Quantum Field Theory and Coalgebraic Logic in Theoretical Computer Science

TL;DR

This paper explores the duality between q-deformed Hopf algebras and coalgebras in quantum field theory within category theory, proposing a coalgebraic framework for modeling dissipative quantum systems and potential quantum computing architectures.

Contribution

It establishes a mathematical duality between algebraic and coalgebraic structures in QFT using category theory, linking thermal parameters to quantum vacua and proposing a new coalgebraic approach to quantum computing.

Findings

Dual equivalence between q-deformed Hopf algebras and coalgebras in QFT.

Quantum vacua labeled by q-parameter relate to thermal states.

Potential for designing universal quantum computers based on coalgebraic QFT.

Abstract

In this paper we suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical \textit{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 $T^{op}$, respectively. The \textit{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.