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Reality as a Vector in Hilbert Space (arxiv.org)
51 points by nabla9 23 days ago | hide | past | favorite | 45 comments

The author is https://en.wikipedia.org/wiki/Sean_M._Carroll

Shortly after Covid19 started last year, he started a series of videos called "The Biggest Ideas in the Universe", playlist here: https://www.youtube.com/watch?v=HI09kat_GeI&list=PLrxfgDEc2N...

I've watched quite a lot of this so-called 'popular' hard science material on youtube, and this series was by far the best I've run across. His approach is a bit different from most others, and his presentation is engaging, approachable but also fairly substantial. Specifically, he doesn't elide all of the relevant math, but doesn't get too deep at the same time. You'll see a lot of integrals, derivatives and graphs.

For example, he talks about an hour on the big idea of 'Change', and another "Space".

Perhaps even more interesting are the followup Q&A videos, which are as long as and perhaps even more detailed than the source.

Do give the linked six minute intro video a watch.

Someone asked in a deleted comment, something along the lines of "Which Hilbert space?", pointing out that a Hilbert space is just a vector space which has an inner product, and is complete under the metric induced by the norm. I wrote a response to this, and while the initial comment was deleted (presumably by the author of the comment), I think my response might still be informative to others who have the same question. It is as follows:

All countably infinite dimensional Hilbert spaces are isomorphically isometric. Pick an orthonormal basis for it, and that gives you an isomorphism with \ell^2(\mathbb{N}) (which preserves the metric).

As such, some people will sometimes just refer to "the Hilbert space", to refer to any countably infinite dimensional Hilbert space over the complex numbers.

If the Hilbert space in question is countably dimensional, then this is "the" one they mean.

I think there are some reasons that some people say one might have to use one with an uncountable dimension? I don't understand these reasons. I think they are connected to quantum field theory? My impression is that it is somewhat controversial whether one with uncountably infinite dimension is ever necessary to model physics. Wikipedia has more to say about this question. Because I don't understand it, you would be better off reading it on wikipedia than reading me trying to summarize something written on Wikipedia which I don't understand.

edit : upon actually reading more of the link, I might see why the comment I meant to reply to was deleted : the paper answers their question, saying that the Hilbert space is determined by the dimension.

So, my comment is a bit redundant.

edit2: Oh, and, furthermore, the article says that in a bounded region, we have reason to expect the dimension of the space to be finite dimensional? This surprises me.

The paper itself says the dimension of the Hilbert space is e^(e^123), so very big but not infinite, countably or otherwise.

That said, I also do not remotely understand this paper. I think I at least grasp the basics of quantum theory itself modeling the very small world as a Hilbert space that collapses to a single real-valued vector upon being measured and the math checks out according to what it predicts and what we actually measure, but after reading this I cannot tell why this should have any consequences for ontology.

I think I understand the basic desire physicists have to include an ontology in their theories. Old-school billiard ball physics and even general relativity have very obvious and intuitive analogies to basic geometric objects we can envision. We definitely get a map with the math and at least a clear allusion to the territory by reifying these geometric analogies. With quantum physics, nothing at all comes out as some clear candidate for a territory. So this guy is just going galaxy brain and proposing maybe there is no territory at all and the most ontologically basic thing from which all observable phenomena we actually experience arises is just the math itself.

I don't really see a justification in the paper, but that's kind of the nature of philosophy. At the most fundamental level, nothing can be justified.

This question is addressed directly by Griffiths in his “Introduction to Quantum Mechanics”. It’s in a footnote at the beginning of chapter 3, “Formalisms”.

Basically he says that when physicists say “Hilbert space”, they’re actually referring to a specific Hilbert space, namely the set of all square integrable functions, sometimes referred to as “L2”.

> Oh, and, furthermore, the article says that in a bounded region, we have reason to expect the dimension of the space to be finite dimensional? This surprises me.

Does it? Iirc a classic basis for a countably infinite Hilbert space is a Fourier series. If you have a finite space to describe, and a maximum frequency you need to represent (which I would say is reasonable to assume), then you only need a finite dimensional space.

I guess the maximum frequency is the surprising thing to me?

Idk, I guess I thought maybe the limited information was, not really limiting the dimension?

Or, maybe I thought because of no clean objective separation between the observable universe, and the rest of it, that that would cause an issue? Not really sure what I was thinking.

I guess it makes sense that it would be finite dimensional.

Hm, actually, I am confused about what counts as the observable universe. Is it that which is in our past light cone, or, that which is in a future light cone of our past light cone, or?

Also, there’s the issue of time in different reference frames. If the state of the observable universe, is supposed to be given by a vector in the same Hilbert space at each time t, uh, well, that appears to depend on the reference frame. Like, the points in space time that are “simultaneous” depends on the reference frame.

So, a vector which describes the state of the universe in a given time slice in a given reference frame, I don’t see how it would also describe it for a different reference frame.

Arisibalt trudged to the kitchen and busied himself at the keg. "I'd better fortify myself," he explained, to no one in particular, "as I am condemned to spend the remainder of the evening drawing light bubbles."

For some reason most of my comment contributions here are references to N.S. Glad to see I'm not the only one who immediately thought of this.

it comes with the username :)

Someone just finished reading Neal Stephenson’s Anathem...

This abstract also describes a central plot point to Greg Egan's "Schild's Ladder".

I make my experimental apparatus. I send an electron through a double slit. Hilbert space tells me it goes through both, and keeps going through both for all eternity. I only see one flash. The whole of the electron is found to have hit that part of the screen, and nowhere else. This interpretation just kicks the can down the road, to the "emergence of consciousness", in order to explain simple experimental results.

Is this a, uh, thing that's going around?

I'm a physics undergrad who published, and the idea of boson field(s) being the source of all physics has been really top of mind. Maybe it's just a consequence of Higg's being discovered finally.

.. the fundamental ontology of the world consists of a vector in Hilbert space evolving according to the Schrödinger equation. The laws of physics are determined solely by the energy eigenspectrum of the Hamiltonian. The structure of our observed world, including space and fields living within it, should arise as a higher-level emergent description ..


It's a statement of pure materialism: Nothing exists outside solutions to that equation. All subjective human "artifacts" we struggle to identify, e.g., consciousness, are emergent properties of their material substrates.

Souls, if they exist (heh), are subject to the same physical laws as bubbles, sand, and frogs.

Emergence is a handwave of colossal proportions that is easily revealed by asking "what is this emergence and where does it come from?" If you're going to be a pure materialist you don't get to just say oh it magically comes from nowhere, you have to show the mechanism. The cellular automata guys can be seen as giving that an intellectually honest shot, but I bet plenty of them are actually dualists anyhow.

Souls? What is this religious idiocy even doing here. There s no if, ofc information doesnt continue to evolve coherently after it's thinking processor disappears.

Why would there be an immortal information processor centered around continuing human thoughts ? So we can debate MAGA politics with Cro-Magnons?

"Souls" psch

didn't read it - and certainly wouldn't understand it if i did - but it does gel nicely with my internal pseudoscientific ponderings about the universe. nice! (this is part tongue-in-cheek and part serious; i often wonder: "what if we are simply tumbling along some "existential gravity" in a higher dimension?")

I love the second quote:

     In Hilbert space, nobody can hear you scream. – Aharonov & Rohrlich (2005)

There's no pictures in the pre-print, how am I supposed to understand this?

I need to see what an eigenspectrum looks like!

Can someone try to ELI5 this for me?

They are attempting to describe physical reality with a single wave equation, and have constructed one with a lot of interesting properties, and their goal is to break the hamiltonian down into independent components to describe physical systems.

The system is a finite-dimensioned Hilbert space (but it's large, like, almost infinite, but being finite is an important feature according to the paper) - and each dimension can be described by an energy spectrum, it's that energy which gives rise to physical reality via fields (and the properties that emerge from those fields).

They propose basically a needle-in-the-haystack approach to finding mathematical solutions to the individual energy dimensions, I guess.

There is a very excellent tutorial on this subject in Anathem, but Neal Stephenson. Great book too; definitely worth reading.

Thanks, I'll check it out!

Just a heads up, the analog in Anathem is called "Hemn spaces"

Relevant XKCD: https://xkcd.com/505/

This is idealism getting lost in the sauce to the highest degree. If you ever find yourself thinking "reality is just math", you must remember: the map is not the territory.

Right, but his whole point is to defend the idea of saying, "what if the math is the reality, what happens then?" It's sort of the same jump that pushed QM forward in the first place of discarding all the assumptions about particles and orbits and just went purely to what was necessary directly in the math.

In the paper he advocates going farther and saying, there's no independent "space" just the vector and space(time) emerges purely from the evolution of that vector.

If your territory is a map, then the map might be the territory.

>what if the math is the reality, what happens then?

Magic. Reversing the relationship between reality and our interpretation of it literally means bending reality with ideas. In this case magic is matrix multiplication.

Yeah. I think the downside is that you end up even further into the "shut-up and compute" paradigm of physics that we are already in.

But if you get predictions from the theory that turn out to be true in the end (like entanglement, Bell's inequality, etc) then it gives you a lot of confidence in your theory.

I don't know how it'll turn out. At this point people are throwing stuff against the wall to see what sticks to get physics out of the local minima it seems to have found itself in.

I don't think Carroll or anyone else is naively wandering into this endeavor from a philosophical standpoint though.

I fully agree to the last point. He had (and probably/hopefully continues to have) very interesting discussions on all kinds of topics ranging from physics to biology to philosophy to psychology to social topics to ... in his mindscape podcast https://www.preposterousuniverse.com/podcast/

Sure, but I think the analogy breaks down in this case.Here the map contains precisely the same amount of information as the territory, no reduction of dimensionality,no loss of resolution. Everything that can be said of the territory can be said of the map and we're left with a case in which the only unique non-shared property of the territory is "physical existence", whatever that is.

Yeah imagine instead of a map you have a 3d model of space time. At a certain degree of granularity the model becomes isomorphic to "reality."

I may be risking sounding like a fool here, but isn't there kind of a real distinction between saying that there are underlying mathematical patterns in a reality that behaves like a state machine, and saying that literally this function and starting condition are isomorphic to the universe?

Like if on a low level it turns out that everything is, I don't know, cellular-automata-based, that would be a more useful frame than wandering vectors in a Hilbert space. But if we get to the end of physics and found that, in fact, you could rederive everything from that one explanation, it wouldn't be unfair to say that the universe is that vector, and perhaps some functions.

I guess really I'm wondering if it matters more how the universe is 'computed' or what it's 'computed on'. In a non-simulated universe, of which it seems there must exist at least one, there would at some point just have to be laws without causes. If those laws corresponded to some simple bit of math, it wouldn't be wrong to merge map and territory.

edit: That's not to say that I actually think the paper is correct. I'm definitely not far enough along in my education to be 100% sure, but there are enough suspicious features and a high enough bar that I'm doubtful. I just meant to mount a general defense of 'what if it's all math' type explanations.

Imagine living in a virtual reality and hearing that the reality is just a computation.

Would it be metaphysical? Not really, because it does not speak about anything but the computation which, by construction, is the reality of such universe. It claims no knowledge of the computer implementation details which are mot reflected in the computation, like its power consumption or the color of the chassis, or even the fact that a chassis might exist.

I think that sticking to intuitive and customary notions of reality when experimental data contradict them is not scientific. Science is all about building a better model that fits more and more experimental data points as tightly as possible. If such model introduces notions that the mundane experience entirely lacks, while still describing these mundane phenomena perfectly, it's time to admit that the reality is indeed "weird", and it's your mundane intuitions which are wrong (or, rather, apply narrowly).

Also, the end result of science is the math that fits the experiments, that is, the model of reality, the language of it is math. You can ask whether there's something behind that math, like you might ask whether there's something behind the computations in a virtual reality universe. I assume we can't see past this border with scientific means, only with metaphysical.

> If you ever find yourself thinking "reality is just math", you must remember: the map is not the territory.

What makes you think your perception of reality is not the map in this dichotomy?

You should read Anathem or Permutation City some time.

I have read Anathem. It's math as occult magic riffing on harry potter.

Pretty much anything can be dismissed with an adlib of the form "It's [boring subject] as [thing I don't like] riffing on [some other thing]." It never captures the nuance, or why X as Y riffing on Z might still be interesting or enlightening.

What will he get from Permutation City? It's a work of fiction, entirely supposition.

But it helps you realize there's nothing obvious about reality. We can question everything, even that maybe we're experiencing reality in a time based manner due to a fluke but theres actually no evolution - we're a static map of space time, and us idiots pretend we evolve and change direction.

Grandiose paper from a joke "scientist". This definitely not science, it's a poorly argued metaphysical proposition.

Sean Carroll is a highly respected physicist. Not sure what you're on about.

Schrödinger equation does not even account for relativistic effects (even though the author pointed out the opposite in the text)... anyone who has read basic texts on philosophy of science knows that this kind of proposition, "it explains reality", is completely "out of reality". it cannot be done by the simple fact that we do not know if any theory will remain as the best explanation for a bunch of experimental facts in the future (in a way, the equation mentioned did not predict even spin or antiparticles). there is just no way to prove this, it is a futile and useless exercise of futurology masked as complex linguistic mumbo-jumbo and not at all impressive physics knowledge...


Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space, CJ Cao, SM Carroll - Physical Review D, 2018 https://journals.aps.org/prd/pdf/10.1103/PhysRevD.97.086003

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