Note:
representations
of Banach *-algebras, that are also defined on Hilbert spaces, are related to
-algebra representations which provide a useful approach to defining quantum space-times.
Quantum Operator Algebras in Quantum Field Theories: QOAs in QFTs Examples of quantum operators are: the Hamiltonian operator (or Schrödinger operator), the position and momentum operators, Casimir operators, Unitary operators, spin operators, and so on. The observable operators are also self-adjoint. More general operators were recently defined, such as Progogine's superoperators. Another development in quantum theories is 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.
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- Mathematical definitions
Definitions:
Let
denote a complex (separable) Hilbert space. A von
Neumann algebra
acting on
is a subset of the algebra of
all bounded operators
such that:
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(1.1) |
If one calls a commutant of a set
the special set of
bounded operators on
which commute
with all elements in
, then this second condition implies that the commutant of the
commutant of
is again the set
.
On the other hand, a von Neumann algebra
inherits a
unital subalgebra from
, and according to the
first condition in its definition
does indeed inherit a
*-subalgebra structure, as further explained in the next
section on C*-algebras. Furthermore, we have notable
Bicommutant Theorem which states that
is a von
Neumann algebra if and only if
is a *-subalgebra of
, closed for the smallest topology defined by continuous
maps
for all
where
denotes the inner product
defined on
. For a well-presented
treatment of the geometry of the state spaces of quantum operator algebras, see e.g. Aflsen and Schultz (2003).
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(1.4) |
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We call
a comultiplication, which is said to be
coasociative in so far that the following diagram commutes
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(1.6) |
There is also a counterpart to
, the counity map
satisfying
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(1.7) |
Now to recover anything resembling a group
structure, we must
append such a bialgebra with an antihomomorphism
,
satisfying
, for
. This map is
defined implicitly via the property :
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(1.8) |
Commutative and noncommutative Hopf algebras form the backbone of quantum `groups' and are essential to the generalizations of symmetry. Indeed, in most respects a quantum `group' is identifiable with a Hopf algebra. When such algebras are actually associated with proper groups of matrices there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.
Recall that a groupoid
is, loosely speaking, a small category
with inverses over its set of objects
. One
often writes
for the set of morphisms
in
from
to
. A topological groupoid consists of a space
, a distinguished subspace
,
called the space of objects of
, together with maps
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(1.9) |
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(1.10) |
Several examples of groupoids are:
(a) locally compact groups, transformation groups , and any group in general (e.g. [59]
(b) equivalence relations
(c) tangent bundles
(d) the tangent groupoid
(e.g. [4])
(e) holonomy groupoids for foliations (e.g. [4])
(f) Poisson groupoids (e.g. [81])
(g) graph
groupoids (e.g. [47, 64]).
As a simple, helpful example of a groupoid, consider (b) above. Thus, let R be an equivalence relationhttp://planetphysics.org/encyclopedia/Bijective.html on a set X. Then R is a groupoid under the following operations:
. Here,
, (the diagonal of
) and
.
So
=
.
When
, R is called a trivial groupoid. A special case of a trivial groupoid
is
. (So every i is equivalent to every j). Identify
with the matrix unit
. Then the groupoid
is just matrix multiplication
except that we only multiply
when
, and
. We do not really lose anything by restricting the multiplication, since the pairs
excluded from groupoid multiplication just give the 0 product in normal algebra anyway.
For a groupoid
to be a locally compact groupoid
means that
is required to be a (second countable) locally compact Hausdorff space, and the product and also inversion maps are required to be continuous. Each
as well as the unit space
is closed in
. What replaces the left Haar measure
on
is a system of measures
(
), where
is a positive regular
Borel measure on
with dense support. In addition, the
â @ Ys are required to vary continuously (when integrated against
and to form an invariant family in the sense that for each x, the map
is a measure preserving homeomorphism
from
onto
. Such a system
is called a left Haar system for the locally compact groupoid
.
This is defined more precisely next.
Let
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(1.11) |
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(1.12) |
The presence of a left Haar system on
has important
topological implications: it requires that the range map
is open. For such a
with a left Haar system, the vector space
is a
convolution *-algebra, where for
:
, with
f*(x)
.
One has
to be the enveloping C*-algebra
of
(and also representations are required to be
continuous in the inductive limit topology). Equivalently, it is
the completion of
where
is the universal representation of
. For
example, if
, then
is just the
finite dimensional algebra
, the span of the
's.
There exists (e.g.[63, p.91]) a measurable Hilbert bundlehttp://planetphysics.org/encyclopedia/HilbertBundle.html
with
and a G-representation L on
. Then,
for every pair
of square
integrable sections of
,
it is required that the function
be
-measurable. The representation
of
is then given by:
.
The triple
is called a measurable
-Hilbert bundle.
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