**Document** "Vacuums and strings" on zenodo.org.

**Comments:**

There exists a volumetric density of force in the so-called Maxwell's vacuum (far from any source).

A part of this density is due to an electromagnetic polarization, the origin of which must be searched in observing the activity of our universe.

In involving the simplest utilization of the theory studying deformed tensor products, and in considering that that density acts "as if" it would be able to deform a piece of string, a whole set of equations of states for that Maxwell's vacuum can be generated. One of them is the celebrated relation:

pressure + density of energy = 0

A modern and simplified version is exposed on the French version of this page.

**Analysis**

In writing Maxwell’s equation (EM) for empty regions of our universe (no mass and no electrical charge) in the dual space E*(3, C), and in introducing then the trivial decomposition for each cross product appearing there, I got an expression for a volumetric density of force.

The mathematical demonstration exhibits seemingly no peculiar difficulty and, in some way, is developed in a totally classical world (17^{th} century) which would be today identified with a universe “à la ADM”; i.e.: as one of the plausible 3 + 1 slices of the full four-dimensional world. Therefore, the result may wake up the attention: “Why is there a force where nothing should happen?”

Thinking more deeply about the circumstances, we can identify two acceptable explanations:

- (i) We intuitively and experimentally know that an observer who is flying in vacuum (for example inside the ISS) will “feel” the influence of relatively far situated sources; recently detected gravitational waves reinforce this knowledge.
- (ii) The mathematics has replaced the supposedly physical space E(3, C) by its mathematical dual representation E*(3, C). Even if mathematics tells us that both spaces are isomorphic to each other: “Is the dual space
*really* the same than the original one, or is it -for example- orthogonal to it?” With different words: “Has the use of this mathematical isomorphism discretely replaced Maxwell’s context where his equations were supposed to act and where the discussion was supposed to be developed by another context?”