Bosonization and Current Algebra of Spinning Strings

We write down a general geometric action principle for spinning strings in $d$-dimensional Minkowski space, which is formulated without the use of Grassmann coordinates. Instead, it is constructed in terms of the pull-back of a left invariant Maurer-Cartan form on the $d$-dimensional Poincar\'e group to the world sheet. The system contains some interesting special cases. Among them are the Nambu string (as well as, null and tachyionic strings) where the spin vanishes, and also the case of a string with a spin current - but no momentum current. We find the general form for the Virasoro generators, and show that they are first class constraints in the Hamiltonian formulation of the theory. The current algebra associated with the momentum and angular momentum densities are shown, in general, to contain rather complicated anomaly terms which obstruct quantization. As expected, the anomalies vanish when one specializes to the case of the Nambu string, and there one simply recovers the algebra associated with the Poincar\'e loop group. We speculate that there exist other cases where the anomalies vanish, and that these cases give the bosonization of the known pseudoclassical formulations of spinning strings.Comment: Latex file, 29 p

Twisted Poincare Invariance, Noncommutative Gauge Theories and UV-IR Mixing

In the absence of gauge fields, quantum field theories on the Groenewold-Moyal (GM) plane are invariant under a twisted action of the Poincare group if they are formulated following [1, 2, 3, 4, 5, 6]. In that formulation, such theories also have no UV-IR mixing [7]. Here we investigate UV-IR mixing in gauge theories with matter following the approach of [3, 4]. We prove that there is UV-IR mixing in the one-loop diagram of the S-matrix involving a coupling between gauge and matter fields on the GM plane, the gauge field being nonabelian. There is no UV-IR mixing if it is abelian.Comment: 11 pages, 3 figure

The quantum sinh-Gordon model in noncommutative (1+1) dimensions

Using twisted commutation relations we show that the quantum sinh-Gordon model on noncommutative space is integrable, and compute the exact two-particle scattering matrix. The model possesses a strong-weak duality, just like its commutative counterpart.Comment: 7 pages, 2 figures, LaTex. References adde

Inequivalence of the Massive Vector Meson and Higgs Models on a Manifold with Boundary

The exact quantization of two models, the massive vector meson model and the Higgs model in the London limit, both describing massive photons, is presented. Even though naive arguments (based on gauge-fixing) may indicate the equivalence of these models, it is shown here that this is not true in general when we consider these theories on manifolds with boundaries. We show, in particular, that they are equivalent only for a special choice of the boundary conditions that we are allowed to impose on the fields.Comment: 14 pages, LATEX File (revised with minor corrections

Covariant Quantum Fields on Noncommutative Spacetimes

A spinless covariant field $\phi$ on Minkowski spacetime \M^{d+1} obeys the relation $U(a,\Lambda)\phi(x)U(a,\Lambda)^{-1}=\phi(\Lambda x+a)$ where $(a,\Lambda)$ is an element of the Poincar\'e group \Pg and $U:(a,\Lambda)\to U(a,\Lambda)$ is its unitary representation on quantum vector states. It expresses the fact that Poincar\'e transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincar\'e transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no covariant Voros fields compatible with *, a result we found earlier. The notion of Drinfel'd twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative.Comment: 20 page

Quantum Geons and Noncommutative Spacetimes

Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime is the Moyal plane. The Poincar\`e group algebra acts on it with a Drinfel'd-twisted coproduct. But the latter is not appropriate for more complicated spacetimes such as those containing the Friedman-Sorkin (topological) geons. They have rich diffeomorphism groups and in particular mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group $S_N$. We generalise the Drinfel'd twist to (essentially) generic groups including to finite and discrete ones and use it to modify the commutative spacetime algebras of geons as well to noncommutative algebras. The latter support twisted actions of diffeos of geon spacetimes and associated twisted statistics. The notion of covariant fields for geons is formulated and their twisted versions are constructed from their untwisted versions. Non-associative spacetime algebras arise naturally in our analysis. Physical consequences, such as the violation of Pauli principle, seem to be the outcomes of such nonassociativity. The richness of the statistics groups of identical geons comes from the nontrivial fundamental groups of their spatial slices. As discussed long ago, extended objects like rings and D-branes also have similar rich fundamental groups. This work is recalled and its relevance to the present quantum geon context is pointed out.Comment: 41 page

The Spin-Statistics Connection in Quantum Gravity

It is well-known that is spite of sharing some properties with conventional particles, topological geons in general violate the spin-statistics theorem. On the other hand, it is generally believed that in quantum gravity theories allowing for topology change, using pair creation and annihilation of geons, one should be able to recover this theorem. In this paper, we take an alternative route, and use an algebraic formalism developed in previous work. We give a description of topological geons where an algebra of "observables" is identified and quantized. Different irreducible representations of this algebra correspond to different kinds of geons, and are labeled by a non-abelian "charge" and "magnetic flux". We then find that the usual spin-statistics theorem is indeed violated, but a new spin-statistics relation arises, when we assume that the fluxes are superselected. This assumption can be proved if all observables are local, as is generally the case in physical theories. Finally, we also show how our approach fits into conventional formulations of quantum gravity.Comment: LaTeX file, 31 pages, 5 figure

UV-IR Mixing in Non-Commutative Plane

Poincar\'e-invariant quantum field theories can be formulated on non-commutative planes if the coproduct on the Poincar\'e group is suitably deformed \cite{Dimitrijevic:2004rf, Chaichian:2004za}.(See also especially Oeckl \cite{Oeckl:1999jun},\cite{Oeckl:2000mar} and Grosse et al.\cite{Grosse:2001mar}) As shown in \cite{Balachandran:2005eb}, this important result of these authors implies modification of free field commutation and anti-commutation relations and striking phenomenological consequences such as violations of Pauli principle \cite{Balachandran:2005eb,Bal3}. In this paper we prove that with these modifications, UV-IR mixing disappears to all orders in perturbation theory from the $S$-Matrix. This result is in agreement with the previous results of Oeckl \cite{Oeckl:2000mar}.Comment: Minor Changes in text and abstract, important references adde

Spin jj Dirac Operators on the Fuzzy 2-Sphere

The spin 1/2 Dirac operator and its chirality operator on the fuzzy 2-sphere $S^2_F$ can be constructed using the Ginsparg-Wilson(GW) algebra [arxiv:hep-th/0511114]. This construction actually exists for any spin $j$ on $S^2_F$, and have continuum analogues as well on the commutative sphere $S^2$ or on $\mathbb{R}^{2}$. This is a remarkable fact and has no known analogue in higher dimensional Minkowski spaces. We study such operators on $S^2_F$ and the commutative $S^2$ and formulate criteria for the existence of the limit from the former to the latter. This singles out certain fuzzy versions of these operators as the preferred Dirac operators. We then study the spin 1 Dirac operator of this preferred type and its chirality on the fuzzy 2-sphere and formulate its instanton sectors and their index theory. The method to generalize this analysis to any spin $j$ is also studied in detail.Comment: 16 pages, 1 tabl

Non-Pauli Transitions From Spacetime Noncommutativity

There are good reasons to suspect that spacetime at Planck scales is noncommutative. Typically this noncommutativity is controlled by fixed "vectors" or "tensors" with numerical entries. For the Moyal spacetime, it is the antisymmetric matrix $\theta_{\mu\nu}$. In approaches enforcing Poincar\'e invariance, these deform or twist the method of (anti-)symmetrization of identical particle state vectors. We argue that the earth's rotation and movements in the cosmos are "sudden" events to Pauli-forbidden processes. They induce (twisted) bosonic components in state vectors of identical spinorial particles in the presence of a twist. These components induce non-Pauli transitions. From known limits on such transitions, we infer that the energy scale for noncommutativity is $\gtrsim 10^{24}\textrm{TeV}$. This suggests a new energy scale beyond Planck scale.Comment: 11 pages, 1 table, Slightly revised for clarity