I looked into

Minkowski space-time for some time now.

\(c^2dt^2-dx^2-dy^2-dz^2=ds^2\)

for years I wondered about the squaring, squared time interval what is the intuition behind it except pure math? dose square time intervals give much intuition at the quantum level?... it is now clear to me that the squaring is needed to get rid of \(v\) in the space-time interval to get it invariant. This is however not needed in the special case where all causal events are linked over distance with the speed of light. This is interesting as under atomism all causal events inside elementary particles are linked over the distance of the Compton length by the speed of light (the speed of the indivisible particle). Now we suddenly get a invariant space-time off the simpler form

\(cdt-dx-dy-dz=0\)

This is linked to the Lorentz transformation. I am only interested in causal events, so then \( t=\frac{L}{v_2}\) where \(v_2\leq c\) for all causal events, in Lorentz Minkowski framework we have

\begin{eqnarray}

c^2t'^2-x'^2&=&\left(\frac{t-\frac{L}{c^2} v}{\sqrt{1-\frac{v^2}{c^2}}}\right)^2c^2-\left(\frac{L -tv}{\sqrt{1-\frac{v^2}{c^2}}}\right)^2 \nonumber\\

&&\left(\frac{\frac{L}{v_2}-\frac{L}{c^2} v}{\sqrt{1-\frac{v^2}{c^2}}}\right)^2c^2-\left(\frac{L -\frac{L}{v_2}v}{\sqrt{1-\frac{v^2}{c^2}}}\right)^2 \nonumber\\

&&\left(\frac{L \frac{c}{v_2}-L \frac{v}{c}}{\sqrt{1-\frac{v^2}{c^2}}}\right)^2-\left(\frac{L -L \frac{v}{v_2}}{\sqrt{1-\frac{v^2}{c^2}}}\right)^2 \nonumber \\

&&\frac{L^2-2L^2\frac{v}{v_2}+L^2 \frac{v^2}{v_2^2}}{1-\frac{v^2}{c^2}}-\frac{L^2 \frac{c^2}{v_2^2}-2L^2\frac{v}{v_2}+

L^2 \frac{v^2}{c^2}}{1-\frac{v^2}{c^2}} \nonumber \\

&&\frac{L^2+L^2 \frac{v^2}{v_2^2}-L^2 \frac{c^2}{v_2^2}-L^2 \frac{v^2}{c^2}}{1-\frac{v^2}{c^2}} \nonumber \\

&&\frac{L^2\left(1-\frac{v^2}{c^2}+\frac{v^2}{v_2^2}-\frac{c^2}{v_2^2}\right)}{1-\frac{v^2}{c^2}} \nonumber \\

&&\frac{L^2\left(1-\frac{v^2}{c^2}\right)\left(1-\frac{c^2}{v_2^2}\right)}{1-\frac{v^2}{c^2}}\nonumber \\

&&L^2\left(1-\frac{c^2}{v_2^2}\right)

\end{eqnarray}

indeed a invariant space-time interval. S

o yes the squaring is needed to get rid of v, without the squaring v dose not disappear and the space-time interval is then not independent

of what frame it is observed from (v is the speed between reference frames, \(v_2\) is the speed that causality is transferred as observed from the rest frame). However in the special case all causal events are linked by the speed of light \(v_2=c\) over a distance, then we do not need to square to get rid of the v, then we get invariant space-time on the simpler form \(cdt-dx-dy-dz=0\).
Minkowski space-time seems to be unnecessarily complex for the deepest quantum level (as it is a top-down theory) where the space-time interval with or without squaring always is zero

\begin{eqnarray}

t'c-x' &=&\frac{\frac{\bar{\lambda}}{c}-\frac{\bar{\lambda}}{c^2} v}{\sqrt{1-\frac{v^2}{c^2}}}c-\frac{\bar{\lambda} -\frac{\bar{\lambda}}{c}v}{\sqrt{1-\frac{v^2}{c^2}}}\nonumber\\

&=&\frac{\bar{\lambda}-\frac{\bar{\lambda}}{c} v}{\sqrt{1-\frac{v^2}{c^2}}}-\frac{\bar{\lambda} -\frac{\bar{\lambda}}{c}v}{\sqrt{1-\frac{v^2}{c^2}}}=0

\end{eqnarray}

No matter the value of v (naturally inside v<c or fully correct \( v\leq c\sqrt{1-\frac{l_p^2}{\bar{\lambda}^2}} )\) this space-time interval is invariant as it always is zero. The key is to understand mass is Compton clocks!

It is also worth mentioning that it is unclear in modern QM if their theory is consistent with Minkowski:

Minkowski Space-Time and Quantum Mechanics
I added a section to my

New QM (some typos in the latest section there I see that I will fix soon)

At the deepest level there is only Planck mass events (lasting for one Planck second), that is collisions between the building blocks of light, and then there is light connecting these Planck mass events, simply indivisibles moving and colliding. So all Planck mass events are connected with the speed of light in this binary world.

The new quantum-space time is a special case of Minkowski where all causality is linked by the speed of light.

The new space-time geometry
\(cdt-dx-dy-dz=0\)

is always invariant. We are talking about the internal structure of elementary particles. In the internal structure of elementary particles that happens over the Compton scale all events are linked with the speed of light (the speed of the indivisible). Still two electrons can naturally move at velocity v relative to each other, my theory take this into account.

Forget the standard QM fantasy of de Broglie wave with infinite wavelength at rest and superluminal velocities that not can carry information, when you work with de Broglie and think it represent something real (and not simply is a derivative) then one get the strangest interpretations. To link a QM theory rooted in de Broglie to Minkowski space-time is no easy task. But to link a Compton wave model to simplified Minkowski is likely consistent.