Perturbative Methods in Field Theory

Philippe A. Martin, FRANÇOIS ROTHEN

Giuseppe Mussardo

Martí Rovira, Robert J. Perry, A. Parreño

In strongly coupled field theories, perturbation theory cannot be employed to study the low-energy spectrum. Thus, non-perturbative techniques are required. We employ the variational method, a rigorous, non-perturbative approach which provides variational upper bounds on the energy eigenstates of the theory. An essential step in the variational method is the choice of trial wave function. In this work, we study the viability of employing a neural network as our variational ansatz. As a first step towards phenomenologically interesting strongly coupled theories like quantum chromodynamics, we study scalar field theories with quartic couplings.

Marco Frasca

We present a full introduction to the recent devised perturbation theory for strong coupling in quantum mechanics. In order to put the theory in a proper historical perspective, the approach devised in quantum field theory is rapidly presented, showing how it implies a kind of duality in perturbation theory, from the start. The approach of renormalization group in perturbation theory is then presented. This method permits to resum secularities in perturbation theory and makes fully algorithmical the resummation, transforming the perturbation calculations in a step by step computational procedure. The general theorem on which is founded a proper application of the strong coupling expansion, based on a result in the quantum adiabatic theory, is then exposed. This theorem gives the leading order of a strong coupling expansion. Then, after the introduction of the principle of duality in perturbation theory that puts in a proper context the quantum field theory method, the resulting theory of the strong coupling expansion and the free picture are presented. An algorithm for the computation of the perturbation series is finally given. This approach has a lot of applications in fields as quantum optics, condensed matter and so on, extending the original expectations of the quantum field theory method. So, we give some examples of application for a class of two-level systems that, in recent years, proved to be extremely important. One of the most interesting concepts that can be obtained in this way is that of a Quantum Amplifier (QAMP) that permits to obtain an amplification to the classical level of the quantum fluctuations of the ground state of a single radiation mode.

Jean Zinn‐Justin

Path integrals have played a fundamental role in emphasizing the profound analogies between Quantum Field Theory (QFT), and Classical as well as Quantum Statistical Physics. Ideas coming from Statistical Physics have then led to a deeper understanding of Quantum Field Theory and open the way for a wealth of non-perturbative methods. Conversely QFT methods are become essential for the description of the phase transitions and critical phenomena beyond mean field theory. This is the point we want to illustrate here. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to an accurate determination of critical exponents of the N-vector model, and more recently of the equation of state of the 3D Ising model. The starting point is the perturbative expansion for RG functions or the effective potential to the order presently available. Perturbation theory is known to be divergent and its divergence has been related to instanton contributions. This has allowed to characterize the large order behaviour of perturbation series, an information that can be used to efficiently "sum" them. Practical summation methods based on Borel transformation and conformal mapping have been developed, leading to the most accurate results available probing field theory in a non perturbative regime. We illustrate the methods with a short discussion of the scaling equation of state of the 3D Ising model. Compared to exponents its determination involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order-dependent mapping.