Frölicher-smooth geometries, quantum jet bundles and BRST symmetry

Abstract

We attempt a clarification of geometric aspects of quantum field theory by using the notion of smoothness introduced by Frölicher and exploited by several authors in the study of functional bundles. A discussion of momentum and position representations in curved spacetime, in terms of generalized semi-densities, leads to a definition of quantum configuration bundle which is suitable for a treatment of that kind. A consistent approach to Lagrangian field theories, vertical infinitesimal symmetries and related currents is then developed, and applied to a formulation of BRST symmetry in a gauge theory of the Yang-Mills type.2010 MSC: 58B99, 81T13, 81T20.

Gauge field theory without groups

“…As an intermediate step, we need the notion of a distributional bundle, that is, a bundle over M whose fibres are distributional spaces. In general, the finite-dimensional geometric structure underlying functional bundles [30,5,7,28,29,14] is that of a two-fibered bundle Z Y X. If Z Y is a vector bundle, then for any x ∈ X one obtains the vector space D x (Y, Z) of all section-distributions [31] Y x → Z x .…”

“…A soldering form, on the other hand, is a fully intrinsic notion. If one chooses an orthonormal frame of H M , then indeed recovers a 'tetrad formalism' which is similar to the usual one 14. If no confusion arises, fibred tensor products will be denoted as plain tensor products.…”

Special generalized densities and propagators: A geometric account

“…Applications to gauge theories and discussions about extension to curved spacetime can be find in previous papers [5,6,7].…”

“…In section 3 we sketch a partly original geometric presentation of free quantum fields introduced in previous papers [6,7], and show how propagators arise from their graded commutators both in the boson and in the fermion case.…”

Natural Extensions of Electroweak Geometry and Higgs Interactions

“…in the sub-sectors proportional to the identity we write Φ ≡ φ ⊗ 1 1 U andΦ ≡φ ⊗ 1 1 U , with φ andφ playing the role of the usual Higgs and anti-Higgs fields Ghosts and anti-ghosts precisely describe the notion of "infinitesimal gauge transformations". They are mutually independent fields, as one sees at once from the different ways in which they appear in the ghost Lagrangian [19], the natural isomorphisms L ∼ = L * notwithstanding. We could also consider extensions of ghost and anti-ghosts similar to the extensions introduced in the case of Higgs fields.…”