Roger Penrose has written some compelling arguments for associating gravitational fields with state vector reduction and entropy. In short, the reasoning goes that Stephen Hawking's demonstration that black holes are associated with entropy, along with the fact that black holes violate Liouville's theorem about the constancy of phase space volume, imply that entropy must be associated with a quantum gravity effect that is due to gravity in general, and not just black holes. But his argument implies that there might also be an effect in the reverse direction. That is, that wave function collapse may modify the effective gravitational field. Furthermore, since it appears to be evident that entropy is as active in the weak gravitational field limit as in the strong (where Penrose makes his case) the place to look for gravitational anomalies due to wave function collapse is in the weak gravitational limit. But there is already experimental evidence of something odd going on with weak gravitational fields due to Mordecai Milgrom's MOND (MOdified Newtonian Dynamics) explanation for the gravitational anomalies of galactic rotation data usually attributed to dark matter. The purpose of this note is to suggest a connection between these theories.

One of the most intriguing effects of state vector reduction is the Quantum Zeno Effect, where repeated measurements of a quantum system are shown to result in a suppression of the time evolution of the wave function. The effect is due to the fact that perturbations in quantum mechanics always show up as the square of the wave function. In order to calcuate the perturbation, one needs to have a discrete set of eigenstates, rather than continuous. In this condition, a perturbational analysis of the time evolution of a wave state that begins as an eigenstate, must show that the eigenstate is abandoned with a time dependency proportional to the square of time rather than (as exponential decay implies) simply proportional to time. It can be shown that as time passes, the perturbational approximation eventually fails and exponential decay follows. In short, the effect depends on (a) discrete eigenstates, and (b) wave function collapse occuring at a rate faster than the rate at which perturbational analysis fails.

To assist the calculation, it helps to imagine an experiment where a test particle is used to calculate the gravitational acceleration of a spherical massive body. If we assume that the test particle is negatively charged, we can apply a negative charge to the massive body that will balance the gravitational acceleration, and then from our knowledge of electricity, we can calculate the acceleration of the massive body. Furthermore, since the massive body is spherical, the cancellation of the gravitaional and electrical force must extend over all space. That is, the cancellation must occur in both the strong field regions of the gravitational field and the weak field regions.

The reason for going about the problem in this way is because we really don't know how to include gravity, a curvature related effect, into quantum mechanics. So instead of beating our heads in guessing at a form for the interaction, we can instead use perturbational analysis off of the states that nature herself provides for an charged particle caught in a gravitational potential. Since we don't know the actual wave function, we cannot calculate the acceleration directly, but we can calcuate the amount of electric field required to accelerate the particle just enough to cancel the gravitational attraction. Since the electric attraction is different depending on the ratios of the charges, the electric calcuation cannot depend on the curvature of space-time around the massive body. What is happening is that we are using electric fields, a situation that we understand at very high levels of accuracy, to calibrate (I want to write "gauge") the gravitational field.

In order for perturbational analysis to apply, we need to have that the unperturbed particle is in an eigenstate. Since the gravitational field is not actually a field, but instead is a curvature effect, the eigenstates of the particle (with the massive body uncharged) will have to correspond to the particle being attracted to the massive body. It is these eigenstates that must be perturbed.

The first requirement for the Quantum Zeno Effect to apply is that the eigenstates be discrete rather than continuous. Of course our particle is bound, therefore its eigenstates must correspond to discrete eigenvalues. What this means in the context of gravitation is unclear, as these eigenstates must correspond to motion towards the gravitating body. It can also be argued, in a way reminiscent of Milgrom's reasoning that the hubble acceleration has something to do with the MOND effect, that the finite size of the universe prevents "box normalization" from being taken all the way to infinity, and that therefore there must be a quantization of momentum similar to that seen in a finite box. I think this last argument is unconvincing, but instead that momenta, or more particularly, velocities are quantized for reasons similar to the quantization of angular momenta, as explained in my recent article: "Ether, Relativity, Gauges and Quantum Mechanics", Carl Brannen, September 2003.

The second requirement of the Quantum Zeno Effect is that the wave function collapse rate has to be fast enough, compared to the rate at which perturbation theory fails. This corresponds to the weak limit for the electric field, and therefore also for the gravitational field. So if our test is done sufficiently far from a sufficiently weak massive body, this criterion will be satisfied as well. But note that this requirement will fail close to the gravitating body, where, therefore, the usual calculations for electric acceleration must equal the usual calculations for gravitational acceleration.

With these two assumptions, the Quantum Zeno Effect applies, and this indicates that in the limit of very weak gravitational fields, the electric field will be able to move the particle from eigenstate to eigenstate with a rapidity only proportional to the square of the field rather than linearly proportional to the field as Newton or Einstein would imply. Thus, if we use the gravitational potential defined by Newton / Einstein, the two fields will be unable to be cancelled with a single electric charge on the massive body.

The acceleration of the electric field depends on the mass and charge of a test particle used to measure it. Because of this fact, it is not possible for the universe to correct for the above discrepancy by making the electric field weak for certain particles in certain gravitational fields. Instead, the universe must maintain its happy equilibrium by making the gravitational field have an effective strength that is sufficient to cancel the effects that the Zeno effect would have in the region of weak attractions.

Equating the field strength of the gravitational
field with the electric field, yields the MOND
relation, but this discussion does not eliminate
the possiblity that *a*_{0} might
depend on the mass of the gravitating body.
For that, either the modified relativity of the
above Brannen article is needed, or perhaps
a better estimate of the bound states based
on an understanding of quantum gravity.