Algebras, Symmetries, Spaces
^{1}
A. Jadczyk^{2}
Inst. of Theor. Physics
University of Wroclaw
Abstract:
After discussing several aspects of noncommutative geometry from a
rather subjective point of view, algebraic techniques are shown to
offer a powerful tool for studying specific manifolds in the realm
of commutative geometry, with possible generalization to infinite
dimensions.
A possible conclusion stemming from the (till now unsuccessful)
experience with relativistic quantum field theory is that the
classical spacetime model breaks down at very small distances
and it has to be replaced by some kind of a 'quantum space'. Thus,
if you 'zoom' several dozen times, you see no space and no time.
No smooth manifold structure ... only deadly noncommutative
'algebra foam'. It may seem that
noncommutative geometry is the way.
In noncommutative geometry sets are first replaced by algebras and
then forgotten. Formally: If
is a map between spaces, and if and are the
respective algebras of functions on these sets, then all the
information about the map is, equivalently, contained in the
induced map between the function algebras
which is an algebra homomorphism:
Notice that the arrow has changed its direction! Thus follows the
Algorithm of noncommutative geometry:
Forget about 's, 's, and 's. Reverse the arrows, and play with
's 's 's alone, without assuming them
commutative. But: do it so as if 's,'s and 's existed.
Applications of the ideas of noncommutative geometry can be found
today in several areas:
 Algebraic geometry (still Abelian)
 Supergeometry (Grassmann algebras): Berezin, Leites, Kostant,
Manin, Penkov^{3}
 "Color" generalizations (arbitrary grading groups): Rittenberg,
Marcinek^{4}
 Simple models: Spera, DuboisViolette, Madore, Kerner, Connes,
quantum groups literature,
 Spacetime out of an algebra: Bannier (see Ref. [5]),
Ocneanu^{5}
One may wonder if
ALGEBRA IS THE ANSWER TO LIFE, UNIVERSE AND
EVERYTHING ?
I am not quite sure that this is indeed the case. I was lucky
enough to be given a chance of learning from Prof. Rudolph Haag,
whom you may know as one of the founders of the algebraic approach
to quantum theory. Once, when in Hamburg, I went to Prof. Haag,
all excited about a new idea concerning algebraic description of
gravity. I was surprised to hear this question: " ... yes, but
why an algebra ?" From this time on I have kept repeating this to
myself: why an algebra ? And also: why a foam ? It
was another idea discussed frequently in Hamburg (usually Detlev
Buchholz and Klaus Fredenhagen were active parties in these
discussions) that, perhaps, at an extreme zoom, at very small
distances, spacetime smoothes out again, and an
"essentially free", conformally invariant field theory may be at
work at this scaling limit. It is at least interesting to
renounce, for a while, the "algebra paradigm", including its
current season's overcoat, the 'noncommutative geometry paradigm'.
When we look for alternatives, we realize that
Reversing the arrows is not what the tigers like
best.
What is then more fundamental, more primitive, than algebras? We
may think of orthomodular lattices (von Neumann, Jauch, Piron, ...
) or operational logics (Randall, Foulis, ... ). One of the
important conclusions that one can arrive at, while analyzing
foundations of quantum theory using these techniques, is
that^{6}
quantization is the result of restriction on the set of available
states.
Thus:
States are more primitive than algebras.
Notice that if
is an algebra homomorphism, then
the induced map between states, goes the reverse way. Thus:
arrows get back their original directions.^{7} The generalization from
"classical" to "quantum" is now encoded in the convex geometry of
the space of states: the set of states is not necessarily a
simplex and the knowledge of pure states does not longer
suffice.^{8} Thus: mixed
states become important^{9},^{10}, and one finds that it is
necessary to study convex and differential geometry of state
spaces^{11}, including the study of manifolds embedded
there^{12}. According to this Philosophy algebras
should be invoked only when they arise as duals of state
spaces.^{13} Till now no really
satisfactory alternative model based on this principle has been
constructed.^{14}
Instead of replacing manifolds by algebras, we will consider here
manifolds embedded in algebras and studied by algebra
techniques.
WHY ILLUSTRATIVE ?
 bundle of C*algebras
 Kähler manifolds naturally embedded into projective Hilbert space of quantum states
 for p=q=2 conformal symplectic manifolds
 give rise automatically to a kind of noncommutative geometry... (reserved for future applications)
References to this Section: See
[15,16,17,18,19,20]
Introduce the following notation:
More precisely, is defined as follows:
Notice that each
defines a
algebra conjugation
which is
the adjoint with respect to the scalar product :
The manifold will be the subject of our study. It
carries the following remarkable predicates:
 HOMOGENEOUS
 IRREDUCIBLE
 HERMITIAN
 SYMMETRIC ^{15}
 COMPLEX
 KÄHLER
 EINSTEIN
 NONCOMPACT
 BOUNDED DOMAIN
It is our aim here to take advantage of the fact that is realized as a particular submanifold in the algebra
and thus allows study by algebraic techniques. In
particular we shall study:
 Riemannian (i.e. positive definite) metric on
 fundamental form
 almost complex structure
 geodesic transport
 boundary projection
If
has signature
then to each dimensional subspace
one associates the operator
:
where is the orthogonal projection^{16} onto . Conversely, for each
the range (= cokernel) subspace of
the projection
is maximal (i.e. dimensional) positive.
is a homogeneous space for the unitary group
the natural action can be also described by
with the isotropy group
Since the central circle group of acts trivially on
we get the isomorphism
Notice that the denominator is the maximal compact subgroup of the
numerator.
First of all, the relations
allow us to
identify the space
of complex tangent vectors
at
with operators such that
Real tangent vectors (
)are characterized by
the extra condition The Lie algebra
can be identified with antiHermitian
operators on They induce fundamental (real)
vector fields on :
A Kählerian metric on is simply
given by^{17}
With this metric becomes a symmetric space: each
is at
the same time in and defines the map
which is a geodesic symmetry at
A natural almost complex structure
on
is given by
Check that maps
into itself:
if and then
Check that :
We also have
The field is parallel:
thus is Kählerian.
The symplectic form is
for tangent at to Both and
are evidently invariant under the action of The
symplectic form is closed
For a symplectic manifold
with a symplectic
action of a Lie group one defines the momentum mapping
(PoincaréCartan form) as a function
satisfying the condition
for all
where for all
the
function is defined by
and is the fundamental vector field
associated to An explicit knowledge of the momentum mapping
is quite useful for a physical interpretation of the geometrical
quantities. We can easily compute the momentum map by using the
introduced algebraic technique. In our case the momentum mapping
is given by a simple formula
where is in and is in
To see that the almost complex structure is covariantly
constant under the LeviCivita connection of it is again
convenient to use the algebraic machine that provides an easy tool
for describing the geodesic parallel transport on
Given two points
the operator
is positive with respect to the p.d. scalar products
The operator
is then unambiguously defined, positive
for both scalar products, and an isometry of we have
Moreover,
and maps the plane onto . The
most interesting property of is that when applied to
tangent vectors to at it maps them into the
tangent vectors at obtained by parallel transport along the
unique geodesic connecting the two points. To see this one uses
the fact that geodesics on are trajectories of
oneparameter subgroups of The transport operators
preserve the almost complex structure on
Assume The Shilov boundary of
is defined as the minimal set on which bounded holomorphic
functions attain their maximum. It consists here of
isotropic planes. Let us fix one such plane denoted
Each being in particular a symmetry in reflects onto another isotropic plane
The map is equivariant with
respect to the stability group at If now an origin is fixed in its image under is called the antipode of or the origin of The Shilov boundary carries a natural (flat) Lorentzian conformal structure.
It is a homogeneous space not only for but also for the stability group of each point
Each point determines a Hilbert space scalar product
to which there correspond the "star":
In this way we produce a bundle of algebras over the fiber over consists of the algebra endowed
with the "star" conjugation One should notice that
the fibers identify here as algebras accidentally owing to the
homogeneity of the geometry.
 1
 A. Jadczyk and D. Kastler, " Liecartan pairs
I," Rep. Math. Phys, 25 151, (1987)
 2
 A. Jadczyk and D. Kastler, "Liecartan pairs II,"
Ann. Phys, 179 (1987) 169200
 3
 R. Coquereaux, A.
Jadczyk, and D. Kastler, "Differential and integral geometry of
grassmann algebras", Rev. Math. Phys. 3, (1991)
63100
 4

W. Marcinek, "Generalized Lie Algebras I,II," Rep. Math.
Phys, (1988)
 5
 U. Bannier, "Allgemeine kovariante algebraische
Quantenfeldtheorie und Rekonstruktion von RaumZeit,", PhD
thesis, University Hamburg, 1987.
 6

H.D. Doebner and W. Lücke, "Quantum logic as a consequence of
realistic measurements on deterministic systems," J. Math.
Phys.
 7
 B. Mielnik, "Theory of filters," Commun.math.
Phys, 15 (1969) 146
 8
 , Ph. Blanchard and A. Jadczyk, "Event Enhanced
Quantum Theory and Piecewise Deterministic Dynamics", Ann.
der Phys 4 (1995), 583599
 9
 H. Araki, "On a characterization of the state space of quantum
mechanics," Commun.math. Phys, 75 (1980) 124
 10
 C. Piron, "New quantum mechanics," In Essays in Honour of W.
Yourgrau, Plenum N.Y., 1983, pp. 345361
 11
 N. Giovannini, "Classical
and quantal systems of imprimitivity", J. Math. Phys.,
22 (1981) 23892403
 12
 F. Reusse, "On classical and quantum relativistic
dynamics", Found. Phys., 9 (1979) 865882
 13
 B. Mielnik,
"Generalized quantum mechanics", Commun. math. Phys.,
37 (1974) 221256
 14
 R. Haag and U. Bannier, "Comments on Mielnik's
generalized (non linear) quantum mechanics", Commuun. math.
Phys. 60 (1975) 16
 15
 A. Odzijewicz, "On reproducing kernels
and quantization of states", Commun. math. Phys., 114
(1988) 577597
 16
 I.T. Todorov, Conformal Description of Spinning
Particles, Springer Verlag, BerlinHeidelberg, 1986
 17
 G.J.
Zuckerman, "Quantum physics and semisimple symmetric spaces", in
Lect. Notes Math., 1077, 1984
 18
 A. Jadczyk, "Geometry of
indefinite metric spaces", Rep. Math. Phys, 1 (1971)
263276
 19
 R. Coquereanx, "Noncommutative geometry and theoretical
physics", J. Geom. Phys 6 (1989) 425490
 20
 R. Coquereanx and A. Jadczyk, "Conformal
theories, curved phase spaces, relativistic wavelets and the
geometry of complex domains", Rev. Math. Phys. 2
(1990) 144
Footnotes
 ... Spaces^{1}
 Talk given at the
8th International Workshop on Mathematical Physics, held at the
Arnold Sommerfeld Institute, Clausthal (Germany), July 1926,
1989. Published in Quantum Groups, H.D. Doebner and J.D.
Hennig (Eds), SpringerVerlag, Berlin 1990 ( Lecture Notes in
Physics, Vol. 370, pp. 426434)
 ...A. Jadczyk^{2}
 email: ajad@ift.uni.wroc.pl
 ... Penkov^{3}
 See also Ref.
[1,2,3]
 ...
Marcinek^{4}
 See [4] and references there
 ...
Ocneanu^{5}
 Private communication
 ...
that^{6}
 See e.g. [6]
 ... directions.^{7}
 Which agrees
with "the natural order of things."
 ...
suffice.^{8}
 See e.g. Ref.[7]
 ... important^{9}
 Anyway they are important for
OPEN systems; and quantum theory of open systems may even become a
necessity if one wants to incorporate equivalence
principle.
 ...,^{10}
 Note added on February 17, 2001: this paper
was written in July 1989. A year later the "Quantum Future"
project began, which resulted in EEQT, "Event Enhanced Quantum
Theory"
[8] , where it was shown that for quantum theory to
describe time series of events, open system
algorithms must necessarily be used
 ...
spaces ^{11}
 in particular the most interesting infinite
dimensional case.
 ...
there^{12}
 E.g. Phase space can be considered in some cases as
a submanifold of the state space, the embedding being implemented
via coherent states.
 ...
spaces.^{13}
 See e.g. Ref.[9]
 ...
constructed.^{14}
 Notice however the discussion in Piron,
Giovannini, Reusse
[10,11,12], and also the discussion of
probabilistic interpretation of the nonlinear Schrödinger
equation: Ref. [13,14]
 ... SYMMETRIC^{15}
 We choose the letter to remind the fact that
is isomorphic to the set of all geodesic symmetries of .
 ... projection^{16}
 'Orthogonal'
with respect to each one of the two relevant scalar products: the
indefinite, and the Hilbert space one obtained by flipping the
sign of the complementary subspace
 ... by^{17}
 The methods apply as well to infinite
dimensions, but in infinite dimensions one has to take a special
care about existence of trace