Roles of Asymptotic Condition and S-Matrix as Micro-Macro Duality in QFT

Definition 4.2 .

Define a relation $\approx$ on $V_{\Lambda}:=\{(x;m):x\in\partial\Lambda,m\in\mathbb{N}^{k}\}$ by: $(x;m)\approx(y;p)$ if and only if

(V1)

$x(m\wedge d(x))=y(p\wedge d(y))$ ; and
(V2)

$m-m\wedge d(x)=p-p\wedge d(y)$ .

Definition 4.3 .

Define a relation $\sim$ on $P_{\Lambda}:=\{(x;(m,n)):x\in\partial\Lambda,m\leq n\in\mathbb{N}^{k}\}$ by: $(x;(m,n))\sim(y;(p,q))$ if and only if

(P1)

$x(m\wedge d(x),n\wedge d(x))=y(p\wedge d(y),q\wedge d(y))$ ;
(P2)

$m-m\wedge d(x)=p-p\wedge d(y)$ ; and
(P3)

$n-m=q-p$ .

Definition 4.5 .

Define $\widetilde{r},\widetilde{s}:\widetilde{P_{\Lambda}}\to\widetilde{V_{\Lambda}}$ by:

\widetilde{r}([x;(m,n)])=[x;m]\qquad\text{and}\qquad\widetilde{s}([x;(m,n)])=[% x,n].

Definition 3.7 .

A Bayesian graphical calculus is called classical if it satisfies the following equivalent conditions:

(a)

modifiers can move through the Frobenius structure:

\raisebox{-8.535827pt}{\psfig{width=276.0pt}}\ .

(47)

(b)

modifiers are of the form:

\raisebox{-5.690551pt}{\psfig{width=126.0pt}}\,,

(48)

Definition 3.14 .

A Bayesian graphical calculus is a Q ${}_{1/2}$ -calculus when modifiers are of the form:

\psfig{width=68.0pt}\,.

(108)

Definizione 1

- A constraint is said:

1.

holonomic, any restriction on the possible configurations [ References ] of the system:

f(\mathbf{q},t)=0

(1)

and it is an integrable relation;

2.

nonholonomic, any restriction on the movements possible [ References ] of the system:

g(\mathbf{q};\dot{\mathbf{q}},t)=0

(2)

and it is not an integrable relation.

If the nonholonomic constraints represents a holonomic constraint, then it is integrable.

Definition 1

A kei or involutory quandle is a set $X$ with a binary operation $\triangleright$ satisfying for all $x,y,z\in X$

(i)

$x\triangleright x=x$ ,
(ii)

$(x\triangleright y)\triangleright y=x$ , and
(iii)

$(x\triangleright y)\triangleright z=(x\triangleright z)\triangleright(y% \triangleright z)$ .

Definition 4.33 .

A quasigroup $(Q,\circ)$ is an $(\alpha,\beta,\gamma)$ -inverse quasigroup if there exist permutations $\alpha,\beta,\gamma$ of the set $Q$ such that

\alpha(x\circ y)\circ\beta x=\gamma y

(50)

for all $x,y\in Q$ [ 51 , 7 , 50 , 96 ] .

Definition 3 (Channel)

The channel outputs the transmitted vector, corrupted by independent and identically distributed (i.i.d.) Gaussian noise:

\displaystyle\mathbf{y}=\mathbf{x}+\mathbf{z}

(6)

where $\mathbf{z}\sim\mathcal{N}(\mathbf{0},N\mathbf{I}^{n\times n})$ for some noise variance $N>0$ .

Definition 3.9 .

A ternary function $g$ is said to be cyclically commutative if it is invariant under the cyclic permutation of variables, i.e.

g(x,y,z)\approx g(y,z,x)\approx g(z,x,y).

Definition 5.1 .

A deformation retract datum of linear factorisations of $W$ consists of

\xymatrix@C+2pc{(L,b)\ar@<-0.8ex>[r]_{i}&(M,b),\ar@<-0.8ex>[l]_{p}}\quad h

where $(L,b)$ and $(M,b)$ are linear factorisations of $W$ , $p$ and $i$ are morphisms of factorisations, and $h$ is a degree one map $M\longrightarrow M$ , which together satisfy the following two conditions:

(i)

$pi=1$ ,
(ii)

$ip=1+bh+hb$ .

Notice that in particular $p$ is a homotopy equivalence with inverse $i$ .

Definition 1

Let $f:V\rightarrow\{1,2\}$

f(v)=\left\{\begin{array}[]{cl}1,&\mbox{ if }v\mbox{ belongs to a component of% type (a) or (b)},\\ 2,&\mbox{ if }v\mbox{ belongs to a component of type (c)},\\ \end{array}\right.

Definition 1.3.6

We say a sesquiad $A$ has no zero divisors , if

ab=0\quad\Rightarrow\quad\left(a=0\ \ \ \text{or}\ \ \ b=0\right).

We call a sesquiad integral , if $1\neq 0$ and

af=bf\quad\Rightarrow\quad\left(a=b\ \ \ \text{or}\ \ \ f=0\right).

An integral sesquiad has no zero divisors, but the converse does not hold in general. A subsesquiad of an integral domain is an integral sesquiad.

Definition 1.2

We set:

q=\ln\rho.

Definition 2 .

A Reed-Solomon-Code $\mathcal{GRS}(q;n,k,\boldsymbol{v})$ with length $n=q-1$ and

\boldsymbol{v}=(\alpha^{0},\alpha^{1},\ldots,\alpha^{q-2})

with $\alpha$ being a primitive element of $\mathbb{F}$ will be referred to as $\mathcal{RS}(q-1,k)$ .
The extended code of length $n=q$ obtained by adding the zero element of $\mathbb{F}$ to the vector $\boldsymbol{v}$ of $\mathcal{RS}(q-1,k)$ , i.e.

\boldsymbol{v}=(0,\alpha^{0},\alpha^{1},\ldots,\alpha^{q-2}),

will be called $\mathcal{RS}^{\ast}(q,k)$ .

Definition 3 (Distance forest)

Let $\mathcal{F}$ be a forest whose leaves are enumerated by $Q$ . Then $\mathcal{F}$ is a distance forest for $M$ if for every $q,p\in Q$ we have

d(q,p)=\begin{cases}\operatorname{level}(\operatorname{lca}(q,p))&\text{ if }% \mathrm{tree}(q)=\mathrm{tree}(p),\\ \infty&\text{ otherwise}.\end{cases}

Definition 4 (Quartet or Four-Point Condition)

The pairwise distances $\{l(i,j)\}_{i,j\in\{a,b,u,v\}}$ for the configuration in Fig. 4 satisfy

l(a,u)+l(b,v)=l(b,u)+l(a,v),

(6)

and the configuration is denoted by $Q(ab|uv)$ .