Marvin Weinstein

A BRST PRIMER

We develop a Hamiltonian formulation of the BRST method for quantizing constrained systems. The rigid rotor is studied in detail and the similarity of this simple quantum system to a gauge theory is explicitly demonstrated. The system is quantized as a gauge theory and then the similarity between BRST and the Gupta-Bleuler approach is displayed. We also apply our formalism to true gauge theories. Both Abelian and non-Abelian gauge theories are studied in detail. Finally, the Hamiltonian treatment of the relativistic and the spinning relativistic particle is presented.

Contractor renormalization group technology and exact Hamiltonian real-space renormalization group transformations

The COntractor REnormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is presented. The method defines a systematic and nonperturbative means of implementing Kadanoff-Wilson real-space renormalization group transformations using cluster expansion and contraction techniques. We illustrate the approach and demonstrate its effectiveness using scalar field theory, the Heisenberg antiferromagnetic chain, and the anisotropic Ising chain. Future applications to the Hubbard and t-J models and lattice gauge theory are discussed.

Dynamic quantum clustering: A Method for visual exploration of structures in data

A given set of data points in some feature space may be associated with a Schrödinger equation whose potential is determined by the data. This is known to lead to good clustering solutions. Here we extend this approach into a full-fledged dynamical scheme using a time-dependent Schrödinger equation. Moreover, we approximate this Hamiltonian formalism by a truncated calculation within a set of Gaussian wave functions (coherent states) centered around the original points. This allows for analytic evaluation of the time evolution of all such states opening up the possibility of exploration of relationships among data points through observation of varying dynamical distances among points and convergence of points into clusters. This formalism may be further supplemented by preprocessing such as dimensional reduction through singular-value decomposition or feature filtering.

Contractor renormalization group method: A new computational technique for lattice systems

The Contractor Renormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is introduced. The method combines contraction and variational techniques with the real-space renormalization group approach. It applies to lattice systems of infinite extent and is ideal for studying phase structure and critical phenomena. The CORE approximation is systematically improvable to any desired degree of accuracy. It is complementary to standard Monte Carlo methods and incorporating dynamical fermions presents no problems. The method is tested using the 1+ l-dimensional Ising model. Submitted to Physical Review Letters. * Work supported by the Department of Energy, contract DE–AC03–76SFO0515. -.

Contractor renormalization group and the Haldane conjecture

The Contractor Renormalization group formalism (CORE) is a real-space renormalization group method which is the Hamiltonian analogue of the Wilson exact renormalization group equations. In an earlier paper\cite{QGAF} I showed that the Contractor Renormalization group (CORE) method could be used to map a theory of free quarks, and quarks interacting with gluons, into a generalized frustrated Heisenberg antiferromagnet (HAF) and proposed using CORE methods to study these theories. Since generalizations of HAFs exhibit all sorts of subtle behavior which, from a continuum point of view, are related to topological properties of the theory, it is important to know that CORE can be used to extract this physics. In this paper I show that despite the folklore which asserts that all real-space renormalization group schemes are necessarily inaccurate, simple Contractor Renormalization group (CORE) computations can give highly accurate results even if one only keeps a small number of states per block and a few terms in the cluster expansion. In addition I argue that even very simple CORE computations give a much better qualitative understanding of the physics than naive renormalization group methods. In particular I show that the simplest CORE computation yields a first principles understanding of how the famous Haldane conjecture works for the case of the spin-1/2 and spin-1 HAF.

p6 - Chiral Resonating Valence Bonds in the Kagome Antiferromagnet

The Kagome Heisenberg antiferromagnet is mapped onto an effective Hamiltonian on the star superlattice by Contractor Renormalization. Comparison of ground state energies on large lattices to Density Matrix Renormalization Group justifies truncation of effective interactions at range 3. Within our accuracy, magnetic and translational symmetries are not broken (i.e. a spin liquid ground state). However, we discover doublet spectral degeneracies which signal the onset of p6 - chirality symmetry breaking. This is understood by simple mean field analysis. Experimentally, the p6 chiral order parameter should split the optical phonons degeneracy near the zone center. Addition of weak next to nearest neighbor coupling is discussed.

ON THE EVOLUTION OF A MASSLESS SCALAR FIELD IN A SCHWARZSCHILD BACKGROUND: A NEW LOOK AT HAWKING RADIATION AND THE INFORMATION PARADOX

We exhibit an explicit foliation of Schwarzschild space–time by spacelike hypersurfaces which extend from Schwarzschild r=0 to r=∞. This allows us to compute the values of a massless scalar field for all space–time points which lie in the future of the surface on which we initially quantize the theory. This is to be contrasted with approaches which start at past null infinity and propagate to future null infinity. One of its virtues is that this method allows us to discuss both asymptotic Hawking radiation and what is happening at finite distances from the black hole. In order to explain the techniques we use, we begin by discussing variants of the flat-space moving mirror9 problem. Then we discuss the canonical quantization of the massless scalar field theory and the geometric optics approximation which we use to solve the Heisenberg equations of motion in the black hole background. Using the example of an infalling mirror, an analogue of the moving mirror problem, we show that, although our spacelike slices extend to r=0, we can avoid discussing an initial state which extends through the horizon. Furthermore, we show that in the same way we avoid having to deal with the singularity at r=0 when we first quantize the system. This discussion naturally leads to a suggestion of how to handle the question of what is happening when the mirror hits the singularity. In the last section of the paper we discuss a discretization of the computation which behaves in the manner we suggest and yet exhibits Hawking radiation.5 This formulation of the problem allows us to discuss all the issues in an explicitly unitary setting. The resulting picture raises some interesting questions about the information paradox.

Detection of nuclear sources in search survey using dynamic quantum clustering of gamma-ray spectral data

In a search scenario, nuclear background spectra are continuously measured in short acquisition intervals with a mobile detector-spectrometer. Detecting sources from measured data is difficult because of low signal-to-noise ratio (S/N of spectra, large and highly varying background due to naturally occurring radioactive material (NORM), and line broadening due to limited spectral resolution of nuclear detector. We have invented a method for detection of sources using clustering of spectral data. Our method takes advantage of the physical fact that a source not only produces counts in the region of its spectral emission, but also has the effect on the entire detector spectrum via Compton continuum. This allows characterizing the low S/N spectrum without distinct isotopic lines using multiple data features. We have shown that noisy spectra with low S/N can be grouped by overall spectral shape similarity using a data clustering technique called Dynamic Quantum Clustering (DQC). The spectra in the same cluster can then be averaged to enhance S/N of the isotopic spectral line. This would allow for increased accuracy of isotopic identification and lower false alarm rate. Our method was validated in a proof-of-principle study using a data set of spectra measured in one-second intervals with sodium iodide detector. The data set consisted of over 7000 spectra obtained in urban background measurements, and approximately 70 measurements of 137Cs and 60Co sources. Using DQC analysis, we have observed that all spectra containing 137Cs and 60Co signal cluster away from the background.

Exploring contractor renormalization : Perspectives and tests on the two-dimensional Heisenberg antiferromagnet

Contractor Renormalization (CORE) is a numerical renormalization method for Hamiltonian systems that has found applications in particle and condensed matter physics. There have been few studies, however, on further understanding of what exactly it does and its convergence properties. The current work has two main objectives. First, we wish to investigate the convergence of the cluster expansion for a two-dimensional Heisenberg Antiferromagnet(HAF). This is important because the linked cluster expansion used to evaluate this formula non-perturbatively is not controlled by a small parameter. Here we present a study of three different blocking schemes which reveals some surprises and in particular, leads us to suggest a scheme for defining successive terms in the cluster expansion. Our second goal is to present some new perspectives on CORE in light of recent developments to make it accessible to more researchers, including those in Quantum Information Science. We make some comparison to entanglement-based approaches and discuss how it may be possible to improve or generalize the method.Contractor renormalization CORE is a numerical renormalization method for Hamiltonian systems that has found applications in particle and condensed matter physics. There have been few studies, however, on further understanding of what exactly it does and its convergence properties. The current work has two main objectives. First, we wish to investigate the convergence of the cluster expansion for a two-dimensional Heisenberg antiferromagnet. This is important because the linked cluster expansion used to evaluate this formula nonperturbatively is not controlled by a small parameter. Here we present a study of three different blocking schemes which reveals some surprises and, in particular, leads us to suggest a scheme for defining successive terms in the cluster expansion. Our second goal is to present some new perspectives on CORE in light of recent developments to make it accessible to more researchers, including those in quantum information science. We make some comparison to entanglement-based approaches and discuss how it may be possible to improve or generalize the method.

CORE: Frustrated Magnets, Charge Fractionalization, and QCD

Abstract.I explain how to use a simple method to extract the physics of lattice Hamiltonian systems which are not easily analyzed by exact or other numerical methods. I will then use this method to establish the relationship between QCD and a special class of generalized, highly frustrated anti-ferromagnets.