"Adhering to the continuum originates with me not in a prejudice, but arises out of the fact that I have been unable to think up anything organic to take its place. How is one to conserve four-dimensionality in essence (or in near approximation) and [at the same time] surrender the continuum?" -- Albert Einstein Einsteins's Reply to Criticisms in relation to Epistemological Problems in Atomic Physics from The Library of Living Philosophers Series (1949)
The science of Mathematics is both the art and science of creating abstract models of the quantifiable. The art is in the creation, while the science is in the focusing of that creation on models that are, within the finite limits of the state of the art, logically consistent. Physics is the art and science of using the vocabulary and paraphernalia of mathematics to create models, again within current state of the art, of the various levels of reality.
The existence of the component of art in physics, and the physical and epistemological limitations of an observer are various lessons learned from both quantum theory and relativity. Yet perhaps the lesson has not been learned well enough. It was with both shock and chagrin when as a student first encountering quantum mechanics that I saw that this new more correct way of understanding a massive, free, point particle required the specification of a complex valued function that was not only continuous and differentiable, approaching zero "fast enough" on approach to infinity, but that this descriptive function had to be well defined, modulo the indefinite phase, over an infinite Euclidean space modeled on the trebled continuum R^3, a mathematical construct of no mean complexity and sophistication; this all given at some instant in an additional continuum of universal time known, fixed and perfectly coordinated at every point of the treble continuum of space. Such a thing could only be arranged, observed, known, or verified by an Infinite, Omnipresent God, whose existence was both before and outside of time and space, and yet in some sense partook of it.
Though I do not wish here to become embroiled in the venerable and ongoing debates concerning the ontological status of the state vector of Quantum Mechanics (QM), the ontological status of the "reduction of the wave packet", the various physical, logical and metaphysical implications of the existing points of view, it is a subject that can hardly be simply ignored. It is probably not unfair to say that most conflicting points of view have merits and that all are in one way or another unsatisfactory. The focus of the problem might well be had in the concluding paragraph of [Bell 1975]:
"The continuing dispute about quantum measurement theory is not between people who disagree on the results of simple mathematical manipulations. Nor is it between people with different ideas about the actual practicality of measuring arbitrarily complicated observables. It is between people who view with different degrees of complacency the following fact; so long as the wave packet reduction is an essential component, and so long as we do not know exactly when and how it takes over from the Schroedinger equation, we do not have an exact and unambiguous formulation of one of our most fundamental theories."
There is, as far as I can see, no immediate grand illumination in these pages on the outstanding perplexities of quantum mechanical interpretation. Since, however, I am suggesting consideration of a quantum theory that is perforce local and finite, and which can have QM as limit, there should be some new interpretational considerations arising from this to add yet more fuel to the fire of already confounding arguments.
Note added, November 25, 2001
Since the above was written years ago, recent developments of the probability interpretation of the algebraic formalism that is the subject of this work indicate strongly that since the "state vector" should be reinterpreted as a "process vector", this vector which is itself then stochastically propagated, should be interepreted as a symbol of genuine ontology and not seen as merely a convenient mathematical device for encoding information. Viz. Origins of Mewtonian Time.
Nonrelativistic QM, in the formulation of Heisenberg, Schroedinger, and Dirac is a global theory in the sense that one does not operate with it naturally in bounded space or time domains (patches). Yet measurement devices and the region over which they can be said to measure are quite clearly finite and bounded. To restrict consideration in QM to a bounded spatial domain, the artifice of an infinite potential barrier or a periodic boundary condition is necessary. Conceptually, QM is done in such a way that a state function is defined simultaneously, in the sense of Newton, on all of space. A first puzzlement that arises from this is the alteration of the state function resulting from a measurement: "the collapse of the wave function" from a linear superposition to a single eigenstate of the observable being measured. If one could find an expression of quantal principles on bounded spacetime patches, perhaps a better understanding of apparent wave function collapse could be achieved. There are other criticisms of current QM possible, some of which are discussed below, that can be related to the lack of a concept of quantum theory in bounded spacetime patches. This concept and a model of it suggested by QM itself are the subject of this paper.
Currently used concepts of space and time are modeled by the continuum of real numbers, implying an ability, in principle, to distinguish conceptually between arbitrarily close "points" thereof: that appropriate topological separation axioms of a continuum are in some sense physically operative. While the energy content of any physically known system is bounded, QM Hamiltonians are characteristically bounded below but not above. In any physically real localized system if enough energy is pumped into it, it is destroyed. Call this simply a phase transition to a different Hamiltonian. Such phase transitions are not characteristic of QM. A straightforward SR principle, (the assertion of the Lorentz invariance of the Minkowski length of the energy-momentum four vector, augmented by the operator substitution procedure taken over from QM) can yield the Klein-Gordan or Dirac equations, which then must be reinterpreted and subjected to a pragmatic second quantization leading to Quantum Field theory (QFT), in order to eliminate difficulties of interpreting negative probabilities. In the passage to a second quantized theory (field quantization), what was probability density can become associated with a charge density.
QM in its Hamiltonian form, clearly distinguishes time as something very different from the spatial extensions; there is no properly defined "time operator" [Carruthers 1968], yet the form of both Special Relativity (SR) and General Relativity (GR) indicates that space and time constitute merely parts of a larger invariant associated with a group of transformations. QM does not seem to allow the implied structural unification of space and time; although, one might argue that the real problem is with the Hamiltonian formalism for even in classical SR in a Hamiltonian context, "No interaction" theorems have been proved [Sudarshan 1974]. Within the context of SR, itself, this is contrary to intuition: SR is the correction to Newtonian mechanics that makes massive particle mechanics consistent with the "mechanics of charge" that is part of electromagnetic (EMT) equations of Maxwell which describe the relations between charges and the electromagnetic field. In hindsight, a classical electromagnetic can exist because the quantum of the field, the photon, has zero mass; moreover, from a quantum mechamical viewpoint, the actual photon field suprisingly turns out to be the 4-potential, and not the antisymmetric 2-form that one thinks of as the classical field. The magnetic and electric potentials were originally invented as sheer physical fictions that made solving the classical equations simpler. That the potentials are physically real is found in confirmations of the Aharanov-Bohm effect. This, like many subtle quantum effects seem to show a nonlocal mechanism, where the final results of experiments depend on the geometry of the physical setup of the experiment, where the geometry is a time invariant structure in the lab frame, extended physically over space. Said that way, maybe it makes quantum phenomena look less crazy than they really are.
A "quantization" of the gravitational field, following the example of the rather calculationally successful quantization of the electromagnetic field, leads (it would seem) inevitably to the metric becoming complex, and hence difficult to interpret as an observable in the usual formalistic sense [DeWitt 1966]. With regard to strong interactions, it might be that we simply do not yet understand the dynamics, (A Hamiltonian), the existence of quantum chromodynamics notwithstanding; but on the other hand, the difficulties in the form of "No Go" theorems, of uniting the unitary symmetries with the concept of Poincare invariance are well known, [McGlinn 1964], [Greenberg 1964], [Michel 1965], [O'Raifeartaigh 1965], [Hermann 1965].
It should be noted that supersymmetry avoids nogo theorems by generalizing Lie algebras to Super-Lie algebras or graded superalgebras with Bosonic and Fermionic sectors.
The Planck length although determinable from dimensional analysis is not fundamental in any workable quantum theory, but indicates that perhaps the continuum model for space and time may not be appropriate and that a concept of "quantized spacetime" should somehow be fundamental [Penrose 1972], [Schroedinger 1956]. Finkelstein has argued that the fundamental quantum should be that of time, the "chronon", [Finkelstein 1981]. See also section XVI. This avoids, perhaps wisely, the question or postulate that determines the dimension of space. Note that very idea of dimension 3 usually postulated depends very much on the models of space and time as continuous structures.
The dimensions 3 and 4 turn out to be interesting mathematically. Many proofs of standard topological theorems fail for dimension 4. Of all possible dimensions n, n=4 yields the greatest variety of regular polytopes. In a way, the most variety is available topologically in dimension 4. Only for n=3 can knot theory exist; in every other dimension it is trivial. This is certainly convenient for the tieing of shoe laces, moorings of boats, etc.
Measurement operators in the form of density matrices can exist that model an apparatus with a possibly unbounded number of pointer positions. Physically this is unreasonable, yet one could argue that such models are idealizations that are approximable by realistic apparati with a finite number of pointer positions. Should it be that such finiteness, in fact, be a matter of principle, as the finitenesses of
0 < h-bar, and 0 < c < infinityhave become? The foregoing is a collection of personal observations and unhappinesses any one of which is not unique to me.
From QM and SR we have learned that there exist certain ideal specifications about a system that we cannot make. In quantum mechanical principle we cannot specify position and momentum "simultaneously": while in special relativistic principle we do not really know what simultaneity is except as a convention, furthermore, this convention is not Lorentz invariant. In considering a simple system like a harmonic oscillator it must be assumed that, in principle, the physics every thing else can be ignored. Such a harmonic oscillator is indestructible, since in QM there is an infinite sequence of energy eigenfunctions with an associated sequence of energy eigenvalues. There is no maximal energy. No matter how hard we kick this thing it remains intact. Physical systems that we know are, however, all destructable. Can we, truly in principle, specify a system that is indestructible and not miss a slightly different and essential physics of destructability? If not, then there must be attached to a model of a realistic physical system, a maximal energy eigenvalue. If we pump more energy into the system that this maximal energy, it must suffer some kind of transformation to another system or collection of systems, the latter suggesting some form of "energetic disassembly". The existence of a maximal energy suggests the appropriateness of finite dimensional Hilbert spaces as the language for such models. These models may then be of open systems.
Jordan has suggested in order to smear the local spacetime points and effectively deal with QFT divergences that use be made of Malcev algebras, nonassociative algebras more general than Lie algebras, to generalize quantum mechanics to a mechanics without dispersion free states [Jordan 1968], [Jordan 1972]. It would seem from the present work that such a generalization is not actually necessary and that the appropriate mechanisms for such a result exist within the context of Lie algebras.
This work is an attempt to find the consequences of a simpler kind of algebraic formalism, that seems to contain the requisite structure for both a "quantum" theory and a "relativistic" theory in bounded domains, and which approximates QM weakly in a manner to be shown. This suggestive formalism touches, on some of the issues raised above.
For a point of reference, I want to review, briefly, some very general and well known mathematical aspects of physical theories. General sources for some of the following are [Sudarshan 1974], chapt. 23, and [Hermann 1965], chapt. 16.
The structures that we call classical, relativistic, and quantum theories are not so much theories as they are metatheories: they tell us about the frame work within which a genuine theory should be cast. They do not tell us whether or not what we construct within these frameworks is, in fact, a model for a physically existing thing. The framework of classical mechanics isolates for us the concept of a phase space, and says that it is a manifold with symplectic structure. Anything further with exception of some technical stuff about continuity and differentiability, must be specified by defining "the specific physical system". The principles of classical mechanics do not tell us what is a priori nonsense, and what corresponds to a physical reality.
An aside: there is an interesting summary of Hamiltonian Mechanics in [Folland 1989], and presentation of the axiomatics of Classical Mechanics in [Bunge 1967].
We define a classical system by first defining the phase space of the system, assigning by a proposed call of measurement procedures a "physical meaning" to the phase space coordinates. The observables are then defined as some class of real valued measurable functions defined on the phase space equipped with a suitable measure (e. g. Lebesgue measure). The set of classical states is then defined as the set of all positive-definite normalized probability distributions on phase space. The observables then form an associative, Abelian algebra over the real field R by linear combinations and pointwise multiplication. The symplectic structure on a phase space with 2n coordinates,
omega = dp_1 v dq_1 + ... + dp_n v dq_n where 'v' is a Grassmann product.is what determines the canonically conjugate pairs of coordinates in phase space. Using this, the definition of Poisson Brackets is added, which define a "derivation" [Appendix B] in the associative algebra. A derivation structure in an associate algebra defines a Lie structure, and the Jacobi identity is fulfilled. The phase space coordinates are relative to an observer. The group of coordinate transformations (change of observer) which preserve the symplectic structure is then the group of canonical transformations. These constitute the most general symmetry group of a Hamiltonian system. The group is not a particularly pleasant one, for as the group of coordinate transformations in GR, it is infinite dimensional. It is one of the infinite Lie pseudogroups, the study of which requires certain topological complications in addition to the usual natural structures considered in the study of finite dimensional Lie algebras and Lie Groups. [Kobayashi 1972], section I.8. I will ignore these complications here. An observable (under certain technical restrictions) generates a one-parameter subgroup of the canonical transformations. [The mapping of observables to generators of one-parameter groups is also typical of QM.] The relation between the pseudo-Lie algebra of observables and the pseudo-Lie group of canonical transformations is not one to one, since an observable f(p,q) = CONST, generates the identity transformation in the group. Since constant functions form the center of the pseudo-Lie algebra of observables, the quotient of the algebra of observables by its center is the pseudo-Lie algebra of the canonical transformations. So far everything has been on a kinematical level. To speak of dynamics, a special observable H(p,q), the Hamiltonian defining the energy of the system must be distinguished. (Note that the Hamiltonian and Lagrangian formulation of classical mechanics are not equivalent.) The energy can be used to define a Riemannian (i. e. metric) structure on the phase space, [Abraham 1967]. In fact, the Hamiltonian function is frequently of the form "kinetic energy + potential energy",
H(q, p) = (1/2) g^(ij) p_i p_j + V(q)where the g^(ij) if nonsingular, can be made into a metric tensor field on phase space.
Because of the normalization of the classical states, any convex combination of classical states is also a state and so the states are a convex set. The analog, of course, exists also in QM. The extremal elements of the convex state set, the "pure" states are those states not expressible as convex combinations of others, and are the analogs of delta function eigenstates (distributions) in QM.
There are different ways of looking at the relationship between Newtonian theories and Relativistic theories. One is that the linear invariance group has changed from the Galilei group to the inhomogeneous Lorentz group. Another is that added to classical mechanics is a constraint which limits observable velocities. QM is then another kind of limitation on the validity of simultaneous measurements of canonically conjugate observables.
The construction of a relativistic Hamiltonian formalism can lead to a few disappointing results, especially with regard to the notion of "interaction". The elements can be found in [Mercier 1959], with further discussion of some of the serious difficulties in [Dirac 1964] and [Sudarshan 1974]. The difficulties are fairly numerous and would, for full exposition, require the reproduction of existing referenced literature for inclusion here. The original literature, best explains itself.
Nonrelativistic Quantum Mechanics
From the algebraic viewpoint, one can formulate QM in the context of C*-algebras, ignoring the technical necessity of treating unbounded operators [Appendix A]. In QM we are immediately, without a proper classical phase space, concerned with the unitary representations of the nilpotent Lie algebra [Appendix B] or Heisenberg algebra abstracted from the Canonical Commutation Relation (CCR). The idea is once again to specify a Hamiltonian, but as an operator in the enveloping algebra of the Heisenberg algebra. The pure states as functions on phase space are replaced by elements of a Hilbert carrier space for a representation of the Heisenberg algebra. These map to idempotent projection operators in the C*-algebra of bounded operators on the Hilbert space. They are the extremal states of the forward cone of positive operators in the algebra. The impure states are contained in the interior of the forward cone and are the density matrices. The observables are represented by Hermitean operators in the algebra, and are generated by polynomials of the canonical operators. The product of two Hermitean (properly, selfadjoint on some domain specific to the operator in the Hilbert space) is not Hermitean, and so the set of observables is not an algebra with respect the associative multiplication of the C*-algebra. We have directly, however, the Lie product as commutator, so the observables do form a kind of Lie algebra, where the notion of commutator has replaced the classical Poisson Bracket as the realization of Lie product. The Lie algebra of alleged observables then defines, through the exponential map, its associated group of unitary transformations acting on the Hilbert space. With the states being generally represented by density matrices, the expectation values are computed by the trace of the product of a density matrix and an Hermitean operator. A significant difference between CM and QM is a QM structure for which there is no CM analog. The superposition principle allows that convex combinations of pure states are also states (in fact, also pure); this being quite distinct from the convex combinations that form impure states. Another significant difference, being the result of the CCR assumption, is that not even a pure state can correspond to a a single determinate point in a phase space (uncertainty relations). In QM there is more of a separate consideration given to the notions of "observable" and "state" than in classical theories. We do not really have a phase space to provide the simultaneous foundation for both as a space on which various measurable functions can be defined.
For discussion and comparison we will need some specifics of standard CCR realizations. Therefore, the following.
A Weyl Pair consists of a pair of strongly continuous 1-parameter unitary groups U(a) and V(b) satisfying
U(a) V(b) = exp( i ab ) V(b) U(a)Such U(a) and V(b) are always representable as exponentials
U(a) = exp( iaQ ) V(b) = exp( ibP )where Q and P are densely defined unbounded selfadjoint operators, defined on a common domain of definition. A formal power series expansion of the exponentials substituted into the Weyl relation yields CCR. This is rigorously valid if both sides of the Weyl relation are taken to act on a suitable common dense subspace.
A Heisenberg Pair (Q, P) is a pair of operators that are densely defined, and selfadjoint. Further there is a dense subspace D of the Hilbert space Hilb, such that
[Q, P] psi = i psi, psi in HilbAny Weyl pair is a Heisenberg pair, but not conversely. The Heisenberg statement of CCR is strictly weaker, and there are Heisenberg pairs that cannot be exponentiated to a Weyl pair.
A Lie algebra defined abstractly by: [Q, P] = E [E, Q] = 0 [E, P] = 0 where [.,.] is the abstract antisymmetric Lie product,is a nilpotent Lie algebra, the abstract Heisenberg algebra, of which CCR is a realization. The realization gives the abstract Lie product as a commutator of operators defined as elements of an associative algebra. Creation and annihilation operators are another kind of realization. The Heisenberg algebra is one of the simplest nilpotent Lie algebras, and as it turns out fundamental in the difficult structure theory of all nilpotent Lie algebras.
Again, there are representations of the Heisenberg algebra that do not exponentiate to the Heisenberg group.
Perhaps surprising is that the Heisenberg algebra is also important in the construction of realizations (that yield infinite dimensional representations) of the classical series of Lie algebras, A_n, B_n, C_n and D_n, i.e., the orthogonal, symplectic and unitary Lie algebras. [Appendix B]
One can also think on the classical level similarly where the canonical Hamiltonian pairs obey a Poison bracket relation, and also realize the structure of the same classical series.
Any disturbance of CCR will also have implications for any "classical" cognate, and so such disturbance must be done with great delicacy.
J. Von Neumann [Neumann 1931] proved that any Weyl pair is unitarily equivalent to the standard example of a Weyl pair whose generators are the position and momentum operators for a free particle. A consequence of this is that any Weyl generator pair must have continuous spectra that extends over the entire real line. If the spectrum of one of the operators is bounded and continuous, a Heisenberg pair can exist but not a Weyl pair. An example of this is the particle in a finite box bounded by infinite potential walls.
Consider uncertainty relations derived from CCR, in one dimension. We omit the standard details, but contrast the Weyl and Heisenberg case. For a Weyl pair all "physical states" are contained in the commutator domain and so for these the uncertainty relation follows unequivocally. For a Heisenberg pair, it is possible for allegedly physical states to be outside the commutator domain and for such states the derivation of the uncertainty relations fails. Those states fall outside of the domain of selfadjointness. [Section III] and [Section XII].
An alternative to failure of the uncertainty relations for certain states is to acknowledge that these states are indeed not physical even though we may want them to be.
The primary kinematic pillar of QM would then seem to be the, Heisenberg Canonical Commutation Relation (CCR) is written formally:
q p - p q = iI (1.1a)with q and p understood to be formally Hermitean. Or, in terms of creation and annihilation operators:
a a! - a! a = I (1.1b)
The other kinematic postulate operative in QM, the Canonical Anticommutation Relation (CAR) is written:
a a! + a! a = I (1.2a)with the condition
a^2 = 0 (1.2b)
In both cases "I" is intended to be an identity within some concrete associative algebra of transformations. From the theorems of [Wielandt 1949], [Wintner 1947], and Olga Taussky [Cooke 1950], it has been clear for some time that the strict satisfaction of CCR within any normed algebra is impossible. Any kind of satisfaction within a finite dimensional algebra is normally disregarded out of hand. On the other hand see [Appendix J], where exact finite dimensional representations of CCR are constructed over finite Galois Fields. On considering representations as Hermitean operators acting on an infinite dimensional separable complex Hilbert space equipped with the usual Euclidean inner product, it turns out that at least one of the operators p and q must be an unbounded operator. Crudely put, the phase space of a quantal system is always unbounded. A consequence of unboundedness is that the operator cannot be defined on the full Hilbert space. The definition problem then becomes one of finding a common dense invariant domain within the Hilbert space upon which both p and q can be defined. The specifics of this problem are directly related to the physical characteristics of the system under consideration. [Reed 1972], When the common domain is established, the previously stated Hermitean form of CCR can be mapped to the Weyl form [Weyl 1928] of the CCR by mapping the Hermitean operators p and q to the one parameter unitary groups:
q -> U(a) = exp( ia q ) (1.3a) p -> V(b) = exp( ib p ) (1.3b)The conditions of Stone's theorem [Stone 1929] being applicable, the unitary Weyl form of CCR is then
V(a) U(b) = exp( iab ) U(b) V(a) (1.4) where a and b are the two real parameters; or V(a) U(b) V-1(a) U-1(b) = exp( iab ) the LHS expressing the "group commutator", which is the formal exponentiation of the algebra commutator.
The key point to be made here is that, in the context of (1.1a), a Hilbert space realization must actually be specified and that the domain of definition must be restricted. A further standard restriction due to interpretational standards rather than to mathematical requirements, is the association of physical states with the equivalence class members of the projective ray space of the Hilbert space. Although one casually speaks of Hilbert space of QM, the space of states (pure) is actually a projective Hilbert space.
More realistically then, write CCR in one of its weak forms, using Dirac notation as
(p q - q p) | psi > = i I | psi > (1.5a) with its implication of the strong operator topology, or < psi | (p q - q p) | psi > = i < psi | I | psi > (1.5b) with its implication of the weak operator topology,where |psi> is in the domain of definition.
In the case of CAR, all of this machinery is not necessary, considering only finite dimensional representations. There exists a single, well known, two dimensional irreducible representation for the strong form of CAR. Any other finite dimensional representation is equivalent to a direct sum of copies of this IRREP. This is also a special case of a similar general result concerning Clifford algberas.
Once we have adopted (1.5) a weak form of CCR, with the understanding that the states |psi> are to be restricted, in some sense, in the representation space, finite dimensional representations (FCCR) are possible. The irksome part of these representations is that the restricted domain of definition is not an invariant domain of the all the operators. That problem will be addressed and resolved later, by showing that the domain need not be restricted, that "dynamics" becomes stochastic, and that the probability interpretation will lean more heavily on transition amplitudes than on expectation values. The partial ordering of the Hilbert space vectors that appears, leads to the consideration of the action of a noncompact pseudounitary group conjugate to SU(n-1, 1).
In the other sections of this work these are investigated. There is no claim that all finite dimensional representations have been found or examined, but we do show a sequence of representations, one for every n>1, and discuss convergence of the sequence. All representations seem to exhibit a spacetime quantization and a dispersion associated with the quantal "eigenpoints".
The desiderata of expressing spacetime notions as quantized, or in terms of intrinsically dispersion free states is not new. A sampling of the various earlier work can be found in [Flint 1948], [Hill 1955], [Jordan 1968], [Jordan 1972], [Rosen 1962], and [Snyder 1947], [Lubkin 1974a] and [Lubkin 1974b] consider quantum formalism in finite dimensional Hilbert spaces. A continuing reason for quantizing space and time is the realization within any program of attempting to quantize the gavitational field, as expressed by the Einstein equations of general relativity, that the coordinate functions must themselves be quantized. The form invariance of the tensorial Einstein equations under general coordinate transformations (general covariance) can be seen as a gauge invariance that appears due to the masslessness of the graviton, similar to the gauge invariance of Maxwell equations due to the masslessness of the photon. The gauge transformations in both situations are connected to the form of interactions of matter with the massless field; if the matter field is quantized, consistency demands that the gauge transformations must also be quantized.
As suggestions for reading this paper it is recommended that [Section VIII] be perused cursorily or skipped altogether on a first reading and used mainly as a reference for technical details. It is a collection of things that must be done but would otherwise break any existing continuity if it were dispersed to various other sections. The material is heavily interdependent and where best to put these things could not be seen to be optimal, no matter the arrangement.
It has been conjectured many times that somehow the spinor concept should be the foundation of local spacetime structure. The FCCR(n) algebraic chain creates such a connection between CAR and the position operator of QM. Within the Clifford-Riemann-Einstein program of geometrizing physics, the internal spaces of particle symmetries, usually involving the Lie algebras SU(n), are not directly wedded to spacetime, but are considered something apart from it. The geometrization program is then not fully developed. FCCR(n) exhibits a clear Q-morphology of "local" spaces whose size is measured by n, and whose natural algebras are SU(n), which is to say that FCCR(n) can be considered as the expression of a Q-theory in the internal (now quantized) spaces of particle physics.
Email me, Bill Hammel at: