# Algebras, Symmetries, Spaces 1

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.

# 1. Noncommutative Geometry

## 1.1 Why quantum space?

A possible conclusion stemming from the (till now unsuccessful) experience with relativistic quantum field theory is that the classical space-time 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.

## 1.2 The idea of noncommutative geometry

In noncommutative geometry sets are first replaced by algebras and then forgotten. Formally: If

 (1)

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
 (2)

which is an algebra homomorphism:
 (3)

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.

## 1.3 Applications

Applications of the ideas of noncommutative geometry can be found today in several areas:

• Algebraic geometry (still Abelian)
• Super-geometry (Grassmann algebras): Berezin, Leites, Kostant, Manin, Penkov3
• "Color" generalizations (arbitrary grading groups): Rittenberg, Marcinek4
• Simple models: Spera, Dubois-Violette, Madore, Kerner, Connes, quantum groups literature,
• Space-time out of an algebra: Bannier (see Ref. [5]), Ocneanu5

## 1.4 And beyond

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, space-time 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 that6

quantization is the result of restriction on the set of available states.
Thus:
States are more primitive than algebras.
Notice that if
 (4)

is an algebra homomorphism, then
 (5)

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 important9,10, and one finds that it is necessary to study convex and differential geometry of state spaces11, including the study of manifolds embedded there12. 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

# 2. Studying manifolds by noncommutative algebra techniques - an easy divertimento and illustrative example

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:

 (6)

More precisely, is defined as follows:
 (7)

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

## 2.1 Relation to Twistors

If has signature then to each -dimensional -subspace one associates the operator :
 (8)

where is the orthogonal projection16 onto . Conversely, for each the range (= co-kernel) subspace of the projection
 (9)

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
 (10)

Notice that the denominator is the maximal compact subgroup of the numerator.

## 2.2 Tangent Spaces

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 anti-Hermitian operators on They induce fundamental (real) vector fields on :
 (11)

## 2.3 Riemannian metric

A Kählerian metric on is simply given by17
 (12)

With this metric becomes a symmetric space: each is at the same time in and defines the map
 (13)

which is a geodesic symmetry at

## 2.4 Almost complex structure

A natural almost complex structure on is given by
 (14)

Check that maps into itself:

if and then

 (15)

Check that :
 (16)

We also have
 (17)

The field is parallel: thus is Kählerian.

## 2.5 Fundamental symplectic form

The symplectic form is
 (18)

for tangent at to Both and are evidently invariant under the action of The symplectic form is closed

## 2.6 The momentum mapping

For a symplectic manifold with a symplectic action of a Lie group one defines the momentum mapping (Poincaré-Cartan form) as a function satisfying the condition

 (19)

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
 (20)

where is in and is in

## 2.7 Geodesic transport formula

To see that the almost complex structure is covariantly constant under the Levi-Civita 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

 (21)

is then unambiguously defined, positive for both scalar products, and an isometry of we have Moreover,
 (22)

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 one-parameter subgroups of The transport operators preserve the almost complex structure on

## 2.8 Boundary map and Cayley transform

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
 (23)

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

## 2.9 C* - algebra bundle

Each point determines a Hilbert space scalar product

 (24)

to which there correspond the "star":
 (25)

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.

## Bibliography

1
A. Jadczyk and D. Kastler, " Lie-cartan pairs I," Rep. Math. Phys, 25 1-51, (1987)

2
A. Jadczyk and D. Kastler, "Lie-cartan pairs II," Ann. Phys, 179 (1987) 169-200

3
R. Coquereaux, A. Jadczyk, and D. Kastler, "Differential and integral geometry of grassmann algebras", Rev. Math. Phys. 3, (1991) 63-100

4
W. Marcinek, "Generalized Lie Algebras I,II," Rep. Math. Phys, (1988)

5
U. Bannier, "Allgemeine kovariante algebraische Quantenfeldtheorie und Rekonstruktion von Raum-Zeit,", 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) 1-46

8
, Ph. Blanchard and A. Jadczyk, "Event Enhanced Quantum Theory and Piecewise Deterministic Dynamics", Ann. der Phys 4 (1995), 583-599

9
H. Araki, "On a characterization of the state space of quantum mechanics," Commun.math. Phys, 75 (1980) 1-24

10
C. Piron, "New quantum mechanics," In Essays in Honour of W. Yourgrau, Plenum N.Y., 1983, pp. 345-361

11
N. Giovannini, "Classical and quantal systems of imprimitivity", J. Math. Phys., 22 (1981) 2389-2403

12
F. Reusse, "On classical and quantum relativistic dynamics", Found. Phys., 9 (1979) 865-882

13
B. Mielnik, "Generalized quantum mechanics", Commun. math. Phys., 37 (1974) 221-256

14
R. Haag and U. Bannier, "Comments on Mielnik's generalized (non linear) quantum mechanics", Commuun. math. Phys. 60 (1975) 1-6

15
A. Odzijewicz, "On reproducing kernels and quantization of states", Commun. math. Phys., 114 (1988) 577-597

16
I.T. Todorov, Conformal Description of Spinning Particles, Springer Verlag, Berlin-Heidelberg, 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) 263-276

19
R. Coquereanx, "Noncommutative geometry and theoretical physics", J. Geom. Phys 6 (1989) 425-490

20
R. Coquereanx and A. Jadczyk, "Conformal theories, curved phase spaces, relativistic wavelets and the geometry of complex domains", Rev. Math. Phys. 2 (1990) 1-44

#### Footnotes

... Spaces1
Talk given at the 8th International Workshop on Mathematical Physics, held at the Arnold Sommerfeld Institute, Clausthal (Germany), July 19-26, 1989. Published in Quantum Groups, H.-D. Doebner and J.-D. Hennig (Eds), Springer-Verlag, Berlin 1990 ( Lecture Notes in Physics, Vol. 370, pp. 426-434)
... Penkov3
... Marcinek4
See [4] and references there
... Ocneanu5
Private communication
... that6
See e.g. [6]
... directions.7
Which agrees with "the natural order of things."
... suffice.8
See e.g. Ref.[7]
... important9
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.
... there12
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]
... SYMMETRIC15
We choose the letter to remind the fact that is isomorphic to the set of all geodesic symmetries of .
... projection16
'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
... by17
The methods apply as well to infinite dimensions, but in infinite dimensions one has to take a special care about existence of trace