Dynamical Locality of the Nonminimally Coupled Scalar Field and Enlarged Algebra of Wick Polynomials

Abstract

We discuss dynamical locality in two locally covariant quantum field theories, the nonminimally coupled scalar field and the enlarged algebra of Wick polynomials. We calculate the relative Cauchy evolution of the enlarged algebra, before demonstrating that dynamical locality holds in the nonminimally coupled scalar field theory. We also establish dynamical locality in the enlarged algebra for the minimally coupled massive case and the conformally coupled massive case. * mtf500@york.ac.ukSince u[P M η s ι −,sÊ − M f ] = 0, we may use χ adv = 1 − χ ret to see that (Ě M η ′ s P M (χ adv ) ′ u)[f ] = u[χ adv P M η s ι +,sÊ + M f ] + u[χ ret P M η s ι −,sÊ − M f ]. (3.19)

Algebraic Quantum Field Theory in Curved Spacetimes

“…where T M M M is the renormalised stress-energy tensor of the theory. 11 Similar computations for other models [136,9,46,64,59] support the view that, in general, the functional derivative of the relative Cauchy evolution may be interpreted as a stress-energy tensor. One is therefore led to regard the relative Cauchy evolution as a proxy for the action.…”

Locally Covariant Quantum Field Theory with External Sources

“…The dynamical locality property has been proven previously for the homogeneous Klein-Gordon theory with non-vanishing mass m = 0 in [16] and for extended algebras of Wick polynomials in [10].…”

“…The problem with this choice is that the affine connection does not dualize to the cotangent bundle and in particular not to the bundle of infinitesimal Weyl algebras A. 10 This issue is sidestepped in [25], by regarding elements of the infinitesimal Weyl algebras as symmetric polynomials acting on the vector space V 0 on which V is modeled, which permits a unique parallel transport between fibres in A to be defined. This could be regarded as a slightly ad hoc mixture of the quantized and classical theories, because V 0 is analogous to the classical solution space of the homogeneous theory.…”

“…The connection thus employed in [25] is much more rigid than that of [9], which does not integrate to a unique parallel transport between fibres (see [9, p. 222]). 11 With these thoughts in mind, the approach in 10 The correct dualization would be to the vector dual bundle of the tangent bundle regarded in the category of affine bundles, but this would lead to the following logical problem: instead of replacing the problem of quantizing affine Poisson spaces by quantizing linear Poisson spaces (which Fedosov's method does, as explained above), the choice of affine connection in [25] replaces the problem of quantizing affine Poisson spaces by quantizing affine Poisson spaces in the fibres. 11 Unique parallel transport depends on theorems on existence and uniqueness of solutions to differential equations that do not necessarily apply to bundles with infinite-dimensional fibres (beyond the Banach case).…”

“…10) for all a ∈ CPA p (M , J ), φ ∈ Sol p (M , J ) and φ ∈ Sol lin p (M ).We notice that the pairing · , · (M ,J) is non-degenerate when act-ing on PS p (M , J ). This means that [(ϕ, α)], φ (M ,J) = 0, for all φ ∈ Sol p (M , J ), implies that [(ϕ, α)] = 0,and, vice versa, that [(ϕ, α)], φ (M ,J) = [(ϕ, α)], φ (M ,J ) , for all [(ϕ, α)] ∈ PS p (M , J ), implies that φ = φ .…”