Noncommutative Geometry: Fuzzy Spaces, the Groenewold-Moyal Plane

In this talk, we review the basics concepts of fuzzy physics and quantum field theory on the Groenwald-Moyal Plane as examples of noncommutative spaces in physics. We introduce the basic ideas, and discuss some important results in these fields. In the end we outline some recent developments in the field.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Quantum Fields on Noncommutative Spacetimes: Theory and Phenomenology

In the present work we review the twisted field construction of quantum field theory on noncommutative spacetimes based on twisted Poincar\'e invariance. We present the latest development in the field, in particular the notion of equivalence of such quantum field theories on a noncommutative spacetime, in this regard we work out explicitly the inequivalence between twisted quantum field theories on Moyal and Wick-Voros planes; the duality between deformations of the multiplication map on the algebra of functions on spacetime $\mathscr{F}(\mathbb{R}^4)$ and coproduct deformations of the Poincar\'e-Hopf algebra $H\mathscr{P}$ acting on~$\mathscr{F}(\mathbb{R}^4)$; the appearance of a nonassociative product on $\mathscr{F}(\mathbb{R}^4)$ when gauge fields are also included in the picture. The last part of the manuscript is dedicated to the phenomenology of noncommutative quantum field theories in the particular approach adopted in this review. CPT violating processes, modification of two-point temperature correlation function in CMB spectrum analysis and Pauli-forbidden transition in ${\rm Be}^4$ are all effects which show up in such a noncommutative setting. We review how they appear and in particular the constraint we can infer from comparison between theoretical computations and experimental bounds on such effects. The best bound we can get, coming from Borexino experiment, is $\gtrsim 10^{24}$ TeV for the energy scale of noncommutativity, which corresponds to a length scale $\lesssim 10^{-43}$ m. This bound comes from a different model of spacetime deformation more adapted to applications in atomic physics. It is thus model dependent even though similar bounds are expected for the Moyal spacetime as well as argued elsewhere

Quantum Fields on the Groenewold-Moyal Plane: C, P, T and CPT

We show that despite the inherent non-locality of quantum field theories on the Groenewold-Moyal (GM) plane, one can find a class of ${\bf C}$, ${\bf P}$, ${\bf T}$ and ${\bf CPT}$ invariant theories. In particular, these are theories without gauge fields or with just gauge fields and no matter fields. We also show that in the presence of gauge fields, one can have a field theory where the Hamiltonian is ${\bf C}$ and ${\bf T}$ invariant while the $S$-matrix violates ${\bf P}$ and ${\bf CPT}$. In non-abelian gauge theories with matter fields such as the electro-weak and $QCD$ sectors of the standard model of particle physics, ${\bf C}$, ${\bf P}$, ${\bf T}$ and the product of any pair of them are broken while ${\bf CPT}$ remains intact for the case $\theta^{0i} =0$. (Here $x^{\mu} \star x^{\nu} - x^{\nu} \star x^{\mu} = i \theta^{\mu \nu}$, $x^{\mu}$: coordinate functions, $\theta^{\mu \nu} = -\theta^{\nu \mu}=$ constant.) When $\theta^{0i} \neq 0$, it contributes to breaking also ${\bf P}$ and ${\bf CPT}$. It is known that the $S$-matrix in a non-abelian theory depends on $\theta^{\mu \nu}$ only through $\theta^{0i}$. The $S$-matrix is frame dependent. It breaks (the identity component of the) Lorentz group. All the noncommutative effects vanish if the scattering takes place in the center-of-mass frame, or any frame where $\theta^{0i}P^{\textrm{in}}_{i} = 0$, but not otherwise. ${\bf P}$ and ${\bf CPT}$ are good symmetries of the theory in this special case.Comment: 18 pages, 1 figure, revised, 2 references adde

Discrete Time Evolution and Energy Nonconservation in Noncommutative Physics

Time-space noncommutativity leads to quantisation of time and energy nonconservation when time is conjugate to a compact spatial direction like a circle. In this context energy is conserved only modulo some fixed unit. Such a possibility arises for example in theories with a compact extra dimension with which time does not commute. The above results suggest striking phenomenological consequences in extra dimensional theories and elsewhere. In this paper we develop scattering theory for discrete time translations. It enables the calculation of transition probabilities for energy nonconserving processes and has a central role both in formal theory and phenomenology. We can also consider space-space noncommutativity where one of the spatial directions is a circle. That leads to the quantisation of the remaining spatial direction and conservation of momentum in that direction only modulo some fixed unit, as a simple adaptation of the results in this paper shows.Comment: 17 pages, LaTex; minor correction

Edge Currents in Non-commutative Chern-Simons Theory from a New Matrix Model

This paper discusses the formulation of the non-commutative Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard star products are not correct for such manifolds, the standard non-commutative CS theory is not also appropriate here. Instead we formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work which has points of contact with ours is due to Lizzi, Vitale and Zampini where the authors obtain a description for the fuzzy disc. The gauge fields in our approach are operators supported on a subspace of finite dimension N+\eta of the Hilbert space of eigenstates of a simple harmonic oscillator with N, \eta \in Z^+ and N \neq 0. This oscillator is associated with the underlying Moyal plane. The resultant matrix CS theory has a fuzzy edge. It becomes the required sharp edge when N and \eta goes to infinity in a suitable sense. The non-commutative CS theory on the strip is defined by this limiting procedure. After performing the canonical constraint analysis of the matrix theory, we find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kac-Moody algebra. Our study shows that there are (\eta+1)^2 abelian charges (observables) given by the matrix elements (\cal A_i)_{N-1 N-1} and (\cal A_i)_{nm} (where n or m \geq N) of the gauge fields, that obey certain standard canonical commutation relations. In addition, the theory contains three unique non-abelian charges, localized near the N^th level. We show that all non-abelian edge observables except these three can be constructed from the abelian charges above. Using the results of this analysis we discuss the large N and \eta limit.Comment: LaTeX, 16 pages and 2 figures. Comments added in sections 4 and 5. A minor error corrected in section 4. Figures replaced for clarity. Typos correcte

The Star Product on the Fuzzy Supersphere

The fuzzy supersphere $S_F^{(2,2)}$ is a finite-dimensional matrix approximation to the supersphere $S^{(2,2)}$ incorporating supersymmetry exactly. Here the star-product of functions on $S_F^{(2,2)}$ is obtained by utilizing the OSp(2,1) coherent states. We check its graded commutative limit to $S^{(2,2)}$ and extend it to fuzzy versions of sections of bundles using the methods of [1]. A brief discussion of the geometric structure of our star-product completes our work.Comment: 21 pages, LaTeX, new material added, minor errors correcte