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

As argued previously, amplitudes of quantum field theories on non-commutative space and time cannot be computed using naive path integral Feynman rules. One of the proposals is to use the Gell-Mann-Low formula with time-ordering applied before performing the integrations. We point out that the previously given prescription should rather be regarded as an interaction-point time-ordering. Causality is explicitly violated inside the region of interaction. It is nevertheless a consistent procedure, which seems to be related to the interaction picture of quantum mechanics. In this framework we compute the one-loop self-energy for a space/time non-commutative $\phi^4$ theory. Although in all intermediate steps only three-momenta play a role, the final result is manifestly Lorentz covariant and agrees with the naive calculation. Deriving the Feynman rules for general graphs, we show, however, that such a picture holds for tadpole lines only.

Space/time non-commutative field theories and causality

We show how the fields and particles of the standard model can be naturally realized in noncommutative gauge theory. Starting with a Yang-Mills matrix model in more than four dimensions, an $SU(n)$ gauge theory on a Moyal-Weyl space arises with all matter and fields in the adjoint of the gauge group. We show how this gauge symmetry can be broken spontaneously down to $SU(3{)}_{c}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}U(1{)}_{Q}$ [resp. $SU(3{)}_{c}\ifmmode\times\else\texttimes\fi{}U(1{)}_{Q}$], which couples appropriately to all fields in the standard model. An additional $U(1{)}_{B}$ gauge group arises which is anomalous at low energies, while the trace-$U(1)$ sector is understood in terms of emergent gravity. A number of additional fields arise, which we assume to be massive, in a pattern that is reminiscent of supersymmetry. The symmetry breaking might arise via spontaneously generated fuzzy spheres, in which case the mechanism is similar to brane constructions in string theory.

Noncommutative gauge theory and symmetry breaking in matrix models

We study the feasibility of detecting noncommutative (NC) QED through neutral Higgs boson (H) pair production at linear colliders (LC). This is based on the assumption that H interacts directly with photon in NCQED as suggested by symmetry considerations and strongly hinted by our previous study on ${\ensuremath{\pi}}^{0}$-photon interactions. We find the following striking features as compared to the standard model (SM) result: (1) generally larger cross sections for an NC scale of order 1 TeV; (2) completely different dependence on initial beam polarizations; (3) distinct distributions in the polar and azimuthal angles; and (4) day-night asymmetry due to the Earth's rotation. These will help to separate NC signals from those in the SM or other new physics at LC. We emphasize the importance of treating properly the Lorentz noninvariance problem and show how the impact of the Earth's rotation can be used as an advantage for our purpose of searching for NC signals.

Pair production of neutral Higgs bosons through noncommutative QED interactions at linear colliders

We study the quantization of the noncommutative selfdual 3 model in 4 dimensions, by mapping it to a Kontsevich model. The model is shown to be renormalizable, provided one additional counterterm is included compared to the 2-dimensional case [1] which can be interpreted as divergent shift of the field . The known results for the Kontsevich model allow to obtain the genus expansion of the free energy and of any n-point function, which is finite for each genus after renormalization. No coupling constant or wavefunction renormalization is required. A critical coupling is determined, beyond which the model is unstable. This provides a nontrivial interacting NC field theory in 4 dimensions.

A nontrivial solvable noncommutative phi 3 model in 4 dimensions

This monograph introduces modern developments on the bound state problem in Schrodinger potential theory and its applications in particle physics. The Schrodinger equation provides a framework for dealing with energy levels of N-body systems. It was a cornerstone of the quantum revolution in physics of the twenties but re-emerged in the eighties as a powerful tool in the study of spectra and decay properties of mesons and baryons. This book begins with a detailed study of two-body problems, including discussion of general properties, level ordering problems, energy level spacing and decay properties. Following chapters treat relativistic generalisations, and the inverse problem. Finally, 3-body problems and N-body problems are dealt with. Applications in particle and atomic physics are considered, including quarkonium spectroscopy. The emphasis throughout is on showing how the theory can be tested by experiment. Many references are provided.

Harald Grosse

Papers

Space/time non-commutative field theories and causality

Noncommutative gauge theory and symmetry breaking in matrix models

Pair production of neutral Higgs bosons through noncommutative QED interactions at linear colliders

A nontrivial solvable noncommutative phi 3 model in 4 dimensions

Particle Physics and the Schrödinger Equation