Noncommutative field theory is the analog of classical field theory on quantum spaces. We discuss both classical as well as quantum aspects of such field theories. In particular, a transparent understanding of the crucial phenomenon of UV/IR mixing is obtained using the novel tool of string modes. These exhibit the stringy nature of noncommutative field theory, and will play an important role in the following. In particular, noncommutative gauge theory is defined via Yang–Mills matrix models.

We discuss the problem of finding the optimal constant in Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities, thereby introducing, in particular, the semiclassical constant and the one-particle constants, which appear in the Lieb–Thirring conjecture. We discuss Keller's problem of minimizing the lowest eigenvalue of a Schrödinger operator among all potentials with a given L^p norm. We present the Aizenman–Lieb monotonicity argument, as well as semiexplicit computations for eigenvalues of the harmonic oscillator (including the counterexample of Helffer and Robert) and the Pöschl–Teller potential. In the one-dimensional case, we present the optimal bounds due to Hundertmark–Lieb–Thomas and Gardner–Greene–Kruskal–Miura. We provide two proofs of the latter bound, namely, the original one based on trace formulas and a more recent one by Benguria and Loss based on the commutation method.

We prove Lieb–Thirring inequalities with optimal, semiclassical constant in higher dimensions by following the Laptev–Weidl approach of "lifting in dimension." We introduce Schrödinger operators with matrix-valued potentials and show how Lieb–Thirring inequalities with semiclassical constants for such operators in one dimension imply the Lieb–Thirring inequality with semiclassical constant in higher dimensions. Subsequently, we prove a sharp Lieb–Thirring inequality in one dimension with exponent 3/2 for Schrödinger operators with matrix-valued potentials. We give a complete proof using the commutation method by Benguria and Loss. We also sketch the original proof by Laptev and Weidl based on trace formula for Schrödinger operators with matrix-valued potentials.

We discuss various independent aspects of sharp Lieb–Thirring inequalities. First, we present an argument of Stubbe which shows that Riesz means of order two and higher approach their semiclassical limit monotonically, thus leading to an alternative proof of sharp Lieb–Thirring inequalities. Next, we discuss the number of negative eigenvalues of Schrödinger operators with radial potentials, following Glaser, Grosse, and Martin. This leads, on the one hand, to a sharp CLR inequality for radial potentials in dimension 4 and, on the other hand, to a counterexample to the Lieb–Thirring conjecture with exponent zero in sufficiently high dimensions. Next, we discuss briefly an approach that disproves the Lieb–Thirring conjecture in a certain range of positive exponents. Finally, we discuss the Lieb–Thirring inequality with exponent one in its dual formulation, also known as kinetic energy inequality, in which it enters in many applications.