Notes on a Space-Time Position Operator

Considering QM on one hand and SR on the other, there is a formal as well as conceptual clash at the level of kinematics: where SR constrains the kinematics by the Lorentzian invariant explicitly containing time, and in such a way to put it on the same level as a spatial coordinate, QM constrains kinematics with CCR not explicitly involving time. The usual formulation of QM in terms of canonically conjugate pairs seems glued to Hamiltonian formalism. Feynman's path integral formulation may yet be an out to this. One might argue that the concept of time enters indirectly through the momentum. But then, at the level of dynamics, time becomes simply the parameter for a one parameter group of unitary transformations. The introduction of a time operator in QM seems problematic. Among the problems is that it should destroy the equivalence between the Heisenberg and SchrX148dinger pictures. This equivalence would seem to depend on the centrality of time: time commutes with everything and can then be represented by a simple parameter. A lesson I learn from this is that the time of QM and the time of SR are conceptually and formally two different things.

In QM, the position operator Q has an eigenbasis |q> with eigenvalues q which correspond to Spatial Points. The formal concept of time does not enter QM as an obvious operator; if conceptually taken as an operator in the algebra of observables it is perforce "central", commuting, as it does, with all observables. Consider, conceptually, a necessary change that is imposed by the relativistic principle on a quantum theory: A relativistic observable cannot be simply "spatial", since a velocity dependent change of coR2|R0rdinate system, will change either the spectral values of such an operator or the expectation values of Q in some admissible state. Think of Lorentzian spatial contraction. For a time operator that may exist in any quantum theory, the argument applies. An essential part of the relativistic principle is that the Space-Time concept is the absolute observable concept, since it is this concept which is relativistically invariant. A space observable and a time observable should apparently be wedded into a single operator associated with an observable. This observable is then not one of "position" but one of "event". Moreover, unless it is obtained universally that for any space\-time separation that may be determined by any observer, and which separation corresponds to the determination of operators Q and T, that

```
[Q, T]  =  0

```

it must be concluded that the "event point" which is to be specified by mathematically simultaneous eigenvalues of Q and T, cannot exist. As CCR coerces the classical nonlinear phase manifold to a linear complex Hilbert space, a lack of commutation between operators Q and T coerces the local Minkowski structure, (and its mapped image in a local region of a curved event manifold) to a linear complex indefinite Hilbert space. From the viewpoint of statistical mechanics, one simulates the uncertainty relations by hashing a classical phase space into a covering by neighborhoods of volume h/5S0nT4: similarly then, recover a conceptual semblance of a classical manifold by a cognate hashing of the spacetime manifold. One cannot take this image too seriously, since it is doubtful that the "absolute spacetime manifold" of SR or GR even exists. In QM, the future and the past of a system can only be determined statistically. With spacetime properly quantized, one can expect that the past and future local geometries and then also the connections of local geometries distributed on a presumed spacelike hypersurface can only be determined statistically. Spacelike related regions are classically disconected in a causal sense, as being outside of each other's causal domain. Even though the invariant speed c can be derived from quanta of space and time, yet can a possible quantum process exceed the speed of light, already even in QFT. A classical precursor to this objection to the global concept of "The Absolute Spacetime" may already be that the local approximation to the curved manifold of GR, given as Minkowski space does not exist within the manifold, but only as a tangent plane attached at a point. If this objection holds within and between say the Planck regime and the Heisenberg regime, it remains to explain the details of the apparent coherence of the universe. The explanation could be anticipated in the model of the limit in QM of "high quantum numbers" or "as h/ goes to zero". The coherence of the universe may also be provided from the GR point of view as a residue of the transluminesent possibilities of process necessary to a correct quantum theoretical accounting.

Without the convenience of the above commutator vanishing, as a generally valid principle or theorem, we are forced to accept the vector of a presumed Hilbert space to represent irreducibly, a "process". The transition from a state picture to a process picture seems inevitable when in possession of a time operator. Once this is transition is made, the quantum sense of world event is replaced with the sense of "eigenprocess". This subtly generalizes the rigidity of the Lorentz group acting on a spacetime composed of event points, while maintaining a causal structure or structure of distinction on the space of process which is the formal Hilbert space. Any possible events or processes become dynamical objects, inextricably bound with a concept of time, that are endowed with a causal relation.

Another lesson from SR, indeed also from GR, is that the time that we measure is told by clocks. An obvious point with some uncomfortable consequences, seen from the Newtonian viewpoint. Clocks are local cyclic contraptions; they are bounded in space, and finite temporally in their cycles and in their precision. This clock measured time is a local concept that can not in principle be wantonly extended thoughout space, in such a way as to provide a universally valid Newtonian time marching forward uniformly, at some given rate. The notion of simultaneity is not relativistically invariant, nor is the notion of the temporal ordering of distantly observed events. QM time is, however, Newtonian in all aspects. A clearly desirable break with with QM is the abandonment of its global nature, and therefore a retreat to a concept of local quantum theory of process, which is the subject of this investigation.

With regard to the everpresent quagmire of ontology in QM, it can be seen that in the context of the present theoretical model, the eigenprocesses of the Hilbert space would appear to be the correct models of physical process, that is to say that they possess a meaningful ontological status. It is not the illusory statistical spacetime that is the proper background for real" events, but rather the Hilbert space of process, and it is the Hilbert space that is endowed with the lightcone distinction specified by G(n) that distinguishes the processes as "spacelike", "lightlike", and "timelike". Moreover, since the a measurement must be considered as an interaction or confluence of process, "preceded" in a lab frame by at least two entering processes. The observed exiting processes although necessarily giving a definite result, do not generally remain in an "eigenstate" as QM would have it, since the eigenstates are actually eigenprocesses. The alleged eigenstates of energy are not stationary, even though they approach stationarity ar n becomes unbounded. Physically then, the collapse of the wave function simply does not happen.

There are two fundamental time concepts that must be distinguished. The first is the "time or clock phase" operator associated with a local system thought of as its own clock. This concept is very similar to the "internal periodic process" introduced by De\ Broglie, and whose straightforward relativistic transformation law, if it is defined by

This is the mathematical cognate of the "physical extension" in space for which we use the position operator. The second has to do with the time that we as macroscopic entities measure according to some local lab clock.

Therefore take as primitive, an operator G whose eigenvectors and eigenvalues are associated with world points. For simplicity, think of 1 spatial dimension and 1 temporal dimension. We expect a Minkowski-like structure to exist on our 2 dimensional spacetime. It is known, that an operator on an infinite dimensional Hilbert space may have a two dimensional spectrum [Putnam\ 1967]. As a rule, such operators are rather unruly. But we want the ability to specify world points with spectral points in a finite dimensional case where the spectrum will always be discrete. One can consider two possibilities immediately: 1) let the spectrum be complex or 2) let World points of the spectrum be on an embedded world line, which is of course 1\-dimensional. Extrapolation to higher spatial dimensions in case 1) seems a bit dicey (coR2|R0rdinating the spacetime content of the various spacetime operators for the other dimensions), but the extrapolation for case 2) seems not so tortured.

```

\\            /
\\        /
\\    /
\\/
/ \\
/     \\
/         \\
/             \

```

Adopt case 2). Then, we need an additional operator G that expresses the Minkowski, light cone, or causal structure. The operator G when transformed by a Lorentz transformation that is a mere rotation in space will not alter the spectral set. A boost transformation will bring the notions of time dilation and Lorentz contraction into play in such a way that the expectation value of the invariant distance will not change. There will be processes that are classified as light cone processes. The additional operator is one that provides a causal structure on the processes space by having as its expectation value, the absolute Minkowski distance associated with a process from the origin of of the absolute 2\-dimensional spacetime. If %10|%00s> a process, is not zero, then \ =\ 0 implies that s is "expected to be on the light cone".

We approach world points given in terms of an invariant Minkowski length which determines a hyperbola, possibly degenerate, and a coR2|R0rdinate determining position on the hyperbola. If %10|%00s> is an eigenprocess, of G, we have a dispersion free process, on the light cone. For consistent connection of FCCR with QM in the limit n%10>%00%10I%00 where Q(%10I%00) is to be a purely spacelike position operator, spacelike is associated with a positive sign of and timelike with a negative sign:

```
< 0 => %10|%00s> is a "timelike process,"
> 0 => %10|%00s> is a "spacelike process,"

```

Geometrically, if %10|%00s> is timelike, the value of is the invariant hyperbolic distance from a hyperbola within the forward light cone, and the expectation values of G on all processes with the same are distributed on the hyperbola. Going to higher spatial dimensions, there are more operators G and still the single operator G to define the causal structure. The hyperbola generalizes as a higher dimensional hyperboloid of revolution about the time axis.

If %10|%00s> is spacelike, the same picture of a hyperbola obtains in the 2 dimensional spacetime, the hyperbola being rotated 90 degrees. In the case of higher spatial dimensions, it generalizes, however, to a pseudo\-Riemannian hypersphere.

The notions of "timelike", "spacelike" and "lightlike" are relative to the origin of an absolute spacetime coR2|R0rdinate system, and describe the expectation value of as a world point on the generalized conic section determined by . The general process %10|%00s> is associated with a probability distribution on the spectrum of G, which smears position over a possibly finite section of a hyperbola.

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