# Ruijsenaars-Schneider model

Post-publication activity

Curator: Simon N. M. Ruijsenaars

The Ruijsenaars-Schneider model describes an arbitrary number $$N$$ of interacting point particles of equal mass $$m$$ on the line. It is an integrable dynamical system both in the classical and in the quantal context. Its defining Hamiltonian $$H$$ is a one-parameter generalization of the Calogero-Moser Hamiltonian $$H_{\rm nr}\ .$$ The extra parameter has a physical interpretation as the speed of light $$c\ .$$ In the limit where $$c$$ goes to infinity (nonrelativistic limit), the Hamiltonian $$H$$ (minus the rest energy $$Nmc^2$$) reduces to the Calogero-Moser Hamiltonian $$H_{\rm nr}\ .$$ Accordingly, the Ruijsenaars-Schneider model is also known as the relativistic Calogero-Moser system.

Just as for $$H_{\rm nr}\ ,$$ the most general form of the interaction is encoded in elliptic functions, and these can be specialized to trigonometric, hyperbolic and rational functions. This yields four distinct regimes, each of which has different physical and mathematical features. In particular, for the rational and hyperbolic regimes the $$N$$ particles move freely for asymptotic times, the interaction being encoded in a solitonic scattering map. That is, the set of particle momenta is conserved and the scattering is factorized as if all particle pairs scatter independently. By contrast, in the trigonometric and elliptic regimes the particles perform oscillations around the freely moving center of mass.

## Definitions

### Classical level

The defining Hamiltonian is given by $H= mc^2 \sum_{j=1}^{N}\cosh(\frac{p_j}{mc})\prod_{1\le k\le N,k\ne j}f(x_j-x_k).$ The special case $$f(x)=1$$ corresponds to $$N$$ particles with positions $$x_1,\ldots,x_N$$ and rapidities $$p_1/mc,\ldots,p_N/mc\ ,$$ moving freely along the line. The generalized pair potential $f^2(x)=a+b\wp(x)$ (with $$a,b$$ positive constants and $$\wp(x)$$ the Weierstrass $$\wp$$-function) is the most general interaction compatible with relativistic invariance, important special cases being $f^2(x) = \left\{ \begin{array}{ll} 1+g^2/m^2c^2x^2 & ({\rm rational}) \\ 1+\sin^2(\nu g/mc)/ \sinh^2(\nu x) & ({\rm hyperbolic}) \\ 1+\sinh^2(\nu g/mc)/ \sin^2(\nu x) & ({\rm trigonometric}). \end{array} \right.$ More specifically, introducing $P= mc \sum_{j=1}^{N}\sinh(\frac{p_j}{mc})\prod_{1\le k\le N,k\ne j}f(x_j-x_k),$ and $B=-m\sum_{j=1}^Nx_j,$ one obtains Poisson brackets $\{ H,P\}=0,\ \ \{ H,B\} =P, \ \ \{ P,B\} =H/c^2,$ which represent the Lie algebra of the inhomogeneous Lorentz group. Thus, $$H,P$$ and $$B$$ represent the time translation, space translation and Lorentz boost generator, respectively. The second and third bracket hold true for an arbitrary $$f\ ,$$ but the first one (encoding space-time translation invariance) yields a severe constraint on $$f^2$$ (for $$N>2$$): Its vanishing amounts to functional equations for $$f^2$$ that hold true for the Weierstrass $$\wp$$-function.

For the above three special cases one easily checks the nonrelativistic limits $\lim_{c\rightarrow \infty }(H-Nmc^2)=H_{\rm nr}, \ \ \ \lim_{c\rightarrow \infty}P=P_{\rm nr},$ where $H_{\rm nr}=\frac{1}{2m}\sum_{j=1}^N p_j^2 + g^2 \sum_{1\le j<k\le N} V(x_j-x_k),\ \ \ P_{\rm nr}=\sum_{j=1}^Np_j,$ with $V(x) = \left\{ \begin{array}{ll} 1/mx^2 & ({\rm rational}) \\ \nu^2/ m\sinh^2(\nu x) & ({\rm hyperbolic}) \\ \nu^2/ m\sin^2(\nu x) & ({\rm trigonometric}). \end{array} \right.$ With a suitable choice of $$a,b$$ for the elliptic case, one again gets these limits, now with $V(x)=\wp(x)/m.$ Furthermore, the limits of the Poisson brackets, namely, $\{ H_{\rm nr},P_{\rm nr}\}=0,\ \ \{ H_{\rm nr},B\} =P_{\rm nr}, \ \ \{ P_{\rm nr},B\} =Nm,$ represent the Lie algebra of the nonrelativistic space-time symmetry group (the Galilei group).

The Hamiltonian $$H_{\rm nr}$$ is the defining Hamiltonian of the nonrelativistic Calogero-Moser system. One obtains a representation of the Galilei group Lie algebra for any (even) $$V(x)\ ,$$ but the $$\wp$$-function choice is essential for $$H_{\rm nr}$$ to give rise to an integrable system. For $$H$$ and $$P\ ,$$ by contrast, this choice is essential to obtain translation invariance, but as a bonus it preserves integrability. Specifically, it is clear that the functions $S_{\pm N}=\exp(\pm \beta\sum_{j=1}^Np_j),\ \ \ \beta=1/mc$ Poisson commute with $$H$$ and $$P\ ,$$ which already yields integrability for $$N=3\ .$$ More generally, it can be shown that the Hamiltonians $S_{\pm l}=\sum_{\stackrel{I\subset \{ 1,\ldots ,N\} }{|I|=l}}\exp (\pm \beta \sum_{j\in I}p_j )\prod_{\stackrel{j\in I}{k\not\in I}}f(x_j-x_k),\ \ \ l=1,\ldots,N$ mutually commute. Observe that one has $S_{-l}=S_{-N}S_{N-l},\ \ \ l=1,\ldots,N-1,$ and $H=(S_1+S_{-1})/2m\beta^2,\ \ \ P=(S_1-S_{-1})/2\beta.$

### Quantum level

The canonical quantization prescription $$p_j\to -i\hbar \partial/\partial x_j,\ j=1,\ldots,N$$ (with $$\hbar$$ Planck's constant) gives rise to unambiguous quantum Hamiltonians $$H_{\rm nr}$$ and $$P_{\rm nr}\ .$$ For $$H$$ and $$P\ ,$$ however, it is not even immediately clear how to define the quantum operators for the free case $$f(x)=1\ .$$ Writing the cosh- and sinh-functions as sums of two exponential functions, the natural definition can be exemplified by $\exp\left( -\frac{\hbar}{mc}i\frac{d}{dx}\right)F(x)=F\left(x-i\frac{\hbar}{mc}\right).$ That is, the quantum counterparts act on functions that have an analytic continuation in $$x_1,\ldots, x_N$$ from the real line to a strip in the complex plane, whose width is (at least) $$2\hbar/mc\ .$$ Thus the quantum Hamiltonians are so-called analytic difference operators.

For the interacting case, where $$f(x)$$ is not constant, the ordering of the noncommuting operators inherited from the classical level would already spoil the commutativity of $$H$$ and $$P\ .$$ To date no general results are known from which it would follow that a different ordering preserving commutativity exists. Even so, this is true in the present case. Specifically, the function $$f(x)$$ can be factorized as $$f_{+}(x)f_{-}(x)\ ,$$ and then the analytic difference operators $S_{\pm l}= \sum_{\stackrel{I\subset \{ 1,\ldots ,N\} }{|I|=l}} \prod_{\stackrel{j\in I}{k\not\in I}}f_{\mp }(x_j-x_k) \exp (\mp i\hbar \beta \sum_{j\in I} \partial_j ) \prod_{\stackrel{j\in I}{k\not\in I}}f_{\pm}(x_j-x_k),\ \ \ l=1,\ldots,N$ do commute. In the elliptic case, this factorization involves the Weierstrass $$\sigma$$-function, and commutativity can be encoded in a sequence of functional equations satisfied by the sigma-function. For the degenerate cases the pertinent factorization is given by $f_{\pm}(x) = \left\{ \begin{array}{ll} ( 1\pm i\beta g/x)^{1/2} & ({\rm rational}) \\ ( \sinh\nu (x\pm i\beta g)/\sinh \nu x)^{1/2} & ({\rm hyperbolic}) \\ ( \sin\nu (x\pm i\beta g)/\sin \nu x)^{1/2} & ({\rm trigonometric}). \end{array} \right.$

The nonrelativistic limit $$c\to\infty$$ of the quantum Hamiltonians $$H$$ and $$P$$ can be determined by expanding $$S_1$$ and $$S_{-1}$$ in a power series in $$\beta =1/mc\ .$$ In this way one obtains the partial differential operators $H_{\rm nr}=-\frac{\hbar^2}{2m}\sum_{j=1}^N\partial_j^2 + g(g-\hbar) \sum_{1\le j<k\le N} V(x_j-x_k),\ \ \ P_{\rm nr}=-i\hbar\sum_{j=1}^N\partial_j.$ Note that instead of the coupling constant dependence $$g^2$$ in the classical potential energy, one gets $$g(g-\hbar)$$ for the quantum potential energy. The extra term arises from the action of the term linear in $$\beta$$ in the expansion of the exponentials on the term linear in $$\beta$$ in the expansion of the functions $$f_{\pm}(x)\ .$$ The term looks peculiar at face value, but it has the consequence that the nonrelativistic quantum eigenfunctions have a maximally simple dependence on $$g\ .$$

## Relation to soliton PDEs and lattices

The classical Ruijsenaars-Schneider $$N$$-particle system with the hyperbolic interaction function $f(x)= [1+\sin^2\tau /\sinh^2(\nu x)]^{1/2},\ \ \ \tau \in (0,\pi /2],$ is intimately related to the $$N$$-soliton solutions of a host of integrable nonlinear partial differential equations and lattices, the most well-known examples being the Korteweg-de Vries and sine-Gordon equations, and the infinite Toda lattice, respectively. (In point of fact, the systems were introduced in 1985 by Ruijsenaars and Schneider with the purpose of replacing the interaction of $$N$$ sine-Gordon solitons by an equivalent interaction between $$N$$ point particles, in the sense that the same scattering occurs.) More specifically, the soliton solutions of the various PDEs and lattices are linked to the particle systems by fixing various parameters and choosing suitable Poisson commuting Hamiltonians as the time and space translation generators. This will now be exemplified for the case of the sine-Gordon equation $(\partial_y^2-\partial_t^2)\phi =\sin (\phi).$

As it stands, this relativistically invariant PDE has no scale parameters. Hence a special choice of the particle parameters $$m,c$$ and $$\nu$$ is necessary, namely, $$m=c=2\nu =1\ .$$ Moreover, just as in most other cases, one needs to choose $$\tau$$ equal to $$\pi/2\ ,$$ so that the pair potential' becomes $f(x)=|\coth(x/2)|.$ The $$N$$-particle phase space is given by $\Omega = \{ (x,p)\in \R^{2N}\mid x\in G\} ,$ where $$G$$ is the configuration space $G= \{ x\in {\mathbb R}^N \mid x_N<\cdots <x_1 \} .$ Denoting a point $$(x,p)\in \Omega$$ by $$u\ ,$$ the point $$u$$ now evolves with the two-parameter Hamiltonian flow $$\exp (tH-yP)\ ,$$ where $$H$$ and $$P$$ have been defined above. This gives rise to a family of space-time dependent particle positions $u_j(t,y)= (\exp (tH-yP)(u))_j,\ \ \ \ j=1,\ldots, N.$ (Thus, in particular, the position $$u_j(0,0)$$ is equal to $$x_j\ .$$)

After these preparations, the connection to sine-Gordon solitons can be detailed: The function $\phi (t,y)= 4\sum_{j=1}^N {\rm Arctg}(\exp [u_j(t,y)]),$ is an $$N$$-soliton solution to the sine-Gordon equation, and all $$N$$-soliton solutions are obtained by letting $$u$$ vary over $$\Omega\ .$$ Furthermore, the requirements $$u_j(t,y)=0,\ j=1,\ldots,N\ ,$$ yield uniquely determined space-time trajectories $y_N(t)<\cdots <y_1(t), \ \ \ \ \forall t\in{\mathbb R},$ for the $$N$$ solitons. Before and after all pair collisions have taken place, these trajectories are located under the soliton maxima exhibited by the $$y$$-derivative of $$\phi(t,y)\ .$$ But the trajectories make it possible to follow the nonlinear interaction during collisions, revealing that the solitons repel each other.

The sine-Gordon equation also has antisoliton solutions; more generally, solutions with an arbitrary number of solitons, antisolitons and their bound states (breathers) exist. Likewise, the $$N$$-particle systems can be generalized to systems with $$N_{+}$$ particles and $$N_{-}=N-N_{+}$$ antiparticles by substituting $$x_j\to x_j-i\pi$$ for $$j=N_{+}+1,\ldots,N\ ;$$ this has the effect that the repulsive pair potential $$|\coth(x/2)|$$ changes to the attractive one $$|\tanh(x/2)|\ .$$ These generalized $$N$$-body systems can now be tied in with the general particle-like' sine-Gordon solutions in the same way as already detailed for the pure soliton solutions.

There is considerable evidence that the classical particle-soliton correspondence turns into a physical equivalence on the quantum level. For the $$N= 2$$ case this has been established beyond doubt. Indeed, the scattering is identical and for the attractive case $$N_{+}=N_{-}=1$$ the bound-state energy spectrum of a quantum relativistic Calogero-Moser particle and antiparticle coincides with the spectrum of quantum sine-Gordon breathers.

## Duality properties

All of the classical Hamiltonians mentioned thus far can be obtained via the so-called Lax matrix $$L\ ,$$ which is an $$N\times N$$ matrix-valued function on the $$N$$-particle phase space. For the relativistic systems it is of the form $L_{jk}=e_jC_{jk}e_k,\ \ \ j,k=1,\ldots,N.$ Here, $$C$$ is a Cauchy type matrix depending only on the positions $$x_1,\ldots,x_N\ ,$$ whereas the quantity $$e_j$$ depends on $$p_j$$ and the positions. For the hyperbolic regime the matrix $$C$$ and vector $$e$$ are given by $C_{jk}=\exp (-\nu (x_j+x_k))\frac{\sinh(i\beta \nu g)}{\sinh\nu (x_j-x_k+i\beta g)},\ \ \ e_j= \exp (\nu x_j+\beta p_j/2)\prod_{l\ne j}f(x_j-x_l)^{1/2}.$ From this it is readily checked that the above Hamiltonian $$S_1$$ is the trace of the Lax matrix. More generally, Cauchy's identity entails that the Hamiltonians $$S_1,\ldots,S_N$$ are the symmetric functions of $$L\ .$$

The Lax matrix is the key tool for obtaining information about the action-angle map, i.e., the canonical transformation after which the Poisson commuting Hamiltonians only depend on generalized momenta---the actions. For the trigonometric, hyperbolic and rational Calogero-Moser systems (both of nonrelativistic and of relativistic type), the construction of this map also involves the so-called dual Lax matrix $$A(x)\ .$$ In the hyperbolic case it is given by $A(x)={\rm diag}(\exp(2\nu x_1),\ldots,\exp(2\nu x_N)).$

For the hyperbolic Ruijsenaars-Schneider systems the two matrices transform into each other under the action-angle map. As such, these systems are self-dual. This remarkable feature can already be gleaned from the commutation relation $\coth(i\beta \nu g)(AL-LA)+(AL+LA)=2e\otimes e,$ which readily follows from the above definitions. Indeed, it is clear that $$A$$ and $$L$$ play symmetric roles in this key relation.

The action-angle map for the rational Ruijsenaars-Schneider systems reveals that the dual systems are the hyperbolic nonrelativistic Calogero-Moser systems. This duality property and its generalization to the self-duality of the hyperbolic regime were obtained first by Ruijsenaars by direct arguments. Since then they been reobtained in various ways via group-theoretic reasoning. The duality properties generalize the self-duality of the rational nonrelativistic Calogero-Moser systems that was first shown by Kazhdan, Kostant and Sternberg.

The natural expectation that these duality properties survive quantization has been confirmed in all cases where it can be checked. In fact, the only case for which this feature cannot be studied yet is the relativistic hyperbolic case with $$N>2$$ and arbitrary coupling. This is because to date the corresponding $$N$$-particle eigenfunctions are not known explicitly. For $$N=2$$ the eigenfunction pertinent to the hyperbolic regime is a one-coupling specialization of a relativistic hypergeometric function that depends on four parameters of coupling type. In terms of these parameters, this function has a stronger self-duality property and symmetries linked to the Weyl group of the Lie algebra $$D_4\ .$$ It can be specialized to the Askey-Wilson polynomials in a similar way as the Gauss hypergeometric function can be specialized to the Jacobi polynomials.

For the trigonometric regime, the quantum eigenfunctions associated with the reinterpretation of the $$N$$ commuting differential/difference operators as self-adjoint operators on a Hilbert space are basically the Jack/Macdonald $$N$$-variable polynomials in the nonrelativistic/relativistic case, respectively. For this regime the respective dual systems are lattice versions of the rational and hyperbolic Ruijsenaars-Schneider systems.