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FQXi FORUM
October 24, 2019

CATEGORY: Questioning the Foundations Essay Contest (2012) [back]
TOPIC: The Illusion of Hilbert Space by Brian Swingle [refresh]
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Author Brian Swingle wrote on Sep. 7, 2012 @ 11:31 GMT
Essay Abstract

I argue that Hilbert space, one of the foundational elements of quantum physics, is unphysical in the context of quantum many-body systems. Physical states reside in a tiny corner of Hilbert and are not best thought of as exponentially long vectors. The important question is how to characterize the space of physical states, and I suggest that it may be useful to take a quantum computer's view of the world. Finally, I apply this reasoning in a specific case to obtain a description of the universal aspects of quantum ground states in terms of an emergent entanglement geometry.

Author Bio

I obtained my PhD from MIT in 2011 and am currently a Simons Fellow at Harvard University.

Download Essay PDF File

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Edwin Eugene Klingman wrote on Sep. 7, 2012 @ 23:04 GMT
Dear Brian Swingle,

I found your essay extremely fascinating and agree with most of your main points. I come to essentially the same conclusion about Hilbert space in my essay,

The Nature of the Wave Function,

i.e., while useful, Hilbert space is not physical. While you focus on spin systems and I look more at atomic orbitals and particle collisions with atoms, we arrive at more or less the same place. I do not use the "quantum computer" analogies that you do, but I find them appropriate.

The key idea is to recognize that Hilbert space is *unphysical* in the space of many-body systems. Your observation that most states in Hilbert space cannot be reached within the age of the universe is very convincing, and supports the idea that

"since most of the states in Hilbert space are unphysical, Hilbert space is a kind of illusion."

I am also happy with your treatment of locality on page 5, and found your discussion of "thinking like a quantum computer" to be very informative. It helped me understand your point about entanglement renormalization. In this sense we complement each other: I focus on historical mis-interpretations while you focus on the very latest conceptual domains of QM.

Brian, I hope that my own essay will be as helpful to you in illuminating key aspects of quantum mechanics as yours was for me. In particular, I hope that entanglement will be 'physical' to you in the context I present. I would very much appreciate your comments on my essay.

Thanks again, and good luck in the competition.

Edwin Eugene Klingman

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Author Brian Swingle replied on Sep. 11, 2012 @ 21:42 GMT
Dear Edwin,

Thank you for reading my essay and for your kind comments. I'm glad you found something interesting in it.

Good luck to you as well.

Best,

Brian

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Andrew Tan wrote on Sep. 7, 2012 @ 23:38 GMT
Is this related to churches of the larger and smaller Hilbert spaces?

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Author Brian Swingle replied on Sep. 11, 2012 @ 21:47 GMT
No, I don't think so.

As far as I know, the churches are related to different perspectives on quantum channels. A quantum channel is just a generalized transformation on density matrices that is supposed to be the quantum analog of a classical noisy channel. I think the churches argue about how seriously to take a particular decomposition of a quantum channel in terms of things called Kraus operators, about the physical meaning of various purifications, and so forth.

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Eckard Blumschein wrote on Sep. 8, 2012 @ 03:21 GMT
Dear Brian Swingle,

Isn't the title "THE illusion ..." misleading? Von Neumann's 1935 disbelief can perhaps not be ascribed to the view that Hilbert space is in some respect too big. You did not even mention Goedel.

I consider my own criticism of Hilbert's program and other views of Hilbert much much less cultivated than your elegant quantum theoretical approach. I would not even firmly believe in quantum computing until it works as promised after so much effort was spent and the first quantum computers were announced some years ago.

Eckard

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Author Brian Swingle replied on Sep. 11, 2012 @ 21:57 GMT
Dear Eckard,

Thank you for reading my essay and for your interesting comments.

My I ask to what your comment about Goedel refers? I'm curious.

I certainly agree that we may be sometime away from a large scale quantum computer, however, I think that quantum error correction has shown that it is in principle possible to build such a computer.

The cool thing from my perspective, as I tried to argue in the essay, is that we can still learn about the physics of quantum many-body systems just by thinking about the theoretical structure of quantum computation.

Best,

Brian

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Eckard Blumschein replied on Sep. 12, 2012 @ 08:37 GMT
Dear Brian,

You certainly know plato.stanford.edu/entries/hilbert-program. What about my own opinion, I would like to mention that I used Hilbert transformation in order to get some MATLAB results shown in previous essays of mine while my present essay contains several utterances critical to Hilbert.

My main objection was best formulated here by Alan Kadin.

Best,

Eckard

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Dean L Waters wrote on Sep. 9, 2012 @ 02:05 GMT
Dear Brian Swingle,

Quite beautiful! The method you follow and the accompanying illustration beautifully map out something I have been attempting to 'see' and draw in my notebooks: the behavior in Hilbert space of a few elements or at other times something like a |giant-mess>!

In short, I have been wrestling with how to intuit and interpret the difference between the huge, highly...

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Author Brian Swingle replied on Sep. 11, 2012 @ 22:02 GMT
Dear Dean,

Thanks for reading my essay and for your kind comments. I like the idea of a Hilbert sea, as it conjures an expansive feeling that I feel is very appropriate

Good luck in the contest.

Best,

Brian

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Jayakar Johnson Joseph wrote on Sep. 9, 2012 @ 12:58 GMT
Dear Brian Swingle,

I think, Hilbert space is much applicable only to describe the dynamics of discrete particles, whereas to describe the volume of finite or infinite tetrahedral-branes assigned in Coherently-cyclic cluster-matter paradigm of universe, the dimensions of each tetrahedral-brane in that volume is defined individually by two intersecting Euclidean planes and in entirety 3-D spatial structure of that volume is expressional.

With best wishes,

Jayakar

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Author Brian Swingle replied on Sep. 11, 2012 @ 22:05 GMT
Dear Jayakar,

Thanks for your interesting comments, although I'm afraid I don't understand much of it.

Good luck in the competition though.

Best,

Brian

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Harlan Swyers wrote on Sep. 9, 2012 @ 23:20 GMT
Brian,

Its an interesting essay, but I can not agree with your position

First, there are a couple confused issues here. It seems that you are starting out with the premise that hilbert space was proposed to be physical in the first place, which I would argue that is not in any way consistent with quantum mechanics to begin with, so its a false argument. Hilbert space is merely...

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Anonymous replied on Sep. 12, 2012 @ 00:06 GMT
Dear Harlan,

Thank you for reading my essay and for your criticisms. I don't know what you mean when you say Hilbert space wasn't physical to begin with since it does capture describe the manifold of physical states for systems of few degrees of freedom.

The comment about keeping Hilbert space around is meant to convey the following. QM remains intact in that I don't formally change the structure of the theory e.g. for few body systems. Instead, I argue that even without modifying the physics of QM, an entirely new structure emerges in many-body systems.

Finally, I certainly agree with you that the universe should not be conceptualized as a classical computer. It was not my intention to convey that. The universe is clearly a quantum computer! So I think you have understood me perfectly if you understood the weakness of "classical thinking". We should think like quantum computers.

Thanks again for your interest and your comments.

Best,

Brian

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Harlan Swyers replied on Sep. 12, 2012 @ 12:47 GMT
Brian,

Thanks for the response.

Hilbert space may be used as a abstract space to put information regarding physical states, but as you pointed out there is ample space for unphysical ones as well. This in itself is not a new insight. Surely you can't suggest that those mathematicians that developed quantum mechanics didn't understand this?!

Physical certainly is a derivative...

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Eckard Blumschein replied on Sep. 12, 2012 @ 14:00 GMT
"we cannot abandon Hilbert space as fundamental to our understanding" ??? Even von Neumann, who had coined the notion Hilbert space, confessed in 1935 in a letter to Birkhoff that he did not believe in Hilbert space any more.

Does the rigged Hilbert space solve that problem? Perhaps not. Let me point to the central role of mathematicians like Hilbert for the development physical theories. Hilbert's disciple von Neumann inherited the monist block universe. Hilbert used to invite those physicists to Goettingen that he considered progressive in the sense they contributed to his ideas. Woldemar Voigt supported Hilbert with money. While Hilbert claimed priority for GR, and Einstein disliked Hilbert's behavior towards Brouwer, they nonetheless altogether supported set theory (ST) and Einstein's SR.

Virtually all those who will judge my essay were trained to adhere ST and SR. Opponents are still blamed to be cranks. I can only try and do my best by revealing and focusing on the perhaps decisive mistakes going back to the late 19th century which were then merely institutionalized in the 20th century.

I quoted Bruhn as an example for lazy prejudices and Spalt as an example for unwelcome careful work.

Judge about Hilbert's program and Goedel's antithesis yourself.

Eckard

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Vladimir F. Tamari wrote on Sep. 10, 2012 @ 02:38 GMT
Dear Brian,

Phew what a relief to read your essay! As a non-academic but serious researcher in physics I was always intimidated by the (i) in Schrodinger's equation, and by its Hilbert space interpretation. It went completely against my intuitions that there is exquisite physical order and local linear causal relations at the smallest scale in Nature. There is no room for Hilbert spaces in my model, and the Born probabilistic interpretation emerges from its lattice geometry rather similar to how it works in your figure. I have attached Figs. 28 & 29 of Beautiful Universe Theory on which I based my fqxi essay Fix Physics! .

I would appreciate it if your take the time to read my essays and give your expert assessment of the model I have presented, qualitative and incomplete as it might be.

Best wishes

Vladimir

attachments: BUFIG29.jpg, 1_BUFIG28.jpg

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Author Brian Swingle replied on Sep. 12, 2012 @ 00:08 GMT
Dear Vladimir,

Thanks for reading my essay and for your kind comments. I shall try to take a look at your essay.

Good luck in the competition.

Best,

Brian

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Member Benjamin F. Dribus wrote on Sep. 10, 2012 @ 05:54 GMT
Brain,

I really enjoyed your essay. It provides a new perspective on some issues I have thought about a good deal. A few questions.

1. Have you thought much about applying ideas like these to the fundamental structure of spacetime? My guess would be yes, since the ideas are rather universal, and since you relate them to general information-theoretic principles like holography.

2. A related question: have you thought about applying things like causal graphs to these ideas? Even in the quantum Ising model, the whole point is causal locality. Ideas like these can be useful even for quantum computing; for instance, the quantum CNOT gate corresponds to an "N-shaped" causal graph, and as you mentioned, choosing also an appropriate single-qubit gate gives you a universal family of gates. Hence, any quantum computer corresponds to a causal graph. Personally, I have thought about trying to turn this around and use quantum computers as "relatively macroscopic models" of the fundamental scale; see for instance the last section of my essay On the Foundational Assumptions of Modern Physics.

3. Have you thought about how this idea applies to Feynman's sum-over-histories method? In his 1948 paper, Feynman rederived Hilbert space, operator algebras, the Schrodinger equation, etc., by summing over "spacetime paths." One of the most vexing problems in QFT is evaluating the resulting path integrals. Information-theoretically, it seems that if you can ignore all but a tiny fraction of the Hilbert space, you might be able to organize the information in the path integrals in a more convenient way as well. Maybe this would produce nothing new, since that's basically the point of a lot of the existing techniques, but it's interesting to consider.

I wish you the best of luck with the contest! Take care,

Ben Dribus

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Member Benjamin F. Dribus replied on Sep. 10, 2012 @ 07:10 GMT
"Brian," I mean, of course... at least "Brain" isn't an insult in this context!

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Author Brian Swingle replied on Sep. 12, 2012 @ 00:13 GMT
Dear Ben,

Thanks for your interest and your kind comments.

1. Yes, I have certainly thought about this issue. I find it a really exciting idea. You can read more about my early attempts in this direction here http://prd.aps.org/abstract/PRD/v86/i6/e065007 or on the arxiv (older version).

2. This sounds quite interesting, but I haven't thought much about it. I shall try to take a look at your essay.

3. This is definitely intriguing. It would be nice if there were a way to sum over much fewer amplitudes and get the same answer. Such a method might have profound computational/numerical consequence.

Best,

Brian

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Alan Kadin wrote on Sep. 10, 2012 @ 13:16 GMT
Dr. Swingle:

I like your title - yours is the only essay with "Hilbert" in the title. And your essay is quite interesting. But I think your challenge to Hilbert Space is not sufficiently fundamental.

In contrast, my essay ("The Rise and Fall of Wave-Particle Duality" http://fqxi.org/community/forum/topic/1296) provides a fundamental challenge to the entire abstract Hilbert Space approach. I suggest that quantum mechanics is not really a theory of all matter, but rather a mechanism that turns a continuous primary field into a discrete localized object (but not a point particle) that follows a classical trajectory. Composite objects (such as nucleons or atoms) are not quantum waves at all, although their primary components are confined quantum waves. This picture is relativistically covariant, logically consistent, and avoids quantum paradoxes. So why has this never been previously considered? The FQXi contest would seem to be an ideal venue to explore such novel concepts, but this has drawn relatively little attention.

Alan M. Kadin, Ph.D. Physics, Harvard 1979

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Eckard Blumschein replied on Sep. 11, 2012 @ 06:55 GMT
I support this objection.

Eckard

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Author Brian Swingle replied on Sep. 13, 2012 @ 19:30 GMT
Dear Dr. Kadin,

Thanks for reading my essay and for your comments.

One thing that interests me about the problem I outline is that you don't have to change the foundations of quantum mechanics to discover an emergent foundational question! For my money, my topic is plenty fundamental, but I respect that not everyone will agree. However, I do believe that if we really understood the structure of the physical states of a many-body system, then we would have a revolution in the physics of quantum matter.

Good luck in the contest.

Best,

Brian

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Vladimir F. Tamari wrote on Sep. 12, 2012 @ 00:52 GMT
Brian and Benjamin,

The sum-over-histories may have a simple physical explanation - this is my rather 'un-brainy' i.e. qualitative intuition: In a locally causal lattice-based theory of the Universe such as Beautiful Universe Theory (BU), lattice nodes transmit angular momentum in units of (h) node-to-node at a speed inversely proportional to the angular velocity of the recipient node- i.e. energy gets bogged (or Higgsed ?) down in dense fields.

Depending on the local energy distribution, this severely constrains the number of possible paths because the energy cannot travel beyond a certain perimeter in the given time interval. If this model is close to how nature works, the calculations reduce to something somewhat similar to those calculating electric potential at a point, derived from the known sources.

Vladimir

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Harlan Swyers wrote on Sep. 12, 2012 @ 20:51 GMT
Brian,

On a completely separate point, I wanted to compliment you on this paper of yours

http://arxiv.org/pdf/1010.4038v1.pdf

coincidentally in these discussions over other papers, I have been digging more into quantum mutual information and find this paper helpful.

best

Harlan

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Author Brian Swingle replied on Sep. 13, 2012 @ 19:31 GMT
Thank you very much, Harlan. I'm glad it was helpful to you.

Best,

Brian

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Hoang cao Hai wrote on Sep. 19, 2012 @ 09:50 GMT
Dear Brian

Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is right.

So let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

A real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you how to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon!

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Hoang cao Hai wrote on Sep. 27, 2012 @ 03:29 GMT
Dear PhD Brian Swingle

Your analysis very interesting .

Would be more attractive if you give your opinion for a real space.

Sorry if my proposal makes you unhappy.

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Sergey G Fedosin wrote on Oct. 4, 2012 @ 03:55 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
and
was the quantity of people which gave you ratings. Then you have
of points. After it anyone give you
of points so you have
of points and
is the common quantity of the people which gave you ratings. At the same time you will have
of points. From here, if you want to be R2 > R1 there must be:
or
or
In other words if you want to increase rating of anyone you must give him more points
then the participant`s rating
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process.

Sergey Fedosin

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S Halayka wrote on Oct. 5, 2012 @ 22:11 GMT
Hi Brian,

Thank you for sharing your essay. It was very well written, and interesting to read.

There is such a thing called signal space in information theory (Shannon, "Communication in the presence of noise"). For instance, where the count of the orthogonal states is 2^n, the signal space has a dimension of n, unlike the state space which has a dimension of 2^n. I wonder if the signal space is generally useful in some way.

- Shawn

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Kamilla Kamilla wrote on Apr. 10, 2016 @ 17:01 GMT
That was an really impressive post and i like that.

192.168.1.1

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Odessa Qamar Sabah wrote on Jan. 19, 2017 @ 08:29 GMT
Thanks for sharing it. I found this Information very interesting and informative! Keep sharing 192.168.1.1 192.168.1.1 192.168.1.1 192.168.1.1

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