Supersymmetric path integrals

The supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the Wiener integral. It is then pushed forward to a compact Riemannian manifold by means of a Malliavin-type construction. The relation to index theory is discussed.Research supported by an NSF postdoctoral fellowship     

Feynman path integrals

Path integrals techniques are derived from a new definition [1] of Feynman path integrals. These techniques are used to establish that Feynman-Green functions for a given physical system are covariances of pseudomeasures suitable for its path integrals. The variance of a pseudomeasure is a more versatile tool than the Feynman-Green function it defines.     

Theory of extreme correlations using canonical Fermions and path integrals

The  t$t$–J$J$  model is studied using a novel and rigorous mapping of the Gutzwiller projected electrons, in terms of canonical electrons. The mapping has considerable similarity to the Dyson–Maleev transformation relating spin operators to canonical Bosons. This representation gives rise to a non Hermitian quantum theory, characterized by minimal redundancies. A path integral representation of the canonical theory is given. Using it, the salient results of the extremely correlated Fermi liquid (ECFL) theory, including the previously found Schwinger equations of motion, are easily rederived. Further, a transparent physical interpretation of the previously introduced auxiliary Greens function and the ‘caparison factor’, is obtained.     

Numerical calculations in elementary quantum mechanics using Feynman path integrals

We show that it is possible to do numerical calculations in elementary quantum mechanics using Feynman path integrals. Our method involves discretizing both time and space, and summing paths through matrix multiplication. We give numerical fesults for various one-dimensional potentials. The calculations of energy levels and wavefunctions take approximately 100 times longer than with standard methods, but there are other problems for which such an approach should be more efficient.     

Thermal ionization in hydrogen plasma simulated using Feynman path integrals

Thermal ionization of hydrogen at temperatures on the order of 10⁴–10⁵ K and densities within 10²⁴–10²⁸ m^?3 has been simulated using Feynman path integrals. This method has been realized for the first time under conditions of a statistical ensemble with fluctuating volume. Multidimensional integrals have been calculated using the Monte Carlo simulation method that was preliminarily tested numerically on a problem of the quantum ground state of a confined hydrogen atom, which admits analytical solution. The position of isolines of the degree of ionization has been determined on the p-T plane of plasma states. The spatial correlation functions for electrons and nuclei are calculated, and the quantum effects in behavior of the electron component are evaluated. It is shown that, owing to the presence of strong Coulomb interactions, plasma retains a substantially quantum character in a broad domain of thermodynamic states, where a formal use of the degeneracy criterion predicts a classical regime. A basically exact stochastic method is developed for calculating the equilibrium kinetic energy of a spatially bounded system of quantum particles free of the dispersion divergence.     

Ordinary superconductivity and path integrals

Summary Functional methods are very powerful in dealing with ordinary superconductivity. The ring geometry is discussed in mean field by means of a Higgs-type Ginzburg-Landau Lagrangian. The presence of a junction in the ring, as in SQUIDs, leads to a θ-vacuum as the ground state. The variable θ is related to the phase difference of the order parameter at the junction and Josephson relations are obtained semi-classically. The system is, to any purpose, in two space dimensions, what can imply exotic statistics. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Jackiw transformation in path integrals

We show that the Jackiw transformation, ωτ=tan^-1(ωt), $\text{R}\text{=r(1+ω}{}^2\text{t}{}^2\text{)}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}$, converts the path integral for a charged particle in a magnetic monopole field into that for a charged oscillator in the same field. We also see this by an exactly soluble example.     

Functional equations for path integrals

We consider the density matrices that arise in the statistical mechanics of the electron-phonon systems. In the path integral representation the phonon coordinates can be eliminated. This leads to an action that depends on pairs of points on a path, that depends explicitly on time differences, and that contains the phonon occupation numbers. The integral is reduced to a standard form by scaling to the thermal length. We use the technique of integration by parts and add specially chosen generating functionals to the action. We set down functional derivative equations for the source-dependent density matrix and for the mass operator. This allows us to develop a series of approximations for the operator in terms of exact propagators. The crudest approximation is a coherent potential approximation applicable at a general temperature.     

Mayer sampling: calculation of cluster integrals using free-energy perturbation methods

Free-energy simulation methods are applied toward the calculation of cluster integrals that appear in diagrammatic methods of statistical mechanics. In this approach, Monte Carlo sampling is performed on a number of molecules equal to the order of the integral, and configurations are weighted according to the absolute value of the integrand. An umbrella-sampling average yields the value of the cluster integral in reference to a known integral. Virial coefficients, up to the sixth for the Lennard-Jones model and the fifth for the SPCE model of water, are calculated as a demonstration.     

Systematically accelerated convergence of path integrals

We present a new analytical method that systematically improves the convergence of path integrals of a generic N-fold discretized theory. Using it we calculate the effective actions S(p) for p< or =9, which lead to the same continuum amplitudes as the starting action, but that converge to that continuum limit as 1/N(p). We checked this derived speedup in convergence by performing Monte Carlo simulations on several different models.     

Space-time transformations in radial path integrals

Nonlinear space-time transformations in the radial path integral are discussed. A transformation formula is derived, which relates the original path integral to the Green function of a new quantum system with an effective potential containing an observable quantum correction ≈?². As an example the formula is applied to spherical brownian motion.     

Brownian motion calculus via path integrals

A stochastic calculus is derived by a path integral expansion of the conditional probability density about the most probable path for a fluctuation. The new term in the stochastic calculus is related to the dispersion in fluctuations from the most probable path and this can be expressed in terms of the excess entropy.     

Exact calculation of one-dimensional irreducible integrals

One-dimensional irreducible integrals (_k) are computed in the form of Mayerf-function polynomials for a general interparticle potential. Obeisance to the exact specification of the irreducible integral definition produces regularities in the interaction of star graphs with the integration process. Tables of _k fork 5 and test solutions are presented.     

Efficient numerical evaluation of Feynman integrals

Feynman loop integrals are a key ingredient for the calculation of higher order radiation effects, and are responsible for reliable and accurate theoretical prediction. We improve the efficiency of numerical integration in sector decomposition by implementing a quasi-Monte Carlo method associated with the CUDA/GPU technique. For demonstration we present the results of several Feynman integrals up to two loops in both Euclidean and physical kinematic regions in comparison with those obtained from FIESTA3. It is shown that both planar and non-planar two-loop master integrals in the physical kinematic region can be evaluated in less than half a minute with O(10~(-3))accuracy, which makes the direct numerical approach viable for precise investigation of higher order effects in multiloop processes, e.g. the next-to-leading order QCD effect in Higgs pair production via gluon fusion with a finite top quark mass.