Talks

This is the central unit of the course - we quantize universal enveloping algebras and their duals. Central discussion is the fact that for the first type of Hopf-algebras the deformation of the coproduct is sufficient and for the second type it is the dual multiplication. This motivates the way quantization is performed in particular and how this gives rise for noncommutativity for the module and comodule spaces that are so interesting for physics. Currently most popular way to quantize universal enveloping algebras is the twisting according to Drinfeld. We discuss how and why this is such a good concept.

Understanding magnetic reconnection is one of the major challenges of plasma physics. It plays an essential role in a wide range of physical systems such as stellar flares, accretion disks, active galactic nuclei, astrophysical dynamos and closer to home, intense magnetic energy releases in the Earth's magnetosphere. It is a phenomena which can be created in the laboratory. Magnetic reconnection occurs when oppositely directed components of field lines are broken and re-connected resulting in destruction of magnetic flux and topological rearrangement of magnetic field lines on very small scales. This can induce the release of magnetic energy on large scales resulting in high speed flows, heating and energetic particle production. There is a strong connection between formation of singular structures in flows and magnetic reconnection which I will discuss and tie this into some of the laboratory and natural physical systems under recent study. In addition to giving a contemporary overview on this subject I will discuss some open questions.

Clifford group as symplectic group, generators of the Clifford group & encoding circuits for stabilizer codes, efficient simulation of Clifford group circuits, efficient simulation of Pauli measurements

We consider N=2 supersymmetric quantum electrodynamics (SQED) with 2 flavors, the Fayet--Iliopoulos parameter, and a mass term $beta$ which breaks the extended supersymmetry down to N=1. The bulk theory has two vacua; at $beta=0$ the BPS-saturated domain wall interpolating between them has a moduli space parameterized by a U(1) phase $sigma$ which can be promoted to a scalar field in the effective low-energy theory on the wall world-volume. At small nonvanishing $beta$ this field gets a sine-Gordon potential. As a result, only two discrete degenerate BPS domain walls survive. We find an explicit solitonic solution for domain lines -- string-like objects living on the surface of the domain wall which separate wall I from wall II. The domain line is seen as a BPS kink in the world-volume effective theory. The domain line carries the magnetic flux which is exactly 1/2 of the flux carried by the flux tube living in the bulk on each side of the wall. Thus, the domain lines on the wall confine charges living on the wall, resembling Polyakov's three-dimensional confinement.

If a large quantum computer (QC) existed today, what type of physical problems could we efficiently simulate on it that we could not simulate on a conventional computer? In this talk, I argue that a QC could solve some relevant physical "questions" more efficiently. First, I will focus on the quantum simulation of quantum systems satisfying different particle statistics (e.g., anyons), using a QC made of two-level physical systems or qubits. The existence of one-to-one mappings between different algebras of observables or between different Hilbert spaces allow us to represent and imitate any physical system by any other one (e.g., a bosonic system by a spin-1/2 system). We explain how these mappings can be performed showing quantum networks useful for the efficient evaluation of some physical properties, such as correlation functions and energy spectra. Second, I will focus on the quantum simulation of classical systems. Interestingly, the thermodynamic properties of any d-dimensional classical system can be obtained by studying the zero-temperature properties of an associated d-dimensional quantum system. This classical-quantum correspondence allows us to understand classical annealing procedures as slow (adiabatic) evolutions of the lowest-energy state of the corresponding quantum system. Since many of these problems are NP-hard and therefore difficult to solve, is worth investigating if a QC would be a better device to find the corresponding solutions.

The mathematical formalism of quantum theory has many features whose physical origin remains obscure. In this paper, we attempt to systematically investigate the possibility that the concept of information may play a key role in understanding some of these features. We formulate a set of assumptions, based on generalizations of experimental facts that are representative of quantum phenomena and physically comprehensible theoretical ideas and principles, and show that it is possible to deduce the finite-dimensional quantum formalism from these assumptions. The concept of information, via an information-theoretic invariance principle, plays a central role in the derivation, and gives rise to some of the central structural features of the quantum formalism.

Universal Enveloping Algebras and dual Algebras of Functions The two most relevant types of Hopf-algebras for applications in physics are discussed in this unit. Most central notion will be their duality and representation.
Motivation: From Quantum Mechanics to Quantum GroupsThe notion of 'quantization' commonly used in textbooks of quantum mechanics has to be specified in order to turn it into a defined mathematical operation. We discuss that on the trails of Weyl's phase space deformation, i.e. we introduce the Weyl-Moyal starproduct and the deformation of Poisson-manifolds. Generalizing from this, we understand, why Hopf-algebras are the most genuine way to apply 'quantization' to various other algebraic objects - and why this has direct physical applications.

Hopf-Algebras and their Representations In order to consolidate the above motivation, we have to introduce Hopf-algebras on a mathematical footing. We define Hopf-algebras, discuss duality and especially we will have a closer look at the question why coproducts induce a multiplication on the dual algebra but not the other why around. With these preparations we close this unit by the discussion of representations and corepresentations - and how these are related for dual Hopf-algebras.

Motivation: From Quantum Mechanics to Quantum Groups The notion of 'quantization' commonly used in textbooks of quantum mechanics has to be specified in order to turn it into a defined mathematical operation. We discuss that on the trails of Weyl's phase space deformation, i.e. we introduce the Weyl-Moyal starproduct and the deformation of Poisson-manifolds. Generalizing from this, we understand, why Hopf-algebras are the most genuine way to apply 'quantization' to various other algebraic objects - and why this has direct physical applications.