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Practically all of the physics equations I've encountered are written in terms of what might be called "remote observables", such as the distances between objects in Euclidean space or between events in Minkowskian space, for example. Even the charges and masses of distant objects would be "remote observables".

This is troubling because, to any given observer, the values of remote observables are not directly measurable or knowable. A local observer can only measure what might be called "local observables": accelerations of his own rest frame, angles of arrival of light rays on his photographic plates, forces on test objects in his own laboratory, and so on. That is, he can only directly measure what happens in a very small volume of space, over a finite interval of time. Of course, he also has the option of sending out probes of one sort or another and observing signals they send back; but the signals are then "local observables" when they arrive. Eventually, from the accumulation of a lot of such observations, he infers what's going on in the rest of the world. I probably am not using the term "local observable" in the normal way; perhaps there is another more appropriate term, but hopefully my meaning is clear.

My question: Have ways been developed to express equations of physics solely in terms of "local observables"?

Edit 1: From @BenCrowell's answer I can see that my meaning of "local observable" is not clearly expressed. Maybe this will help: If we say the Moon is 238,900 miles from the Earth, that is shorthand for saying that it takes about 2.6 seconds for a light to get from a laboratory on the Earth to where we imagine the Moon to be, then back to the laboratory. Or, it might be shorthand for a trigonometric observation that we interpret as a distance to the Moon. We never can measure where the Moon is now; at best we only can measure where it was 2.6 seconds ago. From a record of such measurements we infer that the Moon is orbiting the Earth at a distance of ~1.3 light-seconds.

In my understanding of differential equations, a quantity like $E = E(x_i,t)$ represents the value of E at every point in space and time; and a differential equation in E describes a set of constraints on the relationships between the value of $E$ at different points in space and time.

However, an observer in a laboratory cannot know $E$; anywhere except at his own location. At best, he can know $E$ at his location from the time he starts observing until the time he stops observing, at his own location. That's what I mean by "local observable".

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  • $\begingroup$ Would you consider Maxwell's equations for the EM field to be local? $\endgroup$ – jacob1729 Feb 27 at 16:13
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    $\begingroup$ They are local in the sense that, according to Maxwell's equations, a change at one point in time and space does not result in immediate changes at remote points. But that's not what I mean by "local observables". From a perspective in a laboratory here, it is not possible to measure the values of the EM field over there, when over there is far from here. So the value of $E(x+\delta x, t)$ cannot be known. At best, we can infer the value of $E(x+\delta x, t-\frac{\delta x}{c})$ from a history of measurements made here and extrapolate to $E(x+\delta x, t)$ $\endgroup$ – S. McGrew Feb 27 at 16:46
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The basic laws of classical physics can all be expressed as wave equations, which are differential equations and therefore local. The observables in classical physics can usually be represented as tensors (or tensor densities), which are local.

Nonclassical physics is pretty different. The Schrodinger equation is a wave equation, but the wavefunction isn't an observable. Quantum mechanics doesn't have local realism.

Even the charges and masses of distant objects would be "remote observables".

Charge density, for example, is a local observable. You can't measure it from a distance. It's the timelike component of a tensor (the current density four-vector $\textbf{j}$).

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  • $\begingroup$ Please see my edit for a clarification of what I mean by "local observable". $\endgroup$ – S. McGrew Feb 27 at 16:00
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To simplify the mathematics we postulate that $E(x,t)$ has a definite value at all points $(x,t)$. But at a given point $(x',t')$ we know that $E(x',t')$ can only depend on the values of $E$ within the past light cone of $(x',t')$. So our theories are local in that sense.

And even the postulate of a universal field $E(x,t)$is a local approximation because it does not attempt to model the global geometry of the universe (is the universe finite or infinite ? is it simply connected ?) or whatever sort of boundary may or may not exist at the Big Bang itself.

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  • $\begingroup$ This doesn't really address the question about ways to express the equations of physics in terms of local observables, though your point about the past light cone is well taken. $\endgroup$ – S. McGrew Feb 28 at 5:00

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