Properties of field functionals and characterization of local functionals

Abstract

Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed.The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Yoann Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning.A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre's theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincaré lemma and defining multi-vector fields and graded functionals within our framework. CONTENTS

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

“…Recall that, thanks to [BFR12, Prop. 2.3.12] -see also [BDGR18] -all smooth, local, polynomial functionals depend on a finite number of derivatives of the field configuration ϕ. Remark 47: We observe that the definitions of the functors Γ eq : BkgG → Vec and A reg : BkgG → Alg c generalize slavishly to the case of functionals which depend on the derivatives of the fieldscf.…”

Locality and causality in perturbative algebraic quantum field theory

“…The following definition of locality has been proposed in [BDF09] and refined in [BDLGR18]. LF2 For every ϕ ∈ U, the differential F (1) (ϕ) of F at ϕ has an empty wave front set and the map ϕ → F (1) (ϕ) is Bastiani smooth from U to D.…”

“…if supp(ϕ 1 )∩supp(ϕ 3 ) = ∅. This condition is called partial additivity in [BDLGR18], where it is also shown (by a counter example) that it is strictly weaker than (1). However, for polynomial unit-preserving (i.e.…”

“…In [BDLGR18] it was shown, following the sketch of the proof given in an earlier version of [BFR19], that (1) together with an additional regularity condition that we will recall in section 5.1 is equivalent to saying that F can be written as an integral of some smooth function on the jet space. In the final version of [BFR19], this condition was slightly weakened.…”

“…Theorem 5.3 (VI.3 in[BDLGR18]). Let U ⊂ E and F : U → R be Bastiani smooth.Then, F is local in the sense of definition 5.2 if and only if for every ϕ ∈ U, there is a neighborhood V of ϕ, an integer k, an open subset V ⊂ J k M and a smooth functionf ∈ C ∞ (V) such that x ∈ M → f (j k x ψ) is supported in a compact subset K ⊂ M and F (ϕ + ψ) = F (ϕ) + M f (j k x ψ)dx,whenever ϕ + ψ ∈ V and where j k x ψ denotes the k-jet of ψ at x.…”