94 research outputs found
Electromagnetic Klein-Gordon and Dirac equations in scale relativity
We present a new step in the foundation of quantum field theory with the
tools of scale relativity. Previously, quantum motion equations (Schr\"odinger,
Klein-Gordon, Dirac, Pauli) have been derived as geodesic equations written
with a quantum-covariant derivative operator. Then, the nature of gauge
transformations, of gauge fields and of conserved charges have been given a
geometric meaning in terms of a scale-covariant derivative tool. Finally, the
electromagnetic Klein-Gordon equation has been recovered with a covariant
derivative constructed by combining the quantum-covariant velocity operator and
the scale-covariant derivative. We show here that if one tries to derive the
electromagnetic Dirac equation from the Klein-Gordon one as for the free
particle motion, i.e. as a square root of the time part of the Klein-Gordon
operator, one obtains an additional term which is the relativistic analog of
the spin-magnetic field coupling term of the Pauli equation. However, if one
first applies the quantum covariance, then implements the scale covariance
through the scale-covariant derivative, one obtains the electromagnetic Dirac
equation in its usual form. This method can also be applied successfully to the
derivation of the electromagnetic Klein-Gordon equation. This suggests it rests
on more profound roots of the theory, since it encompasses naturally the
spin-charge coupling.Comment: 14 pages, no figure
Dirac Equation in Scale Relativity
The theory of scale relativity provides a new insight into the origin of
fundamental laws in physics. Its application to microphysics allows to recover
quantum mechanics as mechanics on a non-differentiable (fractal) space-time.
The Schr\"odinger and Klein-Gordon equations have already been demonstrated as
geodesic equations in this framework. We propose here a new development of the
intrinsic properties of this theory to obtain, using the mathematical tool of
Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in
standard physics, is merely postulated. The bi-quaternionic nature of the Dirac
spinor is obtained by adding to the differential (proper) time symmetry
breaking, which yields the complex form of the wave-function in the
Schr\"odinger and Klein-Gordon equations, the breaking of further symmetries,
namely, the differential coordinate symmetry () and the parity and time reversal symmetries.Comment: 33 pages, 4 figures, latex. Submitted to Phys. Rev.
Emergence of complex and spinor wave functions in scale relativity. I. Nature of scale variables
One of the main results of Scale Relativity as regards the foundation of
quantum mechanics is its explanation of the origin of the complex nature of the
wave function. The Scale Relativity theory introduces an explicit dependence of
physical quantities on scale variables, founding itself on the theorem
according to which a continuous and non-differentiable space-time is fractal
(i.e., scale-divergent). In the present paper, the nature of the scale
variables and their relations to resolutions and differential elements are
specified in the non-relativistic case (fractal space). We show that, owing to
the scale-dependence which it induces, non-differentiability involves a
fundamental two-valuedness of the mean derivatives. Since, in the scale
relativity framework, the wave function is a manifestation of the velocity
field of fractal space-time geodesics, the two-valuedness of velocities leads
to write them in terms of complex numbers, and yields therefore the complex
nature of the wave function, from which the usual expression of the
Schr\"odinger equation can be derived.Comment: 36 pages, 5 figures, major changes from the first version, matches
the published versio
Emergence of complex and spinor wave functions in Scale Relativity. II. Lorentz invariance and bi-spinors
Owing to the non-differentiable nature of the theory of Scale Relativity, the
emergence of complex wave functions, then of spinors and bi-spinors occurs
naturally in its framework. The wave function is here a manifestation of the
velocity field of geodesics of a continuous and non-differentiable (therefore
fractal) space-time. In a first paper (Paper I), we have presented the general
argument which leads to this result using an elaborate and more detailed
derivation than previously displayed. We have therefore been able to show how
the complex wave function emerges naturally from the doubling of the velocity
field and to revisit the derivation of the non relativistic Schr\"odinger
equation of motion. In the present paper (Paper II) we deal with relativistic
motion and detail the natural emergence of the bi-spinors from such first
principles of the theory. Moreover, while Lorentz invariance has been up to now
inferred from mathematical results obtained in stochastic mechanics, we display
here a new and detailed derivation of the way one can obtain a Lorentz
invariant expression for the expectation value of the product of two
independent fractal fluctuation fields in the sole framework of the theory of
Scale Relativity. These new results allow us to enhance the robustness of our
derivation of the two main equations of motion of relativistic quantum
mechanics (the Klein-Gordon and Dirac equations) which we revisit here at
length.Comment: 24 pages, no figure; very minor corrections to fit the published
version: a few typos and a completed referenc
Resolution-scale relativistic formulation of non-differentiable mechanics
This article motivates and presents the scale relativistic approach to
non-differentiability in mechanics and its relation to quantum mechanics. It
stems from the scale relativity proposal to extend the principle of relativity
to resolution-scale transformations, which leads to considering
non-differentiable dynamical paths. We first define a complex scale-covariant
time-differential operator and show that mechanics of non-differentiable paths
is implemented in the same way as classical mechanics but with the replacement
of the time derivative and velocity with the time-differential operator and
associated complex velocity. With this, the generalized form of Newton's
fundamental relation of dynamics is shown to take the form of a Langevin
equation in the case of stationary motion characterized by a null average
classical velocity. The numerical integration of the Langevin equation in the
case of a harmonic oscillator taken as an example reveals the same statistics
as the stationary solutions of the Schrodinger equation for the same problem.
This motivates the rest of the paper, which shows Schrodinger's equation to be
a reformulation of Newton's fundamental relation of dynamics as generalized to
non-differentiable geometries and leads to an alternative interpretation of the
other axioms of standard quantum mechanics in a coherent picture. This exercise
validates the scale relativistic approach and, at the same time, it allows to
envision macroscopic chaotic systems observed at resolution time-scales
exceeding their horizon of predictability as candidates in which to search for
quantum-like dynamics and structures.Comment: 30 pages, 4 figure
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