Topics on euclidean classical field equations with unique vacuua
We discuss the real classical field equation (-δ+μ2)φ+λF(φ) = f, where μ2>o, λ≥o, F ∈ c∞(R), and aF (a)≥0 for all a ∈ R, and where the source function f belongs to various function spaces contained in the Sobolev space H−1 (Rd). We review a number of results whose proofs are to appear elsewhere, on existence of a solution φ ∈ Hl(Rd), the correspondences between the function spaces of f and φ (regularity properties), contractivity properties and uniqueness of φ, and analytic dependence on λ and functional differentiability in f. We mention some of the ideas from the proofs, including a few alternate methods we have not discussed elsewhere. We prove a new result on positivity preservingness: if f is a nonnegative measure in H−1(Rd), then 0 ≤ φ ∈ H1(Rd), a result that is also valid for μ2 = 0. Finally, we give an intuitive interpretation of some of the results for the case of d = 1 dimension.
KeywordsPoint Particle Intuitive Interpretation Contraction Mapping Principle Euclidean Path Unique Vacuum
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