Cosmological Applications of Algebraic Quantum Field Theory by Thomas-Paul Hack

By Thomas-Paul Hack

This booklet presents a mostly self-contained and widely obtainable exposition on cosmological functions of algebraic quantum box concept (QFT) in curved spacetime: a basic research of the cosmological evolution in keeping with the normal version of Cosmology; and a primary research of the perturbations in inflation. the 2 primary sections of the e-book facing those functions are preceded by way of sections supplying a pedagogical advent to the topic. Introductory fabric at the building of linear QFTs on common curved spacetimes with and with no gauge symmetry within the algebraic method, bodily significant quantum states on common curved spacetimes, and the backreaction of quantum fields in curved spacetimes through the semiclassical Einstein equation is usually given. The reader must have a easy realizing of normal Relativity and QFT on Minkowski spacetime, yet no historical past in QFT on curved spacetimes or the algebraic method of QFT is required.

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E. on gauge-equivalence classes of on-shell configurations, by Sol/G . [Φ] → O[ f ] ([Φ]) = f, Φ V . Being gauge-invariant, these functionals correspond to meaningful (physical) observables. 1 which is proved in [63]. 2 Let [ f ] ∈ E be arbitrary and let Σ be any Cauchy surface of (M, g). Then, for any bounded neighbourhood O(Σ) of Σ, we can find a g ∈ ker 0 (K † ) with supp g ⊂ O(Σ) and g ∈ [ f ]. 1, which in fact holds for the causal propagator E P of any Green-hyperbolic operator P on an arbitrary vector bundle V , has been crucial.

E. t. Φ1 and ε have spacelike-compact overlapping support. 2 Linear Classical Fields on Curved Spacetimes 33 4. ·, · Sol may be computed as a suitable integral over an arbitrary but fixed Cauchy surface Σ of (M, g) with future-pointing normal vector field N and induced measure dΣ. If there exists a ‘current’ j : Γ (V ) × Γ (V ) → T ∗ M such that ∇ μ jμ (Φ1 , Φ2 ) = Φ1 , PΦ2 V − Φ2 , PΦ1 V for all Φ1 , Φ2 ∈ Γ (V ), then Φ1 , Φ 2 Sol dΣ N μ jμ (Φ1 , Φ2 ) . = Σ 5. For all Φ ∈ Sol and all h ∈ ker 0 (K † ), E P h, Φ Sol = h, Φ V .

To see the full singularity structure of u at x0 , we therefore need to consider all test functions which are non-vanishing at x0 . With this in mind, one first defines the wave front set of distributions on Rm and then extends it to curved manifolds in a second step. 10 A neighbourhood Γ of k0 ∈ Rm is called conic if k ∈ Γ implies λk ∈ Γ for all λ ∈ (0, ∞). Let u ∈ Γ0 (Rm ). A point (x0 , k0 ) ∈ Rm × (Rm \ {0}) is called a regular directed point of u if there is an f ∈ Γ0 (Rm ) with f (x0 ) = 0 such that, for every n ∈ N, there is a constant Cn ∈ R fulfilling | f u(k)| ≤ Cn (1 + |k|)−n for all k in a conic neighbourhood of k0 .

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