# quantum operator algebras in quantum field theories

## 0.2 Quantum operator algebras in quantum field theories: QOA Role in QFTs

### 0.2.1 Quantum groups; quantum operator algebras and related symmetries.

Quantum theories adopted a new lease of life post 1955 when von Neumann beautifully re-formulated quantum mechanics (QM) and quantum theories (QT) in the mathematically rigorous context of Hilbert spaces and operator algebras defined over such spaces. From a current physics perspective, von Neumann’ s approach to quantum mechanics has however done much more: it has not only paved the way to expanding the role of symmetry   in physics, as for example with the Wigner-Eckhart theorem and its applications, but also revealed the fundamental importance in quantum physics of the state space geometry of quantum operator algebras.

## 0.3 Basic mathematical definitions in QOA:

### 0.3.1 Von Neumann algebra

Let $\mathcal{H}$ denote a complex (separable) Hilbert space. A von Neumann algebra $\mathcal{A}$ acting on $\mathcal{H}$ is a subset of the algebra of all bounded operators   $\mathcal{L}(\mathcal{H})$ such that:

If one calls a commutant of a set $\mathcal{A}$ the special set of bounded operators on $\mathcal{L}(\mathcal{H})$ which commute with all elements in $\mathcal{A}$, then this second condition implies that the commutant of the commutant of $\mathcal{A}$ is again the set $\mathcal{A}$.

On the other hand, a von Neumann algebra $\mathcal{A}$ inherits a unital subalgebra   from $\mathcal{L}(\mathcal{H})$, and according to the first condition in its definition $\mathcal{A}$, it does indeed inherit a $*$-subalgebra structure  as further explained in the next section on C* -algebras. Furthermore, one also has available a notable bicommutant theorem which states that: “$\mathcal{A}$ is a von Neumann algebra if and only if $\mathcal{A}$ is a $*$-subalgebra of $\mathcal{L}(\mathcal{H})$, closed for the smallest topology defined by continuous maps $(\xi,\eta)\longmapsto(A\xi,\eta)$ for all $$ where $<.,.>$ denotes the inner product  defined on $\mathcal{H}$ ”.

For a well-presented treatment of the geometry of the state spaces of quantum operator algebras, the reader is referred to Aflsen and Schultz (2003; [AS2k3]).

### 0.3.2 Hopf algebra

First, a unital associative algebra consists of a linear space $A$ together with two linear maps:

 $\displaystyle m$ $\displaystyle:A\otimes A{\longrightarrow}A~{},~{}(multiplication)$ (0.2) $\displaystyle\eta$ $\displaystyle:\mathbb{C}{\longrightarrow}A~{},~{}(unity)$

satisfying the conditions

 $\displaystyle m(m\otimes\mathbf{1})$ $\displaystyle=m(\mathbf{1}\otimes m)$ (0.3) $\displaystyle m(\mathbf{1}\otimes\eta)$ $\displaystyle=m(\eta\otimes\mathbf{1})={\rm id}~{}.$

This first condition can be seen in terms of a commuting diagram :

 $\begin{CD}A\otimes A\otimes A@>{m\otimes{\rm id}}>{}>A\otimes A\\ @V{{\rm id}\otimes m}V{}V@V{}V{m}V\\ A\otimes A@ >m>>A\end{CD}$ (0.4)

Next suppose we consider ‘reversing the arrows’, and take an algebra $A$ equipped with a linear homorphisms $\Delta:A{\longrightarrow}A\otimes A$, satisfying, for $a,b\in A$ :

 $\displaystyle\Delta(ab)$ $\displaystyle=\Delta(a)\Delta(b)$ (0.5) $\displaystyle(\Delta\otimes{\rm id})\Delta$ $\displaystyle=({\rm id}\otimes\Delta)\Delta~{}.$

We call $\Delta$ a comultiplication, which is said to be coasociative in so far that the following diagram commutes

 $\begin{CD}A\otimes A\otimes A@<{\Delta\otimes{\rm id}}<{} (0.6)

There is also a counterpart to $\eta$, the counity map $\varepsilon:A{\longrightarrow}\mathbb{C}$ satisfying

 $({\rm id}\otimes\varepsilon)\circ\Delta=(\varepsilon\otimes{\rm id})\circ% \Delta={\rm id}~{}.$ (0.7)

A bialgebra  $(A,m,\Delta,\eta,\varepsilon)$ is a linear space $A$ with maps $m,\Delta,\eta,\varepsilon$ satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism $S:A{\longrightarrow}A$, satisfying $S(ab)=S(b)S(a)$, for $a,b\in A$ . This map is defined implicitly via the property :

 $m(S\otimes{\rm id})\circ\Delta=m({\rm id}\otimes S)\circ\Delta=\eta\circ% \varepsilon~{}~{}.$ (0.8)

We call $S$ the antipode map.

A Hopf algebra is then a bialgebra $(A,m,\eta,\Delta,\varepsilon)$ equipped with an antipode map $S$ .

Commutative   and non-commutative Hopf algebras form the backbone of quantum ‘groups’ and are essential to the generalizations  of symmetry. Indeed, in most respects a quantum ‘group’ is closely related to its dual Hopf algebra; in the case of a finite, commutative quantum group    its dual Hopf algebra is obtained via Fourier transformation  of the group elements. When Hopf algebras are actually associated with their dual, proper groups of matrices, there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.

### 0.3.3 Groupoids

Recall that a groupoid ${\mathsf{G}}$ is, loosely speaking, a small category with inverses        over its set of objects $X=Ob({\mathsf{G}})$ . One often writes ${\mathsf{G}}^{y}_{x}$ for the set of morphisms   in ${\mathsf{G}}$ from $x$ to $y$ . A topological groupoid consists of a space ${\mathsf{G}}$, a distinguished subspace  ${\mathsf{G}}^{(0)}={\rm Ob(\mathsf{G)}}\subset{\mathsf{G}}$, called the space of objects of ${\mathsf{G}}$, together with maps

 $r,s~{}:~{}\hbox{}$ (0.9)