"Quantum field theory and coalgebraic logic in theoretical computer science
Abstract
In this paper we suggest that in the framework of the Category Theory it is possible to demonstrate
the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in QFT, interpreted as a thermal field
theory. Each pair algebra-coalgebra characterizes, indeed, a QFT system and its mirroring thermal
bath, respectively, so to model dissipative quantum systems persistently in far-from-equilibrium
conditions, with an evident significance also for biological sciences. The q-deformed Hopf Coalgebras
and the q-deformed Hopf Algebras constitute two dual categories because characterized by the
same functor T , related with the Bogoliubov transform, and by its contravariant application T*
respectively. The q-deformation parameter, indeed, is related to the Bogoliubov angle, and it is
effectively a thermal parameter. Therefore, the different values of q identify univocally, and then
label, the vacua appearing in the foliation process of the quantum vacuum. This means that, in the
framework of Universal Coalgebra, as general theory of dynamic and computing systems (“labelled
state-transition systems”), the so labelled infinitely many quantum vacua can be interpreted as the
Final Coalgebra of an “Infinite State Black-Box Machine”. All this opens the way to the possibility
of designing a new class of universal quantum computing architectures based on this coalgebraic
formulation of QFT, as its ability of naturally generating a Fibonacci progression demonstrates.