2 research outputs found
The Pauli equation in scale relativity
In standard quantum mechanics, it is not possible to directly extend the
Schrodinger equation to spinors, so the Pauli equation must be derived from the
Dirac equation by taking its non-relativistic limit. Hence, it predicts the
existence of an intrinsic magnetic moment for the electron and gives its
correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon
and Dirac equations have been derived from first principles as geodesics
equations of a non-differentiable and continuous spacetime. Since such a
generalized geometry implies the occurence of new discrete symmetry breakings,
this has led us to write Dirac bi-spinors in the form of bi-quaternions
(complex quaternions). In the present work, we show that, in scale relativity
also, the correct Pauli equation can only be obtained from a non-relativistic
limit of the relativistic geodesics equation (which, after integration, becomes
the Dirac equation) and not from the non-relativistic formalism (that involves
symmetry breakings in a fractal 3-space). The same degeneracy procedure, when
it is applied to the bi-quaternionic 4-velocity used to derive the Dirac
equation, naturally yields a Pauli-type quaternionic 3-velocity. It therefore
corroborates the relevance of the scale relativity approach for the building
from first principles of the quantum postulates and of the quantum tools. This
also reinforces the relativistic and fundamentally quantum nature of spin,
which we attribute in scale relativity to the non-differentiability of the
quantum spacetime geometry (and not only of the quantum space). We conclude by
performing numerical simulations of spinor geodesics, that allow one to gain a
physical geometric picture of the nature of spin.Comment: 22 pages, 2 figures, accepted for publication in J. Phys. A: Math. &
Ge