DAVID RITZ FINKELSTEIN

ELEMENTARY PROCESSES

Abstract. Since non-semisimple groups are unstable, every non-semisimple group in physics will probably

be the locus of a major conceptual reformation. We carry out such reformations of the canonical commut­

ation relations , the space-time continuum, and the interface between experimenter and system. The canon­

ical reformation introduces a large orthogonal group and its Clifford algebra and spinors . The space-time

reformation eliminates the differential calculus from basic physical laws, introducing a generic elementary

quantum process and a fundamental time X. The interface reformation leads to a non-operational cosmic

quantum theory that treats the physical experimenter as a quantum system, recovering the operational quan­

tum theory by a systematic quotient process. We propose a real form of Fermi statistics for the elementary

processes, in a Clifford algebra generated from the real line (which represents the empty set) by iterated

quantifi cation .We use this quantum set language to program the spin and space-time variables and equations

of motion of a Dirac particle.

1. REFORMATIONS

The three main evolutionary leaps of twentieth century physics have a suggestive fam­ily resemblance. Special relativity introduced a non-commutativity of boosts. Generalrelativity and gauge theory introduced a non-commutativity of infinitesimal transla­tions. Quantum theory introduced a non-commutativity of observations.

Do all historic radical changes in the foundations of physics introduce non-commu­tativities ? Is this the form the next such change will likely take?

The analyses of Segal (1951) and Inonu and Wigner (1953) suggest that the an­swer to these questions is yes. Groups and Lie algebras that are not semisimplewe call compound . A compound group is unstable with respect to small changesin its structure and is almost certainly but an approximation to a more physical stablesemisimple Lie algebra, subject to verification when measurements become more ac­curate. Non-commutativity is what stabilizes and simplifies an unstable compoundgroup, as curvature stabilizes a sail.

The compound group generally respects an absolute A that is presumably false:an idol in the sense of Francis Bacon. Dethroning such an idol is a radical conceptualchange.

What shall we call this inverse process to group contraction? "Revolution" is thecommon name but is inappropriate, since the deformed theory has the utmost respectfor the old, incorporates it as a limiting case, stays as close to it as possible, and

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A. Ashtekar et at. (eds.), Revisiting the Foundations ofRelativistic Physics , 479--497.© 2003 Kluwer Academic Publishers.

480 DAVID RITZ FINKELSTEIN

never supersedes it entirely. (Centuries after Copernicus we still speak of sunrise.)"Deformation" is too general; this deformation must begin from a singular limitingcase and remove its instabil ity.

Since the process is a deformation that repairs the theory let us call it reformation .The general prescription for reformation is clear.1. Select a compound group of the existing theory. The compound group generally

has an idol.2. Find the physical effects that allegedly demonstrate the existence of the idol.

They show what the idol couples into.f3. If A couples into B, couple B into A reciprocally, thereby introducing a small

constant coefficient, the reformation parameter. If A couples to nothing, eliminate it.It was unnecessary.

A is then no longer an absolute of the theory.Reformation always inaugurates such a reciprocity. This implies that physical

groups ultimately couple their variables reciprocally. Such a "reciprocity principle"was applied to the coupling between space-time geometry and matter by Mach andEinstein . Since some of our habits of thought still harbor idols that violate recipro­city, we must expect future reformations as radical as the reformations that introducedquantum theory, special relativity, and general relativity.

This principle does not tell us which reformation is next, or predict the size ofa reformation parameter, or tell us which of several possible reformations is mostrelevant to current experiment. Since there are many idols in circulation today, theprobability of a proposed reformation is small. On the other hand the probability ofthe unreformed theory is O.

The standard example of reformation for Segal and for In6nii and Wigner is Ein­stein's replacement of the compound Galilean group of rotations and boosts by thesimple Lorentz group. This historic process eliminated the idol of absolute time.When Galileo transforms from dock to ship, he couples time into space (x' = x - V t)but not space into time (t' = t). The Galileo space/time algebra is not stable/semi­simple because there is this one-way time-to-space coupling under transformationsfrom one observer to another.

The stability principle is not devoid of discrimination. For example, it does notlead to string theory which leaves the all compound groups of present physics stillcompound . Indeed, if the string moves in a flat space-time, perhaps one of manydimensions, as originally proposed, then string theory is a counter-reformation, re­establishing an idol that general relativity dethroned.

Special relativity, general relativity, gauge theory, and quantum theory are allreformations. These retrodictive successes suggest that we test the predictive con­sequences of Segal's principle.

Any theory starts by assuming some physical invariants that couple to physicalvariables without back-coupling. Therefore the reformation strategy promises a never­ending succession of reformations, each introducing a new fundamental constant asthe value of a reformation parameter after the examples of c, It and G.

ELEMENTARY PROCESSES

2. THE CANONICAL REFORMATION

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The first stage of reformation is idol-detection.The most conspicuous current idol is the quantum imaginary Iii of the canonical

commutation relations

[q, p] = Iii, [p , i] = 0, [q , i] = 0 (1)

(Segal 1951). The algebra of (p,q, i) is isomorphic to a subalgebra (ax,bx,ad of theGalileo algebra where ax and bxrespectively translate and boost infinitesimally in thex direction. Heisenberg's group is just as likely to be wrong as Galileo 's, being just asunstable. It would be a miracle - more precisely, an event of probability 0 - if thecanonical commutation relations were exact.

The reformed canonical commutation relations are

[q, p] = Iii, [p,i] = Xq, [q, i] = J-Lp (2)

with small new fundamental Segal constants A, J-L :I 0 supplementing Planck 's con­stant. We can make a dimensionless combination N = AJ-L from them, and a time Xdefined below.

This reformation relativizes Heisenberg's Iii and suspends its commutativity withpand q.

Relativizing Heisenberg's absolute i is isomorphic to relativizing Galileo's abso­lute time.

This reformation, however, triggers an avalanche of many secondary ones, farmore than its Einsteinian prototype. Different particles have different space-time co­ordinates, momenta, and boosts, but the same Iii couples to all canonical pairs . Unify ­ing each pair with Iii unifies them all with each other. This fuses a great many groupsSO(3) into a vast SO(N) group and replaces the p's, q's and Iii by spin operators ofthat group.

Then the same instability attacks the differential calculus relation [d /dx, x] = 1.Its reformation eliminates the differential calculus from all basic laws, replacing dif­ferential operators too by spin operators. This incidentally makes the theory finite.

Nevertheless, this is a conservative reconstruction of physic s. Any reformationmakes changes in the predictions of today 's working physics that can be as small aswe like. We can always return as closely as we like to the predictions of the old theoryby choosing the reformation parameter small enough. The prediction, however, isthat eventually experiment will favor the reformed theory over the unstable one andprovide a non-zero value for the reformation parameter. Until that happens, we mightthink that this reformation is merely a way of regularizing the present singular theory.

There is a note out of tune. The isomorphism between Segal's reformation ofHeisenberg and Einstein's reformation of Galileo seems imperfect: Einstein used onlyone constant c to reach special relativity from Galileo relativity. Why does Segal needtwo?

The difference is that Segal set out to simplify a group and Einstein did not. ThePoincare group of special relativity still is not simple, thanks to its translation sub­group. Had he set out to simplify the Galileo group, Einstein would have gone all the

482 DAVID RITZ FINKELSTEIN

way to the de Sitter group in the first step, introducing a second constant, the radius ofthe universe, besides lightspeed c. The Galilean reformation leads at once not only tospecial relativity but also to the expanding universe.

After the canonical reformation i is no longer a c number but a dynamical variable.Therefore we start from a quantum kinematics that lacks it, a real quantum theory. Avariable i once provided a Higgs field (in quaternionic gauge theory, (Finkelstein et al.1963». It seems likely to do so again.

The orthogonal group of the reformed Heisenberg algebra has a single finite­dimensional defining spin representation from which all the physical representationscan be built algebraically. The usual infinite-dimensional representation of momen­tum-energy by differentiation and position by multiplication gives way to a finite­dimensional representation of all the components of space-time-momentum-energy­ni by sums of products of spin operators. The ultimate factors correspond to finiteelementary quantum processes, chronons.

3. THE INTERFACE REFORMATION

The classic prototype of a theory ripe for radical change, Galileo's space-time, is abundle with time as base and space as fiber. The reformation unifies fiber and base.

All bundle theories share the instability of Galilean space-time and are as much inneed of reformation. The group of the bundle couples the base into the fiber but notreciprocally. We must eventually unify the fibers and the base of the gauge theories.The unification of p and x that Segal proposed is another special case, the cotangentbundle of space-time.

In modern field theory, however, the space-time base is classical, part of the exosys­tem, and the fiber is quantum. To unify them we must also unify experimenter andquantum system. We cannot eliminate experimenters but we can relativize the abso­lute experimenter of operational quantum theory, thus eliminating another idol.

One does this most often by imaging a cosmic viewpoint that covers all possibledivisions into experimenter and system. This results in what claims to be a quantumtheory of the cosmos, a concept that Heisenberg and Bohr resisted strongly at first,but which was finally accepted by Bohr as a necessary step. Unity and operationalityultimately part, and we pursue unity. We assume - like Laplace - a metaphoric orvirtual cosmic experimenter (CE) who can input, transduce, and outtake the entirecosmos including all the experimenters in it. Quantum field theory has implicitlytaken the cosmic perspective from the start, since the actual experimenter is made ofthe same field as the system under experiment.

A maximally informative experiment has three stages corresponding to the factorsin the transition amplitude (31211). Therefore the experiments of the CE also havethree parts . The CE inputs the cosmos in the remote past in an initial mode 11), sets upa medial dynamical development 2, and outtakes the cosmos in the distant future witha final dual vector (31, experimentally testing the cosmic transition amplitude (31211).

One should not be misled by the revealing miscount of two experimental interven­tions given in (von Neumann 1932). One might make the same miscount today by

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counting as interventions, say, only what goes on at CERN in the target and detectorbuildings, overlooking the accelerator facility that produces the input beam.

Some do not express quantum theory operationally but in terms of collapsiblestates; others speak of actual operations and actions . We call these formulations onticand praxic respectively. Ontists naively imagine that a unitary operator represents anindividual action faithfully, like a classical rotation matrix. To give the operator some­thing definite to act on they count the input mode vector 11) as a state. Then they countas interventions only the dynamical process 2 and the outtake action (31. Praxists re­cognize that an operator representing an action is a matrix of transition amplitudesreferring to ensembles of many experiments. They count all three actual stages of anexperiment.

The ontic formulation of the quantum theory is an atavistic vestige of the discred­ited wave theory of quantum phenomena, but it gets to the same physical results as themore accurate praxic formulation by making two errors that cancel. The first error iscalling the first action a state of the system. Actually we cannot extract informationspecifying the first action from the system, according to the amplitude formula. It maybe called a state, but it is not of the system but of the exosystem, just as a probabilitydistribution is not of one case but of its context, a source or an ensemble of cases.Calling the last action a collapse to a new state is a similar error that compensates forthe first. The same artifice works just as well for cosmic quantum theory.

Any operational theory, classical or quantum, divides the cosmos into a system thatis treated dynamically and an exosystem that is not. The exosystem always includesthe experimenter. We call this division the (system-exosystem) interface. The twovertical bars in the amplitude formula (31211) may be used to represent the interface,since the dynamics operates only inside them. The interface is sometimes called thequantum or Heisenberg cut but it is not peculiar to quantum theory, accept insofar asquantum theory dispelled some of the naivety that permitted most people most of thetime to overlook the interface in pre-quantum physics.

Introducing a cosmic experimenter and a cosmic dynamical law does not eliminatethe interface but shifts it from between system and exosystem to between cosmos andthe CEoIt does not reduce the number of interventions but shifts them from the actualexperimenter to the virtual CEo

4. THE DYNAMICAL REFORMATION

The Hamiltonian giving the dynamical law of the system under study is an idol ofthe current quantum theory, acting on the system with no reciprocal action . As step2 of the reformation process we do thought experiments in which we determine thedynamical law for a system under maximal quantum resolution.

In the quantum theory of Heisenberg and Bohr, one postulates the transition amp­litude formula A = (31211). This familiar assumption ofquantum theory is approxim­ate on several counts. It splits the experiment into three stages represented by the threefactors in A. In stages 11) and (31the system development is totally dominated by itsinteraction with the exosystem and in stage 2 the system is totally decoupled from thecxosystem and develops according to a known Hamiltonian operator H . The expres-

484 DAVID RITZ FINKELSTEIN

sion of H in terms of primitive system variables is idealized as completely knowable,hence classical, even though it governs a quantum system, because it summarizes aninfinite number of quantum experiments, not one. The three stages are assumed to beindependently variable.

Our experiments, however, are not infinitesimal in size and infinite in number,but necessarily finite in size and number, for several reasons, including their quan­tum structure, their gravitational fields, and their cosmological limits . Therefore ourknowledge of H is incomplete in principle.

Since we measure H on an ensemble, we should treat H as a quantum variableof the ensemble, with maximal descriptions combining by quantum superposition asusual.

Furthermore, if we make the dynamics a quantum variable then we probably needno others. Newton and Mach already raised the possibility that the dynamical law isvariable. Here we further propose that it is the sole variable. To be sure it is typically ahighly composite variable, but its parts too are dynamical processes This correspondsto taking the variable metric of general relativity to be the sole dynamical variable, asin Einstein's unified field theory.

In some earlier efforts we took the causal relation between events as the primevariable, and sought the dynamics that governs it. Now the dynamics is the sole in­dependent quantum variable, with its own algebra. We are to find all other physicalvariables within that algebra. The causal relation and the metric tensor give a reducedstatistical description of the dynamics, adequate for a planet but not for a proton.

A sharp description of the dynamics is a spinor of SO(N) that we call a dynamicsmode. The relation between spinor and dynamics is like that between a wave functionand an atom. The spinor describes the dynamics sharply but is not the dynamics. Theaddition of operators to form a Hamiltonian is a classical process distinct from thequantum superposition of spinors.

The high constancy of the dynamics over the sidereal cosmos suggests that it is allone order-domain of the dynamics, and that some basic cell serves as order parameter,repeating its structure throughout our ambient vacuum and defining the dynamics ofour cosmos. This unit cell is then the molecule of dynamics, composed of somenumber of chronons that remains fixed as the total number N varies.

The space-time metric tensor is a partial description of dynamics. As the Lag­rangian of a massive point particle, it defines, and is operationally defined by, howa test body - a smallish system whose tides hardly effect its orbit - moves undergravity. The theory of the space-time metric can therefore give useful hints toward atheory of the dynamics in general. The usual quantum assumption of a fixed dynam­ics for arbitrary initial and final actions is a quantum analogue of the pre-relativityassumption of a flat space-time for arbitrary space-time content.

In classical thought the path description varies from experiment to experiment onthe same system, while the action function is fixed. The specification of the dynamicsleaves a set of possible paths, not a sharp description of the path. The initial data thencomplete the determination of the path.

In the quantum theory, however, the dynamics assigns a probability amplitude toeach path, and is therefore identical with a superposition of path descriptions, up to a

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duality; or to a highly entangled description of one q path. Such a path description iscomplementary to the specification of the end-points or for that matter of any point onthe path.

In the c theory the endpoint specification is supplementary to the dynamics de­scription, and completes it. In the q theory the endpoint data is complementary to thedynamics description and violates it. The classical theory describes its dynamics onlycrisply, as a set of many possible paths. The existing quantum theory describes itsdynamics sharply, as a single entangled path. We follow the quantum model ratherthan the classical in setting up a chronon dynamics .

Smolin (1992, 1997) has suggested that the dynamics not only varies but under­goes Darwinian evolution, like genes, without proposing a microscopic locus for thedynamics. The development of the dynamics may be like crystal growth, guided bythe entire unit cells, or like cell multiplication, guided by specialized code structuresburied within each cell. In either case we require a higher-level dynamics to controlthe lower-level dynamics under consideration, as Einstein's gravitational equationscontrol the metric.

We do not yet understand well enough how the approximation of a fixed dynamicscan work so well if the dynamics is as highly variable as we claim. This dynamicalfact corresponds to the chronometrical fact that the flat space-time metric is a goodapproximation for much of physics , despite the variability of the space-time metric.

The chronon dynamics must contract both to a standard q field theory and to gen­eral relativity, in two appropriate contractions with X --t °and N --t 00 . It wouldsuffice, for example, if in the limit Ii --t 0, X --t 0, N X --t 00 we recover classicalspace-time and general relativity; and in the limit li = constant, X --t 0, N --t 00,

N X = t = constant, we return to the continuum-based quantum field theory.Our main variable is no longer a causal relation as in our first efforts (Finkel­

stein 1969). A causal relation is a weak statement about what dynamics is pos­sible. It is stronger and sharper, hence more vulnerable , to give the actual dynamical­development operator D for the universe. This contains the causal connections andmore. The space-time metric is a reduced statistical description of D .

5. THE SPACE-TIME REFORMATION

The usual quantum theory assumes an underlying fundamental flat space-time, ignor­ing Einstein locality. But the space-time in the experimental chamber can be determ­ined operationally from the dynamics, and so should not be postulated. We assumeneither flatness nor even any definite dimension or signature under high resolution.These all come from how the atomic elements of the dynamics are interconnected ina particular mode.

We consider the simplest possibility only: that the atomic elements of the dynam­ics - chronons henceforth - have no parts, cannot be factored . We expect that thestandard classical space-time points are the correspondence limit of many such quan­tum events.

All field theories are ripe for reformation . Field theories, including string andmembrane theories, exhibit the same symptom of impending failure as Galilean clas-

486 DAVID RITZ FiNKELSTEIN

sical mechanics, a basic one-way coupling, formerly from time to space, now fromspace-time to field, resulting in a non-semisimple transformation group, presumptiveevidence of an algebra contraction in the Segal-Inonu-Wigner sense (Marks et al.1999). If such a theory is valid in some domain, it is expected to be a contraction of asimpler theory of greater domain.

The bundle basic to all field physics is the tangent bundle of space-time. Thesymmetry between p and x that the canonical commutation relations imply, even be­fore their reformation by Segal, is violated by any bundle over space-time. Simplicityrequires us to unify the field variables and the space-time variables.

The differential-calculus limit 6.t -t 0 results in the canonical commutation rela­tions [ax,z] = 1, which have a non-semisimple group. To deform this group to SO(3)(or SO(2, 1) we cut off the limit 6.t -t 0 at a finite value 6.t = X, introducing a finiteelementary dynamical process - the chronon - and a fundamental constant X givingthe time-scale of the chronon.

We suppose that the dynamical process is an aggregate of finite elementary quan­tum processes, chronons. We will specify the structure of this aggregate after wedevelop necessary concepts of Clifford-algebraic quantum theory and quantum statis­tics.

The usual operator-algebraic quantum kinematics lacks space-time concepts. Oneappends these as necessary. Frequently this disturbs simplicity by creating a fiberbundle. and a continuous time coordinate. Then one can analyze the dynamics intoever finer transformations and never reach elementary or atomic transformations. Thisassumption leads to infinities and is likely wrong. The atomism principle cuts this pro­cess off, and yet does not conflict with exact Lorentz invariance on sub-cosmologicalscales of time and length. Snyder space-time is an early example of such a Lorentz­invariant quantum space-time with discrete space, though it still has continuous time.

A crystal has many characteristic times with different physical meanings. So doesthe vacuum, experimentally speaking.

The space-time analogue of the crystal cell-size l is the fundamental time-scale Xof the chronon. The cell of the q space-time is an assembly of several chronons.

The vacuum must also have a coherence number Nc , analogous to the pure number(Acll)D for a superconductor, where AC is the Ginzburg-Landau coherence length,l is the the cell size of the crystal , and D = 2 or 3 is the dimensionality of thesuperconductor.

Further, each mode that can not propagate in a crystal has a penetration depthlv - Similarly each mode to which the space-time q crystal is not transparent has apenetration depth lp. Penetration depths may manifest experimentally as Comptonwavelengths defining masses, for example of gauge fields.

A well-known elementary qualitative argument equates the Planck mass to a quan­tum black-hole mass. Then the Planck length is not the cell size but the penetrationlength of a high mode of the network.

Standard relativistic quantum field physics incorporates an absolute locality prin­ciple that has been enormously fertile, producing electromagnetic field theory, generalrelativity, and the gauge revolution of recent decades. The chronon replaces infin­itesimal locality by a finite locality . This act of desperation calls for several excuses

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and a promise to make restitution. We must restore infinitesimal locality by a suitablecontraction of the finite locality.

Originally we dropped infinitesimal locality just for the sake of finiteness. Presentcontinuum-based field theory is a kind of science-fiction, postulating acts that areimpossible in principle, exploring some consequences of this fantasy, and carefullyignoring others. A better quantum theory would postulate only acts that are possiblein principle. Such acts cannot be infinitesimals. We retain a finite locality, in whichelementary interactions couple events separated by X > O.

Nowadays we further justify this renunciation on Segal's grounds of stability andalgebra simplicity. They couple x and p into one simple orthogonal-group algebra.Since x has local matrix elements rv <5(x - x' ) and p has slightly nonlocal onesrv <5'( x - x' ), mixing x and p breaks locality.

To put it bluntly: One measures x with a small pinhole and p with a large diffrac­tion grating. The reformed theory cannot tell its diffraction grating from its pinhole.

The standard quantum theory is local but not simple; we seek a quantum theorythat is simple, hence not local. The c correspondents of our q events are now pointsof at least an 8-dimensional space, combining x and p, not a 4-dimensional one. The8-dimensional quadratic space 8lR. = {( xl-' ,Pl-')} = 4lR. EB 4lR. has possible signatures(6 - 2), (4 - 4), and (2 - 6), based on the Minkowski signature (3 - 1). In section10 we opt for the neutral quadratic form. Then time t and momentum are composedof Clifford units with signature +1 and space x and energy E with signature - 1.

6. CLIFFORD t ALGEBRA

A roman capital designates a quantum entity: X stands for the chronon and D for thedynamics much as H stands for hydrogen. Each system S has a quadratic vector spaceV = Vs of quantum modes or channels and an operator algebra A = A s = Endo Vrepresenting possible actions upon the system. Cliff(N+, N _) and CLIFF V are realClifford algebras with an adjoint t taken to be the main anti -automorphism. CLIFF

is the Clifford functor from real t spaces to their free Clifford algebras. Cliff is theClifford map V --+ C = CLIFF V , linearly mapping each vector v E V to a Cliffordelement Cliff v E C . By a real Clifford t algebra over a quadratic space V withquadratic form gab - possibly indefinite or singular - we mean a real t algebra Cgenerated by lR. and an isomorph of V in which the Clifford relations hold in the form

(Vv E V)(Cliffv)2 = - llvll. (3)

The Clifford algebra is free (or universal) if all its relations follow from these pos­tulates. We define the anti-automorphism t : C --+ C by postulating that (Cliff v)t =- Cliff v for all v E V. The top unit of a basis {i a } of V is

Xi := IT Xa = XN·· · XI·a

(4)

If A = CLIFF V is a Clifford algebra and A rv Endo V ' then V ' is afree spinor spacefor A and we write V' = JA. J is a kind of square root, extracting a vector space of

488 DAVID RITZ FINKELSTEIN

dimension D from an algebra of dimen sion D 2• Dimension grows exponentially from

V to A : Dim A = 2Di m- .

7. QUANTIFICATION

What is the statistics of the chronon?Before we describe our latest model , we recapitulate the concept of statistics in

general.One usually defines a statistics for indistinguishable particles by representing the

permutations of individuals in the aggregate algebra. This creates a semantic problem:What physical operations can be meant by an exchange of indistinguishable objects?

We evade this problem if we consider a unified system of a variable number ofquantum entities, with number ranging from none to very many. Then an exchange ofa pair may act trivially on the pair but non-trivially on each of its members by itself.We therefore define an exchange as an automorphism of the many-body algebra thatexchanges specified elements of the one-body algebra. We define a statistics so thatit provides a Lie homomorphism t. : A -7 A' of the l-quantum endomorphism Liealgebra A into the many-quantum algebra A', describing how aggregates transformunder transformations of the individual.

For a quantum statistics we further impose a coherence condition: that cbe inducedby an operator-valued form Q t : V -7 A' here called the quantification.form so thatfor all a E A, b(a) = QtaQ .

For example if ¢> is any boson state vector and Q t = Q; is the boson quantifi ca­tion form then Q2t ¢> is the boson annihilation operator associated with mode ¢>. Forcomplex Fermi statistics, the quantification form b = b l maps a state vector 'Ij; to afermion creation operator.

Any quantification functor Q fits into a commutative diagram

E NDO

Q

V ECa

1

VECa

ALGa

1

ALGa

(5)

V ECa designates the category of neutral mode vector spaces, A LGa the category ofendomorphism algebras of such spaces. The horizontal arrows E ND O lead from tspaces describing quanta to their endomorphism algebras . The vertical arrows Qlead from individuals to collectives. The resultant morphism Q E NDO = E NDO Qtransforms the individual chronon state-vector space to the aggregate dynamics al­gebra. Associated with each mode of quantification Q and each t space V there is anoperator-valued form Q t : V -7 A' = QV that we call the quantification form. Itsdual is an operator-valued vector Q. Since A = V 0 Co V, Q induces a mappingc : A -7 QA,a I----> QtaQ that we require to be a Lie homomorphism, respectingcommutation relations [a,b] = c. The mapping t. generalizes the braces {... } of settheory and the successor function b of Peano.

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We tag the real Fermi, complex Fermi, and Bose quantification operators and unit­izers with a subscript 0, 1,2 counting the independent imaginaries in the coefficientnumber system of the associated classical group.

We may imbed permutations in the orthogonal group by applying them to the axesof a frame in the input vector space of the individual system. Then relative to anyframe in VI, Q also defines a representation of the permutation group SN C O(N),thus subsuming the usual concept of statistics.

In the standard quantum theory the elements of a time-translation process are infin­itesimal, distinguishable, and obey Maxwell-Boltzmann statistics . The true chrononsmust have an i-less irreducible (simple) statistics. Maxwell statistics is not simple,Bose is neither real nor stable. Neither will do. This seems to favor Fermi-Diracstatistics . Finkelstein (1996) was based on Fermi statistics .

But Fermi statistics is usually expressed in a Grassmann algebra. Grassmann al­gebra is unstable; its reformation is a Clifford algebra. Finkelstein and Rodriguez(1981, 1984) proposed such a reformed Fermi statistics. Here we use a Fermi statis­tics formulated from the start within a stable Clifford algebra (Plymen and Robinson1995).

The global dynamics D = QoX is a q set of microscopic elementary dynamicalprocesses or chronons X. Qo is a special quantification functor defining the real Fermi­Dirac statistics of the chronon, discussed in the next section.

The number N of X' s in D reflects the space-time measure of the process wechoose to study. We take N to be finite, leaving any limit N ---> 00 for last. Inordinary quantum experiments N » 1.

Since we seek an analysis into elementary processes, we suppose that the basicdynamics describes only one step in history, from one moment to the next, a step of Xin cosmic time. For autonomou s systems, the whole story is constructed by iteratingD.

To describe Fermi-Dirac spin 1/2 quanta we assume that the spinor space V toois a real quadratic space with a neutral bilinear form t . Then its automorphi sm groupis an orthogonal group O(N,N ); its Clifford algebra CLIFF V is a fermionic algebragenerated by creators and annihilators of spin 1/2 excitation s; and the spinors of Vrepresent assemblies of such processes.

All actions on a set of chronons can be expressed algebraically in terms of chrononpermutations. Our real Fermi statistics is thus more expressive than the usual, whereno (nontrivial) system variables can be expressed in terms of quantum permutations(= ± 1).

It is well-known in principle how to construct approximate bosons out of manyfermions.

In the early days of quantum physics, the 2-valued representations of the rotationgroup were overlooked for a time because they do not occur in a tensor product ofspinless quanta. We overlooked the much-studied 2-valued representations of the per­mutation group and 2-valued statistics until recently for much the same reason: Thespinor space does not occur in a tensor product of the vector spaces of individualquanta.

Our main constitutive assumption about the dynamics can now be expressed:

490 D AVID RITZ FINKELS TEIN

The dynamical process D is a real Fermi-Dirac aggregate of finitely many chro­nons.

A spinor D of Cliff(N+ , N _) maximall y describes the unified quantum space­time-matter-dynamics D. We expect a suitable D to define both the action function alsof q field theory and gravity in appropriate contraction limit s.

Quantum theories that rest on Hilbert space, with its definite sesquilinear form, arenot sufficient for our purpo se. Our spinors have real neutral quadratic forms. They areneutral so that they can split relati vistically into space-time and energy-momentumforms, which in turn are indefinite so that they can split relativi stically into spacelikeand timelike parts..

8. MULTIPLE QUANTIFICATION

Standard quantum physics uses "00 + 1" levels of quantification: the 00 classicalquantifications of classical set theory to produce the underlying classical space-timecontinuum from the empty set, and one quantification in the quantum theory to pro­duce fermionic or boson ic algebras from a l-quantum theory.

For example to build up the hierarchy of points, lines, surfaces, volumes, . .. of thespace-time manifold . Two levels that occur in the quantum theory are those of thespace of one-fermion wave functions of Dirac's first electron theory, and its quantifi c­ation, the many-fermi on field theory.

Since we are unifying the experimenter with the system, we must unify their logics,including their quantification processes.

To constrain the problem we suppose that all quantifications are of one kind . Thisis clearly a vestige of set theory but it might still be right. We seek one quan tifica­tion whose iteration success ively produ ces space-time, a one-fermi on system, and themany-fermion system.

We have defined the quantification functor so that it can be iterated . We may iterateQ and ENDO and extend the diagram (5) downward and to the right as necessary, j ustas we iterate the power set functor in set theory. For example, Q2 = QQ producesan aggregate of aggregates with a well-defined algebra and vector space. These haveinternal structures resulting from the preceding aggregation.

9. SPINS AS AGGREGATES

The most appropriate statistics for the chronon, we propo se, is a real generalizationof Fermi statistics that describes the many-fermion system by a spinor space of state­vectors and a Clifford t algebra of observables.

It is already well-known that spinors arise from statistics without reference to spin(Wiman 1898; Schur 1911; Hoffman and Humphreys 1992; Nayak and Wilczek 1996;Wilczek 1997; Wilczek 1998; Wilczek 1999) and can be used to describe compl exFermi ensembles (Plymen and Robin son 1995). We give such a spinorial descriptionfor real quantum statistics .

We define the real Fermi quantification functor Qo (and a real Fermi statistics) by

V ' = QoV = ..;CLIFF(V EB Co V) (6)

ELEMENTARY PROCESSES 491

Here Co V is the t space of real forms on V.This spinorial representation for Fermi-Dirac statistics is essentially that of Dirac

(1974) and Plymen and Robinson (1995) . It supersedes Finkelstein (1996) whereFermi quantification was represented by a different functor

V' = QGV = VEC GRASS V , (7)

in which GRASS V is the Grassmann algebra over V and VEC is the forgetful functorthat remembers only the t vector space of a t algebra. It happens that both quantifica­tions produce vector spaces V' of the same dimension 2D , where D is the dimensionof V. But Qo has a larger invariance group . Finkelstein (1996) was written before weencountered Segal (1951).

To verify that the new formulation captures the essential features of the usual Fermiquantification, consider orthonormal basis vectors 'l/Jn E V and dual basis vectors'l/Jn t

E Co V with {'l/Jn, 'l/Jmt = 8~. Because the natural metric on W = V EB Co Vis neutral and has V and Co V as maximal isotropic subspaces, the correspondingClifford generators in CLI FFW indeed obey Fermi-Dirac relations

(8)

The exchange X : 'l/Jl +--> 'l/J2 of two orthogonal unit vectors in V induces anexchange pair 'l/Jl +--> 'l/J2' 'l/Jlt +--> 'l/J2t in CLIFF(V EB Co V).

Thus the real Fermi statistics can represent one swap Xl +--> X2) = (12) project­ively in several ways: by the reflection Xl - X2, the reflection Xl + X2, or the 1r

rotation 1 ± X1X2 . Read (2002) uses some of these in a comparison of several statis­tics, including braid statistics . These operators agree on the rays of Xl and X2 but noton other rays in the X1X2 plane. To swap two fermions , moreover, we must swap boththe hermitian parts xt +--> xt and the anti-hermitian parts Xl +--> X2 of their creationoperators.

We shall assign positive indices n to the real parts 'l/Jn of the complex vectorsIn) and negative indices - n to the corresponding imaginary parts 'I/J-n. To avoidambiguity we do not use the index n = O. Then the creation and annihilation operatorsfor the complex mode 'l/Jn with n > 0 are (up to a possible normalization factor)respectively

"t( 'l/Jn + 'I/J- n)('l/Jn - 'l/J-n)t"

Xn + X- n ,Xn - X-n (9)

where Xn = Cliff 'l/Jn E CLIFF W. The fermion swap too can be done in several waystoo, such as (xt - xt)(XI - X2)'

Since all of these representations are in the same algebra with the same phys­ical interpretation, they belong to one physical theory and we propose to regard themas consequences of one statistics, the real Fermi statistics . We assign the followingnames:

Let m , n > O. A semi-swap projectively represents the finite transformation

492 DAVID RITZ FINKELSTEIN

(mn) : V ~ V,'l/Jm f---* 'l/Jn,'l/Jn f---* 'l/Jm;

(Vk 1= m, n) (mn) : 'l/Jk f---* 'l/Jk, (10)

by the first-grade Clifford operator Xm - Xn.A swap projectively represents the finite transformation (mn) by the second-grade

operator (Xm - Xn)(X-m - X- n)Spin represents the infinitesimal orthogonal transformation

Lmn : 'l/Jm'l/Jn

(Vk 1= m , n) L mn: 'l/Jk (11)

by the Clifford operator !(XmXn + X-mX-n) .Swap is conceptually more primitive than (Lorentz) spin as permutation is more

primitive than rotation, invoking no spatial concepts of length or angle but only iden­tity. We propose to reduce all physical actions and gauge transformations, includingthe dynamical development, to swaps of elementary events, as figures in a tapestrymay be created, moved, and annihilated by interweaving its threads.

The Grassmann algebra of QG is the standard representation of the spinors ofV EEl Co V in the form given by Chevalley (1954). The spinor representation has theinvariance group SO(W) of dimension D(2D + 1), mixing creators and annihilators,while the Grassmann representation reduces that to the group GL(D) of dimensiononly D2 .

Furthermore the spinor spaces V' = QV resulting from Qo have a natural neut­ral metric. This enables us to iterate the functor Qo to make a hierarchy of nestedstructures.

We shall call the Grassmann theory the restricted Fermi statistics and the spinortheory the real Fermi statistics .

Since the spinors that describe spins maximally are isomorphic to the spinors thatdescribe real Fermi-Dirac aggregates maximally, we shall identify spins of SO(N, N)with real Fermi-Dirac ensembles of 2N elements .

Because intrinsic spin is so much simpler than orbital angular momentum and theother variables of quantum field theory, many have considered spinors to be morefundamental than space-time vectors. Schur's theory of spinors - preceding Cartan's- shows that this is wrong. Spinors over W clearly describe sets of quanta describedindividually by vectors in W .

This is consistent with the fact that spinors are an irreducible representation of theLorentz group. Mode vectors and the entities they describe are categorically dual, andbehave dually under decomposition. An irreducible tensor may describe a compos­ite object. That vectors are quadratic in spinors is actually primafacie evidence thatthe spinor describes a larger aggregate than a vector describes. The spinor providesenough information to define a vector, but not conversely. The spinor therefore de­scribes a more complex structure than the vector does.

ELEMENTARY PROCESSES 493

When the space-time melts down into its quantum elements and the classicalLorentz group loses meaning, spin loses physical meaning, but swap has meaningas long as identity does.

It is natural to suppose that all spin derives from swap, the more primitive conceptof the two. Then every 2-valued spin representation signals a deeper 2-valued statis­tics. Quanta of spin 1/2 are likely aggregates of at least four fermionic entities.

Empirically, the standard spin-statistics linkage applies to quanta and not to space­time points. For example the classical space-time points of standard quantum theoryhave M-B statistics, not Bose or Fermi. In chronon dynamics too the spin-statisticsconnection must hold for excitations on the top level, that of dynamics, and not on thelower level of space-time.

Earlier we proposed that the spin-statistics correlation arose from the fact thatprocesses of 271" rotation and exchange are homotopic for skyrmeons (Finkelstein andMisner 1959). It seems now that its source is still deeper; that spin is correlatedwith statistics, and 271" rotation with pair exchange, because fundamentally rotation isexchange, a permutation operation.

10. TIlE DIRAC REFORMATION

In a theory of space-time quanta, the dimension and signature of space-time are orderparameters of the vacuum condensate. Algebra contraction must produce the usualMaxwell-Boltzmann aggregate of dimension 4 and signature 2 or -2, and a complexquantum theory.

In Fermi statistics, squads of eight , octads, are special, thanks to the octal period­icities of the spinorial clock (Atiyah, Bott, and Shapiro 1964) and the spinorial chess­board (Budinich and Trautman 1988). A set (Fermi-Dirac aggregate) of 8N fermionsof neutral signature is algebraically isomorphic to a sequence (Maxwell-Boltzmannaggregate) of N octads of neutral signature:

Octad lemma

Cliff(4N, 4N) ~ Cliff(4, 4) 0 .. . 0 Cliff(4, 4) 13) = [Cliff(4, 4)I3)JN. (12)

This lemma would not hold with any squad smaller than the octad. It would not holdwith signature ±(6 - 2). Therefore we choose the neutral signature for the octad. Thisextracts the Maxwell-Boltzmann statistics that we need for space-time from Fermistatistics.

Therefore we suppose that the formation of space-time spontaneously reducessymmetry from SO(4N, 4N) to SO(4, 4)N in the way described by the octad lemma.Fermionic chronons assemble into maxwellonic octads . The resulting quantum M-Bspace includes a Bose subspace in which further Bose condensation can proceed.

If the chronon is the atom of time, the octad might be the molecule or unit cell.Two classical 8-dimensional spaces vie for election to be the correspondent of the

1R(4,4) underlying each quantum octad: the complex Minkowski space-time C I3) M 4

and the symplectic Minkowski tangent-cotangent space M EI1 M t. Conveniently,

494 DAVID RITZ FINKELSTEIN

we have already inferred from Segal simplicity that under higher resolution the twobecome the same.

We can now obtain the infinite-dimensional algebra A = A (xl-' ,8m ) of the space­time manifold, generated by the variables x'", 8m , by contracting an atomistic Cliffordalgebra, as required by Fermi statistics .

Feynman (1971) proposed that the space-time coordinate-difference operator is thesum of many mutually commut ing tetrads of Dirac vectors:

(13)

It is a quantum form of the proper-time Heisenberg-Dirac equation dx!' / dr = ,I-'. Weincorporate and flesh out this proposal in the chronon dynamics that follows.

Our reformation requires a larger Clifford algebra than Feynman's, because we re­form the canonical commutation relations . For the present we use the Clifford algebraC := Cliff(3N, 3N) with defining relations

(14)

where f-l = 1,2,3,4 indexes 4 chronons in a Dirac tetrad, f-l 5,6 indexes twochronons in a reformed complex plane, and T = 0,1 , . . . , N - 1 indexes N neutralhexads.

We write the top Clifford element of each hexad in C as

XT,r := X6,rX5,r'" X2,rXl ,r (15)

To reform the canonical commutation relations we suppose that each hexad contributes

(16)

to the space-time coordinate,(17)

to momentum-energy, and(18)

to the complex i ; we drop the hexad index T for the moment and use! Ii = c = X =1 (chronon units). These are the antihermitian generators associated with the usualobservables. At the end we must supply a factor i to make a hermitian observable.

By duality, when we sum contributions to the total space-time coordinate we av­erage contributions to the momentum-energy. To approximate a square root of - 1 wealso average the contributions b.i, supposing that they all align in the vacuum.

Then in this quantum space-time the reformed manifold-algebra commutation re­lations are for each hexad:

! [b.x V, b.PI-']

t [b.i, b.x~]2 [b.PI-' , b.2]

(19)

ELEMENTARY PROCESSES 495

The reformed i generates the symplectic symmetry between x and p of classical mech­anics, as Segal (1951) proposed.

The sum over the hexads is

(20)

Here we restore x to seconds and p to joules, leaving c = 1.The time unit X implies discreteness for space-time-coordinates resulting from

single measurements, but not for expectation values like centroids. Averages oversufficiently many measurements may still be arbitrarily small.

There are far too many i's for comfort in this Clifford algebra. There is only onei in the canonical algebra, but there is a LliT for each of the 2N octads in C. Withinthis Maxwell-Boltzmann algebra of hexads there is however a Bose subalgebra, andin this Bose subalgebra there is a projector describing a condensation of all the locali 's into one global average i.

The leading 4 x 4 components of

(21)

are infinitesimal generators of the Lorentz group. The 6 x 6 totality of the L's infin­itesimally generate the reformed Poincare group, which is the de Sitter group.

To produce locally commuting squads of fermions from globally anticommutingones, we have linearly ordered them, by a kind of cosmic-time parameter T. Thiscosmic time seems have no correspondent in field theory. It is analogous to proper­time parameters of Feynman and Schwinger and is similarly helpful. It permits oneto formulate a Lorentz-invariant concept of dynamics operator, as an operator thatgenerates a unit increase of T . It permits us to identify a canonical conjugate to T withrest mass M rv liiOT (Schwinger 1951). It makes possible and natural a space-timequantization much like Feynman 's (13)

and extends it to momentum-energy space.It is straightforward to reform the one-particle Dirac equation within this Clifford

algebra (Galiautdinov and Finkelstein 2002).The reformation of the usual quantum field theory of a spinor field requires an­

other stage of quantification. The reformation of the one-graviton theory of spin 2resembles that of four fermions of spin 1/2. The reformation of gravitation theorycombines these two extensions. These are now under study. The reformation of theDirac equation required only two new reformation parameters, a chronon time X anda chronon number N . More will likely be needed.

We have reconsidered yet again the statistics of these finite elementary processes.Real Fermi statistics is the most plausible we have tried so far. It produces 2-valuedspinor representations of rotations, the Dirac spin operators and the fermionic cre­ation operators of the Dirac equation . It leads to an Maxwell-Boltzmann sequenceof octads like the M-B assembly of the usual space-time points, with four space-time

496 DAVID RITZ FINKELSTEIN

dimensions. It lends itself to a hierarchy of multiple quantifications that might unifyspace-time and fields in one algebraic theory.

ACKNOWLEDGEMENTS

This contribution includes work done in collaboration with James Baugh, AndreiGaliautdinov, Michael Gibbs, William Kallfelz, Dennis Marks, Tony Smith and ZhongTang. I am grateful for stimulating discussions with Giuseppe Castagnoli, ShlomitRitz Finkelstein, Raphael Sorkin, and Frank Wilczek. I thank the state of Georgia, theInstitute for Scientific Interchange, the Elsag-Bailey Corporation, and the M. and H.Fcrst Foundation for their generous support.

Georgia Institute ofTechnology

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