Why on earth would anybody be interested in such a concept as noncommutative geometry - other than the usual reason for doing unnatural things to strange objects: it hasn't been done, and it looks possible? What would a noncommutative geometry look like? How does the notion of noncommutative geometry relate to quantizing or "superizing"?

Noncommutative geometry is not a neat package of mathematics as e.g., Complex Analysis: analytic functions of a single complex variable. It is more like a disparate collection of various and sundry - much like Real Analysis. It connects with many branches of mathematics, and extends further the existing connections between geometry and algebra by including noncommutative algebra in that general connection. [Classical Geometry & Physics Redux]

The designation seems focused, but there are various approaches and various definitions that are all connected with a wide range of other mathematics: topology, measure theory, algebraic topology, topological algebra - quantum anything and super anything.

In the general sense of noncommutativity, "quantum" is more general, but less well defined than the rather more specific "super" designation. Quantum always implies noncommutative (not necessarily conversely), but how that might be restrictedly well defined is not clear. Super, however, defines a specific and relatively simple form of noncommutativity (and quantization) - that happens to be related to mathematical "quantum" notions derived from the mathematics of physical quantum theory predicated on intrinsic uncertainty relations that restrict, in principle, what can be physically measured or known.

The idea of generalizing concepts is indigenous to mathematics; the history of numbers and abstract algebra is built on the idea.

But, as the seemingly abstract extensions from Euclidean geometry to hyperbolic and spherical gemoetries by Lobachewsky and Bolyai and then to more general Riemannian geometries have proved not only interesting in their own right, but also useful in physical theory, so also the seemingly abstract notion of noncommutative geometry has use in physical theory. Apart from the relatively simple process of generalization in mathematics, there are also compelling motivations from physics to consider the general notion of noncommutative geometry, and these can be talked about a bit.

The relativistic theory of gravity called General Relativity (GR) due to Einstein finds its expression in Riemannian geometry (slightly extended because of the hyperbolic metrical signature of spacetime).

Perhaps the major and continuing problem of theoretical physics is to make coherent sense of Einstein's theory of gravity, as a classical field theory in the context of the principles of quantum theory. These two metatheoretic constructs of Relativity (R) and Quantum (Q) are the great pillars of the physics of the 20th century, but, they seem to dislike each other intensely - so intensely that they are, as customarily stated, mutually contradictory and incompatible. To theorists, this is more than annoying, and more than interesting.

The first attempts to weld these concepts together followed the agonized and semisuccessful construction that quantized the electromagnetic (EM) field as a field theory described by the classical Maxwell equations.

The quantization of the EM field itself provided a number of necessary new understandings. One, which was a bit startling, is that the EM 4-vector potential A_μ, classically thought to be merely a convenient fiction that combined the earlier fictions of a 3-vector magnetic potential and a scalar electric potential was actually fundamental rather than fictitious. The relativistic classical potential A_μ turned out to be the real quantum field cognate to the classical field.

From this Q new perspective, A_μ is the operative field while the classical field (a second rank tensor field, antisymmetric in its indices)

	Fμν  :=  ∂μ  Aν - ∂ν Aμ

It is A_μ that shows the vectorial (spin-1) nature of the photon as quantum of the EM field, while the gauge invariance of the equation that A_μ satisfies asserts it masslessness, and linear dependence of its components.

With this more geometric understanding, what happens with quantizing the EM field in hand, there comes an idea of quantizing GR, recognizing that the gravimetric field g_μν (a second rank tensor field, symmetric in its indicies) generalizes the Newtonian gravitational potential, and that its Riemann-Christoffel curvature tensor is associated with the classical field responsible for classical forces. The gauge invariance in this case turns out to be the general coordinate invariance of GR.

The tensorial nature of g_μν tells that the underlying quantum of the gravitational field has spin-2, and the form invariance of the tensorial equations that are the Einstein equations tells that this underlying quantum of the gravitational field, the "graviton", is massless. [I have fudged over a few complications here as anyone might discover by counting unfettered degrees of freedom of a symmetric tensor field g_μν on a four dimensional manifold. This is just a story, not the mathematical physics itself, which is considerbly more complicated.]

Early on in the development of quantum theory, it was thought a possibly happy fact that g_μν was a symmetric tensor field while F_μν was an antisymmetric field. That was the source of many attempts at a classical "unified field theory" that combined what were then the only two physical forces that were known.

The idea of Einstein's Unified Field Theory was to learn how these two fields must unite to form a consistant physical theory, as the originally separate electric and magnetic theories came together in a theory of electromagnetism described by Maxwell's equations

Unfortunately, Maxwell's equations are not yet known to be complete or fully consistant, or even correct at very high energies. Nor has the correct unification of these equations with the, also supposedly correct, Einstein equations ever ben found or established.

Lurching forward into the business of field quantization, however, many unhappy facts were soon discovered to be true - and truly problematic.

First, quantum theory, generally, is linear: it is a story of linear operators operating on an infinite dimensional vector space. There is happy harmony with the Maxwell equations on that score because the Maxwell equation are a hyperbolic system of (again) linear partial differential equations, making their classical solutions relatively straightforward in principle if not always in deed.

The Einstein equations, however, are a hyperbolic system of *nonlinear* partial differential equations, making their classical solutions practically impossible to obtain in general, and *very* difficult after some simplifying symmetries have been applied. With these equations, one is happy to have *any* nontrivial solutions.

The nonlinearities of GR especially frustrate the Cauchy Problem (loosely spoken, with given initial conditions, propagate them in time) associated with hyperbolicity. The problem is that the with the specifics of the nonlinearities, initial conditions cannot be freely given, but are in fact constrained by the time developed solution which one seeks to calculate. This is a classic mathematical Catch-22.

Another bone of contention between GR and quantization is related to the fact that Q theory is also based in complex numbers, transition amplitudes whose absolute values squared become transition probabilities.

The linearity of the Maxwell equations allows complex entities to be formed and decomposed into real entities readily, and a "real part" of an entity does not thereby become a confused concept.

The nonlinearity of the Einstein equations detroys this freedom and distinction in such a way that classically real entities become in a quantization, confused and not consistently interpretable. Probabilities and the gravimetric field become complex, and there is no clear way to interpret these. Another way of stating the problem that is more obvious but perhaps not quite so revealing is that the nonlinearity destroys the essential quantum principle of superposition.

Another consequence of this exploration is that consistency requires the gauge fields also to be quantized, and this means in the context of GR, quantizing the coordinate functions that are supposed to be laid on spacetime in order to designate numerically, spacetime points. If the coordinates are always quantized, this effectively creates an intrinsic uncertainty of order of the Planck units in the underlying space of spacetime points. This needs to be expressed in the coordinate functions being mapped (locally) to operators that do not commute, and there, finally, is a physical motivation for wanting to think about noncommutative geometry in an important physical context.

Haag and Kastler [Haag 1964] first developed the idea of a C^*-algebra composed of local C^*-algebras in the context of quantum field theory and quantum statistical mechanics. [Emch 1972]

That this mapping to operators is local also suggests that the noncommutative geometry could and possibly should be further constrained to be bounded and finitistic in terms of the operator spectra.

Robert Geroch suggested the idea of (nonlinear) Einstein Algebras whose elements are solutions of the Einstein equations [Geroch 1972] There, the combining of solutions to produce other solutions would have to be nonlinear operations unlike the usual addition and multiplication of linear operators; therein lieth exactly the difficulty in developing this idea.

Combining the ideas of Haag, Kastler and Geroch, one might develop a mathematical context for QG of Noncommutative C^*-manifolds, i.e., "almost complex manifolds" like structures whose local complex tangent spaces are instead noncommutative C^*-algebras of finitistic type. One entrance to this might be though the theory of "noncommutative schemes".

Roughly speaking, a classical phase space of states of a physical system associated with the canonical formalism thereof is a 2n dimensional manifold equipped with a symplectic structure foliated by a one parameter family of hypersurfaces of constant energy. In quantization, the points of the phase space are mapped to elements of a projective Hilbert space, and it is the Hilbert space which replaces the phase space. As in classical mechanics, the phase space is still only a space of kinematically allowed states; motions, or flows on the phase space are necessary to describe the dynamics of a system and this will involve functions on the phase space.

While at first this seems like a magical transformation, it turns out that one can formulate classical mechanics in terms of Hilbert spaces by understanding that the observables are really measurable functions defined on classical phase space, and that a Hilbert space of measurable functions can be defined. The Hilbert space is not actually the point or quantum theory; rather, the structure of the algebra of quantum mechanical observables derived from the appropriate Heisenberg algebra is the point.

All countably infinite dimensional Hilbert spaces, by virtue of standard theorems of functional analysis are isometrically isomorphic, and are not therefore sufficient to distinguish different physical systems.

QM itself also gives an even simpler example of a noncommutative geometry with the coordinates of its phase space Q_k, P_k, all of which do not commute, and this relates back to the idea of noncommutative coordinates generally.

The symplectic structure given by fundamental Poisson brackets of a Hamiltonian formalism of classical mechanics maps, in canonical quantization to the symplectic structure provided by the Heisenberg algebra.

The "easier" aspect of QM is that its phase space, while being noncommutative is also always flat, while classical Hamiltonian mechanics admits curved phase spaces as well. If the QM phase space were not flat, the sets of operators {Q_k}, and {P_k} would not each be separately commutative, and the algebra would not be a Heisenberg Lie algebra.

One can see the flat (physical) Q-space and the separate flat (momentum) P-space simply by giving each set in the product of spectral representations of mutually commuting operators. The spaces are the products of the operator spectra, i.e., the product of the Sp(Q_k) and the product of the Sp(P_k).

There is a perfectly nicely defined commutative Euclidean geometry in the {Q_k} sector described according to Felix Klein's "Erlanger program" by the inhomogeneous invariance group IO(n), n being the number of Q_k, and similarlity in the {P_k} sector. It is when they get together that they are classically up to no good.

In CM (Hamiltonian/Canonical formulation), the "observables" are represented by measurable functions defined on the whole phase space. Poisson brackets that define the kinematics are then differential operators defined on a subspace of the measurable functions that are also differentiable.

There is even a simpler set of examples of noncommutative geometry surrounding quaternions, which can be either real quaternions, isomorphic to the Lie algebra gl(2, R) or complex quaternions, isomorphic to the Lie algebra gl(2, C). As extensions of the algebraically complete field of complex numbers, quaternions retain all the properties of fields, except commutativity, losing that property to become skew-fields. Their n-fold topological products are then specific noncommutative generalizations of the Euclidean real and complex spaces Rⁿ and Cⁿ.

The free floating notion of "geometry" automatically contains the notions of metricity and translations, though technically it need not. The general concept of point-set topology transcends and eliminates both of those concepts in its generalization of geometry.

The approach to QG by quantization of the Einstein equation, through the nonlinearization of an essentially linear quantum theory, actually turns out to be remarkably messy and, my guess is that its messiness arises from the approach being too specifically tied to the Einstein equations. While the specific details that would have to be used complicates the concept burdensomely, considerations of geometry from a more abstract point of view make the notion of noncommutative topology more approachable.

Generalizing from geometry to topology makes for one interesting begining in a duality expressed by the Gelfand Representation for commutative C^*-algebras, saying that commutative C^*-algebras are dual to compact Hausdorff spaces. [Ref both Wiki and Rickart here] [Rickart 1960], [Appendix A]

Modulo the difficulties that QM always require unbounded operators, the bounded linear operators that act on a complex Hilbert space is a C^*-algebra. One can be more general than this abstractly in defining a C^*-algebra, but the Gelfand-Naimark theorem, giving the noncommutative analog of the Gelfand representation, will boil the situation down to just this representation of the abstract algebra by bounded linear operators on a complex Hilbert space. This subject is then more properly called noncommutative topology, rather than noncommutative geometry.

In both commutative and noncommutative cases, it is the duality in the constructions of the theorems which provides the representations of the algebras. Mathematics is filled with various kind of dualities, but the kind of duality spoken of in this context of linear spaces is the dual space of linear functionals on a linear space. For a general Banach space, a normed linear space complete in its norm topology, the space of its linear functionals can be either larger or smaller than the Banach space. A Hilbert space can be seen a special Banach space that is self dual, i.e., the Hilbert space is isometrically isomorphic to its dual space of linear functionals.

The positive linear functionals of norm 1, of a normed linear space are also called its states. These constitute the core of the linear functionals since all the other linear functionals are merely scalings of the states. For a Hilbert space, they are easy to describe, but for a C^*-algebra they are a little more interesting.

First notice that the Hilbert space can be injected into its C^*-algebra by the map of states of the Hilbert space to projection operators of the algebra,

	|v>  →  |v><v|  :=  M(v)

	<v|v>  =  Tr( M(v) )

	<u|v>  =  Tr( M(u) M(v) )

	 Σ pk M(vk)
	 k
   where
	 Σ pk  =  1
	 k

	   <A>M  :=  Tr( A M )

   [Neumann 1932]

[It is often not made clear in physics literature that pure states are states for the Hilbert space, while density matrices as impure states are states for the C^*-algebra. More confusing, perhaps, is that in Hilbert space context, the phrase "pure state" is often applied to an "eigenstate", while the complex linear combinations thereof are called mixed states, which also happens to be a terminology to describe density matrices which are primarily invoked within quantum statistical mechanics.]

The tricks for similarly doing a quantum theory that will handle the necessary unbounded operators of QM are messy, not to the point, and I avoid them here. They exist - but forget anything here that depends on transition amplitudes. In standard Q theory, they exist conceptually, but are often infinite, a problem that is almost always not generally rectifiable in that context.

This section is merely an interesting excursion relative to what follows and can be skipped on a first reading without deletorious affect.

An earlier approach to noncommutative geometry (the first rigourous approach I think) is due to Alain Connes, and then also to Yuri Mannin. The notions spring fundamentally from homology, or more specifically from generalizing De Rham cohomology to a noncommutative condition.

Blah - blah - blah.

Within the theory of functions of several complex variables, Bergman's kernel provides a duality between a compact Kaehler manifold as "complex domain" (think analog of a compact Hausdorff space in the context of the commutative C^*-algebras) and a Hilbert space of complex valued (analytic) functions defined on it. The kernel function relates the metric field of the Kaehler manifold to the inner product of the Hilbert space of complex valued (analytic) functions.

Instead of making life more messy and complicated, let us make it a little simpler and easier by considering noncommutative *finite* dimensional C^*-algebras algebras, which are simply algebras of linear operators acting on a finite n dimensional complex vector space.

Such "geometries" will necessarily be not only noncommutative, but by their finitism, also, quantized exactly in the spirit, and necessities of a generalized quantum theory. What provides a geometric interpretation is the set of algebraic relations that generate the algebra.

Begin with the simplest of all possible C^*-algebras M(2, C) the 2x2 complex matrix algebra, which happens to be spanned by the 8 real dimensional Lie algebra gl(2, C). As a topological alone, it has the topology of a complex space C⁴. gl(2, C) also has a lattice of sub Lie algebras (with isomorphisms)

	su(2) = so(3) = sp(1)           gl(2, R)

	su(1,1) = so(2,1) = sl(2,R)     sl(2,C) = so(3,1)    gl(2,C)

		iso(2)

Algebraic structures that related to Lie algebraic structures are the Superlie algebraic structures classified by Marc Kac, and these are built on the notion of supergeometry that combines a commutative Euclidean geometry with an anticommutative Grassmannian geometry into a space with two sectors of these different types.

The physical motivation for doing this is the idea of supersymmetry. Particle spin is a property that can either be introduced ad hoc, or seen as a natural consequence of relativistic axioms. It is similar in form to angular momentum which can be described both classically and quantally by a Lie algebra so(3) of the special orthogonal group SO(3), but which also acknowledges the Lie algebraic isomorphism of so(3) and su(2), and the ultimate ascendency of su(2).

The former so(3) quantizes angular momentum to have, by the values of its single Casimir operator in an irreducible representation, values that are integral multiples of ħ.

The latter su(2) also allows these values to assume the same values, but also half-odd integral multiples of ħ, (ħ/2, 3ħ/2, 5ħ/2, ... ). SU(2) is the universal covering group of SO(3): SO(3) is *locally* homomorphic to SU(2); they have different global topologies, specifically, SO(3) is doubly connected, and SU(2) is singly connected.

While angular momentum has a symmetry group SO(3) is connected with its conservation, spin has a symmetry group SU(2) connected with its conservation. Since the SU(2) symmetry does not seem to be reducible to any explanation in terms of E³ spatial gemeomtery, spin symmetry is said to be an internal symmetry, i.e., something somehow outside of space an time. Nevertheless, particles of integral spin and particles of half-odd integral spin are predicted to, and do have different physical properties. These differences lead to differences in their statistical behavior in quantum statistical theory. The "integral spin" particles called Bosons, obey what are called Bose-Einstein statistics, and the half-odd integral spin particles, called Fermions, obey Fermi-Dirac statistics. There are classic theorems of relativistic quantum field theory that make these connections (Spin-Statistics Theorems), and empirically, they seem to hold.

More interestingly, the Fermions appear to be the particles that compose physical substance (electrons, quarks, protons, neutrons, ...), while the Bosons are the mediators of forces (photon, Weak bosons, mesons, graviton(?), gluon, (?)) according to the QFT model.

There is then partition of the theoretical reality of particle physics into two parts that are structurally related. The idea of supersymmetry is that these partitions are really exposing two aspects of the "same" thing, and that there is eventually in high enough energy (i.e., early enough in "time") a situation where these symmetries are united into a supersymmetry.

A real supersymmetry would make equivalent by the supersymmetry group, forces and material substances. Supergeometry upon which supersymmetry is based is a specific and simple, though not trivial, noncommutative geometry.

   If a finite set S of cardinality n is quantized by a mapping

	   S → L:
		   ak → |ak>

   where in the image, k indexes an orthonormal basis of an n
   dimensional complex Hilbert space, L, then the map

	L  →  CL(L):
		{a}  →  γk
   with

	{γk, γj}  :=  γk γj  +  γk γj  =  2 δkj

	L x L*  →  CL(L) x CL*(L)

   where * is the dual map (Hermitean conjugation).

If one sticks to a "unitary nature" of QT generally, and there is no apparent way of saying which point is which, a unitary transformation of bases in L leaves "the quantization of the set S" invariant.

Without worrying about the axioms just yet, a topology on S, which is first a collection of subsets of S, can be specified by a characteristic function char(s) on its power set, which takes the value +1 on a subset if it is in T, and 0 otherwise.

If the number elements of T is m, 0 < m =< 2ⁿ, and without any loss of generality char(s) for any topology can be normalized so that

	 Σ char(s)  =  1
	s ∈ 2S

	|t)  :=   Σ cA γA
	          A

   with cA being a set of complex numbers.  Dually,

	(t|  :=   Σ (cA)* (γA)!
	          A

   If |t) and |t') are two such quantum topologies, then

	(t'|t)  =   Σ  Σ (cA)* (γA)! cBB γB
	            A  B

	(t'|t)*  =  (t|t')

and some sort of "time" is necessary for any concept of dynamics. No *specific* time concept is specified by this, except as one existing that is implied by an apparent partial ordering of t. I don't see that, physically, there is any choice in interpretion of complex conjugation.

This is, of course, a notational sleight of hand since the label "t" is not a label for a value of some time parameter but in fact a label for a quantum topology. However, that t can be interpreted as a more general concept of time makes sense if the quantized set and its quantum topologies are looked at as a quantum clock together with its states. It is by the sequence of its states that clock reveals the "time it tells".

This describes merely the arena of a dynamics. A model, which will be much more complicated, involves being able to relate quantum topologies to classical topologies and their structures, where I will simply guess topological algebra will come in handy, probably in a form like or related to simplicial homology.

It will also involve being able to compute explicitly, at least, "weighted transition trees" of quantum topologies, and with a bit of cleverness (think of cancellations in either Fourier theorem or in Feynman path integrals) quasiclassical paths through the space of quantum topologies.

The linear operators on the space of quantum topologies, CL(L) of S can be represented by elements of the algebra CL(L) x CL^*(L), the projection operators of which can be written as a direct product of elements

		|t)(t|

ignoring the algebraic structure of CL(L), and thinking only of its structure as a complex vector space.

The density operators are then defined as the convex combinations of these extremum states lying on the boundary of forward cone of all "states" on the space of quantum topologies. Such a general state (density matrix) is given then by

	(τ)  :=  Σ pa |ta)(ta|
	         a

   where {|ta)} is an arbitrary collection of quantum topologies,
   pa ≥ 0, for all a, and

	         Σ pa  =  1
	         a

Felix Klein sought to understand the fundamentals of geometry through the action of a group on a set together with those entities defined on the set left invariant under the action of the group. That is the essence of his Erlanger Program.

In the context of noncommutative geometry, something I think Klein did not envisage, the set becomes a linear space. "Quantizing" things tends to make them more complicated in that the number of basic concepts tends to increase. In the case of an underlying set, the idea of "point" becomes more complicated. The original set S corresponds in this quantization to the elements of some orthonomal basis of a Hilbert space. But the necessary allowance of complex linear combinations of quantum "states" in quantum theory allows that there are "fictitious" points in the quantum domain that do not exist classically.

In standard QM where position is given by an operator Q whose spectrum is always taken as continuous, this peculiarity is masked to the point of being ignorable. If the geometry is essentially discrete in its eigenvalues however, the superposition provides a mathematical construction whereby the background discreteness can give an appearance of continuity commensurate with the usual continuity of the scalar field of the linear spaces and algebras involved.

The set of the Erlanger then becomes the complex linear space, L, as above, and a group acts on L, preserving certain structures defined on L, and these defined structures are usually given by the algebrai structure of an algebra of operators acting on L. This algebra is new; it is an addition and extension of the concepts of the Erlanger program, yet it fits neatly into it.

The nonlinearities of classical geometries have been relieved by a kind of embedding in a higher dimensional linear space. Quantizing a set of cardinality n, changes the context to an (n-1) dimensional simplex, since and n dimensional simplex is defined by n+1 points. From this understanding, the idea that standard QM with position operator Q having a real line spectrum requires an infinite dimensional Hilbert space is not surprising.

	Q1² + Q2² + Q3²

	[Qi, Qj]  =  0

   one posits a so(3) or su(2) algebra so the position operators obey

	[Qi, Qj]  =  i εijk Qk

Another split that quantization (or "noncommutativization") causes in the Erlanger program is that while a classical geometry in Klein's sense is a unique global item, quantized geometries can be defined with mathematical refinement to differ in a nesting of algebras simply by the addtion or subtraction of a single point. This process has an algebraic routineness about it that it is no way connected to the classical topologic/geometric ideas of compactification or projectivization.

A Clifford algebra interpreted as a set quantization of S does not provide for the dimension of a space in which the set is contained; the algebra dimension comes from the cardinality of S.

The "vector subspace" of the Clifford algebra corresponds to the Hilbert space L from which it is derived and its elements are linear combinations of the singleton sets, as quantized sets. Such a dimension must be expressed first in the C^*-algebra of linear operators on L (an m-dimensional Heisenberg algebra in QM serves such purpose by having an invariant commutative Lie subalgebra of position operators {Q_k}), and then "lifted" to a similar C^*-algebra of linear operators acting on CL(L), i.e, lifted to an element of CL(L) x CL^*(L), another C^*-algebra.

	Σ Σ cAB (γA)! x γB
	A B

This is merely a geometric aspect of such a construction. The physics of such modeling seems to require in addition the specification of a kinematics contained in a symplectic structure of the sort defined by the Poisson brackets of classical Hamiltonian formalism which are preserved in a "quantization map". The quantization map itself is not fundamental; it is just a way of getting to the right place when we have no idea what we are doing. It seems, however, that the symplectic kinemtical structure *is* fundamental to any further definition of dynamics. Unitary Lie groups (hence also algebras) happen to have a natural symplectic structure.

The symplectic structure is necessary to define the kinematical context (that which is physically possible) of a system of "points" which are to be considered as physically dynamical (picking out from what is possible, the probable) states or processes of the objects that are the points of the set that is to be quantized.