理论计算机科学中的量子场论与余代数逻辑
Quantum Field Theory and Coalgebraic Logic in Theoretical Computer Science
摘要：
在本文中，我们建议，在范畴论的框架下，它是可能的，以证明数学和逻辑之间的\textit{dual equivalence}的$q$ -变形的Hopf余代数的范畴和QFT的$q$ -变形的Hopf代数的范畴，解释为热场论。每一对代数-余代数分别刻画了QFT系统和它的镜像热浴，从而在远离平衡的条件下持续地模拟耗散量子系统，这对生物科学也具有明显的意义。$q$ -变形的Hopf余代数和$q$ -变形的Hopf代数构成了两个对偶范畴，因为它们分别由与Bogoliubov变换有关的同一函子$T$和它的逆变应用$T^{op}$刻画.实际上，\textit{q} -变形参数与Bogoliubov角有关，并且它是有效的热参数。因此，$q$的不同值可以明确地识别并标记出量子真空的叶理过程中出现的真空。这意味着，在泛余代数的框架下，作为动力学和计算系统（“标记状态转移系统”）的一般理论，这样标记的无限多个量子真空可以被解释为“无限状态黑箱机”的最终余代数。所有这些都为基于QFT的这种共代数公式设计一类新的通用量子计算架构开辟了道路，因为它自然生成斐波那契级数的能力证明了这一点。
英文摘要：
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.
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