Four-Fermion Theory and the Conformal Bootstrap

We employ the conformal bootstrap to re-examine the problem of finding the critical behavior of four-Fermion theory at its strong coupling fixed point. Existence of a solution of the bootstrap equations indicates self-consistency of the assumption that, in space-time dimensions less than four, the renormalization group flow of the coupling constant of a four-Fermion interaction has a nontrivial fixed point which is generally out of the perturbative regime. We exploit the hypothesis of conformal invariance at this fixed point to reduce the set of the Schwinger-Dyson bootstrap equations for four-Fermion theory to three equations which determine the scale dimension of the Fermion field $\psi$, the scale dimension of the composite field $\bar{\psi}\psi$ and the critical value of the Yukawa coupling constant. We solve the equations assuming this critical value to be small. We show that this solution recovers the fixed point for the four-fermion interaction with $N$-component fermions in the limit of large $N$ at (Euclidean) dimensions $d$ between two and four. We perform a detailed analysis of the $1/N$-expansion in $d=3$ and demonstrate full agreement with the conformal bootstrap. We argue that this is a useful starting point for more sophisticated computations of the critical indices.Comment: 31pp, text and figures both in Latex, UBCTP 92-3

N=2* Super-Yang-Mills Theory at Strong Coupling

The planar N=2* Super-Yang-Mills (SYM) theory is solved at large 't Hooft coupling using localization on S(4). The solution permits detailed investigation of the resonance phenomena responsible for quantum phase transitions in infinite volume, and leads to quantitative predictions for the semiclassical string dual of the N=2* theory.Comment: 34 pages, 9 figures; v2: the name of one author change

A new approach to understanding the frequency response of mineral oil

Dielectric spectroscopy is non-invasive diagnostic method and can give information about dipole relaxation, electrical conduction and structure of molecules. Since the creation of charge carriers in mineral oil is not only from dissociation but also injection from electrodes, the injection current cannot be simply ignored. The polarization caused by the charge injection has been studied in this paper. Based on our research, if the mobility of the injected charge carriers is fast enough so that they can reach the opposite electrode, the current caused by the injection will contribute only to the imaginary part of the complex permittivity and this part of the complex permittivity will decrease with the frequency with a slope of -1 which is in a good agreement with the experimental result. The classic ionic drift and diffusion model and this injection model will be combined to make an improved model. In this paper, the frequency responses of three different kinds of mineral oils have been measured, and this modified model has been used to simulate the experiment result. Since there is only one unknown parameter in this improved model, a better understanding of the frequency response in mineral oil can be achieve

Frequency domain state-space system identification

An algorithm for identifying state-space models from frequency response data of linear systems is presented. A matrix-fraction description of the transfer function is employed to curve-fit the frequency response data, using the least-squares method. The parameters of the matrix-fraction representation are then used to construct the Markov parameters of the system. Finally, state-space models are obtained through the Eigensystem Realization Algorithm using Markov parameters. The main advantage of this approach is that the curve-fitting and the Markov parameter construction are linear problems which avoid the difficulties of nonlinear optimization of other approaches. Another advantage is that it avoids windowing distortions associated with other frequency domain methods

A comparison of spike time prediction and receptive field mapping with point process generalized linear models, Wiener-Voltera kernels, and spike-triggered averaging methods

Poster presentation: Characterizing neuronal encoding is essential for understanding information processing in the brain. Three methods are commonly used to characterize the relationship between neural spiking activity and the features of putative stimuli. These methods include: Wiener-Volterra kernel methods (WVK), the spike-triggered average (STA), and more recently, the point process generalized linear model (GLM). We compared the performance of these three approaches in estimating receptive field properties and orientation tuning of 251 V1 neurons recorded from 2 monkeys during a fixation period in response to a moving bar. The GLM consisted of two formulations of the conditional intensity function for a point process characterization of the spiking activity: one with a stimulus only component and one with the stimulus and spike history. We fit the GLMs by maximum likelihood using GLMfit in Matlab. Goodness-of-fit was assessed using cross-validation with Kolmogorov-Smirnov (KS) tests based on the time-rescaling theorem to evaluate the accuracy with which each model predicts the spiking activity of individual neurons and for each movement direction (4016 models in total, for 251 neurons and 16 different directions). The GLMs that considered spike history of up to 35 ms, accurately predicted neuronal spiking activity (95% confidence intervals for KS test) with a performance of 97.0% (3895/4016) for the training data, and 96.5% (3876/4016) for the test data. If spike history was not considered, performance dropped to 73,1% in the training and 71.3% in the testing data. In contrast, the WVF and the STA predicted spiking accurately for 24.2% and 44.5% of the test data examples respectively. The receptive field size estimates obtained from the GLM (with and without history), WVF and STA were comparable. Relative to the GLM orientation tuning was underestimated on average by a factor of 0.45 by the WVF and the STA. The main reason for using the STA and WVF approaches is their apparent simplicity. However, our analyses suggest that more accurate spike prediction as well as more credible estimates of receptive field size and orientation tuning can be computed easily using GLMs implemented in Matlab with standard functions such as GLMfit

Kernels and point processes associated with Whittaker functions

This article considers Whittaker's function $W_{\kappa ,\mu }$ where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu-1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linear system with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. The Hankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight $w$. The operation of translating $\varphi$ is equivalent to multiplying $w$ by an exponential factor to give $w_\varepsilon$. The determinant of the Hankel matrix of moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e's transcendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski (Comm. Math. Phys. 211 (2000), 335--358)