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Nima Moshayedi
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These notes were inspired by the course Quantum Field Theory from a Functional Integral Point of View given at the University of Zurich in Spring 2017 by Santosh Kandel. We describe Feynmans path integral approach to quantum mechanics and quantum field theory from a functional integral point of view, where the main focus lies in Euclidean field theory. The notion of Gaussian measure and the construction of the Wiener measure are covered. Moreover, we recall the notion of classical mechanics and the Schrodinger picture of quantum mechanics, where it shows the equivalence to the path integral formalism, by deriving the quantum mechanical propagator out of it. Additionally, we give an introduction to elements of constructive quantum field theory.
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A comprehensive study is performed of general massive, tensor, two-loop Feynman diagrams with two and three external legs. Reduction to generalized scalar functions is discussed. Integral representations, supporting the same class of smoothness algorithms already employed for the numerical evaluation of ordinary scalar functions, are introduced for each family of diagrams.
We provide new insight into the analysis of N-body problems by studying a compactification $M_N$ of $mathbb{R}^{3N}$ that is compatible with the analytic properties of the $N$-body Hamiltonian $H_N$. We show that our compactification coincides with the compactification introduced by Vasy using blow-ups in order to study the scattering theory of N-body Hamiltonians and with a compactification introduced by Georgescu using $C^*$-algebras. In particular, the compactifications introduced by Georgescu and by Vasy coincide (up to a homeomorphism that is the identity on $mathbb{R}^{3N}$). Our result has applications to the spectral theory of $N$-body problems and to some related approximation properties. For instance, results about the essential spectrum, the resolvents, and the scattering matrices of $H_N$ (when they exist) may be related to the behavior near $M_Nsetminus mathbb{R}^{3N}$ (i.e. at infinity) of their distribution kernels, which can be efficiently studied using our methods. The compactification $M_N$ is compatible with the action of the permutation group $S_N$, which allows to implement bosonic and fermionic (anti-)symmetry relations. We also indicate how our results lead to a regularity result for the eigenfunctions of $H_N$.
Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. In this monograph, using a new coend definition of the Boardman-Vogt construction of a colored operad, we define homotopy algebraic quantum field theories and homotopy prefactorization algebras and investigate their homotopy coherent structures. Homotopy coherent diagrams, homotopy inverses, A-infinity-algebras, E-infinity-algebras, and E-infinity-modules arise naturally in this context. In particular, each homotopy algebraic quantum field theory has the structure of a homotopy coherent diagram of A-infinity-algebras and satisfies a homotopy coherent version of the causality axiom. When the time-slice axiom is defined for algebraic quantum field theory, a homotopy coherent version of the time-slice axiom is satisfied by each homotopy algebraic quantum field theory. Over each topological space, every homotopy prefactorization algebra has the structure of a homotopy coherent diagram of E-infinity-modules over an E-infinity-algebra. To compare the two approaches, we construct a comparison morphism from the colored operad for (homotopy) prefactorization algebras to the colored operad for (homotopy) algebraic quantum field theories and study the induced adjunctions on algebras.
A new formalism is introduced to treat problems in quantum field theory, using coherent functional expansions rather than path integrals. The basic results and identities of this approach are developed. In the case of a Bose gas with point-contact interactions, this leads to a soluble functional equation in the weak interaction limit, where the perturbing term is part of the kinetic energy. This approach has the potential to prevent the Dyson problem of divergence in perturbation theory.
Boundary conditions in relativistic QFT can be classified by deep results in the theory of braided or modular tensor categories.
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