Necessary and sufficient conditions for the existence of a Lagrangian in field theory II. Direct analytic representations of tensorial field equations

By using a characterization of the concept of analytic representation and a variational approach to self-adjointness introduced in a preceding paper, we prove a theorem, according to which a necessary and sufficient condition for a class $\text{C}$², regular, tensorial, quasi-linear system of field equations to admit an ordered direct analytic representation in terms of the Lagrange equations in a region R of its variables is that the system is self-adjoint in R. We point out as a first corollary that if the ordering requirement is removed from the definition of analytic representation, then the condition of self-adjointness of the field equations is only sufficient for the existence of a Lagrangian density. We then provide as a second corollary a methodology for the computation of the Lagrangian density for the representation of self-adjoint quasi-linear tensorial field equations. This methodology is also particularized for ordinary semilinear systems of tensorial field equations through a third corollary. The above results are interpreted from the viewpoint of interactions. We first recover, through a fourth corollary, the conventional structure of the total Lagrangian density L_Tot = Σ_{1 a}ⁿL_Free^(a) + L_Int for the semilinear form of the field equations, and then introduce through a fifth corollary a generalized structure of the type L_Tot = Σ_{1 a}ⁿL_{Int, I}^(a)L_Free^(a) + L_Int.II for t representations of the field equations in the quasi-linear form. Therefore, our analysis seems to indicate that a general form of representing interacting fields is characterized by (n+1)-interaction terms in the Lagrangian: n multiplicative terms and one additive term to the Lagrangian for the free fields.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. III. Generalized analytic representations of tensorial field equations

In this paper we first study the equivalence transformations of class C², regular, tensorial, quasi-linear systems of field equations which (a) preserve the continuity, regularity, and quasi-linear structure of the systems; and (b) occur within a fixed system of Minkowski coordinates and field components. We identify, among the transformations of this class, those which either induce or preserve a self-adjoint structure of the field equations and we term them genotopic and isotopic transformations, respectively. We then give the necessary and sufficient conditions for an equivalence transformation of the above type to be either genotopic or isotopic. By using this methodology, we then extend the theorem on the necessary and sufficient condition for the existence of ordered direct analytic representations introduced in the preceding paper to the case of ordered indirect analytic representations in terms of the conventional Lagrange equations; we introduce a method for the construction of a Lagrangian, when it exists, in this broader context; and we explore some implications of the underlying methodology for the problem of the structure of the Lagrangian capable of representing interactions within the framework of the indirect analytic representations. Some of the several aspects which demand an inspection prior to the use of this analytic approach in actual models are pointed out. In particular, we indicate a possible deep impact in the symmetries and conservation laws of the system generated by the use of the concept of indirect analytic representation. As a preparatory step prior to the analysis of these problems, we study some methodological aspects which underlie the generalized Lagrange equations postulated in the first paper of this series for the case when they are regular, namely, when they are simple equivalence transformations of the conventional Lagrange equations. We first introduce a generalization of the action principle capable of inducing the generalized as well as the conventional equations. In this way we establish that the former equations are “bona fide” analytic equations. Finally, as our most general analytic framework for the case of unconstrained field equations, we work out the necessary and sufficient condition for the existence of ordered direct analytic representations of quasi-linear systems in terms of the generalized analytic equations and study their relationship to the conventional representations.     

Necessary and sufficient conditions for the existence of a lagrangian. The case of quasi-linear field equations

Necessary and sufficient conditions for the existence of the Lagrangian associated with given field equations of motion are investigated. For the quasi-linear equations A_ab^μν(x^λ, φ^c, φ_?^c)φ_μν^b + B_a(x^μ, φ^b, φ_ν^b) = 0, the complete necessary and sufficient conditions are obtained, resorting to the formalism of an exterior derivative. It is emphasized that, to find expressions of these conditions, the anti-symmetric parts of the second derivatives of a Lagrangian, $\text{R}{}_{\mathrm{\mu \nu }}{}^{\mathrm{ab}}\text{= (}\text{?}{}^2\text{L}\text{?φ}{}_{\mathrm{\mu }}{}^{\mathrm{a}}\text{?φ}{}_{\mathrm{\nu }}{}^{\mathrm{b}}\text{?}\text{?}{}^2\text{L}\text{?φ}{}_{\mathrm{\nu }}{}^{\mathrm{a}}\text{?φ}{}_{\mathrm{\mu }}{}^{\mathrm{b}}\text{)/2}$, which disappear in the field equations, take an important role. The procedure to construct the Lagrandian associated with the field equations is also presented.     

Necessary and sufficient conditions for existence of exponential dichotomy of impulsive equations

Necessary and sufficient conditions for the existence of an exponential dichotomy of impulsive differential equations in a Hilbert space are found.     

Necessary and sufficient conditions for trivial solutions in supergravity

We determine the conditions necessary for a solution of the supergravity field equations with infinitesimal spin-3/2 field to be a pure gauge transformation of an Einstein vacuum field. The analysis depends on the Petrov classification of the curvature tensor and uses two-component spinor calculus. For general type I, the type II, and typeD, the necessary conditions found are also shown to be sufficient, and the explicit form of the gauge transformation can be given.Work supported in part by the Einstein Memorial Foundation.     

Variational formalism and field equations of tetrad gravitation theory in the Weyl-Cartan space

A variational formalism of tetrad gravitation theory is developed in the Weyl-Cartan space with independent variations in the tetrad coefficients, metric tensor components, and affine connectivity coefficients that considers the Weyl condition imposed on the nonmetricity based on the method of undetermined Lagrange multipliers. The gravitational field equations are derived for the Lagrangian comprising all possible quadratic convolutions of curvature, torsion, and nonmetricity tensors in addition to the linear component. Differential identities are obtained for the general gravitational Lagrangian in the Weyl-Cartan space. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 56–59, June, 2006.     

Decay of correlations. IV. Necessary and sufficient conditions for a rapid decay of correlations

We considerN-particle systems whose probability distributions obey the master equation. For these systems, we derive the necessary and sufficient conditions under which the reducedn-particle (n) probabilities also obey master equations and under which the Ursell functions decay to their equilibrium values faster than the probability distributions. These conditions impose restrictions on the form of the transition rate matrix and thus on the form of its eigenfunctions. We first consider systems in which the eigenfunctions of theN-particle transition rate matrix are completely factorized and demonstrate that for such systems, the reduced probabilities obey master equations and the Ursell functions decay rapidly if certain additional conditions are imposed. As an example of such a system, we discuss a random walk ofN pairwise interacting walkers. We then demonstrate that for systems whoseN-particle transition matrix can be written as a sum of one-particle, two-particle, etc. contributions, and for which the reduced probabilities obey master equations, the reduced master equations become, in the thermodynamic limit, those for independent particles, which have been discussed by us previously. As an example of suchN-particle systems, we discuss the relaxation of a gas of interacting harmonic oscillators.Supported in part (grants to D.B. and K.E.S.) by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018, and in part (grant to I.O.) by the National Science Foundation.     

Necessary and sufficient conditions for the presence of global potentials in space

It is shown that either the presence of a magnetic flux in a finite spatial region or jumps in the field strength are necessary to generate a global potential. The problem of detecting the magnetic Aharonov—Bohm effect is discussed.     

Lagrangian equations of motion for a massive conductor in a magnetic field

The motion of a nonferromagnetic massive conductor is considered in a quasi-inhomogeneous, varying magnetic field. It is shown that the equations of motion can be written in Lagrangian form.SUNTs. M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 63–69, July, 1994.     

Variational approach to the Hamiltonian structure of a new hierarchy of nonlinear equations

We determine the symplectic Hamiltonian structure associated with the nonlinear evolution equations obtained from two new isospectral problems. We follow the method of variation with respect to the field variables. An explicit example is given to demonstrate the new class of equations that are generated.     

Necessary and sufficient conditions for current conservation in the framework of Relativistic Quantum Field Theory

Necessary and sufficient conditions for the conservation of vector currents covariant under translations and of the Conformal currents, are formulated and proven. Only the weak spectrum condition is assumed, thus the theorems serve as generalizations of a well-known theorem due to Coleman. This is done under the axioms of Relativistic Quantum Field Theory due to Wightman.     

Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points

Given a set of Wightman functions one would like to associate to it a field on Euclidean space admitting a simultaneous diagonalization. We investigate when this can be done in such a way that the Schwinger functions are the expectation values of this commutative field with a bounded metric operator commuting with the field. This requires as a tool the characterization of those linear functionals on the symmetric tensor algebra over a space of test functions which can be represented by complex measures on the corresponding space of distributions.     

On necessary and sufficient conditions for the existence of time and entropy operators in quantum mechanics

It is shown that time and entropy operators may exist as superoperators in the framework of the Liouville space provided that the Hamiltonian has an unbounded absolutely continuous spectrum. In this case the Liouville operator has uniform infinite multiplicity and thus the time operator may exist. A general proof of the Heisenberg uncertainty relation between time and energy is derived from the existence of this time operator.     

Symplectic approach to the theory of quantized fields. I

Our aim in this paper the first one of a series concerned with the problem of field quantization starting from the symplectic structure underlying the classical theory, is to build up the variational theory necessary to all further constructions. The basic notions are the vertical bundle and thestructure 1-form used to define thegeneralized infinitesimal contact transformation which allows us to state and solve the variational problem related to field physics.Giving a system of modulevalued differential forms of different degree on the vertical bundle which solutions are the stationary cross sections is the main result in the paper. In this scheme the Euler-Lagrange classical equations are the expressions induced by such a system of differential forms on any cross section of the vertical bundle. This gives us a complete linearization of the Euler-Lagrange equations and, starting from it, a natural globalization of these equations. Finally, the notion of variational problem invariant by a Lie group is defined in this scheme, Noether's theorem related to such invariant problem is formulated and an intrinsic version of the so-called Noether invariants of classical variational calculus is obtained.This work has been realized in the Seminar of Mathematical Physics, directed by ProfessorJ. Sancho, in the Faculty of Science at the University of Barcelona (Spain).     

Necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the Cramer-Rao lower bound. Part I. General approach with an application to time-delay and Doppler shift estimation.

Analytic expressions for the first order bias and second order covariance of a general maximum likelihood estimate (MLE) are presented. These expressions are used to determine general analytic conditions on sample size, or signal-to-noise ratio (SNR), that are necessary for a MLE to become asymptotically unbiased and attain minimum variance as expressed by the Cramer-Rao lower bound (CRLB). The expressions are then evaluated for multivariate Gaussian data. The results can be used to determine asymptotic biases. variances, and conditions for estimator optimality in a wide range of inverse problems encountered in ocean acoustics and many other disciplines. The results are then applied to rigorously determine conditions on SNR necessary for the MLE to become unbiased and attain minimum variance in the classical active sonar and radar time-delay and Doppler-shift estimation problems. The time-delay MLE is the time lag at the peak value of a matched filter output. It is shown that the matched filter estimate attains the CRLB for the signal's position when the SNR is much larger than the kurtosis of the expected signal's energy spectrum. The Doppler-shift MLE exhibits dual behavior for narrow band analytic signals. In a companion paper, the general theory presented here is applied to the problem of estimating the range and depth of an acoustic source submerged in an ocean waveguide.