Lagrangian Pairs III

What we have done so
far is to follow the standard procedure used to define derivatives of the Lagrangian. In this example, we show how to construct the Hamiltonian using the derivatives as terms.

In our case, we are obliged to define 2 separate but, in a sense, interacting Lagrangians.

The argument for this pairing up is trivial. In the absence of any observer there is no velocity and no spatial or temporal extent.

A particle 'alone' in some void cannot experience spacetime. Spacetime is as a consequence of observation. A particle that is spontaneously decaying can experience spacetime because the decayed material become external observers.
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The reason why I start my blog with this idea is because if we consider Newton's law which can be summised as saying that...

a body in a state of motion continues in that state unless otherwise disturbed by an external influence.

Moreover, if we add to that, Einstein's idea that can be summised as...
'a body in freefall is not aware of its own motion.'

We can conclude that there is no way a moving body can tell that it is moving unless there is an external reference point. Such an object, in a sense, cannot spontaneously collapse its own wavefunction. Here, I classify spontaneous emissions (virtual particle pairs) as external observers. So that we can suggest that the Uncertainty Principle arises from the presence of external observer(s). But...why is this important?

In my opinion this implies that a freely moving particle alone in spacetime does not know that it is moving. Nor does it know that it is stationary if it cannot establish that it is moving. That is to say, according to such a particle there is no such thing as velocity. I mean that to such a particle there is no such thing as velocity(v), not even v = 0, because there is no such thing as velocity.
So, it means that velocity exists iif there is an external observer whose presence can provide a reference frame.
If we have two observers Alice(A) and Bob(B), my suggestion is that A assigns velocity to B and vice-versa.

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In the picture to the right we are shown the interaction, as it were, between the 2 observers in Lagrangian terms.

By substituting (ii) into (i) we make plain the fact that the observer is an integral part of any Lagrangian.

In this brief discussion we want to counter the argument that our Lagrangian is prone to 'instantenous action at a distance' notions by saying that we can think of such Lorentz-invariance breaking terms as geometrical artefacts rather than as actual physical transformations. Much like a gauge theory.