Many topics of modern probability have counterparts in mathematical physics and quantum mechanics. For example, the study of the parabolic Anderson model is related to Anderson localization; interacting particle systems and spin systems are related to quantum spin systems and quantum many-body theory; and the Gaussian free field as well as Malliavin calculus connect to (Euclidean) quantum field theory.
The aim of this course is to give an introduction to quantum mechanics for mathematicians with a probability background, providing basic intuition and a dictionary facilitating access to the mathematical physics literature. The focus will be on connections with probability, notably Markov process theory, rather than partial differential equations and spectral theory.
The main blocks will be (A) the basic Hilbert space set-up of quantum mechanics and the Feynman-Kac formula, (B) quantum statistical mechanics (pure states vs. mixed states, density matrices, KMS/Gibbs states), © quantum many-body theory and quantum spin systems (operator algebras, creation and annihilation operators, fermions and bosons), and (D) a glimpse at quantum field theory.
this is a course for advanced master students or PhD-students
linear algebra, probability, Markov processes; basic knowledge of functional analysis (Hilbert spaces, spectral theory of self-adjoint operators, Fourier transforms) useful but not necessary.
M. Reed, B. Simon: Methods of modern mathematical physics. Vol I. Functional analysis, Vol II: Fourier analysis, self-adjointness, Academic Press 1972/1975. J. Glimm and A. Jaffe: Quantum physics. A functional integral point of view, Springer-Verlag, 1981.