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This chapter discusses the perturbative calculation of correlation or vertex functions expressed in terms of field (functional) integrals. The successive contributions to the perturbative expansion are Gaussian expectation values which can be calculated with the help, for example, of Wick′s theorem and which have a representation in the form of Feynman diagrams. It illustrates diagrammatically the relations between the first connected correlation functions and the corresponding vertex functions. It shows that the calculation of a field integral by the steepest descent method organizes the perturbative expansion as an expansion in the number of loops in the Feynman diagram representation. Finally, it defines here more generally dimensional continuation and introduces dimensional regularization. Exercises are provided at the end of the chapter.

This chapter uses the assumptions introduced in Chapter 9 to show that it is indeed possible to find a non-Gaussian fixed point in dimension d = 4 - e, both in models with reflection and rotation symmetries. It briefly introduces the field theory methods that will be described more thoroughly in the following chapters. Finally, it presents a selection of numerical results concerning critical exponents and some universal amplitude ratios.

This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.

The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric wave functions of bosons. In the case of fermion systems, however, one faces the problem that fermion wave functions, or fermion correlation functions (or Green functions) are antisymmetric with respect to the exchange of a fermion pair. Thus, the construction of generating functions requires the introduction of an antisymmetric or Grassmann algebra of ‘classical functions’. It is then possible to generalize to Grassmann algebras the notions of derivatives and integrals, yielding quite parallel formalisms for bosons and fermions, in particular, to define a path integral for fermion systems, analogous to the holomorphic path integral for bosons. This chapter discusses differentiation and integration in Grassmann algebras, Gaussian integrals and perturbative expansion, partition function, and quantum Fermi gas.

This chapter studies in detail the tensor model with an arbitrary quartic interaction. Using the Hubbard–Stratonovich intermediate field representation, the quartic model can be reformulated in terms of a model of edge colored maps. In this new formulation, it is shown that the 1/N expansion of the model holds non perturbatively. It is shown that the cumulants are Borel summable in the coupling constant, uniformly in N. The perturbative 1/N consists in showing that each term in a formal power series expansion in the coupling constants of a cumulant obeys a scaling bound in 1/N. Establishing the 1/N expansion non perturbatively consists in proving that any cumulant is an analytic function of the coupling constant in a certain domain in the complex plane and showing that, in this domain, the cumulant writes as a sum of explicit terms in 1/N plus a rest term, which is analytic and obeys an appropriate scaling bound in 1/N.

In quantum field theory (QFT), the main analytic tool to calculate physical quantities is the perturbative expansion. Following, Dyson's intuitive argument, the divergence of perturbative series was demonstrated in some models of quantum mechanics (QM) with polynomial potentials, using the Schrödinger equation. Later, it was proposed to study the problem within a path integral formulation. A systematic method in field theory was proposed by Lipatov, using the field integral representation of the φ⁴₄ field theory and instantons. It can be shown that the ground-state energy of the quartic anharmonic oscillator is analytic in a cut-plane. The imaginary part of the energy on the cut is related to barrier penetration. The behaviour of the perturbative coefficients at large orders is related to the behaviour of the imaginary part for small and negative coupling and can be obtained by instanton methods. The method has been generalized to the class of potentials for which (in general complex) instanton contributions have been calculated. The same method can be readily applied to boson field theories, while the extension to field theories involving fermions, like Quantum QED, requires additional considerations. The general conclusion is that, in QFT, all perturbative series, expanded in terms of a loop-expansion parameter, are divergent series.

This chapter presents the perturbative expansion of invariant tensor measures in terms of Feynman graphs. It is shown that, assuming that the perturbation and the Gaussian part scale at the same rate with N, the moments of such measures admit (as formal power series in the coupling constants) a 1/N expansion indexed by the degree and that (still in the perturbative sense) all such measures are properly uniformly bounded. In the second part of the chapter the continuum limit of random tensor models and their Schwinger–Dyson equations are discussed.

This chapter presents an alternative approach to the quantization of fields, an approach that will be critically important for the development of quantum field theory in curved space, which is the subject of the second part of the book. It starts by providing a description of a functional integral in quantum mechanics, concentrating on the representation of an evolution operator. It then considers the functional representation of the Green functions and the generating functional in quantum field theory, including for fermionic theories. After that, perturbative calculations of the generating functionals and their general properties are formulated. The chapter ends with a brief description of ζ‎-regularization as a technique for defining functional determinants.