quantum operator algebras in quantum field theories

quantum operator algebras in quantum field theories

0.1

Introduction

This is a topic entry that introduces

quantum operator algebras

and presents concisely the important
roles they play in

quantum field theories

.

Definition 0.1

.

Quantum operator algebras

(QOA) in quantum field theories are
defined as the

algebras

of observable

operators

, and as such, they are also related to the

von Neumann algebra

;
quantum operators are usually defined on

Hilbert spaces

, or in some QFTs on Hilbert space bundles
or other

similar

families of spaces.

Remark 0.1

.

Representations

of Banach

*

-algebras (that are defined on Hilbert spaces) are closely related
to C* -algebra representations which provide a useful approach to defining

quantum space-times

.

0.2

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

Important examples of quantum operators are: the

Hamiltonian operator

(or Schrödinger operator), the position and momentum operators,

Casimir operators

,

unitary operators

and spin operators. The observable operators are also

self-adjoint

. More general operators were recently defined, such as Prigogine’s superoperators.

Another development in

quantum theories

was the

introduction

of Frechét

nuclear spaces

or ‘rigged’ Hilbert spaces (Hilbert

bundles

). The following

sections

define several types of quantum operator algebras that provide the

foundation

of modern quantum field theories in mathematical physics.

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:

•

Von Neumann algebra

•

Hopf algebra

•

Groupoids

•

Haar systems

associated to measured groupoids or

locally compact groupoids

.

•

C*-algebras and

quantum groupoids

entry (attached).

0.3.1

Von Neumann algebra

Let

ℋ

denote a complex (

separable

) Hilbert space. A

von Neumann algebra

𝒜

acting on

ℋ

is a subset of the algebra of all

bounded operators

ℒ

⁢

(

ℋ

)

such that:

•

(i)

𝒜

is

closed under

the

adjoint

operation

(with the
adjoint of an

element

T

denoted by

T

*

).

•

(ii)

𝒜

equals its bicommutant, namely:

𝒜

=

{

A

∈

ℒ

(

ℋ

)

:

∀

B

∈

ℒ

(

ℋ

)

,

∀

C

∈

𝒜

,

(

B

C

=

C

B

)

⇒

(

A

B

=

B

A

)

}

.

(0.1)

If one calls a

commutant

of a set

𝒜

the special set of
bounded operators on

ℒ

⁢

(

ℋ

)

which commute with all elements in

𝒜

, then this second condition implies that the commutant of the
commutant of

𝒜

is again the set

𝒜

.

On the other hand, a von Neumann algebra

𝒜

inherits a

unital

subalgebra

from

ℒ

⁢

(

ℋ

)

, and according to the
first condition in its definition

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: “

𝒜

is a von
Neumann algebra if and only if

A

is a

*

-subalgebra of

L

⁢

(

H

)

, closed for the smallest

topology

defined by

continuous
maps

(

ξ

,

η

)

⟼

(

A

⁢

ξ

,

η

)

for all

<

A

ξ

,

η

)

>

where

<

.

,

.

>

denotes the

inner product

defined on

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

:

m

:

A

⊗

A

⟶

A

,

(

m

⁢

u

⁢

l

⁢

t

⁢

i

⁢

p

⁢

l

⁢

i

⁢

c

⁢

a

⁢

t

⁢

i

⁢

o

⁢

n

)

(0.2)

η

:

ℂ

⟶

A

,

(

u

⁢

n

⁢

i

⁢

t

⁢

y

)

satisfying the conditions

m

⁢

(

m

⊗

𝟏

)

=

m

⁢

(

𝟏

⊗

m

)

(0.3)

m

⁢

(

𝟏

⊗

η

)

=

m

⁢

(

η

⊗

𝟏

)

=

id

.

This first condition can be seen in terms of a commuting

diagram

:

A

⊗

A

⊗

A

→

m

⊗

id

A

⊗

A

id

⊗

m

↓

↓

m

A

⊗

A

⁢

@

>

m

≫

A

(0.4)

Next suppose we consider ‘reversing the arrows’, and take an
algebra

A

equipped with a linear homorphisms

Δ

:

A

⟶

A

⊗

A

, satisfying, for

a

,

b

∈

A

:

Δ

⁢

(

a

⁢

b

)

=

Δ

⁢

(

a

)

⁢

Δ

⁢

(

b

)

(0.5)

(

Δ

⊗

id

)

⁢

Δ

=

(

id

⊗

Δ

)

⁢

Δ

.

We call

Δ

a

comultiplication

, which is said to be

coasociative

in so far that the following diagram commutes

A

⊗

A

⊗

A

←

Δ

⊗

id

A

⊗

A

id

⊗

Δ

↑

↑

Δ

A

⊗

A

⁢

@

<

Δ

≪

A

(0.6)

There is also a counterpart to

η

, the

counity

map

ε

:

A

⟶

ℂ

satisfying

(

id

⊗

ε

)

∘

Δ

=

(

ε

⊗

id

)

∘

Δ

=

id

.

(0.7)

A

bialgebra

(

A

,

m

,

Δ

,

η

,

ε

)

is a linear space

A

with maps

m

,

Δ

,

η

,

ε

satisfying the above properties.

Now to recover anything resembling a group structure, we must
append such a bialgebra with an

antihomomorphism

S

:

A

⟶

A

,
satisfying

S

⁢

(

a

⁢

b

)

=

S

⁢

(

b

)

⁢

S

⁢

(

a

)

, for

a

,

b

∈

A

. This map is
defined implicitly via the property :

m

⁢

(

S

⊗

id

)

∘

Δ

=

m

⁢

(

id

⊗

S

)

∘

Δ

=

η

∘

ε

.

(0.8)

We call

S

the

antipode map

.

A

Hopf algebra

is then a bialgebra

(

A

,

m

,

η

,

Δ

,

ε

)

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

𝖦

is, loosely speaking, a

small
category

with

inverses

over its set of objects

X

=

O

⁢

b

⁢

(

𝖦

)

. One
often writes

𝖦

x

y

for the set of

morphisms

in

𝖦

from

x

to

y

.

A topological groupoid

consists of a space

𝖦

, a distinguished

subspace

𝖦

(

0

)

=

Ob

⁢

(

𝖦

)

⊂

𝖦

,
called

the space of objects

of

𝖦

, together with maps

r

,

s

:

(0.9)

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