# AN UNDERSTANDING OF BRAIN DISSIPATION USING HOPF ALGEBRA

### Abstract

In quantum field theory, we consider systems that are open and are in constant communication with a bigger system (a coupling to the bigger system) that is known as the thermal bath. The openness of the system enables t h i s c o m m u n i c a t i o n , a l b e i t i n t e r a c t i o n . T h i s communication or interaction is captured in one worddissipation. In seeking to articulate the dynamics of the system's dissipation, we compute the details of the dynamics responsible for such dissipation, that is to say the Hamiltonian describing the system, the thermal bath and the system-thermal bath coupling. This is for the sake of attaining to a formalism of the system. We know that it is difficult, albeit impossible to compute a system that is open, this is because the exchange of energy, and fluctuation of forces between the systems, provides for an erroneous reading. Therefore in order to capture this dynamics, the dissipation processes are enabled between the system and the bath, in such case, the system is generically closed. In such wise, we preserve the canonical structure of the canonical commutation relations (CCR). The issue we encounter in that which pertains to brain system, is in providing a formalism that will match the dynamics expressed in the usage of quantum field theory (QFT). We posit that an understanding of the brain system, can be better represented by borrowing excerpts from the formalisms as furnished by Hopf Algebra as used in Category Theory. We start off, by indicating that the dissipation is an integral aspect of human brain systems, and then we show that QFT is better adapted to providing an explanation to the aforementioned system. We then bring in the formalism provided by Hopf Algebra for a fitting conclusion to this.