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I am toying for a while with the idea of an hysteretic space-time.
Hysteresis is the dependence of the state of a system on its history.
Though it could make for some interesting twists in the world, I am not sure up to which point it could be made self-consistent.
Therefore my question is:
All the rest being the same, in a space-time that shows hysteresis, could elementary particles like bosons and fermions exist?
You could do that with some sort of extra particles or "hidden variables" (for example an electron, once removed from near a proton and brought back, would experience a slightly different electric force - its charge has become hysteretic).
There is something like this (but not so radical) in James P. Hogan's Entoverse.
But the hysteresis hypothesis has tremendous consequences, since you have, in effect, introduced a possibly nonzero curl everywhere - the electron above would experience its own hysteresis as a time variation of the local electric field. On a closed loop, this translates to total work being nonzero, i.e. energy can be destroyed or created from nothing. In a larger sense, no vector field can now satisfy the irrotationality criteria, immediately doing away with any form of conservative field.
No conservativeness for gravitational fields means no stable orbits (the consequences at the micro scale for the possibility of life are probably just as profound, but I haven't the knowledge to even begin imagining them).
A hysteretic space could probably exist just like some sort of dense quantum soup.
I was looking over the conversation in the comments (which mentions things like "revision counters" and "reverse-hysteresis"), and then it struck me:
Doesn't the universe work like that anyway?
Think about causality. "Every action proceeds from another", as my high-school philosophy teacher used to say. Now, I get that philosophy is rather outdated, but the basic concept applies here: Everything has a past, which has effected its current present.
For example, consider the soot in a chimney. At one point that soot was some combination of wood, paper, and other flammable substances. If it had been left alone, it would stay exactly the way it was due to Newton's first law of motion.
However, that's not what happened. At some point in its history the constituent components had excessive heat applied to them. As a result, the carbon, tar, and a few other components sublimated, becoming the soot in that chimney.
I argue that our universe does, in fact, work that way; the current state of each and every thing is derived from the events in its past.
That being said, I don't have a physics degree, so I could be misunderstanding the question.
One example that might be food for thought here is that of the wave equation in even spatial dimensions, which (unlike the cases of odd spatial dimensions such as the familiar one or three) exhibits a kind of hysteresis. Quoting Wikipedia (emphasis mine):
… the solution at a given point $P$, given $(t, x, y, z)$ depends only on the data on the sphere of radius $ct$ that is intersected by the light cone drawn backwards from $P$. It does not depend upon data on the interior of this sphere. Thus the interior of the sphere is a lacuna for the solution. This phenomenon is called Huygens' principle. It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure. It is not satisfied in even space dimensions.
Thus in even dimensions, provided the propagation of light and sound is still governed by the wave equation, one would in principle be seeing/hearing not only what is there "now" (really: the short moment of time ago that it took the wave to reach the observer) but also afterimages of what was there earlier. (It's possible this would lead to such blurring that sight and sound are rendered essentially useless as senses; I haven't checked how the math works out in that regard, but would expect interference to be important.) Since the Klein–Gordon equation is in a sense just a modification of the wave equation obtained by adding a term with just the function (rather than second partial derivatives thereof), one would expect something similar for that, and the Dirac equation is (to put it loosely) a square root of the Klein–Gordon equation, so again we have "waves" as solutions, and one would expect even/odd dimensionality to have an impact on what they look like.
Going to four spatial dimensions is probably too hard for just a story, but one could reasonably pull off a setting where there besides normal physics also are extra interactions (magic, ley lines, maybe something like axions, or whatever seems appropriate for the setting) that rather exhibits hysteresis — maybe because of obeying an even-dimensional wave equation analog, or maybe because there is a lasting imprint on materials as in magnetic hysteresis. One could argue that there for humans would be an evolutionary pressure against using the non-normal physics for senses (since such senses would report data that is no longer current), and hence these effects wouldn't be immediately apparent, but some animals could have other priorities; likewise genetics or proper training might give some individuals a certain sensitivity to it.
But as to your precise question, I too believe "yes, why not" to be a reasonable answer.
To add to The Daleks' answer:
Let's consider electrons and quarks, which are elementary particles.
A neutron is composed of two down quarks and one up quark; a free neutron (i.e.: not associated to an atomic nucleus) decays in a little less than fifteen minutes, resulting in the appearance of a new electron in our universe. For that electron to exist, the neutron had to be freed from a nucleus; without that event, the electron would not be.
Besides that, one of the down quarks in the proton becomes an up quark in the process (converting the neutron into a proton). That quark changing flavor is also dependent on the system history; it would remain a down quark if nothing had happened.
I therefore extrapolate this to all particle physics and propose that any universe with at least one time-like dimension is hysteretic.
