arXiv subjects
Publications and source records attributed to Jacek Wosiek.
An Euclidean representation of bosonized Majorana fermions, prior to imposing constraints, is derived in three space-time dimensons. The difference with the standard three dimensional Ising system is epmhasized. The mild sign problem, does not preclude Monte Carlo simulations in intermediate volumes. Implementing constraints is briefly outlined.
We present a bosonization procedure which replaces fermions with generalized spin variables subject to local constraints. It requires that the number of Majorana modes per lattice site matches the coordination number modulo two. If this condition is not obeyed, then bosonization introduces additional fermionic excitations not present in the original model. In the case of one Majorana mode per site on a honeycomb lattice, we recover a sector of Kitaev's model. We discuss also decagonal and rectangular geometries and present bosonization of the Hubbard model. For geometries with a boundary we find that certain fermionic edge modes naturally emerge. They are of different nature than edge modes encountered in topological phases of matter. Euclidean representation for the unconstrained version of a spin system of the type arising in our construction is derived and briefly studied by computing some exact averages for small volumes.
Contrary to the common wisdom, local bosonizations of fermionic systems exist in higher dimensions. Interestingly, resulting bosonic variables must satisfy local constraints of a gauge type. They effectively replace long distance exchange interactions. In this work we study in detail the properties of such a system which was proposed a long time ago. In particular, dependence of the constraints on lattice geometry and fermion multiplicity is further elaborated and is now classified for all two dimensional, rectangular lattices with arbitrary sizes. For few small systems the constraints are solved analytically and the complete spectra of reduced spin hamiltonias are shown to agree with the original fermionic ones. The equivalence is extended to fermions in an external Wegner $Z_2$ field. It is also illustrated by an explicit calculation for a particular configuration of Wegner variables. Finally, a possible connection with the recently proposed web of dualities is discussed.
Many research programs aiming to deal with the sign problem were proposed since the advent of lattice field theory. Several of these try to achieve this by exploiting properties of analytic functions. This is also the case for our study. There auxiliary complex variables are introduced and desired weight is obtained after integrating them out. In this note we clarify conceptual difficulties with this procedure encountered in previous works. In the process we observe an exciting connection with thimbles and discover an interesting hidden symmetry present in the problem. Problem of negative eigenvalues of the action will be revisited and considered from a different perspective. As a byproduct we perform simulations of simple quantum systems directly in Minkowski time.
After a short review of one of proposals to avoid complex stochastic processes in Complex Langevin studies, the recent progress in the former is reported. In particular, the new developments allow now to construct positive and normalizable representations for gaussian quantum mechanical, as well as field theoretical, path integrals directly in the Minkowski time. A relation to the idea of thimbles is also discussed.
We study the problem of constructing positive representations of complex measures. In this paper we consider complex densities on a direct product of $U(1)$ groups and look for representations by probability distributions on the complexification of those groups. After identifying general necessary and sufficient conditions we propose several concrete realizations. Finally we study some of those realizations in examples representing problems in abelian lattice gauge theories.
A positive representation for a set of complex densities is constructed. In particular, complex measures on a direct product of U(1) groups are studied. After identifying general conditions which such representations should satisfy, several explicit realizations are proposed. Their utility is illustrated in few concrete examples representing problems in abelian lattice gauge theories.
A simple integral relation between a complex weight and the corresponding positive distribution is derived by introducing a second complex variable. Together with the positivity and normalizability conditions, this sum rule allows to construct explicitly equivalent pairs of distributions in simple cases. In particular the well known solution for a complex gaussian distribution is generalized to an arbitrary complex slope. This opens a possibility of positive representation of Feynman path integrals directly in the Minkowski time. Such construction is then explicitly carried through in the second part of this presentation. The continuum limit of the new representation exists only if some of the additional couplings tend to infinity and are tuned in a specific way. The approach is then successfully applied to three quantum mechanical examples including a particle in a constant magnetic field -- a simplest prototype of a Wilson line. Further generalizations are shortly discussed and an amusing interpretation of new variables is briefly mentioned.
By introducing a second complex variable, the integral relation between a complex density and the corresponding positive distribution is derived. Together with the positivity and normalizability conditions, this sum rule allows to construct explicitly equivalent pairs of distributions in simple cases discussed here. In particular the well known solution for a complex gaussian distribution is generalized to an arbitrary complex inverse dispersion parameter. This opens a possibility of positive representation of Feynman path integrals directly in the Minkowski time.
Recently found positive representation for an arbitrary complex, gaussian weight is used to construct a statistical formulation of gaussian path integrals directly in the Minkowski time. The positivity of Minkowski weights is achieved by doubling the number of real variables. The continuum limit of the new representation exists only if some of the additional couplings tend to infinity and are tuned in a specific way. The construction is then successfully applied to three quantum mechanical examples including a particle in a constant magnetic field -- a simplest prototype of a Wilson line. Further generalizations are shortly discussed and an intriguing interpretation of new variables is alluded to.
We discuss how to implement, in lattice gauge theories, external charges which are not commensurate with an elementary gauge coupling. It is shown that an arbitrary, real power of a standard Wilson loop (or Polyakov line) can be defined and consistently computed in lattice formulation of non-abelian, two dimensional gauge theories. However, such an observable can excite quantum states with integer fluxes only. Since the non-integer fluxes are not in the spectrum of the theory they cannot be created, no matter which observable is chosen. Also the continuum limit of above averages does not exist unless the powers in question are in fact integer. On the other hand, a new continuum limit exists, which is rather intuitive, and where above observables make perfect sense and lead to the string tension proportional to the square of arbitrary (non necessary commensurate with gauge coupling) charge.
In this Letter we point out that the analytic solution of the two dimensional U(1) gauge theory, on a finite lattice, reveals in the continuum limit the renowned Manton's spectrum of topological electric fluxes together with their effective hamiltonian and wave functions. We extend this result for the system with strings and external charges providing also a novel interpretation of the theta parameter. Some further generalizations are also outlined.
In this paper we present results of computations for the ground energy of weakly coupled double well potential in quantum mechanics. We give a numerical evidence for cancelation of imaginary contributions to energy coming from Borel resummation and multi-instanton terms. We also estimate several higher coefficients of the multi-instanton expansion which are not given in the literature.
We consider the light-cone (LC) gauge and LC quantization of the dimensional reduction of super Yang Mills theory from four to two dimensions. After integrating out all unphysical degrees of freedom, the non-local LC Hamiltonian exhibits an explicit ${\cal N}=(2,2)$ supersymmetry. A further SUSY-preserving compactification of LC-space on a torus of radius $R$, allows for a large-$N$ numerical study where the smooth large-$R$ limit of physical quantities can be checked. As a first step, we consider a simple, yet quite rich, "Coulomb approximation" that maintains an ${\cal N}=(1,1)$ subgroup of the original supersymmetry and leads to a non-trivial generalization of 't Hooft's model with an arbitrary --but conserved-- number of partons. We compute numerically the eigenvalues and eigenvectors both in momentum and in position space. Our results, so far limited to the sectors with 2, 3 and 4 partons, directly and quantitatively confirm a simple physical picture in terms of a string-like interaction with the expected tension among pairs of nearest-neighbours along the single-trace characterizing the large-$N$ limit. Although broken by our approximation, traces of the full ${\cal N}=(2,2)$ supersymmetry are still visible in the low-lying spectrum.
Supersymmetric Yang-Mills quantum mechanics (SYMQM) results from the dimensional reduction of the Yang-Mills field theory in $D$ space-time dimensions to a single point in the $D-1$ dimensional space. It can be also viewed as the effective quantum mechanics of zero momentum modes of the original theory. These systems were first considered in 80's as simple models with supersymmetry. Independently, zero-volume field theories (especially pure Yang-Mills) were employed as the starting point of the small volume expansion, which is an important theoretical tool complementary to early lattice calculations. Later the models attracted a new wave of interest after the hypothesis of the equivalence, between the $D=10, SU(\infty)$ SYMQM and a M-theory of D0 branes. In this talk some new results obtained for the D=4 system with SU(2) gauge symmetry will be presented.
A new, recursive method of calculating matrix elements of polynomial hamiltonians is proposed. It is particularly suitable for the recent algebraic studies of the supersymmetric Yang-Mills quantum mechanics in any dimensions. For the D=2 system with the SU(2) gauge group, considered here, the technique gives exact, closed expressions for arbitrary matrix elements of the hamiltonian and of the supersymmetric charge, in the occupation number representation. Subsequent numerical diagonalization provides the spectrum and restricted Witten index of the system with very high precision (taking into account up to $10^5$ quanta). Independently, the exact value of the restricted Witten index is derived analytically for the first time.
We present our recent results on the odderon intercept in perturbative QCD, obtained through the solution of the Baxter equation and investigation of the spectrum of the relevant constant of motion.
We present our recent results on the odderon intercept in perturbative QCD, obtained through the solution of the Baxter equation and investigation of the spectrum of the relevant constant of motion.