The Illusion of Hilbert Space by Brian Swingle

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.

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 [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] 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

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