Not know if you have read my essay. In the essay, I show, that must exist four solutions for positive total energy. And these solutions all involve energy and momentum components, imaginary. The existence of these unobservable states, virtual, giving the minimum value of Heisenberg uncertainty.
Similarly, this implies the wave-particle duality. This dual state, while not measured, explains the double path followed by an electron, for example, in the experiment of the slit.
[math]E^{2}=m^{2}c^{4}+p^{2}c^{2}= (imc^{2}+pc)(-imc^{2}+pc)=E_{1}^{2}
[/math]
[math]E^{2}=m^{2}c^{4}+p^{2}c^{2}= (imc^{2}-pc)(-imc^{2}-pc)=E_{2}^{2}
[/math]
[math]E^{2}=m^{2}c^{4}+p^{2}c^{2}= (mc^{2}+ipc)(mc^{2}-ipc)=E_{3}^{2}
[/math]
[math]E^{2}=m^{2}c^{4}+p^{2}c^{2}= (-mc^{2}+ipc)(-mc^{2}-ipc)=E_{4}^{2}
[/math]
[math]E_{T}=\sqrt{E_{1}^{2}+E_{2}^{2}+E_{3}^{2}+E_{4}^{2}}=2E
[/math]
[math]E_{T}=\sqrt{E_{1}^{2}+E_{2}^{2}+E_{3}^{2}+E_{4}^{2}}=2E\rightarrow min(2\triangle p\triangle x\geq\hbar)
[/math]
Note, as the solutions are paired Terms of positive-negative energy, and negative-positive momentum
fulfilled:
[math]\left(\begin{array}{cc}
imc^{2}-pc & -imc^{2}+pc\\
imc^{2}+pc & -imc^{2}-pc
\end{array}\right)=0[/math]
[math]\left(\begin{array}{cc}
mc^{2}+ipc\; & mc^{2}-ipc\\
-mc^{2}+ipc\; & mc^{2}-ipc
\end{array}\right)=0[/math]
An example, found experimentally, the quantum entanglement. This effect is where the key to quantum mechanics. It is recognized that the change in the particle interlaced, to measure her mate is independant of the distance and occurs instantly. But this last assertion is an assumption, not been demonstrated either experimentally or mathematically. On the contrary, if it is allowed that virtual particles exceed the speed of light, in their ephemeral existence. But what happens to a virtual graviton?
A virtual graviton, which has a typical energy of 1.5 x 10^-43 J
This implies, that his time of existence, according to the uncertainty principle, it would:
[math](\hbar/2)/1.5\cdot10^{-43}=351557484\, s[/math]
At this time, at least can travel a distance of: c351557484= 1.05 x 10^17 m
In other words, the perturbation produced in space-time, to perform the measurement on particle A, is transmitted, meaning through grid quantified spacetime, the particle B
After doing experiments with such long distances such as those between one satellite-laboratory and laboratory on earth could possibly measure the change delay.
Generally, the existence of mathematical singularities in physical theory; often tell us that the theory is incomplete. I am referring to a zero time in changing the particle B
But as nature exibe extraordinary behavior; must be demonstrated both theoretical, and experimentally.
Finally: an example of the existence of these dual virtual states, is the Casimir force. Force produced in the vacuum between two conducting plates
[math]\frac{F_{c}}{A}=\frac{\hbar c\pi^{2}}{240d^{4}}
[/math]
Regards