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Mathematical Physics

Guide for MΦ

The Mathematical Physics program

Master Program

In the master program Mathematical Physics there are many possible courses you can participate beside the two mandatory lectures. The following list collects some of the relevant lecture courses if you are interested in a master thesis at Chair X. Moreover, some relations between these courses are indicated. Beside the lectures there are also RiGs of interest: here Chair X offers regularly a broad variety of RiGs. If interested in a master thesis at Chair X one should have participated in a reasonable selection of the following courses. In addition we expect a regular interest in the seminar Deformation Quantization.

The following lectures will be offered either in English or in German, depending on the lecturer and the requests by the participants.If you are interested in one of these lectures being held in English it is advisable to contact the lecturer early in advance.

The lecture Algebra and Dynamics of Quantum Systems in the master program is one of the two mandatory courses. On the physics side the focus is on quantum theory with particular emphasis on conceptual questions: the structure of a quantum theory is determined by its observable algebra, states are a secondary concept arising from the algebra and representations of the abstract observable algebra by explicit operators have to be studied in detail.

In quantum physics, observables are described by a non-commutative *-algebra. Already on the purely algebraic side one can define states as positive functionals and describe representation on pre-Hilbert spaces. However, to obtain physically reasonable spectra and spectral measures one needs a non-trivial amount of analysis. Here several analytical requirements on the observable algebras can be asked, culminating in the definition of a C*-algebra. For  such algebras one has a useful notion of a spectrum in combination with a continuous calculus. To understand representations one needs a more detailed understanding of operators on a Hilbert space. Also  an extension of the continuous calculus to a measurable one is essential. Unbounded operators on a Hilbert space are then the starting point to define a dynamics in quantum theory by means of the Schrödinger equation. Here self-adjointness is of crucial importance.

This lecture can be seen as a particular continuation of Functional Analysis from the Bachelor program. The results from that lecture on Banach and Hilbert spaces are re-used and refined, including in particular the study of operators on such spaces. Moreover, some techniques from  Complex Analysis and Topology are needed.

The mathematical techniques from the theory of C*-algebras and operator algebras are used in axiomatic approaches to quantum theory and in particular in axiomatic quantum field theory (AQFT). Here the Haag-Kastler axioms provide a very general framework for the notion of quantum field theories on Minkowski space. In the recent years this point of view has been substantially refined and transferred to arbitrary globally hyperbolic spacetimes.

At Chair X the algebraic aspects of observable algebras are studied in detail. One important aspect is the actual construction of the quantum mechanical observable algebra starting from classical data. Here we use the techniques of deformation of algebras. In a second step questions about functional analytical properties of the algebras arise since the algebras are not or not directly C*-algebras. For that reasons also other types of topological algebras are considered.

The lecture Analysis and Geometry of Classical Systems is the second mandatory course in the master program on mathematical physics. The topic is now the  analytic and geometric aspects of systems from classical physics, though with substantial variations from year to year. One central theme is classical field theories. On the one hand one needs to find geometric techniques and formulations which stay meaningful on general spacetimes. On the other hand, one needs sophisticated analytic methods to understand the solutions of the field  equations which are typically partial differential equations.

As field theories one can consider systems from continuum mechanics of fluid dynamics. There, the field equations describe the movements and deformations of solid state physics of the flow behavior of liquids, respectively. In the second variant of this course one considers field equations from particle physics, in particular the classical Yang-Mills  equations thus providing the classical starting point for the usual quantum field theories in a differential  geometric context. On the mathematical side, building on the results of Differential Geometry, the theory of principal fiber bundles and their connections need to be developed. There the main focus is on the geometry side of the story. Analytical questions on the solution theory prove to be highly non-trivial. A third version of the course considers (linear) wave equations on general spacetimes, i.e. on  Pseudo Riemannian Manifolds with Lorentz signature as they are needed in general relativity. For sufficiently nice spacetimes a global solution can be found. Depending on the pre-knowledge of the participants aspects of differential geometry, Lie groups and spacetimes are recalled at the beginning of the lecture.

The master course Differential Geometry is a continuation of the Bachelor lectures Geometric Analysis and Elementary Differential Geometry. Here abstract manifolds  are the main focus.  A very useful prerequisite for this lecture is point set topology.