Narrow Results

We compute several identities relating curvature actions to second-order differential operators on twistors and sections of related bundles. Based on these formulas, we establish various estimates and vanishing theorems.

We introduce a notion of ${\varphi }_1$-coordinated module twistor for a ${\varphi }_1$-coordinated module of a nonlocal vertex algebra. Furthermore, we study twisted tensor products and L-R-twisted tensor products of ${\varphi }_1$-coordinated modules of nonlocal vertex algebras and we unify these two constructions by ${\varphi }_1$-coordinated module twistors.

Consistency conditions for the local existence of massless spin-3/2 fields have been explored to find the facts that the field equations for massless helicity-3/2 particles are consistent if the space–time is Ricci-flat, and that in Minkowski space–time the space of conserved charges for the fields is its twistor space itself. After considering the twistorial methods to study such massless helicity-3/2 fields, we show in flat space–time that the charges of spin-3/2 fields, defined topologically by the first Chern number of their spin-lowered self-dual Maxwell fields, are given by their twistor space, and in curved space–time that the (anti-)self-duality of the space–time is the necessary condition. Since in N=1 supergravity torsions are the essential ingredients, we generalize our space–time to that with torsion (Einstein–Cartan theory), and investigate the consistency of existence of spin-3/2 fields in this theory. A simple solution to this consistency problem is found: The space–time has to be conformally (anti-)self-dual, left-(or right-) torsion-free. The integrability condition on α-surface shows that the (anti-)self-dual Weyl spinor can be described only by the covariant derivative of the right-(left-)handed torsion.

We study the Hamiltonian structure and constraints of twistor-like super p-brane which preserves 3/4 of D = 4, N = 1 supersymmetry using the conversion method. Classical and quantum realizations of the BRST charge, the unified superalgebra of symmetries of the p-brane action, including generalized global superconformal OSp(1|8) symmetry together with Virasoro and Weyl symmetries, are presented. We prove nilpotency of the Hermitian quantum BRST charge of the p-brane and closure of the quantized superalgebra OSp(1|8) of its global symmetry.

In this paper, we show that the mass-shell constraints in the gauged twistor formulation of a massive particle given in [Deguchi and Okano, Phys. Rev. D 93, 045016 (2016) [Erratum 93, 089906(E) (2016)]] are incorporated in an action automatically by extending the local $U(2)$ transformation to its inhomogeneous extension denoted by $IU(2)$. Therefore, it turns out that all the necessary constraints are incorporated into an action by virtue of the local $IU(2)$ symmetry of the system.

A twistor model of a free massless spinning particle in four-dimensional Minkowski space is studied in terms of space–time and spinor variables. This model is specified by a simple action, referred to here as the gauged Shirafuji action, that consists of twistor variables and gauge fields on the one-dimensional parameter space. We consider the canonical formalism of the model by following the Dirac formulation for constrained Hamiltonian systems. In the subsequent quantization procedure, we obtain a plane-wave solution with momentum spinors. From this solution and coefficient functions, we construct positive-frequency and negative-frequency spinor wave functions defined on complexified Minkowski space. It is shown that the Fourier–Laplace transforms of the coefficient functions lead to the spinor wave functions expressed as the Penrose transforms of the corresponding holomorphic functions on twistor space. We also consider the exponential generating function for the spinor wave functions and derive a novel representation for each of the spinor wave functions.

A twistor framework for foundations of physics is outlined, in which (1) spacetime is secondary, derived from twistor constructions; (2) the primary physical agents are described by elements of sheaf cohomology defined on twistor space, which encode all information about the spacetime entities (fields and particles); (3) the dynamics of spacetime entities is encoded in the holomorphic structures of twistor space and revealed through Penrose transform and its generalizations (twistor diagrams); (4) the nonlinearity of gravity and nonabelian gauge interactions is a manifestation of the deformation of twistor space effected by the self-interacting twistor agents. A consistent framework of quantum gravity is attainable within twistor formalism because not only can gravity be derived from the primary physical agents of the formalism but also, and most importantly, the formalism itself can be proven intrinsically quantal without the necessity of imposing a quantum postulate upon it extrinsically through a quantization procedure. Indeed, the Planck constant and the whole quantum edifice built thereupon can be derived from formalism. Thus, the conflicts between the principles of general relativity and quantum theory are dissolved because both theories are consequences of twistor formalism, and the century-long debate concerning the possibility of a consistent framework of quantum gravity is closed. Many technical and conceptual issues must be clarified before the twistor formalism can be turned into a practical research framework. But the foundational pillars of the framework provided by twistor formalism are rock solid.

Twistor spaces are certain complex three-manifolds, which are associated with special conformal Riemannian geometries on four-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler–Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theory of classical mechanic. This paper aims to introduce Euler–Lagrage partial differential equations (mathematical modeling, the equations of motion according to the time) for the movement of objects on twistor space and also to offer a general solution of differential equation system using the Maple software. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn.

Dirac electron theory and QED do not take into account gravitational field, while the corresponding Kerr-Newman solution with parameters of electron has very strong stringy, topological and non-local action on the Compton distances, polarizing space-time and deforming the Coulomb field. We discuss the relation of the electron to the Kerr's microgeon model and argue that the Kerr geometry may be hidden beyond the Quantum Theory. In particular, we show that the Foldi-Wouthuysen ‘mean-position’ operator of the Dirac electron is related to a complex representation of the Kerr geometry, and to a complex stringy source. Therefore, the complex Kerr geometry may be hidden beyond the Dirac equation.