In both relativistic and quantum type physical models, space and time appear modeled on the continuum of real numbers. This would not be a problem but for a fundamental difference between the mathematical model and physical reality. The space and time of a pseudoeuclidean manifold is an intrinsically inert thing, while a physical spacetime is apparently an intrinsically energetic and chaotic thing, taking into account the general properties that should be possessed by an as yet to be constructed quantum theory of gravity (QG). There is every reason to suppose that on a fundamental level, the very "points" of spacetime should be regarded as quantum entities endowed with kinematic and dynamic properties, and that the underlying topological structure is possessed of quantum fluctuations as is the gravimetric field.
Given both the deterministic evolution extracted from General Relativity (GR), and the inherent and indispensible uncertainty which must be possessed by "quantum points" in a valid Q theory, the principle of GR and the principle of Q are at odds.
I vote more in favor of a Q principle, which at its essence is the principle of indeterminacy. The standard algebraic statements of the Canonical Commutations Relations (CCR) and the Canonical Anticommutations Relations (CAR) are not the only ways by which this essential principle can be asserted. Cf. FCCR which happens to provide an uncertainty principle and connects both CCR and CAR.
If one is going to have a theory of Quantum Gravity (QG), which demands quantization of the very coordinate functions that specify or locate spacetime points, and leaves the coordinate system unspecified, this implies to me that the coordinate systems are not the issue; the issue is coming to grips with the concept of "quantum point" and then by conceptual processes of collection and distinction, the concept of "quantum set", which will necessarily be a concept more free and uninhibited than that of a classical set. Properties and structures will necessarily appear which do not appear in classical set theory.
Along with the indeterminacy principle comes the other indispensible superposition principle which asserts a linearity of a Q theory. If one is to abandon this linearity, what might replace it would have to be a general state composition principle that includes linearity as a special case. I don't know what this might be, and for that reason alone am loathe to dispense with superposition which provides the intrinsically quantum interference effects. As Einstein did not now how to abandon the continuum, though he knew it to be necessary, I do not know how to abandon superposition.
My picture is not that the universe evolves according to the utterly deterministic laws of of GR, but rather evolves fundamentally according to the indeterministic Q principle, which is not to say that the evolution is completely random:
There are large (long range) forces at work that statistically form the overall evolution, but not the small; it is thus that human beings have come to exist.
The universe as well as our own bodies and genetic material that are subject to "randomizers" without which Darwinian evolution would be impossible, without which our own immune systems would be non functional.
Adaptation requires randomizers to provide alternatives. The indeterminacy principle is the source and ultimate randomizer of all existence; without it there could be no universe as we know it; there could be no evolution of anything.
It would be a dead thing in which beings such as we and all other creatures, living and extinct could never exist, in an ascending hierarchy of structure that allows for the "winking in" and "winking out" of various alternative structures. Even the long range statistical forces are the product of most probable quantum evolution.
From the viewpoint of GR, the metric tensor is associated with, among other things, the gravitational potential energy distributed over the points of an existing spacetime modeled as a continuous and differentiable pseudoriemannian manifold of four dimensions, that for technical reasons is also assumed to be paracompact.
With a nonvanishing cosmological constant, the cosmological term "Λ gkj" can be interpreted as an addendum to the stress-energy-momentum tensor, representing a kind of vacuum state potential energy in the absence of all material nongravitational sources. The energy of the spacetime vacuum is enough, mathematically, to support the existence of the spacetime itself; perhaps in analogy to the picture of EM radiation wherein the magnetic component and the electrical component appear to "cause" or support each other's existence.
From the viewpoint of standard quantum mechanics (QM), there are practical observational limitations on point distinguishability by the uncertainty relation that implies ever increasing energy required for ever decreasing distinguishable distances. A fundamental concept in the statement of QM is the notion of position determined according to some fiduciary coordinate system overlaying a given E3.
From the viewpoint of quantum field theory (QFT) there are energetic limitations by pair production (electron-positron production, for example, if one is using photons for observation purposes.)
Here, one has difficulties separating or distinguishing points, or more accurately, particles at distinct positions. This difficulty arises in a regime already twenty orders of magnitude greater than the the Planck scale, say 10(-13) cm, where the Planck scale is 10(-33) cm. From an epistemological standpoint, the proper regime of a quantized spacetime then covers these 20 orders of magnitude; it is not an insignificantly sized regime. As Feynman said, "There's plenty of room at the bottom."
Returning to the viewpoint of the theory general relativity (GR), the fundamental concept of absolute position in QM is declared meaningless, since any coordinate system is as good as any other. The scalars of physical importance are the invariants constructed from "geometric objects" [Niejenhuis 1952] whose components transform under the infinite dimensional pseudogroup of general coordinate transformations. The geometric objects are usually tensorial, meaning that they are transformed under a general coordinate transformation by a specific representation of this pseudogroup, which is not faithful. Tensors are defined at a point, and it is the only the local (at a point) aspects of the coordinate transformations that contribute to the tensorial transformation.
In order to consider a concrete group of coordinate transformations, the global topology of the manifold must be known; but, this is either a dynamical aspect of GR or a question of initial conditions at the moment of creation, and more certainly a more dynamical aspect of a theory of QG. Concentration on local (neighborhood of a point) aspects of geometry, topology and gravimetrics is almost a direct implication of general covariance, and completely consistent with the view of general covariance as a gauge covariance.
From the viewpoint of a putative QG theory, there are energetic limitations by pair production of gravitons, quanta of the metric field, determined by the Planck limit. Here one has difficulties with the very ontology of points and coordinates, in the classical mathematical sense, not just measurement difficulties that can be waved away as a technological or engineering deficiency. An absolute limit to the applicability of a continuum model is implied by the Planck regime/limit.
The existence of an absolute bottom to point distinguishability leads to a fundamentally atomic view of spacetime. The range over 20 orders of magnitude gives plenty of room for many complications in structures that may occur as a result of putting the spacetime "atoms" together. The existence of an absolute limit means also that one cannot take local Minkowskian geometry, hence the Lorentz Group and its covering SL(2, C) all that seriously at such a fundamental level. Within the QST regime, Lorentz covariance must be abandoned, as an a priori assumption; an unfamiliar territory is this.
From every direction, it is clear that although the continuum model is a useful, approximate model for spacetime in the regime that is not approaching the Planck length, it is ultimately, theoretically and logically incorrect. A minor miracle is that the current formalism finds a way around the spatial uncertainties that arise at the energy level of electom positron pair production. This must be eventually explained.
When I say "theoretically incorrect" I mean that a physical theory constructed with continuum as a premise will, perforce, predict unphysical consequences, i.e., apparent paradoxes when the regime suitable for its assumption is assumed nevertheless. When a singularity is predicted by or elicited from any physical theory, the logical inference to draw is that the theory fails in the locus of the singularity: singularities are not real and physical things, but rather, mathematical things.
A singularity is a form of infinity, and related by multiplicative inverse to the existence of infinitesimals. Neither infinities nor infinitesimals are observed, nor can they be. It is ultimately futile, though often instructive and sleekly practical in some regimes, to allow truly fundamental physics to be modeled in terms of continua, precisely because the appearance of singularities is a clear indicator of failure of theory in the regime within which the singularity appears.
A truly fundamental physics that starts with a notion of "the background", a vacuum conceived in the spirit of quantum field theoretic vacua, must start with notions that are finite, discrete, and algebraic. The concept that is most logically and physically primitive and attractive is that of set. From the physical point of view the sets become more specifically sets of spacetime points, or events, and not just points of an unstructured mathematical space.
Although it makes more logical sense to start with some notion of quantum point and proceed in axiomatic fashion, from a practical standpoint it seems to me to make better sense to start from both classical and quantum positions and work toward a middle, i.e., a proper connection. That proper connection does not actually exist since neither theoretical constructs respect their boundaries, nor the boundaries of the other: both pretend in themselves to universal applicability. This, unfortunately, is a completely typical aspect of all physical theories, which most probably stems from a fundamental aspect of a common psychological nature of their creators.
We are after all, macroscopic and essentially classical entities; the quantum regime is not within our direct experience, though it is inherent in our chemical stucture, but only our indirect and statistical experience.
In our thought patterns, the classical modes come first; hence we invent and try to make precise a notion of "quantization", which is some formal process that takes a classical model into the cognate quantized model. The process of quantization has, from experience, its peculiar limitations.
One classical model can seemingly give rise to a multiplicity of possible quantized models which cannot at present all be disambiguated so as to discern a correct one. What is classically commutative is not, quantally; therein lieth the major rub.
The ordering of operator products in the construction of a Hamiltonian is irrelevant in the classical case, and quite relevant in the quantum case. Moreover, there appear to be possibilities of ontology in the quantum regime that do not have proper classical analogs. Particle spin is a readily available example. Various attempts, over decades, to formulate the spin of the electron as some kind of rotational motion have all failed.
On the other hand, see a later historical and mathematical peroration Classical Geometry & Physics Redux showing that the concept of spin is neither neither relativistic, nor quantum mechanical, but instead, purely classical in the sense of Cartesian geometry. The failure of explaining spin classically then turns out to be mostly a matter of having gotten the pictures of logically Euclidean geometry wrong.
Consider, therefore, the simplest possible system, that of spacetime points from both classical and quantum viewpoints, taking into account what would seem to be the essential features of theories that would be classical and the essential features of theories that would be quantum.
Quantum theory has an essential linear feature, superposition, the idea that the state of the system is generally representable by a complex linear combination of mutually exclusive alternatives. The cardinality of the set of mutually exclusive alternatives, becomes the dimension of a complex linear space whose elements represent exhaustively the states of the system. That the space is complex and not simply real is what allows that the alternatives may interfere with one another. But this realization actually comes later. The complex space is endowed with an inner product, through which one defines the concept of mutual exclusivity. With the inner product, the complex space becomes a Hilbert space. The Hilbert spaces used have been from dimension two, (spin-1/2 and CAR, the canonical anticommutation relation) then, in integer valued dimensions, to the infinite dimensional separable Hilbert spaces of QM that arise from theoretical models of m particles in an n-dimensional space, and on to the nonseparable Hilbert spaces that arise in the constructions of quantum field theories. We must, naturally restrict our fundamental constructions to those involving finite dimensional complex spaces.
That the complex space need be linear may not be an essential feature, linearity of the space provides the certainly essential feature of superposition.
If the complex space is not linear, things become more complicated and there are more decisions to be made, and further structures to define. The complex space now has curvature and some recipe must be provided to replace linear superposition that allows at very least, for the mapping of a collection of arbitrary points, of the now curved complex manifold to points of the manifold, thus maintaining a concept of the interfering combination of alternatives.
So long as we have abandoned linearity, there is no particular reason not to abandon also the restriction that the space be complex. Consider rather an "almost complex space" where the local complex structures are not integrable to a single complex structure that covers the entire manifold.
This would enhance the concept of phase interference when the state combination recipe is used: the state of two "close but inseparable" things need not have the same phase. The idea of things "at a quantum point" having phase interference begins to make both mathematical and physical sense. Notice that we are now starting to think in terms of "real" geometric objects as being represented by objects in the Hilbert space (or more properly in the C*-algebra, and not as elements in the underlying space over which the Hilbert space of functions is built.
Presumably the inner product of the linear space might be replaced most generally by an almost Hermitean metric on the almost complex manifold.
Although this kind of model might be taken as a primitive nonlinear quantum theory, it is fraught with so many new concepts, and interpretational problems, that we forego further investigation of its possibilities and retain linearity of the n-dimensional complex space, and restrict attention to finite dimensional complex Hilbert spaces denoted H(n).
Using the standard bra(c)ket notation of Dirac and the representation of the Euclidean inner product by the Kronecker δ, an orthonormal basis |k> of H(n).
<k|j> = δkj
is the expression that the basis is associated with an exhaustive
set of mutually exclusive alternatives.
That any two states |φ>, |ψ>
are mutually exclusive is expressed by their orthogonality.
<ψ|φ> = 0
Another feature of QM that would seem essential is the probability interpretation for which the transition from H(n) to a projective Hilbert space PH(n) is generally affected. In this case, a ray of the Hilbert space is associated with a physical state and not a vector. Any vector |ψ> representing a physical state may be multiplied by an arbitrary phase factor exp( i ω ) and still represent the same physical state. That QM states are ambiguous as to overall phase is a consequence of the normalization condition
<ψ|ψ> = 1
necessary for the standard probability interpretation of the quantities
<ψ| A |ψ>
being expectation values of observable A when the system is in the
state |ψ>.
Mathematically, the representatives of the states of a QM system are elements, not of a Hilbert space H(n), but, of an associated projective Hilbert space PH(n).
For a time independent Hamiltonian in QM, the time dependent eigenstates have the form
exp( -i Ek t /h-bar ) |Ek>
where
H |Ek> = Ek |Ek>
so that although a time dependent eigenstate has this time dependent
phase factor, the physical eigenstate is actually independent of time.
It is only when general states, which are linear combinations of the
time dependent eigenstates are formed, that these phase factors finally
cause the interference phenomena so typical of quantum theory.
If one insists that the physical state is actually represented by an element of H(n) and not by an element of PH(n), then the time dependent energy eigenstates are truly time dependent, and cannot be considered truly stationary. It is the probability interpretation, the reality of the energy eigenvalues demanding that the Hamiltonian be Hermitean, and from this the unitarity of the operator of time translation (dynamical evolution) and therefore the maintenance of state normalization (conservation of probability) that makes PH(n) the desirable and consistent space of physical states in QM.
If probability is not dynamically conserved, it must mean that in some secret way not represented in the supposedly exhaustive system formalism, that probability measure is being lost (old states with previously vanishing probabilities, but not accounted for, are becoming available) or that probability measure is being gained (old states with previously nonvanishing probabilities, but not accounted for, are becoming unavailable). In either case there would have to be states that have not been accounted for; this is modeled by an expanding or contracting system. I will return to this idea after considering simple classical notions of spacetime point set theory, mentioning here that a local Hilbert space in energetic or thermodynamic contact with neighboring Hilbert spaces should describe an open system where the total probabilitiy measure may be sujected to changes, and which cannot be then really Hamiltonian in structure.
Prior Remarks on Classical Spaces:
The inclination of our species to count, stems from the more fundamental neurological inclination to make distinctions: "this" and "not this", and thus, to construct mental concepts of sets and patterns, one of which is the pattern of number in the sense of counting. We take this quirk of our nervous systems so seriously and tightly, that alternative models of a reality are rarely considered. In the witticism of Leopold Kronecker (1823-1891), "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk." (Dear God made the whole numbers, everything else is the work of Man.)
Consider the extreme distinction giving rise to the null set {}. The "universe" is exterior to it, while its interior is empty - or, we cannot know what is in it. The distinction is that which separates interior from exterior, this from that. The concept of replication of the null set is enough to generate the natural numbers conceptually by considering collections of null sets
{ {}, {}, ..., {} }
The two concepts necessary, and seemingly sufficient to the natural
numbers are then distinction and replication.
From these two allegedly "God given" or "Platonically given" concepts
assumed to be universally and unshakably valid physical
concepts the rest of mathematics can be built - with extraordinary labor.
One can, of course, replicate the symbolic process, even further by letting
{{n}} := { {}, {}, ..., {} }
define the set of n null sets, and so also define sequences of them
{{{Sm}}} := {{n0}, {{n1}}, ..., {{nm}} }
construed as a sequence of natural numbers.
A strange assumption appears in this language that we can,
in fact, distinguish null sets and label them even though
we have no real or defined properties by which to distinguish them at all.
It is an assumption of the applicability of language itself to both language, and to that to which it refers. This tacit assumption is worth paying attention to.
In perhaps a strange way, this construction indicates the abstraction of counting numbers as being a logical system independent of that which is counted by showing it to be a counting of "nothings". The counting depends, more strangely, on the ability to distinguish one nothing from an identical nothing, which does not make very good logical sense. Classically, this is a fine weirdness; quantally, it become more weird.
On the concept of replication:
What do we mean by replication?
The recreation of an entity from a given entity, in such a way that
the recreated entity is indistinguishable from the given entity,
which is to say that the given entity and its recreation are in some
equivalence relation.
The temptation is to say that they are "identical" in the Aristotelian
sense A=A.
If this were to be claimed, then we would have a difficult time trying
to explain how the physical replication could be known.
Replication involves two other concepts: indistinguishability
and equivalence relation (i.e., a relation that is
symmetric and transitive). How do we make the distinction that
allows the replication to be known?
By introducing yet a third concept of separation, which is
the distinction.
Physicists would automatically think of separation in space,
and separation in time.
A given thing (its existence as 'event' in the spacetime context) is here;
its replicant is there (a different event), but which is somehow
indistinguishable from "another" event.
Simply for the concept of natural numbers, we need fundamental concepts of: distinction, indistinguishability, and equivalence relation; separation being a derived distinction. These concepts arise naturally from the behavior of our neural structures, and are so ingrained that to be intellectually delivered from any one of them is not an easy task.
To continue from simply counting null sets, to set theory, one needs to add a concept or operation of labeling, so that a point or element of a set can be intrinsically distinguished from all others.
The types of models of reality that human beings have made, indeed the massive hierarchical structure of mathematics is very much dependent on the neural structures of human beings. The suggestion that "reality" was created precisely for human beings to model according to structural dictates of their own neural systems seems a trifle vain, and certainly not a suggestion that should be taken very seriously; it is, at core, a religious assumption based on faith: "the belief in a proposition for which there is either no evidence or evidence to the contrary". These kinds of assumptions have no place in science or mathematics, and yet, such things do indeed exist in the existing scientific process. There is no particular reason to suppose that an intelligent species of some other planet would have the same perceptual apparatus or the same neural structure, if even a neural structure as we know it. Nevertheless, all our scientific concepts are rooted in the neurology of our species dependent perceptions.
In dealing with a conjectured "quantum set theory", the fundamental concepts leading to classical set theory may have to be eliminated replaced or augmented. The resulting mathematical structure will, in all likelihood, not be any easy conceptual variation on classical set theory. All this having been said, return to considerations of spacetime of SR.
Classical spacetime, we consider without anything so sophisticated and mysterious as mass. The only events that can be considered as primitive then are lightlike. So, take as classically primitive a finite set S of lightlike events and its power set 2S. Thinking of these events as points on the lightcone in a Minkowski space or 4-vectors from the origin to those points, any two lightlike events uμ, vμ can be either lightlike related (collinear)
vμ = λ uμ
timelike related,
or spacelike related and therefore "causally disjunct".
If vμ and uμ are collinear,
one can be said to precede the other, in a given reference frame,
by comparing their time components.
Even in the classical sense of SR or GR, there is an implied dynamical aspect, that of lightlike propagation to an event.
If only one event is present, it is trivial and must be the origin point of Minkowski space, the zero vector. For two events, these are possibly lightlike connected, timelike or spacelike connected, and there are two ways for each of these connection types:
X X X X
\ / / <
X X
timelike lightlike spacelike
X X
/ \ \ >
X X X X
In QM, the points of a one dimensional physical space appear as the eigenvectors of a linear Hermitean operator Q acting on the Hilbert space of states. The spectral structure of Q consists of the necessarily orthogonal eigenbasis of the Hilbert space given by these eigenvectors and the spectrum Sp( Q ), i.e., the set of eigenvalues each of which is attached to an eigenvector. The spectrum provides a measure for the separation between any two "eigenpoints" of space, and in effect defines a metric on the coordinate space. Picturesquely, lay the classical coordinate line along the diagonal of an infinite matrix. Using a representation where the position operator is diagonal is commonly called the Schroedinger representation.
Although one starts with the model and image of single Euclidean spatial coordinate (a "straight line") in passing to the coordinate operator, there is no mathematical requirement in the operator structure that the coordinate structure being "represented" is indeed a straight line. The Euclidean assumption only appears when there are additional operators for additional coordinates and quadratic forms are assumed for representing the usual Euclidean invariants of distances and angles; which is to say, when the nature of the dynamics of massive particles is considered. Only then is it clear that a Euclidean geometry is being subtly imposed. By itself a single position operator entails no such assumption, and one might consider Q as any curvilinear coordinate.
In abandoning the notion that the spectrum of Q corresponds to points of a line, but still retaining the idea that the eigenvalues are real and correspond to distances from some fiducial point, the "points" that are being represented by the eigenstates might just as well be considered to be on semihyperspheres of dimension (m-1) in some space of m-dimensions, where m is arbitrary. There is a positive semihypersphere and a negative semihypersphere separated by an (m-1)-dimensional hyperplane. This is to say that the points can still have a considerable number of degrees of freedom and are not uniquely specified, or that the semihyperspheres are the things actually being represented. In fact, there is no particular reason to keep to hyperspheres, and we might replace them with (m-1)-dimensional hyperboloids. Complex linear superpositions of hyperspheres or hyperboloids are no more mathematically peculiar than the superpositions of points.
Considering hyperspheres and hyperboloids as being the objects represented by eigenstates of an operator suggests also considering that the m-dimenional space might have a Lorentzian type inner product, and that the points might be on (m-1)-dimensional lightspheres (cross sections of the light cone) where the eigenvalues correspond to points of a cross section passing through the cognate time coordinate.
Within the context of QM, there is no readily available "time operator" that complements the position operator(s); time remains, a simple continuous parameter of the dynamic evolution group exactly as it was in classical mechanics. If QM could easily be made consistent with SR, one would think that both spatial coordinates and time would exist as defined operators. As pointed out in the previous paragraph, a concept of time can be imputed to the space that contains the objects represented by the eigenvectors of a reinterpreted position operator.
To simplify discussion, the technical details implied by the fact that in most model situations of QM the operator Q is unbounded with continuous spectrum, will be ignored. Furthermore, I actually want to speak of the situation where there is a similar operator, but which has bounded and discrete spectrum.
Q(n) |qk> = qk |qk>
k = 0, 1, ..., (n-1)
It should be noted here, that we are interested in a theory of
"quantized points" (Qpoints), not in mirroring a coordinate of
a classical Euclidean space space; so, it makes sense to analyse
and question
the attributes and structure normally ascribed to a position
operator in QM.
The structural assumption in QM is that an "observable" is represented by a Selfadjoint operator; this covers both bounded and unbounded operators. In finite dimensions or the case of boundedness, simple Hermiticity will do: there are no domain problems to consider.
For Hermitean Q(n)
Q!(n) = Q(n)
where '!' means Hermitean conjugation, i.e., complex conjugation
of all matrix elements and matrix transpose.
These two operations commute.
This condition guarantees that all the eigenvalues of Q(n) are
real and also that every eigenspace is of dimension 1, and that
the set of eigenvectors of Q(n) form an orthonormal basis for
the linear space upon which Q(n) acts, which then
says that Q(n) is also diagonalizable. For Q(n) to be a proper
coordinate operator, its spectrum must also be well ordered.
The only property of Q(n) that we are interested in first is that it have an orthogonal eigenbasis, since each eigenvector corresponds to (is represented by) a Qpoint. The minimal requirement on Q(n) would be that it be a normal operator:
Q!(n) Q(n) = Q(n) Q!(n)
in which case one has
Q(n) |qk> = qk |qk>
and
Q!(n) |qk> = qk* |qk>
with '*' being complex conjugation, as the eigenvalues may now be
complex. From this, it is fairly clear that any normal operator N
can be written as the sum of two Hermitean operators A and B
N = A + i B
where
[A, B] := AB - BA = 0
The eigenvectors of a normal operator can be associated with a set
of points in the complex plane, in which the eigenvalues may have
some specified meaning. That meaning must, in fact, be specified.
An interesting question arises surrounding the notion of distinctness of Qpoints. On one hand, I claimed above that distinctness of points should be recognized mathematically by orthogonality of the eigenvectors. In QM, the spectrum of the Q operator automatically is such that there is a 1-1 correspondence between eigenvalues and eigenvectors. This particular structuring has slipped away, and to fill the gap one might immediately impose the requirement that the spectrum of Q(n) be nondegenerate: there are no two eigenvalues that are the same. If this condition is not imposed it would mean that Qpoints could exist with multiple occupancy, i.e., we can have distinct Qpoints "at the same place", or that one may distinguish points on one of the (m-1) dimensional hypersurfaces. This is a concept that is almost forced when we consider next the superposition of Qpoints. Elements of the linear space on which Q(n) acts can be represented as linear combinations of the Q(n) eigenbasis. This is an essential characteristic of a Q type theory.
Before continuing with the "superposition principle", let's look again at the now normal operator Q(n). It represents a set of points (Qpoints) together with the inescapable eigenvalues associated with eigenvector. In QM, a mostly obscured assumption is that the eigenvalues are interpreted as coordinate values of a straight line. Alone such an operator simply represents a collection of Qpoints, none of which lie on any given undefined and unspecified line. What, then to make of eigenvalues? In QM the eigenvalues are measures of distance from a given zero point which may or may not be included in the set. Let that still be so. Then let Q(n) represent a "universe" of Qpoints where the eigenvalues are interpreted as "distances" (possibly complex) from an assumed zero point. The points will be assumed to lie in some Rm in Q(n) is Hermitean, and in some Cm if Q(n) is merely normal. We do not know the real dimension of the space in which the set lies, but we do know that m is bounded above by (n-1), when Q(n) is Hermitean. [Think of the (n+1) points that determine an n-simplex] Further, for Q(n) normal and not Hermitean the strict upper bound for the real dimension is (2n-2).
But, real eigenvalues of Q(n) can be either positive, negative or zero. The zero is no problem as the zero reference point is then part of the set. With Positive and negative eigenvalues one must think of a radial variable with positive and negative values so that -r is the antipodal point of +r. If the set is construed as fitting into an m-dimensional space, where m > 1, Q(n) is not sufficient to make the classical geometrical form of the set unique.
With complex eigenvalues, the radial Qpoint coordinate "distances" must be further extended to be complex, say
qk = rk exp( i θk )
leaving the classical geometry of the set still not unique.
Extension of real Q(n) to a complex one can be done by considering
Q(n) → Q(n) V(n)
where V(n) is unitary and [Q(n), V(n)] = 0, and can also be done by
letting Q(n) be the real part of the extension:
Q(n) → Q(n) + iZ(n)
where Z(n) is Hermitean and [Q(n), Z(n)] = 0.
There is no immediate reason to choose any specific method of complexification.
Missing still is the collection of distances between the points. In this context also, the "problem" of eigenvalue degeneration meaning that multiple points may coexist "at the same place" remains a conceptual nonproblem as it was for real Q(n). The multiple points can be at the same distance from the zero point, but in different directions. The directions have not been specified and their lack is related to the missing distances between points.
To understand a concept, it is often helpful to understand its negation; it is also helpful to understand a thing by understanding that which contains it or is contained within it. We seem only to be able to understand, and even this to a limited extent, what is contained in spacetime. Aristotle understood that the concept of space without time has no meaning and time without space has no meaning. Only with SR was the unified concept formally made. Still, this formalism, a model based on a mathematical continuum, whose limitations are clear from the quantum theory that has been built upon it.
We do not know what spacetime really "is", nor even whether asking the question is meaningful. All we can do is attempt to make better models. Metaphorically, we speak of the fabric of spacetime as if were somehow made of some substance.
To understand physical substances, physicists traditionally think in terms of analysis, of subdividing and progressively getting at the structure of smaller and smaller parts: substance to chemical compounds, to molecules, to atoms, to electrons and nuclei, nuclei to hadrons and mesons, these to quarks and gluons, and possibly deeper yet. Though we have not yet gotten to the irreducible analytic bottom, a synthetic physics is already begun by "government science". God help us all.
In analyzing spacetime, the brick wall is the Planck level. The ultimate theoretical limitation appears to be the Planck length, below which concepts like "point of a continuum" and "coordinate" have no meaning; similarly for the Planck time. Presumably then that which we see model as 4 dimensional smooth manifold, the classical spacetime of GR is the result of some collective phenomenon arising from these indivisible Planck scale entities which one expects to look something like a very small 4-tube in spacetime of radius equal to the Planck length and height equal to the Planck time. The 4-tube is a spatial 3-sphere pushed through a Planck time, which is to say something like a string.
Using the idea of a 3-sphere may not turn out not to be a very bright idea, since we cannot know what goes on within the spatial region. Within a Planck time we can't know whether the beginning of the interval is also its end. We could have a little string, open or closed.
The "points" of GR spacetime are, physically, specifications of and event, and must not be confused with a spatial point. Consider the closed form of a hypertorus, remembering that we are considering an event.
From the context of the collective phenomenon that we see as spacetime, the closing of a timelike interval on itself forms a time unto itself, divorced from the macroscopic concept of time. A cyclic entity is then created which persists through a statistically evolving macroscopic time, which has at least one property of a particle that persists for some macroscopic time.
The concept of time becomes a plurality of concepts. The concepts are distinguished by the sizes of structures. To confuse or fail to make formal distinctions between them is to invite unnecessary paradox.
Once the time concept has been broken into two distinct conceptual entities, the question of the existence of other distinct temporal concepts arises.
One of the truly vexing things about GR as a physical theory and its associated thought patterns is the assumed absolute ontology of the spacetime manifold.
Whether predictable or not, both past and present (in the time of a Ricci scalar, which is not a Newtonian time) are completely determined; there is no room for chance in anything, unless the Ricci scalar itself is somehow Q inteterminate, but also a highly peaked Q "observable"; this is necessary in order for theory to conform with what we see consistently.
We are not willing to be subjected to Bohr's philosophical foolishness, that an "objective reality" does not exist. It may not be what we imagine it is, but it exists, and must exist for any serious discussion of anything. Ontology is simply not a psychological decision.
Wheeler, long ago, gave us a foam picture for what a quantized spacetime might look like near the Planck level: semiparticulate, with dynamics of topology as well of its particulate structure.
If a Qpoint is a dynamical object, it has both momentum and energy with respect to the collective foam phenomenon, and is in some respects like a particle within the wave of the present, a disturbance that is a kind of local spacelike hypersurface, with multiplicities of time concepts in the surface.
One can look at structure at different levels of size and see entirely different kinds of things. Assuming that there *is* a bottom, and that it is indeed the Planck level, the first level is that of Qpoint as a dynamical quantum object which is truly atomic (indivisible) in the sense of Democritus since it cannot be subdivided. This is not to say that it cannot have structure (e.g., see open and closed), or even a structure that depends on its "kinematics" or "dynamics". An event seems always to be an energetic event, and there is possible sense to microdynamics according to microtimes.
As the wave of the present progresses with quantum dispersion into the unpredictable future, one can even envision Qpoints being pulled in pairs from a vacuum. The space of future events has not yet been created, the "future" is not completely determined, the past does not really exist except as memory; the absolute spacetime continuum therefore does not really exist. Only the local present can have ontological status.
A necessary property of the foundations of a proper theory of "quantum gravity" would have to be the essential indeterminacy of the future, as well as of any indeterminable but fuzzily remembered past. Such a property is so completely at odds with the determinism of the Einstein equations, that the idea of "quantizing the classical gravitational field" seems foolish on the the face of it, instructive though these historical attempts have been.
Though we do not know the details of the air molecules in a gas, we can from statistical mechanics derive the standard equations for thermodynamic systems. Though there is indeterminacy in the particulars of dynamics of a quantized spacetime, there should not be any big surprises in its evolution as determined GR for its scale. It may be mistake to try to "quantize GR", as it may be missing a small and crucial correction in the quantization procedure, and better to derive it as a kind of thermodynamics from a completely elementary beginning.
After looking at the possible structure and selfcontained quantum nature of a Qpoint, the next step is to start putting them together in a quantum theoretical way, which is to say start making a theory of quantum sets. In doing so, we understand the difference between the creation of an abstract axiomatic system, and one that should be tailored to a model, specifically designed to represent the ultimate constituents of a physical quantum spacetime, not just the points of a static mathematical space. At the same time the time concept first expressed should then be of local Q aspect and not anything like a newtonian time. Any dynamic law relative to such a local Q time would not then be one that is necessarily invariant under translations in a derived Newtonian time. A Newtonian time is derived in such a way that the Newtonian dynamic is, in fact, invariant under such translations.
Making the most general abstraction from the idea of a quantum position operator to a finite quantum set of discete quantum points, without trying to put any physics into these ideas, the mathematical model becomes simply that a quantum point corresponds to an element of a canonical basis of some vector space V(S), and the set S corresponds to the entire basis. Allowing that linear combinations of points are possible, the set structure takes on the character of an algebraic simplex, so that there is an injection
S → V(S)where the cardinality Card( S ) = Dim( V(S) ). This induces an injection of the power set 2S of all subsets into the vector space of all subspaces of V(S). A subset of cardinality m is mapped into a subspace of V(S) with dimension m. A collection of points of S has a surjection to a collection of elements of the canonical basis of V(S). In the induced map of the 2S, one is lead immediately to the space of subspaces of V(S) spanned by all possible collections of the canonical basis, {aj : j = 0, 1, 2, ..., n-1}.
In this most general abstraction, no inner product has been assumed, as would be the case for a Hilbert space, or pseudohilbert space (a space with a pseudo inner product is not positive definite); the field of the vector space has not been assumed, and no symmetry properties have been imposed on the collection of basis elements that is the image of any given subset of S. We have no angles between vectors except a zero angle; any two vectors are either linearly dependent or they are not. If multiplying a basis element by an element of the field does not change the associated point in S; if a zero vector is not associated with any point in S, it is then clear that the injection of S should not be into V(S), but rather into the projective or ray space:
S → PV(S)
Interestingly, the concept of a projective space as being appropriate enters more primitively that one might have guessed since it usually appears in the explanation of QM as being the result of requiring the existence of a probability distribution associated with a state vector and the conservation of probability.
The first obvious thought is that points of S have acquired a richer structure in the passage to V(S) than they had in S, since now linear combinations of points make sense, and these linear combination seem to generate new kinds of points of S.
How should the mathematical form of a quantum set theory be considered? If one believes that the fundamental notions of quantum theory and that its algebraic representation is essentially correct, then points should at least be mapped to elements of a vector space.
What do we want from a quantum set theory? One should be able to calculate numbers (transition amplitudes/probabilities) and talk about conditions (states/processes) of a set, in a way similar to the way one can speak of a quantum gas. It won't be the same, of course, since one of the aspects that would good to describe would be something like a quantum version of topology.
A preliminary notion of physical point combinations and interactions based on a quantal notion of the impossibility of absolute stasis, of fluctuations, suggests the defining of a binary operation on V(S), which will give V(S) and algebraic structure
General: a binary operation defined by <ai, ak> = cikj aj, defined on the basis of V(S) extends by linearity to all of V(S). turns V(S) into an algebra, cikj being the algebra's structure constants, subsumes Grassmann, Clifford, Lie, Cayley, Jordan and Malcev algebras. Little is required to extend to a superalgebra. Eventually, a subject of investigation called "quantum algebraic topology" should arise.
The algebraic structure is equivalent to a "classical" statement of what transformations are possible; we still do not have a notion of probability measures on the tranformations, i.e. a mechanism where what is likely to happen/be is discussable.
The binary operation <., .> can be symmetrized or antisymmetrized, the general binary operation being written as a sum of both
<ai, ak> = (1/2) [ai, ak] + (1/2){ai, ak}
isolating a Lie part and a Grassmann-Clifford (or Jordan) part,
assuming that "(1/2)" makes sense in the field of the vector space.
A Fourier transform gives rise to both complex structure for V(S) and point kinematics, introducing notions of momentum, energy and time
Consider Fourier transforms on a vector space over the Galois fields GF[p] and GF[pm]. The fields will have to contain n-th roots of unity in order to define a Fourier transform.
In the future sections, various impositions of these structures will be investigated; attempts will be made to correlate and interpret them in terms of a quantum theory of dynamical physical points.
In short form, a classical physical system described in canonical formalism has a space of states that is a cotangent bundle over the configuration space; a bit perversely this is called phase space, while an older designated "state space" might be seen as a phase space bundle over a one dimensional space associated with time.
The observables are measurable functions of the fundamental canonical variable pairs (p, q), and states are measurable functions on the phase space, which also has a matural symplectic structure defined by the fundamental Poisson backets.
While phase space is generally a nonlinear manifold, the space of measurable functions on it can be fashioned into a Hilbert space, making the connection with quantum theory and its constant emphasis on Hilbert space, a construction not very much emphasized in quantum theory, a bit more sensible and intimate.
The major structural difference between classical and quantum mechanics is that while the Hilbert space of states in classical mechanics is made from functions on a strightforward symplectic manifold, the analogous functions in quantum theory cannot be so defined because the underlying phase space has become noncommutative: the q-sector variables do not commute with the p-sector variables.
One way of realizing the Hilbert space in quantum theory is to define complex valued functions on the classically commuting q-sector; another way would be similarly to define functions on the p-sector. We talk routinely about Configuration space representation and Momentum space representation. They are connected by the usual Fourier kernel, a unitary transformation on the (projective) Hilbert space, that is involutory of order 4.
In classical mechanics, the combining of systems is formally affected though the topological product of phase spaces, and the direct products of the algebras of observables.
In quantum mechanics, the combining of systems is formally affected similarly by products of Hilbert spaces and nominally at least of C*-algebras. Since the spaces are linear, the cognate combining principle is that of direct product.
In classical mechanics, there is duality in the interpretation of a phase space of identical free particles between particle number v. configuration space dimension in classical mechanics for a single particle. Similarly, in quantum mechanics, the space for n free one dimensional particles is the same as that for a single particle in an n dimensional space, the state defined as a function in the Hilbert space, of n monadic (in the irreducible sense of Leibnitz) particles in one dimension is equivalent to a state of a single n-state particle.
In classical mechanics, constructing multiparticle mechanics from single particle mechanics, there is no problem in telling even identical particles apart, and each particle can in principle be tagged to keep track of it. There is no requirement as to how the topological products can be taken, and no symmetry imposed on the permutations of the particles, except the permutation symmetry group itself.
In quantum mechanics, in systems of identical quantum objects, this labeling and distinguishing of particles is not possible, and this impossibility allows that some states that were classically different are now identical. No Labeling and symmetry
Superselection and spin for example.
Alternative interpretations revisted
Application to the idea of a dynamical set. Classical dynamic set Quantum dynamical set Set as a direct product of points Distinguishability.
Consider a singleton set, and the classical proposition, "The singleton set occupied by point a". If we have n identical candidate points, we can label them as ak, and consider the set of mutually exclusive propositions "The singleton set occupied by point ak".
In the cognate quantum situation, the labeling is impossible and the best we can do is talk about the occupation of the singleton set, and most generally we talk about the probability that that singleton in occupied.
[Added April 4, 2006]
By all means, also check out the delightful and provoking article on
nth quantization
by John Baez, referencing David Finkelstein.
Baez seems to have a wonderful sense of humor about all
this that my writings seem to lack.
The ideas explained in Baez's article are very much the same as those
expressed here, mostly because they are
rather natural connections and extensions of prior mathematics
and physics.
What I speak of here is what Finkelstein and Baez call
"first quantization".
Nth quantization is the iteration of application of the same map here (functor in the context of Category theory [Wikipedia]). If anybody has first dibs on the general idea, it it probably Feynman and Finkelstein, who were speaking of such things back in the late 1960s. As far as I can remember, the idea that physics could be understood as the computational process of a computer originated with Feynman, and the actual concept of a quantum set theory originated with David Finkelstein, and J. A. Wheeler.
Conceptually, one needs to decide what concept the phrase "quantum set theory" is supposed to denote. That may seem, at first, to be a silly thing to say, but "points" have specific contexts, even in pure mathematics. What kind of space might these points be members of? A topological space? If so, is it Hausdorff? Is is a metrizable space? Affine? If they are to be elements of some sort of "geometrical space", how, precisely, do you define the geometry? Are they elements of a category? The questions, and answers do matter.
The context I have chosen is that of quantum mechanics, not more generally, quantum theory, but quantum mechanics - and even a quantum mechanics that does not exist in the commonly understood garden of physics: quantum mechanics simply does not allow this kind of finite discretation, except for the finite dimensional representations of the anticommutation relations. Standard quantum mechanics depends on an essential view of the world in terms of continua. So this entire subject is either a rather general matter of mathematical possibilities, or a refinement of the physics of "physical geometry", and then on a level of theoretical difficulty rivaling that of quantum gravity.
Email me, Bill Hammel at