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this is my first question here. A Universe in my setting is made of at least a countably infinite-dimensional real coordinate space; and I want to know how our physics (quantum field theory/string theory) would function within it. As far as I know, Hilbert spaces are essential to quantum mechanics in some way; but I haven't gotten anything helpful online. Can anyone help with this?

L.Dutch
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初音ミク
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    Welcome 初音ミク. Please take our [tour] and refer to the [help] for our guidelines. There's a related question: Two time dimensions and thermodynamics. The answer there seems to address your question at least in part. – Escaped dental patient. Oct 10 '22 at 02:18
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    Hello 初音ミク, welcome to [worldbuilding.se]. Asking how physics (a very broad subject) would operate in an infinite-dimensional universe (we've never seen one, unless we live in one, but we can't prove it either way) might violate our [help/dont-ask] book rule, which means it's likely too broad for our Stack. This might actually be a better question for [physics.se]. Can you be more specific about what you need to know? – JBH Oct 10 '22 at 05:01
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    @JBH: It wouldn't necessarily need a book length answer. In fact, the answer is very short. Our physics, or anything like it, cannot work in an infinitely dimensional universe, for the trivial reason that calculus doesn't work. I would strongly suggest the querent to first work out how to compute simple things, such as, for example, kinetic energy in such a universe; or, just as simple, to show how conservation of momentum would work. – AlexP Oct 10 '22 at 08:13
  • You are dealing with aspects of physics here which are not even fully understood by the best people in the field. Do you really think your audience would care about scientific plausibility on that level? – Philipp Oct 10 '22 at 11:04
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    @AlexP The [help/dont-ask] states, "If you can imagine an entire book that answers your question, you’re asking too much." I can easily imagine the question as asked at the time of my comment being answered with a library of books. If we use "wouldn't necessarily" as the criteria, then no question ever fails the book test. – JBH Oct 10 '22 at 15:14
  • I'm with @AlexP on this one. The answer is simple: can't be done. I put said answer on the list. It's all good. – Robert Rapplean Oct 10 '22 at 19:07
  • Physics won't work in an "infinite" dimensional universe. All concepts of measurement go out the window. A simple distance metric fails because you have to take the "infinite root" of an "infinite series," so the very concept of "space" itself breaks down.

    Without length, you have no speed (v = d/t). Without speed you have no energy (Ke = 1/2 mv^2).

    Infinities are VERY bad in any physical model because they destroy the model's predictive ability. With an infinity, literally anything can happen, so the theory is useless.

     – stix Oct 10 '22 at 19:24
  • @RobertRapplean Given that what an "Infinite dimensional universe" is an undefined term, I'd argue that this question is more unanswerable, than anything else. In your answer you're not saying "there won't be physics" you're saying that some of the phenomena will not work as expected with more than 3 physical dimensions. Which isn't what OP asks. – sphennings Oct 10 '22 at 19:32
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    @AlexP Joining JBH for closure, because the querent asked "how to do it", not "can we do it". Saying that it cannot (or can) is answering the latter, not the first . Indeed and as you explained, answering "How" would require to think up quite a lot of new systems outside our knowledge to make it worky work ^^". And even if it was the "can it" question, I fear that to fully grasp why it couldn't be induced you'd need to explain quite a lot of things :). – Tortliena - inactive Oct 10 '22 at 21:19
  • @ 初音ミク Unfortunately I don't know in details exactly how you can develop this limitless dimensional universe, this goes quite beyond my expertise. However, I do know that cutting things into smaller parts really helps. AlexP's advise is a good start : What would you like to see happen through your story? For instance if you want battles, focus on kinetics and newtonian physics stuff. If you want the story set in multi-dimensional seas and pirates, then look at fluid dynamics. Hopefully you'll manage to sort this very, very complex stuff out ;). – Tortliena - inactive Oct 10 '22 at 21:25

2 Answers2

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Having more time dimension or more spatial dimensions seems to be leading to the same problems. Quoting verbatim from my answer, and just look at the chart with more spatial dimensions, a universe with 1 time dimension and infinite spatial dimensions seems to be unstable:

Scientists have been exploring the consequences of having more dimensions, both on the space-side N and on the time-side T.

According to this wikipedia page

If T differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) N = 1 and T = 3 has the peculiar property that the speed of light in a vacuum is a lower bound on the velocity of matter; all matter consists of tachyons

enter image description here

It seems therefore that, generally speaking, we cannot know. Even if we can come with some differential equations, those won't help to predict the story and the evolution of the modeled system.

Also this paper seems to suggests that in certain conditions first and second law of thermodynamics can be derived, but the second law cannot be satisfied.

In this work, we have considered the Vaidya spacetime in null radiating fluid with perfect fluid in higher dimension and have found the solution for barotropic fluid. We have shown that the Einstein's field equations can be obtained from Unified first law i.e., field equations and unified first law are equivalent. The first law of thermodynamics has also been constructed by Unified first law. From this, the variation of entropy function has been derived on the horizon. The variation of entropy function inside the horizon has been derived using Gibb's law of thermodynamics. So the total variation of entropy function has been constructed at apparent and event horizons both. If we do not assume the first law, then the entropy on the both horizons can be considered by area law and the variation of total entropy has been found at both the horizons. Also the validity of generalized second law (GSL) of thermodynamics has been examined at both apparent and event horizons by using the first law and the area law separately. When we use first law of thermodynamics and Bekenstein-Hawking area law of thermodynamics, the GSL for apparent horizon in any dimensions are satisfied, but the GSL for event horizon can not be satisfied in any dimensions.

L.Dutch
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Can't be done. Here's why.

  1. Energy and field propagation. In our three dimensional universe, gravity, light, and electromagnetism propagate based on the square of the distance. That's the number of physical dimensions - 1. If you had an infinite number of dimensions, then propagation would be essentially non-existent. All things would either be dissociated or a singularity. It occurs to me that the entire universe would act like dark matter.

  2. No orbits. In three dimensions, we have stable orbits around a gravitation generating object. In five dimensions, there is no such thing. Orbits are quasi-stable. The orbiting objects either fly off or merge with the object they're orbiting. With infinite dimensions, even if you do solve the propagation issue, the math won't allow orbits.

Thus, you have a situation that is entirely unimaginable, and certainly unrelatable. The factors that go into a story (persons or settings, for instance) would be incapable of existing.

Robert Rapplean
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  • Seems like physics still works in infinite spatial dimensions but some common phenomena that we experience in 3 dimensions won't occur. Remember that OP asked how the physics would work, not would orbits, or energy and field propagation. I'm surprised that you think that a story must be bounded by what is physically possible. There's a romance in many dimensions that starts out describing a 2 dimensional world that is entirely impossible with our current understanding of physics, but somehow the story still exists. As worldbuilders things are only impossible when we allow them to be so. – sphennings Oct 11 '22 at 03:59
  • @sphennings, A story only needs to be bounded by what is physically possible if it calls itself science fiction. Most stories are bound by what is plausible. It might be a measure of good fantasy if it makes particularly bizarre circumstances plausible. Honestly, I have more fun making the mythological into the mundane than I do making the incomprehensible into something entertaining. – Robert Rapplean Oct 11 '22 at 04:05
  • That's fine but it has nothing to do with OP's question. – sphennings Oct 11 '22 at 04:35
  • It has everything to do with the OP's question. He asked how physics would work, and this is the world building forum, so that is a presumed context. How it works for world building is that no building of any kind is possible. Anything that entered an infinite dimension space would immediately disintegrate. If he wanted the physics of how it would work, he should have posted his question on a physics forum. The quantum mechanics answer would be that particles would be impossible, too. – Robert Rapplean Oct 11 '22 at 15:13
  • Your thesis, "it can't be done" is not an answer to the question "How will physics work?" if OP was asking "Can a human survive?" or "Would it be recognizable?" sure. But that's not what OP asked. Note how your explanation for why 3 dimensional constructs will not work, are describing the physics of an infinitely dimensional space. Physics will work. It may be alien to us. It may be boring or uninteresting for reasons you've described. But, by definition there will be physics. While I do think the question is too broad, we do cater to questions about non-standard physics on this site. – sphennings Oct 11 '22 at 16:06
  • I'm sure what you're looking for is "instantly destructive to normal matter." That answer would be correct for higher dimensions, but I think you underestimate the problem with infinite dimensions. Try to perform the Pythagorean theorem with an infinite number of terms. It's like doing math with a singularity, and produces the kind of impossibility that this forum usually discards as resulting in "opinion based" answers.

    Generally speaking, I dislike turning people away with a flip negation, and prefer to provide information that allows people to wrap their head around the problem.

     – Robert Rapplean Oct 11 '22 at 19:15
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    Physics, as we define it, literally does not work with infinite dimensions. Our math breaks if you try. – Robert Rapplean Oct 11 '22 at 19:19
  • Maybe describe how the math breaks then. I suspect that you'll have more strange behaviors and less it never works, since you definitely can get infinite series that converge to finite values. For instance (0, 0, ... ,0) and (1, 0, ... , 0) are exactly 1 unit apart. – sphennings Oct 11 '22 at 21:09
  • From my perspective, I did describe the math, but in a way that most people could relate to. Throwing a pile of numbers and formulas tends to shut peoples' brains off before they can absorb it. Your concept of having serieses that converge to finite values doesn't work when you're talking gravity, fields, or quantum effects. If you want to talk about "infinite dimensions available, but we only use 3" then it's indistinguishable from having three. I actually have a significant portion of my next book dedicated to "4 dimensions, but we only use 3," so that's demonstrably story related. – Robert Rapplean Oct 11 '22 at 22:27