where [??] is the d'Alembertian
operator in four dimensions.
We denote by [??] the D'Alembertian
operator [??] = [[partial derivative].sub.tt] - [[partial derivative].sub.xx].
Keywords: D'Alembertian, homotopy argument, eigenvalues.
where [SIGMA] is a bounded domain in [R.sup.N](N [greater than or equal to] 1) with boundary [partial derivative][SIGMA], Q = x (0, 2[pi]),  = [partial derivative]2/[partial derivative][t.sup.2]-[DELTA] is the D'Alembertian, h is a given function in [L.sup.2](Q) and g : [SIGMA] x R x R [right arrow] R is a Caratheodory function and 2[pi]-periodic in t .
* Dirac's master wave equation can be factorized--essentially by taking the square-root of the d'Alembertian
operator applied to a Majorana 2-spinor wavefunction--to obtain not only Dirac's famous electron equation (in the common 4-spinor formalism), but also the equations for more exotic spinning particles (including the Proca equation, the Duffin-Kemmer equation for spins 0 and 1, and the Rarita-Schwinger equations for spin 3/2).
Then, noninteger powers of d'Alembertian
are considered in different works (for example, see Section 28 in  and [12-14]).
where [[nabla].sup.2] is Laplacian and [[partial derivative].sub.[mu]] [[partial derivative].sup.[mu]] is d'Alembertian
where  = [[nabla].sup.i][[nabla].sub.i] is the d'Alembertian
There is another difficulty: there is no general covariant d'Alembertian
which, being in its clear form, could be included into the Einstein equations.
In this limit we must fully face the nonlocal aspect of a minimal length theory, which in the proposed algebra is signaled by the presence of the d'Alembertian
in the denominator.
where [??] - [g.sup.[alpha][beta][[nabla].sub.[alpha] [[nabla].sub.[beta] is the covariant d'Alembertian
. Using (23) in (22) gives
Marino, "Canonical quantization of theories containing fractional powers of the d'Alembertian
operator" Journal of Physics A: Mathematical and General, vol.