-dual point the duality transformation just interchanges momenta and coordinates in such a"Hamiltonian": $\widetilde T(\vec p, \vec q) = T(\vec q, \vec p)$. For integrable system with several coupling constants the corresponding"Hamiltonians"$T_i(\vec p, \vec q)$ satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants: [ d/dg_a - T_a(p,q), d/dg_b - T_b(p,q) ] = 0. Some explicit formulas are given for harmonic oscillator and for Calogero-Ruijsenaars-Dell system.


Introduction
The study of non-perturbative quantum phenomena revealed the real role of integrable systems in physics: τ -functions of integrable hierarchies appear to describe partition functions of quantum theories, with time-variables identified with coupling constants [1,2]. Broadening understanding of this relation, together with the growing interest to peculiarities of essential (non-perturbative) quantum physics, stimulates a new attention to the basics of integrability theory, which brings one back to the very foundations of classical Hamiltonian mechanics that were not sufficiently investigated in the past. The present paper is devoted to one of such basic subjects, to the description of integrable systems in terms of the Hamiltonian evolution in the space of coupling constants. This approach can help one to describe the entire variety of integrable systems and to reveal the meaning of peculiar relations, like dualities, between different such systems.

Variation of couplings as canonical transform -peculiar feature of integrable systems
Integrable system with N coordinates q i and N momenta p i is characterized by existence of N Poissoncommuting Hamiltonians H i ( p, q), ∂ q = 0. For such a system one can consider a canonical transformation, treating these Hamiltonians as new momenta-like or coordinatelike variables. This transformation (of which the infinitesimal version is a certain Hamiltonian flow) will be the subject of the present paper.
To make the problem precise, let us consider a one-parametric family of integrable models, parameterized by a single coupling constant g such that the model is free when g = 0. This means that at g = 0 the Hamiltonians H k ( p) = I:|I|=k i∈I e p i for p-rational and p-trigonometric models respectively and some elliptic functions of p for their elliptic generalizations.
The adequate definition of the new canonical variables P g = P ( p, q|g) and Q g = Q( p, q|g) 1 is [3] where H i ( p, q|g) define the Hamiltonians of the dual integrable system (for other discussions of this duality see [4] and [5]). Note that in (1) they depend on the "dressed" variables P and Q and, moreover, P and Q are interchanged. The shape of the dual Hamiltonians is dictated by the requirement that the new variables P and Q are canonical, i.e.
and the Poisson brackets are

Infinitesimal version of canonical transforms -the Hamiltonian flows
Relations (1) define P g as functions of p, q and the coupling constant g. Of special interest and importance is the infinitesimal version of this canonical transformation, considered as a Hamiltonian flow along the g direction in the space of coupling constants. Such transformation is generated by a new Hamiltonian T ( p, q; g), according to the rule: This new Hamiltonian does not commute with the old ones, but it converts them into a new set of commuting Hamiltonians.
Moreover, instead of considering a pre-given family of integrable systems, one can use any function T ( p, q|g) to define a whole one-parametric family of integrable systems, though explicit construction of the corresponding Poisson-commuting Hamiltonians is rarely possible. Further, a multi-parametric family of integrable systems is similarly generated by any collection of functions T a ( p, q|g) satisfying the compatibility (Whitham) equations: 2 The coupling constant variation for the dual integrable system is governed by the dual Hamiltonian T ( p, q|g): for (and H (0) Relation between T ( p, q|g) and T ( p, q|g) becomes especially simple at any self-dual point (where H i ( p, q |g SD ) = H i ( p, q |g SD )): As immediate consequence, any symmetric function T ( p, q|g) = T ( q, p|g) defines a one-parametric family of self-dual integrable systems.
Note that it is not that immediate calculation to restore the Hamiltonian starting from arbitrarily given T ( p, q|g). Say, one can build a perturbative series in the coupling constant g. At the self-dual point for the system with one degree of freedom one takes the expansion T (p, q|g) The result ultimately is that the symmetric part of ∂ p Φ i (p, q) = ∂ 2 q,p T i (p, q)+ is given independently at any order, while the antisymmetric parts are fixed by Φ i 's at lower orders: 2 As mentioned in the abstract to this paper, after quantization of the original system these Ta( p, q|{g b }) become a family of operatorsTa(ˆ p,ˆ q|{g b }), satisfying the zero-curvature condition in the space of coupling constants: This can serve as an explanation of emergency of classical integrability (the zero-curvature equations) in the study of quantum integrable systems.
where all the derivatives are taken w.r.t. to the first variable. A trivial solution to these equations (9) is Φ i (p, q) = p n i +1 q n i with arbitrary {n i }. This implies that the Hamiltonian H(p, q) = pf (pq) is self-dual with arbitrary function f . Indeed, one easily checks this is the case (note that for such a Hamiltonian pq = P Q). We discuss later in detail the rational Calogero Hamiltonian that gets to this class of Hamiltonians.

Generating functions of canonical transforms
Along with the Hamiltonians T ( P , Q|g) one can also consider the generating functions of canonical transformations in question, like S( Q, q|g) or its Legendre transform F ( P , q|g) = P Q − S( Q, q|g), such that and Of course, for canonical transformation as implied by (10), but one should be more careful when drawing similar conclusion from (11): the second g derivatives satisfy eq.(5), because g-derivative is taken at constant P g and Q g , which themselves depend on g. Similarly, At self-dual points F ( P , q|g SD ) = F ( q, P |g SD ) is a symmetric function.
Note that (with p and Q constant), i.e. T is, in a sense, a more invariant quantity than S and F , which does not depend on the choice of independent variables.
Another way to see it is to consider a flow from an integrable system with coupling constant g 1 to the same system with coupling constant g 2 . Then, T depends only on g 2 , but not on g 1 , while the generating functions depend on both g 1 and g 2 .

Generating functions, evolution operator and eigenfunctions
Basically, if reexpressed in terms of Q and q (instead of P and Q) the Hamiltonian T = ∂S/∂g. Therefore, e iS can be considered as a kind of an evolution operator (kernel) in the space of coupling constants, which performs a canonical transformation from the free system to the integrable one. After quantization it can be symbolically represented as where |ψ λ and |ψ (0) λ are eigenfunctions of the system and c λ are some coefficients, depending on the spectral parameter λ. The evolution operator satisfies the quantum version of eq. (1), Since eigenfunctions |ψ (0) λ (Q) of a free system, are just exponents of the spectral parameter λ, the dressed eigenfunctions |ψ λ (q) are basically Fourier transforms of the evolution operator e iS(Q,q) : Similarly, Note that there is a freedom in solutions of eq.(2) to shift {Q i } by an arbitrary function of {P i }. This shift is quite complicated in terms of the generating function S, but in F it is just an addition of the term depending only on {P i }. In the quantum case, this ambiguity in the definition of F is just a matter of normalization of the eigenfunction.
One can also consider the quantum counterpart of T (p, q|g) which is a Hamiltonian that gives rise to a Schrödinger equation w.r.t. the coupling constant Therefore, the wavefunction ψ can be also realized as a path integral over the phase space variables P (g), Q(g).
The generating functions S(Q, q), F (P, q) and T (p, q) satisfy the quasiclassical versions of these relations (when multiple derivatives of S, F and T are neglected). Exact definition of quantum evolution operators, including effects of discrete spectra and identification of the spectra at different values of coupling constants, i.e. precise definition of the spectral parameter λ in such a way that eq.(15) is diagonal in λ, will be considered elsewhere.

Some conceptual questions for further examination
The previous section explained possible needs to study integrable systems from the non-conventional perspective: in terms of the auxiliary Hamiltonians T ( p, q|g), describing the associated Hamiltonian flows in the space of coupling constants. Some immediate problems were also mentioned, like (i) criteria, distinguishing the Hamiltonians T associated with conventional integrable systems at the entire space of such Hamiltonians; (ii) structure of algebra, generated by all the Hamiltonians H i and T a , in which the H i 's generate a commutative (Cartan) subalgebra; (iii) duality relation between T and T ; (iv) relation between integrable systems and flat connections over the space of coupling constants encoded in (5).
One can also add to this list: (v) identification of the flows and Hamiltonians T a in terms of Seiberg-Witten theory, [6,2]; (vi) treatment of T 's in terms of the theta-function formalism of refs. [3,5]; (vii) the "gauge-equivalence" of the Hamiltonians T a , corresponding to different choices of the constant-p, q sections over the moduli space (i.e. with different quantities kept constant when the couplings g a are varied); (viii) the role of the distinguished "flat-modulus" a [6], for which the WDVV equations [7] hold; (ix) further development of the notion of generalized τ -function, making use of the evolution operator e S , introduced in the previous section (see also the concluding section 5 below).
Since the subject is relatively new, we do not think it is time now to get into further discussions of these (and many other) problems. Instead in the remaining part of this paper we consider some simple examples: the Hamiltonians T ( p, q; g), associated with the flows in the coupling constant space for some well-known integrable systems like harmonic oscillator and Calogero-Ruijsenaars-Double-elliptic (DELL) 3 family.

Particular examples of Hamiltonian flows 4.1 Harmonic oscillator
Perhaps surprisingly, even for this simple model one obtains rather sophisticated formulas.
For harmonic oscillator H(p, q|ω) = 1 2 (p 2 + ω 2 q 2 ). Let the frequency ω play the role of the coupling constant so that H (0) (p) = 1 2 p 2 . Then Note that in this case the ω-evolution possesses "the conservation law" The inverse transformation looks like p = P cos ωQ P , q = P ω sin ωQ P i.e. the Hamiltonian of the dual flow is (Note that with this definition H (0) (p) = p and also note that p, q-duality does not respect conventional dimensions of p and q so that it should not be a surprise that ωq/p in H got substituted by ω p/ q in H. Of course, p and q are nothing but Q ω and P ω .) The generator of ω-evolution such that is For the dual system we similarly have and In order to obtain the generating function S(Q, q|ω) one needs first to reexpress p and P through q and Q. Unfortunately, the formulas are transcendental. Instead, it is easy to express p and Q through q and P , this involves only elementary functions The corresponding generating function so that the relation (18) in quasiclassical approximation is where λ 2 2ω = ν + 1 2 and the Hermite polynomials, satisfying are given by inverse Laplace transform of the generating function e −x 2 /4+t/x Taking this integral in the saddle point approximation, one gets which, at large ν, is in accordance with eq.(30).

Generic Schrödinger-like systems
For a single particle any dynamics is integrable, and any Hamiltonian is canonically equivalent to the free one. Therefore, it may make sense to look at the corollaries of our formalism in such context. This can help to see the difference between models which have and have not simple integrable generalizations to the multiparticle case.
For H = p 2 + g 2 V (q) the Hamiltonian T can be found in the form of a perturbation expansion: where There is no way to integrate explicitly these equations in the general case. However, for harmonic oscillator exact answer exists for the entire expansion, eq.(25). Similarly, exact answers are available for potentials of the rational and trigonometric Calogero models (see below). However, for elliptic Calogero model with V (q) ∼ sn −2 (q|k) eq.(35) implies that the generator T (p, q|g) involves the integral elliptic tangent (i.e. the Weierstrass zeta-function), T n(q|k) = dξ sn 2 (ξ|k) ∼ ζ(ξ|τ ). 4 Higher orders of the g expansion contain multiple integrals of T n(ξ). This reflects a more sophisticated group structure of elliptic models.
The simplest way to derive formulas (34)-(35) is to construct the Hamiltonian q = H(P, Q) dual to p 2 + g 2 V (q). The canonicity condition then gives Therefore, the dual Hamiltonian can be obtained via solving the equation where q(P, Q) is given by (37). One easily obtains the expansion of these two formulas in powers of g, in particular, reproducing (34)-(35).
The difference between the algebraic and integral tangents is that ∂ ∂ξ) tn(ξ|k) = dn(ξ|k) cn 2 (ξ|k) while ∂ ∂ξ) T n(ξ|k) = 1 cn 2 (ξ|k) One can express the dressed variables P = P g , Q = Q g through the bare ones, p, q The inverse transform is Note that, similarly to the case of harmonic oscillator, the g-flow possesses a conservation law The flow is generated by the Hamiltonian This is obviously a symmetric function of p and q, thus, T D (p, q) = T (p, q) as it should be for the self-dual system.
In order to obtain the evolution operator e iS , one should rewrite T in terms of the q, Q-variables. The p, P -variables are then given by The symmetricity (integrability) condition (12) is true and one obtains In this case, S is a simple linear function of g, and ∂S/∂g = S/g.

Similarly,
F (P, q) = P 2 q 2 + g 2 + g 2 log P 2 q 2 + g 2 − g P 2 q 2 + g 2 + g , ∂F (P, q|g) ∂g = T (p(P, q), q|g) One can also consider the transformation from the system given at (g 1 , Q 1 , P 1 ) to that given at (g 2 , Q 2 , P 2 ). The corresponding generating function S(Q 1 , Q 2 , g 1 , g 2 ) = g 1 log is the function of both g 1 and g 2 , while is the function of only g 2 .
For other members of the Calogero-Ruijsenaars family we just list some relevant formulas, parallel to those in the rational-rational case.

The rational-trigonometric case (trigonometric Calogero model)
In this case S is still a simple linear function of g S = g arcsinh sinh Q sinh q (52)

4.3.3
The trigonometric-rational case (rational Ruijsenaars model) In this case, S is no longer a simple linear function of the coupling constant ǫ.

SU(N) rational Calogero model
To give an illustration of what happens in the multiparticle case we start here with the simplest case of the rational Calogero model. More profound examples will be considered elsewhere.
The Hamiltonians H i ( p, q) = T rL i are defined as traces of powers of the Lax matrix L ij = p i δ ij + g 1 q ij . Accordingly H 0 i ( p) = j p i j . The Hamiltonian flow within the family of rational Calogero models (along the g direction) is governed by Technically, higher order terms can be done in the following way. As the first step, one needs to find dual Hamiltonians. Note that dual Hamiltonians technically each time emerge as necessary ingredient in constructing T . Moreover, with these intermediate objects, constructing T becomes well-defined although often complicated procedure and reduces to finding solutions of some partial differential equations. Indeed, first of all the dual Hamiltonians emerge as solutions to partial differential canonicity equations and then T is obtained by integration the following set of equations: These equations just follow from the fact that the dual Hamiltonians H k are nothing but independent of the coupling constant g coordinates {q i }.
Now, having H k calculated, one can easily construct a recursive procedure of calculating T as series in g: T = g 2k+1 T k . Indeed, note that, in the leading order in g, H k are just simple functions of Q i , say, just Q k which do not depend on g. This means that plugging the power expansion of T into (56) gives a system of N linear algebraic equations for the derivatives D (1) i ≡ ∂T 1 /∂P i . Solving these equations, one should just integrate D (1) i , finally obtaining T 1 . Then one can use this calculated T 1 in order to find T 2 etc. Note that each time there emerge non-trivial integrability conditions Thus, the most difficult part of the calculation is to find dual Hamiltonians. In the rational Calogero system, when the system is self-dual, the calculation is very immediate. Similarly, the calculation can be easily done for any member of the Calogero-Ruijsenaars-Dell family.

Conclusion
A peculiar feature of integrable systems is that interaction is introduced by a canonical transformation of a free system. In other words, the corresponding flows in the space of coupling constants are Hamiltonian ones. This observation should provide new insights into the structure of exact (non-perturbative) partition functions, which are always (generalized) τ -functions of some integrable hierarchies [1,2].
Let us remind that the partition function, arising after functional integration over fields, is essentially unsensitive to the choice of original action A(ϕ), and the equivalence classes (of actions) can be represented by integrable systems, i.e. by the systems where the partition function is a matrix element of a group elementĜ of some group [8]. Our evolution operator in the space of coupling constants, e iŜ = P exp i {k} Ĥ {k} dt {k} , which performs a canonical transformation of integrable system into the free one, as discussed in s.2 should also be a group element. The moduli ϕ 0 , parameterizing the boundary conditions and vacuum states of the original theory are related to the (eigen)values of the Hamiltonians H {k} (i.e., in our simple examples, the dressed momenta P ). The couplings (perturbations) t {k} (i.e. our g) become time-variables of the τ -function. The fact that the space of relevant Hamiltonians (the moduli space) is an orbit of the canonical transformation group, which can be alternatively considered as changing the couplings can further clarifies the relation between the times-and moduli-dependencies of the partition functions [9] and bring us closer to the long-awaited group-theoretical treatment of entire τ -function, its time-dependencies being associated with the commutative (Cartanian) part of the algebra (generated by the Hamiltonians H i ), while moduli-dependencies -with its non-Cartanian part (generated by T a and their commutators among themselves and with H i ). See refs. [10] for related approaches to this problem.