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Quantum Fields and Probability

Periodic Reporting for period 3 - QFPROBA (Quantum Fields and Probability)

Reporting period: 2020-10-01 to 2022-03-31

Quantum Field Theory (QFT) has become a universal framework to study physical systems with infinite number of degrees of freedom. It was created by the pioneers of Quantum Mechanics in the 30’s for the study of the interaction of the electromagnetic field with charged matter. With Quantum Electrodynamics and later with the Standard Model all known fundamental forces were united in a QFT. QFT ideas spread to condensed matter physics in the 50’s and lead to a spectacular success in the 60’s by the explanation of universality in the phenomena of second order phase transitions using the Renormalization Group (RG), a powerful method to study scale invariant problems. QFT and RG ideas were then successfully applied to noisy systems, dynamical systems, non-equilibrium systems and many other problems.

Early on QFT was also acknowledged as an interesting problem for mathematicians as its rigorous formulation and study was challenging. This lead first in the 50’s to axiomatic characterizations of what sort of mathematical object QFT is and in then in the 60’s to Constructive QFT that set as its goal to produce concrete examples of QFT satisfying the axioms.
New ideas from probability theory lead to its biggest achievements by the 80’s. These techniques were then used in rigorous statistical mechanics, disordered systems and many other problems.

In the end of 90’s a new interaction between probability theory and statistical physics started with Oded Schramm’s SLE (Schramm-Loewner Evolution), a new approach to two dimensional critical phenomena and QFT via random curves (phase boundaries). Subsequent work by Duplantier and Sheffield on random surfaces and of Smirnov in scaling limit of percolation and the Ising mode were landmarks in these new approaches. In the field of Stochastic Partial Differential Equations QFT ideas on renormalization were applied by Hairer to great effect.

The present project grows out of these developments. Its aim is to bring new ideas from probability theory to constructive QFT. It aims to a probabilistic understanding of Liouville QFT which has been a central element in many developments in the physics of string theory and quantum gravity as well as the mathematics of random surfaces and representation theory. It also aims to developing the renormalization group as a probabilistic tool to understand singular dynamical problems. The overall aim is to bring the communities of probabilists and theoretical physicists working on these problems to a closer collaboration.
Our main results so far are the proofs of two celebrated conjectures were made by physicists on the Liouville Conformal Field Theory (LCFT):

1. An exact expression due to Dorn, Otto, Zamoldchicov and Zamoldchicov for the three-point correlation function of LCFT on the two-sphere, the DOZZ formula.

2. The conformal bootstrap formula for the four and higher point functions on the 2-sphere.

These are major breakthroughs in the field.

Other work done during the first period (detailed in project achievements) is:

-Virasoro representation theory for LCFT
-Analytic continuation of LCFT correlation functions
-Local limit of critical Ising model on random planar triangulations
-A new probabilistic construction of the Phi^4_2 theory
-A new proof Cardy's formula for the critical percolation model
-Asymptotics of the determinant of the discrete Laplacian on surfaces
The proofs of the DOZZ formula and the bootstrap are beyond state of the art. They have already sparkled followup work by several authors. We expect these methods to extend to LCFT in other surfaces and possibly shed new light to major questions in theoretical physics around the Alday-Gaiotto-Tachicawa conjecture. This will be our main line of research in the near future.
Quantum sphere courtesy Francois David