Quantum Field Theory

Two major pillars of modern physics are quantum mechanics and the theory of relativity. 

An unavoidable consequence of quantum mechanics and of the special theory of relativity is the theory of relativistic quantum fields. A quantum field is a system of infinitely many degrees of freedom, in which a quantal degree of freedom is assigned to each point in space. These fields permeate the entire space at every given moment, and their quantum fluctuations materialize as particles (or quanta of radiation). Thus,  Quantum Field Theory (QFT) is capable of explaining the creation and annihilation of particles and their interactions in between in the most natural way. Indeed, The Standard Model of elementary particle physics, namely, the theory which describes all known “elementary” particles and their interactions  under the electro-weak and the strong forces (i.e., all known forces but gravity), is a quantum field theory. This QFT was vindicated by experimental checks to very high precision, down to distances of the order of 10^{-17} cm  (one ten-thousandth the size of a proton). There is strong evidence for physics beyond the Standard Model, but even these deviations from the Standard Model seem to be described by local relativistic QFT down to length scales of 10^{-18} cm. Thus, QFT is a major framework for high energy physics,  and to this date, and with the current capabilities of experimental particle physics, it is the only theoretical framework of high energy physics confirmed by experiment.

Aspects of quantum field theory pursued here include non-perturbative aspects of quantum field theory such as vacuum (i.e., ground state) structure and the related issues of dynamical symmetry breaking, the effects of external conditions (such as boundary conditions and the ensuing Casimir effect, or the influence of external background fields) on the physical contents of quantum field theories, color confinement, extended objects (solitons and instantons, QCD strings, random  surfaces and their random geometries) and fluctuations around them.

An important arena in which many of these issues may be studied are the so-called “large N” vector and matrix models. Large N models are quantum field theories whose fields are vectors and/or matrices with a very large number of components. These components are arranged such that there is large amount of symmetry in these models, which enables exact solutions of these quantum field theories as N tends to infinity. Examples include  random non-hermitian matrix models (which are kind of lobotomized field theories living in zero space and zero time dimensions – i.e., at a point), which became popular in recent years. Another example is the formation of self- consistent solitons (a.k.a. “fermion bags”) in certain interacting field theories, which can serve as a concrete model to  formation of particles of finite size and internal structure (such as the proton, for example), out of point-like elementary excitations of the quantum fields involved (the so called quarks and gluons, in the case of the proton).

Quantum field theory is quite robust: its methods are applicable in other physical disciplines, and most notably – in condensed matter physics. 


String Theory

Our understanding of the fundamental forces and particles in nature is based on a framework called quantum field theory (QFT). This framework however does not seem to be suitable to describe the gravitational force as a quantum field theory. Both technical and conceptual issues arise when one tries to use the QFT machinery to describe gravity. String theory is a framework in which quantum gravity can be described through. However since our understanding on string theory was limited to a perturbative description it was hard to study many questions. This changed with the discovery of the AdS/CFT correspondence. This amazing idea claims that a string theory in a background space time called Anti-de-Sitter space (called the bulk)  is in fact alternatively described by a normal QFT (in this case a conformal field theory) that lives in one lower dimension (called the boundary). This correspondence is also called a holographic theory, since the space-time the string theory lives in (our world, roughly speaking) and all of the local physics in it , is emergent from a QFT description that lives in one lower dimension.

Bulk Reconstruction 

How exactly is the physics of the bulk  encoded in the boundary theory. One long time theme is the construction of local bulk operators from the fundamental degrees of freedom of the boundary theory. This in fact is tightly connected to an old problem in quantum gravity, which is how does one construct semi-local physical observables in a theory of gravity. This is also connected to the so called problem of time in quantum gravity. Apparently using ideas from AdS/CFT one can answer some of these long standing questions, and the inquiry into these through this lens have uncovered interesting connections between different objects appearing in conformal field theories, and also interesting connections with quantum information.

Bootstrap methods

Conformal field theories (CFTs) describe physical systems at their critical point, undergoing second order phase transitions. A remarkable aspect of CFTs is that there are methods to solve them even at strong coupling, due to their high degree of symmetry. This is the goal of the conformal bootstrap program, which makes use of numerical and analytical tools to solve the crossing equations of a CFT. 

One of the important new tools of the conformal bootstrap are dispersion relations. Dispersion relations enable to construct correlation functions from their singularities. Dispersion relations can be used to relate the high energy (UV) physics with the low energy (IR) physics of a theory. Moreover, dispersive techniques enable to constrain effective field theories of gravity, and as a result they have recently played a role in the swampland program.


Another interesting aspect of the holographic ideas is the ability to compute certain properties of strongly coupled quantum field theories through the study of the classical properties of the dual string theory. Certain interesting condensed matter phenomena such as super-conductivity, super-fluidity , fractional quantum Hall states, strongly coupled anyons, etc , can be studied with new tools.