Two-spinor Formalism Research Articles

Polarization Spin-Tensors in Two-Spinor Formalism and Behrends–Fronsdal Spin Projection Operator for D-Dimensional Case

In the work, the recurrent differential relations that connecting the polarization spin-tensor of the wave function of a free massive particle of an arbitrary spin for $D=4$ and new formula of the $D$-dimensional Behrends-Fronsdal spin projection operator are found.

Quaternion Electromagnetism and the Relation with Two-Spinor Formalism

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary “i” should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation.

Unitary Representations of the Wigner Group ISL(2, ℂ) and A Two-Spinor Description of Massive Particles With An Arbitrary Spin

Based on Wigner unitary representations for the covering group ISL(2,ℂ) of the Poincare group, we obtain spin-tensor wave functions of free massive particles with an arbitrary spin that satisfy the Dirac–Pauli–Fierz equations. In the framework of a two-spinor formalism, we construct spin-polarization vectors and obtain conditions that fix the corresponding density matrices (the Behrends–Fronsdal projection operators) determining the numerators in the propagators of the fields of such particles. Using these conditions, we find explicit expressions for the particle density matrices with integer (Behrends–Fronsdal projection operators) and half-integer spin. We obtain a generalization of the Behrends–Fronsdal projection operators to the case of an arbitrary number D of space–time dimensions.

Two-spinor description of massive particles and relativistic spin projection operators

On the basis of the Wigner unitary representations of the covering group ISL(2,C) of the Poincaré group, we obtain spin-tensor wave functions of free massive particles with arbitrary spin. The wave functions automatically satisfy the Dirac–Pauli–Fierz equations. In the framework of the two-spinor formalism we construct spin-vectors of polarizations and obtain conditions that fix the corresponding relativistic spin projection operators (Behrends–Fronsdal projection operators). With the help of these conditions we find explicit expressions for relativistic spin projection operators for integer spins (Behrends–Fronsdal projection operators) and then find relativistic spin projection operators for half integer spins. These projection operators determine the numerators in the propagators of fields of relativistic particles. We deduce generalizations of the Behrends–Fronsdal projection operators for arbitrary space–time dimensions D>2.

The decoupling problem of the Proca equation; and treatment of Dirac, Maxwell and Proca fields on the resulting pp-wave spacetimes

In this work we take a formal approach to the problem of decoupling Proca equations in curved space-times. We use Newman-Penrose (NP) two-spinor formalism to represent the Proca vector by one complex and two real scalars. We show that a decoupled second order differential equation for one of the real scalars can be derived if and only if the background space-time admits a covariantly constant null vector. Thus, the background space-time must be a pp-wave vacuum. We evaluate the separability of Proca, Maxwell and Dirac equations on the resulting pp-wave background.

LORENTZ VIOLATION DISPERSION RELATION AND ITS APPLICATION

We derive a modified dispersion relation (MDR) in the Lorentz violation extension of the quantum electrodynamics (QED) sector in the standard model extension (SME) framework. Based on the extended Dirac equation and corresponding MDR, we observe the resemblance of the Lorentz violation coupling to spin–gravity coupling. We also develop a neutrino oscillation mechanism induced by the presence of nondiagonal terms of Lorentz violation couplings in two-flavor space in a two-spinor formalism by explicitly assuming neutrinos to be Marjorana fermions. We also obtain a very stringent bound (∽ 10-25) on one of the Lorentz violation parameters by applying the MDR to the ultrahigh energy cosmic ray (UHECR) problem.

Two-spinor Formulation of First-Order Gravity Coupled to Dirac Fields

A two-spinor formalism for the Einstein Lagrangian is developed. The gravitational field is regarded as a composite object derived from soldering forms. Our formalism is geometrically and globally well-defined and may be used in virtually any 4m-dimensional manifold with arbitrary signature as well as without any stringent topological requirement on space-time, such as parallelizability. Interactions and feedbacks between gravity and spinor fields are considered. As is well known, the Hilbert–Einstein Lagrangian is second order also when expressed in terms of soldering forms. A covariant splitting is then analysed leading to a first-order Lagrangian which is recognized to play a fundamental role in the theory of conserved quantities. The splitting and thence the first-order Lagrangian depend on a reference spin connection which is physically interpreted as setting the zero level for conserved quantities. A complete and detailed treatment of conserved quantities is then presented.

Recursive numerical integration of multi-particle phase space with peripheral matrix element

On the basis of the Wigner unitary representations of the covering group ISL(2,C) of the Poincaré group, we obtain spin-tensor wave functions of free massive particles with arbitrary spin. The wave functions automatically satisfy the Dirac–Pauli–Fierz equations. In the framework of the two-spinor formalism we construct spin-vectors of polarizations and obtain conditions that fix the corresponding relativistic spin projection operators (Behrends–Fronsdal projection operators). With the help of these conditions we find explicit expressions for relativistic spin projection operators for integer spins (Behrends–Fronsdal projection operators) and then find relativistic spin projection operators for half integer spins. These projection operators determine the numerators in the propagators of fields of relativistic particles. We deduce generalizations of the Behrends–Fronsdal projection operators for arbitrary space–time dimensions D>2.