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 space 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.
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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(0.1) |
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
, 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: “
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, the reader is referred to Aflsen and Schultz (2003; []).
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(0.4) |
Next suppose we consider `reversing the arrows', and take an
algebra
equipped with a linear homorphisms
, satisfying, for
:
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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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(0.6) |
There is also a counterpart to
, the counity map
satisfying
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(0.7) |
A bialgebra
is a linear space
with maps
satisfying the above properties.
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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(0.8) |
We call
the antipode map.
A Hopf algebra is then a bialgebra
equipped with an antipode
map
.
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.
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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(0.9) |
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(0.10) |
Thus, as it is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalise bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to Brown (2006).
Several examples of groupoids are:
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
.
Therefore,
=
.
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
's 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 in the next subsection.
Let
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(0.11) |
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(0.12) |
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
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 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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