The origin of special relativity has much to do with the origin of dynamics.
Useful axioms or theorems that could act as fundaments for physics exist, but they treat only the static relational structure of physical items and physical fields. These law sets consist for a part of traditional quantum logic as it is defined by Birkhoff, von Neumann, Jauch, Piron and others. Quantum logic says nothing about dynamics and it says nothing about physical fields. The static configuration of fields is treated in the Helmholtz and Hodge theorems. Dynamics couple these static ingredients. When dynamics occurs in steps, then after each step the static status quo is reestablished with new conditions. The laws governing the static status quo are reinstalled as well. In this picture nature bumps in universe wide steps from one static condition to the next one. With other words there exists a global progression parameter. This is not our common notion of (coordinate) time, because that has only local validity. In the new condition the actual configuration of the fields determine what will happen with the influenced physical items in the next step. The physical items are sources (charges) of the fields and when they move they form the currents that cause still other kinds of fields. Investigation of inertia shows that a movement that differs from a uniform movement and a non-uniform distribution of charges or currents cause forces that control the next step. In this way dynamics emerges.
All physical items in universe cooperate in the implementation of inertia. Because electrical charges can compensate each other while masses cannot do that inertia shows up for gravitational fields and not for electromagnetic fields. However, both fields take part in the emergence of dynamics.
The axioms that define the static interrelations between physical items stem from quantum logic. The sketched picture implies that the lattice of propositions about physical systems is controlled by influences that act as a sticky resistance against altering the relational structure of the set of propositions. In nature this sticky resistance is implemented by physical fields. Locally the action of the fields alter the observed properties of the physical items. It is done such that the combination of space and local time gets a Minkowski structure. In this way we get two new notions of time: coordinate time and proper time.
The global progression parameter is not a dimension of a space with Minkowski signature. Thus with this parameter the fields do not encounter a maximum speed of information transfer.
The former statement means that fields are not living in a Minkowski space. How is it then possible that our observable world has a Minkowski signature? It is only possible when during the step a conversion takes place that reshapes the expectation value of the position observable into something that has a Minkowski signature. Something that is capable of this is an infinitesimal transform of a hyper complex number q by a hyper complex number u that is close to unity:
[math]\textit{u}\approx 1+ \Delta \textit{s}[/math]
The transform runs:
[math] \Delta \textit{q} = \textit{u}\cdot \textit{q}/\textit{u} - \textit{q} [/math]
The infinitesimal steps of s and q are imaginary and orthogonal. The rectangular triangle can be closed with an infinitesimal step in the form of the hypotenuse. The corresponding variable is the coordinate time t. Now holds:
[math] \Delta \textit{s}^{2} = \Delta \textit{t}^{2} - \Delta \textit{q}^{2} [/math]
This construct has a Minkowski signature. In order to achieve this, q and u must be quaternions or higher dimensional hyper complex numbers. During the process the real part of q is not changed. It is not participating in the q step. The empty place is filled by the coordinate time t.
If this story is true then the space variable q and the coordinate time t have little in common. They do not form a spacetime construct. However the dynamically changed imaginary part of q forms together with t a Minkowskian spacetime.
Einstein left us his formulas for special relativity, but he never gave an explanation WHY these formulas work. The isomorphism of the lattice of Hilbert subspaces with the lattice of quantum logical propositions suggests that properties of physical items are represented by eigenvalues or as expectation values of normal operators in Hilbert space. Spacetime, with its Minkowski signature does not fit as an eigenvalue of a normal operator. Above an explanation is given for the fact that hyper complex numbers that can act as eigenvalues are capable of converting hyper complex numbers into constructs that indeed have a Minkowski signature.
The transform is indirectly done by a thing called redefiner. In quantum logic it redefines quantum logical propositions and its equivalent in Hilbert space redefines the subspaces that represent both the proposition and the physical item that is described by that proposition. The redefiner does not touch the Hilbert vectors. Thus it does not touch eigenvectors of operators. It represents the action of the Schrödinger picture. In the Heisenberg picture a set of parallel trails of unitary operators cooperate in order to shift the Hilbert subspace that corresponds with the considered physical item. The unitary transforms also shift the eigenvectors of operators. Since Hilbert space acts as an affine space for the change of the representation of the physical item, in both pictures the expectation value of the observed property of the physical item must be the same. In each step the set of unitary transforms cause an infinitesimal transform of the expectation value of q in the form
[math] \Delta \textit{q} = \textit{u}\cdot \textit{q}/\textit{u} - \textit{q} [/math]
as sketched above.
More details can be found in http://www.scitech.nl/English/Science/Ontheoriginofdynamics.pdf.
