## 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.