It's a simple enough idea:

For a fundamental particle, QM says that the particle's effective position has a degree of uncertainty to it. It ought to be smeared out over the surrounding region of spacetime as a probability field. Its energy and momentum are smudged. Although any attempts to sample that energy and momentum are going to give quantised results that jump about a lot, if we average the results of a large number of similar possible measurements to produce a smoothed, idealised map of the probability-field for the particle's distribution of massenergy, we end up with a density-field surrounding the particle's notional position that expresses the distribution of massenergy through space that – functionally and definitionally – would appear to have the properties of a gravitational field (because that's effectively what a massenergy field is). The momentum is similarly smudged, giving us a polarised field-distortion component that expresses the particle's state of motion. The shape of the field tells us the probability-weightings for the likelihood of our being able to make certain measurements at given locations.

One interpretation might be that the underlying stochastic processes are truly random, and that the shape and dimensionality of classical physics appears as an emergent feature.

Another might be that the shape represents classical physics principles operating below the quantum threshold, but being drowned out by signal and sampling noise (until we average out the noise to reveal the underlying structure).

Trouble is, if we take these QM averaged-field descriptions seriously, they imply that the correct classical geometrical model for particle interactions isn't flat spacetime – the existence and state of motion of a particle corresponds to a deviation from flat spacetime, and the greater the relative velocity between particles, the more significant the associated gravitomagnetic curvature effects become.

With this approach to quantum mechanics, observer-dependence doesn't have to be some "spooky" effect where the same experiment has physically-diverging results depending on how the observer looks at it, due to reality having an odd, probablistic multiple-personality disorder ... we get different predictions when we change the position and speed of our observers because the presence and properties of those observers physically alters the shape of the experiment, in ways that can cause quantised measurements of the geometry to come out differently. It's a non-Euclidean, nonlinear problem. At those scales there's no such thing as a perfect observer, so to describe how the experiment should play out in isolation, and then to repeat it with different embedded observers charging across the playing-field, is to carry out different experiments.

It's not a difficult argument, but "particle physics" people have a tendency to argue in favour of special relativity by saying that we know for a fact that curvature plays no measurable role in high-energy physics, and mathematicians have a habit of trusting physicists when they say what their experiments show ... so I didn't know of any examples of QM people discussing velocity-dependent curvature when I wrote the book.

Anyhow, earlier this year I stumbled across one. Funny story: I was looking at my Google Analytics statistics, and obsessing about how nobody from Mongolia seemed to have been visiting my website, and then I happened to visit a "citation statistics" site that included a world-map, so naturally I zoomed in to find out what fundamental theoretical research had been coming out of Mongolia. As one does. The search only gave one result, so I clicked on it.

And then I choked into my coffee as this thing came up:

"Space-time structure near particles and its influence on particle behavior"

International Journal of Theoretical Physics [23] 1031-1041 (1984)

Kh. Namsrai, Institute of Physics and Technology, Academy of Sciences, Mongolian People's Republic, Ulan-Bator, Mongolia

Abstract: "An interrelation between the properties of the space-time structure near moving particles and their dynamics is discussed. It is suggested that the space-time metric near particles becomes a curved one ... "DOI 10.1007/BF02213415

The paper also appears as a chapter towards the end of Namsrai's rather expensive book, Nonlocal Quantum Field Theory and Stochastic Quantum Mechanics (Springer, 1985)

11.8.2:

" ... Physically this relationship means that by knowing the space-time structure near the particle we can calculate its velocity (generalized) and, on the contrary, by the value of the particle velocity one can try to build the space-time structure near the moving particle. Thus, it seems, there exists a profound connection between these two concepts and they enter as a single inseparable entity into our scheme. "The "stochastic" approach looks at QM from a "shotgun" perspective, superimposes the result of a large number of potential measurements ... and arguably generates a spacetime that's "curved" in the vicinity of a "moving" particle in such a way as to describe the particle's velocity. The curvature generates the velocity, and the velocity generates the curvature. Which was kinda what I'd been saying. But with a lot more advanced math to back it up.

Of course, the irony here is that Namsrai's paper and book describe the emergent classical properties of apparently random processes ... and I only found the piece (which is exactly what I needed to find), by using an apparently random method.

Spooky! ;)

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