Quantum field theory, integrable models and beyond

Harald Grosse, Edwin Langmann A, Ernst Raschhofer

Gesualdo Delfino A, Giuseppe Mussardo, P. Simonetti

Gábor Takács

<!--HTML-->I give an introduction the Truncated Conformal Space Approach, which was originally invented by Yurov and Zamolodchikov in the framework of the scaling Lee-Yang model. Since then it has become a very powerful tool to study not only finite volume spectra, but also form factors, resonances, phase transitions and renormalization group flows in one-dimensional quantum field theories, both integrable and non-integrable. Its scope has been extended to theories with boundaries and impurities, and also generalized to settings beyond perturbed conformal field theories. In this talk, beyond explaining the basics of the approach I also give a brief overview of the recently introduced renormalization methods to deal with cut-off dependence, which led to a major improvement in precision and also widened the potential scope of applications.

Carl A. Brannen

In 2014 Steven Weinberg noted that quantum mechanics can avoid various difficulties, such as the many worlds hypothesis, by taking the quantum states to be density matrices without reference to state vectors. An immediate consequence of Weinberg's idea is that electron spin can be taken to follow SO(3) instead of SU(2). This radical departure from present understanding motivates our exploration of density matrices as a method of going beyond the Standard Model. An important tool for Standard Model calculations is the Feynman path integral formulation of quantum field theory. When the path integral is Wick rotated from time to imaginary time or temperature it becomes a method of cooling down density matrices. While this does not show that one goes beyond the Standard Model by Wick rotation, it does show a close relation between this method of cooling density matrices and quantum field theory of the Standard Model. We explore these ideas and exhibit toy models with particle content and symmetry similar to the Standard Model.

Hector J. de Vega

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.