Constructability for the Banach-Tarski Theorem

v. 6 n. 33

NOTICE

1. Cover image: Schematic representation. https://www.shutterstock.com/search/abstract-construction
2. The complete archive to these Letters is not available in the "newsletter" category but can be accessed in the "posts" category.

This is a continuation of recent Letters about the mutual benefits to quantum physics, cosmology and the pure mathematics of the Banach-Tarski Theorem. This Theorem is often referred to as a paradox because it is counter intuitive. In a common form it states that a solid ball in three-dimensional space can be disassembled into a finite number of pieces, then reassembled into two solid balls of the same size with the same pieces with no gaps or distortion of the pieces after some translations and rotations of the pieces. Another example is the original ball can be disassembled/reassembled into a larger ball, where the product may be any size; this implies the size of the visible Universe.

In this Theorem volume is not conserved; the product(s) may be larger or smaller than the original. Ordinarily there is no mention of mass; it is pure abstract mathematics that has no relationship with the real world -- physics; it is said to be "non-constructable" and apart from reality. But even mathematicians might be wary of this Theorem because of this divorce from reality. This could be problematic for mathematics because, the Theorem and its like are widely employed in the proofs of other theorems, possibly placing much of mathematics in question.

The trend these days is for physics to be led by such mathematics, unlike earlier when it was more a tool. But the paradoxical nature of this prominent mathematical Theorem may reflect back on physics to show how precarious the structure of mathematics and mathematical physics might be. If this Theorem can be bolstered, physics in its present conventional abstract guise might be seen as on firmer ground as well.

The attempt in recent writings is to bring this Theorem into the real world by introducing the concept of mass. This is unusual because the infinities involved with the Theorem do not bode well with real masses being substituted for infinite mathematical points. Further, each subpart in these items contains infinite mathematical points, and assigning even infinitesimal mass to infinite points can result in infinite masses, collapsing the Theory. The paradox arises because one item is transformed into two identical items, which is more like magic than science. This is why the Theorem is in the pure mathematical realm. But its possible utility to physics was suggested in recent Letters, so it is worth the effort to further attempt blending this mathematics and physics.

Picture a hand-held three-dimensional sphere in real space -- in the ground-state quantum vacuum or the continuum of spacetime -- coincident with the continuum of three-dimensional mathematical space.

Within this sphere are an infinite number of mathematical points; any finite dimension, no matter how small has an infinite number of zero-dimensional points. The energy density of the zero-point quantum vacuum is about 10^-8 ergs/cm^3. New virtual particles are continually coming into and going out of existence in this real space. Since there are no time constraints in the Theorem, any degree of one-to-one pairing of mathematical point to virtual particle accuracy may be reached even if there is only one virtual particle per second in the original volume. The time constraint is entirely dependent on the physical initial condition.

In this case the product must have a reduced volume because the energy density of the product cannot be lower than the ground state vacuum energy density (the volume of the product(s) cannot be larger than the original), thus exemplifying the reduction of volume for the product, instead of the common example of doubling the volume of the original. This exemplifies physical realism for the Theorem; only certain products are permitted, depending on initial conditions of the environment. This could fulfill the desire for constructability in the Theorem for sake of the mathematics as well as the physics.

As usual, volume is not conserved but mass-energy is conserved (at zero, see below), while at the same time there is the required one-to-one matching of points to "masses," and avoidance of the paradox. Further, the masses of virtual particles are not real, immeasurable, as mathematical points and subsets are immeasurable; energy is borrowed but given back. The pool of virtual particles is as amorphous as the treatment of mathematical points in sets. Each pool, mathematical or physical, is effectively infinite in accord with the constraints of the Theorem; the points may pair with the ground state virtual particles as they wish, one virtual particle at a time, over a time span that is not specified in the Theorem rather from the physical initial conditions. There is a conservation of energy at zero eventually because the energy employed is borrowed and immeasurable.

The Theorem is guided by transferring temporarily borrowed energy, but on final viewing it is taken away; during the procedure the pairing was as "real" as points are (un)real and as virtual particles are (un)real. However, the energy density of space is actually real and conventionally calculated to be 10^-8 ergs/cm^3, and the points have dipped into this real pool one by one, and have dealt with reality as long as it took to form one smaller sphere from the original, where the total volume of the product is half that of the original in this case. After construction no energy is apparent; it was only borrowed during the construction process, and therefore ultimately conserved at zero. The uncertainty principle and observational physics serve as a catalyst for physical constraint on a mathematical Theorem, then put back on the shelf, along with any energy involved.