The Illusion of Hilbert Space by Brian Swingle

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

"Brian," I mean, of course... at least "Brain" isn't an insult in this context!

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

I support this objection.

Eckard

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

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

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

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 of physics, so if we go back to James C. Maxwell's book on physics, http://books.google.com/books?id=noRgWP0_UZ8C&printsec=titlepage&dq=matter+and+motion&hl=en#v=onepage&q=matter%20and%20motion&f=false

We find the first chapter is a discussion of matter and motion, which is also the simplest case described. It is here that our classical notion of physical states is ultimately derived. Hilbert space is fundamentally different in this regard. Physical states can be represented in Hilbert space, these are classical understood as relating to particles and forces, however there are plenty of other states admissible in Hilbert space that have absolutely no description in classical terms. These some would call unphysical. These are perfectly acceptable in the broader Hilbert space.

So perhaps its a matter of definition, in any case, hilbert space is an abstraction and not physical in the sense that it is a space where we our classical concepts reside. We might find a projection of hilbert space into a more classical space, but there is no there there.

As far as imagining the universe as a quantum computer, I think there is some validity as we restrict ourselves to physics above the plank scale, however, this is where we start running into some interesting problems, and it is likely that the complexity of physics below the plank scale are even too great for a quantum computer. This may also be where what might be considered unphysical states become physical. We simply don't know without sufficient evidence. However, one thing is certain, we cannot abandon Hilbert space as fundamental to our understanding, we might be able to find even more fundamental concepts, especially as we abstract into algebraic structures, but Hilbert space provides a key intersection to our understanding that is critical to the endeavor.

"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

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

Eckard,

Thanks for this insight. There are points you bring up in your paper that are indeed puzzling. Rigged Hilbert spaces are of course immensely useful, and it is the notion of convolution that is perhaps the most attractive feature. Indeed, when I speak of Hilbert space I speak of it in a general sense, I probably could use better language because Rigged Hilbert space is treated as a distinct entity.

I really like the notion of convolution, since the proof of the central limit theorem using convolution is one of my favorite proofs (see min 32:00 in link:

http://www.youtube.com/watch?feature=player_embedded&v=LA4Uv6PMRTM#!

I harbor a private prejudice that the proof of the central limit theorem, showing that recursive convolution results in a Gaussian distribution, is key to the understanding of the unification of gravity to quantum mechanics.