Periodic Reporting for period 4 - QFPROBA (Quantum Fields and Probability)
Reporting period: 2022-04-01 to 2024-03-31
Summary of the context and objectives
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.
Conclusion of the action
There are two complementary approaches in physics to Quantum Field Theory : Feynman's Path Integral and Conformal Bootstrap. This project has for the first time in the history of mathematical QFT been able to reconcile these approaches in the context of the Liouville QFT. Starting from a rigorous probabilistic construction of the path integral the axioms of the bootstrap approach have been verified and used to obtain a complete solution of the LCFT.
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.
Conclusion of the action
There are two complementary approaches in physics to Quantum Field Theory : Feynman's Path Integral and Conformal Bootstrap. This project has for the first time in the history of mathematical QFT been able to reconcile these approaches in the context of the Liouville QFT. Starting from a rigorous probabilistic construction of the path integral the axioms of the bootstrap approach have been verified and used to obtain a complete solution of the LCFT.
Our main results are the proofs of celebrated conjectures that 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 spectrum of critical exponents of LCFT.
3. Using 1 and 2 to verify conformal bootstrap axioms and to obtain explicit expressions for LCFT correction functions on all Riemann surfaces.
These are major breakthroughs in the field.
Other major results of the project are:
-A probabilistic construction of the Virasoro algebra symmetry of LCFT by constructing a a new family of Markovian dynamics associated to holomorphic vector fields defined in the disk. Using this we have also determined explicitly the scattering matrix of the theory.
-Solution of the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian Multiplicative Chaos measures.
-Solution of the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature and computation of the critical perimeter exponents
-Use of singular stochastic PDE methods and variational and stochastic control method to proving sInvariance of phi^4 QFT measure under nonlinear wave and Schrödinger equations on the plane, stochastic quantization of Sinh-Gordon QFT, construction of phi^4 QFT model in infinite volume and construction of invariant Gibbs measure for Anderson Non Linear Wave equation
-Regularity results for singular SPDES such as Stochastic Schauder-Tychonoff type theorems, stability and moment estimates and Sobolev regularity of occupation measures and paths
- Self similar long time asymptotic for coagulation equations with applications to atmospheric physics and population dynamics- proving self similar behaviour, long time asymptotic and also the existence of anomalous self similarity. Franco also applied renewal equations to population dynamics of physiologically structured populations.
The results of the project have been disseminated in journal articles and conference talks. They were also featured in a Quanta magazine article and selected for the main mathematics breakthroughs in 2022. The proof of the DOZZ formula was awarded the Georg Polya prize by SIAM.
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 spectrum of critical exponents of LCFT.
3. Using 1 and 2 to verify conformal bootstrap axioms and to obtain explicit expressions for LCFT correction functions on all Riemann surfaces.
These are major breakthroughs in the field.
Other major results of the project are:
-A probabilistic construction of the Virasoro algebra symmetry of LCFT by constructing a a new family of Markovian dynamics associated to holomorphic vector fields defined in the disk. Using this we have also determined explicitly the scattering matrix of the theory.
-Solution of the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian Multiplicative Chaos measures.
-Solution of the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature and computation of the critical perimeter exponents
-Use of singular stochastic PDE methods and variational and stochastic control method to proving sInvariance of phi^4 QFT measure under nonlinear wave and Schrödinger equations on the plane, stochastic quantization of Sinh-Gordon QFT, construction of phi^4 QFT model in infinite volume and construction of invariant Gibbs measure for Anderson Non Linear Wave equation
-Regularity results for singular SPDES such as Stochastic Schauder-Tychonoff type theorems, stability and moment estimates and Sobolev regularity of occupation measures and paths
- Self similar long time asymptotic for coagulation equations with applications to atmospheric physics and population dynamics- proving self similar behaviour, long time asymptotic and also the existence of anomalous self similarity. Franco also applied renewal equations to population dynamics of physiologically structured populations.
The results of the project have been disseminated in journal articles and conference talks. They were also featured in a Quanta magazine article and selected for the main mathematics breakthroughs in 2022. The proof of the DOZZ formula was awarded the Georg Polya prize by SIAM.
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 several other Conformal Field Theories such as the Wess-Zumino-Witten CFT and are working to that end.