Bell Inequalities in Four Dimensional Phase Space and the Three Marginal Theorem

We address the classical and quantum marginal problems, namely the question of simultaneous realizability through a common probability density in phase space of a given set of compatible probability distributions. We consider only distributions authorized by quantum mechanics, i.e. those corresponding to complete commuting sets of observables. For four-dimensional phase space with position variables qi and momentum variables pj, we establish the two following points: i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2) and (p1,p2), there does not always exist a positive phase space density rho({qi},{pj}) reproducing them as marginals; this settles a long standing conjecture; it is achieved by first deriving Bell-like inequalities in phase space which have their own theoretical and experimental interest. ii) given instead at most three compatible probabilities, there always exist an associated phase space density rho({qi},{pj}); the solution is not unique and its general form is worked out. These two points constitute our ``three marginal theorem''.Comment: 21 pages, Latex, no figure

The quantum anharmonic oscillator in the Heisenberg picture and multiple scale techniques

Multiple scale techniques are well-known in classical mechanics to give perturbation series free from resonant terms. When applied to the quantum anharmonic oscillator, these techniques lead to interesting features concerning the solution of the Heisenberg equations of motion and the Hamiltonian spectrum.Comment: 18 page

Marginal distributions in (2N)(\bf 2N)-dimensional phase space and the quantum (N+1)(\bf N+1) marginal theorem

We study the problem of constructing a probability density in 2N-dimensional phase space which reproduces a given collection of $n$ joint probability distributions as marginals. Only distributions authorized by quantum mechanics, i.e. depending on a (complete) commuting set of $N$ variables, are considered. A diagrammatic or graph theoretic formulation of the problem is developed. We then exactly determine the set of ``admissible'' data, i.e. those types of data for which the problem always admits solutions. This is done in the case where the joint distributions originate from quantum mechanics as well as in the case where this constraint is not imposed. In particular, it is shown that a necessary (but not sufficient) condition for the existence of solutions is $n\leq N+1$. When the data are admissible and the quantum constraint is not imposed, the general solution for the phase space density is determined explicitly. For admissible data of a quantum origin, the general solution is given in certain (but not all) cases. In the remaining cases, only a subset of solutions is obtained.Comment: 29 pages (Work supported by the Indo-French Centre for the Promotion of Advanced Research, Project Nb 1501-02). v2 to add a report-n

Joint Probabilities Reproducing Three EPR Experiments On Two Qubits

An eight parameter family of the most general nonnegative quadruple probabilities is constructed for EPR-Bohm-Aharonov experiments when only 3 pairs of analyser settings are used. It is a simultaneous representation of 3 Bohr-incompatible experimental configurations valid for arbitrary quantum states.Comment: Typo corrected in abstrac

Exchange operator formalism for N-body spin models with near-neighbors interactions

We present a detailed analysis of the spin models with near-neighbors interactions constructed in our previous paper [Phys. Lett. B 605 (2005) 214] by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials.Comment: RevTeX, 31 pages, no figures; important additional conten

Bell Inequalities in Phase Space and their Violation in Quantum Mechanics

We derive ``Bell inequalities'' in four dimensional phase space and prove the following ``three marginal theorem'' for phase space densities $\rho(\overrightarrow{q},\overrightarrow{p})$, thus settling a long standing conjecture : ``there exist quantum states for which more than three of the quantum probability distributions for $(q_1,q_2)$, $(p_1,p_2)$, $(q_1,p_2)$ and $(p_1,q_2)$ cannot be reproduced as marginals of a positive $\rho(\overrightarrow{q},\overrightarrow{p})$''. We also construct the most general positive $\rho(\overrightarrow{q},\overrightarrow{p})$ which reproduces any three of the above quantum probability densities for arbitrary quantum states. This is crucial for the construction of a maximally realistic quantum theory.Comment: 11 pages, latex, no figure

Infrared Fixed Point Structure in Minimal Supersymmetric Standard Model with Baryon and Lepton Number Violation

We study in detail the renomalization group evolution of Yukawa couplings and soft supersymmetry breaking trilinear couplings in the minimal supersymmetric standard model with baryon and lepton number violation. We obtain the exact solutions of these equations in a closed form, and then depict the infrared fixed point structure of the third generation Yukawa couplings and the highest generation baryon and lepton number violating couplings. Approximate analytical solutions for these Yukawa couplings and baryon and lepton number violating couplings, and the soft supersymmetry breaking couplings are obtained in terms of their initial values at the unification scale. We then numerically study the infrared fixed surfaces of the model, and illustrate the approach to the fixed points.Comment: 16 pages REVTeX, figures embedded as epsfigs, replaced with version to appear in Physical Review D, minor typographical errors eliminated and references reordered, figures correcte

Analytical Study of Non-Universality of the Soft Terms in the MSSM

We obtain general analytical forms for the solutions of the one-loop renormalization group equations in the top/bottom/$\tau$ sector of the MSSM. These solutions are valid for any value of $\tan \beta$ as well as any non-universal initial conditions for the soft SUSY breaking parameters and non-unification of the Yukawa couplings. We establish analytically a generic screening effect of non-universality, in the vicinity of the infrared quasi fixed point, which allows to determine sector-wise a hierarchy of sensitivity to initial conditions. We give also various numerical illustrations of this effect away from the quasi fixed point and assess the sensitivity of the Higgs and sfermion spectra to the non-universality of the various soft breaking sectors. As a by-product, a typical anomaly-mediated non-universality of the gaugino sector would have marginal influence on the scalar spectrum.Comment: Latex, 18 pages, 3 figure