264,190 research outputs found

    Finding Exponential Product Formulas of Higher Orders

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    In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves important symmetries of the system dynamics. We focuse on two algorithms of constructing higher-order exponential product formulas. The first is the fractal decomposition, where we construct higher-order formulas recursively. The second is to make use of the quantum analysis, where we compute higher-order correction terms directly. As interludes, we also have described the decomposition of symplectic integrators, the approximation of time-ordered exponentials, and the perturbational composition.Comment: 22 pages, 9 figures. To be published in the conference proceedings ''Quantum Annealing and Other Optimization Methods," eds. B.K.Chakrabarti and A.Das (Springer, Heidelberg

    Optimized Negative Dimensional Integration Method (NDIM) and multiloop Feynman diagram calculation

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    We present an improved form of the integration technique known as NDIM (Negative Dimensional Integration Method), which is a powerful tool in the analytical evaluation of Feynman diagrams. Using this technique we study a ϕ3ϕ4% \phi ^{3}\oplus \phi ^{4} theory in D=42ϵD=4-2\epsilon dimensions, considering generic topologies of LL loops and EE independent external momenta, and where the propagator powers are arbitrary. The method transforms the Schwinger parametric integral associated to the diagram into a multiple series expansion, whose main characteristic is that the argument contains several Kronecker deltas which appear naturally in the application of the method, and which we call diagram presolution. The optimization we present here consists in a procedure that minimizes the series multiplicity, through appropriate factorizations in the multinomials that appear in the parametric integral, and which maximizes the number of Kronecker deltas that are generated in the process. The solutions are presented in terms of generalized hypergeometric functions, obtained once the Kronecker deltas have been used in the series. Although the technique is general, we apply it to cases in which there are 2 or 3 different energy scales (masses or kinematic variables associated to the external momenta), obtaining solutions in terms of a finite sum of generalized hypergeometric series de 1 and 2 variables respectively, each of them expressible as ratios between the different energy scales that characterize the topology. The main result is a method capable of solving Feynman integrals, expressing the solutions as hypergeometric series of multiplicity (n1)(n-1) , where nn is the number of energy scales present in the diagram.Comment: 49 pages, 14 figure

    Monopole Condensation in Lattice SU(2) QCD

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    This is the short review of Monte-Carlo studies of quark confinement in lattice QCD. After abelian projections both in the maximally abelian and Polyakov gauges, it is seen that the monopole part alone is responsible for confinement. A block spin transformation on the dual lattice suggests that lattice SU(2)SU(2) QCD is always ( for all β\beta) in the monopole condensed phase and so in the confinement phase in the infinite volume limit.Comment: Contribution to Confinement '95, March 1995, Osaka, Japan. Names of figure files are corrected. 8 page uuencoded latex file and 10 ps figure

    Generalization of the Lie-Trotter Product Formula for q-Exponential Operators

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    The Lie-Trotter formula eA^+B^=limN(eA^/NeB^/N)Ne^{\hat{A}+\hat{B}} = \lim_{N\to \infty} (e^{\hat{A}/N} e^{\hat{B}/N})^N is of great utility in a variety of quantum problems ranging from the theory of path integrals and Monte Carlo methods in theoretical chemistry, to many-body and thermostatistical calculations. We generalize it for the q-exponential function eq(x)=[1+(1q)x](1/(1q))e_q (x) = [1+ (1-q) x]^{(1/(1-q))} (with e1(x)=exe_1(x)=e^x), and prove eq(A^+B^+(1q)[A^B^+B^A^]/2)=limN[e1(1q)N(A^/N)][e1(1q)N(B^/N)]Ne_q(\hat{A}+\hat{B}+(1-q) [\hat{A}\hat{B}+\hat{B}\hat{A}] /2) = \lim_{N\to \infty} {[e_{1-(1-q)N}(\hat{A}/N)] [e_{1-(1-q)N}(\hat{B}/N)]}^N. This extended formula is expected to be similarly useful in the nonextensive situationsComment: 5 pages, no figure

    Momentum distribution and correlation of two-nucleon relative motion in 6^6He and 6^6Li

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    The momentum distribution of relative motion between two nucleons gives information on the correlation in nuclei. The momentum distribution is calculated for both 6^{6}He and 6^6Li which are described in a three-body model of α\alpha+NN+NN. The ground state solution for the three-body Hamiltonian is obtained accurately using correlated basis functions. The momentum distribution depends on the potential model for the NN-NN interaction. With use of a realistic potential, the 6^6He momentum distribution exhibits a dip around 2 fm1^{-1} characteristic of SS-wave motion. In contrast to this, the 6^6Li momentum distribution is very similar to that of the deuteron; no dip appears because it is filled with the DD-wave component arising from the tensor force.Comment: 14 pages, 9 figure