Duality and Hidden Symmetries in Interacting Particle Systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.     

Self-Duality of Markov Processes and Intertwining Functions

We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric linear combination of them, is the generator of a Markov process. The theorem is applied to a series of examples, including Markov processes with a discrete state space (e.g. interacting particle systems) and Markov processes with continuous state space (e.g. diffusion processes). In the examples we use explicit representations of Lie algebras that are unitarily equivalent. As a consequence, in the discrete setting self-duality functions are given by orthogonal polynomials whereas in the continuous context they are Bessel functions.     

Duality for Stochastic Models of Transport

We study three classes of continuous time Markov processes (inclusion process, exclusion process, independent walkers) and a family of interacting diffusions (Brownian energy process). For each model we define a boundary driven process which is obtained by placing the system in contact with proper reservoirs, working at different particle densities or different temperatures. We show that all the models are exactly solvable by duality, using a dual process with absorbing boundaries. The solution does also apply to the so-called thermalization limit in which particles or energy is instantaneously redistributed among sites. The results shows that duality is a versatile tool for analyzing stochastic models of transport, while the analysis in the literature has been so far limited to particular instances. Long-range correlations naturally emerge as a result of the interaction of dual particles at the microscopic level and the explicit computations of covariances match, in the scaling limit, the predictions of the macroscopic fluctuation theory.     

Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions

We consider quantum field theoretical models inn dimensional space-time given by interaction densities which are bounded functions of an ultraviolet cut-off boson field. Using methods of euclidean Markov field theory and of classical statistical mechanics, we construct the infinite volume imaginary and real time Wightman functions as limits of the corresponding quantities for the space cut-off models. In the physical Hilbert space, the space-time translations are represented by strongly continuous unitary groups and the generator of time translationsH is positive and has a unique, simple lowest eigenvalue zero, with eigenvector , which is the unique state invariant under space-time translations. The imaginary time Wightman functions and the infinite volume vacuum energy density are given as analytic functions of the coupling constant. The Wightman functions have cluster properties also with respect to space translations.     

3-dimensionalSU(2) lattice gauge theory in terms of gauge invariant variables

As a first step towards a duality transformation for theSU(2) lattice gauge theory in 3 dimensions, the integration over all gauge variant variables is performed explicitly after introducing gauge invariant auxiliary variables. The resulting new Hamiltonian is complex and involves a sum over closed loops. Each of these loops is confined to an elementary cube of a dual lattice. Like in a previous investigation for theO(4) symmetric Heisenberg ferromagnet Rühl's boson representation is used to derive the result.     

Correlation Inequalities for Interacting Particle Systems with Duality

We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other—a process which we call here the symmetric inclusion process (SIP)—or repel each other—a generalized version of the well-known symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction,—the so-called Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process, which is fermionic. Finally, we consider a boundary driven version of the SIP for which we prove duality and then obtain correlation inequalities.     

Identifying an extraZ 0 boson with a new model

According to the principle of minimality, we find a newSU(6) model. ThisSU(6) model, and other models, can be identified as a theoretical origin of an extra Z⁰ boson. We apply the strategy of Boudjemaet al. (BLRV) which is very effective in identifying the theoretical origin of an extra Z⁰ boson in the newSU(6) model, and compare the model with six other models.     

The Entropy Production of Diffusion Processes on Manifolds and Its Circulation Decompositions

In non-equilibrium statistical mechanics, the entropy production is used to describe flowing in or pumping out of the entropy of a time-dependent system. Even if a system is in a steady state (invariant in time), Prigogine suggested that there should be a positive entropy production if it is open. In 1979, the first author of this paper and Qian Min-Ping discovered that the entropy production describes the irreversibility of stationary Markov chains, and proved the circulation decomposition formula of the entropy production. They also obtained the entropy production formula for drifted Brownian motions on Euclidean space R ⁿ (see a report without proof in the Proc. 1st World Congr. Bernoulli Soc.). By the topological triviality of R ⁿ , there is no discrete circulation associated to the diffusion processes on $R^n$. In this paper, the entropy production formula for stationary drifted Brownian motions on a compact Riemannian manifold M is proved. Furthermore, the entropy production is decomposed into two parts – in addition to the first part analogous to that of a diffusion process on R ⁿ , some discrete circulations intrinsic to the topology of M appear! The first part is called the hidden circulation and is then explained as the circulation of a lifted process on M×S ¹ around the circle S ¹. The main result of this paper is the circulation decomposition formula which states that the entropy production of a stationary drifted Brownian motion on M is a linear sum of its circulations around the generators of the fundamental group of M and the hidden circulation. Received: 4 November 1998 / Accepted: 7 April 1999     

The Continuum Directed Random Polymer

Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter β, and for a given β and realization of the noise the path is a Markov process. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all β>0 and for almost all realizations of the white noise the path measure has the same Hölder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).     

Monte carlo simulation of a nucleon interacting with a neutral scalar boson field

A recently proposed Monte Carlo algorithm to solve a Schrödinger equation expressed in Fock-space representation, suitable for the case of hamiltonians describing problems in one-dimensional discrete momentum space, is now extended to the one-, two- and three-dimensional continuous k-spaces. This extension is tested by employing it for an analytically solvable hamiltonian. For this purpose the ‘static source’ limit of the hamiltonian corresponding to the interaction between a nucleon and a neutral, scalar boson field is simulated. The results of the Monte Carlo procedure reproduce very well the exact solution.     

Deductive inference of nuclear boson models (IBM, TQM, FQM) based on SU(6) dynamical symmetry

A single deductive inference of Schwinger realization (= interacting boson model—IBM), Holstein-Primakoff realization (= truncated quadrupole phonon model—TQM) and Dyson realization (= finite quadrupole phonon model—FQM) of dynamical SU(6) quadrupole collective algebra (QCA) is presented with a full scope of their isomorphism on the level of representations. Dyson realization of QCA is explicitly constructed by using holomorphically parametrized generalized coherent state and explicit form of root vectors. Utilizing appropriate orthogonalizing operators Holstein-Primakoff realization of QCA has been derived from the Dyson realization. The carrier spaces of Schwinger and Holstein-Primakoff realizations are investigated on the same footing and Marshalek's boson is rigorously derived. The intertwining operator which connects Schwinger and Holstein-Primakoff realizations is constructed and its domain and image are determined. It is shown that the intertwining operator has well-defined inverse in a definite factor space of the IBM basis space which is proved to be isomorphic to the physical subspace of the TQM basis space, meaning equivalence of IBM and TQM on level of representations.     

The<Emphasis Type="Italic">SU</Emphasis>(1,1) coherent states related to the affine group wavelets

The Perelomov coherent states ofSU(1,1) are labeled by elements of the quotient ofSU(1,1) by its rotation subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parameterize the coherent states by elements of that group. Such a formulation permits to find new properties of theSU(1,1) coherent states and to relate them to affine wavelets. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

Field representations of the conformal group with continuous mass spectrum

The discrete series of the conformal groupSU(2, 2) is realized on a Hilbert space of holomorphic functions over a bounded domain or the field theoretic tube domain. The boundary values of these functions form Hilbert spaces of distributions. For the realization over the tube domain the boundary distributions transform like classical spinorial fields with a continuous mass spectrum extending from zero to infinity. The reduction of these field realizations of the whole discrete series into unitary irreducible representations of the inhomogeneous Lorentz group is explicitly given.     

Electroweak Sudakov logarithms and real gauge-boson radiation in the TeV region

Electroweak radiative corrections give rise to large negative, double-logarithmically enhanced corrections in the TeV region. These are partly compensated by real radiation and, moreover, affected by selecting isospin-non-invariant external states. We investigate the impact of real gauge boson radiation more quantitatively by considering different restricted final state configurations. We consider successively a massive abelian gauge theory, a spontaneously broken SU(2) theory and the electroweak Standard Model. We find that details of the choice of the phase space cuts, in particular whether a fraction of collinear and soft radiation is included, have a strong impact on the relative amount of real and virtual corrections.     

Quantum Dynamical Algebra SU(1,1) in One-Dimensional Exactly Solvable Potentials

In this paper, we establish the underlying quantum dynamical algebra SU(1,1) for some one-dimensional exactly solvable potentials by using the shift operators method. The connection between SU(1,1) algebra and the radial Hamiltionian problems is also discussed. PACS numbers: 03.65.Ge     

Factorized Duality,Stationary Product Measures and Generating Functions

We find all self-duality functions of the form

$$\begin{aligned} D(\xi , \eta )= \prod _{x} d(\xi _x, \eta _x) \end{aligned}$$

for a class of interacting particle systems. We call these duality functions of simple factorized form. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion processes, as well as duality and self-duality functions for their continuous counterparts. The approach is based on, firstly, a general relation between factorized duality functions and stationary product measures and, secondly, an intertwining relation provided by generating functions. For the interacting particle systems, these self-duality and duality functions turn out to be generalizations of those previously obtained in Giardinà et al. (J Stat Phys 135:25–55, 2009) and, more recently, in Franceschini and Giardinà (Preprint, arXiv:1701.09115, 2016) . Thus, we discover that only these two families of dualities cover all possible cases. Moreover, the same method discloses all simple factorized self-duality functions for interacting diffusion systems such as the Brownian energy process, where both the process and its dual are in continuous variables.

     

The fluctuation-dissipation theorem for contracted descriptions of Markov processes

It is shown that if the Onsager-Casimir relations and the fluctuationdissipation theorem are valid for a stationary, Gaussian, Markov process in anN-dimensional space, then these relations are valid when the process is projected into a subspace of the original space. Both time-reversal-even and time-reversal-odd variables are allowed. Previous derivations of the fluctuation-dissipation theorem for Brownian motion from fluctuating hydrodynamics are special cases of the present result. For the Brownian motion problem, the fluctuation-dissipation theorem is proven for the case of a compressible, thermally conducting fluid with a nonlocal equation of state. Arbitrary slip boundary conditions are considered as well.     

Deformation of nuclei as a function of angular momentum in the U(6) ⊃ SU(3) model

In the framework of a hybrid rotational model, proposed recently by Moshinsky as a consequence of a comparison between the Gneuss and Greiner extension of the Bohr and Mottelson model and the interacting boson model, we study the shape of nuclei by calculating the average of the expectation value of the square of the deformation parameter β with respect to the rotational states with the same angular momentum belonging to a given irreducible representation of SU(3). This work generalises to three dimensions the corresponding analysis carried out in two dimensions by Chacón, Moshinsky, and Vanagas. We use the canonical chain for U(3), i.e., the chain U(6) ? U(3) ? U(2) ? U(1), to obtain an analytical formula for the quantity studied. We bring out the overall stretching effect of the angular momentum on the shape of nuclei. The influence of other parameters, such as the boson number and the irreducible representation of SU(3), is also studied.     

Duality relations for asymmetric exclusion processes

We derive duality relations for a class ofU _q [SU(2)]-symmetric stochastic processes, including among others the asymmetric exclusion process in one dimension. Like the known duality relations for symmetric hopping processes, these relations express certainm-point correlation functions inN-particle systems (Nm) in terms of sums of correlation functions of the same system but with onlym particles. For the totally asymmetric case we obtain exact expressions for some boundary density correlation functions. The dynamical exponent for these correlators isz=2, which is different from the dynamical exponent for bulk density correlations, which is known to bez=3/2.     

Stochastic evolution of Yang-Mills connections on the noncommutative two-torus

We study quantum stochastic parallel transport processes where the noise terms arise from quantum Brownian motion in Fock space and the connection is chosen to minimize the Yang-Mills functional on a Heisenberg module over the smooth algebra of the noncommutative two-torus. Each such process yields a dilation of a quantum dynamical semigroup whose action on components of the connection induces a family of transformations of the moduli space. From a physical point of view, this describes a highly singular interaction between quantized Yang-Mills fields and the free boson field.