36 research outputs found
Dirac equation: Representation independence and tensor transformation
We define and study the probability current and the Hamiltonian operator for
a fully general set of Dirac matrices in a flat spacetime with affine
coordinates, by using the Bargmann-Pauli hermitizing matrix. We find that with
some weak conditions on the affine coordinates, the current, as well as the
spectrum of the Dirac Hamiltonian, thus all of quantum mechanics, are
independent of that set. These results allow us to show that the tensor Dirac
theory, which transforms the wave function as a spacetime vector and the set of
Dirac matrices as a third-order affine tensor, is physically equivalent to the
genuine Dirac theory, based on the spinor transformation. The tensor Dirac
equation extends immediately to general coordinate systems, thus to
non-inertial (e.g. rotating) coordinate systems.Comment: 28 pages, standard LaTeX. v3: matches version accepted in the
Brazilian Journal of Physics: minor wording improvements, refs updated. v2:
Intro and Conclusion improved (novelty more emphasized). Uniqueness and
positive definiteness extended to any admissible affine coordinates. 10 new
ref
The Hamiltonian structure of Dirac's equation in tensor form and its Fermi quantization
Currently, there is some interest in studying the tensor forms of the Dirac equation to elucidate the possibility of the constrained tensor fields admitting Fermi quantization. We demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors
Nonlinear modes of the tensor Dirac equation and CPT violation
Recently, it has been shown that Dirac's bispinor equation can be expressed, in an equivalent tensor form, as a constrained Yang-Mills equation in the limit of an infinitely large coupling constant. It was also shown that the free tensor Dirac equation is a completely integrable Hamiltonian system with Lie algebra type Poisson brackets, from which Fermi quantization can be derived directly without using bispinors. The Yang-Mills equation for a finite coupling constant is investigated. It is shown that the nonlinear Yang-Mills equation has exact plane wave solutions in one-to-one correspondence with the plane wave solutions of Dirac's bispinor equation. The theory of nonlinear dispersive waves is applied to establish the existence of wave packets. The CPT violation of these nonlinear wave packets, which could lead to new observable effects consistent with current experimental bounds, is investigated
Measuring a Kaluza-Klein radius smaller than the Planck length
Hestenes has shown that a bispinor field on a Minkowski space-time is
equivalent to an orthonormal tetrad of one-forms together with a complex scalar
field. More recently, the Dirac and Einstein equations were unified in a tetrad
formulation of a Kaluza-Klein model which gives precisely the usual
Dirac-Einstein Lagrangian. In this model, Dirac's bispinor equation is obtained
in the limit for which the radius of higher compact dimensions of the
Kaluza-Klein manifold becomes vanishingly small compared with the Planck
length. For a small but finite radius, the Kaluza-Klein model predicts velocity
splitting of single fermion wave packets. That is, the model predicts a single
fermion wave packet will split into two wave packets with slightly different
group velocities. Observation of such wave packet splits would determine the
size of the Kaluza-Klein radius. If wave packet splits were not observed in
experiments with currently achievable accuracies, the Kaluza-Klein radius would
be at least twenty five orders of magnitude smaller than the Planck length
General reference frames and their associated space manifolds
We propose a formal definition of a general reference frame in a general
spacetime, as an equivalence class of charts. This formal definition
corresponds with the notion of a reference frame as being a (fictitious)
deformable body, but we assume, moreover, that the time coordinate is fixed.
This is necessary for quantum mechanics, because the Hamiltonian operator
depends on the choice of the time coordinate. Our definition allows us to
associate rigorously with each reference frame F, a unique "space" (a
three-dimensional differentiable manifold), which is the set of the world lines
bound to F. This also is very useful for quantum mechanics. We briefly discuss
the application of these concepts to G\"odel's universe.Comment: 14 pages in standard 12pt format. v2: Discussion Section 4
reinforced, now includes an application to G\"odel's universe
Basic quantum mechanics for three Dirac equations in a curved spacetime
We study the basic quantum mechanics for a fully general set of Dirac
matrices in a curved spacetime by extending Pauli's method. We further extend
this study to three versions of the Dirac equation: the standard
(Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which
are based on the recently proposed linear tensor representations of the Dirac
field (TRD). We begin with the current conservation: we show that the latter
applies to any solution of the Dirac equation, iff the field of Dirac matrices
satisfies a specific PDE. This equation is always satisfied for
DFW with its restricted choice for the matrices. It similarly
restricts the choice of the matrices for TRD. However, this
restriction can be achieved. The frame dependence of a general Hamiltonian
operator is studied. We show that in any given reference frame with minor
restrictions on the spacetime metric, the axioms of quantum mechanics impose a
unique form for the Hilbert space scalar product. Finally, the condition for
the general Dirac Hamiltonian operator to be Hermitian is derived in a general
curved spacetime. For DFW, the validity of this hermiticity condition depends
on the choice of the matrices.Comment: 35 pages (standard 12pt format). v3: Introduction reinforced, a few
wording improvements in the body, former appendix removed and made into a
paper, arXiv:1003.3521. v2: a few additional informations, e.g. regarding the
similarity transformations that are allowabl
Hestenes' Tetrad and Spin Connections
Defining a spin connection is necessary for formulating Dirac's bispinor
equation in a curved space-time. Hestenes has shown that a bispinor field is
equivalent to an orthonormal tetrad of vector fields together with a complex
scalar field. In this paper, we show that using Hestenes' tetrad for the spin
connection in a Riemannian space-time leads to a Yang-Mills formulation of the
Dirac Lagrangian in which the bispinor field is mapped to a set of Yang-Mills
gauge potentials and a complex scalar field. This result was previously proved
for a Minkowski space-time using Fierz identities. As an application we derive
several different non-Riemannian spin connections found in the literature
directly from an arbitrary linear connection acting on Hestenes' tetrad and
scalar fields. We also derive spin connections for which Dirac's bispinor
equation is form invariant. Previous work has not considered form invariance of
the Dirac equation as a criterion for defining a general spin connection