Pilitron dynamics

Pilitron dynamics is the theory of how pilitrons interact.

Basics
Position is described by a position descriptor, $$\tilde{x}$$. This descriptor could be anything, so that there is no dependence on the coordinate system used. It could for example be a vector, if you choose such system of coordinates.

Direction is described by a direction descriptor, $$\tilde{e}$$, which, again, could be anything you pick it to be.

There are some rules though:


 * $$\tilde{x}+\tilde{e}$$ = the position descriptor which describes the position a point is placed in when it moves from $$\tilde{x}$$ in direction $$\tilde{e}$$.
 * $$n\tilde{e}$$ = the length $$n$$ in direction $$\tilde{e}$$.
 * $$|\tilde{a} - \tilde{b}|$$, when $$\tilde{a}$$ and $$\tilde{b}$$ are two position descriptors, is the distance between them.
 * $$\Omega(\tilde{a},\tilde{b})$$, when $$\tilde{a}$$ and $$\tilde{b}$$ are points, gives the unit direction descriptor from $$\tilde{a}$$ to $$\tilde{b}$$.
 * $$\Delta(\tilde{a}, \tilde{b})$$ = the distance between two positions.
 * $$\tilde{a} + \tilde{b}$$, when a and b are directions, is like adding together vectors and then normalizing.

The field function, $$F(\tilde{x})$$ gives the force direction vector at the specified point in the field, multiplied by the speed of light $$c$$.

The energy function, $$E(\tilde{x})$$ gives the energy at the specified position.

Fundamental Law
The fundamental law of pilitron dynamics can be written as follows:


 * $$\tilde{x}^\prime = \tilde{x} + \int_{A}^{B} F(\tilde{x}) \mathrm{d}t = \tilde{x} + \left(\theta(B) - \theta(A)\right)$$

Where $$A$$ and $$B$$ describe a length in time, and $$\theta(x)$$ is the antiderivative of $$F(x)$$, so:


 * $$\theta^\prime(x) = F(x)$$

Dynamic Symmetry
Pilitron dynamics sais that all motion is symmetrical - if we add up all velocities in the Universe (including direction) we get 0:


 * $$\sum_{a} \sum_{b} F_a(\tilde{x}_b) = 0$$

Momentum
Momentum is a very important property, and in pilitron dynamics is given by the momentum function, $$M(\tilde{x})$$, which is defined as follows:


 * $$M(\tilde{x}) = \frac{E(\tilde{x}) - \mu}{F(\tilde{x})}$$

where $$\mu$$ is the dark energy constant, which depends on the particular field.

Action & Lagrangian
In pilitron dynamics, we define action as being the work required to move an object from a reference position, $$\mathbf{R}$$ to its current position $$\mathbf{P}$$, times the time it takes for that movement to occur ($$a$$ to $$b$$):


 * $$\mathcal{S} = \int_{t=a}^{t=b} \mathcal{L} dt$$

The pilitron lagrangian, $$\mathcal{L}$$, is therefore the work required to make an immediate change to the system. Since the energy that moves the object (the kinetic energy) is the momentum times the velocity, then the Lagrangian without fields is given by:


 * $$\mathcal{L} = pv$$

The Lagrangian can be different if a particle interacts with some kind of field.

Space
The Lagrangian at a given position in space can be given by:


 * $$\mathcal{L}(\tilde{x}) = M(\tilde{x})F(\tilde{x})$$

Relativistic Quantum Mechanics
Relativistic Quantum Mechanics (RQM) is a theory that merges quantum mechanics with relativity, in the framework of pilitron dynamics.

Basics & Analogy
One of the central rules of quantum mechanics is that things are uncertain. A particle can be in many positions at the same time. One of the central rules of relativity is that things appear differently depending on your frame of reference. Yet another rule of quantum mechanics is that when something is observed, its wave function collapses.

If you were on the Sun, you could see the Earth orbiting it, but on Earth, it appears as though it was the Sun orbiting the Earth. Relativistic Quantum Mechanics takes a similar assumption with the quantum world. If you were a positively-charged particle, [+], and a negative particle, [-], was orbiting you, then because of quantum mechanics, [-] would appear everywhere on the orbit at the same time. But what if you were to look from [-]'s perspective? It would look as if [+] was orbiting you, and since you don't actually have a specific position in relation to [+], it would look as if [+] was everywhere on the orbit at the same time. Therefore, we can say 2 things:


 * An "observed" quantum system is whatever you pick to be your frame of reference.
 * The relative position and velocity of quantum particles is a distribution, not just a single value.