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

Field theories on deformed spaces suffer from the IR/UV mxing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this desease by adding one more marginal operator. We review these ideas, show the application to φ 3 models and use heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a θ-deformed space and derive noncommutative gauge actions.

Renormalization and Induced Gauge Action on a

We study the graded derivation-based noncommutative differential geometry of the $Z_2$-graded algebra ${\bf M}(n| m)$ of complex $(n+m)\times(n+m)$-matrices with the ``usual block matrix grading'' (for $n\neq m$). Beside the (infinite-dimensional) algebra of graded forms the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show, that ${\bf M}(n|m)$ is a ``noncommutative graded manifold'' in a stricter sense: There is a natural body map and the cohomologies of ${\bf M}(n|m)$ and its body coincide (as in the case of ordinary graded manifolds).

Graded Differential Geometry of Graded Matrix Algebras

One can prove and extend some results about the order and spacings of energy levels for the Klein–Gordon equation with vector‐like and scalar‐like potentials.

Order and spacings of energy levels for the Klein–Gordon equation

Over many years, we developed the construction of the ϕ4-model on four-dimensional Moyal space The solution of the related matrix model $\mathcal {Z}[E,J]=\int d{\Phi } \exp (\text {tr}(J{\Phi }- E{\Phi }^{2} -\frac {\lambda }{4} {\Phi }^{4}))$
 is given in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of E The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group Locality is fulfilled The Schwinger 2-point function is reflection positive in special cases

/pdf/how-prof-zeidler-supported-our-research-on-exact-solution-of-4xjxjvk12w.pdf

Harald Grosse

Papers

Renormalization and Induced Gauge Action on a

Graded Differential Geometry of Graded Matrix Algebras

Order and spacings of energy levels for the Klein–Gordon equation

How Prof. Zeidler Supported Our Research on Exact Solution of Quantum Field Theory Toy Models

Algebraic theory of Coulomb scattering