Is Spacetime Countable? by Sean Gryb

Sean,

Good to see another fine submission from you! I didn't have time to participate in the present contest, but I enjoyed your essay. Good luck with the contest, and take care,

Ben

Thanks, Sean. I won't pretend to speak for Carlo Rovelli, though I think I do understand what he means by degrees of freedom in an n-particle state space (I would say n-dimension state space and use the Hilbert space) where things happen discontinuously because the field assumes t = 1 and time coordinates are therefore constant (static).

In a continuous spacetime field, time changes with space -- dynamically, as you say. So I think what Carlo is saying, is that because a continuum theory such as shape dynamics cannot associate a definite dimensionless integer to a point of the space evolving in time, it is just as, if not more, meaningful to speak of the algebraic (Fock space) degrees of freedom as to speak of finite degrees of freedom in your continuously evolving system -- because the two descriptions, absent a simple time parameter of reversible trajectory, are equivalent.

Personally, I don't agree with Carlo, though I understand the Fock space construction. At the end of the day, I think that if shape dynamics is mathematically complete, it allows relative degrees of freedom; i.e., time reversibility that guarantees conservation of information. That's what the 't Hooft and Corda references are all about.

All best,

Tom

Thanks Sean. I know it's extremely late, but even if you don't get a chance to look at it before the deadline I would appreciate you reviewing it at some point in the future. You seem to have an open mind, so I would value your thoughts.

Best of luck to you in the finals.

Ralph

Hi Sean,

It's just the difference between algebra and analysis. If one speaks of a continuous energy spectrum, there's no point in the spectrum at which any element (any measured state at any instant) is discontinuous from the line. If one speaks of a state of particle interactions ("state" means "static") all degrees of freedom are discontinuous from the line, thus discrete.

The distinction is nontrivial, because finite degrees of freedom define particle states and predict an historical path (Feynman path integral). The ensemble transformation of a continuous measurement function such as you describe results, as you say, in the absence of fundamental discreteness -- so I think what Carlo is asking, is how can anything that is not fundamentally discrete be countable? One would apparently have to either eliminate the measurement function (and thus eliminate the physics), or allow that countable bits of spacetime are already described by evolution of the state vector in standard quantum theory.

I'm on your side of the question, with a caveat: one must choose which element of continuous spacetime -- time or space -- is finite. General relativity conventionally chooses time (the universe has a finite beginning and a finite end). If one chooses space, which surely shape dynamics does, infinite varieties of time-dependent shapes of a finite space potentially describe a universally entangled wavefunction of continuous measurement values, without assuming discrete particles with discrete degrees of freedom.

All best,

Tom

Dear Sean Gryb:

I am an old physician, and I don't know nothing of mathematics and almost nothing of physics, but after the common people your discipline is the one that uses more the so called "time" than any other.

I am sending you a practical summary, so you can easy decide if you read or not my essay "The deep nature of reality".

I am convince you would be interested in reading it. ( most people don't understand it, and is not just because of my bad English). Hawking, "A brief history of time" where he said , "Which is the nature of time?" yes he don't know what time is, and also continue saying............Some day this answer could seem to us "obvious", as much than that the earth rotate around the sun....." In fact the answer is "obvious", but how he could say that, if he didn't know what's time? In fact he is predicting that is going to be an answer, and that this one will be "obvious", I think that with this adjective, he is implying: simple and easy to understand. Maybe he felt it and couldn't explain it with words. We have anthropologic proves that man measure "time" since more than 30.000 years ago, much, much later came science, mathematics and physics that learn to measure "time" from primitive men, adopted the idea and the systems of measurement, but also acquired the incognita of the experimental "time" meaning. Out of common use physics is the science that needs and use more the measurement of what everybody calls "time" and the discipline came to believe it as their own. I always said that to understand the "time" experimental meaning there is not need to know mathematics or physics, as the "time" creators and users didn't. Instead of my opinion I would give Einstein's "Ideas and Opinions" pg. 354 "Space, time, and event, are free creations of human intelligence, tools of thought" he use to call them pre-scientific concepts from which mankind forgot its meanings, he never wrote a whole page about "time" he also use to evade the use of the word, in general relativity when he refer how gravitational force and speed affect "time", he does not use the word "time" instead he would say, speed and gravitational force slows clock movement or "motion", instead of saying that slows "time". FQXi member Andreas Albrecht said that. When asked the question, "What is time?", Einstein gave a pragmatic response: "Time," he said, "is what clocks measure and nothing more." He knew that "time" was a man creation, but he didn't know what man is measuring with the clock.

I insist, that for "measuring motion" we should always and only use a unique: "constant" or "uniform" "motion" to measure "no constant motions" "which integrates and form part of every change and transformation in every physical thing. Why? because is the only kind of "motion" whose characteristics allow it, to be divided in equal parts as Egyptians and Sumerians did it, giving born to "motion fractions", which I call "motion units" as hours, minutes and seconds. "Motion" which is the real thing, was always hide behind time, and covert by its shadow, it was hide in front everybody eyes, during at least two millenniums at hand of almost everybody. Which is the difference in physics between using the so-called time or using "motion"?, time just has been used to measure the "duration" of different phenomena, why only for that? Because it was impossible for physicists to relate a mysterious time with the rest of the physical elements of known characteristics, without knowing what time is and which its physical characteristics were. On the other hand "motion" is not something mysterious, it is a quality or physical property of all things, and can be related with all of them, this is a huge difference especially for theoretical physics I believe. I as a physician with this find I was able to do quite a few things. I imagine a physicist with this can make marvelous things.

With my best whishes

Héctor