Canonical Reduction Research Articles

DIRAC FIELDS ON SPACELIKE HYPERSURFACES, THEIR REST-FRAME DESCRIPTION AND DIRAC OBSERVABLES

Grassmann-valued Dirac fields together with the electromagnetic field (the pseudo-classical basis of QED) are reformulated on spacelike hypersurfaces in Minkowski space–time and then restricted to Wigner hyperplanes to get their description in the rest-frame Wigner-covariant instant form of dynamics. The canonical reduction to the Wigner-covariant Coulomb gauge is done in the rest frame. It is shown, on the basis of a geometric inconsistency, that the description of fermions is incomplete, because there is no bosonic carrier of the spin structure describing the trajectory of the electric current in Minkowski space–time, as it was already emphasized in connection with the first quantization of spinning particles in a previous paper.

Exact Solutions for Self-dual Yang–Mills Equations

The (constrained) canonical reduction of four dimensional self-dual Yang–Millstheory to 2, (2+1) dimensional sine-Gordon theory and 2 dimensional Liouvilles theory areconsidered. The Bäcklund transformations (BTs) areimplemented to obtain new classes of exact solutions for the reduced 2 dimensional sine-Gordonand Liouville models. Another transformation is developed and used to obtain exact solution forthe 2+1 and the original 3+1 sine-Gordon models.

Canonical reduction of two-dimensional gravity for particle dynamics

We develop the formalism for canonical reduction of (1 + 1)-dimensional gravity coupled with a set of point particles by eliminating constraints and imposing coordinate conditions. The formalism itself is quite analogous to the (3 + 1)-dimensional case; however in (1 + 1) dimensions an auxiliary scalar field is shown to have an important role. The reduced Hamiltonian is expressed as a form of spatial integral of the second derivative of the scalar field. Since in (1 + 1) dimensions there exists no dynamical degree of freedom of the gravitational field (i.e. the transverse-traceless part of the metric tensor is zero), the reduced Hamiltonian is completely determined in terms of the particles' canonical variables (coordinates and momenta). The explicit form of the Hamiltonian is calculated both in post-linear and post-Newtonian approximations.

On the canonical reduction of spherically symmetric gravity

In a thorough paper Kuchar has examined the canonical reduction of the most general action functional describing the geometrodynamics of the maximally extended Schwarzschild geometry. This reduction yields the true degrees of freedom for (vacuum) spherically symmetric general relativity (SSGR). The essential technical ingredient in Kuchar's analysis is a canonical transformation to a certain chart on the gravitational phase space which features the Schwarzschild mass parameter , expressed in terms of what are essentially Arnowitt - Deser - Misner variables, as a canonical coordinate. (Kuchar's paper complements earlier work by Kastrup and Thiemann, based mostly on Ashtekar variables, which has also explicitly isolated the true degrees of freedom for vacuum SSGR.) In this paper we discuss the geometric interpretation of Kuchar's canonical transformation in terms of the theory of quasilocal energy - momentum in general relativity given by Brown and York. We find Kuchar's transformation to be a `sphere-dependent boost to the rest frame', where the `rest frame' is defined by vanishing quasilocal momentum. Furthermore, our formalism is general enough to cover the case of (vacuum) two-dimensional dilaton gravity. Therefore, besides reviewing Kuchar's original work for Schwarzschild black holes from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten black-hole geometrodynamics. Finally, addressing a recent work of Louko and Whiting, we discuss some delicate points concerning the canonical reduction of the `thermodynamical action', which is of central importance in the path-integral formulation of gravitational thermodynamics.

Jaynes-Cummings model diagonalization based on symmetry

We illustrate a new systematic diagonalization procedure through its application to the well known Jaynes-Cummings model. Our method, starting from the consideration of peculiar invariance properties of the Hamiltonian, singles out specific constants of motion by which we succeed in introducing appropriate unitary transformations accomplishing a canonical reduction of the Hamiltonian into a diagonal form. The potentialities of the treatment is briefly discussed.

Canonical reduction of self-dual Yang-Mills theory to sine-Gordon and Liouville theories

We present the (constrained) canonical reduction of four dimensional self-dual Yang-Mills theory to two dimensional sine-Gordon and Liouville field theories: The Dirac brackets in the latter two theories are derived from the reduction of the Poisson brackets in the former, at the classical level.

Canonical correlation and reduction of multiple time series

It is shown that a degenerate rankd-variate stationary time series can be reduced to a full rank time series of lower dimension via an orthogonal transformationT provided that ρ, the canonical correlation between past and future of the time series is strictly less than one. Procedures for estimation of rank of the multiple time series,T and testing ρ=1 are outlined, the latter is related to testing the unit root hypothesis in ARMA models.

A canonical reduction of order for the Kepler problem

Using a new redudion of order technique we obtain four conserved quantities of the darsical planar Kepler problem via the symmetries of the equations of motion and an extra vector field. By passing to a suitable quotient manifold of the evolution space we deduce that negative energy orbits are closed and periodic, without having to solve the differential equation. In this paper we look at the classical Kepler problem through the application of a new reduction of order technique (l). We believe that this technique provides a systematic means of producing first integrals from group actions more general than symmetries without relying on the existence of a variational principle. Moreover, it is a powerful tool for investigating global properties such as periodicity. The Kepler problem is a good vehicle for a demonstration of these features. The planar equations of motion are defined by a second-order differential equation field on five-dimensional evolution space. There are three point symmetries which take solutions of the equation of motion into other solutions: time translation, which leaves the solution curves invariant but alters their starting times; rotation about the origin, which again leaves the curves invariant but alters the initial angular position; and the transformation associated with the Runge-Leuz vector which allows the orientation and eccentricity of the orbits to remain the same but changes the semi-latus rectum. We wish to show that conserved quantities of the system, i.e. the objects which remain invariant along solution curves, can be obtained using alternative techniques to Noether's theorem (2) or solving the differential equation (3, 41. Lie's method (2) provides us with the three point symmetries mentioned above. To effect our technique a fourth vector field is required which is a symmetry of the integrable distribution made up of the second-order differential equation field and its three symmetries. To find this field we modify the usual reduction of order via symmetry method by taking successive quotients of the extended phase space by the three actions (for example, we project down to a four-dimensional manifold by setting the time coordinate to a constant, thus identifying all the orbits which only differ by a time translation) and we reduce our differential equation field to first order on a two-dimensional quotient.

A geometric approach to stochastic model reduction by canonical variables

The reduced-order representation of possibly nonstationary observations of Markovian processes with additive measurement noise is considered. A canonical variable model reduction scheme is derived in a geometric setting.

A Schur decomposition for Hamiltonian matrices

A Schur-type decomposition for Hamiltonian matrices is given that relies on unitary symplectic similarity transformations. These transformations preserve the Hamiltonian structure and are numerically stable, making them ideal for analysis and computation. Using this decomposition and a special singular-value decomposition for unitary symplectic matrices, a canonical reduction of the algebraic Riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution. After presenting some real decompositions for real Hamiltonian matrices, we look into the possibility of an orthogonal symplectic version of the QR algorithm suitable for Hamiltonian matrices. A finite-step initial reduction to a Hessenberg-type canonical form is presented. However, no extension of the Francis implicit-shift technique was found, and reasons for the difficulty are given.

The Identification of Regional Forecasting Models Using Space: Time Correlation Functions

Two broad classes of aggregate regional forecasting model are defined: autoprojective models belonging to the general family of space-time autoregressive-moving average processes, and explanatory (including leading indicator) models that constitute the class of space-time regression or transfer function processes. A procedure based on the estimation of space-time correlation functions is developed for identifying (selecting) operational models from these classes. For a given body of data, both the degree of non-stationarity and the type and order of representative model can be decided by studying the shape of the appropriate correlation functions. Using artificial and actual data series, the procedure is shown to perform well. The space-time autocorrelation functions give a clear indication of the sort of autoprojective process generating the data. Although less straightforward and often requiring data pre-whitening, the space-time cross-correlation function appears to be quite successful in identifying explanatory (transfer function) models that are plausible approximations to the dynamic relationships underlying the observed series. THE need to construct empirical spatial forecasting models in the context of urban and regional analysis is becoming increasingly recognized (see, for example, Chisholm, Frey and Haggett, 197 i). An appropriate class of formal model for describing space-time data series and generating optimal forecasts is provided by linear stochastic difference equations. The process of modelbuilding is concerned with relating such a class of statistical models to the data at hand and involves much more than model fitting. Forecasting procedures could be seriously deficient if these models were either inadequate or unnecessarily excessive in the use of parameters. Thus an essential stage in the model-building process is that of model identification. By identification we mean preliminary analysis of the data to suggest what particular kind of model might be worthy of further investigation.' The specific aim is to derive aggregate models possessing maximum simplicity and the smallest number of parameters consonant with representational accuracy. That is, the objective is to obtain adequate but parsimonious models (Tukey, 1961; Box and Jenkins, I970). The identification of spatial-temporal forecasting models (and spatial models in general) is basically a 'best subset problem' of the sort frequently encountered in multivariate analysis (for example, Beale, Kendall and Mann, 1967). A variety of techniques is available for use in searching for parsimonious model structures, for example multiple step-wise regression (Newbold, 1972; Payne, I973) and methods of canonical reduction (Daling and Tamura, 1970; Hawkins, 1973). However, in this paper we develop an identification technique based on the estimation of space-time correlation functions. This procedure has the advantage of providing insight into the nature of the process generating the space-time series for a given variable as well as suggesting a 95 This content downloaded from 207.46.13.111 on Sun, 07 Aug 2016 05:15:22 UTC All use subject to http://about.jstor.org/terms R.L.MARTIN AND J.E.OEPPEN sub-class of models appropriate for that process. It should be stressed at the outset that such identification is not exact and offers only a guide to the choice of tentative models that can then be subjected to rigorous estimation and diagnostic checking. In the first section we propose a broad class of space-time forecasting models structured in terms of linear difference equations. The second section develops a general framework for model identification in terms of space-time autocorrelation and cross-correlation functions. The application of these procedures to artificially generated and actual data series is discussed in the third and fourth sections. SOME AGGREGATE SPACE-TIME FORECASTING MODELS The various components of a space-time forecasting model include time dependence and spatial dependence lagged in time, and the possibility of (repeated) differencing in time and space to allow for temporal and spatial non-stationarity. Assume that data are available for i = 1,2, ... n zones over t = I,2,. .., T time periods, plus any earlier lagged values that may be required. We use the term 'zone' in the wide sense to define an areal aggregate such as a county, enumeration district, employment exchange area etc., or a point location, such as a town. Suppose the variable of interest in zone i at time t is Yi,, which takes on values denoted by yi,. The models under consideration are linear equations which study the expected value of Y,, defined as y*,, conditional upon the observed values of Y in zone i and certain other zones in previous time periods and/or upon the lagged values of one or more relevant explanatory variables Xi, X2, etc. The use of such an equation for forecasting is then obvious. To assist in the models that follow, we first make the following definition: Let L be a spatial lag operator such that L?yit = Yit and (I) Lyit = E ij Yjt for any s > jeJs where Js is the set of zones at spatial lag s from zone i, and the {wij} are weights scaled so that

Letter: Ideal diet.

Melatonin and its analog 5-MCA-NAT (5-methylcarboxyamino-<i>N</i>-acetyl tryptamine) are active compounds reducing intraocular pressure (IOP). This action is mediated through MT₂ and the putative MT₃ melatonin receptor, producing a transient reduction of IOP that lasts for a few hours and has not yet been characterized. The use of melatonin and its analog are causing a decrease in chloride efflux from rabbit nonpigmented epithelial cells (NPE), possibly explaining the decrease in IOP. Melatonin and 5-MCA-NAT inhibited rabbit NPE chloride release in a concentration-dependent manner, whereas the pD₂ values were between 4.5 ± 1.2 and 4.4 ± 1.0, respectively. Melatonin hypotensive action was enhanced by the presence of MT₂ antagonists, such as DH97 (<i>N</i>-pentanoyl-2-benzyltryptamine) and 4-P-P-DOT (4-phenyl-2-propionamidotetralin) and by the nonselective melatonin receptor antagonist luzindole. Prazosin (1.5 <i>µ</i>M) partially reverses the melatonin action by acting as a selective MT₃ antagonist. However, at 15 nM it acts as an <i>α</i>-adrenergic receptor antagonist, enhancing the melatonin effect. Regarding the intracellular pathways triggered by melatonin receptors, neither phospholipase C/protein kinase C pathway nor the canonical reduction of intracellular cAMP was responsible for melatonin or 5-MCA-NAT actions. On the contrary, the application of these substances produced a concentration-dependent increase of cAMP, with pD₂ values of 4.6 ± 0.2 and 4.9 ± 0.7 for melatonin and 5-MCA-NAT, respectively. In summary, melatonin reduces the release of chloride concomitantly to cAMP generation. The reduction of Cl⁻ secretion accounts for a decrease in the water outflow and therefore a decrease in aqueous humor production. This could be one of the main mechanisms responsible for the reduction of IOP after application of melatonin and 5-MCA-NAT.

THE MASSIVE SPIN–TWO FIELD

The energy of the free massive spin-two field is exhibited as a manifestly positive-definite function of its canonical variables. The indefiniteness of the form recently obtained by Adler is shown to be only apparent in nature. A summary of the canonical reduction of the theory is given in the Appendix.

On te Algebra of All Local Observables

4) has a canonical reduction into factors according to the spectrum of its center. We prove that the canonical reduction of the algebra of all local observables also reduces the translation unitary operators. A vacuum sector is defined to be any reduced space in this reduction which contains at least one translationally invariant vector. We prove that the algebra of all local observables is a factor of type I with a unique vacuum state in any vacuum sector. Our main physical conclusion is essentially that the unique vacuum and irreducible algebra of all local observables are the most general situation of interest as far as the existence of a vacuum in the theory is desired.

The canonical property and asymptotic reduction procedures

The connection between asymptotic reduction techniques and the canonical structure of interacting fields is examined. The expression for theS-matrix has additional terms when derivative-coupled theories are introduced. It is shown that in the PS(PV) pion-nucleon interaction and meson electrodynamics these additional terms do not produce contributions to theS-matrix. However, we propose a modification of the reduction formula for the case of the interactions with the derivative coupling in such a way to preserve the canonical commutation properties, requiring the canonical commutation relations for artificial free fields defined by eq. (11).

The general theory of canonical correlation and its relation to functional analysis

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations wherepandqare not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

Fixation of Physical Space-Time Coordinates and Equation of Motion of Two-Body Problem

The consistency of the canonical reduction of gravitational field coupled with point particles is investigated. It is shown that the Hamiltonian, obtained by eliminating constraints and imposing coordinate condition, gives exactiy the same equations of motion as obtained by Einstein, Infeld, and Hoffnnan. Furthermore, the derivation of the approximate (up to the fourth order of velocity of particle) equation of motion is presented without tedious calculation. (auth)

Consistency of the Canonical Reduction of General Relativity

The question of consistency of the canonical reduction of general relativity (obtained by eliminating constraints and also imposing coordinate conditions in the action or generator) is examined. It is shown that the equations of motion obtained from this ``reduced'' formalism agree with the original Einstein equations. Agreement is also established for the generators of space-time translations. In order to establish consistency, it is necessary to discard certain well-defined divergence terms in the original Lagrangian. These would otherwise appear as nondivergences in the reduced Lagrangian, incorrectly altering the equations.

Linear Differential Equations and Functional Analysis, II. Equations with Periodic Coefficients

In a previous paper [9], to be quoted henceforth as I, the authors discussed linear differential equations (or rather, systems) from a very general point of view, by use of methods of functional analysis. In this paper we intend to supplement the discussion on the same lines for the particular case of periodic equations, or at least equations with a periodic homogeneous part. It might be thought that this discussion could be subsumed under the case of almost periodic equations with prescribed module discussed in I, Section 6. There are two reasons why this is not the case: in the first place, almost-periodicity presupposes continuity, whereas periodicity does not; this difficulty might be overcome by extending the discussion to Stepanoff-almost-periodic functions (which constitute, in the terminology of I, the closure of A in M) but the theory of this class of functions is by no means so rich as to yield all the interesting properties, nor do the (S)-almost-periodic functions seem objects of sufficient interest at present in connection with differential equations to warrant such an investigation on so slight a motive. It is nevertheless true that the methods used for almost periodic equations in I may be applied to periodic equations to yield similar results; however, and this is the second and main reason for a separate discussion, the periodic case is by far the richer in structure. Some of the results and methods of I are of interest even in the finitedimensional case; for periodic equations, however, there is in that case a well-known canonical reduction to equations with constant coefficients, due essentially to Floquet [2], and the behavior of the latter type of equations in the finite-dimensional case is exhaustively known. The fact that such a Floquet representation does not obtain in general in infinitedimensional spaces, a matter discussed in Section 2, requires us to seek results by other methods. Section 3 is devoted to the discussion of periodic and bounded solutions from different points of view, and forms the main part of this paper. Section 4 sketches familiar applications, in the style of I, to non-linear equations and to the description of the 88

Coverings of Bipartite Graphs

For the purpose of analysing bipartite graphs (hereinafter called simply graphs) the concept of an exterior covering is introduced. In terms of this concept it is possible in a natural way to decompose any graph into two parts, an inadmissible part and a core. It is also possible to decompose the core into irreducible parts and thus obtain a canonical reduction of the graph. The concept of irreducibility is very easily and naturally expressed in terms of exterior coverings. The role of the inadmissible edges of a graph is to obstruct certain natural coverings of the graph.