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Calogero-Moser system
Francesco Calogero (2008), Scholarpedia, 3(8):7216. | doi:10.4249/scholarpedia.7216 | revision #91096 [link to/cite this article] |
Calogero-Moser dynamical system is a one-dimensional many-body problem that can be explicitly solved.
Contents |
The CM system
The dynamical system that is generally called Calogero-Moser (hereafter CM) is the model characterized by the Hamiltonian
\tag{1}
H\left( \underline{p},\underline{q}\right) =\frac{1}{2}\sum_{n=1}^{N}\left(p_{n}^{2}+\omega ^{2}q_{n}^{2}\right) +g^{2}\sum_{m,n=1;m\neq n}^{N}\left(q_{n}-q_{m}\right) ^{-2}~.
Here and hereafter N is an arbitrary positive integer, the two (real) N-vectors \underline{p}\equiv \left( p_{1},...,p_{N}\right) [respectively \underline{q}\equiv \left( q_{1},...,q_{N}\right)] feature as their components the N canonical momenta p_{n} [respectively the N canonical coordinates q_{n}], g^{2} is a positive "coupling constant" characterizing the strength of the interparticle two-body interaction and \omega^{2} is a nonnegative constant characterizing the strength of the interaction with an external "harmonic oscillator" potential. This Hamiltonian describes the (nonrelativistic) one-dimensional N-body problem of N equal particles (whose mass has been set to unity) interacting pairwise via a repulsive force singular at zero separation and with a common, confining, external "harmonic oscillator" potential (absent if \omega =0). An analogous model is characterized by the Hamiltonian
\tag{2}
H\left( \underline{p},\underline{q}\right) =\frac{1}{2}\sum_{n=1}^{N}p_{n}^{2}+\sum_{m,n=1;m\neq n}^{N}\left[ \frac{\omega ^{2}}{2N}\left( q_{n}-q_{m}\right) ^{2}+g^{2}\left( q_{n}-q_{m}\right) ^{-2}\right] ~.
It has the merit – in contrast to (1) – of being translation-invariant, and yet to yield, in its center-of-mass system, an almost identical dynamics to that yielded by the Hamiltonian (1): the difference among the two models is that the center-of-mass oscillates harmonically in the first case, (1), while it moves freely in the second case, (2).
In the classical context the Newtonian equations of motion yielded by (1) read
\tag{3}
\ddot{q}_{n}+\omega ^{2}q_{n}=2g^{2}\sum_{m=1;m\neq n}^{N}\left( q_{n}-q_{m}\right) ^{-3}~,
where of course q_{n}\equiv q_{n}\left( t\right) \ , the (real) independent variable t is the time and superimposed dots denote time differentiations.
In the quantal context the stationary Schrödinger equation corresponding to the Hamiltonian (1) reads
\tag{4}
\left[ -\frac{1}{2}\Delta +\frac{1}{2}\omega ^{2}\sum_{n=1}^{N}x_{n}^{2}+g^{2}\sum_{m,n=1;m\neq n}^{N}\left(
x_{n}-x_{m}\right) ^{-2}\right] \Psi =E\Psi ~,
The interest of this model lies in its exact solvability, both in the classical and quantal contexts.
In the classical context, the Hamiltonian model (1) is completely integrable – N integrals of motion in involution can be explicitly exhibited – indeed superintegrable – 2N-1 functionally independent integrals of motion can be explicitly exhibited – and algebraically solvable: the solution of the initial-value problem of the Newtonian equations of motion (3) can be performed by algebraic operations, specifically by computing the N eigenvalues of an N\times N matrix explicitly known in terms of the initial data q_{n}\left( 0\right) \ , \dot{q}_{n}\left( 0\right) and of the time t\ . In the confined case (\omega >0), the solution is isochronous: completely periodic, q_{n}\left( t+T\right) =q_{n}(t)~,~~~n=1,...,N~,
In the not confined case (\omega =0) the time evolution of the N particles features the following neat asymptotic relation among their positions and velocities p_{n}=\dot{q}_{n}\ , in the remote past and future:
\tag{6}
q_{n}\left( t\right) =p_{n}^{\left( \pm \right) }\,t+q_{n}^{\left( \pm \right) }+O\left( t^{-1}\right) ~~~\text{as}~~~t\rightarrow \pm \infty ~,
p_{n}^{\left( +\right) }=p_{N+1-n}^{\left( -\right) }~,~~~n=1,...,N~,
with the ordering of these velocities corresponding of course to the ordering of the particles on the line, say p_{1}^{\left( -\right)}>p_{2}^{\left( -\right) }>...>p_{N}^{\left( -\right) } and p_{1}^{\left( +\right) }<p_{2}^{\left( +\right) }<...<p_{N}^{\left( +\right) } corresponding to q_{1}\left( t\right) <q_{2}\left( t\right) <...<q_{N}\left( t\right) \ : note that the ordering of the particles on the line cannot change throughout the time evolution due to the singular nature of the two-body interaction at zero separation, forbidding them from overtaking each other. This outcome, (6), is independent of the value of g^{2} (provided g^{2}>0). It is congruent with what happens for extremely small g^{2} when the particles move freely except for colliding elastically; but of course for arbitrary (nonvanishing) g^{2} the motion of each particle is influenced by the motion of every other particle, indeed significantly so since this is a many-body problem with long-range two-body interactions. It is indeed instructive to try and make for oneself a graph of the positions q_{n}\left( t\right) of the N particles as functions of the time t reflecting the features described above, for small and large values of g^{2}\ .
In the quantal context (which, remarkably, was solved firstly: see below), in the confined case (\omega ^{2}>0) the spectrum of the stationary Schrödinger equation (4) is discrete and equispaced, being given by the simple formula
\tag{8}
E_{k}=\omega \left[ k+\frac{N}{2}+\frac{N\,\left( N-1\right) }{2}\left( a+\frac{1}{2}\right) \right] ~~~~,~k=0,1,2,...~,
a=\frac{1}{2}\left( 1+2g^{2}\right) ^{1/2}~,
In the not confined case (\omega =0), the scattering matrix can be computed in closed form, and it vanishes unless the initial and final momenta, p_{n} and p_{n}^{\prime }, satisfy the restrictions, p_{n}^{\prime }=p_{N+1-n}\ , corresponding to the classical outcome, see (6) ("no diffraction").
The solvable character of the CM model is relevant in a physical context inasmuch as it provides a toy many-body problem with interparticle pair forces singularly repulsive at short-distance (as are the forces in typical, molecular or nuclear, many-body problems), both when the system is confined due to the external harmonic oscillator potential (\omega ^{2}>0) – mimicking a container, although one with soft walls – and as well when there is no containment (\omega^{2}=0) and the key physical phenomena are then scattering processes (in the freely moving center-of-mass system). The main shortcoming with respect to physical reality is the one-dimensional character of this model, and the special form of the two-body interaction, causing the peculiarly neat phenomenology outlined above. Indeed – rather than as a toy physical many-body problem (although originally invented with this application in mind) – this model turned out to be more relevant in other areas of theoretical and mathematical physics and in pure mathematics: as one of the first nontrivial examples of integrable dynamical system, and because of its connection with various (old and new) developments in several fields. (Two assessments: "The last decade has witnessed a true explosion of activities involving Calogero-Moser-Sutherland models, and these now play a role in research areas ranging from theoretical physics (such as soliton theory, quantum field theory, string theory, solvable models of statistical mechanics, condensed matter physics, and quantum chaos) to pure mathematics (such as representation theory, harmonic analysis, theory of special functions, combinatorics of symmetric functions, dynamical systems, random matrix theory, and complex geometry)" [see(vDV2000)]; "Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc." [see(E2005)]. Digiting Calogero Moser on Google (Monday 19 June and Thursday 29 June, 2008) yielded (approximately) 40,600 and 41,000 links; Calogero-Moser, 37,000 and 37,500; Calogero Moser system, 26,900 and 27,300; Calogero Moser dynamical system, 10,500 and 15,700; Calogero Sutherland, 40,400 and 41,700 (whatever this means: Moser Sutherland yielded 157,000 and 148,000 links, but the second and fourth were a YouTube site with a video featuring both (Edda) Moser and (Joan) Sutherland singing a Mozart arietta...).
Additional results
There are several variations on, and generalizations of, the CM system, and connections with other problems (in addition to those mentioned above). In this section they are outlined quite tersely: the reader whose interest is stimulated by these hints will have no difficulty to pursue the specific matter via the web (possibly also using as guidance some of the names quoted below), or by consulting the textbooks listed at the end of the following section.
An extension of the CM model – which can be arrived at by considering the CM model with \omega =0 on a circle rather than a straight line, via the identity \sum_{\ell =-\infty }^{\infty }\left( q+\frac{\pi \ell }{a}\right) ^{-2}= \left[ \frac{\sin \left( aq\right)}{a}\right]^{-2}
Here a is an arbitrary real constant. This model was introduced by Bill Sutherland. In this case the particle coordinates q_{n} are of course defined mod ( \pi /a )\ . If instead a is replaced by ia (with i the imaginary unit, i^{2}=-1) this Hamiltonian becomes
\tag{10}
H\left( \underline{p},\underline{q}\right) =\frac{1}{2} \sum_{n=1}^{N}p_{n}^{2}+g^{2}\,a^{2}\sum_{m,n=1;m\neq n}^{N}\left\{ \sinh \left[ a\left( q_{n}-q_{m}\right) \right] \right\} ^{-2}~,
and may be considered a more appropriate toy model for the nuclear many-body problem, inasmuch as the pair potential is now short-range. Of course for a=0 both these Hamiltonians reduce back to the CM Hamiltonian (1).
A more general extension of the CM model (with \omega =0) is characterized by the Hamiltonian H\left( \underline{p},\underline{q}\right) =\frac{1}{2} \sum_{n=1}^{N}p_{n}^{2}+g^{2}\,a^{2}\sum_{m,n=1;m\neq n}^{N}\wp \left[a\left( q_{n}-q_{m}\right) ;\omega _{1},\omega _{2}\right] ~,
In the classical context all these Hamiltonian models are integrable: they are also solvable, at least in the trigonometric and hyperbolic cases. In the trigonometric ("Sutherland") case all motions are multiply periodic (all solutions are a linear superposition of N trigonometric functions the periods of which depend on the initial data). In the hyperbolic case the asymptotic outcome (6) obtains again, but now with (7) replaced by q_{n}^{\left( +\right) }=q_{N+1-n}^{\left( -\right) }+\sum_{m=1;m\neq n}^{N}\Delta \left( p_{m}^{\left( -\right) }-p_{n}^{\left( -\right) };g;a\right) ~,
In the quantal context the spectrum is discrete (a quadratic function of the integer quantum numbers) in the trigonometric case (9) ("no bands", in spite of the periodic character of the potential), and the scattering matrix is very simple (again, "no diffraction") in the hyperbolic case (10). The treatment of the quantal elliptic case is, again, much more complicated: key contributions have been made by many researchers including Edwin Langmann and (also in a more general context, see below) Simon Ruijsenaars and Felipe van Diejen.
Another type of extension consisted in the insertion of more general "external potentials" than the simple harmonic one featured by the Hamiltonian (1); the more general such potential compatible with the preservation of the integrable character of the model is a quartic polynomial, with arbitrary coefficients.
Another type of generalization led to the treatment of more than one type of particles: for instance a (still exactly treatable) generalization of the model (10) – obviously of some interest as a toy many-body problem – includes arbitrary numbers of equal mass particles of two types, with the repulsive pair potential g^{2}a^{2}/\sinh ^{2}\left( ax\right) acting among equal particle, and the attractive potential -g^{2}a^{2}/\cosh ^{2}\left( ax\right) acting among different particles – hence also giving rise to two-body bound states ("molecules"), with the possibility to investigate exactly the phenomenology of interactions among all such objects – both in the classical and quantal contexts.
Interesting connections have been uncovered (by Martin Kruskal, Jürgen Moser and others) among the motions of the particles described, in the classical context, by the Newtonian equations of motion (3) – but with the coordinates q_{n}=z_{n} considered as complex numbers – and the time evolution of the poles of certain solutions of the Korteweg-de Vries PDE (prototypical integrable nonlinear evolution equation in 1+1 variables: one-dimensional space and time); likewise interesting connections have also been uncovered (mainly by Igor Krichever) among CM-type models and the Kadomtsev-Petviashvili equation (prototypical integrable nonlinear evolution PDE in 2+1 variables). The connection among the positions z_{n}\left( t\right) of particles evolving according to integrable many-body models and the poles or zeros of the solution \psi (z,t) of (integrable nonlinear, or linear) PDEs has indeed been an important tool to identify several (classes of) new integrable many-body problems and to investigate their time evolutions [see(C1978)].
Another line of generalization has associated internal degrees of freedom ("spin") to the particles whose time evolution is described by CM-type models.
An extension that led to many mathematical developments emerged from the replacement of the quantity q_{n}-q_{m} in the above equations (see for instance (2)) with \underline{\alpha }\cdot \underline{q} where the N-vector \underline{\alpha } characterizes a "root system".
An extremely interesting extension of the CM system (including its trigonometric, hyperbolic and elliptic generalizations, see above) has resulted in a class of RS models (see below), it would probably be better to specify it here </review> that can be considered as a "relativistic" generalization of it – to the extent this is possible for many-body models describing evolutions as functions of an independent variable interpreted as (absolute) time. This RS extension features an additional constant, playing the role of "speed of light", and reproduces the CM models when this constant is set to infinity. The justification for considering this a relativistic extension is because the Galilei invariance of the CM model gets replaced by the Poincaré invariance of the extended model.
Another class of extensions obtains from the replacement – in the classical context – of the Hamiltonian (1) with H\left( \underline{p},\underline{z}\right) =\frac{1}{2}\sum_{n=1}^{N}\left( p_{n}^{2}+\omega ^{2}z_{n}^{2}\right) +\sum_{m,n=1;m\neq n}^{N}g_{nm}^{2}\,\left( z_{n}-z_{m}\right) ^{-2}~.
Finally, as we just indicated, it is possible – and generally appropriate and interesting – to extend the study of all the models mentioned above from the real to the complex, by considering the motion of the "particles", identified by their coordinates z_{n}\left( t\right) , to take place in the complex z-plane rather than on the real line. It is thereby also possible to go from real one-dimensional models ("motions on the line") to real two-dimensional models ("motions in the plane"). And it is even possible to do so compatibly with some kind of rotation-invariant (or at least covariant, in terms of two-vectors) character for these latter models. The models thereby obtained are interesting inasmuch as they are no more constrained by the time-independent ordering that characterizes the standard CM model on the line: particles can now go around each other. Hence this extension generally yields many-body models, whose Hamiltonian and solvable characters are preserved, but which feature richer dynamics.
Historical notes
The first paper introducing and solving, in a quantal context, the CM system on the line with arbitrary N is [C1971] (for N=3 this problem had been solved a bit earlier [see(C1969)]; the analogous model on the circle – or equivalently with the rational (inverse-square) potential in (1) replaced by the trigonometric (inverse-sine-square) potential, see (9) – was treated, essentially simultaneously and certainly independently, by Bill Sutherland [cf.(S1971)]. Jürgen Moser [cf.(M1975)] proved that the CM model is, in the classical context, integrable indeed solvable, for arbitrary N (for N=3 this problem had been solved somewhat earlier by Carlo Marchioro, and to some extent – much earlier – by Carl Jacobi [cf.(J1866)] – although Jacobi's contribution had been forgotten and was only rediscovered later, by Askol'd Perelomov). Legend has it that Moser was initially sceptical about the conjectures on the classical behavior of the CM system proffered in [C1971]: his input was then instrumental to prove their validity. His contribution fitted into the bloom of investigations on integrable systems following the discovery by Morikazu Toda of the first instance of integrable one-dimensional many-body problem (with nearest-neighbor exponential interactions) [cf.(T1967)] and by Martin Kruskal and his collaborators of the solvable character of the Korteweg-de Vries equation [cf.(GGKM1967)], and profited (as most of the subsequent investigations) from the fundamental contribution by Peter Lax (the notion of "Lax pairs" [cf.(L1968)]). The early papers related to the CM model began to label it as "Calogero Moser" or (especially in the trigonometric variant) by using the label "Sutherland" (alone or in conjunction with other names). The extension to elliptic interactions was introduced, in the classical context, in [C1975]; this involved the introduction and solution of certain functional equations, which subsequently gave rise to a mathematical trend of its own [cf.(BC1990)]. In this same paper [cf.(C1975)] the model featuring particles of different types (see above) was also introduced. The more general results on the CM model with an external (in fact quartic) potential are due to Vladimir Inozemtsev and Stefan Wojciechowski. The original idea of the extension to other root systems is due to Misha Olshanetsky and Askol'd Perelomov, who also wrote two seminal review papers on the classical and quantal aspects of these models [cf.(OP1981), (OP1983)]; many contributed to subsequent developments, including Ryu Sasaki. Another seminal paper for mathematical developments was (KKS1978). The generalization from CM to RS ("Ruijsenaars-Schneider" or "relativistic") many-body models was introduced in [RS1986] and pursued by Simon Ruijsenaars and others in many subsequent papers. For the treatment of isochronous systems see the book with this title listed below.
Apologies are offered to all those who made important contributions to developments connected with the CM model and are not mentioned here: they are too many to fit into a Scholarpedia entry.
Acknowledgements
It is a pleasure to thank Mario Bruschi, Jean-Pierre Françoise and Matteo Sommacal for having accepted to be co-curators of this Scholarpedia entry and for helpful feedbacks on this article.
References
[BC1990] M. Bruschi and F. Calogero: General analytic solution of certain functional equations of addition type, SIAM J. Math. Anal. 21(1990), 1019-1030
[C1969] F. Calogero: Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196
[C1971] F. Calogero: Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436 ("Erratum", ibidem 37 (1996), 3646)
[C1975] F. Calogero: Exactly solvable one dimensional many-body problems, Lett. Nuovo Cimento 13 (1975), 411-416
[C1978] F. Calogero: Motion of Poles and Zeros of Special Solutions of Nonlinear and Linear Partial Differential Equations, and Related "Solvable" Many Body Problems, Nuovo Cimento 43B (1978), 177-241
[CS2002] F. Calogero and M. Sommacal: Periodic solutions of a system of complex ODEs. II. Higher periods, J. Nonlinear Math. Phys. 9 (2002), 483-516
[vDV2000] J. F. van Diejen and L. Vinet, editors: Calogero-Moser-Sutherland Models, CRM Series in Mathematical Physics, Springer, New York (2000)
[E2005] P. Etingof: Lectures on Calogero-Moser Systems, delivered at the ETH (Zurich) in the spring and summer 2005, MIT preprint, http://www-math.mit.edu/~etingof/zlecnew.pdf.
[GGKM1967] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M Miura: Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095-1097
[J1866] C. Jacobi: Problema trium corporum mutuis attractionibus cubis distantiarium inverse proportionalibus recta linea se moventium, in Gesammelte Werke, Berlin, 4 (1866), 533-539
[KKS1978] D. Kazhdan, B. Kostant and S. Sternberg: Hamiltonian group actions and dynamical systems of Calogero type, Commun. Pure Appl. Math. 31 (1978), 481-507
[L1968] P. D. Lax: Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 28 (1968) 141-188
[M1975] J. Moser: Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220
[OP1981] M. A. Olshanetsky and A. M. Perelomov: Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), 313-400
[OP1983] M. A. Olshanetsky and A. M. Perelomov: Quantum integrable systems related to Lie algebras, Phys. Rep. 94(1983), 313-404
[RS1986] S. N. M. Ruijsenaars and H. Schneider: A new class of integrable systems and its relation to solitons, Ann. Phys. (NY) 170(1986), 370-405
[S1971] B. Sutherland: Exact results for a quantum many-body problem in one dimension. I & II", Phys. Rev. A4 (1971), 2019-2021 (1971) & A5(1972), 1372-1376
[T1967] M. Toda: Vibration of a chain with a nonlinear interaction, J. Phys. Soc. Japan 22 (1967), 431-436
Internal references
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Martin Gutzwiller (2007) Quantum chaos. Scholarpedia, 2(12):3146.
- David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
Recommended Reading
- Perelomov, A. M. 1990. Integrable systems of classical mechanics and Lie algebras. Birkhauser, Basel.
- Hoppe, J. 1992. Lectures on Integrable Systems. Springer, Berlin.
- Calogero, F. 2001. Classical many-body problems amenable to exact treatments, Lecture Notes in Physics Monograph 'm66'. Springer, Berlin.
- Babelon, O., Bernard D. and Talon M. 2003. Introduction to classical integrable systems. Cambridge University Press, Cambridge.
- Etingof, P. 2007. Calogero-Moser Systems and Representation Theory. Zurich Lectures in Advanced Mathematics, European Mathematical Society Publishing House.
- Calogero, F. 2008. Isochronous systems. Oxford University Press, Oxford.