Operator product expansion algebra

We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean $\varphi^{4}$-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of arXiv:1105.3375, that the 3-point OPE, $< O_{A_1} O_{A_2} O_{A_3} > = \sum_{C} \cal{C}_{A_1 A_2 A_3}^{C}$, usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity $\cal{C}_{A_1 A_2 A_3}^{B}=\sum_{C}\cal{C}_{A_1 A_2}^{C} \cal{C}_{C A_3}^{B}$ is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for non-coinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such `consistency conditions' and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.Comment: 53 pages, v2: typos removed, minor corrections, v3: typos removed, minor changes in introduction, v4: Note added in proo

The Paradox of the Drug Elimination Program in New York City Public Housing

In this study, we examine the effects of the DEP intervention at three levels of complementary theoretical and practical relevance: the public housing development itself, the neighborhood in which public housing is situated, and the police precinct where the tract is located. From surveys of residents, observations of program activities, and analyses of NYCHA\u27s program records, we compiled detailed information on the components of DEP and the reactions of public housing residents to each type of intervention. We then analyzed panel data from 1985-1996 to estimate the effects of DEP on crime rates in and around the city\u27s public housing projects before and after the implementation of DEP. We used alternate measures of DEP interventions to provide robust estimates of the contributions of DEP to the decline in the city\u27s crime rates beginning in 1991, a year after the onset of DEP. The results show that while DEP efforts were linked to declines in crime rates in the census tracts and police precincts surrounding public housing, there were no measurable declines in crime in the public housing projects themselves. We draw lessons from theories of procedural justice and social norms to explain why harsh treatment of public housing residents by police and prosecutors may have generated resistance among public housing residents to the legal norms that were the focus of DEP efforts. We begin in Section I with a detailed description of the DEP Program implemented by the New York City Housing Authority. Section II provides an overview of NYCHA\u27s Drug Elimination Program as implemented in the City\u27s public housing projects. Section III discusses the details of the research protocols and analytic methods. Section IV presents the results for each of the three spatial dimensions of program effects: the projects themselves, the census tracts where public housing developments are sited, and the police precincts where DEP and other law enforcement strategies were implemented and managed. Section V discusses lessons for the theory, policy, and practice of drug control

The Bustle of Horses on a Ship: Drug Control in New York City Public Housing

For decades, violence, drugs and public housing have been closely linked in political culture and popular imagination. In 1990, the Department of Housing and Urban Development (HUD) made funds available to public housing authorities to combat drug and crime problems. This program, the Drug Elimination Program (DEP) combined several strategies under one administrative umbrella: police enforcement, drug treatment, drug prevention, youth and gang outreach, community organizing, integrated health and social service agencies, and tenant mobilization projects. In New York, the Housing Authority spent $165 million on DEP in its 330 public housing sites between 1990 and 1996. Yet there has been little research on this large investment, either in New York City or nationwide. In this study, we examined the effects of the DEP intervention at three levels of complementary theoretical relevance: the public housing development itself, the neighborhood in which public housing is situated, and the police precinct that surrounds each public housing project. We used spatial analyses and hierarchical regressions to estimate DEP effects on drug and crime in public housing sites and their surrounding neighborhoods. We show that crime and drug problems were reduced significantly in the immediate neighborhoods and police precincts surrounding the public housing sites, but crime and drug problems in public housing sites were unaffected by DEP interventions. The absence of effects within public housing reflects the details of the DEP strategies, with its disproportionate allocation of funds to policing strategies compared to demand reduction and informal social control programs. DEP police efforts were nominally focused on public housing sites, but in reality were diffused in the NYPD\u27s broader administrative units to provide resources that benefited law enforcement generally. Seen this way, DEP was an important and strategically valuable supplement to the NYPD\u27s strategic response to a particularly acute violence and crime epidemic, but did little to alter the basic social organization of crime and drugs within public housing sites. We argue for an intervention model that promotes collective action between residents and legal actors, interactions that promote citizen compliance and cooperation with police. The police depend heavily on the voluntary cooperation of citizens to fight crime, and DEP created disincentives for cooperation. This social norms approach would invite policing of drug problems in the context of a legitimacy-focused approach that promotes citizen-based regulation of crime and disorder

Observation of Motion Dependent Nonlinear Dispersion with Narrow Linewidth Atoms in an Optical Cavity

As an alternative to state-of-the-art laser frequency stabilisation using ultra-stable cavities, it has been proposed to exploit the non-linear effects from coupling of atoms with a narrow transition to an optical cavity. Here we have constructed such a system and observed non-linear phase shifts of a narrow optical line by strong coupling of a sample of strontium-88 atoms to an optical cavity. The sample temperature of a few mK provides a domain where the Doppler energy scale is several orders of magnitude larger than the narrow linewidth of the optical transition. This makes the system sensitive to velocity dependent multi-photon scattering events (Dopplerons) that affect the cavity field transmission and phase. By varying the number of atoms and the intra-cavity power we systematically study this non-linear phase signature which displays roughly the same features as for much lower temperature samples. This demonstration in a relatively simple system opens new possibilities for alternative routes to laser stabilization at the sub 100 mHz level and superradiant laser sources involving narrow line atoms. The understanding of relevant motional effects obtained here has direct implications for other atomic clocks when used in relation with ultranarrow clock transitions.Comment: 9 pages (including 4 pages of Supplemental Information), 6 figures. Updated to correspond to the published versio

Non-linear Spectroscopy of Sr Atoms in an Optical Cavity for Laser Stabilization

We study the non-linear interaction of a cold sample of strontium-88 atoms coupled to a single mode of a low finesse optical cavity in the so-called bad cavity limit and investigate the implications for applications to laser stabilization. The atoms are probed on the weak inter-combination line \lvert 5s^{2} \, ^1 \textrm{S}_0 \rangle \,-\, \lvert 5s5p \, ^3 \textrm{P}_1 \rangle at 689 nm in a strongly saturated regime. Our measured observables include the atomic induced phase shift and absorption of the light field transmitted through the cavity represented by the complex cavity transmission coefficient. We demonstrate high signal-to-noise-ratio measurements of both quadratures - the cavity transmitted phase and absorption - by employing FM spectroscopy (NICE-OHMS). We also show that when FM spectroscopy is employed in connection with a cavity locked to the probe light, observables are substantially modified compared to the free space situation where no cavity is present. Furthermore, the non-linear dynamics of the phase dispersion slope is experimentally investigated and the optimal conditions for laser stabilization are established. Our experimental results are compared to state-of-the-art cavity QED theoretical calculations.Comment: 7 pages, 4 figure