A novel approach to non-commutative gauge theory

We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter $\Theta(x)$, which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, $\Theta\to 0$, the standard $U(1)$ gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, $-i[f,g]_\star\approx\{f,g\}$. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra $[\delta_f,\delta_g]A=\delta_{\{f,g\}}A$, and the NC field strength ${\cal F}$, covariant under these transformations, $\delta_f {\cal F}=\{{\cal F},f\}$. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, ${\cal F}$, should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, $S=\int {\cal F}^2$. As guiding example, the case of $su(2)$-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.Comment: 16 pages, no figures. Minor correction

Noncommutative RdR^d via closed star product

We consider linear star products on $R^d$ of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with the one which is closed with respect to some trace functional, $Tr( f\star g)= Tr( f\cdot g)$. We introduce the derivative operator on the algebra of the closed star product and show that the corresponding Leibnitz rule holds true up to a total derivative. As a particular example we study the space $R^3_\theta$ with $\mathfrak{su}(2)$ type noncommutativity and show that in this case the closed star product is the one obtained from the Duflo quantization map. As a result a Laplacian can be defined such that its commutative limit reproduces the ordinary commutative one. The deformed Leibnitz rule is applied to scalar field theory to derive conservation laws and the corresponding noncommutative currents.Comment: published versio

Alternative Canonical Formalism for the Wess-Zumino-Witten Model

We study a canonical quantization of the Wess--Zumino--Witten (WZW) model which depends on two integer parameters rather than one. The usual theory can be obtained as a contraction, in which our two parameters go to infinity keeping the difference fixed. The quantum theory is equivalent to a generalized Thirring model, with left and right handed fermions transforming under different representations of the symmetry group. We also point out that the classical WZW model with a compact target space has a canonical formalism in which the current algebra is an affine Lie algebra of non--compact type. Also, there are some non--unitary quantizations of the WZW model in which there is invariance only under half the conformal algebra (one copy of the Virasoro algebra).Comment: 22 pages; UR-133

Dynamical Aspects of Lie--Poisson Structures

Quantum Groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at $SU(2)$ and $SU(1,1)$, as submanifolds of a 4--dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of the motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure.Comment: 17 pages, figures not include

Land Degradation in the Sahel: An Application of Biophysical Modeling in the Optimal Control Setting

Low-input farming practices in many parts of the developing world have pushed cultivation onto marginal lands. Sustainability of already fragile ecosystems is threatened. Farmers place a high priority on satisfying subsistence food needs with on-farm production. Population pressure is high throughout much of Sub-Saharan Africa. Farmers in those regions are challenged by the need to put continually more food on their table over the coming years. An optimal control model was developed to investigate alternative farming practices within this setting. Namely, whether farmers would choose continued land expansion of if they would adopt crop intensive practices. The model included an environmental subcomponent to estimate the degradation costs from continued expansion onto marginal areas. The modeling activities from the Sahel of West African reinforce farmers' observed propensity to clear new land in lieu of crop intensification. Model activities suggest an important role for crop intensification under adequate policy conditions as well as the need to introduce new technology before degradation erodes its potential.Land Economics/Use,

NLO Renormalization in the Hamiltonian Truncation

Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is a numerical technique for solving strongly coupled QFTs, in which the full Hilbert space is truncated to a finite-dimensional low-energy subspace. The accuracy of the method is limited only by the available computational resources. The renormalization program improves the accuracy by carefully integrating out the high-energy states, instead of truncating them away. In this paper we develop the most accurate ever variant of Hamiltonian Truncation, which implements renormalization at the cubic order in the interaction strength. The novel idea is to interpret the renormalization procedure as a result of integrating out exactly a certain class of high-energy "tail states". We demonstrate the power of the method with high-accuracy computations in the strongly coupled two-dimensional quartic scalar theory, and benchmark it against other existing approaches. Our work will also be useful for the future goal of extending Hamiltonian Truncation to higher spacetime dimensions.Comment: 28pp + appendices, detailed version of arXiv:1706.0612