Arjen Dijksman, 2003

There are at least two ways to restore determinism in the probability wavefunction of a particle:

- Having point-like or small spherical singularities guided by a wave, like Bohmian or De Broglie pilot-wave and double solution theories.
- Considering particles as linear segments that oscillate, vibrate or rotate. The wavefunction is then related to the periodic behavior of the particle. Among them there is Superstring theory, but also the Rotating Vector Model (RVM) for fundamental particles.

The particularity of the RVM theory is that inertia proceeds from pure mechanical considerations on massless particles, whereas other theories have to presuppose that some particles are inertial and some may be massless. The RVM theory describes a fundamental particle as a straight-line segment of constant length that rotates. This rotation transmits a probability wavefunction to the particle.

To illustrate that particularity, let AB and CD be two segment-like particles of length L that collide. For simplicity, it is sufficient to consider the special case that:

- the rotational speed of CD is zero,
- AB is rotating in a plane perpendicular to CD, with a rotational speed w,
- the collision point is the center of each segment,
- the translational velocity
**V**of CD with respect to AB is perpendicular to AB at the instant of collision.

As the particles are massless, they depart from each other at a fixed constant velocity c. The velocity due to the rotation at the extremities of AB is wL/2 .

For rotational speeds w < p c/L, the interaction is instantaneous, both particles just touch and go.

Figure 1 shows the collision sequence in the special case that the velocity at the extremity of AB is equal to pc/2, i.e. w = p c/L. The two particles first touch and depart from each other but recollide as the B-extremity of AB reaches the intersection of the circle and the path of CD. This affects the final direction of the departing velocity.

For w > p c/L, the second collision contact, located somewhere between the center of AB and the extremity B, is non instantaneous, as that point rotates faster than c about the geometrical center of AB: AB pushes CD. The two particles glide on each other for a little while. During this interaction, the contact point progresses in the direction of one of the extremities. As soon as the contact point has reached that extremity, both particles depart from each other at the velocity c. This non-instantaneity of the interaction may be interpreted as inertia, especially when the extremely rapidly rotating segment, an electron say, has to plough its way through a swarm of slowly rotating particles, photons or neutrinos say. In that case, the electron is constantly hindered in its progress through space. The electron is also constantly projecting encountered photons: it sets up an electromagnetic field varying with its acceleration or deceleration.

In the given interpretation, it appears that the measure of inertia depends on w. The measure of inertia may be assimilated to mass and the rotational speed w of the particle may be assimilated to internal energy. The greater the internal energy, the greater the mass. Further investigation is needed to verify if this dependence is of the form E = mc˛.

Figure 1: Collision
sequance between AB and CD seen in the reference
framework of AB, with w = p c/L. |

Figure 1a: First step. Vector
particles AB and CD before collision. |

Figure 1b: Second step. Vector
particles AB and CD at the instant of collision. |

Figure 1c: Third step. Vector
particles AB and CD just after the instant of collision. |

Figure 1d: Fourth step. Vector
particles AB and CD at the instant of recollision. |

Dijksman A., 1999. Fusion 75 (in french), translated at http://materion.free.fr/physique/reconsider.htm.