The tetrahexahedric Calogero model

We consider the spherical reduction of the rational Calogero model (of type A _n-1, without the center of mass) as a maximally superintegrable quantum system. It describes a particle on the (n = 2)-sphere in a very special potential. A detailed analysis is provided of the simplest non-separable case, n = 4, whose potential blows up at the edges of a spherical tetrahexahedron, tesselating the two-sphere into 24 identical right isosceles spherical triangles in which the particle is trapped. We construct a complete set of independent conserved charges and of Hamiltonian intertwiners and elucidate their algebra. The key structure is the ring of polynomials in Dunkl-deformed angular momenta, in particular the subspaces invariant and antiinvariant under all Weyl reflections, respectively.     

The conformal points of the generalized Thirring model II

In the large-N limit, conditions for the conformal invariance of the generalized Thirring model are derived, using two different approaches: the background field method and the hamiltonian method based on an operator algebra, and the agreement between them is established. A free field representation of the relevant algebra is presented, and the structure of the stress tensor in terms of free fields (and free currents) is studied in detail.     

supersymmetric 3-particles Calogero model

We constructed the most general N=4 superconformal 3-particles systems with translation invariance. In the basis with decoupled center of mass the supercharges and Hamiltonian possess one arbitrary function which defines all potential terms. We have shown that with the proper choice of this function one may describe the standard, A₂ Calogero model as well as G₂ and BC₂ Calogero models, which, by construction, possess N=4 superconformal symmetry. The main property of all these systems is that even with the coupling constant equal to zero they still contain nontrivial interactions in the fermionic sector. In other words, there are infinitely many non-equivalent N=4 supersymmetric extensions of the free action depending on one arbitrary function. We also considered quantization and explicitly showed how the supercharges and Hamiltonian are modified. In the quantum case the constructed systems exhibit only invariance with respect to N=4 Poincaré supersymmetry.     

Multicenter Higgs oscillators and Calogero model

We show that the spherical part of N-particle Calogero model describes, after exclusion of the center of mass, the motion of the particle on (N − 2)-dimensional sphere interacting with N(N − 1)/2 force centers with Higgs oscillator potential. In the case of four-particle system these force centers are located at the vertexes of cuboctahedron. The geometry of the five-particle case is also investigated.     

Fun and frustration with Calogero model

We review some algebraical (oscillator) aspects of N-body single-species and multispecies Calogero models in one dimension. We treat them as a particular cases of deformed harmonic oscillators and discuss the corresponding Fock spaces. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

The spherical sector of the Calogero model as a reduced matrix model

We investigate the matrix-model origin of the spherical sector of the rational Calogero model and its constants of motion. We develop a diagrammatic technique which allows us to find explicit expressions of the constants of motion and calculate their Poisson brackets. In this way we obtain all functionally independent constants of motion to any given order in the momenta. Our technique is related to the valence-bond basis for singlet states.     

Deformed Heisenberg algebras, a Fock-space representation and the Calogero model

We describe generally deformed Heisenberg algebras in one dimension. The condition for a generalized Leibniz rule is obtained and solved. We analyze conditions under which deformed quantum-mechanical problems have a Fock-space representation. One solution of these conditions leads to a q-deformed oscillator already studied by Lorek et al., and reduces to the harmonic oscillator only in the infinite-momentum frame. The other solution leads to the Calogero model in ordinary quantum mechanics, but reduces to the harmonic oscillator in the absence of deformation. Received: 27 April 2000 / Published online: 8 September 2000     

Permutation invariant algebras, a Fock space realization and the Calogero model

We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the -extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of a mapping from the Calogero oscillators to the free Bose oscillators and vice versa. Received: 26 July 2001 / Revised version: 9 January 2002 / Published online: 12 April 2002     

Constants of motion of the four-particle Calogero model

We present the explicit expressions of the complete set of constants of motion of four-particle Calogero model with excluded center of mass, i.e. of the A ₃ rational Calogero model. Then we find the constants of motion of its spherical part, defining two-dimensional 12-center spherical oscillator, with the force centers located at the vertexes of cuboctahedron.     

Quantization in generalized coordinates III-Lagrangian formulation

It is shown that if one incorporates the generalized coordinate quantum velocitiesQ ¹ as given byQ ¹=l[H,Q ¹](h=1) into the generalized classical Lagrangian for a free particle (the total energy),L=1/2Q ¹ g _tk Q ^k one does not obtain (no matter what ordering of the operatorsq ^l ,q ^k andg _lkwe choose the correct quantum Lagrangian operator which is a transformation from -1/2V² to generalized coordinates (Gruber, 1971, 1972).q ^l as given byq ^l=i[H,q ^l] turns out to be the Hermitian part of a more generaiized operator which we call the total generalized velocity operator similar to the notation in ear previous articles (Gruber, 1971, 1972). This total velocity operator really determines the fundamental structure governing our system in the Lagrangian formulation. We show that ft is through the total velocity operator that we make the transition from classical to quantum mechanics and through our procedure we arrive at the correct quantum Lagrangian operator.     

Quantization of the Maxwell field in linear conformal invariant gauges

It is shown that the generalized Lorentz gauges provide all linear conformal invariant gauges, i.e. gauges such that A =0.     

Density waves in the Calogero model - revisited

The Calogero model bears, in the continuum limit, collective excitations in the form of density waves and solitary modulations of the density of particles. This sector of the spectrum of the model was investigated, mostly within the framework of collective-field theory, by several authors, over the past 15 years or so. In this work we shall concentrate on periodic solutions of the collective BPS-equation (also known as “finite amplitude density waves”), as well as on periodic solutions of the full static variational equations which vanish periodically (also known as “large amplitude density waves”). While these solutions are not new, we feel that our analysis and presentation add to the existing literature, as we explain in the text. In addition, we show that these solutions also occur in a certain two-family generalization of the Calogero model, at special points in parameter space. A compendium of useful identities associated with Hilbert transforms, including our own proofs of these identities, appears in Appendix A. In Appendix B we also elucidate in the present paper some fine points having to do with manipulating Hilbert-transforms, which appear ubiquitously in the collective field formalism. Finally, in order to make this paper self-contained, we briefly summarize in Appendix C basic facts about the collective field formulation of the Calogero model.     

Quantization,coherent states and geometric phases of a generalized nonstationary mesoscopic RLC circuit

We present an alternative quantum treatment for a generalized mesoscopic RLC circuit with time-dependent resistance, inductance and capacitance. Taking advantage of the Lewis and Riesenfeld quantum invariant method and using quadratic invariants we obtain exact nonstationary Schrödinger states for this electromagnetic oscillation system. Afterwards, we construct coherent and squeezed states for the quantized RLC circuit and employ them to investigate some of the system’s quantum properties, such as quantum fluctuations of the charge and the magnetic flux and the corresponding uncertainty product. In addition, we derive the geometric, dynamical and Berry phases for this nonstationary mesoscopic circuit. Finally we evaluate the dynamical and Berry phases for three special circuits. Surprisingly, we find identical expressions for the dynamical phase and the same formulae for the Berry’s phase.     

Quantization of conformal gravity: Another approach to the renormalization of gravity

The non-renormalizability of quantum gravity poses a great problem to the construction of any unified field theory of all known interactions. Normally, we start with a unitary theory of gravity and investigate its renormalization properties. This is the first of a series of papers where we start with the opposite approach, beginning with a renormalizable theory and investigating its unitarity structure. In particular, we study non-perturbative approaches to the quantization of conformal gravity. Using ADM coordinates, we perform the canonical quantization of the Weyl action C_μναβ², which is renormalizable and is also local scale invariant. Although this theory is certainly not unitary in perturbation theory, we speculate that unitarity may be restored when we approach this theory non-perturbatively, by examining the possibility of different phase transitions.     

Collective field formulation of the multi-species Calogero model and its duality symmetries

We study the collective field formulation of a restricted form of the multi-species Calogero model, in which the three-body interactions are set to zero. We show that the resulting collective field theory is invariant under certain duality transformations, which interchange, among other things, particles and antiparticles, and thus generalize the well known strong-weak coupling duality symmetry of the ordinary Calogero model. We identify all these dualities, which form an Abelian group, and study their consequences. We also study the ground state and small fluctuations around it in detail, starting with the two-species model, and then generalizing to an arbitrary number of species.