The Illusion of Hilbert Space by Brian Swingle

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

Is this related to churches of the larger and smaller Hilbert spaces?

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

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 entangled, potentially universe-spanning kets and the relatively 'simple examples' developed mathematically in EPR-type experiments. Nothing in the actual math seemed to intuitively help sort this out!

I'm going to have to go back through your development of the method in your essay before I fully grasp or internalize it, but I am basically the darned kid who has been trying to stack 'balls' and you walked up and said, "Using balls is silly. Here, try these blocks insteadI You can build all sorts of things with these: connect them, take them apart ... They can actually be made stable if you are careful!"

Stunned boy looks up, "Wow! Cool! Thanks!"

I've been thinking a lot lately about how foundational issues might become resolvable if we give into the *highly* parallel behavior of Nature and consider the |kets> to *be* quantum computers--self-stabilizing boats in a sea of entanglement ... Hilbert Boats floating on a Hilbert Sea with a lot going on in the depths!

What also emerges from your essay is a way to handle the strong sense I have that it is unwise to ignore the *efficiency* that is implied by the principals of least action, and how Nature is relentlessly parsimonious, and what that might mean in terms of the huge symmetry structures discussed *and* entanglements and superpositions and such.

It seems that if a single all-encompassing and entirely-expanded Hilbert space description is so danged huge, then Nature might be parsimonious in that context as well. If identical particles automatically entangle, then it seems there is already evidence of Nature *using* some form of parsimony in this context, and the kind of patterned-deconstruction that you illustrate seems like a step in the right direction.

My own essay (1527) is largely about taking the |ket> nature of the wavefunction seriously, and how we might benefit from considering entanglements as basically *structural* elements of Hilbert space, much like the nodes in your illustration, not active in normal-space the way 'particles' are.

By doing so, the implied *relationships* between entangled |Alice> and |Bob> remain, but this is *not* a connection through ordinary space so it may be possible to that no causally violable distance is actually involved. |Alice> connects to the relationship at zero distance. |Bob> connects to the relationship at zero distance. |Alice> and |Bob> are not *functionally* connected to each other at all, just connected to a support structure that exists only in some kind of Hilbert-utilizing space. If |Alice> disconnects from that support structure (a photon upon detection for instance) then |Bob> loses-access-to the superposed state, |Alice> carries away her half and |Bob> is stuck with the other. (Oversimplified, but I'm just trying to get the non-classical nature across).

Also, I didn't use the Hilbert Sea metaphor in my essay, but after looking at the nodes at the top of your illustration, and the connections in the 'depth' dimension below, this seems like an apt metaphor to capture the qualitative difference between the 'particles' (stuck on the surface), and some kind of sorting and simplifying mechanism that is consolidating the accounting that goes on beneath the surface.

Very exciting and inspirational stuff.

Nice job.

Dean

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

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 where the wave function resides, which itself is not a physically real entity in any correct interpretation of quantum mechanics.

The second issue is the cherry picking approach to the problem. On one hand you argue with getting rid of hilbert space, and on the other you argue that this isn't an essay against quantum mechanics. How can one not see that these are contradictory positions?

From the paper:

"If so, we're in trouble. Ignoring causality and the lack of materials, even if we filled up our entire Hubble volume, the whole visible universe, with our best classical storage device, we could only store the quantum state of a few hundred spins using this huge classical memory. Suddenly the illusory nature of

Hilbert space is brought into focus."

So the argument being presented is that we should somehow think that the universe should be confined into some classical computer? It is not the case that the universe needs to follow our prejudices. If anything this observation is an argument about the weaknesses of classical thinking.

If we follow the process outlined, it appears to be a discussion about decoherence, states are becoming less entangled as things become more course grained. However there is this point that is brought up:

"The idea of thinking like a quantum computer, that is thinking in terms of quantum processes and dynamics, is a powerful way to give a physically meaningful notion of quantum states. Hilbert space thus survives but only as a stage which is anyway mostly unused."

So the point is that Hilbert space is fundamental to how we describe processes, that we have difficulty in imagining how to model it is a secondary issue, but that we would bias our thinking is a risky proposition. One has to keep in mind that these issues of emergence have existed for a long time, and there has been an understanding that the dynamics that we associate with classical motion are not part of the wave function of a single particle, but are only apparent when there is a larger quantum apparatus (referred to as a "classical object" in the Quantum Mechanics text of Landau and Lifshitz).

That an apparatus is sometimes referred to as a classical object should not be seen as undermining the quantum nature of the larger system.

There is more discussion, to be had here, but giving up hilbert space is antithetical to the whole of QM.

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

VladimirAttachment #1: BUFIG29.jpgAttachment #2: 1_BUFIG28.jpg

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