This article reviews the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, on both the classical and the quantum level.

Noncommutative field theory

Quantum field theory on noncommutative spaces

Thank you very much for reading methods of modern mathematical physics. Maybe you have knowledge that, people have look numerous times for their favorite novels like this methods of modern mathematical physics, but end up in harmful downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they are facing with some infectious virus inside their desktop computer. methods of modern mathematical physics is available in our digital library an online access to it is set as public so you can download it instantly. Our books collection saves in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the methods of modern mathematical physics is universally compatible with any devices to read.

Methods Of Modern Mathematical Physics

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.

/pdf/the-quantum-structure-of-spacetime-at-the-planck-scale-and-1sm3koeaki.pdf

The quantum structure of spacetime at the Planck scale and quantum fields

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

A new method is presented to study supersymmetric quantum mechanics. Using relative scattering techniques, basic relations are derived between Krein’s spectral shift function, the Witten index, and the anomaly. The topological invariance of the spectral shift function is discussed. The power of this method is illustrated by treating various models and calculating explicitly the spectral shift function, the Witten index, and the anomaly. In particular, a complete treatment of the two‐dimensional magnetic field problem is given, without assuming that the magnetic flux is quantized.

/pdf/witten-index-axial-anomaly-and-krein-s-spectral-shift-2cfpwklcw8.pdf

Witten index, axial anomaly, and Krein's spectral shift function in supersymmetric quantum mechanics

We reformulate the concept of connection on a Hopf–Galois extension B⊆P in order to apply it in computing the Chern–Connes pairing between the cyclic cohomology HC

 2

 n
 (B) and K

 0 (B). This reformulation allows us to show that a Hopf–Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz–Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern–Connes pairing for the super line bundles associated to a super Hopf fibration.

/pdf/strong-connections-and-chern-connes-pairing-in-the-hopf-3uaoiw7tvy.pdf

Strong Connections and Chern–Connes Pairing¶in the Hopf–Galois Theory

We present the main ideas and techniques of the proof that the duality-covariant four-dimensional non-commutative Φ4-model is renormalisable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalisation byflow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces

Harald Grosse

Papers

Witten index, axial anomaly, and Krein's spectral shift function in supersymmetric quantum mechanics

Strong Connections and Chern–Connes Pairing¶in the Hopf–Galois Theory

Renormalisation of ?4-Theory on Non-Commutative $mathbb{R}(4) $ to All Orders

Field theory on the q-deformed fuzzy sphere I

Renormalization of the noncommutative ϕ3 model through the Kontsevich model