arXiv subjects

Mayeul Arminjon

Publications and source records attributed to Mayeul Arminjon.

At least 37 records · Page 2Linked to original sources

arXiv2013

Comment on "Spin in an arbitrary gravitational field" arXiv:1308.4552

The authors of that work [Phys. Rev. D 88, 084014 (2013)], arXiv:1308.4552, derive quantum-mechanical equations valid for the covariant Dirac equation by restricting the choice of the tetrad field through the use of the "Schwinger gauge". Yet it has been shown previously that this gauge leaves space for a physical ambiguity of the Hamiltonian operator. It is shown here precisely how this ambiguity occurs with their settings. There is another ambiguity in the Foldy-Wouthuysen Hamiltonian, for the time-dependent case which is relevant here. However, their equations of motion for classical spinning particles are unambiguous.

physics.gen-ph↗

arXiv2013

On reference frames and the definition of space in a general spacetime

First, we review local concepts defined previously. A (local) reference frame $\mathrm{F}$ can be defined as an equivalence class of admissible spacetime charts (coordinate systems) having a common domain $\mathrm{U}$ and exchanging by a spatial coordinate change. The associated (local) physical space is made of the world lines having constant space coordinates in any chart of the class. Second, we introduce new, global concepts. The data of a non-vanishing global vector field $\,v\,$ defines a global "reference fluid". The associated global physical space is made of the maximal integral curves of that vector field. Assume that, in any of the charts which make some reference frame $\mathrm{F}$: (i) any of those integral curves $l$ has constant space coordinates $x^j$, and (ii) the mapping $l\mapsto (x^j)$ is one-to-one. In that case, the local space can be identified with a part (an open subset) of the global space.

gr-qc↗

arXiv2013

Should there be a spin-rotation coupling for a Dirac particle?

It was argued by Mashhoon that a spin-rotation coupling term should add to the Hamiltonian operator in a rotating frame, as compared with the one in an inertial frame. For a Dirac particle, the Hamiltonian and energy operators H and E were recently proved to depend on the tetrad field. We argue that this non-uniqueness of H and E really is a physical problem. We compute the energy operator in the inertial and the rotating frame, using three tetrad fields: one for each of two frameworks proposed to select the tetrad field so as to solve this non-uniqueness problem, and one proposed by Ryder. We find that Mashhoon's term is there if the tetrad rotates as does the reference frame --- but then it is also there in the energy operator for the inertial frame. In fact, the Dirac Hamiltonian operators in two reference frames in relative rotation, but corresponding to the same tetrad field, differ only by the angular momentum term. If the Mashhoon effect is to exist for a Dirac particle, the tetrad field must be selected in a specific way for each reference frame.

gr-qc↗

arXiv2013

A simpler solution of the non-uniqueness problem of the covariant Dirac theory

Although the standard generally-covariant Dirac equation is unique in a topologically simple spacetime, it has been shown that it leads to non-uniqueness problems for the Hamiltonian and energy operators, including the non-uniqueness of the energy spectrum. These problems should be solved by restricting the choice of the Dirac gamma field in a consistent way. Recently, we proposed to impose the value of the rotation rate of the tetrad field. This is not necessarily easy to implement and works only in a given reference frame. Here, we propose that the gamma field should change only by constant gauge transformations. To get that situation, we are naturally led to assume that the metric can be put in a space-isotropic diagonal form. When this is the case, it distinguishes a preferred reference frame. We show that by defining the gamma field from the "diagonal tetrad" in a chart in which the metric has that form, the uniqueness problems are solved at once for all reference frames. We discuss the physical relevance of the metric considered and our restriction to first-quantized theory.

math-ph↗

arXiv2013

On the non-uniqueness problem of the covariant Dirac theory and the spin-rotation coupling

Gorbatenko & Neznamov [arXiv:1301.7599] recently claimed the absence of the title problem. In this paper, the reason for that problem is reexplained by using the notions of a unitary transformation and of the mean value of an operator, invoked by them. Their arguments actually aim at proving the uniqueness of a particular prescription for solving this problem. But that prescription is again shown non-unique. Two Hamiltonians in the same reference frame in a Minkowski spacetime, only one of them including the spin-rotation coupling term, are proved to be physically non-equivalent. This confirms that the reality of that coupling should be checked experimentally.

gr-qc↗

arXiv2013

Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved spacetime. This canonical form is needed to apply the Whitham Lagrangian method. The latter method, unlike the WKB method, places no restriction on the magnitude of Planck's constant to obtain wave packets, and furthermore preserves the symmetries of the Dirac Lagrangian. We show using canonical Dirac fields in a curved spacetime, that the probability current has a Gordon decomposition into a convection current and a spin current, and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck's constant. We further discuss the classical-quantum correspondence in a curved spacetime based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved spacetime are a direct consequence of Whitham's Lagrangian method, and not just a physical hypothesis as introduced by Einstein and de Broglie, and by many quantum mechanics textbooks.

gr-qc↗

arXiv2012

Summary of a non-uniqueness problem of the covariant Dirac theory and of two solutions of it

We present a summary of: 1) the non-uniqueness problem of the Hamiltonian and energy operators associated, in any given coordinate system, with the generally-covariant Dirac equation; 2) two different ways to restrict the gauge freedom so as to solve that problem; 3) the application of these two ways to the case of a uniformly rotating reference frame in Minkowski spacetime: we find that a spin-rotation coupling term is there only with one of these two ways.

gr-qc↗

arXiv2011

Gravitation as a pressure force: a scalar ether theory

If the presence of a gravitational field breaks the Lorentz symmetry valid for special relativity, an "absolute motion" might be detectable. We summarize a scalar theory of gravity with a such "ether", which starts from a tentative interpretation of gravity as a pressure force. The theory also admits that our physical standards of space and time are affected by gravitation similarly as they are affected by a uniform motion. General motion is governed by an extension of Newton's second law to the curved space-time which is thus obtained. Together with the scalar field equation of the theory, this leads to a true conservation equation for the total energy. The law of motion also leads to an alternative 4-component equation governing the dynamics of a continuum in terms of its energy-momentum tensor. That new equation implies that mass conservation is obtained as a limiting behaviour for a weak and slowly varying gravitational field and/or at a low pressure. In the presence of the Lorentz force field, the new dynamical equation gives the second group of the gravitationally-modified Maxwell equations in the investigated theory. This is consistent with the geometrical optics of the theory as governed by the proposed extension of Newton's second law. The theory has the correct Newtonian limit; it predicts Schwarzschild's exterior metric of general relativity and geodesic motion in the static situation with spherical symmetry. A post-Newtonian approximation of this theory shows that no preferred-frame effect occurs for photons at the (first) post-Newtonian approximation. It is argued that the existence of preferred-frame effects in celestial mechanics, comparable in magnitude with the "relativistic" effects, does not a priori invalidate the theory.

physics.gen-ph↗

arXiv2011

Four-vector vs. four-scalar representation of the Dirac wave function

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.

gr-qc↗

arXiv2011

Classical-quantum correspondence and wave packet solutions of the Dirac equation in a curved spacetime

The idea of wave mechanics leads naturally to assume the well-known relation $E=\hbar ω$ in the specific form $H=\hbar W$, where $H$ is the classical Hamiltonian of a particle and $W$ is the dispersion relation of the sought-for wave equation. We derive the expression of $H$ in a curved spacetime with an electromagnetic field. Then we derive the Dirac equation from factorizing the polynomial dispersion equation corresponding with $H$. Conversely, summarizing a recent work, we implement the geometrical optics approximation into a canonical form of the Dirac Lagrangian. Euler-Lagrange equations are thus obtained for the amplitude and phase of the wave function. From them, one is led to define a 4-velocity field which obeys exactly the classical equation of motion. The complete de Broglie relations are then derived exact equations.

gr-qc↗

arXiv2011

A solution of the non-uniqueness problem of the Dirac Hamiltonian and energy operators

In a general spacetime, the possible choices for the field of orthonormal tetrads lead (in standard conditions) to equivalent Dirac equations. However, the Hamiltonian operator is got from rewriting the Dirac equation in a form adapted to a particular reference frame, or class of coordinate systems. That rewriting does not commute with changing the tetrad field $(u_α)$. The data of a reference frame F fixes a four-velocity field $v$, and also fixes a rotation-rate field $\MatΩ$. It is natural to impose that $u_0=v$. We show that then the spatial triad $(u_p)$ can only be rotating w.r.t. F, and that the title problem is solved if one imposes that the corresponding rotation rate $\MatΞ$ be equal to $\MatΩ$ - or also, if one imposes that $\MatΞ=\Mat{0}$. We also analyze other proposals which aimed at solving the title problem.

gr-qc↗

arXiv2011

A non-uniqueness problem of the Dirac theory in a curved spacetime

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four-dimensional, noncompact, and admits a spinor structure. We find that, for each among the three versions of the equation, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. In particular, we prove that the Dirac energy spectrum is not unique. This non-uniqueness of the energy spectrum comes from an effect of the choice of coefficients, and applies in any given coordinates.

gr-qc↗

arXiv2010

Representations of the Dirac wave function in a curved spacetime

The Dirac wave function in a curved spacetime is usually defined as a quadruplet of scalar fields. It can alternatively be defined as a four-vector field. We describe these two representations in a common geometrical framework and we prove theorems that relate together the different representations and the different choices of connections. In particular, the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the Dirac equation for a vector wave function, in the same spacetime.

gr-qc↗

arXiv2010

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 Gödel's universe.

gr-qc↗

arXiv2010

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 $γ^μ$ satisfies a specific PDE. This equation is always satisfied for DFW with its restricted choice for the $γ^μ$ matrices. It similarly restricts the choice of the $γ^μ$ 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 $γ^μ$ matrices.

gr-qc↗

arXiv2010

Non-uniqueness of the Dirac theory in a curved spacetime

We summarize a recent work on the subject title. The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. In this paper, we focus on the standard version of the gravitational Dirac equation, but the non-uniqueness applies also to our alternative versions. We find that the changes which lead to an equivalent operator H, or respectively to an equivalent operator E, are determined by initial data, or respectively have to make some point-dependent antihermitian matrix vanish. Thus, the vast majority of the possible coefficient changes lead neither to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. We show that even the Dirac energy spectrum is not unique.

gr-qc↗

arXiv2008

Dirac-type equations in a gravitational field, with vector wave function

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the locally-geodesic systems, corresponding to the Levi-Civita connection. Another class, thus another connection, emerges if a preferred reference frame is available. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon equations and two distinct Dirac-type equations in a general metric, depending on the connection used. Each of these two equations is generally-covariant, transforms the wave function as a four-vector, and differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One obeys the equivalence principle in an often-accepted sense, whereas the DFW equation obeys that principle only in an extended sense.

gr-qc↗

arXiv2008

Quantum mechanics for three versions of the Dirac equation in a curved spacetime

We present a recent work on the Dirac equation in a curved spacetime. In addition to the standard equation, two alternative versions are considered, derived from wave mechanics, and based on the tensor representation of the Dirac field. The latter considers the Dirac wave function as a spacetime vector and the set of the Dirac matrices as a third-order tensor. Having the probability current conserved for any solution of the Dirac equation gives an equation to be satisfied by the coefficient fields. A positive definite scalar product is defined and a hermiticity condition for the Dirac Hamiltonian is derived for a general coordinate system in a general curved spacetime. For the standard equation, the hermiticity of the Dirac Hamiltonian is not preserved under all admissible changes of the coefficient fields.

gr-qc↗