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Register a new userSpacetime algebra as a powerful tool for electromagnetism
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We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field.
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We investigate certain $Z_3$-graded associative algebras with cubic $Z_3$-invariant constitutive relations. The invariant forms on finite algebras of this type are given in the low dimensional cases with two or three generators. We show how the Lorentz symmetry represented by the $SL(2, {bf C})$ group emerges naturally without any notion of Minkowskian metric, just as the invariance group of the $Z_3$-graded cubic algebra and its constitutive relations. Its representation is found in terms of Pauli matrices. The relationship of this construction with the operators defining quark states is also considered, and a third-order analogue of the Klein-Gordon equation is introduced. Cubic products of its solutions may provide the basis for the familiar wave functions satisfying Dirac and Klein-Gordon equations.
We give some evidences which imply that W(1+infinity) algebra describes the symmetry behind AGT(-W) conjecture: a correspondence between the partition function of N=2 supersymmetric quiver gauge theories and the correlators of Liouville (Toda) field theory.
We test in $(A_{n-1},A_{m-1})$ Argyres-Douglas theories with $mathrm{gcd}(n,m)=1$ the proposal of Songs in arXiv:1612.08956 that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamats Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual $(A_{n-1},A_{m-1})$ theories in the large $m$ limit, and then provide evidence for Songs proposal to hold (at least) in some simple modules (including the vacuum module) at finite $m$. We also discuss some observed mismatch in our approach.
In research problems that involve the use of numerical methods for solving systems of ordinary differential equations (ODEs), it is often required to select the most efficient method for a particular problem. To solve a Cauchy problem for a system of ODEs, Runge-Kutta methods (explicit or implicit ones, with or without step-size control, etc.) are employed. In that case, it is required to search through many implementations of the numerical method and select coefficients or other parameters of its numerical scheme. This paper proposes a library and scripts for automated generation of routine functions in the Julia programming language for a set of numerical schemes of Runge-Kutta methods. For symbolic manipulations, we use a template substitution tool. The proposed approach to automated generation of program code allows us to use a single template for editing, instead of modifying each individual function to be compared. On the one hand, this provides universality in the implementation of a numerical scheme and, on the other hand, makes it possible to minimize the number of errors in the process of modifying the compared implementations of the numerical method. We consider Runge-Kutta methods without step-size control, embedded methods with step-size control, and Rosenbrock methods with step-size control. The program codes for the numerical schemes, which are generated automatically using the proposed library, are tested by numerical solution of several well-known problems.
The inverse problem in optics, which is closely related to the classical question of the resolving power, is reconsidered as a communication channel problem. The main result is the evaluation of the maximum number $M_epsilon$ of $epsilon$-distinguishable messages ($epsilon$ being a bound on the noise of the image) which can be conveyed back from the image to reconstruct the object. We study the case of coherent illumination. By using the concept of Kolmogorovs $epsilon$-capacity, we obtain: $M_epsilon ~ 2^{S log(1/epsilon)} to infty$ as $epsilon to 0$, where S is the Shannon number. Moreover, we show that the $epsilon$-capacity in inverse optical imaging is nearly equal to the amount of information on the object which is contained in the image. We thus compare the results obtained through the classical information theory, which is based on the probability theory, with those derived from a form of topological information theory, based on Kolmogorovs $epsilon$-entropy and $epsilon$-capacity, which are concepts related to the evaluation of the massiveness of compact sets.
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