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Register a new userOn the Moyal Star Product of Resurgent Series
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We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series (a subspace of $1$-Gevrey formal series in $ihbar/2$ with coefficients in $C{q,p}$), which we show is stable under Moyal star product.
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Using the formalism of quantizers and dequantizers, we show that the characters of irreducible unitary representations of finite and compact groups provide kernels for star products of complex-valued functions of the group elements. Examples of permutation groups of two and three elements, as well as the SU(2) group, are considered. The k-deformed star products of functions on finite and compact groups are presented. The explicit form of the quantizers and dequantizers, and the duality symmetry of the considered star products are discussed.
A system of coupled kinetic transport equations for the Wigner distributions of a free variable mass Klein-Gordon field is derived. This set of equations is formally equivalent to the full wave equation for electromagnetic waves in nonlinear dispersive media, thus allowing for the description of broadband radiation-matter interactions and the associated instabilities. The standard results for the classical wave action are recovered in the short wavelength limit of the generalized Wigner-Moyal formalism for the wave equation.
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Let $V$ be a finite dimensional inner product space over $mathbb{R}$ with dimension $n$, where $nin mathbb{N}$, $wedge^{r}V$ be the exterior algebra of $V$, the problem is to find $max_{| xi | = 1, | eta | = 1}| xi wedge eta |$ where $k,l$ $in mathbb{N},$ $forall xi in wedge^{k}V, eta in wedge^{l}V.$ This is a problem suggested by the famous Nobel Prize Winner C.N. Yang. He solved this problem for $kleq 2$ in [1], and made the following textbf{conjecture} in [2] : If $n=2m$, $k=2r$, $l=2s$, then the maximum is achieved when $xi_{max} = frac{omega^{k}}{| omega^{k}|}, eta_{max} = frac{omega^{l}}{| omega^{l}|}$, where $ omega = Sigma_{i=1}^m e_{2i-1}wedge e_{2i}, $ and ${e_{k}}_{k=1}^{2m}$ is an orthonormal basis of V. From a physicists point of view, this problem is just the dual version of the easier part of the well-known Beauzamy-Bombieri inequality for product of polynomials in many variables, which is discussed in [4]. Here the duality is referred as the well known Bose-Fermi correspondence, where we consider the skew-symmetric algebra(alternative forms) instead of the familiar symmetric algebra(polynomials in many variables) In this paper, for two cases we give estimations of the maximum of exterior products, and the Yangs conjecture is answered partially under some special cases.
We construct a version of the complex Heisenberg algebra based on the idea of endless analytic continuation. In particular, we exhibit an integral formula for the product of resurgent operators with algebraic singularities. This algebra would be large enough to capture quantum effects that escape ordinary formal deformation quantisation.
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