99,763 research outputs found

    Quantization of the Nonlinear Sigma Model Revisited

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    We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional assumptions on the target homogeneous space, in agreement with the work of Friedan. We carry out some explicit computations for the O(N)O(N)-model. Finally, we show how a suitable notion of the renormalization group establishes the Ricci flow as the one loop renormalization group flow of the nonlinear sigma model.Comment: 51 page

    Dense-choice Counter Machines revisited

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    This paper clarifies the picture about Dense-choice Counter Machines, which have been less studied than (discrete) Counter Machines. We revisit the definition of "Dense Counter Machines" so that it now extends (discrete) Counter Machines, and we provide new undecidability and decidability results. Using the first-order additive mixed theory of reals and integers, we give a logical characterization of the sets of configurations reachable by reversal-bounded Dense-choice Counter Machines

    Classically Time-Controlled Quantum Automata: Definition and Properties

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    In this paper we introduce classically time-controlled quantum automata or CTQA, which is a reasonable modification of Moore-Crutchfield quantum finite automata that uses time-dependent evolution and a "scheduler" defining how long each Hamiltonian will run. Surprisingly enough, time-dependent evolution provides a significant change in the computational power of quantum automata with respect to a discrete quantum model. Indeed, we show that if a scheduler is not computationally restricted, then a CTQA can decide the Halting problem. In order to unearth the computational capabilities of CTQAs we study the case of a computationally restricted scheduler. In particular we showed that depending on the type of restriction imposed on the scheduler, a CTQA can (i) recognize non-regular languages with cut-point, even in the presence of Karp-Lipton advice, and (ii) recognize non-regular languages with bounded-error. Furthermore, we study the closure of concatenation and union of languages by introducing a new model of Moore-Crutchfield quantum finite automata with a rotating tape head. CTQA presents itself as a new model of computation that provides a different approach to a formal study of "classical control, quantum data" schemes in quantum computing.Comment: Long revisited version of LNCS 11324:266-278, 2018 (TPNC 2018

    Finite-size scaling above the upper critical dimension revisited: The case of the five-dimensional Ising model

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    Monte Carlo results for the moments of the magnetization distribution of the nearest-neighbor Ising ferromagnet in a L^d geometry, where L (4 \leq L \leq 22) is the linear dimension of a hypercubic lattice with periodic boundary conditions in d=5 dimensions, are analyzed in the critical region and compared to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod. Phys. C (1998)]. We show that this finite-size scaling theory (formulated in terms of two scaling variables) can account for the longstanding discrepancies between Monte Carlo results and the so-called ``lowest-mode'' theory, which uses a single scaling variable tL^{d/2} where t=T/T_c-1 is the temperature distance from the critical temperature, only to a very limited extent. While the CD theory gives a somewhat improved description of corrections to the ``lowest-mode'' results (to which the CD theory can easily be reduced in the limit t \to 0, L \to \infty, tL^{d/2} fixed) for the fourth-order cumulant, discrepancies are found for the susceptibility (L^d ). Reasons for these problems are briefly discussed.Comment: 9 pages, 13 Encapsulated PostScript figures. To appear in Eur. Phys. J. B. Also available as PDF file at http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm

    The Fubini-Furlan-Rossetti Sum Rule Revisited

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    The Fubini-Furlan-Rossetti sum rule for pion photoproduction on the nucleon is evaluated by dispersion relations at constant t, and the corrections to the sum rule due to the finite pion mass are calculated. Near threshold these corrections turn out to be large due to pion-loop effects, whereas the sum rule value is closely approached if the dispersion integrals are evaluated for sub-threshold kinematics. This extension to the unphysical region provides a unique framework to determine the low-energy constants of chiral perturbation theory by global properties of the excitation spectrum.Comment: 12 pages, 7 postscript figures, EPJ style files include
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