Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the N=1 super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well.

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Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

Journal of Geometric Analysis ◽

10.1007/s12220-021-00702-4 ◽

2021 ◽

Author(s):

Andreas Bernig ◽

Dmitry Faifman ◽

Gil Solanes

Keyword(s):

Riemannian Manifolds ◽

Riemannian Geometry ◽

Riemannian Space ◽

Space Forms ◽

Type Formula ◽

Isometric Embeddings ◽

Curvature Measures ◽

Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

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A relativistic basis of the quantum theory.—II

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1934.0126 ◽

1934 ◽

Vol 145 (855) ◽

pp. 645-656 ◽

Cited By ~ 1

Keyword(s):

Quantum Mechanics ◽

Wave Equation ◽

Quantum Theory ◽

General Expression ◽

Matrix Equation ◽

Covariant Derivative ◽

Riemannian Geometry ◽

Order Equation ◽

Second Order ◽

Space Curvature

The paper is a continuation of the last paper communicated to these 'Proceedings.' In that paper, which we shall refer to as the first paper, a more general expression for space curvature was obtained than that which occurs in Riemannian geometry, by a modification of the Riemannian covariant derivative and by the use of a fifth co-ordinate. By means of a particular substitution (∆ μσ σ = 1/ψ ∂ψ/∂x μ ) it was shown that this curvature takes the form of the second order equation of quantum mechanics. It is not a matrix equation, however but one which has the character of the wave equation as it occurred in the earlier form of the quantum theory. But it contains additional terms, all of which can be readily accounted for in physics, expect on which suggested an identification with energy of the spin.

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The spectral theorem for well-bounded operators

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700031827 ◽

1993 ◽

Vol 54 (3) ◽

pp. 334-351 ◽

Cited By ~ 1

Author(s):

Ian Doust ◽

Qiu Bozhou

Keyword(s):

Functional Calculus ◽

Continuous Functions ◽

Weak Compactness ◽

Compact Interval ◽

Spectral Theorem ◽

Type B ◽

Bounded Operators ◽

Absolutely Continuous Functions ◽

Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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A local functional calculus

Integral Equations and Operator Theory ◽

10.1007/bf01195008 ◽

1986 ◽

Vol 9 (2) ◽

pp. 218-236 ◽

Cited By ~ 1

Author(s):

Paul McGuire

Keyword(s):

Functional Calculus ◽

Local Functional

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Functional calculus for sesquilinear forms and the purification map

Reports on Mathematical Physics ◽

10.1016/0034-4877(75)90061-0 ◽

1975 ◽

Vol 8 (2) ◽

pp. 159-170 ◽

Cited By ~ 159

Author(s):

W. Pusz ◽

S.L. Woronowicz

Keyword(s):

Functional Calculus ◽

Sesquilinear Forms

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WEYL GEOMETRY AS CHARACTERIZATION OF SPACE-TIME

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194511001176 ◽

2011 ◽

Vol 03 ◽

pp. 87-97 ◽

Cited By ~ 5

Author(s):

F. P. POULIS ◽

J. M. SALIM

Keyword(s):

General Relativity ◽

Scalar Field ◽

Riemannian Geometry ◽

Axiomatic Approach ◽

Space Time ◽

Weyl Geometry ◽

Test Particles ◽

Variational Formalism ◽

Physical Fields

Motivated by an axiomatic approach to characterize space-time it is investigated a reformulation of Einstein's gravity where the pseudo-riemannian geometry is substituted by a Weyl one. It is presented the main properties of the Weyl geometry and it is shown that it gives extra contributions to the trajectories of test particles, serving as one more motivation to study general relativity in Weyl geometry. It is introduced its variational formalism and it is established the coupling with other physical fields in such a way that the theory acquires a gauge symmetry for the geometrical fields. It is shown that this symmetry is still present for the red-shift and it is concluded that for cosmological models it opens the possibility that observations can be fully described by the new geometrical scalar field. It is concluded then that this reformulation, although representing a theoretical advance, still needs a complete description of their objects.

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