We develop a gauged Wess-Zumino model in noncommutative Minkowski superspace. This is the natural extension of the work of Carlson and Nazaryan, which extended N = 1/2 supersymmetry written over deformed Euclidean superspace to Minkowski superspace. We investigate the interaction of the vector and chiral superfields. Noncommutativity is implemented by replacing products with star products. Although, in general, our star product is nonassociative, we prove that it is associative to the first order in the deformation parameter C. We show that our model reproduces the N = 1/2 theory in the appropriate limit, namely when the deformation parametersCαβ = 0. Essentially, we find the N = 1/2 theory and a conjugate copy. As in the N = 1/2 theory, a reparameterization of the gauge parameter, vector superfield and chiral superfield are necessary to write standard C-independent gauge theory. However, our choice of parameterization differs from that used in the N = 1/2 supersymmetry, which leads to some unexpected new terms.a similar paper has been submitted to JHEP testbetter@yahoo.com 1

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Let A denote an n-dimensional associative algebra over R. This paper gives an introductory exposition of calculus over A. An A-differentiable function f : A → A is one for which the differential is right-A-linear. The basis-dependent correspondence between right-A-linear maps and the regular representation of real matrices is discussed in detail. The requirement that the Jacobian matrix of a function fall in the regular representation of A gives n 2 −n generalized A-CR equations. In contrast, some authors use a deleted-difference quotient to describe differentiability over an algebra. We compare these concepts of differentiability over an algebra and prove they are equivalent in the semisimple commutative case. We also show how difference quotients are ill-equipt to study calculus over a nilpotent algebra.Cauchy Riemann equations are elegantly captured by the Wirtinger calculus as ∂f ∂ z = 0. We generalize this to any commutative unital associative algebra. Instead of one conjugate, we need n − 1 conjugate variables. Our construction modifies that given by Alvarez-Parrilla, Frías-Armenta, López-González and Yee-Romero in [16].We derive Taylor's Theorem over an algebra. Following Wagner, we show how Generalized Laplace equations are naturally seen from the multiplication table of an algebra. We show how d'Alembert's solution to the wave-equation can be derived from the function theory of an appropriate algebra. The inverse problem of A-calculus is introduced and we show how the Tableau for an A-differentiable function has a rather special form. The integral over an algebra is also studied. We find the usual elementary topological results about closed and exact forms generalize nicely to our integral. Generalizations of the First and Second Fundamental Theorems of Calculus are given. This paper should be accessible to undergraduates with a firm foundation in linear algebra and calculus. Some background in complex analysis would also be helpful.

Let A denote an associative unital real algebra of finite dimension. We discuss the structure of of sequences and numerical series over A. We also study convergence of power series over A. The ratio, root and geometric series results are modified due to both the submultiplicativity of the norm and the calculational novelty of zero-divisors. The termby-term differentiation theorem for power series on A is established and we use power series to construct and analyze exponential, trigonometric and hyperbolic functions on arbitrary A.

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