Fleeting thoughts: Actual infinity as an experimental subject in quantum theory

Actual infinity seems sometimes as an especially speculative idea, to which even many mathematicians object as the source of contradictions in the foundation of mathematics. It adds to the conception of infinity to be considered not only as a process, but as a completed whole.

Nevertheless quantum mechanics and information as well as the gauge theories underlying the dominating Standard model cannot exclude to be interpreted implicitly or explicitly in terms of actual infinity thus introducing it as a possible element in a physical theory and as a subject of experiments. That experimentally accessible actual infinity, its philosophical context and admissibility is discussed here.

Quantum mechanics is the first physical theory, which involves infinity as essential element in its basic mathematical model: It utilizes Hilbert space, an infinitely dimensional vector space, thus requiring convergence for the length of any vector in it to be finite. The true notion of limit of an infinite convergent series supposes it to be considered as a whole by or in its limit: however no way that sequence to be differentiated experimentally from a finite one just for its convergence.

The gauge idea conjectures that Hilbert space is “inserted” in any space-time point and that a series of elements in Hilbert space, designated by successive space-time points of a trajectory, can be “gauged” by the limit, to which they converge. Thus the weak, strong and electromagnetic interaction as the Standard model manifests turn out unified as those limits and in fact actual infinity is used as a necessary conception under which these three interactions can be unified. All experiments in favor of the Standard model including those, which confirmed the Higgs boson with sufficient accuracy, can be interpreted as the indirect experimental proof of actual infinity.  

Another ways for that experimentally confirmable actual infinity are all phenomena of entanglement studied by quantum information. Being defined by the cases of a wave function of a quantum system tensorially nonfactorizable to those of its subsystems, entanglement can be interpreted as a direct interaction of coherent states of these quantum subsystems. The coherent states involve actual infinity immediately: The Kochen – Specker theorem excludes any “hidden variable” and thus any well-ordering in coherent state before measurement. However it turns out to be well-ordered as the results of measurement: So the measurement in quantum mechanics requires the well-ordering theorem equivalent to the axiom of choice, and one cannot help but introduce actual infinity. The phenomena of entanglement can be directly deduced from the necessity of actual infinity in quantum mechanics and information: So the entanglement can be considered as an experimentally observable effect from the physical actual infinity involved in quantum mechanics by means of “coherent state”.  

Quantum mechanics and information is the first and single theory, experimentally very well confirmed, which can be interpreted as based on actual infinity and thus accepting it as a physical entity: A domain of unification, transition or even identification between physics and mathematics starts to be seen...