Generalized Noether's theorem. I. theory

This article is a theoretical investigation of generalized Noether's theorem, which, though unconcerned with considerations such as coordinate transformations, symmetry, and invariance, is the basic mechanism of conventional Noether's theorem, its extensions, and its inverse. The generalized theorem is a completely new approach to the subject—formally, conceptually, and practically. It is an association, for a set of field equations, of field variations with conserved currents. The theorem is stated from two points of view and analyzed with regard to its interpretation and its formal and conceptual relation to conventional Noether's theorem and extensions, transformation groups, and Hamilton's principle. The inverse theorem is also treated. The role of coordinate transformations in conventional Noether's theorem is analyzed.     

Mixing of operators in Wilson expansions

The tree-loop theorem of 't Hooft and Veltman is used in Fermi type gauges to show that gauge invariant operators in Wilson expansions mix in general with non-gauge invariant operators. Background field gauges are proposed in which this does not occur. Calculations of anomalous dimensions in these gauges are discussed.     

XIII. Exact S-matrices and form factors in 1 + 1 dimensional field theoretic models with soliton behaviour

A review is given of the derivation of exact S-matrices in field theoretic models with soliton behaviour, that means models obeying infinitely many conservation laws which imply the factorization of the S-matrix. Form factors of various operators are calculated exactly by means of Watson's theorem. The exact value of the finite Sine-Gordon wave function renormalization constant is determined.     

Some applications of geometry is continuum mechanics

Some contemporary ideas from differential geometry are applied to continuum mechanics. The Lie derivative is used to clarify the notion of “objective rates”, an intrinsic treatment of Piola transformations is described, a simplified proof of Vainberg's theorem for potential operators is given by way of the Poincaré lemma on infinite dimensional manifolds, and a new derivation of the basic equations of continuum mechanics is presented which is valid in a general Riemannian manifold setting.     

A new approach to Gaussian path integrals and the evaluation of the semiclassical propagator

The expansion of path variations in terms of solutions of Morse's boundary problem is applied in order to evaluate Gaussian path integrals. Together with a recently discovered theorem on infinite products of eigenvalues of Sturm-Liouville type operators this yields an expression for the most general semiclassical propagator. The properties of the latter are investigated in the light of the Morse theory. The general methods developed here are illustrated by the example of a charged particle moving in a homogeneous magnetic field.     

On wave properties of identical particles

A generalization of de Finetti's theorem is used to show that certain quantities which usually are interpreted as evidence for the wave aspect of quantum phenomena may be explained from a statistical dependence induced by the indistinguishability of the particles under consideration.     

Egorov’s Theorem for Transversally Elliptic Operators on Foliated Manifolds and Noncommutative Geodesic Flow

The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations. Mathematics Subject Classifications (2000) 58J40, 58J42, 58B34.     

On converse to Koopman's Lemma

Koopman's Lemma states that if a flow T_t is measure preserving for a measure μ on a constant energy surface Ω, then the flow generates a one parameter family of unitary operators U_t on L² (Ω, μ). We show here a converse, namely that under certain (physically motivated) conditions a unitary operator family U_t can be made to generate a corresponding underlying family T_t of point transformations. This result comes out of questions of independent interest in the study of relationships between reversibility and irreversibility, and has application to the foundations of statistical mechanics. In particular, it establishes the principle often used intuitively in chemistry that a forward moving (e.g., Markov) process that loses information cannot be reversed. In a different setting, it provides the answer to a question in the representation theory of isometries on L^p spaces a Banach-Lamperti theorem). These results also allow an interesting reformulation of Ornstein's isomorphism theorem on Bernoulli systems.     

Generalized Noether's theorem. II. Application

We consider the application of generalized Noether's theorem, a completely new approach — formally, conceptually, and practically — to Noether's theorem, its extensions, and its inverse. We discuss application in general and present specific applications: to the Dirac field, to creation/annihilation currents, and to zilch.     

Symmetry transformations in indefinite metric spaces: A generalization of Wigner's theorem

We formulate and prove a generalization to indefinite metric spaces of Uhlhorn's version of Wigner's theorem.     

Wigner's function and other distribution functions in mock phase spaces

This review deals with the methods of associating functions with quantum mechanical operators in such a manner that these functions should furnish conveniently semiclassical approximations. We present a unified treatment of methods and results which usually appear under expressions such as Wigner's function, Weyl's association, Kirkwood's expansion, Glauber's coherent state representation, etc.; we also construct some new associations.Section 1 gives the motivation by discussing the Thomas-Fermi theory of an atom with this end in view.Section 2 introduces new operators which resemble Dirac delta functions with operator arguments, the operators being the momenta and coordinates. Reasons are given as to why this should be useful. Next we introduce the notion of an operator basis, and discuss the possibility and usefulness of writing an operator as a linear combination of the basis operators. The coefficients in the linear combination are c-numbers and the c-numbers are associated with the operator (in that particular basis). The delta function type operators introduced before can be used as a basis for the dynamical operators, and the c-numbers obtained in this manner turn out to be the c-number functions used by Wigner, Weyl, Kirkwood, Glauber, etc. New bases and associations can now be invented at will. One such new basis is presented and discussed. The reasons and motivations for choosing different bases is then explained.The copious and seemingly random mathematical relations between these functions are then nothing else but the relations between the expansion coefficients engendered by the relations between the different bases. These are shown and discussed in this light. A brief discussion is then given to possible transformation of the p, q labels.Section 3 gives examples of how the semiclassical expansions are generated for these functions and exhibits their equivalence.The mathematical paraphernalia are collected in the appendices.     

MacDonald's theorem and Milatz's theorem for multivariate stochastic processes

MacDonald's theorem, which expresses the spectral density of a randomly fluctuating variable α(t) in terms of the finite time average of that variable, α_θ(t), is generalized for multivariate processes. For purely random processes, having a white spectrum, this also yields the corresponding generalization of Milatz's theorem.     

Bell's inequality and non-contextual dispersion-free states

With a view to exploring the possibility of wider implication of Bell's theorem, we argue that Bell's inequality is derivable as a general consequence of non-contextual hidden-variable theories. We formulate a new type of gedanken example indicating incompatibility of Bell's inequality with quantum mechanical predictions concerning simultaneous measurement of commuting observables associated with a system having no spatially separated components, where the locality condition is not at issue. Significance of this example is pointed out.     

The classical limit of temperature Green's function expansions

Rules are obtained for calculating the classical limit of Green's function diagrammatic expansions. The classical cluster expansion is derived by calculating the classical limit of the exact Green's function. Other operators of interest in linear response theory may be calculated in the classical limit. The retarded real-time spin density correlation function, proportional to the magnetic susceptibility, is shown to be exactly proportional to the density in this limit. The relation of this work to other approaches is discussed.     

A theorem for time-dependent dynamical systems

A generalization of a theorem of Reid and Cullen is given. The limitations of using only point transformations are discussed and a procedure for the use of Noether's theorem with velocity-dependent transformations promoted.     

Eine phasenraumdarstellung für spinsysteme

We start with the definition of two mapping operators, one of them is the projection operator onto coherent spin states. With the help of these operators we derive a mapping theorem which defines a correspondence between the operators in spin space andc-number functions of a certain class. It is shown that this correspondence is one-to-one. The quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase space average of classical statistical mechanics. We also derive a product theorem which allows us to transcribe the equations of motion for operators into equivalent equations for thec-number functions. As an illustration of the theory, some examples are discussed.     

The renormalized σ model

A renormalization procedure of the boson σ model based on the finite field equations of Brandt-Wilson is given. We first show that the current operators appearing in the field equations, which are finite local limit of sums of nonlocal field products and suitable subtraction terms, can be chosen to be the same form as the one given for the symmetric limit except for the symmetry breaking constant source term itself. The set of integral equations derived from the field equations is shown to be equivalent to the usual Bogoliubov-Parasiuk-Hepp renormalization theory, and gives us immediately all the renormalized Green's functions in each order of perturbation theory in clear and straightforward fashion. We then analyze the structures of the model in detail. In particular, Ward identities are shown to be satisfied to all orders of perturbation theory. The Goldstone theorem is a particular consequence of these identities.     

Dissipative forces and the algebra of operators in stochastic quantum mechanics

Davidson's construction of a Hilbert space and of quantum operators on the basis of the Fényes-Nelson stochastic mechanics is extended to the case in which a dissipative force linear in the velocity is present. The hamiltonian becomes a nonlinear operator but the position and linear momentum operators are the same as in ordinary quantum mechanics.     

Adler's theorem in finite massless QED and possible extensions to non-Abelian gauge theories

A new proof of Adler's theorem in finite massless QED is given and generalizations to non-Abelian gauge theories are studied.