Functional Integration on Paracompact Manifods: Functional Integration on Manifold

Pierre Grange 1, * E. Werner 2, 3, *
Abstract : In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette (CDWM) focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. CDWM’s central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures. This is just the mathematical content of the Taylor-Lagrange Scheme (TLRS) developed by the authors in a recent past. In this scheme fields are living in metric Schwartz-Sobolev spaces, subject to open coverings with subordinate partition-of-unity (PU) test functions. In effect these PU, through the convolution operation, lead to smooth C ∞ field functions on an extended Schwartz space. Thereby, first, the basic assumption of differential geometry -that fields live on differentiable manifolds- is validated. And, second, we show that it is just the notion of convolution in the theory of distributions which leads to a sound definition of CDWM’s Laplace-Stieltjes transform, stemming from the existence of an isometry invariant Hausdorff measure in the space of fields. In gauge theories the construction of C ∞ smooth vector fields on a curved manifold is established, as required for differentiable vector fibre bundles. The choice of a connection on a principal bundle P is discussed for pure Abelian and Yang-Mills cases in relation with the presence of PU test-functions. A connection on P being a unique separation of the tangent space TuP into vertical and horizontal sub-spaces, it further separates in the functional Hausdorff integration measure a finite Gaussian integration over the gauge parameter which factors out and plays no physical role. For non-Abelian SU (2) and SU (3) Yang-Mills gauge theories the properties of the Vilkowisky-DeWitt connection and of the Landau-DeWitt covariant back-ground gauge result in very simple calculation rules.
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Pierre Grange, E. Werner. Functional Integration on Paracompact Manifods: Functional Integration on Manifold. Theoretical and Mathematical Physics, Consultants bureau, 2018, pp.1-29. ⟨hal-01942764⟩

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