Constructing conformally invariant equations using the Weyl geometry

We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. This method that we call 'Weyl-to-Riemann' is based on two features of the Weyl geometry. (i) Weyl space is defined by the metric tensor and the Weyl vector W; it...

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Bibliographic Details
Published in:Classical and quantum gravity 2013-06, Vol.30 (11), p.115005-12
Main Author: Faci, Sofiane
Format: Article
Language:English
Subjects:
algebraic methods
conformal invariance
Construction
Differential equations
Equivalence
gauge field theories
Invariants
Mathematical analysis
Quantum gravity
Scalars
Vectors (mathematics)
Weyl geometry
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Summary:We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. This method that we call 'Weyl-to-Riemann' is based on two features of the Weyl geometry. (i) Weyl space is defined by the metric tensor and the Weyl vector W; it is equivalent to the Riemann space when W is a gradient. (ii) Any homogeneous differential equation written in the Weyl space by means of the Weyl connection is conformally invariant. The Weyl-to-Riemann method selects those equations whose conformal invariance is preserved when reducing to the Riemann space. Applications to scalar, vector and spin-2 fields are presented, which demonstrate the efficiency of this method. In particular, a new conformally invariant spin-2 field equation is exhibited. This equation extends Grishchuk-Yudin's equation and fixes its limitations since it does not require the Lorenz gauge. Moreover, this equation reduces to the Drew-Gegenberg and Deser-Nepomechie equations in Minkowski and de Sitter spaces, respectively.
ISSN:0264-9381
1361-6382
DOI:10.1088/0264-9381/30/11/115005