Quantum Spacetimes in the Year 1

We review certain emergent notions on the nature of spacetime from noncommutative geometry and their radical implications. These ideas of spacetime are suggested from developments in fuzzy physics, string theory, and deformation quantisation. The review focuses on the ideas coming from fuzzy physics. We find models of quantum spacetime like fuzzy $S^4$ on which states cannot be localised, but which fluctuate into other manifolds like $CP^3$ . New uncertainty principles concerning such lack of localisability on quantum spacetimes are formulated.Such investigations show the possibility of formulating and answering questions like the probabilty of finding a point of a quantum manifold in a state localised on another one. Additional striking possibilities indicated by these developments is the (generic) failure of $CPT$ theorem and the conventional spin-statistics connection. They even suggest that Planck's `` constant '' may not be a constant, but an operator which does not commute with all observables. All these novel possibilities arise within the rules of conventional quantum physics,and with no serious input from gravity physics.Comment: 11 pages, LaTeX; talks given at Utica and Kolkata .Minor corrections made and references adde

Skyrmions, Spectral Flow and Parity Doubles

It is well-known that the winding number of the Skyrmion can be identified as the baryon number. We show in this paper that this result can also be established using the Atiyah-Singer index theorem and spectral flow arguments. We argue that this proof suggests that there are light quarks moving in the field of the Skyrmion. We then show that if these light degrees of freedom are averaged out, the low energy excitations of the Skyrmion are in fact spinorial. A natural consequence of our approach is the prediction of a $(1/2)^{-}$ state and its excitations in addition to the nucleon and delta. Using the recent numerical evidence for the existence of Skyrmions with discrete spatial symmetries, we further suggest that the the low energy spectrum of many light nuclei may possess a parity doublet structure arising from a subtle topological interaction between the slow Skyrmion and the fast quarks. We also present tentative experimental evidence supporting our arguments.Comment: 22 pages, LaTex. Uses amstex, amssym

Dirac operator on the q-deformed Fuzzy sphere and Its spectrum

The q-deformed fuzzy sphere $S_{qF}^2(N)$ is the algebra of $(N+1)\times(N+1)$ dim. matrices, covariant with respect to the adjoint action of \uq and in the limit $q\to 1$, it reduces to the fuzzy sphere $S_{F}^2(N)$. We construct the Dirac operator on the q-deformed fuzzy sphere-$S_{qF}^{2}(N)$ using the spinor modules of \uq. We explicitly obtain the zero modes and also calculate the spectrum for this Dirac operator. Using this Dirac operator, we construct the \uq invariant action for the spinor fields on $S_{qF}^{2}(N)$ which are regularised and have only finite modes. We analyse the spectrum for both $q$ being root of unity and real, showing interesting features like its novel degeneracy. We also study various limits of the parameter space (q, N) and recover the known spectrum in both fuzzy and commutative sphere.Comment: 19 pages, 6 figures, more references adde

Fuzzy Nambu-Goldstone Physics

In spacetime dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu-Goldstone modes described by fields with values in G/H. In two-dimensional spacetimes as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their fuzzy analogues for G=SU(N+1), H=S(U(N-1)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy versions of continuum models on S^2 when the target spaces are Grassmannians and flag manifolds described by (N+1)x(N+1) projectors of rank =< (N+1)/2. These fuzzy models are finite-dimensional matrix models which nevertheless retain all the essential continuum topological features like solitonic sectors. They seem well-suited for numerical work.Comment: Latex, 18 pages; references added, typos correcte

Bosonic Description of Spinning Strings in 2+12+1 Dimensions

We write down a general action principle for spinning strings in 2+1 dimensional space-time without introducing Grassmann variables. The action is written solely in terms of coordinates taking values in the 2+1 Poincare group, and it has the usual string symmetries, i.e. it is invariant under a) diffeomorphisms of the world sheet and b) Poincare transformations. The system can be generalized to an arbitrary number of space-time dimensions, and also to spinning membranes and p-branes.Comment: Latex, 12 page

Non-Linear Sigma Model on the Fuzzy Supersphere

In this note we develop fuzzy versions of the supersymmetric non-linear sigma model on the supersphere S^(2,2). In hep-th/0212133 Bott projectors have been used to obtain the fuzzy CP^1 model. Our approach utilizes the use of supersymmetric extensions of these projectors. Here we obtain these (super) -projectors and quantize them in a fashion similar to the one given in hep-th/0212133. We discuss the interpretation of the resulting model as a finite dimensional matrix model.Comment: 11 pages, LaTeX, corrected typo

The Gauss Law: A Tale

The Gauss law plays a basic role in gauge theories, enforcing gauge invariance and creating edge states and superselection sectors. This article surveys these aspects of the Gauss law in QED, QCD and nonlinear $G/H$ models. It is argued that nonabelian superselection rules are spontaneously broken. That is the case with $SU(3)$ of colour which is spontaneously broken to $U(1)\times U(1)$. Nonlinear $G/H$ models are reformulated as gauge theories and the existence of edge states and superselection sectors in these models is also established.Comment: Published version. 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