Reality as a Vector in Hilbert Space
arxiv.org
arxiv.org
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.
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.
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.
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”.
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.
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.
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.
In Hilbert space, nobody can hear you scream. – Aharonov & Rohrlich (2005)I need to see what an eigenspectrum looks like!
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.
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.
Magic. Reversing the relationship between reality and our interpretation of it literally means bending reality with ideas. In this case magic is matrix multiplication.
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.
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.
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.
What makes you think your perception of reality is not the map in this dichotomy?
Wha¿
Souls, if they exist (heh), are subject to the same physical laws as bubbles, sand, and frogs.
Why would there be an immortal information processor centered around continuing human thoughts ? So we can debate MAGA politics with Cro-Magnons?
"Souls" psch
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