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

In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated (fuzzy) sphere. The corresponding field theory actions conserve all basic symmetries of the standard commutative version (space isometries and global chiral symmetry), but due to the noncommutativity of the space the fields are regularized and they contain only finite number of modes.

Topologically nontrivial field configurations in noncommutative geometry

Observables of supersymmetric quantum mechanics are coded by taking the antisymmetric tensor product with anticommuting parameters. Next the authors define superunitary transformations, which mix bosonic and fermionic degrees of freedom, in order to construct automorphisms of the canonical (anti)commutation relations (C(A)CR). Conversely, every automorphism of the C(A)CR is implemented by an essentially unique superunitary transformation.

Groups of automorphisms of the canonical commutation and anticommutation relations

Statistical mechanics of polyacetylene (CH) x

Equivalence of the Z4 and the Z2 × Z2 lattice gauge theories

A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the differencial calculus of non-commutative geometry. The regulated theory involves only finite number of degrees of freedom and is manifestly supersymmetric.

Harald Grosse

Papers

Topologically nontrivial field configurations in noncommutative geometry

Groups of automorphisms of the canonical commutation and anticommutation relations

Statistical mechanics of polyacetylene (CH) x

Equivalence of the Z4 and the Z2 × Z2 lattice gauge theories

Field Theory on a Supersymmetric Lattice