From Classical to Quantum Mechanics: "How to translate physical ideas into mathematical language"

In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the question of a "true quantization" with "h" must be seen as an independent problem not directly related with quantum formalism. Moreover, this non-standard formulation of Classical Mechanics exhibits a new kind of operation with no classical counterpart: this operation is related to the "quantization process", and we show why quantization physically depends on group theory (Galileo group). This analytical procedure of quantization replaces the "correspondence principle" (or canonical quantization) and allows to map Classical Mechanics into Quantum Mechanics, giving all operators of Quantum Mechanics and Schrodinger equation. Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We find also that this approach gives a natural semi-classical formalism: some exact quantum results are obtained only using classical-like formula. So this procedure has the nice property of enlightening in a more comprehensible way both logical and analytical connection between classical and quantum pictures.Comment: 47 page

Superconformal field theories from IIB spectroscopy on AdS5×T11AdS_5\times T^{11}

We report on tests of the AdS/CFT correspondence that are made possible by complete knowledge of the Kaluza-Klein mass spectrum of type IIB supergravity on $AdS_5 \times T^{11}$ with T^{11}=SU(2)^2/U(1). After briefly discussing general multiplet shortening conditions in SU(2,2|1) and PSU(2,2|4), we compare various types of short SU(2,2|1) supermultiplets on AdS_5 and different families of boundary operators with protected dimensions. The supergravity analysis predicts the occurrence in the SCFT at leading order in N and g_s N, of extra towers of long multiplets whose dimensions are rational but not protected by supersymmetry.Comment: 11 pages, To appear in the proceedings of the STRINGS '99 conference, Potsdam (Germany), 19-25 July 199

Closedness of star products and cohomologies

We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.Comment: 16 page

Symplectic connections and Fedosov's quantization on supermanifolds

A (biased and incomplete) review of the status of the theory of symplectic connections on supermanifolds is presented. Also, some comments regarding Fedosov's technique of quantization are made.Comment: Submitted to J. of Phys. Conf. Se

Finite N Index and Angular Momentum Bound from Gravity

We exactly compute the finite N index and BPS partition functions for N=4 SYM theory in a newly proposed maximal angular momentum limit. The new limit is not predicted from the superconformal algebra, but naturally arises from the supergravity dual. We show that the index does not receive any finite N corrections while the free BPS partition function does.Comment: 14 pages, v2: minor revisions, published versio

Anterior cruciate ligament reconstruction with bone-patellar tendon-bone autograft versus allograft in young patients

Objectives: Traditionally, bone-patella tendon-bone (BTB) autograft has been the gold standard graft choice for younger, athletic patients requiring ACL reconstruction. However, donor site morbidity, post-operative patella fracture, and increased operative time have led many surgeons to choose BTB allograft for their reconstructions. Opponents of allografts feel that slower healing time, higher rate of graft failure, and potential for disease transmission makes them undesirable graft choices in athletic patients. The purpose of this study is to evaluate the clinical outcomes, both subjective and objective, of young patients that who have undergone either BTB autograft or allograft reconstructions with a minimum of 2-year follow-up. Methods: One hundred and twenty patients (60 autograft, 60 allograft), age 25 and below at time of surgery, were contacted after being retrospectively identified as patients having an ACL reconstruction with either a BTB allograft or autograft by one senior surgeon. Patients were administered the Lysholm Knee Scoring Scale and IKDC Subjective Knee Evaluation questionnaires. Fifty (25 BTB autograft and 25 BTB allograft) of the 120 returned for physical examination as well as completion of a single leg hop test and laxity evaluation using a KT-1000 arthrometer evaluation. Of the 120 patients contacted, there were a total of 7 failures (5.8%) requiring revision, 6 in the allograft group (86%) and 1 in the autograft group (14%). Results: The average Lysholm scores were 89.0 and 89.56 and the average IKDC scores were 90.8 and 92.1 in the autograft and allograft groups respectively. The differences in the Lysholm scores and the IKDC scores were not significant. The single leg hop and KT-1000 scores were also not significantly different. One autograft patient had a minor motion deficit. Three allograft patients had a grade 1 Lachman and pivot glide. One autograft patient and two allograft patients had mild patellafemoral crepitus. There was no significant difference in anterior knee pain between the two groups Conclusion: There is no significant difference in patient-rated outcome between ACL reconstructions using BTB autografts versus allografts. However, the overall study group did reveal an increased failure rate requiring revision in the allograft group. © The Author(s) 2015

Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation

We consider a generalized Dirac-Fock type evolution equation deduced from no-photon Quantum Electrodynamics, which describes the self-consistent time-evolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by Chaix-Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989), and recently established by Hainzl-Lewin-Sere, we prove the existence of global-in-time solutions of the considered evolution equation.Comment: 12 pages; more explanations added, some final (minor) corrections include

Realizations of the Lie superalgebra q(2) and applications

The Lie superalgebra q(2) and its class of irreducible representations V_p of dimension 2p (p being a positive integer) are considered. The action of the q(2) generators on a basis of V_p is given explicitly, and from here two realizations of q(2) are determined. The q(2) generators are realized as differential operators in one variable x, and the basis vectors of V_p as 2-arrays of polynomials in x. Following such realizations, it is observed that the Hamiltonian of certain physical models can be written in terms of the q(2) generators. In particular, the models given here as an example are the sphaleron model, the Moszkowski model and the Jaynes-Cummings model. For each of these, it is shown how the q(2) realization of the Hamiltonian is helpful in determining the spectrum.Comment: LaTeX file, 15 pages. (further references added, minor changes in section 5