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.     

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.     

Generalized no-broadcasting theorem

We prove a generalized version of the no-broadcasting theorem, applicable to essentially any nonclassical finite-dimensional probabilistic model satisfying a no-signaling criterion, including ones with "superquantum" correlations. A strengthened version of the quantum no-broadcasting theorem follows, and its proof is significantly simpler than existing proofs of the no-broadcasting theorem.     

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.     

Generalized Kochen-Specker theorem

A generalized Kochen-Specker theorem is proved. It is shown that there exist sets of n projection operators, representing n yes-no questions about a quantum system, such that none of the 2 possible answers is compatible with sum rules imposed by quantum mechanics. Namely, if a subset of commuting projection operators sums up to a matrix having only even or only odd eigenvalues, the number of yes answers ought to he even or odd, respectively. This requirement may lead to contradictions. An example is provided, involving nine projection operators in a 4-dimensional space.Dedicated to Professor Max Jammer on the occasion of his 80th birthday.I am grateful to N. D. Mermin for patiently explaining to me that ref. 11 was a Kochen-Specker argument, not one about locality, as I had wrongly thought. This work was supported by the Gerard Swope Fund, and the Fund for Encouragement of Research.     

The Jost-Schroer theorem and Wilson's fixed point condition for the renormalized coupling constant

On the basis of the Jost-Schroer-Pohlmeyer-Strocchi theorems it is proved that Wilson's renormalization group approach leads to the fixed point condition α^fix_Λ = 0, unless the skeleton theory is a nonlocal theory with a non-analytic dependence on mass and coupling parameter. This result resolves a paradoxon in connection with the Adler-Schwinger-Bell-Jackiw anomaly.