Dirac equation: representation independence and tensor transformation

Arminjon, Mayeul; Reifler, Frank

doi:10.1590/s0103-97332008000200007

Cited by 18 publications

(63 citation statements)

References 30 publications

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“…To reach this goal, in particular to study the influence of the possible choices for the field of Dirac matrices γ µ (X), it is necessary to be able to use any possible choice of the latter. This is achieved by using the "hermitizing matrix" A of Bargmann [8] and Pauli [33,34], which allows one to define the current and the scalar product for a generic (ordered) set (γ µ ) of Dirac matrices [4]. To our knowledge, until now, the possible choices for the fields γ µ (X) and the Hilbert space scalar product have not been systematically investigated, even for the standard (DFW) equation.…”

Section: Aim Of This Workmentioning

confidence: 99%

“…In a Minkowski spacetime in both Cartesian and affine coordinates, the TRD theory with constant Dirac matrices has been proved [4] to be quantum-mechanically fully equivalent to the genuine Dirac theory. See also Subsect.…”

Section: Status Of the Two Alternative Dirac Equationsmentioning

confidence: 99%

“…The proof of this theorem in Ref. [4] is directly valid for TRD, because that proof uses the tensor transformation of the Dirac matrices, Eq. (9) here, but this is not in an essential way.…”

Section: A Common Tool: the Hermitizing Matricesmentioning

confidence: 99%

“…Using this fact, it is straightforward to modify the proof of Theorem 6 in Ref. [4] so that it applies to DFW.…”

Section: A Common Tool: the Hermitizing Matricesmentioning

confidence: 99%

“…For TRD, the components A µν of matrix A build a covariant second-order tensor [4]. For DFW, we may choose A = A .…”

Section: A Common Tool: the Hermitizing Matricesmentioning

confidence: 99%

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Basic quantum mechanics for three Dirac equations in a curved spacetime

Arminjon¹,

Reifler²

2010

Braz. J. Phys.

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124

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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.

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Section: Aim Of This Workmentioning

confidence: 99%

Section: Status Of the Two Alternative Dirac Equationsmentioning

confidence: 99%

Section: A Common Tool: the Hermitizing Matricesmentioning

confidence: 99%

“…Using this fact, it is straightforward to modify the proof of Theorem 6 in Ref. [4] so that it applies to DFW.…”

Section: A Common Tool: the Hermitizing Matricesmentioning

confidence: 99%

“…For TRD, the components A µν of matrix A build a covariant second-order tensor [4]. For DFW, we may choose A = A .…”

Section: A Common Tool: the Hermitizing Matricesmentioning

confidence: 99%

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Basic quantum mechanics for three Dirac equations in a curved spacetime

Arminjon¹,

Reifler²

2010

Braz. J. Phys.

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124

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

2011

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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.

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

Arminjon

2011

Annalen der Physik

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International audienceIn 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 $(u_\alpha)$. The data of a reference frame F fixes a four-velocity field $v$, and also fixes a rotation-rate field $\Mat{\Omega}$. 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 $\Mat{\Xi}$ be equal to $\Mat{\Omega}$ - or also, if one imposes that $\Mat{\Xi}=\Mat{0}$. We also analyze other proposals which aimed at solving the title problem

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Dirac equation: representation independence and tensor transformation

Cited by 18 publications

References 30 publications

Basic quantum mechanics for three Dirac equations in a curved spacetime

Basic quantum mechanics for three Dirac equations in a curved spacetime

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

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

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