1. What are the key properties of causal perturbation theory in Quantum Field Theory (QFT)?
Causal perturbation theory in Quantum Field Theory (QFT) is based on rigorous axioms and properties that contribute to its conceptual clarity. These properties include: 1. The time-ordered product (T-product) is defined by axioms, with causality being the most important. 2. The interaction is adiabatically switched off, separating the infrared (IR) problem from the ultraviolet (UV) problem. The adiabatic limit is performed only at the end of the construction, typically for observable quantities. 3. The T-product is constructed in position space, using induction on the number of factors. 4. Renormalization is the mathematically well-defined problem of extending known distributions from D' (Mn - n) to D' (Mn), where M is the space-time manifold and n is the thin diagonal in Mn. 5. The observables are constructed as formal power series in the coupling constant, without considering the convergence of the series. 6. The EG-construction yields all solutions of the axioms, with the set of solutions being the orbit of the Stuckelberg-Petermann renormalization group. 7. Troubles with overlapping divergences do not appear due to the inductive procedure in the construction of the T-product. 8. It applies to nonrenormalizable interactions, such as perturbative quantum gravity. 9. Fields in causal perturbation theory are 'off-shell', meaning they are not restricted by any field equation. The algebra of classical fields is given by the pointwise product of functionals. Quantization of the free theory is obtained by deformation of this product, with the star product containing information about the free theory. The main references for causal perturbation theory are Epstein and Glaser (EG), R. Stora, G. Scharf, K. Fredenhagen, and the book by [9].
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2. What is the purpose of the wave front set condition in the definition of the Poisson bracket?
The purpose of the wave front set condition in the definition of the Poisson bracket is to ensure the existence of pointwise products of distributions which appear in the definition of the Poisson bracket. This condition is important for the algebraic structure of classical fields and their interactions. By imposing the wave front set property, it guarantees that the Poisson bracket is well-defined and satisfies the necessary properties for a Lie bracket, such as skew-symmetry, the Leibniz rule, and the Jacobi identity. This allows for the study of the dynamics and symmetries of classical fields in the context of field theory and mathematical physics.
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3. What are the main techniques to compute extension D ' (R k \ {0}) t 0 - t D ' (R k)?
The main techniques to compute extension D ' (R k \ {0}) t 0 - t D ' (R k) include direct extension and W-extension. In the proof of Thm.4.4, the extension t is constructed using the equation [5, 10], where kh C (R k ) is defined such that 0 <= kh(x) <= 1, kh(x) = 0 for |x| <= 1, and kh(x) = 1 for |x| >= 2. The notation kh r (x) := kh(rx) is used. The expression t 0, kh r h exists, and it is shown that the limit (4.19) exists, defining a distribution t D ' (R k). For practical computations, the formula (4.19) means that the direct extension t is given by the same formula as t 0. When sd(t 0) >= k, the unique extension t o to D ' o satisfying sd(t o) = sd(t 0) - k is obtained. The relations t W , h = t o , W h = t o , h = t 0 , h show that t W is an extension of t 0. The proof of sd(t W ) = sd(t 0) is more elaborate.
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4. What are the disadvantages of using a W-extension for practical computations?
The disadvantages of using a W-extension for practical computations include the lack of manifest Lorentz covariance, as there is no Lorentz covariant function D(Rk). Additionally, computing the W-extension explicitly requires explicit formulas for the functions (wa), making the computation unhandy. Due to these factors and the Taylor expansion in m >= 0 of t0, in practice, one is left with the problem of finding an extension tD' (Rk) that scales almost homogeneously with degree D. This presents a challenge in finding a distribution f0 D' (Rk\{0}) that satisfies the conditions (4.25). While a general method to solve this problem is not known, differential renormalization has been successfully applied to various concrete examples, providing a potential solution to the problem.
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