Multi-Dimensional Hermite Polynomials in Quantum Optics

We study a class of optical circuits with vacuum input states consisting of Gaussian sources without coherent displacements such as down-converters and squeezers, together with detectors and passive interferometry (beam-splitters, polarisation rotations, phase-shifters etc.). We show that the outgoing state leaving the optical circuit can be expressed in terms of so-called multi-dimensional Hermite polynomials and give their recursion and orthogonality relations. We show how quantum teleportation of photon polarisation can be modelled using this description.Comment: 10 pages, submitted to J. Phys. A, removed spurious fil

Accurate light-time correction due to a gravitating mass

This work arose as an aftermath of Cassini's 2002 experiment \cite{bblipt03}, in which the PPN parameter $\gamma$ was measured with an accuracy $\sigma_\gamma = 2.3\times 10^{-5}$ and found consistent with the prediction $\gamma =1$ of general relativity. The Orbit Determination Program (ODP) of NASA's Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression for the gravitational delay which differs from the standard formula; this difference is of second order in powers of $m$ -- the sun's gravitational radius -- but in Cassini's case it was much larger than the expected order of magnitude $m^2/b$, where $b$ is the ray's closest approach distance. Since the ODP does not account for any other second-order terms, it is necessary, also in view of future more accurate experiments, to systematically evaluate higher order corrections and to determine which terms are significant. Light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics; Fermat's action functional at its minimum is just the light-time between the two end points A and B. A new and powerful formulation is thus obtained. Asymptotic power series are necessary to provide a safe and automatic way of selecting which terms to keep at each order. Higher order approximations to the delay and the deflection are obtained. We also show that in a close superior conjunction, when $b$ is much smaller than the distances of A and B from the Sun, of order $R$, say, the second-order correction has an \emph{enhanced} part of order $m^2R/b^2$, which corresponds just to the second-order terms introduced in the ODP. Gravitational deflection of the image of a far away source, observed from a finite distance from the mass, is obtained to $O(m^2)$.Comment: 4 figure

The Maslov index and nondegenerate singularities of integrable systems

We consider integrable Hamiltonian systems in R^{2n} with integrals of motion F = (F_1,...,F_n) in involution. Nondegenerate singularities are critical points of F where rank dF = n-1 and which have definite linear stability. The set of nondegenerate singularities is a codimension-two symplectic submanifold invariant under the flow. We show that the Maslov index of a closed curve is a sum of contributions +/- 2 from the nondegenerate singularities it is encloses, the sign depending on the local orientation and stability at the singularities. For one-freedom systems this corresponds to the well-known formula for the Poincar\'e index of a closed curve as the oriented difference between the number of elliptic and hyperbolic fixed points enclosed. We also obtain a formula for the Liapunov exponent of invariant (n-1)-dimensional tori in the nondegenerate singular set. Examples include rotationally symmetric n-freedom Hamiltonians, while an application to the periodic Toda chain is described in a companion paper.Comment: 27 pages, 1 figure; published versio

Deconstructing Non-Abelian Gauge Theories at One Loop

Deconstruction of 5D Yang-Mills gauge theories is studied in next-to-leading order accuracy. We calculate one-loop corrections to the mass spectrum of the non-linear gauged sigma-model, which is the low energy effective theory of the deconstructed theory. Renormalization is carried out following the standard procedure of effective field theories. The relation between the radius of the compactified fifth dimension and the symmetry breaking scale of the non-linear sigma-model is modified by radiative corrections. We demonstrate that one can match the low lying spectrum of the gauge boson masses of the effective 4D gauged non-linear sigma-model to the Kaluza-Klein modes of the 5D theory at one-loop accuracy

A Scaling Method and its Applications to Problems in Fractional Dimensional Space

A scaling method is proposed to find (1) the volume and the surface area of a generalized hypersphere in a fractional dimensional space and (2) the solid angle at a point for the same space. It is demonstrated that the total dimension of the fractional space can be obtained by summing the dimension of the fractional line element along each axis. The regularization condition is defined for functions depending on more than one variable. This condition is applied (1) to find a closed form expression for the fractional Gaussian integral, (2) to establish a relationship between a fractional dimensional space and a fractional integral, (3) to develop the Bochner theorem, and (4) to obtain an expression for the fractional integral of the Mittag–Leffler function. Some possible extensions of this work are also discussed

Vacuum correlations at geodesic distance in quantum gravity

The vacuum correlations of the gravitational field are highly non-trivial to be defined and computed, as soon as their arguments and indices do not belong to a background but become dynamical quantities. Their knowledge is essential however in order to understand some physical properties of quantum gravity, like virtual excitations and the possibility of a continuum limit for lattice theory. In this review the most recent perturbative and non-perturbative advances in this field are presented. (To appear on Riv. Nuovo Cim.)Comment: report U.T.F. 332, July 94. Plain TeX, 67 pp. (+ 1 table and 7 figures, available from the author