Generalized Noether theorems in canonical formalism for field theories and their applications

A generalization of Noether's first theorem in phase space for an invariant system with a singular Lagrangian in field theories is derived and a generalization of Noether's second theorem in phase space for a noninvariant system in field theories is deduced. A counterexample is given to show that Dirac's conjecture fails. Some preliminary applications of the generalized Noether second theorem to the gauge field theories are discussed. It is pointed out that for certain systems with a noninvariant Lagrangian in canonical variables for field theories there is also a Dirac constraint. Along the trajectory of motion for a gauge-invariant system some supplementary relations of canonical variables and Lagrange multipliers connected with secondary first-class constraints are obtained.     

Generalized Noether identities for nonlocal transformations

Starting from the transformation properties of an action integral of a system under local and nonlocal transformations, we derive the generalized Noether identities for a variant system under those transformations. The applications of the theory to the Yang-Mills field with higher order Lagrangian is presented under the Coulomb gauge condition, a new conserved PBRS charge is found which differs from the BRS conserved charge, and another conserved charge connected with nonlocal transformation is also obtained.     

Generalized chaos synchronization theorems for bidirectional differential equations and discrete systems with applications

Two constructive generalized chaos synchronization (GCS) theorems for bidirectional differential equations and discrete systems are introduced. Using the two theorems, one can construct new chaos systems to make the system variables be in GCS. Five examples are presented to illustrate the effectiveness of the theoretical results.     

Generalized Noether identities and application to Yang-Mills field theory

We derive generalized Noether identities for a system with noninvariant action integral under an infinite continuous group and deduce the string conservation laws of the system. We give a preliminary application to field theory and discuss the strong conservation laws for the BRS transformation and the weak conservation laws of Yang-Mills fields. The Dirac constraint of the system is examined.     

Generalized Noether theorem and Poincaré invariant for nonconservative nonholonomic systems

We generalize Noether's theorem and the Poincaré invariant to conservative and nonconservative systems with nonlinear nonholonomic constraints. The conservation laws of such systems are illustrated.     

Generalized Goldberg-Sachs theorems for pseudo-Riemannian four-manifolds

It has been recently observed that the generalized Goldberg-Sachs theorem in general relativity as well as some of its corollaries admit appropriate Riemannian versions. In this paper we use the formalism of spinors to give alternative proofs of these results clarifying the analogy between positive Hermitian structures of oriented Riemannian four-manifolds and shear-free congruences of oriented Lorentzian four-manifolds. We also prove similar results for oriented pseudo-Riemannian four-manifolds when the metric is of zero signature. This allows us to describe compact oriented four-manifolds possibly admitting a pseudo-Riemannian Einstein metric of zero signature whose positive Weyl tensor has two distinct eigenvalues corresponding to non-isotropic eigenspaces.     

Generalized lorenz-mie theory and applications

Many optical sizing techniques rely on particle/laser interactions. The classical Lorenz-Mie theory describing sphere/plane wave interactions is therefore misleading when designing instruments and processing data when the particle size is not small enough with respect to beam diameters. In such cases the use of the generalized Lorenz-Mie theory is required. After summarizing essential features of the generalized Lorenz-Mie theory for sphere/arbitray wave interactions, this paper describes applications of the theory with some emphasis on the analysis of phase-Doppler anemometers.     

Generalized Lorenz-Mie theory and applications

The basic formulas of generalized Lorenz-Mie theory are presented, and are applied to scattering of a focused Gaussian laser beam by a spherical particle. Various applications of focused beam scattering are also described, such as optimizing the rate at which morphology-dependent resonances are excited, laser trapping, particle manipulation, and the analysis of optical particle sizing instruments. Each of these applications requires either special positioning the beam with respect to the particle or illumination of only part of the particle by the beam.     

Generalized entropy production fluctuation theorems for quantum systems

Based on trajectory-dependent path probability formalism in state space, we derive generalized entropy production fluctuation relations for a quantum system in the presence of measurement and feedback. We have obtained these results for three different cases: (i) the system is evolving in isolation from its surroundings; (ii) the system being weakly coupled to a heat bath; and (iii) system in contact with reservoir using quantum Crooks fluctuation theorem. In Case (iii), we build on the treatment carried out by H T Quan and H Dong [arXiv/cond-mat:0812.4955], where a quantum trajectory has been defined as a sequence of alternating work and heat steps. The obtained entropy production fluctuation theorems (FTs) retain the same form as in the classical case. The inequality of second law of thermodynamics gets modified in the presence of information. These FTs are robust against intermediate measurements of any observable performed with respect to von Neumann projective measurements as well as weak or positive operator-valued measurements.     

Generalized lattice fermion actions and applications

We derive the general form of lattice fermion action consistent with the requirements of gauge invariance, translation invariance, reflection positivity and invariance under 90° rotations, and involving only bilinear, nearest neighbour couplings. The meaning of the parameters occuring in the action is discussed analyzing the spectrum, the symmetries and the axial Ward identities of the theory, and their renormalization is studied within the Migdal-Kadanoff approximation. In particular we give the relation between the dependence of the vacuum energy density on the CP phases appearing in the action and the mean topological density and susceptibility.     

Adiabatic theorems and applications to the quantum hall effect

We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic limit of quantum mechanics: 1. For slow time variation of the Hamiltonian, the time evolution reduces to spectral subspaces bordered by gaps. 2. The eventual tunneling out of such spectral subspaces is smaller than any inverse power of the time scale if the Hamiltonian varies infinitly smoothly over a finite interval. Except for the existence of gaps, no assumptions are made on the nature of the spectrum. We apply these results to charge transport in quantum Hall Hamiltonians and prove that the flux averaged charge transport is an integer in the adiabatic limit.     

Fluctuation-dissipation theorems for systems in non-thermal equilibrium and applications

The linear response theory of systems obeying Fokker-Planck equations is discussed under the assumption that the principle of detailed balance is satisfied. This theory is used to obtain fluctuation-dissipation theorems for systems in a non-thermal equilibrium state corresponding to the steady state solution of the Fokker-Planck equation. Some of the aspects of the threshold region of a single mode laser (without detuning) are discussed. In particular it is shown that, contrary to the results established recently by others, the “linear susceptibility” χ(Ω) is acontinuous function of the pump parameterp. The results are then specialized to Gaussian Markov process in which case an alternate form of the fluctuation-dissipation theorems is also given. Finally few other generalizations of the fluctuation-dissipation theorems are also briefly discussed.     

Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.     

Generalized gauge fields and applications to weak interactions

Yang-Mills' field is generalized to possess a nontrivial scalar part. The most general transformations for such a field under the 3-parameter isotopic gauge transformation is obtained. Using this generalized gauge field, a gauge invariant Lagrangian is constructed within the framework of the quark model. Interactions for spin-1 as well as for spin-0 are generated. As a further application a weak interaction theory mediated by the generalized gauge (boson) field is formulated. The entire weak interactions are generated in two halfs; the hadron-boson interaction is generated according to Yang-Mills' trick using the generalized gauge field and the other half (boson-lepton, etc.) is then generated by making use of the scalar part of the gauge fields according to the conventional pion gauge principle. The effective Lagrangian is then found to be mediated by the effective propagators which fall off as p⁻² at high momenta; the unitarity of the theory can thereby be insured. Universality in weaker sense than the usual one is applied to the intermediate bosons; our theory for β-decay then reduces to Cabibbo's at low energy.     

Generalized metric spaces for correlation fringes and applications

A novel conception and its applications, the generalized metric for correlation fringes, are presented, which can be used to develop and analyze the relevant algorithms. The relationship between the metric and the correlation fringes representation is examined and the generalized metric is defined. The theoretical analysis of traditional algorithms for correlation fringes indicates that all of them can be regarded as the generalized metrics. In addition, the generalized metric for correlation fringes is applied in developing a new algorithm. A novel generalized metric for correlation fringes based on cross entropy is presented and is proved by theoretical derivation and experimental verification. All the investigations show that it is reasonable and valid to define the generalized metric for correlation fringes in digital speckle pattern interferometry (DSPI). The fringes obtained by cross-entropy metric can be used to represent the same desired information by subtracting. There are different vector function structures among the generalized metrics, which directly influence the quality of fringes and the difficulty in post-processing.     

Quantum Noether method

We present a general method for constructing perturbative quantum field theories with global symmetries. We start from a free non-interacting quantum field theory with given global symmetries and we determine all perturbative quantum deformations assuming the construction is not obstructed by anomalies. The method is established within the causal Bogoliubov-Shirkov-Epstein-Glaser approach to perturbative quantum field theory (which leads directly to a finite perturbative series and does not rely on an intermediate regularization). Our construction can be regarded as a direct implementation of Noether's method at the quantum level. We illustrate the method by constructing the pure Yang-Mills theory (where the relevant global symmetry is BRST symmetry), and the N = 1 supersymmetric model of Wess and Zumino. The whole construction is done before the so-called adiabatic limit is taken. Thus, all considerations regarding symmetry, unitarity and anomalies are well defined even for massless theories.