Stephen L. Adler

Bio: Stephen L. Adler is an academic researcher from Princeton University. The author has contributed to research in topics: Quantum field theory & Gauge theory. The author has an hindex of 43, co-authored 225 publications receiving 11925 citations. Previous affiliations of Stephen L. Adler include Institut des Hautes Études Scientifiques & Institute for Advanced Study.

Axial vector vertex in spinor electrodynamics

Abstract: Working within the framework of perturbation theory, we show that the axial-vector vertex in spinor electrodynamics has anomalous properties which disagree with those found by the formal manipulation of field equations. Specifically, because of the presence of closed-loop "triangle diagrams," the divergence of axial-vector current is not the usual expression calculated from the field equations, and the axial-vector current does not satisfy the usual Ward identity. One consequence is that, even after the external-line wave-function renormalizations are made, the axial-vector vertex is still divergent in fourth- (and higher-) order perturbation theory. A corollary is that the radiative corrections to ${\ensuremath{ u}}_{l}l$ elastic scattering in the local current-current theory diverge in fourth (and higher) order. A second consequence is that, in massless electrodynamics, despite the fact that the theory is invariant under ${\ensuremath{\gamma}}_{5}$ tranformations, the axial-vector current is not conserved. In an Appendix we demonstrate the uniqueness of the triangle diagrams, and discuss a possible connection between our results and the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ and $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ decays. In particular, we argue that as a result of triangle diagrams, the equations expressing partial conservation of axial-vector current (PCAC) for the neutral members of the axial-vector-current octet must be modified in a well-defined manner, which completely alters the PCAC predictions for the ${\ensuremath{\pi}}^{0}$ and the $\ensuremath{\eta}$ two-photon decays.

Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation

Abstract: We consider two simple field-theoretic models, (a) spinor electrodynamics and (b) the $\ensuremath{\sigma}$ model with the Polkinghorne axial-vector current, and show in each case that the axial-vector current satisfies a simple anomalous divergence equation exactly to all orders of perturbation theory. We check our general argument by an explicit calculation to second order in radiative corrections. The general argument is made tractable by introducing a cutoff, but to check the validity of this artifice, the second-order calculation is carried out entirely in terms of renormalized vertex and propagator functions, in which no cutoff appears.

Quaternionic quantum mechanics and quantum fields

Abstract: PART I: INTRODUCTION AND GENERAL FORMALISM 1: Introduction 2: General Framework of Quaternionic Quantum Mechanics 3: Further General Results in Quaternionic Quantum Mechanics PART II: NON-RELATIVISTIC QUATERNIONIC QUANTUM MECHANICS 4: One-Particle Quantum Mechanics--General Formalism 5: Stationary State Methods and Phase Methods 6: Scattering Theory and Bound States 7: Methods for Time-Development 8: Single Channel Time-Dependent Formal Scattering Theory 9: Multi-Particle and Multi-Channel Methods 10: Further Multi-Particle Topics PART III: RELATIVISTIC QUATERNIONIC QUANTUM MECHANICS 11: Relativistic Single Particle Wave Equations Spin-0 and Spin-1/2 12: More on Relativistic Wave Equations: The Spin-1 Gauge Potential, Lagrangian Formulations, and the Poincare Group 13: Quaternionic Quantum Field Theory 14: Outlook Appendix A: Proof of the Jacobi Identity for the Generalized Poisson Bracket Appendix B: Derivation of Gaussian Integral Formulas

Einstein Gravity as a Symmetry Breaking Effect in Quantum Field Theory

Abstract: This article gives a pedagogical review of recent work in which the Einstein-Hilbert gravitational action is obtained as a symmetry-breaking effect in quantum field theory. Particular emphasis is placed on the case of renormalizable field theories with dynamical scale-invariance breaking, in which the induced gravitational effective action is finite and calculable. A functional integral formulation is used throughout, and a detailed analysis is given of the role of dimensional regularization in extracting finite answers from formally quadratically divergent integrals. Expressions are derived for the induced gravitational constant, for the induced cosmological constant, and for quantized matter theories on a background manifold, and a strategy is outlined for computing the induced constants in the case of an SU(n) gauge theory. By use of the background field method, the formalism is extended to the case in which the metric is also quantized, yielding a derivation of the semiclassical Einstein equations as an approximation to quantum gravity, as well as general formulas for the induced (or renormalized) gravitational and cosmological constants.

Pattern Recognition and Machine Learning

Abstract: Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Information Theory, Inference and Learning Algorithms

Abstract: Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.