# Guide for M-Phi

## Guide MΦ

In the study program Mathematical Physics one has great freedom, both in the bachelor and the master program, to choose lectures. To guarantee a reasonable choice and a meaningful order one may need a guiding hand, in particular, to have the needed mathematical background for physics lectures at the right time.

### Bachelor Program

The following list gives a selection of lectures in the bachelor program and indicates their mutual relations as well as their relations to lectures in the physics part of the bachelor program in mathematical physics. It should be understood from the point of view of Chair X, nevertheless it might be useful in general. If you are interested in a bachelor thesis at Chair X the following lectures are very helpful.

Note, however, that the following lectures will mainly be offered in German only. For a more detailed description we refer to the German version.

### 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.

At Chair X the differential geometric results of this lecture are needed, used and developed further in various ways. On the one hand, the theory of principal fiber bundles is not only the starting point for Yang-Mills field theories but also the basis for a general study of group actions of Lie groups. Such actions occur also in situations in Geometric Mechanics. But aspects of wave equations on globally hyperbolic spacetimes also play an increasingly important role. Finding solutions with global Green operators is the starting point for many quantization methods in field theory.

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.

Differentiable manifolds are defined as specific topological spaces with additional properties, a differentiable structure, which allows one to speak

intrinsically (i.e. without referring to an embedding as a subset of Euclidean space) about differentiability and tangent spaces. For this purpose, a not inconsiderable conceptual effort is required first. But then you get a very powerful theory to formulate geometric questions in a coordinate-free way.

In mathematical physics, there are maybe two really important trends in terms of using modern mathematical concepts: on the one hand, the functional-analytic techniques, which are needed in any form of quantum theory. On the other hand, there is a geometrization of physical theories in almost every field. Here, of course, the general theory of relativity is only a particularly prominent example, but also the Yang-Mills theories for the interactions in particle physics as well as the question of spinors lead in the end to new geometric formulations and insights. Topological properties also play an increasingly important role, for example in solid-state physics. For this reason, differential geometry has to be considered as one of the important pillars of any modern physical theory and therefore

it should not be missing in any study of mathematical physics.

At Chair X, a basic knowledge of differential geometry is probably almost indispensable. Even for the more algebraic or functional-analytical questions at Chair X, differential geometry always offers the decisive examples, the motivation and, not at least, the intuition of what should actually be done. In particular, there are also the continuing lectures on Geometric Mechanics or Pseudo-Riemannian and Riemannian Geometry but also the Analysis und Geometry of Classical Systems , which are of importance and interest for the work at the Chair X. Finally, the differential geometry forms the starting point for various working groups, seminars and not at least the master's theses at Chair X.

The lecture *Geometric Mechanics* is concerned with the task of formulating Lagrangian and especially Hamiltonian mechanics in the language of differential geometry. In the case of Hamiltonian mechanics this goal is achieved using the notion of symplectic manifolds. Various statements about classical mechanics known from physics lectures can then be considered in a differential geometric context. Often, this allows us to give easier and more conceptual proofs. This makes the lecture to a logical successor of the bachelor lecture on classical mechanics with obvious relevance in mathematical physics.

Beside these aspects there are strong mathematical reasons to deal with symplectic geometry. There are various applications in pure and applied mathematics, from efficient numerical algorithms in the theory of Ordinary Differential Equations, like symplectic integrators, to deep applications in representations theory of Lie algebras and Lie groups. Furthermore, Kähler geometry provides a connection from symplectic geometry to other areas of geometry as for example Riemannian or complex differential geometry.