arXiv subjects

Gerald Guralnik

Publications and source records attributed to Gerald Guralnik.

12 recordsLinked to original sources

arXiv2019

Holography, Fractals and the Weyl Anomaly

We study the large source asymptotics of the generating functional in quantum field theory using the holographic renormalization group, and draw comparisons with the asymptotics of the Hopf characteristic function in fractal geometry. Based on the asymptotic behavior, we find a correspondence relating the Weyl anomaly and the fractal dimension of the Euclidean path integral measure. We are led to propose an equivalence between the logarithmic ultraviolet divergence of the Shannon entropy of this measure and the integrated Weyl anomaly, reminiscent of a known relation between logarithmic divergences of entanglement entropy and a central charge. It follows that the information dimension associated with the Euclidean path integral measure satisfies a c-theorem.

hep-th↗

arXiv2014

On exact statistics and classification of ergodic systems of integer dimension

We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This definition generalizes the construction of Hamiltonian systems by a Hamiltonian and a symplectic form. Some low dimensional examples are given, as well as a discretized field theory with a large number of degrees of freedom and a local nearest neighbor interaction. We also evaluate unequal-time correlations of these systems without direct numerical simulation, by Padé approximants of a short-time expansion. We briefly speculate on the possibility of constructing chaotic dynamical systems with non-integer dimension and exactly known statistics. In this case there is no probability density, suggesting an alternative construction in terms of a Hopf characteristic function and a two-form.

nlin.CD↗

arXiv2012

On the Asymptotics of the Hopf Characteristic Function

We study the asymptotic behavior of the Hopf characteristic function of fractals and chaotic dynamical systems in the limit of large argument. The small argument behavior is determined by the moments, since the characteristic function is defined as their generating function. Less well known is that the large argument behavior is related to the fractal dimension. While this relation has been discussed in the literature, there has been very little in the way of explicit calculation. We attempt to fill this gap, with explicit calculations for the generalized Cantor set and the Lorenz attractor. In the case of the generalized Cantor set, we define a parameter characterizing the asymptotics which we show corresponds exactly to the known fractal dimension. The Hopf characteristic function of the Lorenz attractor is computed numerically, obtaining results which are consistent with Hausdorff or correlation dimension, albeit too crude to distinguish between them.

nlin.CD↗

arXiv2011

Gauge Invariance and the Goldstone Theorem

This manuscript was originally created for and printed in the "Proceedings of seminar on unified theories of elementary particles" held in Feldafing Germany from July 5 to 16 1965 under the auspices of the Max-Planck-Institute for Physics and Astrophysics in Munich. It details and expands upon the Guralnik, Hagen, and Kibble paper that shows that the Goldstone theorem does not require physical zero mass particles in gauge theories and provides an example through the model which has become the template for the unified electroweak theory and a main component of the Standard Model.

hep-th↗

arXiv2011

Dynamics of the chiral phase transition from AdS/CFT duality

We use Lorentzian signature AdS/CFT duality to study a first order phase transition in strongly coupled gauge theories which is akin to the chiral phase transition in QCD. We discuss the relation between the latent heat and the energy (suitably defined) of the component of a D-brane which lies behind the horizon at the critical temperature. A numerical simulation of a dynamical phase transition in an expanding, cooling Quark-Gluon plasma produced in a relativistic collision is carried out.

hep-th↗

arXiv2009

Effective Potential for Complex Langevin Equations

We construct an effective potential for the complex Langevin equation on a lattice. We show that the minimum of this effective potential gives the space-time and Langevin time average of the complex Langevin field. The loop expansion of the effective potential is matched with the derivative expansion of the associated Schwinger-Dyson equation to predict the stationary distribution to which the complex Langevin equation converges.

hep-lat↗

arXiv2008

Remarks on Power Spectra of Chaotic Dynamical Systems

We develop novel methods to compute auto-correlation functions, or power spectral densities, for chaotic dynamical systems generated by an inverse method whose starting point is an invariant distribution and a two-form. In general, the inverse method makes some aspects of chaotic dynamics calculable by methods familiar in quantum field theory. This approach has the numerical advantage of being amenable to Monte-Carlo parallel computation. We demonstrate the approach on a specific example, and show how auto-correlation functions can be computed without any direct numerical simulation, by Pade approximants of a short time expansion.

nlin.CD↗

arXiv2008

Complex Langevin Equations and Schwinger-Dyson Equations

Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger-Dyson equations of the associated quantum field theory. Specific examples in zero dimensions and on a lattice are given. Relevance to the study of quantum field theory phase space is discussed.

hep-th↗

arXiv2007

Complexified Path Integrals and the Phases of Quantum Field Theory

The path integral by which quantum field theories are defined is a particular solution of a set of functional differential equations arising from the Schwinger action principle. In fact these equations have a multitude of additional solutions which are described by integrals over a complexified path. We discuss properties of the additional solutions which, although generally disregarded, may be physical with known examples including spontaneous symmetry breaking and theta vacua. We show that a consideration of the full set of solutions yields a description of phase transitions in quantum field theories which complements the usual description in terms of the accumulation of Lee-Yang zeroes. In particular we argue that non-analyticity due to the accumulation of Lee-Yang zeros is related to Stokes phenomena and the collapse of the solution set in various limits including but not restricted to, the thermodynamic limit. A precise demonstration of this relation is given in terms of a zero dimensional model. Finally, for zero dimensional polynomial actions, we prove that Borel resummation of perturbative expansions, with several choices of singularity avoiding contours in the complex Borel plane, yield inequivalent solutions of the action principle equations.

hep-th↗

arXiv2000

Fermions, Gauge Theories, and the Sinc Function Representation for Feynman Diagrams

We extend our new approach for numeric evaluation of Feynman diagrams to integrals that include fermionic and vector propagators. In this initial discussion we begin by deriving the Sinc function representation for the propagators of spin-1/2 and spin-1 fields and exploring their properties. We show that the attributes of the spin-0 propagator which allowed us to derive the Sinc function representation for scalar field Feynman integrals are shared by fields with non-zero spin. We then investigate the application of the Sinc function representation to simple QED diagrams, including first order corrections to the propagators and the vertex.

hep-ph↗

arXiv1999

Fast Evaluation of Feynman Diagrams

We develop a new representation for the integrals associated with Feynman diagrams. This leads directly to a novel method for the numerical evaluation of these integrals, which avoids the use of Monte Carlo techniques. Our approach is based on based on the theory of generalized sinc ($\sin(x)/x$) functions, from which we derive an approximation to the propagator that is expressed as an infinite sum. When the propagators in the Feynman integrals are replaced with the approximate form all integrals over internal momenta and vertices are converted into Gaussians, which can be evaluated analytically. Performing the Gaussians yields a multi-dimensional infinite sum which approximates the corresponding Feynman integral. The difference between the exact result and this approximation is set by an adjustable parameter, and can be made arbitrarily small. We discuss the extraction of regularization independent quantities and demonstrate, both in theory and practice, that these sums can be evaluated quickly, even for third or fourth order diagrams. Lastly, we survey strategies for numerically evaluating the multi-dimensional sums. We illustrate the method with specific examples, including the the second order sunset diagram from quartic scalar field theory, and several higher-order diagrams. In this initial paper we focus upon scalar field theories in Euclidean spacetime, but expect that this approach can be generalized to fields with spin.

hep-ph↗

arXiv1999

The Sinc Function Representation and Three-Loop Master Diagrams

We test the Sinc function representation, a novel method for numerically evaluating Feynman diagrams, by using it to evaluate the three-loop master diagrams. Analytical results have been obtained for all these diagrams, and we find excellent agreement between our calculations and the exact values. The Sinc function representation converges rapidly, and it is straightforward to obtain accuracies of 1 part in 10^6 for these diagrams and with longer runs we found results better than 1 part in 10^{12}. Finally, this paper extends the Sinc function representation to diagrams containing massless propagators.

hep-ph↗