A Riccati type PDE for light-front higher helicity vertices

This paper is based on a curious observation about an equation related to the tracelessness constraints of higher spin gauge fields. The equation also occurs in the theory of continuous spin representations of the Poincar\'e group. Expressed in an oscillator basis for the higher spin fields, the equation becomes a non-linear partial differential operator of the Riccati type acting on the vertex functions. The consequences of the equation for the cubic vertex is investigated in the light-front formulation of higher spin theory. The classical vertex is completely fixed but there is room for off-shell quantum corrections.Comment: 27 pages. Updated to published versio

Towards Unifying Structures in Higher Spin Gauge Symmetry

This article is expository in nature, outlining some of the many still incompletely understood features of higher spin field theory. We are mainly considering higher spin gauge fields in their own right as free-standing theoretical constructs and not circumstances where they occur as part of another system. Considering the problem of introducing interactions among higher spin gauge fields, there has historically been two broad avenues of approach. One approach entails gauging a non-Abelian global symmetry algebra, in the process making it local. The other approach entails deforming an already local but Abelian gauge algebra, in the process making it non-Abelian. In cases where both avenues have been explored, such as for spin 1 and 2 gauge fields, the results agree (barring conceptual and technical issues) with Yang-Mills theory and Einstein gravity. In the case of an infinite tower of higher spin gauge fields, the first approach has been thoroughly developed and explored by M. Vasiliev, whereas the second approach, after having lain dormant for a long time, has received new attention by several authors lately. In the present paper we briefly review some aspects of the history of higher spin gauge fields as a backdrop to an attempt at comparing the gauging vs. deforming approaches. A common unifying structure of strongly homotopy Lie algebras underlying both approaches will be discussed. The modern deformation approach, using BRST-BV methods, will be described as far as it is developed at the present time. The first steps of a formulation in the categorical language of operads will be outlined. A few aspects of the subject that seems not to have been thoroughly investigated are pointed out.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Liberal Arts Inspired Mathematics: A Report OR How to bring cultural and humanistic aspects of mathematics to the classroom as effective teaching and learning tools

This is the report of a project on ways of teaching university-level mathematics in a humanistic way. The main part of the project recounted here involved a journey to the United States during the fall term of 2012 to visit several liberal arts colleges in order to study and discuss mathematics teaching. Several themes that came up during my conversations at these colleges are discussed in the text: the invisibility of mathematics in everyday life, the role of calculus in American mathematics curricula, the is algebra necessary?\u27\u27 discussion, teaching mathematics as a language, the transfer problem in learning, and the relationship between humanistic mathematics and mathematics as taught in liberal arts contexts

Counterterms in Gravity in the Light-Front Formulation and a D=2 Conformal-like Symmetry in Gravity

In this paper we discuss gravity in the light-front formulation (light-cone gauge) and show how possible counterterms arise. We find that Poincare invariance is not enough to find the three-point counterterms uniquely. Higher-spin fields can intrude and mimic three-point higher derivative gravity terms. To select the correct term we have to use the remaining reparametrization invariance that exists after the gauge choice. We finally sketch how the corresponding programme for N=8 Supergravity should work.Comment: 26 pages, references added, published versio

Patients with shoulder impingement remain satisfied 6 years after arthroscopic subacromial decompression: A prospective study of 46 patients

Background Although arthroscopic subacromial decompression (ASD) is a common procedure for treatment of shoulder impingement, few long term results have been published. In this prospective study, we determined whether the high degree of patient satisfaction at 6 months postoperatively reported by us earlier remained at the 6-year follow-up. Patients and methods We originally reported high patient satisfaction 6 months after ASD for shoulder impingement in 50 prospectively studied patients using the Disability of the Arm Shoulder and Hand questionnaire (DASH) and the Visual Analog Scale (VAS). Patients with associated shoulder disorders were excluded. The surgeons were experienced shoulder arthroscopists. 6 years after surgery, the DASH questionnaire and the VAS were sent to these 50 patients. 2 patients had other medical problems of the upper extremity that affected the DASH and VAS scores, 1 patient was lost to follow-up, and another refused to participate. Thus, 46 patients with a mean age of 55 (33-78) years were included in this 6-year evaluation. Results The considerable improvement in both the DASH score and the VAS that was observed 6 months after surgery persisted or had even improved 6 years after surgery. Interpretation Properly selected patients with shoulder impingement treated with ASD remain satisfied 6 years after surgery

Structure of Higher Spin Gauge Interactions

In a previous paper, higher spin gauge field theory was formulated in an abstract way, essentially only keeping enough machinery to discuss "gauge invariance" of an "action". The approach could be thought of as providing an interface (or syntax) towards an implementation (or semantics) yet to be constructed. The structure then revealed turns out to be that of a strongly homotopy Lie algebra. In the present paper, the framework will be connected to more conventional field theoretic concepts. The Fock complex vertex operator implementation of the interactions in the BRST-BV formulation of the theory will be elaborated. The relation between the vertex order expansion and homological perturbation theory will be clarified. A formal non-obstruction argument is reviewed. The syntactically derived sh-Lie algebra structure is semantically mapped to the Fock complex implementation and it is shown that the recursive equations governing the higher order vertices are reproduced. Global symmetries and subsidiary conditions are discussed and as a result the tracelessness constraints are discarded. Thus all equations needed to compute the vertices to any order are collected. The framework is general enough to encompass all possible interaction terms. Finally, the abstract framework itself will be strengthened by showing that it can be naturally phrased in terms of the theory of categories.Comment: A few changes and additions made in the Introduction. Three references added. Typos corrected. Text agrees with published version in J. Math. Phys. except for minor journal specific proof-reading changes. 61 page