Tangleoids with quantum field theories in biosystems

Ozel, Cenap; Albeladi, Hadeel; Linker, Patrick

doi:10.1515/cmb-2024-0007

Full Article

1

Introduction

The quantum effects were reported for certain biological systems such as the magnetic orientation sense in migratory birds. In quantum field theories it is possible to derive effective quantum field theories (EQFTs) that describe composite particles based on the dynamics of microscopic particles these composite particles are made of. For example, one can derive dynamics of protons and neutrons when using the dynamics of their consistent particles (here the quarks) and therefore the underlying theory, quantum chromodynamics. One might go one step further and derive quantum field theories for a system of particles that are usually described by classical dynamics. These might be parts of biological cells responsible for signalling within an organism. So, in that case, when quantum electrodynamics, the theory of electrically charged particles like electrons, and quantum hydrodynamics, the second quantized version of classical hy- drodynamics, are combined to make up an EQFT, one can derive a theory for parts of a biological cell. Novel effects can be described within this theory. The previous papers [2,6] showed that quasiparticles can be categorized by compositeness and by their occurrence in a time-dependent process. The latter is also interesting because it is able to capture far-from-equilibrium effects that also occur in biological systems. Category theory allows us to predict the structure of quasiparticles and their general behavior. Quasiparticle models are also useful to describe unexplained phenomena like ectoplasm materialization, where human-shaped forms arise temporarily from tissue.

The article is organized as follows: In Section 2 we review the oriented welded tangleoid categories including braid relations. Then in Section 3 we give quantum field formalism with tangleoids including braid relations in biosystems. Finally, we present our conclusions in Section 4.

2

Unoriented welded tangleoids

In this section, we review the definition of welded tangleoids categories defined in [1]. A monoidal category (see for example [5]) of unoriented welded tangleoids has been defined by giving a presentation by using the presentation of slideable $\frac{1}{2}$ \frac{1}{2} -monoidal categories [1].

Definition 2.1

[1, definition 7.2.1] Consider the monoidal graph $β = (N, E (β), \otimes_{0}, 0, δ_{1}, δ_{2}),$ \beta =\left({\mathbb{N}},E\left(\beta ),{\otimes }_{0},0,{\delta }_{1},{\delta }_{2}), where for all $m, n \in N$ m,n\in {\mathbb{N}} , $m \otimes_{0} n = m + n$ m{\otimes }_{0}n=m+n , and $E (β) = {X_{+}, X_{-}, X, \cup, \cap, ¡,!},$ E\left(\beta )=\left\{{X}_{+},{X}_{-},X,\cup ,\hspace{0.33em}\cap ,\hspace{0.1em}\text{¡}\hspace{0.1em},\!\right\}, the incidence maps $\begin{matrix} δ_{1} X_{+} & = & 2, & δ_{2} X_{+} = 2, & δ_{1} X_{-} = 2, & δ_{2} X_{-} = 2, \\ δ_{1} X & = & 2, & δ_{2} X = 2, & δ_{1} \cup = 0, & δ_{2} \cup = 2, \\ δ_{1} \cap & = & 2, & δ_{2} \cap = 0, & δ_{1} ¡ = 1, & δ_{2} ¡ = 0, \\ δ_{1}! & = & 0, & δ_{2}! = 1 . \end{matrix}$ \begin{array}{rcllll}{\delta }_{1}{X}_{+}& =& 2,& {\delta }_{2}{X}_{+}=2,& {\delta }_{1}{X}_{-}=2,& {\delta }_{2}{X}_{-}=2,\\ {\delta }_{1}X& =& 2,& {\delta }_{2}X=2,& {\delta }_{1}\cup =0,& {\delta }_{2}\cup =2,\\ {\delta }_{1}\cap & =& 2,& {\delta }_{2}\cap =0,& {\delta }_{1}\hspace{0.1em}\text{¡}\hspace{0.1em}=1,& {\delta }_{2}\hspace{0.1em}\text{¡}\hspace{0.1em}=0,\\ {\delta }_{1}\!& =& 0,& {\delta }_{2}\!=1.& & \end{array}

These generators can be presented geometrically as

Consider the path category, see for example [4], over $β^{*}$ {\beta }^{* } , the extent of the monoidal graph $β$ \beta . $P (β^{*}) = (N, {hom}_{P (β^{*})} (n, m), •, ϕ_{_}) .$ P\left({\beta }^{* })=\left({\mathbb{N}},{\hom }_{P\left({\beta }^{* })}\left(n,m),\bullet ,{\phi }_{\_}). Therefore $Ω (β) = (P (β^{*}), \otimes_{0}, 0,_{n} #, #_{m})$ \Omega \left(\beta )=\left(P\left({\beta }^{* }),{\otimes }_{0},0{,}_{n}\#,{\#}_{m}) is a $\frac{1}{2}$ \frac{1}{2} -monoidal category, whose set of objects is the set of natural numbers, where for all $n, m, k \in N$ n,m,k\in {\mathbb{N}} ; ${}_{n}#_{m} (k) = n \otimes_{0} k \otimes_{0} m = n + k + m,$ {}_{n}\#_{m}\left(k)=n{\otimes }_{0}k{\otimes }_{0}m=n+k+m, and for all generating morphism $(f : k \to k') \in E (β)$ (f:k\to k^{\prime} )\in E\left(\beta ) , we have ${}_{n}#_{m} (f) = n + k + m \overset{n Θ f Θ m}{⟶} n + k' + m .$ {}_{n}\#_{m}(f)=n+k+m\hspace{0.33em}\mathop{\longrightarrow }\limits^{n\Theta f\Theta m}\hspace{0.33em}n+k^{\prime} +m. Then we have the free- $\frac{1}{2}$ \frac{1}{2} -monoidal category-triple $(β, Ω (β), δ) .$ \left(\beta ,\Omega \left(\beta ),\delta ).

Definition 2.2

(Unoriented welded tangleoids category) The unoriented welded tangleoids category $U W T C$ UWTC is the strict monoidal category formally presented by $F (Ω (β) ∕ \bar{\underset{̲}{W}}),$ {\mathfrak{F}}(\Omega \left(\beta )/\overline{\underline{W}}), where $Ω (β)$ \Omega \left(\beta ) defined in [1, Section 7.2] and $\bar{\underset{̲}{W}}$ \overline{\underline{W}} is the $\frac{1}{2}$ \frac{1}{2} -monoidal closure of the congruence template $W$ W that is defined as follows.

Given $m, n \in N$ m,n\in {\mathbb{N}} , then $W_{m, n}$ {W}_{m,n} is the relation in ${hom}_{P (β^{*})} (m, n)$ {\hom }_{P\left({\beta }^{* })}\left(m,n) , defined as (the picture will follow).

In ${hom}_{P (β^{*})} (1, 1)$ {\hom }_{P\left({\beta }^{* })}\left(1,1) , we have the only relations

$[W T_{1}]$ \left[W{T}_{1}] : $({id}_{1} \otimes \cap) (X \otimes {id}_{1}) ({id}_{1} \otimes \cup) \sim_{W_{1, 1}} {id}_{1} \sim_{W_{1, 1}} (\cap \otimes {id}_{1}) ({id}_{1} \otimes X) (\cup \otimes {id}_{1})$ \left({{\rm{id}}}_{1}\otimes \cap )\left(X\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes \cup )\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}{{\rm{id}}}_{1}\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}\left(\cap \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes X)\left(\cup \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1}) .
$[W T_{2}] : ({id}_{1} \otimes \cap) (X_{+} \otimes {id}_{1}) ({id}_{1} \otimes \cup) \sim_{W_{1, 1}} {id}_{1} \sim_{W_{1, 1}} ({id}_{1} \otimes \cap) (X_{-} \otimes {id}_{1}) ({id}_{1} \otimes \cup)$ \left[W{T}_{2}]:\left({{\rm{id}}}_{1}\otimes \cap )\left({X}_{+}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes \cup )\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}{{\rm{id}}}_{1}\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}\left({{\rm{id}}}_{1}\otimes \cap )\left({X}_{-}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes \cup ) .
$[W T_{3}] : (\cap \otimes {id}_{1}) ({id}_{1} \otimes X_{-}) (\cup \otimes {id}_{1}) \sim_{W_{1, 1}} {id}_{1} \sim_{W_{1, 1}} (\cap \otimes {id}_{1}) ({id}_{1} \otimes X_{+}) (\cup \otimes {id}_{1}) .$ \left[W{T}_{3}]:\left(\cap \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{-})\left(\cup \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1})\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}{{\rm{id}}}_{1}\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}\left(\cap \otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{+})\left(\cup \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1}).
$[W T_{4}] : (\cap \otimes {id}_{1}) ({id}_{1} \otimes \cup) \sim_{W_{1, 1}} {id}_{1} \sim_{W_{1, 1}} ({id}_{1} \otimes \cap) (\cup \otimes {id}_{1})$ \left[W{T}_{4}]:\left(\cap \otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes \cup )\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}{{\rm{id}}}_{1}\hspace{0.33em}{ \sim }_{{W}_{1,1}}\hspace{0.33em}\left({{\rm{id}}}_{1}\otimes \cap )\left(\cup \otimes {{\rm{id}}}_{1}) .

In

{hom}_{P (β^{*})} (2, 2)

{\hom }_{P\left({\beta }^{* })}\left(2,2)

, we have the only relation

$[W T_{5}] : X_{-} X_{+} \sim_{W_{2, 2}} {id}_{2} \sim_{W_{2, 2}} X_{+} X_{-}$ \left[W{T}_{5}]:{X}_{-}{X}_{+}\hspace{0.33em}{ \sim }_{{W}_{2,2}}\hspace{0.33em}{{\rm{id}}}_{2}\hspace{0.33em}{ \sim }_{{W}_{2,2}}\hspace{0.33em}{X}_{+}{X}_{-} .

In

{hom}_{P (β^{*})} (3, 3)

{\hom }_{P\left({\beta }^{* })}\left(3,3)

, we have the only relations

$[W T_{6}] : (X_{+} \otimes {id}_{1}) ({id}_{1} \otimes X_{+}) (X_{+} \otimes {id}_{1}) \sim_{W_{3, 3}} ({id}_{1} \otimes X_{+}) (X_{+} \otimes {id}_{1}) ({id}_{1} \otimes X_{+})$ \left[W{T}_{6}]:\left({X}_{+}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{+})\left({X}_{+}\otimes {{\rm{id}}}_{1}){ \sim }_{{W}_{3,3}}\left({{\rm{id}}}_{1}\otimes {X}_{+})\left({X}_{+}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{+}) .
$[W T_{7}] : (X_{+} \otimes {id}_{1}) ({id}_{1} \otimes X) (X \otimes {id}_{1}) \sim_{W_{3, 3}} ({id}_{1} \otimes X) (X \otimes {id}_{1}) ({id}_{1} \otimes X_{+})$ \left[W{T}_{7}]:\left({X}_{+}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes X)\left(X\otimes {{\rm{id}}}_{1}){ \sim }_{{W}_{3,3}}\left({{\rm{id}}}_{1}\otimes X)\left(X\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{+}) .
$[W T_{8}] : (X \otimes {id}_{1}) ({id}_{1} \otimes X_{+}) (X_{+} \otimes {id}_{1}) \sim_{W_{3, 3}} ({id}_{1} \otimes X_{+}) (X_{+} \otimes {id}_{1}) ({id}_{1} \otimes X)$ \left[W{T}_{8}]:\left(X\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{+})\left({X}_{+}\otimes {{\rm{id}}}_{1}){ \sim }_{{W}_{3,3}}\left({{\rm{id}}}_{1}\otimes {X}_{+})\left({X}_{+}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes X) .

In

{hom}_{P (β^{*})} (3, 1)

{\hom }_{P\left({\beta }^{* })}\left(3,1)

, we have the only relations

$[W T_{9}] : (\cap \otimes {id}_{1}) ({id}_{1} \otimes X_{-}) \sim_{W_{3, 1}} ({id}_{1} \otimes \cap) (X_{+} \otimes {id}_{1})$ \left[W{T}_{9}]:\left(\cap \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{-}){ \sim }_{{W}_{3,1}}\left({{\rm{id}}}_{1}\otimes \cap )\left({X}_{+}\otimes {{\rm{id}}}_{1}) .
$[W T_{9}]' : (\cap \otimes {id}_{1}) ({id}_{1} \otimes X_{+}) \sim_{W_{3, 1}} ({id}_{1} \otimes \cap) (X_{-} \otimes {id}_{1})$ \left[W{T}_{9}]^{\prime} :\left(\cap \otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes {X}_{+}){ \sim }_{{W}_{3,1}}\left({{\rm{id}}}_{1}\otimes \cap )\left({X}_{-}\otimes {{\rm{id}}}_{1}) .
$[W T_{9}]'' : (\cap \otimes {id}_{1}) ({id}_{1} \otimes X) \sim_{W_{3, 1}} ({id}_{1} \otimes \cap) (X \otimes {id}_{1})$ \left[W{T}_{9}]^{\prime} ^{\prime} :\left(\cap \otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes X){ \sim }_{{W}_{3,1}}\left({{\rm{id}}}_{1}\otimes \cap )\left(X\otimes {{\rm{id}}}_{1}) .

In

{hom}_{P (β^{*})} (1, 3)

{\hom }_{P\left({\beta }^{* })}\left(1,3)

, we have the only relations

$[W T_{10}] : ({id}_{1} \otimes X_{+}) (\cup \otimes {id}_{1}) \sim_{W_{1, 3}} (X_{-} \otimes {id}_{1}) ({id}_{1} \otimes \cup)$ \left[W{T}_{10}]:\left({{\rm{id}}}_{1}\otimes {X}_{+})\left(\cup \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1}){ \sim }_{{W}_{1,3}}\left({X}_{-}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes \cup ) .
$[W T_{10}]' : ({id}_{1} \otimes X_{-}) (\cup \otimes {id}_{1}) \sim_{W_{1, 3}} (X_{+} \otimes {id}_{1}) ({id}_{1} \otimes \cup)$ \left[W{T}_{10}]^{\prime} :\left({{\rm{id}}}_{1}\otimes {X}_{-})\left(\cup \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1}){ \sim }_{{W}_{1,3}}\left({X}_{+}\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes \cup ) .
${[W T_{10}]}^{″} : ({id}_{1} \otimes X) (\cup \otimes {id}_{1}) \sim_{W_{1, 3}} (X \otimes {id}_{1}) ({id}_{1} \otimes \cup)$ {\left[W{T}_{10}]}^{^{\prime\prime} }:\left({{\rm{id}}}_{1}\otimes X)\left(\cup \hspace{0.33em}\otimes \hspace{0.33em}{{\rm{id}}}_{1}){ \sim }_{{W}_{1,3}}\left(X\otimes {{\rm{id}}}_{1})\left({{\rm{id}}}_{1}\otimes \cup ) .

In

{hom}_{P (β^{*})} (1, 0)

{\hom }_{P\left({\beta }^{* })}\left(1,0)

, we have the only relation

$[W T_{11}] : \cap ({id}_{1} \otimes!) \sim_{W_{1, 0}} ¡ \sim_{W_{1, 0}} \cap (! \otimes {id}_{1})$ \left[W{T}_{11}]:\cap \left({{\rm{id}}}_{1}\otimes \!){ \sim }_{{W}_{1,0}}\hspace{0.33em}\hspace{0.1em}\text{¡}\hspace{0.1em}{ \sim }_{{W}_{1,0}}\hspace{0.33em}\cap \left(\!\otimes {{\rm{id}}}_{1}) .

In

{hom}_{P (β^{*})} (0, 1)

{\hom }_{P\left({\beta }^{* })}\left(0,1)

, we have the only relation:

$[W T_{12}] : ({id}_{1} \otimes ¡) \cup \sim_{W_{0, 1}}! \sim_{W_{0, 1}} (¡ \otimes {id}_{1}) \cup$ \left[W{T}_{12}]:\left({{\rm{id}}}_{1}\otimes \hspace{0.1em}\text{¡}\hspace{0.1em})\cup { \sim }_{{W}_{0,1}}\hspace{0.33em}\hspace{0.33em}\!{ \sim }_{{W}_{0,1}}\left(\hspace{0.1em}\text{¡}\hspace{0.3em}\otimes {{\rm{id}}}_{1})\cup .

In

{hom}_{P (β^{*})} (2, 1)

{\hom }_{P\left({\beta }^{* })}\left(2,1)

, we have the only relations

$[W T_{13}] : (¡ \otimes {id}_{1}) X_{+} \sim_{W_{2, 1}} {id}_{1} \otimes ¡$ \left[W{T}_{13}]:\left(\hspace{0.1em}\text{¡}\hspace{0.3em}\otimes {{\rm{id}}}_{1}){X}_{+}{ \sim }_{{W}_{2,1}}{{\rm{id}}}_{1}\otimes \hspace{0.1em}\text{¡}\hspace{0.1em} .
$[W T_{13}]' : ({id}_{1} \otimes ¡) X_{-} \sim_{W_{2, 1}} ¡ \otimes {id}_{1}$ \left[W{T}_{13}]^{\prime} :\left({{\rm{id}}}_{1}\otimes \hspace{0.1em}\text{¡}\hspace{0.1em}){X}_{-}{ \sim }_{{W}_{2,1}}\hspace{0.1em}\text{¡}\hspace{0.1em}\otimes {{\rm{id}}}_{1} .
$[W T_{14}] : (¡ \otimes {id}_{1}) X \sim_{W_{2, 1}} {id}_{1} \otimes ¡$ \left[W{T}_{14}]:\left(\hspace{0.1em}\text{¡}\hspace{0.1em}\otimes {{\rm{id}}}_{1})X{ \sim }_{{W}_{2,1}}{{\rm{id}}}_{1}\otimes \hspace{0.1em}\text{¡}\hspace{0.1em} .
$[W T_{14}]' : ({id}_{1} \otimes ¡) X \sim_{W_{2, 1}} ¡ \otimes {id}_{1}$ \left[W{T}_{14}]^{\prime} :\left({{\rm{id}}}_{1}\otimes \hspace{0.1em}\text{¡}\hspace{0.1em})X{ \sim }_{{W}_{2,1}}\hspace{0.1em}\text{¡}\hspace{0.1em}\otimes {{\rm{id}}}_{1} .

Note that we do not impose that in ${hom}_{P (β^{*})} (2, 1)$ {\hom }_{P\left({\beta }^{* })}\left(2,1) :

$(¡ \otimes {id}_{1}) X_{-} ≁_{W_{2, 1}} {id}_{1} \otimes ¡$ \left(\hspace{0.1em}\text{¡}\hspace{0.1em}\otimes {{\rm{id}}}_{1}){X}_{-}{\nsim }_{{W}_{2,1}}{{\rm{id}}}_{1}\otimes \hspace{0.1em}\text{¡}\hspace{0.1em} .

These relations can be present geometrically as (note we read the diagram from bottom to top)

https://etheses.whiterose.ac.uk/31190/1/Albeladi_Hadeel_Mathematics_PhD_2022.pdf.

3

Quantum field theories in biosystems

The open question of whether quantum field theories can be used in biological systems will be addressed in this article. Quantum effects were reported for certain biological systems like the magnetic orientation sense in migratory birds. In quantum field theories, it is possible to derive EQFTs that describe composite particles based on the dynamics of microscopic particles where these composite particles are made of. For example, one can derive the dynamics of protons and neutrons when using the dynamics of their consistent particles (here the quarks) and therefore the underlying theory, quantum chromodynamics. One might go one step further and derive quantum field theories for a system of particles usually described by classical dynamics. These might be parts of biological cells responsible for signalling within an organism. So, in that case, when quantum electrodynamics, the theory of electrically charged particles like electrons, and quantum hydrodynamics, the second quantized version of classical hydrodynamics, are combined to make up an EQFT, one can derive a theory for parts of a biological cell. Novel effects can be described within this theory. The previous papers [2,6] showed that quasiparticles can be categorized by compositeness and by their occurrence in a time-dependent process. The latter is also interesting because it can capture far-from-equilibrium effects that also occur in biological systems. Category theory allows us to predict the structure of quasiparticles and their general behavior. Quasiparticle models are also useful to describe unexplained phenomena like ectoplasm materialization, where human-shaped forms arise temporarily from tissue [7,8].

The quantum field theory formalism used to describe the scattering of elementary particles can also be applied to highly composite systems like polymeric molecules [3]. To achieve this, we will use a quantum field theory dependent on fermion fields $Ψ, \bar{Ψ}$ \Psi ,\bar{\Psi } , where the bar denotes the Hermitian conjugation and bosonic fields $A_{μ}$ {A}_{\mu } . The index $μ$ \mu runs from 0 (time component) to 4. The spatial components have indexes 1, 2, and 3. We denote with $\partial_{μ}$ {\partial }_{\mu } the differentiation by the coordinate $x_{μ}$ {x}_{\mu } , with $γ_{μ}$ {\gamma }_{\mu } the Dirac matrices and with $m$ m the mass of electrons. The action $S$ S for quantum electrodynamics reads: (A) $S = \int d^{4} x (- i \bar{Ψ} γ_{μ} (\partial^{μ} + i e A^{μ}) Ψ + F_{μ ν} F^{μ ν}), F_{μ ν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} .$ S=\int {d}^{4}x\left(-i\bar{\Psi }{\gamma }_{\mu }\left({\partial }^{\mu }+ie{A}^{\mu })\Psi +{F}_{\mu \nu }{F}^{\mu \nu }),\hspace{1.0em}{F}_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.

To derive the partition function $Z$ Z to the above action, one calculates the following integral over possible configurations (also called Feynman’s path integral): (B) $Z = \int D [Ψ] \int D [\bar{Ψ}] \int D [A_{μ}] e^{i S} .$ Z=\int D\left[\Psi ]\int D\left[\bar{\Psi }]\int D\left[{A}_{\mu }]{e}^{iS}.

To create a many-body state, one considers the expectation value of the product of elementary quantum fields, e.g. $\frac{\int D [Ψ] \int D [\bar{Ψ}] \int D [A_{μ}] Ψ (x_{1}) \dots Ψ (x_{n}) e^{i S}}{Z} = ⟨ Ψ (x_{1}) \dots Ψ (x_{n}) ⟩,$ \frac{\int D\left[\Psi ]\int D\left[\bar{\Psi }]\int D\left[{A}_{\mu }]\Psi \left({x}_{1})\ldots \Psi \left({x}_{n}){e}^{iS}}{Z}=\langle \Psi \left({x}_{1})\ldots \Psi \left({x}_{n})\rangle , $⟨ X ⟩$ \langle X\rangle is the same as $(\frac{\int D [Ψ] \int D [\bar{ψ}] \int D [A_{μ}] X e^{i S}}{Z})$ \left(\frac{\int D\left[\Psi ]\int D\left[\bar{\psi }]\int D\left[{A}_{\mu }]X{e}^{iS}}{Z}) for $n$ n distinct spacetime points. Let $J$ J be a space-time coordinate dependent auxiliary field. We want to derive correlation functions by deriving (B) by the field $J$ J . To achieve this, we add to the action (A) the auxiliary action (C) $S_{a} = \int d^{4} x (J \bar{Ψ} (x_{1}) \dots \bar{Ψ} (x_{n}) + \bar{J} Ψ (x_{1}) \dots Ψ (x_{n})) .$ {S}_{a}=\int {d}^{4}x\left(J\bar{\Psi }\left({x}_{1})\ldots \bar{\Psi }\left({x}_{n})+\bar{J}\Psi \left({x}_{1})\ldots \Psi \left({x}_{n})). This yields the partition function dependent on auxiliary fields: (D) $Z [J, \bar{J}] = \int D [Ψ] \int D [\bar{Ψ}] \int D [A_{μ}] e^{i (S + S_{a})} .$ Z\left[J,\bar{J}]=\int D\left[\Psi ]\int D\left[\bar{\Psi }]\int D\left[{A}_{\mu }]{e}^{i\left(S+{S}_{a})}.

As an example, we may derive $Z$ Z by J one time and by $\bar{J}$ \bar{J} the other time, at different spacetime points. This leads to (E) $\begin{matrix} \frac{\partial^{2} Z [J, \bar{J}]}{\partial J \partial \bar{J}} & = & \int D [Ψ] \int D [\bar{ψ}] \int D [A_{μ}] Ψ (x_{1}) \dots Ψ (x_{n}) \bar{Ψ} (x_{1}^{'}) \dots \bar{Ψ} (x_{n}^{'}) e^{i (S + S_{a})} \\ = & Z G [x_{1}, \dots, x_{n}, x_{1}^{'}, \dots, x_{n}^{'}] . \end{matrix}$ \begin{array}{rcl}\frac{{\partial }^{2}Z\left[J,\bar{J}]}{\partial J\partial \bar{J}}& =& \displaystyle \int D\left[\Psi ]\displaystyle \int D\left[\bar{\psi }]\displaystyle \int D\left[{A}_{\mu }]\Psi \left({x}_{1})\ldots \Psi \left({x}_{n})\bar{\Psi }\left({x}_{1}^{^{\prime} })\ldots \bar{\Psi }\left({x}_{n}^{^{\prime} }){e}^{i\left(S+{S}_{a})}\\ & =& ZG\left[{x}_{1},\ldots ,{x}_{n},{x}_{1}^{^{\prime} },\ldots ,{x}_{n}^{^{\prime} }].\end{array} We can define $G [x_{1}, \dots, x_{n}, x_{1}^{'}, \dots, x_{n}^{'}]$ G\left[{x}_{1},\ldots ,{x}_{n},{x}_{1}^{^{\prime} },\ldots ,{x}_{n}^{^{\prime} }] as the many-particle Green function associated with incoming spacetime points $x_{1}, \dots, x_{n}$ {x}_{1},\ldots ,{x}_{n} and outgoing spacetime points $x_{1}^{'}, \dots, x_{n}^{'}$ {x}_{1}^{^{\prime} },\ldots ,{x}_{n}^{^{\prime} } . We proceed with Taylor expansion in terms of interaction term $e \bar{ψ} γ_{μ} A^{μ} ψ$ e\bar{\psi }{\gamma }_{\mu }{A}^{\mu }\psi and auxiliary action. The partition function (D) then yields: (F) $\begin{matrix} Z [J, \bar{J}] & = & \int D [Ψ] \int D [\bar{Ψ}] \int D [A_{μ}] e^{\int d^{4} x [\bar{ψ} γ_{μ} \partial^{μ} ψ + i (\partial_{μ} A_{ν} - \partial_{ν} A_{μ}) (\partial^{μ} A^{ν} - \partial^{ν} A^{μ})]} \\ \times \sum_{k = 0}^{\infty} \frac{1}{k!} {\int d^{4} x i e \bar{ψ} γ_{μ} A^{μ} ψ)}^{k} \sum_{l = 0}^{\infty} \frac{1}{l!} {(i S_{a})}^{l} . \end{matrix}$ \begin{array}{rcl}Z\left[J,\bar{J}]& =& \displaystyle \int D\left[\Psi ]\displaystyle \int D\left[\bar{\Psi }]\displaystyle \int D\left[{A}_{\mu }]{e}^{\displaystyle \int {d}^{4}x\left[\bar{\psi }{\gamma }_{\mu }{\partial }^{\mu }\psi +i\left({\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu })\left({\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu })]}\\ & & \times \mathop{\displaystyle \sum }\limits_{k=0}^{\infty }\frac{1}{k\!}{\left(\displaystyle \int {d}^{4}xie\bar{\psi }{\gamma }_{\mu }{A}^{\mu }\psi \right)}^{k}\mathop{\displaystyle \sum }\limits_{l=0}^{\infty }\frac{1}{l\!}{\left(i{S}_{a})}^{l}.\end{array}

The partition function (F) is a Gaussian integral over all interaction vertices and the product of quantum fields that define the composite particles. It can be evaluated by considering all possible contractions. Conventionally, this is done by drawing a Feynman diagram. But for now, the arrangements of quantum fields in space and time matters. Multiple factors in the products may be defined on equal time points, thus the set of these particles span a subcategory of welded tangleoids. When considering factors belonging to different time points, the subcategory will be extended to the general welded tangleoid category. Remarkably, composite particles can not only be formed by collecting all relevant elementary particles they are consisting of but also extra particles they may encounter as a dynamic process progresses.

4

Conclusion

In general, a polymer like a protein molecule will not only consist of its atoms and the spatial alignment they make up, but also the extra molecules they may pick up during a non-equilibrium process. Mathematically, this can be formulated as a categorical extension. Concerning the formation of human-like shapes from basic tissue: While human tissue remains physically the same in a near-equilibrium situation, it will be significantly altered if it picks up other particles in a non-equilibrium process including particles from other tissue not belonging to the organism. So, for an amount of time, quasiparticle clusters will be formed at the tissue after the non-equilibrium process. These will further scatter and may exhibit structures picked up from before until the thermalization, i.e., the natural tendency to get back to an equilibrium state, occurs. Such a phenomenon is called “ectoplasm materialization.” It was never explained scientifically due to its strong complexity linked to far-from-equilibrium processes that are still not well understood. Spiritualistic and parapsychological theories were proposed to explain the materialization of human-like shapes for a moment. The exact causes of this phenomenon are still unclear, but this article will shed light on how it might be caused.

We observe that the generators of the tangleoid category that will depict the Feynman graphs are $X$ X for a scattering vertex. This arises after carrying out the integral over $A_{μ}$ {A}_{\mu } in the partition function (D) that will lead to a four-valent interaction vertex generated by a quartic term in the fermionic fields. With $X_{+}$ {X}_{+} and $X_{-}$ {X}_{-} we will depict order changes like that regarding time orderings. Ordinary propagators are depicted by $\cup$ \cup and $\cap$ \cap . Finally, the generators $!$ \! and $¡$ \hspace{0.1em}\text{¡}\hspace{0.1em} come into play if other foreign fields are picked up.

Concerning the formation of human-like shapes from basic tissue: While human tissue remains physically the same in a near-equilibrium situation, it will be significantly altered if it picks up other particles in a non-equilibrium process including particles from other tissue not belonging to the organism. So, for an amount of time, quasiparticle clusters will be formed in the tissue after the non-equilibrium process. These will further scatter and may exhibit structures picked up from before until the thermalization, i.e., the natural tendency to get back to an equilibrium state, occurs. Such a phenomenon is called “ectoplasm materialization.” It was never explained scientifically due to its strong complexity linked to far-from-equilibrium processes that are still not well understood. Spiritualistic and parapsychological theories were proposed to explain the materialization of human-like shapes for a moment. The exact causes of this phenomenon are still unclear, but this article will shed light on how it might be caused.

Tangleoids with quantum field theories in biosystems

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