Banach-Tarski Paradox: A Physical Interpretation (updated)

NOTICE

• NEVER TRUST THE FOLLOWING
• Almost irrelevant economic argument in an early version of this post is now divided to another post
• Addendum: some corrections due to misunderstandings to measure theory (Aug 29 2013)

Paradoxical Decomposition

Having eyes, see ye not? and having ears, hear ye not? and do ye not remember? When I brake the five loaves among five thousand, how many baskets full of fragments took ye up? They say unto him, Twelve. And when the seven among four thousand, how many baskets full of fragments took ye up? And they said, Seven. And he said unto them, How is it that ye do not understand? *1

• Banach-Tarski (1924) stated: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces
• Such pieces must have measure zero (CORRECTED Aug 29 2013: do not have any measure) to make the ball recomposed from certain congruence transformations (motions) of them having arbitrary positive measure
• von Neumann (1929) further stated a 2-dimensional analogue of above
• Some said that just blame the (abuse of) axiom of choice, which assure zero-measured (CORRECTED Aug 29 2013: unmeasurable) subsets of the continuum having same cardinality of the continuum
• Others think that the concept of continuum is an idealization beyond physical reality

What do we mean to say "zero-measured" in QM/QFT?

• Measure is (classical) volume
• Measure is also (quantum) probability
• So "it's zero-measured" just means "physically such state can't be realized"
• But it is still possible to think mathematically about direct sum decomposition (hence diagonalization) of a fermion into several bosonic fields which can not actually exist with interactions
• Such bosonic fields might be, if realized, like holographic images projected with extremely high energy coherent light sources
• Even if such projection were (approximately) possible, you would not see even one of such images
• It would need just sub-femto seconds for extreme pressure of coherent light to make everything around evaporated

Addendum (Aug 29 2013)

• I was confusing "measure zero" with "unmeasurable"
• Unmeasurable sets in standard ZFC set theory are non-member of any σ-algebra, whereas zero-measured (null) sets are valid member
• Why unmeasurable subsets could be constructed? Probably, because of lack of sufficient determinacy - under ZF+AD (implies countable choice, but not full AC), no such subset exists. ZFC+certain large cardinal axioms also works somewhat alike.
• For any decomposition to be definite, we need strong orthogonality far beyond the degree that standard ZFC set theory assures
• The full story should be written: wait for me or publish your much better conception

Footnotes

1. Mark 8:18-21