One rather general way of looking at the problem of quantum gravity (QG), which has eluded theoretical physicists now for over 80 years, is to consider it as the problem of creating a mathematical model that embodies simultaneously the *fundamental* concepts of general relativity (GR) and the fundamental concepts of quantum theory (QT).

The unification of these two essential pillars of modern physics has proved to be extremely vexing in that all the simple attempts at *formal* unification result in conceptual quagmires that seem not to be interpretable in any reasonble way.

It is possible to be suspicious of either, or both of the components that are to be united, and then to suspect that perhaps the formalisms already overburden the concepts by structural assumptions that are conceptually unnecessary.

The first subproblem then is to decide just what the fundamental concepts of these mathematical models are in relation to the real world.

This may sound like a simple question to which the answers are already known, in the sense that the mathematical and interpretational aspects of QM and GR have already been axiomatized. That context, however, begs the question by avoiding it. For example, in [Bunge 1967], the formalisms are each separately made primary, and simply put on a mathematically logical basis; the conceptual basis of the overall physics is avoided.

One thinks immediately of the extensive and sophisticated mathematical formalism in which GR and current notions of QT are expressed, and that these are the fundamental concepts. The expression of QT is currently restricted to "methods of quantization", of which there many besides the ever popular canonical quantization that begins with the Hamiltonian formulation of classical mechanics. Such a quantization method starts with a mathematical formulation of the kinematics and dynamics of a classical system and then applies quantization rules of transformation to express the quantized version of the kinematics and dynamics of the same system. I would suggest here that this may be shortsighted and too rigid a point of view. I would rather take the viewpoint that these formalisms are specific instantiations of underlying metaprinciples of GR and QT, and that the standard instantiations are simply not appropriate, nor amenable to a consistant combination of the two, metaprinciples; the key, again, being that the formalism overburdens the concepts by making unnecessary, and actually wrong, assumptions.

This problem of quantum gravity might be called the primal Gordian knot of current theoretical physics, one that has resisted the theoretical analog of "percussive engineering" for over eighty years.

We have two highly complicated physical theories each extremely succesful in their regimes of discourse. Yet, in subtle and not so subtle ways they seem to contradict one another. The existing mathematical formalisms can, with even more complication and sophistication, be slammed together; the result is an unintepretable mess. The full word on recent techniques of quantizing the gravitational field using quantum electrodynamics as a model is not yet in. However clever and sophisticated the attempts are, personally, I do not believe they will solve the problem, though something can often be learned even in failure. That is the exciting life of any scientist: most often, they are wrong, as the following may also turn out to be.

In appoaching the QG problem, with the understanding that no current combination of relativistic and quantum formalisms solve the problem, there are three obvious areas of consideration in an attempt to understand the basic nature of the problem itself:

I allow that neither QT nor GR are exactly correct formalistically, but that they may be adjusted to permit their consistent combining; but the very first thing to do is to examine the problem areas paying particular attention to philosophical and interpretational aspects of theory.

Prior General Considerations On The Nature of Relativistic (R) and Quantum (Q) Theories:

As they are customarily considered and formulated mathematically, the metaprinciples of quantum theory and relativistic theory are mutually inconsistent.

The current orthodox QG scene consists of two streams of thought: First, attempt to "quantize the gravitational field", and second, start at some supposed quantum level and construct a theory that gives rise to massless spin-2 particles that would then take the role of the putative graviton; the latter stream being, of course, the string theory approach.

In the first stream, it eventually becomes clear that the very coordinates themselves must, as a matter of consistency, be quantized; nobody seems to know what to do with this, or how to do such a quantization correctly - or, there's a perfectly good theory someplace, and I know nothing about it. Though the coordinates represent points of a putative spacetime manifold, they are not the points. There is good reason to believe that the points of the manifold as events cannot be specified with arbitrary precision and that they are intrinsically limited by the existence of a physically real Planck regime which conceptually resists any continuum model which declares to exist what, in fact, does not exist. There is also good reason to believe that our geometrical and topological pictures of unstructured points, lines, etc. are actually in disagreement with known mathematics.
Classical Geometry & Physics Redux

In the string theory, the second stream, a background problem exists and shows that a string theory can be built only when an a priori classical spacetime is given.

Success in either the first or second stream cannot lead to any truly fundamental physical theory, though they may, in some sense, be said to work, provide enlightenment, or approximation techniques.

Considerations, More Specifically:

Here are some of my considerations, a bit more explicitly, taking the above streams of thought into account:

Putting all of this together, one comes to the seemingly lame cul du sac: there are strongly correct aspects of what I will loosely call orthodox Q and R principles; it is also apparent that these principles are seriously defective for any fundamental physics, each as they stand in their current formulations, as well as in any attempted concurrent validity.

Since it seems to me most certain that Q principles are necessarily far more dominant in fundamental theory than R principles, it may turn out that R theory, which is usually constructed on the level of continuum models, describe an emergent symmetry that is not present on the Planck level.

A serious axiomatic problem that exists in any R theory is the fundamental assumption of the existence of an absolute spacetime (ST), and in particular, the existence of the temporal extension that is still connected to the Newtonian time that flows not only uniformly, but coherently and synchronously at every point of space. Newton himself was bothered by having to make this assumption; in retrospect, at that time, he had little choice.

The advances of QM and R theories have done little but propagate this fairly mystical idea of Newtonian time in physical theories. In a relativistic spacetime, there is no motion. The idea of thinking in terms of dynamics of a relativistic theory is artificial at best, and misleading at worst. We we call these theories "relativistic", "absolute" would be more accurate and certainly less misleading. Relativity theory is not about things being relative, it is about different absolutisms, of a four dimensional space and group theoretic (geometrical) invariants constructed from everything that it contains.

One can see consequences of this relativistic fiction in any attempt to create the Hamiltonian formulation of Electromagnetic Theory (EMT); this results in a "frozen formalism" that can be not quite be avoided by inventing yet another ad hoc dynamical parameter. [Mercier 1959].

Either one enters an infinite regress of temporal concepts, or finally determines a necessarily Q origin for an apparent, local, Newtonian time that serves for the non R dynamical descriptions of systems in Q and Newtonian regimes. QM qua QM is not about to serve up such an explanation.

One can see consequences yet again in the "No Interaction Theorems" in Special Relativity due to Mukunda and Sudarshan [Sudarshan 1974].

Modern physics has not yet addressed this fundamental mystery of what time actually is, how we get away with the Newtonian mystery, what space actually is, and why, locally and at least down to about a hadron radius we get away with a Euclidean model. We'll look at these mysteries a bit later on.

Despite what seem to be essential failures in the two approaches criticized above in meeting the problems of quantum gravity, there is a third approach that seems to avoid the problem that an a priori spacetime background needs to be given. That approach is through loop quantum gravity which may be allied with the present approach consisting of a fundamental generalization of the basic kinematical constraint of quantum theory.

While loop quantum gravity is still an approach that is a quantization procedure with a goal of constructing a projective Hilbert space of states, its two main features are that it does avoid the assumption of a "background metric", and concommitantly that field on which attention is focused is an affine connection. In GR, while the metric is associated with a gravitational potential, the affine connection whose components are the Riemann-Christoffel symbols, are associated with gravitational forces.

The differences between a "standard" classical field, prototypically the EM field, that occur with the gravitational field are that the classical field is being defined on some given metric spacetime with a gauge group (as a consequence of the masslessness of the field) that is not directly concerned with geometry, whereas the gravitiational field is the metric field and the gauge group is the infinitely parametered group of coordinate transformations, the Bondi-Metzner-Sachs group of diffeomorphisms. In its abstraction to the theory of differential forms [Flanders 1963] , classical ElectroMagnetic Theory (EMT) is seen to be a purely topological theory (not requiring a metric), so much so that it can be reformulated without much difficulty in terms of homological algebra (actually a dual cohomological algebra), without even continuity, much less differentiability requirements.

In attempts to quantize the GR, these different aspects get in the way, regarding both formal mathematical manipulation and regarding the concepts of space, time and spacetime themselves.

The notion of QT in its most general form is not clear cut, even though we possess a fair number of its aspects by an examination of the methods of quantization. There is a QT expressed by the quantization of a known classical system, or standard nonrelativistic quantum mechanics (QM), [Merzbacher 1961], [Messiah 1965]. Even there, however, as is well known, there is the ambiguity in the quantizing map from classical to quantum formalism in the ordering of products of operators that can only sometimes be resolved by symmetrizing the product.

There is QT expressed by the quantization of the electromagnetic field, quantum electromagnetics QEM, often considered as the quantization of an infinite number of classical oscillators. Add to QEM an interaction with the electron as described by the Dirac equation and one has the still current crown jewel of theoretical physics, quantum electrodynamics (QED).

One cannot think of QT simply as some process of quantizing a classical system. Logically, QT comes first; the classical system is some kind of an average and/or most probable result. From this view, the process of quantization is a bit primitive and silly. Quantization is not a proper QT, but, it surely gives some good clues. Mathematical physicists are now busy creating quantum groups, quantum polynomials, quantum geometry, noncommutative geometry, and supersymmetrizing every mathematical thing in sight. Supersymmetry and quantum groups, however, are other stories for another time, though their connection with a genuine QT is probably not nil.

One should logically start with an unknown metatheory that is QT, and from this derive its representations or instantiations. Thus, "quantization of a classical system" is doing things backwards.

That there should be possible such a metatheory as QT is suggested by an incompleteness or gap in that there exist quantum systems with no classical analogs. The obvious examples are particles with spin, particularly those of half-odd integral spin. Models of spin-1/2 as an actual particle spinning in space, appear periodically and prodigeously in the physics literature; none ultimately seem satisfactory or convincing, although there classical suggestions to spin-1/2 in Euler angles of classical mechanics, the "belt trick" that illustrates a topological property of E³, that shows itself even with regard to macroscopic objects like belts that are embedded in it. The notion of spin, however, is not just some special property of E³. As Lie group SU(2) happens to be the universal covering group of the doubly connected Lie group SO(3), every SO(n) for n > 2 is doubly connected and has a similar SPIN(n) group as its universal covering group.

More generally and specifically, there is a spin covering group for any pseudoorthogonal Lie group SO(p, q). The spin covering groups are gotten at through Clifford algebras, of which the Pauli matrices and the Dirac matrices are special cases. So the existence of spin can be had in any Eⁿ, and then "locally at a point" in any manifold of n dimensions.

It is sometimes said that spin is a quantum phenomenon, others point to the derivation of the Dirac equation to say that it is a relativistic phenomenon. Spin, once seen as arising simply from the factorization of quadratic forms is neither inherently Q or R in nature; it can be either, and spin can be derived from the nonrelativistic Schrödinger equation, as it was by Feshbach and Villars simply by factorising the quadratic Laplacian operator that appears there. The mystery reduces to why it is that quadratic forms permeate physical theory.

The only available remark I have is that quadratic forms happen to be mathematical tools with which to express magnitudes associated with complicated mathematical objects. Symmetry groups are often defined as being those that leave such magnitudes invariant. In a sense we can blame the central appearance of quadratic forms in mathematics on the old theorem on right triangles attributed to Pythagoras.

An significantly additional insight is provided, however, by prof. James G. Gilson of Queen Mary College ( http://www.maths.qmul.ac.uk/~jgg/) in his work on stochastic quantum theory.

Spin seems now to have origin in subtle geometric properties of E³; and yet, that very subtlety makes the explanation not very satisfying. What would be satisfying? Probably a geometric rather than algebraic understanding, a picture that shows "what is going on". Surely, it is this kind of satisfaction that is being sought in the many models that try to envision spin-1/2 as a spinning particle. In a very real way, to the human mind, to understand is to geometrize. We do this, knowing full well that there are and must be limits to validity of such an approach to understanding.

If there is not a clear picture of geometry at the basis of geometry of our picture of macroscopic spacetime, assuming there is still a mathematical model, it will be undoubtedly algebraic.

An important conceptual crux of the problem of inventing QG is that the existing theories imply that the very fabric of spacetime, at least as defined by any system of coordinatizing functions, must be Q in their nature.

From a different point of view, one can see that the general covariance which requires a form invariance under coordinate substitution has a pattern much like that of Yang-Mills gauge fields, and involves the introduction of a gauge field that is the massless gravitational field obeying the Einstein equation, again implying the necessity of coordinate quantization since the gravitational field is the gauge field of general covariance. [Kaempffer 1965] gives a concise, and I think clear, exposition on this point.

We know an essential aspect of quantum systems is that there is dispersion: as a physically realizable QM state propagates in time, what will be measured at some future time will be less determinate that what one measures at the point of state preparation. Put another way, the wave function speads in time.

The exceptions to this spreading is are "stationary" energy eigenstates. The prediction of the future as well as retrodiction of the past is, however, not generally determinate; it turns out that this rule must be conceptually applied to the fabric of spacetime, yet GR and SR starts with the assumption of a given spacetime manifold meaning that the entire history of the universe is already determined at the moment of its first existence. Either the fundamental mathematical assumptions of both SR and GR are manifest nonsense, or the standard interpretations of the fundamental mathematical formalism is nonsense. In either case, there is nonsense in the air of theory, and it would be a good idea instead to be speaking sensibly.

Concerning the incompleteness of quantum mechanics (QM), a particular aspect of QT, which is for one particle and does not contain the principles of GR, or SR, there are at least two schools of thought: the Copenhagen school (after Niels Bohr) believes that QM is complete and absolutely true, the other more skeptical school which had Einstein in its camp believes that QM is essentially incomplete.

I suppose it is worth remarking, but perhaps gratuitous, that a question of physical incompleteness is distinct from sensibility, and also distinct from mathematical incompleteness. Most physically theories are necessarily physically incomplete simply because all of physical reality has not yet been completely, absolutely and successfully modelled; to assume that such a thing is even possible requires a bit of arrogance. Interestingly, Lenin said categorically that it was not.

Logically, and theoretically, however, things are a bit more subtle when it comes to QT and Relativistic Theory (RT), simply because these are misnomers. They are not theories, but metatheories which govern the construction of theories. While it is not too difficult to understand when a physical theory is incomplete, understanding the physical incompleteness of a physical metatheory is not so easy exactly because the metatheory is another level removed from any physical modelling and from physical measurements.

However, the way in which an incompleteness is expressed or thought of varies. Einstein once had the idea that underlying QM was a theory of "hidden variables" behaving in a deterministic way but whose statistical behavior gave rise to the indeterminism of QM that is expressed in the Uncertainty Principle. Others see an incompleteness in terms of the two processes of QM, 1. the evolution of a quantum state by the partial differential equation called the Schrödinger equation and 2. the discontinuous process of measurement. QM does not provide a connection between them, and does not tell us what happens during a measurement process.

In this sense it may be also called incomplete. Various modifications of the Schrödinger equation have been investigated that are nonlinear in an attempt to obtain such a picture; no satisfactory alternative to the Schrödinger equation has emerged from this. Johann v. Neumann [Neumann 1932], went so far as proving a theorem that said that hidden variables were impossible. That theorem, though interesting, is generally considered to be discredited and off the mark, especially since David Bohm has constructed a type of hidden variable theory that agrees with all the predictions of QM, while being a hidden variable theory. This theory has been known and around for many years. More attention should probably be paid to it, as John Bell [Bell 1975], has suggested.

While Einstein expected that a perfectly good classical statistics of these hidden variables existed to explain the quantum statistics, Bohm's theory has an underlying classical behavior that is guided by an essentially quantum statistics of a "pilot wave". The peculiarities of the quantum probability theory involving a complex structure are what seem unavoidable. Cf. [Mackey 1968], at 3.8, "Why The Hilbert Space Is Complex".

The notion of GR in its most general form is not clear cut. The second rank symmetric gravimetric tensor field that generalizes the gravitational potential of Newtonian gravity theory can be seen, modulo locally operative coordinate transformations, as determining 1) a local lightcone structure and 2) a conformal factor.

Although there are other gravitational theories that cannot be ruled out besides that of Einstein, e.g., Brans-Dicke, here I will consider only the Einstein theory with cosmological constant as the classical theory of gravitation to which a concept of quantum gravity should relate. The cosmological constant in even this classical theory represents a "vacuum energy" or zero point energy which will be even more important in a quantum context. That a vacuum energy appears in the classical concept despite the masslessness of the graviton is a result of the specifics of nonlinearity of the Einstein equations.

What are the essential conceptual bases of GR?
This is an easier question simply because GR is not a quantized theory. The essence, irrespective of the specific gravitational equations of Einstein, is the principle of General Covariance which is not so much a statement of theory as it is of metatheory. It says how the laws of physics should be cast as geometrical objects of a curved spacetime manifold. It says that real measureable physical objects are those that transform as tensors under the infinite dimensional pseudogroup of general coordinate transformations (the Bondi-Metzner-Sachs group) of the manifold and that the laws of physics should be form invariant (expressed tensorially) under that pseudogroup. The spacetime manifold should, in this case, possess, at very least, the smoothness of a differentiable or C¹ manifold, where C⁰ is merely continuous. With Einstein's equations, it should be a C² manifold, with the added technicality of paracompactness. [Hawking 1973]

The notion of general covariance appears to be just a little simpler than it actually is. While the equations of any physical theory may have a very large group of symmetries of their form invariance, the solutions to these equations may be constrained in such a way that the group preserving the equations' form is too large to be allowed by a given solution. E.g., the famous spheristatic Schwarzschild solution of the Einstein equations, with vanishing cosmological constant will not allow general all general coordinate transformations without breaking the topological structure of the underlying manifold. It was exactly by introducing such a topology breaking coordinate transformation that David Finkelstein interpreted the singular surface at r = 2m of the Schwarzschild solution as a semipermeable membrane, and in so doing also provided the basis for wormhole structures in GR, for which the writers for Star Trek should be eternally grateful.

If you insist that allowed coordinate systems on a manifold be singularity free, i.e., the Jacobian of the transformation changes be bounded and nonzero everywhere, where certain global topologies admit certain kinds of coordinate systems, others will not. The Euclidean plane will admit global rectangular coordinate systems, while a sphere will not. A sphere will, however, admit two partitioning patches where a non singular coordinate system can be defined on each. Try wrapping the plane onto a sphere and see how a singularity must develop.

Interestingly, the torus, T² admits a nonsingular cartesian (i.e., rectilinear) coordinate system, as do surfaces with more punctures: for two punctures, picture two doughnuts surgically connected by a cylinder. Iterate the obvious extensions, and note the posibilities of ring structures and intersecting ring structures resulting. E.g., replace the verticies of a tetrahedron with tori, and its 6 lines with connecting tubes.

We know that, ultimately, general covariance cannot remain on the planck level and so carrying the covariance principle or an abstraction thereof to the Planck level is not trivial. If there is a suitable abstraction, then it must somehow lead back to general coordinate invariance in a limit where the Planck length is taken to zero.

Is general covariance as gauge transformation helpful? Is cone plus scaling factor useful? Starting with SR as prior picture (locality) makes more sense. But can Lorentz invariance of a fundamental law be maintained at the Planck level? Lorentz structure, or rather its algebraic abstraction to its Lie algebra of generators turns out to be an almost spanning structure of the very simplest of general quantum structures. In plexus of such simplest algebraic structures, it would be rather impossible to find something other than the structure that is there.

The Hilbert space associated with such a structure is of 2 complex dimensions, i.e., 4 real dimensions. Suppose one had such a plexus of minimally energetic little quantized volumes of space - and necessarily an algebraic Q time that provides for the expression of an energetically induced "transience". On average, over the plexus or substantial pieces of it, an overall structure that exhibits a local Lorentz or SL(2, C), should be exactly what is to be expected. From this viewpoint, the essential structure of a local spacetime as a Minkowski space is a result of the being of the foregoing very cool plexus of Q existence. A Minkowski space is the, as it has long been interpreted from the viewpoint of the Einstein equations, a vacuum (lowest energy state) of quantum existence. This interpretation is a bit more structured and specific, however, since the the vaccum as lowest energy state in the Einstein equations is merely a trivial solution to the equations, while as the lowest energy state of a plexus of Hilb(2), there is an intrinsic energy, together with possible energy fluctuations, that has as analog the zero point energy of an harmonic oscillator in QM. One avoids the infinities of QFT by understanding both the finiteness of the local Hilb(2) structure and the finiteness of its replications that fill what would be called a "vacuum universe", which, of course, is merely a possible state of a universe.

The idea of the Cauchy Problem

What are the fundamental metaprinciples?
Heisenberg, according some folklore (it may be documentable) understood the uncertainty principle from data and developed his matrix mechanics as an abstract system that would yield it. It would appear to me that in, truth, the fundamental principle of QT is the Uncertainty Principle. A mathematical formalism that yields the an uncertainty principle seems, necessarily, to involve a set of noncommutative operators. Such objects appear in noncommutative rings and algebras. Structures of these types which, in a sense, measure the noncommutativity of the operators (a point of physical interest) are Lie rings and Lie algebras.

That a set of quantum operators can be found corresponding to the generalized coordinates of a classical system may be true, but the converse is clearly not true. This is to say that, logically, quantization methods form a proper subset of instantiations of QT. One way of expanding this proper subset so that it is no longer proper is to drop unnecessary assumptions. One assumption that cannot be dropped is the uncertainty principle since it is almost the funamental essence of quantum theory. Let's see what can be dropped and probably should.

Can the Cauchy problem be done without?
In classical Hamiltonian physics, a particle state is given by (q, p) its generalized coordinate q together with its generalized momentum. The solution of the Cauchy problem is the time evolution of this state. The generator of the time evolution is the energy or Hamiltonian function of the (q, p). In quantum physics the state becomes either an element of a projective Hilbert space or a density matrix while (q, p) becomes a set of operators acting on the Hilbert space. Again a Hamiltonian (operator this time) is the generator of time evolution of the state, and the evolution is just as determinate as the classical state. (This is the Schrödinger picture.) The difference is what is being evolved. In classical physics the values (q, p) can, in principle, be precisely specified; in quantum physics they may not. In classical physics the readability or observation of the values of (q, p) is assumed selfevident; in quantum physics the connection observations and formalism is through expectation values of observables that are represented by operators. These expectation values are the quantum mechanical connections with classical concepts. These connections are what give quantum physics its interpretational meaning. The temporal evolution of the quantum state implies a temporal evolution of expectation values. So, here is the point of this paragraph: If the Cauchy problem is not valid in its usual sense in QM, we will not have temporal evolution of expectation values, which is to say, the classical connection is lost and with it interpretational meaning.

Suppose that an Hermitean "time operator" T exists which does not correspond to any specific macrotime parameter but is associated with some property of the system which makes it into a clock. No reasonably defined time operator can exist with the confines of standard QM, so obviously some adjustments are going to have to be made.

Suppose further that the eigenvectors (I've eschewed "eigenstate" deliberately.) of an Hermitean energy operator H are not stationary: T and H do not commute, but that nevertheless one retains the concept that "Energy is generator of time translations", in the same way that for noncommuting Q and P operators, "momentum is the generator of spatial translations." Dually, of course, the coordinate operator Q is the generator of translations in momentum space.

With regard to relativistic quantum structures derived from combinatorics, see Relativitic Feynman Path Integrals & Random Flight

For a genuine exponential limit to exist that will agree with standard QM, it will be necessary to take n to infinity also in such a way that s(n) becomes a continuous variable, or at least that it have a bounded, nonzero limit. This is necessary to avoid contradiction with the very general theorems of [Wielandt 1949], [Wintner 1947], and Olga Taussky [Cooke 1950], which say that the CCR are not representable by bounded operators in any normed algebra over the real or complex numbers. See, e.g., the discussion at the end of [Appendix J].

If we take n to be very large and consider a large number m < n of transitions, the propagator will behave like the exponential propagator of QM closely enough to restore the predictions of QM at the Fermi regime of about 10^(-13) cm., which is roughly the radius of a Hadron. This would require approximately that n = 10²⁰, a rather large number for a vector space dimension indicating that on a hadronic level, a continuum approximation will, for many purposes, be good.

Now, let us interpret the m-click propagator structure by its parts:

First notice that the propagator, call it V, is an operator valued function of H(n), the supposed energy operator that motivates a translation in "time" (as so defined), and so is indirectly a function of n, as well as m, so write

In a Markov process, the probability matrices multiply to determine the final state. Here, the probability amplitude matricies multiply, and in either case, one does not have a final state, but a set of probabilities, one for any possible final state, given a possible initial one. Evolution of the quantum state itself is now stochastic. It turns out that to normalize for a proper probability distribution, one need only define the probabilities by

As the m-click propagator V(m) acts on Hilb(n) also acts by the dual map as V^†(m) on Hilb(n), in a Schrödinger picture, one may also then, in a Heisenberg picture, express propagation as an action on the C^*-algebra with symbolic structure Hilb^†(n) X Hilb(n), so for any operator A representing an "observable" property of the system, propagation is expressed as

Since for large values of n and m, which is to say with great spatial extension and for long waiting times measured in absolute Planck clicks, V(m) is well approximated by the standard unitary time propagator exp( -i H/ħ s ), for these same conditions, the standard expression in QM of propagation also appears, for which one can use the Baker Campbell Hausdorff formula. This shows that the for the conditions of large n and m, one may approximate in the propagator,

Take a brief excursion from QM to GR and then back to QM:
In attempts to quantize the gravitational field as was done with the EM field, or simply noting the existence and meaning of the Planck units for length and time, it becomes clear that in quantizing gravity it will be necessary for the basic notions of space and time embodied in the coordinate functions also to be quantized in some way. At the Planck level, the continuum model of space and time structure breaks down. Once consequence of this is that also at this level the distinction between these two classical concepts breaks down. If a mathematical model of space and time exists and we have not reached ultimate area of impossibility, it should reflect, at least, operators that correspond to these "degrees of freedom". Such operators may be said to correspond to the classical degrees of freedom, but of course, they will have nonclassical properties that involve noncommutativity. The question is how should this noncommutativity be expressed mathematically. This depends on how one expects the associated uncertainties to behave.

The above is a natural consideration, coming directly from quantum mechanical notions, of local space, time and spacetime with intrinsic uncertainty regarding points of the space that need not go beyond the context of representations of Lie algebras.

? Consider the relativistic invariant

Our notion of time as a statistical parameter would certainly be in accordance with the observed connection between the arrow of time and entropy (as first, I believe put forth by I. Prigogine), and the second law of thermodynamics: with increasing time the entropy of a closed system does not decrease, and rather tends to increase. So the story goes that the universe, being by definition a closed system, has its time statistically determined by the increase in entropy.

I would like to see or do an analysis of "dynamics" of a general spacelike hypercell space (with thickness measured by a local "i=sqrt(-1)" representing the logically necessary local quantum time extension expectation value, together with a global definition of the entropy of the hypercell space that determines what the hypercell surface does next. The locality of i suggests that the hypercell space may be expressed as having an "almost complex extension", and is not restricted to being a "complex extension" that is global. Again, one should include or expect as a theorem that the probability distribution of s will be highly peaked. As a question, this is not nearly enough well posed; it is merely an idea of a mathematical model that needs to be worked out.

On a fundamental level, time as we understand it experientially and within the formalism of current physics, does not exist. I am not talking here about a concept of psychological time, which has very much to do with brain's capacity for memory. This concept is very difficult to get out of; dealing with time in SR is a good start. The future based on a time of a fundamental level is indeterminate and does not exist as some spacelike slice in an assumed a priori given spacetime continuum; the past, is equally indeterminate and also, in the same sense, does not exist. Time travel, is then impossible, except possibly as a process that involves some kind of quantum tunneling, that is statistical and short ranged.

The idea of time travel into the future in sense of H. G. Wells' "Time Machine" fantasy is just plain silly. The future is the future precisely because it hasn't yet happened, and this as true for QM wave functions as it is for classical states; moreover, it is not predictable in any determinate sense except in a statistical manner. What we do expect however is that in most important circumstances a classical future will be predicted with a highly peaked statistical distribution. That can be expected simply on the basis that for certain regimes with which we are familiar the predictive powers of classical physics are not complete nonsense.

In a very real sense, there are many physical senses of time, each of which are connected regimes of spatial magnitude: the time by which the age of the universe is measured is not a Newtonian time. This is also rather distinct from a "time coordinate" in GR. The time coordinate of SR is not the "uniformly progressing" time of Newton. These are funny types of formal temporal constructions that are not things that one can measure with any sort of directness. For directness, ultimately, we always fall back on clocks of various sizes which are defined by different things. We may use a fiduciary ammonia clock, or we may use the cycle of the earth in its orbit about the sun. We may use the cycles of the moon, or something completely different. In any case, the clock is physically bounded, cyclic, and we must in some way be able to read it.

The indeterminateness of QT expressed in its essential dispersion is true for the very fabric of spacetime. Only a "local now" can be said to exist on a fundamental level. Therefore, consider a quantum "space" (the counterpart of a specific spacelike hypersurface segment in GR) which is a nexus of quantum spacetime cells each of which has within it a structure of a local quantum time which cannot be conceptually or even ontolgically divorced from the local quantum space. One can no more consider arbitrarily contracting points of space to a point, than one consider squeezing time to a point, and conversely. So this quantum spacelike hypersuface made of nexi must have a local thickness in local quantum times and in local quantum space. These quantum cells evolve according to their own quantum time, but the cells do not all evolve with complete independence, but are influenced by those to which they are "connected", suggesting that the concept of local evolution itself is not determinate. The question of how the entities of this grand nexus relate is key.

A naturally intuitive idea is that interaction is by direct contact or more generally by proximity. This is, of course, no grand intuition as it turns out that the very concepts of interaction and proximity are not independent so that the idea of their connection is at least partially if not completely tautologous. An exception is that "spooky" action at a distance encountered in entangled states of QM such as those of the EPR configuration.

The quantum statistical behavior of these nexial entities then determines the evolution of the quantum "spacelike" hypersurface of local "nows" and in doing so draws out a definition of the macroscopic parameters that we call time and space. There is no given a priori spacetime manifold, and so a basic premise underlying GR is inoperative on a fundamental level in a fairly specific way. Space and time as we would normally think of them arise statistically from quantum extensions that are represented by elements of local algebras. The alternative, of accepting this assumption of a dynamically emergent, apparent spacetime continuum, is to accept an absolute determinism that is physically real: the past and the future are already given, exist, and have ontological status. Everything down to every particle decay, creation and interaction is completely determined. That would not be consistent with the necessary aspects of quantum theory. Given the essential role of randomizers of some sort in so many regimes of existence, I am unwilling to accept the absolute determinism of GR, and with it, the implied denial of any component of free will.

It seems to me that then one must, in a fundamental physics, ultimately abandon the existence of even a macroscopic given spacetime continuum and the diffeomorphism group of the general covariance associated with it. This is to say that a "correct" thing to do does not entail a canonical quantization of the Einstein equations, however mathematically instructive it might be to wrestle with.

This is not quite as dire as it might appear since one can argue that the probabilities of evolution become highly peaked on average and that in the large the spacetime continuum is a valid approximation as is the principle of general covariance when speaking "in general", that is, without putting too much detail into the universe. The quantum details are what provides non determinism while the peakedness of the distribution in transition amplitudes is what provides the damping of excessive randomness.

In the context of the Aharanov-Bohm effect, the phase of the QM state is identified with a path integral of vector potential, A for the EM field F. In the absence of any EM field, the integral should be path independent, which means the phase is determinate, or singlevalued. This makes particular sense when F is recognized as essentially the curvature tensor derived from A as a connection on spacetime.

Another way of defining a single valued function in the presence of an EM field is to allow that the space on which the the state function is defined is multiply connected (multiply sheeted). The appearance of indeterminate phase is then a consequence of collapsing the sheets. One might conjecture that the effect of the EM field on the quantum level is to disrupt the single sheetedness of classical spacetime giving a picture not unlike the spacetime foam picture first proposed by Wheeler.

The minimal coupling of the EM field to a quantum particle is defined by allowing that the gauge invariance of the the EM field be identified with the phase invariance of the QM state; it may be that the gauge invariance (coordinate choice) of the gravitational field may be actually associated with the free will of choice. A choice of coordinates is, in fact, a choice of free will which does not seem to be associated with a physical field.

A Remark on essential differences between tensorial objects of opposing symmetries:
Recalling the association of multivalued functions on one manifold with those functions made univalent on a the "same" manifold that has been surgically altered by punctures in connection with a gemetrical interpretation of of Aharanov-Bohm effect that is connected with an antsymmetric EM field tensor, this antsymmetric tensor field gives information of a topological nature.

On the other hand, tensor fields with symmetric indical symmetry seem to provide metrical/interaction information, protypically given by the fundamental, symetric metrical tensor field g_μν of GR.

The curvature tensor derived from g_μν, having both indicial symmetries and antisymmetries is then possessed of both topological and metrical information. Cf. Appendix B to FCCR presentation on Lie Groups

Further considerations:
Masslessness and masiveness
Masslessness <=> gauge invariance (a symmetry)
Massiveness <=> gauge noninvariance (local hidden or broken symmetry)
Masslessness is primary in an embryonic universe

This is merely a conceptual outline of what I see as a correct approach to QG based primarily on minimal necessary properties of a QT that is not yet clearly formed. This QT takes precedence over GR, as it should when we are speaking of ultimately small structures. GR is a classical phenomenon, and should be retrieved as classical mechanics is retrieved by a concept of "most probable" or by an averaging technique. A known point that should be emphasized is that an expectation value and a most probable value are not necessarily equal.

One will have the same conundrum with approximating GR out this kind of GQ as one has in understanding where and how systems pass from Q-type behavior to a C-type behavior. This problem can be expressed in the question: why don't baseballs behave like elementary particles? Try doing a Young double slit experiment with baseballs. A resulting diffraction pattern is not likely to be found. It's not exactly clear that the experiment can even be done!

The affine structure of a space is given by its geodesics. Consider, simply as an example a space of two dimensions with the T² toplogy: a doughnut ..., cyclic in two coordinates. Cut, make a half twist and reglue. Space then has torsion. If we shrink the major radius to zero ... a projective ball with antipodal points identified, from the exterior, it has the physical appearance of no volume, regardless of its radius, it behaves as it were a point: anything that goes in, appears immediately at the antipodal point.

Now consider a space T³, 3 dimensional and cyclic in three coordinates.