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result(s) for
"Dirac equation."
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Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data
In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \\mathbb{R}^{1+d} (d\\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.
eBook
The relativistic tunneling flight time may be superluminal, but it does not imply superluminal signaling
Wavepacket tunneling, in the relativistic limit, is studied via solutions to the Dirac equation for a square barrier potential. Specifically, the arrival time distribution (the time-dependent flux) is computed for wavepackets initiated far away from the barrier, and whose momentum is well below the threshold for above-barrier transmission. The resulting distributions exhibit peaks at shorter times than those of photons with the same initial wavepacket transmitting through a vacuum. However, this apparent superluminality in time is accompanied by very low transmission probabilities. We discuss these observations, and related observations by other authors, in the context of published objections to the notion that tunneling can be superluminal in time. We find that many of these objections are not consistent with our observations, and conclude that post-selected (for transmission) distributions of arrival times can be superluminal. However, the low probability of tunneling means a photon will most likely be seen first and therefore the superluminality does not imply superluminal signaling.
Journal Article
Thin Layer Quantization Method for a Spin Particle on a Curved Surface
Using the fundamental framework of the thin-layer quantization method, we discuss the non-relativistic limit of the Schrödinger-Dirac equation for a particle constrained to move on a curved surface. We show that the inclusion of spin connections in the formalism give rise to scalar terms which provide a new scalar geometric potential. The coupling between the spin connections determined by the geometry of the curved surface and the spin of the particle can generate bound states even for the repulsive case of this obtained geometric potential. The developed procedure is applied to a surface of axial symmetry. We give three interesting examples of surface confinement, namely cylindrical, spherical and conical, and we explicitly deduce the energy levels for each case.
Journal Article
Gravidynamics, spinodynamics and electrodynamics within the framework of gravitational quantum field theory
By noticing the fact that the charged leptons and quarks in the standard model are chirality-based Dirac spinors since their weak interaction violates maximally parity symmetry though they behave as Dirac fermions in electromagnetic interaction, we show that such a chirality-based Dirac spinor possesses not only electric charge gauge symmetry U(1) but also inhomogeneous spin gauge symmetry WS(1,3)=SP(1,3)⋊W
1,3
, which reveals the nature of gravity and spacetime. The gravitational force and spin gauge force are governed by the gauge symmetries W
1,3
and SP(1,3), respectively, and a biframe spacetime with globally flat Minkowski spacetime as base spacetime and locally flat gravigauge spacetime as a fiber is described by the gravigauge field through emergent non-commutative geometry. The gauge-geometry duality and renormalizability in gravitational quantum field theory (GQFT) are carefully discussed. A detailed analysis and systematic investigation on gravidynamics and spinodynamics as well as electrodynamics are carried out within the framework of GQFT. A full discussion on the generalized Dirac equation and Maxwell equation as well as Einstein equation and spin gauge equation is made in biframe spacetime. New effects of gravidynamics as extension of general relativity are particularly analyzed. All dynamic equations of basic fields are demonstrated to preserve the spin gauge covariance and general coordinate covariance due to the spin gauge symmetry and emergent general linear group symmetry GL(1,3,R), so they hold naturally in any spinning reference frame and motional reference frame.
Journal Article
Lorentz covariance of optical Dirac equation and spinorial photon field
In a recent paper (2014 New J. Phys. 16 093008) Barnett discussed the so-called optical Dirac equation and referred to the involved wave function as a spinor. But as he claimed explicitly, he did not really associate that wave function with a true spinor. Here we show that if the optical Dirac equation is interpreted as the dynamical equation for the photon in conventional quantum mechanics, the wave function, called the photon field, does transform under the Lorentz transformation as a spinor. For the optical Dirac equation to be Lorentz covariant, however, a constraint on the photon field is required, which can be cast into the form of Maxwell’s divergence equations. It is found that the spinorial photon field not only satisfies the principle of locality but also has the right dimensionality as is required by conventional quantum mechanics.
Journal Article
The Dirac equation across the horizons of the 5D Myers–Perry geometry: separation of variables, radial asymptotic behaviour and Hamiltonian formalism
We analytically extend the 5D Myers–Perry metric through the event and Cauchy horizons by defining Eddington–Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster–Röken and Stone’s formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green’s functions of the radial ODE.
Journal Article
Revisiting the Schrödinger–Dirac Equation
In flat spacetime, the Dirac equation is the “square root” of the Klein–Gordon equation in the sense that, by applying the square of the Dirac operator to the Dirac spinor, one recovers the equation duplicated for each component of the spinor. In the presence of gravity, applying the square of the curved-spacetime Dirac operator to the Dirac spinor does not yield the curved-spacetime Klein–Gordon equation, but instead yields the Schrödinger–Dirac covariant equation. First, we show that the latter equation gives rise to a generalization to spinors of the covariant Gross–Pitaevskii equation. Then, we show that, while the Schrödinger–Dirac equation is not conformally invariant, there exists a generalization of the equation that is conformally invariant but which requires a different conformal transformation of the spinor than that required by the Dirac equation. The new conformal factor acquired by the spinor is found to be a matrix-valued factor obeying a differential equation that involves the Fock–Ivanenko line element. The Schrödinger–Dirac equation coupled to the Maxwell field is then revisited and generalized to particles with higher electric and magnetic moments while respecting gauge symmetry. Finally, Lichnerowicz’s vanishing theorem in the conformal frame is also discussed.
Journal Article
Solutions of Umbral Dirac-Type Equations
The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated by the normalized systems.
Journal Article
Series Representation of Solutions of Polynomial Dirac Equations
In this paper, we consider the polynomial Dirac equation
D
m
+
∑
i
=
0
m
-
1
a
i
D
i
u
=
0
,
(
a
i
∈
C
)
, where
D
is the Dirac operator in
R
n
. We introduce a method of using series to represent explicit solutions of the polynomial Dirac equations.
Journal Article