PARTS: THE Quantum Gravity PROBLEM AN EXTEREMLY BRIEF HISTORY OF SPACE SPACETIME VARIABLES QM & ITS DISCONTENTS GR & ITS DISCONTENTS FCCR

NOTE: this file contains no extended ascii characters; the extended work paper also uses no extended characters of the US_ASCII set. The notation there may appear slightly different, but the translations should be quite intuitive. The notation, generally, is perfectly disgusting, but, as far as I can invent without acceptable alternative. Cf. Notation.

The magic word 'XYZZY' appears as a marker,
primarily for me, at points that need additional material
that I haven't gotten around to.

This little discourse deals in an oblique way with notions that precede those of of "Quantum Gravity". No nonexistent thing has ever generated so much talk, writing or thought. I've been thinking about it for years. Physicists, generally, have been thinking about it for about 80 years now, and nobody seems to have actually done anything crazy enough to make the whole idea sane.

I'm not really sure that my toy "solves the problem", but it looks like maybe it's a start. I have to define or better "redefine" the problem, or at least its source. This way I can indicate how the toy fits in.

My apologies if this seems rather terse and dense; it is an attempted summary of the more detailed working document of over 600 pages. I'm sure I've omitted some interesting results and some serious fretting over physical interpreatations as well as over some things that I suspect to be true but cannot prove.

As a preliminary overview (if the math & physics lingo is familiar, and even if it isn't) and a historical perspective to the following,

The Cyclotron Notebooks by Phillip Gibbs is highly reccommended. To save a few bytes, I'm going to use a set of abbreviations, and so give the table to start with. ABBREVIATIONS: CAR = Canonical Anticommutation Relation CCR = Canonical Commutation Relation CM = Classical Mechanics (Newton) CR = Commutation Relation FCCR = Finite Canonical Commutation Relation GR = General Relativity (Einstein's Field Equations) HM = Canonically Formulated (Hamiltonian) CM HOR = Harmonic Oscillator Representation of CCR IRREP = IRreducible REPresentation LM = Lagrange Formulation of CM Q = "The Quantum Priciple" QED = Quantum Electrodynamics QEMT = Quantum Electromagnetic Theory QFT = Quantum Field Theory QM = Standard nonrelativistic quantum mechanics QST = QuantumSpaceTime (whatever that is) QT = Some most general quantum principle (unknown) R = "The Relativity Priciple" SR = Special Relativity ST = SpaceTime NOTATION: The Hilbert space notation is that of Dirac. := borrowed from Pascal programming language is a definition: the LHS is defined by the RHS. ^ is an exponent (chosen over the Fortran "**") X= is not equal to. ! Hermitean conjugation THE SOURCE OF THE PROBLEM: Way back in kindergarten, when we first learn QM it's presented in the Schroedinger representation as a Hilbert space (nobody says Hilbert) of complex valued functions over the standard Newtonian space and time. The functions must also be differentiable in their arguments and vanish at infinity so the square of their modulus is finitely integrable over the spatial domain. Otherwise we can't define a probability density. Relax differentiablity but demand continuity where a potential may be discontinuous. Otherwise we would miss out on the quantization of energy values altogether. We learn "to quantize" by making the holy operator substitutions in a classical expression for the energy, to wit, the Hamiltonian function. Somehow, the ambiguity of operator ordering in products of the canonical q, and p variables is quietly avoided. Behold we have "derived" the Schroedinger Equation from the classical equations of motion. Black Magic? You bet! If we're really mindwarped by this, we eventually find Dirac's Principles of Quantum Mechanics. Things take a slightly more elegant turn and it is found out that Schroedinger's Representation of QM is equivalent to Heisenberg's (which nobody told us about anyhow). Those of us with a mathematical bent are illuninated by having the mathematical arena clearly and elegantly defined as an infinite dimensional complex Hilbert space. Vectors yea! Later comes the bitter truth that this is almost as bad as the Schroedinger Stuff. No wonder, since J. v. Neumann proved a theorem that says that any formulation of quantum mechanics is equivalent to the Schroedinger equation. So we are stuck with the seemingly lovely rose CCR with all its attendent thorns. [Q, P] := QP - PQ = i h-bar I where i = sqrt( -1 ), I is the identity operator, and h-bar is Planck's constant divided by 2(pi). It turns out that early papers of Wielandt, Wintner and Taussky in the '40s proved that this expression cannot exist an any normed algebra. Eventually the notion of "unbounded operator" comes into being. It is mathematically necessary for at least one of the pair {P, Q} to be unbounded for CCR to be satisfied, and it is possible to find in the Hilbert space upon which they act a "dense common invariant domian" for the pair, so that if we restrict the action of the pair to this domain then CCR can be "weakly" satisfied. Symbolically, [Q, P] |D> = i h-bar |D> where |D> represents any vector in the restricted domain. The universe is saved; our little "grey cells" are so completely fried /-) having been pounced upon by unbounded operators no less!) that we would never question CCR and its folklore of unboundedness again. But, take a look at the HOR to be found in any decent QM text. Messiah, e.g. Q is an infinte matrix zero everywhere but for the lines of elements just above and just below the diagonal. The upper offdiagonal is the same as the lower offdiagonal. Starting from the top and counting down the k-th element of either offdiagonal has the value sqrt( k / 2 ) P has a similar structure, with the upper & lower diagonal elements being respectively -i sqrt( k / 2 ) and +i sqrt( k / 2 ) The HOR also happens to be the number representation of the creation and annihilation operators used in QFT, and B = (1/srqt(2)) (Q + iP) B! = (1/srqt(2)) (Q - iP) where ! indicates Hermitean conjugation (transpose, complex conjugate) and inversely Q = (1/srqt(2)) (B! + B) P = (i/srqt(2)) (B! - B) The number operator is defined by N := B! B which has an infinite integer spectral set {0, 1, 2, ...} and the B operators obey [B!, B] = h-bar I There is a clear sense (that of the existence infinitesimals and infinities) in which QM must be fundamentally wrong below some scale of length, but its obvious quantitative and qualitative correctness above that length makes attempting to change its foundations an operation that is even more delicate that lovemaking among porcupines. PURPOSE (abstract): What I would like to show is that the traditional folklore about any quantum theory having to be done in infinite dimensional spaces with unbounded operators is at best highly dubious and at worst, simply wrong; furthermore, that a good number of QM algebraic properties can be retained by a similar model in finite dimensional complex Hilbert spaces (finite dimensional C*-algebras) and that the apparent restriction to finite dimensions actually allows further algebraic detail to emerge which has the effect of: defining a concept of *Local* *Quantum* *Theory* within which there exists a genuine time operator not present in QM; automatically quantizing space and time such that position and time operators do not, in general, commute; maintaining an uncertainty relation that is just as sensible as that provided by QM; providing a Fourier relation between Q and P (In QM the technical source of the uncertainty relation); adding an additional Fourier transform between N and a time operator T which cannot properly exist in QM; enriching the background structure for physical events to be a linear space containing the concepts of position, momentum, energy and time, which space is also provide with a natural indefinite quadratic form; making the interesting geometrical objects (Spectra of selected operators) quantized loop-like objects (the effective topology of the spectra is that of a circle); creating an algebraic bridge between CAR (n=2) and CCR (n unbounded); showing time to be "created" in the process of algebraically necessary transitions or "quantum jumps" of a complex Markov type process; indicating a 'dynamics without dynamics' (in J. A. Wheeler parlance), where the state itself does not have a determinate evolution so that the emphasis of physical descriptor shifts from *state* to *process* (The vectors of the Hilbert space and the density matrices of the algebra then represent process not state); indicating a shift of perspective from expectation values to transition amplitudes and then transition probabilities, suggesting the relevance of a complex or quantum Markov process as *the* dynamical process; not actually making the life of theorists any easier, but *possibly* a little more sensible in terms of eliminating actual infinitesimals and singularities, magnifying the complexity of algebraic possibilities. The material presented here is not without problems and questions, the answers to which may ultimately damn it to the nether regions. It is probably incomplete from a relativistic perspective as it is seemingly incapable, as it stands, of accomodating negative energy states, (which may after all be introduced by a simple finite subtraction or "renormalization"). My "feeling" at the moment is that a simple subtraction is not sufficient. THE "QUESTION": What happens in the Oscillator Realization & Number Representation, if we simply truncate the matrices to be square at some dimension n? Writing [Q(n), P(n)] = i h-bar G(n) With definitions of B(n), B!(n) as above, it follows that [B!(n), B(n)] = h-bar G(n) N(n) := B!(n) B(n) The Trace Tr( Q(n) P(n) ) = Tr( P(n) Q(n) ), so Tr( G(n) ) = 0, which is clearly true for the trace of any commutator of finte matrices. Perfoming the matrix multiplications, it isn't hard to see that G(n) has "almost" the form of the indentity matrix: all ones down the diagonal up to the last one, which must have the value -(n-1), since the trace vanishes. Since most of the proofs are long and detailed, from here on, I'm simply going to state some results with no proof and little, if any argument. Trust me :-). The following hold N(n) = Diag[0, 1, 2, ..., (n-1)] [N(n), B(n) ] = -B(n) [N(n), B!(n)] = +B!(n) [N(n), B^k(n) ] = -k B^k(n) [N(n), B!^k(n)] = +k B!^k(n) and [N(n), Q(n)] = -i P(n) [N(n), P(n)] = +i Q(n) (The carat symbol '^' indicating a power.) It is clear that we are working in the Number Operator Representation: all vectors are represented as linear combinations of the eigenvectors of N(n), |n, k> or just |k> more briefly, where k = 0, 1, 2, , (n-1). Assume the |n, k> (or just |k> when we can) constitute the "canonical basis". f(n) := exp( i (pi)/2 N(n) ) is a discrete Fourier transform, clearly unitary and idempotent of order 4. Also f(n) Q(n) f!(n) = +P(n) f(n) P(n) f!(n) = -Q(n) which is all algebraically *exactly* at it is with CCR. So, f^2(n) appears to be a spatial inversion. Complex conjugation would also appear to be a "time inversion"; this operation, however, is not invariant under general basis changes so such appearances can be deceiving. To see the f(n) properties, use the preceding CR's of N(n) with Q(n) and P(n), together with the Baker-Campbell-Hausdorff formula to show that more generally for a continuous parameter 'a', exp( +i a N(n) ) Q(n) exp( -i a N(n) ) = + Q(n) cos a + P(n) sin a and exp( +i a N(n) ) P(n) exp( -i a N(n) ) = - Q(n) sin a + P(n) cos a clearly expressing a rotation in the Q-P plane in the operator algebra. Let a = (pi)/2 and then f(n) Q(n) f!(n) = +P(n) f(n) P(n) f!(n) = -Q(n) falls out as a special case. If 'a' varies continuously the above relations describe the behavior of the dynamical evolution (Heisenberg picture) of the SHO in QM, where a is then interpreted as a time parameter. AN ASIDE: It's always bothered me (and apparently a few other people) that CCR (with necessarily infinite representations involving unbounded operators) and CAR {A!, A} := A! A + A A! = I (with only a single 2x2 irreducible representation) should both be the algebraic cornerstones of QM, differ only by a sign, and be apparently unrelated and so wildly dissimilar algebraically. In the commutation relation for the B(n), B!(n), if we take n=2, they satisfy CAR. There is now an ALGEBRAIC BRIDGE: CAR -> CCR in the limit as n becomes unbounded. A technical sense in which B(n) -> B is in the strong operator topology. I have taken to referring to this bridge as Finite Canonical Commutation Relations or FCCR, or when necessary FCCR(n). This bridge does not seem to have any connection to the parastatistics that between CCR and CAR introduced by Greenberg about 20 years ago. Nor does it seem to have any relation to the so called q-CCR which depend on a real parameter q: -1=> q <= +1, where QP + q PQ = I There might be a connection with superalgebras or a generalization thereof: both commutator and anitcommutator can be looked upon as bilinear 2-forms on an underlying algebra. (commutator) (anticommutator) | 0 +1 | | 0 +1 | | -1 0 | | +1 0 | XYZZY MORE PROPERTIES & DETAILS of FCCR: Both Q(n) and P(n) are Hermitean and have real eigenvalues distributed symmetrically about zero. In fact, Q(n) being also real, is symmetric hence diagonalizable by a real orthogonal transformation. (See below.) Its eigenvectors, taking the |k> eigenbasis of N(n) to be "canonical" have real components. The eigenvalues of Q(n) and therefore also of P(n) are the roots of the n-th Hermite polynomial, which as n becomes unbounded become dense in the real number field. The spectrum is nondegenerate for every n. Only when n is odd is zero in the spectrum. An approximation for the eigenvalues can be given: q(n, k) = DELTAq(n) ( (n-1)/2 - k ) where the uniform eigenvalue spacing is approximated fairly well by DELTAq(n) = sqrt( 6/(n+1) ) even for relatively small values, say 5 or 6, of n. Although asymptotically for large n one calculates DELTAq(n) = (pi/(2 sqrt(n)) Operator algebraic differences also occur depending on n being even and odd. Every Hilbert(n) supports an IRREP of SU(2) and of SL(2, C). If n is odd, these are also IRREPS of SO(3) and SO(3, 1) respectively. The eigenvectors of Q(n) and P(n) are all G-null <q| G(n) |q> = 0 <p| G(n) |p> = 0 Though, [G(n), N(n)] = 0 of course, commutators of G(n) with Q(n), P(n) are not zero: [Q(n), G(n)] X= 0 [P(n), G(n)] X= 0 These two commutators close in mutual commutation on an algebra conformal to the SU(2) Lie algebra. Including Q(n), P(n), G(n) in the set to be closed pushes SU(2) -> SU(3), once again then SU(3) - SU(4) and ultimately the operators close on and generate by commutation the full set of nxn Hermitean matrices, i.e. SU(n). This is the cognate property of irreducibility of the Schroedinger Operators proved by v. Neumann. (I'm told there is now an English translation of this paper; I've not seen it.) This implies that any operator of the full nxn matric algebra is representable (not necessarily uniquely) as a complex polynomial in Q(n) and P(n). From the viewpoint of SU(n), G(n) is merely an element of the Cartan subalgebra, and not otherwise distinguished. An interesting step that I've not yet made (takers anyone?) is an investigation of the usefulness here of a second quantization by way of the "quantum su(n)" group. XYZZY? In the large and on a classical level of GR, we expect given the finite speed of light that the seeable and knowable universe is finite, and probably unbounded, i.e. spatially, a 3 dimensional space of roughly constant positive curvature. If FCCR is a legitimate usurper of CCR, n should have a finite, though enormously large value if the universe can be almost completely covered with one patch. How many Planck lengths do we need to measure the radius of the known universe? On the order of 10^62. Is this a valid application of FCCR? Maybe, maybe not. SUPERPOSITIONS OF STATES: The eigenstates |q> and |p> lie on the G-null cone in Hilbert(n). Any *real* linear combination |q> is again real and also lies within the G-null cone. <phi| G(n) |phi> < 0 In a complex space with indefinite quadratic form, the forward cone is continuously connected (by the phases) to the backward cone. XYZZY (charge density?) DIAGONALIZING TRANSFORMATIONS OF Q(n) & P(n): The diagonalizing transformations XI(n) of Q(n) and PI(n) of P(n) do *not* leave G(n) invariant. (Neither do general curvilinear transformations leave the Minkowski metric invariant.) The matrix elements of XI(n) are given by [XI(n)](kj) = d^(-1/2)(j) H( k, q(j) ) where d(k) := SUM(j) H( k, q(j) ) H( k, q(j) ) and H(k, z) is the normalized k-th Hermite polynomial in z q(j) is the j-th root of H( n, z ), where q(0) is maximal and q(j+1) < q(j) for all j=0, 1, ..., n-1. The cognate energy functions are <k|q(j)>. Even for small values of n, one can see the central preaking of <0|q(j)> that is characteristic of the Gaussian nature of the groundstate energy function. For the "highest energy", eigenstate, <n-1|q(j)> the distribution is essentially flat with alternating phases of 0 and pi; the signs alternate. In a limit of unbounded n, if one informally takes the "value at a point" to be an average over a small neighborhood of the point, the function appraoches a zero function. By expressing XI!(n) XI(n) = I(n) in components, one obtains an orthogonality relation for the Hermite polynomials on a discrete set. SUM(i) H(k, q(i)) H(j, q(i)) = delta(kj) objects being defined as above. I suspect that this is a special case of a general theorem valid for a certain proper subclass of orthogonal polynomials, but I've neither seen the more general theorem, nor stated it precisely, much less proved it. SOME MATRIX VALUES: <q(k)| P(n) |q(j)> = 0, k = j = -i exp( i(pi)(k-j) )/( q(k) - q(j) ), k X= j <q(k)| G(n) |q(j)> = kronecker-delta(k, j) - exp( i(pi)(k-j) ) UNCERTAINTIES: The proof of the uncetainty relation goes through in QM textbook fashion. [See e.g. Merzbacher's "Quantum Mechanics"] Interestingly, it has the same point of failure as exists in QM. In slapping a bra on the left and the conjugate ket on the right of CCR, these must not be eigenvectors of either Q or P. In QM, this case fails because these eigenvectors happen to lie outside the common domian of selfadjointness for the operator pair. In FCCR, the proof fails because all these eigenvectors are G-null, making the RHS of FCCR statement vanish. If the uncertainty in Q(n) vanishes, there is still an uncertainty in P(n) which is very approximately 25% of the full spectral diameter of P(n); the percentage is independent of n. Gratuitously said, the uncertainty in P(n) in this case become infinite when n becomes infinite. In QM, we have had to consider certain states to be unphysical, particularly the eigenstates of Q and P. They are not strictly elements of Hilbert space and not strictly normalizable. The Dirac delta-function is not strictly a function. Unlike a general Banach space a Hilbert space is automorphically equivalent to its space of linear functionals. To make mathematical sense of manipulation of these useful eigenstates it becomes necessary to invent the theory of distributions and split the Hilbert space into a smaller space (a space of test functions) and a larger space of distributions. Symbolically, test functions < H < distributions Then a expression (test-f)*(distib) adequately replaces the expression fg, with f and g both in H. In FCCR one must either do something similar, or accept that the vectors |psi> for which <psi| G(n) |psi> = 0 actually have physcial significance. There is a definite problem with the first choice: Since by commuation FCCR(n) generates the full SU(n) algebra defining IRREP, there appears to be no way of consistently eliminating vectors of the Hilbert space and defining an *invariant* and proper subdomain. It would then appear that one must take the G-null states seriously from a physical viewpoint. We have had experience with a structure that instrinsically incorporates similar null-vectors on a cone and that structure is SR. The suggestion is immediate that the G-null vectors of FCCR be associated with "the local light cone". The Q(n) and P(n) eigenvectors should then said to be on the light cone. Since they clearly involve a sense of time they must be understood as representing aspects of *lightlike* *process* rather than that of "state". The eigenvalues of Q(n) may be considered to be either measures of length or measures of time and likewise the eigenvalues of P(n) may be considered to be either measures of momentum or measures of Energy. Linear combinations of G-null vectors are generally not G-null. XYZZY THE TIME OPERATOR: In QM, for the SHO or the creation and annihilation operators, there have been numerous attempts at defining a "time operator"; this should be associated with a phase operator (Hamilton-Jacobi theory) conjugate to the number operator. The problem is that strictly speaking (the unboundedness problem again) the phase operator doesn't exist. An unbounded operator can be recognized by an unbounded spectrum. Clearly the number operator N of CCR is unbounded. When n is finite, however, one can decompose B(n) and B!(n) as the product of an Hermitean operator and a unitary operator thus: B(n) = C(n) N^(1/2)(n) B!(n) = N^(1/2)(n) C!(n) where C(n) is the cyclic operator in the eigenbasis of N(n) such that C(n) |k> = |k-1> where N(n) |k> = k |k> k = 0, 1, 2, ..., (n-1) and the index of the basis set k is understood cyclically, that is |n-k> = |k> Now since C(n) is unitary and idempotent of order n, it has as eigenvalues the complex n-th roots of unity, and has a generator T(n), so it can be written C(n) = exp( i 2(pi)/n t(n) T(n) ) or C(n) = exp( i w(n) T(n) ) where w(n) is a parameter depending on n and having the physical dimension of (1/TIME) and T(n) is an operator with eigenvalues t k where t is some "fundamental time quantum" and k is an integer. Then T(n) is unitarily equivalent to (t N(n)), and there is a a unitary diagonalizing transformation for T(n) U(n) T(n) U!(n) = t N(n) which also happens to be a Fourier transform, that relates N(n) and T(n). U(n) is unitary and idempotent of order 4. Its matrix elements can be given by <k| U(n) |j> = sqrt( 1/n ) exp( i (2pi/n) kj ) The eigenvectors of T(n) can be expressed as *equally* *weighted* superpositions of the N(n) states |k> states, which represent the pointer positions of an optimal or best clock. See [Wigner Rev. Mod. Phys. 29, 255-268 (1957)] for such a construction in the context of QM. Actually having a "time operator" operator, however is like the dog chasing the volkwagen actually catching it. Now what do we do? The position operator Q(n) does not commute with T(n). In fact T(n) doesn't seem to commute with anything. The commutators [N(n), T(n)], [G(n), T(n)] don't have any easy algebraic relation and the first one is definitely not equal to iG(n). The fact that G(n) seems to become an indefinite bilinear form on the finite dimensional Hilbert space that one may have to take seriously as an inner product, appears troublesome. It is true that if n is even, [f^2(n), U^2(n)] = 0 How might it be possible to interpret this algebra relying on what's come and gone before. On G(n) not being the identity operator. In QM one talks about measurements of Q and P at the same time being incompatible, the incompatibility being quantified by CCR and Planck's constant. FCCR says "You can't talk about these measurements being at the same time." So G(n) must have something to say about time measurements. A major change in thinking about NM happened with the introduction of HM. Prior to HM, the "background" of physical events was space. In canonical formalism, the background of physical events becomes the phase space with a metric derived from the Hamiltomian energy function. This loosening of the background concept is enhanced & reinforced by QM where the phase space is replaced by a projective Hilbert space. SR loosens the background concept in a completely different direction leaving the absolute space and absolute time behind for the looser concept of and absolute spacetime. GR further pulls stress, energy, and momentum together into one tensorial object. According to FCCR, it would seem that position, time, momentum and energy are all "Q-incompatible" and one can chose only one to diagonalize. If we take G(n) to be "the metric" by which one computes probabilities, negative probabilities will result. It may be proper to take G(n) as an operator invariant under a group of symmetry transformations, but it is not proper to replace <psi|psi> with <psi|G(n)|psi> and "normalize a probability density" by it. Once can, of course, fix the value of <psi|G(n)|psi> = +1 in imitation of the QM, but then all the |psi> conforming to this constraint lie on a hyperbolic complex manifold embedded in Hilbert(n). This hyperbolic manifold is not closed under linear combinations (neither is the G-null cone) and one is then stuck with relinquishing the superposition property, and possibly finding some other nonlinear rule for the combination of vectors in Hilbert(n). This sounds like an interesting line to investigate, but I haven't gotten around to it. An alternative to holding on to the QM interpretation of the mathematical formalism is to abandon the dominance of probabilities and allow that the fundamental computable numbers of physical interest are *Transition amplitudes* (shades of Feynman path integrals), and not *expectation* *values*. Concomitantly, the fundamental physical concept ceases to be that of "quantum state" and become that of "quantum process". The consideration of process rather than state is already made necessary by T(n) not commuting with anything. If G(n) is construed as a metric on the background which is the local finite dimensional C*-algebra, it is a metric on a space which unifies: P(n) E(n) Q(n) T(n) moreover it is an indefinite metric not unlike the indefinite metric on (x, t) space [contravariant] and on (p, E) [covariant] put together in one space. The conclusion is that FCCR may provide a correct, if somewhat wierd, coalescence of R and Q that is consistant. The group of transformations that preserve G(n) is conjugate equivalent in GL(n, C) to SU(n-1, 1), and of course, has a maximal compact subgroup SU(n-1). When n=2, it is exactly SU(1, 1), the isotropy or "little" subgroup of SL(2, C) which leaves invariant a spacelike vector. THE REAL 'STATE' IS THERMODYNAMIC & IN THE ALGEBRA, NOT THE HILBERT SPACE: A local FCCR algebra pertains only to a patch of the universe; A QT of a local *open* system which presumably is in thermal contact with "the rest of the universe" as a thermodynamic reservoir. This is to say that a local "state" or process is entangled with its other, and that a realistically considered process is necessarily impure. The impure processes should then be represented by density matrices. To decompose such density matrices into a probabilistic distribution (in the classical sense) of pure states is not uniquely defined. The Hilbert space can be absorbed into the algebra in the usual way by the map |psi> -> |psi><psi| to projection operators which will be extremal in the forward cone of density matrices *if* the inner product on the algebra is given by Tr( Y Z ) with Y, Z elements of the algebra. No details on the alternative inner product Tr( Y G(n) Z ) are given here. DYNAMICS WITHOUT DYNAMICS: The "state" does not have a determinate evolution. It evolves statistically by transition amplitudes as Markovian process. It is this process which appears to create time. If one looks at the values |<qk|tj>|^2 numerically, they appear to have maximal values that describe a sinusoidal curve. I have tried to prove this analytically and not been successful. It is not hard to see that these transition probabilities form a doubly stochastic matrix. If one looks at the values of <tk| E(n) |tj> where E(n) = (1/2)( Q^2(n) + P^2(n) ) Oscillator energy = N(n) + (1/2)G(n) taking still as axiomatic that energy is the generator time translations, then for n=2 and n=3 the "clock pointer positions" evolve chaotically, for n > 3 the transitions tk -> t(k-1) and tk -> t(k+1) dominate. As n increases further these adjacent transitions dominate even more, so that as n inceases the n-point oscillator behaves more and more like a smoothly running clock. Remember that unlike QM, the energy eigenstates are *not* stationary, since the energy does not commute with the T(n) operator. There is another aspect of the transition amplitudes for the n-point SHO that is important. The transition amplitude for no transition (expectation values) does not vanish, so there is an intrinsic speed of the clock that can be statistically calculated. A rate of transition is determined also not just the expected rate of the clock. Time would then seem to appear almost magically as a consistantly determined effect, independent of any prior measure except that of a given quantum. An observed speed of the clock can slow if the change of reference frame increases relatively the probability of stasis. Darwinism and the essential "randomizer". XYZZY MULTIPLE CONCEPTS and FORMAL TIME STRUCTURES: XYZZY THE DIMENSION OF ST, QST & ALL THAT: Fixed v. Floating Dimension. Various definitions of "dimension" pre/times Primacy of Lightcone (G-nullity) & Penrose Twistors construction of superalgebra and quantum group XYZZY EXTENDING TO 3-DIMENSIONAL "SPACE" The identity and integrity of a quantum system is recognized by irreducibility of its maximal Abelian operator set acting on its space of states. The rule for combining systems in QM is the direct product. It is also the usual rule for extending spatial dimensions. All infinite dimensional separable Hilbert spaces are essentially equivalent, what distinguishes them in QM is the structure of the set of operators that we set up as the "observables of the system". All finite dimensional Hilbert spaces are not essentially equivalent; they are partitioned into equivalence classes labeled by their topological invariant dimension. The size of a "Qset" measured by the n of an Hilbert(n) is distinguished from the dimension of the Qset that is determined by the algebraic structure of the fundamental operator set associated with the definition of a physical system. It would seem appropriate to seek the extension of spatial dimension in the extension of the operator set rather than resorting to extensions by the direct product operation. The Q(n)-P(n) pair or the B(n)-B!(n) pair already generate by commutation, the full SU(n) algebra (in defining representation) space over the reals, and then that of SL(n, C) by complex extension. If one extends the FCCR(n) set to include other Q-P pairs then the different Q's and the different P' will not commute among themselves, producing and explict model of a noncommutative geometry. Let W(n) := (n-1)I(n) - N(n) (which looks like a number operator turned upside down). Let S(1, n) := (1/2)( B!(n) W^(1/2)(n) + W^(1/2)(n) B(n) ) S(2, n) := (i/2)( B!(n) W^(1/2)(n) - W^(1/2)(n) B(n) ) S(3, n) := (1/2)( W(n) - N(n) ) then [S(a, n), S(b, n)] = i epsilon(a, b, c) S(c, n) where a,b,c = 1,2,3 expresses the CR's for SU(2) algebra. In the basis |q> in which Q(n) is diagonal define an SU(2) IRREP with the same numerical values as the S(a, n) preceding, calling the operators SQ(a, n) so that [Q(n), S(3, n)] = 0 exponentiate the SQ's for the rotation operators R(1,3; n) := exp( i (pi/2) SQ(2, n) ) R(2,3; n) := exp( i (pi/2) SQ(1, n) ) [don't worry about the sign ambiguities when n is even.] and let Q(1, n) := R(1,3; n) Q(n) R!(1,3; n) Q(2, n) := R!(2,3; n) Q(n) R(2,3; n) Q(3, n) := Q(n) which are to be taken as a set 3 noncommuting position operators. The differences between successive eigenvalues of the SQ(a, n) is exactly 1, uniformly (excepting at a cyclic wraparound. The same differences for the Q(a, n) are asymptotically and uniformly DELTAq(n) = sqrt( (pi/2n) ), which is to say that the Q(a, n) are conformal to the SQ(a, n) and that asymptocially one has [Q(a, n), Q(b, n)] =~ i DELTAq(n) epsilon(a, b, c) Q(c, n) as n goes to infinity, [Q(a, n), Q(b, n)] -> 0 Now make a similar construction in the |p> eigenbasis with giving a new IRREP of SU(2): SP(a, n) perform similar rotations on P(n) to define P(a, n) which will by the same reasoning obey [P(a, n), P(b, n)] =~ i DELTAp(n) epsilon(a, b, c) P(c, n) We also happen to know that DELTAp(n) = DELTAq(n), and that as n goes to infinity, [P(a, n), P(b, n)] -> 0 Then define 3-space operator metric G(a,b; n) by [Q(a, n), P(b, n)] = i G(a,b; n) But SIGMA Q^2(a, n) = SIGMA P^2(a, n) = (n^2 - 1)/4 I(n) showing the Q's and P's as operator 3-vectors to be on a "quantum sphere" of asymptotic radius (n/2) Additionally, SQ(a, n) is unitarily equivalent to SP(a, n) and so there exists a V(n) with V(n) V!(n) = I(n) which maps one IRREP into the other and then asymptotically maps V(n) Q(a, n) V!(n) = P(a, n) Is it necessarily a Fourier transform? It is not difficult to show that V(n) = XI!(n) PI(n) = XI!(n) f(n) XI(n) f!(n) and V!(n) = f(n) XI!(n) f!(n) XI(n) V^2(n) = XI!(n) f(n) XI(n) f!(n) XI!(n) f(n) XI(n) f!(n) XYZZY If one attempts to make a simple minded SHO by summing over the E(n) = SIGMA (1/2)( Q^2(a, n) + P^2(a, n) ) for the three values of 'a', the energy becomes completely degenerate (this is the sum of Casimir operators for two IRREPS) with E(n) = 2 (n^2 - 1)/4 = (n^2 - 1)/2 A proper energy for such a system would more likely be some operator quadratic form such as E(n) = SIGMA G(a,b; n) Q(a, n) Q(b, n) + SIGMA G(a,b; n) P(a, n) P(b, n) modulo the usual problem of operator ordering. I have not yet looked into this structurally. Here again the thought pattern is definitely nonrelativistic. There is no guarantee that any quantum behavior has a classical limit, but if it (is supposed to, etc, ...) then the standard prescription of symmetrizing of all possible orderings may be appropriate. This is all just model playing for the purpose of demonstrating that such games can be played with. Since we are finite dimensional spaces there is room for serious numerical playing around. To generate simple position operators Q(a, n), I've simply used what amounts to a property of SU(2) and a tacit assumption of isotropy, i.e. a local symmetry that appears to be not unreasonable. Just the idea of symmetries expressed by the algebra attached to some Lie group is probably a good start for playing. Some metaplaying: In CM the thing to define to characterize a system is the energy or Hamiltonian as function defined on phase space. Why? Because almost all the essential laws of physics are basically statements of the conservation of energy E (or some loosening generalization thereof). In SR particle mechanics, E becomes merely a component of a 4-vector and it is (E^2 - p^2) that is conserved under the group of Lorentz transformations (Poincare actually) and even the conformal group, ..., blah, blah. Relativistic systems with interactions are hard to deal with. Try making really good sense with the angular momentum of the electromagnetic field. SR implies the necessity of fields for the interactions precisely because it denies action at a distance (Hoyle & Narlikar notwithstanding). Hamiltonian formalism which works so elegantly in CM becomes problematic in SR systems: 'Frozen Formalism' [See e.g. A. Mercier, "Analytical and Canonical Formalism in Physics", Dover Pubs.] - A brief but enlightening book. XYZZY In (x^ - c^2 t^2), transforming the x away by a Lorentz boost to the comoving frame. If there is some time operator T(n) gotten by magikal incantation (no energy operator having been defined), then there should be (so say I) an action of a boost transformation (hyperbolic rotation): "inner action" "outer action" A(theta) Q A!(theta) = (cosh theta) Q(n) + (sinh theta) T(n) = Q'(n) A(theta) T A!(theta) = (sinh theta) Q(n) + (cosh theta) T(n) = T'(n) XYZZY GR equates two forms of energy [Einstein tensor] [Stress-energy-momentum tensor] G(mu, nu) = k T(mu, nu) where LHS is the Einstein Tensor expressing "energy" in terms of geometrical curvature and the RHS is the k the gravitational constant times the the stress-energy-momentum tensor expressing energy in terms of matter, fields and whatever is not strictly geometry. To build a cognate of the Riemann curvature tensor, or even the Christoffel symbols of affine connection from a metric field it is necessary to be able to take a "derivative with respect to coordinate". But the coordinates here are operators. I have been thinking about definitions of operations that take the appropriate kind of derivative but am not ready to speak yet. Nevertheless, under the restriction to a space of constant curvature, the Riemann tensor becomes expressible purely algebraically in terms of the metric (equivalent to a metric induced on bivectors). Classically it is written XYZZY A MODEL OF STATISTCALLY DETERMINED DIMENSION: In any Q theory in an algebraic context it is difficult to get away from the notion of some fixed dimension for space time that is basically the standard axiomatized Brouwer-Uhryson dimension. Somehow it always seems to be put in, sometimes covertly, "by hand". Yet there are models that can be built where a dimension (not necessarily integer valued) is determined "thermodynamically" by a statistics of graphs that are formed by a set of verticies that may or not be connected by lines all of the same (interaction) length. The idea starts by defining the dimension at a vertex for such graphs as the number of connecting lines. A fully connected set of (n+1) verticies is an n-simplex; to each of the n+1 verticies n lines are connected. This (classical) model envisions the universe as essentially crystalline in nature and appropriately predicts a very low temperature for the current universe. I have tried to see a connection between FCCR(n) and such graph statistics but have not been successful. Either this is nonsense or I haven't thought long or deeply enough. Definitely a topic for the future. CAVEATS ON APPARENT SIMPLICITY: Replication, combininig, Interaction, Linking are going to be messy. Harmonicity is not a relativistically invariant concept (except on the light cone), this, despite many people writing down what would appear to be relativistic equations for a simple harmonic oscillator. The seas are not necessarily simple because they are "merely" H-O-H. I envision FCCR as being the metaphorical cognate of of a water molecule. Higher structures must arise from the ways in which replications assemble themselves. THE 'IFS' OF FCCR: Either FCCR(2) is the extent of the cosmological initial condition, or the initial condition is specified by some ensemble of them. Then, why not an ensemble of FCCR(k) for all k < n, for some n. XYZZY ON PUTTING LOCAL ALGEBRAS TOGETHER & THE QST CONCEPT: The general QT method of combining systems is direct product. FCCR(k) generates by commutation, the defining IRREP of the Lie algebra SU(k). For fixed k, the direct product of m such IRREPS decomposes by "mj-symbols" into a direct sum of IRREPS of SU(k). The direct product SU(n) X SU(m) is a subalgebra of SU(nm) algebra. XYZZY: for k>2 SU(k) has k-1 "Fundamental Representations" that are *not* unitarily equivalent. SU(2) is distinguished by having only 2-1=1 such fundamental representation. SU(3) has two fundamental representations (usually denoted as 3 and 3*. The SU(k) Lie algebras are the only ones that have the property that the fundamental IRREP direct product [adjoint IRREP] [singleton IRREP] k X k* = (k^2 - 1) + 1 The adjoint IRREP (also called the vector representation) of any Lie algebra is the representation constructed from the structure constants. In QM one sees space emerge from the spectra of position operators. Each eigenvector |q> of Q is associated with a point; never mind that the vector is not strictly in the Hilbert space; any vector of the projective Hilbert space can be represented by a linear combination of the |q>'s. One might think that garden variety QM is already telling that there is already something wrong with "point". It is possible, of course, to interpret the failure of the proof in QM to be nothing more than a mathematical inconvenience; however, failure, even in the very similar mathematical model FCCR, might suggest that one take the failure a bit more seriously in terms of actual physcial significance. Continue this association nonetheless. The smallest set of quantum points contains at least two points (n=2) at a maximal separating distance. As n increases, the Spectral radius of Q(n) increases as sqrt( n ) and the distance btween the eigenvalues of Q(n) decrease as sqrt( 1/n ). More allowed space implies more allowed energy and with higher and higher energies the resolution of points becomes better. If we want to link local algebras together to form a universe, first think in terms of the Hilbert spaces, which should be linked by sharing subspaces: the quantum equivalent of set theoretic intersection. Then the power set or set of all subsets of nonquantal points becomes the complex Clifford algebra of all subspaces of the Hilbert space. The C*-algebra then being the direct product of Hilbert space with its conjugate space, we must take the direct product of the Clifford algebra with its conjugate to express the cognate quantum set theory. An obvious question is one of the statistics of quantum points. Do they have Fermionic or Bosonic properties, or both? Are "quantum points" the ultimate dynamical objects? The pertinant objects here seem rather to be the mixed background sets that constitute the local algebras. Space, Time, Momentum and Energy become tangled up, which is actually proper if one considers the flat spacetime that we see locally to be itself a statistical average of of some underlying quantum dynamics. In this view, the "presubsets" of spacetime correspond to the local C*-algebras: subalgebra -> subset. When n is even half-odd integral spin (fermion) IRREPS of SU(2) are supported, and when n is odd, integral spin (boson) IRREPS of SO(3) are supported. There is no other unitary group that has exactly one IRREP for for every n. All the finite dimensional (nonunitary) IRREPS of SL(2, C) can be obtained as operators of homogeneous polynomials of two complex variables (or essentially putting together two IRREPS of SU(2). It might be correct therefore, to consider a combined (direct product?) pair of FCCR's instread of just one. A MORE GENERAL PASSAGE: LOCAL -> GLOBAL C*-bundles C*-algebraic presheaves XYZZY PROBLEM 1: Untangle the multiplicity of time concepts. PROBLEM 2: The dimension problem. There seem to be (only?) two approaches to an explanation of of why spacetime has on the average (in the semilarge) its manifest 3+1 structure. The first is to explain it from a highly peaked average of underlying statistical events. This is in accord with our picture of the flat Minkowski space representing a local notion of vacuum, and a violence of its attendant quantum fluctuations. The second is to understand dimension as a necessary algebraic constraint. Although, to me personally, the first, statistical approach is more conducive, FCCR would seem to lean in favor of the algebraic constraint. The question, in the this second case would still remain: why the algebraic contraint? Are we not simply putting the phenomenology of classical dimension back in by hand? PROBLEM 3: Develop an local addition on algebras attached Presheaves of finite dimensional C*-algebras. PROBLEM 4: Since the rotated Q(n) operators are asymptotically conformal to an n dimensional IRREP of SU(n) is there a connection here with the machinery of "spin networks" introduced by Roger Penrose? The connection being in the interconnections of local algebras. PROBLEM 5: Define Field operators on such a local QST patch as this. PROBLEM 6: Explain how the "expansion of the universe" comes about. The process should apparently involve an evolution of the sort FCCR(n) -> FCCR(n+1) mirrored in SU(n) -> SU(n+1) REMARKS: Although this may all seem a bit crazy, but I see some virtue in eliminating from physical theory, nasty and unobservable things like infinitessimals and singularities. For large n, behavior approximates QM, and for any finite n, however large, there are no singularities. If sanity is the criterion, consider the sometimes crazy consequences of QM, especially with Copenhagen interpretation appended. For years people have attempted to find some picture of spin by using some analog of classical angular momentum. All such attempts that I've seen are unstatisfactory and somewhat tortured messes leaving one just as geometrically confused as before. Although this enterprise seems reasonable and natural enough at first, perhaps a better geometrical understanding of spin might be had by considering it as an aspect of harmonic oscillator of the type presented here. There is logical virtue in "explaining CCR" and therefore QM in terms of a limite of finite algebra. QM is normally arrived at through the rather perverse (ill defined, and backward) process of "quantization" that is supposed to map a classical system to a quantum system. The logical perversity here is the underlying presumption that somehow the classical picture is primary and that QT is merely an inconvenient afterthought. Without question the shoe is on the wrong foot. Starting with an already quantized everything is logically preferable to forcing classical mechanics on a QT, whatever form that might ultimately take. Presented with the infinitely many degrees of freedom in CM it is quite impossible to say which of all mathematically possible systems indeed exist physicallly. We basically make models up that seem to fit the classical phenomenology. If, on the other hand, one starts with some axiomatic scheme with a finite number of degrees of freedom, the separation, if indeed a separation is necessary, of the physically possible (and therefore compulsory) systems from the impossible systems seems to become tractable as a problem or statement. Any comments, connections, remarks or constructive criticism :-), gratefully accepted and acknowledged. --------------------------------------------------------------------------- A Brief History (M. Jammer) Greeks Primitive Continuum of Space Atomists: Democritus, Leucippus Paradoxes - Zeno The Matter-Space dichotomy mirroring the Mind-Matter dichotomy. The Continuum Formalized Dimension as a toplogical invariant and alternative definitions (Hausdorff) Peano space filling curves. Concept of "Dynamical Dimension" from a statistical graph model. Space and time concepts to be distinguished Phenomenological Operational Formal Mathematics Clifford-Riemann-Einstein Program of geometrization. Classical extension of "the background" space -> phase space Hamiltonian Mechanics. SR extension of the absolute space & absolute time to a looser "absolute spacetime". SR is not about "relativism" but about absolutism and absolute invariants under a group of transformations, and has therefore a most unfortunate monicker with which we are stuck. Adding SR + Hamiltonian -> Frozen formalism Space Time Momentum Energy Paradox of time: we use it to measure it. What's going on here? The paradox of space measured by rulers and clocks. The assumption (and it should be a valid one) must be made that local rulers and clocks of some sort exist. Periodic local structures. Rosenfeld and the point concept in QEMT The formal correction of QM to agree with SR and then with GR is a disaster. The parallel of Einstein correcting CM to agree with EMT. Which is wrong? A physical Dubiousness: SR in GR is in tangent and cotangent spaces "at a point!", not actually "in" the spacetime manifold. Under a condition of high curvature at P, the tangent space is not a very good approximation. So what does it mean to say "GR approximates SR 'locally'"? The only really good physical 'locally' is at a point - but Rosenfeld .... The ambiguity of quantization as a general map. The mutual contradictions of R and Q: Q is an operator, t is a parameter R indicates an absolute determinism, Q on the other hand implies an indeterminism of both past and future: by the "time" I hear you say "now", "now" is already past and our illusory notion of time doesn't really exist. When does the indeterminism become classically determinate? (The "collapse of the wave function" problem.) Although in the context of QM one can say many experimentally sensible things, one can also say many crazy things that are not even really crazy enough. The artificial and unnecessarily inserted observer implicated in most texts on QM is clear nonsense. Einstein's question to Bohr: "Do you really believe that if I don't look at the sun it doesn't exist?" Then there's all this psychophysical parallelism (v. Neumann) stuff. There has to be a middle ground which means one cannot correct R as formalized to agree with Q, nor the other way around. Both the Q principle and the R principle must currently be formalized badly, or too specifically formalized. FCCR loosens this specificity yet continues the results and consequences of CCR that are necessary for essential physical correctness of QT as presently understood. It is clear (to me at least) that the entire *formal* business as currently set up for both R and Q is patent nonsense, and that something truly insane is necessary to make things sane. Probably a mathematics that can accomodate both a local C*-algebraic structure as well as a manifold-like structure. The two immediately obvious candidates are manifolds with complex or almost complex structures, and C*-bundles where the fibres are C*-algebras and the cross sections of the bundle have to do with spacetime structure. One can think formally of QM and QFT in terms of C*-bundles where the base space is a classical (Newtonian or Minkowskian) spacetime and the fibres are C*-algebras. Unfortunately, both of these nifty if rather sophisticated structures bring one back immediately to the problem of the single point. The linear complex spaces (tangent spaces) of the manifold are not in the manifold and locality once again is formally only a point. With a C*-bundle one is really back with a picture of a "standard" QFT: the objects of interest are operator valued functions defined on some *continuous* space. No one has ever seen nor measured directly or indirectly an infinity or an infinitesimal; it seems most unlikely that anyone ever will. On a fundamental level, spacetime must be discrete but clearly not in a classical but quantum sense. A proper foundation of physics must begin formally with finite algebra (it looks like Lie algebras) and counting. Darwinism and the randomizer. Natura salutus facit sempre - the source of the randomizing QM = CM + CCR Outline of the messiness of CCR No time operator. Hilbert Folklore of QM. Theorems of Wielandt & ? Finite dimensions are out. Since in their proper regimes both QM and most particularly QED have amazing success, more than one could hope for from such formal nonsense, try to take the essence of QM call it QT and the essence of the "Relativity Principle", the essence of ST and form a logically consistent theory. There, that ought to about do it. [And now I sit down :-)] FCCR

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