Are point mass amplitudes really nonsense here? It's common to view them as distributions, and then they can be differentiated like anything else. Is that not doable with Schrodinger's equation?

First, minor editing thingie: The bit at the top, that's supposed to be a link to "classical configuration spaces" is just text, not a link.

Second, I think you're wrong about "If you had a point mass of amplitude, an infinitely sharp spike in the quantum arena, the amplitude distribution would not be twice differentiable and the future evolution of the system would be undefined. The known laws of physics would crumple up like tinfoil. Individual configurations don't have quantum dynamics; amplitude distributions do."

Specifically, I do... (read more)

They do not describe, even in principle, how one configuration develops into another configuration.

I'm going to just clarify this point, which I disagree with as written (not strictly wrong, but it overlooks something important). You can make a minor extension to quantum mechanics that does describe how one configuration develops into another. That extension is Bohmian mechanics, which is empirically equivalent to orthodox QM.

Basically, you postulate that in addition to the wavefunction, there is a configuration, and it obeys a certain law of motion whi... (read more)

Regarding the top diagram:

Today, it's taking us 3 dimensions just to visualize an amplitude distribution over the position of one 1-dimensional particle at a single moment in time.

To clarify the meaning of the above diagram, the left-to-right direction is the position of A.

Which of the following is true:

A) In the real universe there is a particle located in one specific position in that diagram. (In that case it is not clear to me what the amplitude distribution represents.)

OR

B) No particle exists in the real universe but all you have is an amplitude distribution over several positions giving the illusion that there is a particle in one position.

Chris, in case you didn't see me ask you last time...

http://www.overcomingbias.com/2008/04/philosophy-meet.html#comment-110472438

do you know of a good survey of decoherence?

Eliezer --

I'm loving this latest series of posts. I'm in the midst of exams right now, so I've only been able to skim them, but I will definitely go over them later. Keep up this important work!

• anonymous o.b. reader

Eliezer,

Very minor quibble/question. I assume you mean 2^Aleph_0 rather than Aleph_1. Unless one is doing something with the cardinals/ordinals themselves, it is almost always the numbers Aleph_0, 2^Aleph_0, 2^2^Aleph_0... that come up rather than Aleph_n. You may therefore like the convenient Beth numbers instead, where:

Beth_0 = Aleph_0 Beth_n+1 = 2^Beth_n

Jess: ah, thanks. Wait, how does that identify position space as "special"?

But wait, wouldn't that still imply there's no unique way to slice something into a positional configuration space? ie, given a lorenzian reference frame, you can then define a configuration space and amplitudes over them, but different frames would produce different configuration spaces, so there probably isn't a "single true" set of positional configurations. So one can freely sum and reslice them along different lines anyways. I'm guessing that, if anything, c... (read more)

But wait, wouldn't that still imply there's no unique way to slice something into a positional configuration space?

The space-time interval is a conserved quantity under Lorentz transforms. The position or time interval separatedly are not conserved quantities. Since the spacetime intervals are conserved, the configuration space is unique and consistent.

Max: I'm aware of that. That's kinda my point.

No single unique way to slice it into positional configurations. Different inertial frames, different spacial slices. Configurations over spacetime instead of over space would probably work. But that's not what's being talked about here.

Unless, maybe we kind of extend the same trick over to configuration space? Take configuration space, add a time dimension, then make angled slices through that configuration-time (analogous to spacetime)?

In classical configuration spaces, you can take a single point in the configuration space, and the single point describes the entire state of a classical system. So you can take a single point in classical configuration space, and ask how the corresponding system develops over time. You can take a single point in classical configuration space, and ask, "Where does this one point go?"

The development over time of quantum systems depends on things like the second derivative of the amplitude distribution. Our laws of physics describe how amplitude... (read more)

I have a question on locality. If we consider multiple particles, then the Laplacian is the sum of Laplacian operators corresponding to each particle. This means that the wave function evolution as a whole depends on the local environment around every particle in the configuration. Seemingly this would allow changes in the neighborhood of particle 1 to affect the evolution of particle 2. Yet we know this does not happen, or at least, there are limits to the range of effects which can occur, because of relativity. I don't know how this locality constraint is effectively enforced, in a multi-particle configuration space, or in fact in what sense QM is local when we consider multiple particles.

Psy-Kosh: Position-space is special because it has a notion of locality. Two particles can interact if they collide with each other traveling at different speeds, but they cannot interact if they are far from each other traveling at the same speed.

The field, defined everywhere on the 4-D spacetime manifold, is "reality" (up until the magical measurement happens, at least). You can construct different initial value problem (e.g. if the universe is such-and-such at a particular time, how will it evolve?) by taking different slices of the spacetim... (read more)

Eliezer: why uncountably infinite? I find it totally plausible that you need an infinite-dimensional space to represent all of configuration space, but needing uncountability seems, at least initially, to be unlikely.

Of course, it would be the mathematician who asks this question...

Maybe because there are uncountably many possible positions for the particle, with an amplitude associated with each one? Just guessing.

Wait... I think I've got it (note, am going on no sleep here, so I apologise in advance if this isn't as clear as it could be)

Pretty much any "reasonable" transform of a quantum state is unitary, right?

That is, the time evolution would be a repeated unitary transform (if considered in discrete steps) or an integrated one (if considered continuously)

Unitary transforms have inverses, right?

So pretty much no matter what the "true" way of slicing states and evolving them is, there will be some transformation to a different set of orthognal ... (read more)

I have the same questions for Eliezer as Jadagul and Toby Ord, namely:

• Why would the space of amplitude distributions have uncountable dimension? Unless I've misunderstood, it sounds like it would be something like L^2, which is separable (has countable orthogonal dimension). (Of course, maybe by "dimension" you just meant the cardinality of a Hamel basis, in which case you're right -- there's no Hilbert space with Hamel dimension aleph_0. However, "dimension" in the context of Hilbert spaces nearly always refers to orthogonal dimensio

Psy-Kosh: I haven't done the math out myself, but others have shown that all the predictions of QED are self-consistent under special relativity. You can change all the non-Lorentz-invariant numbers (eg, absolute position, absolute velocity) by moving to a different reference frame, but all the actual predictions are Lorentz-invariant.

Tom: I wasn't denying that. I was simply trying to figure out that if so, what's the "actual reality"? ie, there wouldn't be a single unique set of privaliged positional configuration spaces, since different reference frames will work just fine.

I think I may have been very very unclear in the question/confusion I brought up.

(This is a repost of a comment I made a few days ago under the topic "Distinct Configurations", but if someone could address this, I would really appreciate it.)

So I guess I get how [configurations being the same as long as all the particles end up in the same place] works in theory, but in practice, doesn't a particle going from A-B have SOME kind of effect that is different than if it went from B-C, even without the sensitive thingy? I don't know if it would be from bouncing off other particles on the way, or having some kind of minute gravitat... (read more)

Dave, see my comment on Distinct Configurations.

Hey Eliezer, this is a great post. I just have one question: HOW ARE YOU SO AWESOME?! Seriously, these posts are incredible.

I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable

Well, the number of points in a Hilbert space of dimension 2 is uncountable, and yet the space has dimension 2!

I suspect the source of the confusion here is that you're trying to think of the values of a function as its "coordinates". But this is wrong: the "coordinates" are the coefficients of a Fourier series expansion of the function.

The confusion is understandable, given that the two concepts coincide in the finite-... (read more)

Also:

I'd say that I was assuming the continuum hypothesis, except that I'm an infinite set atheist.

Not that again! Let's not mix up the map and the territory. You may not think there are any infinite sets out there in the territory, but mathematics is about the map -- or, rather, mapmaking in general. So it's a category error to jump from the conviction that the "real world" contains only finitely many things to Kroneckerian skepticism about mathematical objects.

For what it's worth, the cardinality of the set of reals is 2^aleph_0. The continuum ... (read more)

Amazingly great post. But I'm still confused on one point.

Say we want to set up the quantum configuration space for two 1-dimensional particles. So we have a position coordinate for each one, call them x and y. But wait, the two particles aren't distinguishable, so we really need to look at the quotient space under the equivalence (x,y) ~ (y,x). But this is no longer a smooth manifold is it? At the moment I'm at a loss for a proof that it isn't, but I certainly can't find a smooth structure for it. And if it's not smooth then what the heck do second derivatives of amplitude distributions mean?

Nick - thanks for the link. I admit I tend to glaze over the comments as many of them are frankly over my head. I re-read yours and it makes more sense to me.

Larry, this link helped explain some aspects of multiparticle wave functions to me: http://galileo.phys.virginia.edu/classes/252/symmetry/Symmetry.html. They seem to deal with the full space of x and y positions, with the derivatives defined analytically, and then they impose either symmetric or anti-symmetric conditions on possible solutions depending on whether the particles are bosons or fermions. I'm not sure if this will fully answer your question but perhaps it will shed some light.

"I was simply trying to figure out that if so, what's the "actual reality"?"

There is none, at least not in those terms. There is no "actual positional configuration space", any more than there's an "actual inertial reference frame" or "actual coordinate system"; they are all equivalent in the experimental world. Feel free to use whichever one you like.

"I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable."

The number of functions of... (read more)

Eliezer: the Hilbert space of QM is generally believed to have a countable basis

It would be more accurate to say that when a Hilbert space is used, it has countable dimension. But the Hilbert space of a quantum field, naively, ought to have uncountable dimension, because there are continuum-many degrees of freedom. In practice, quantum field theory is done using a lot of formalism which only formally refers to an underlying Hilbert space - sum over histories, operator algebras - and even the simplest real-life quantum field theories (i.e. those used in par... (read more)

Tom McCabe: The category of Hilbert spaces includes spaces of both finite and infinite dimension, so it presumably includes both countable and uncountable infinities.

mitchell porter: But the Hilbert space of a quantum field, naively, ought to have uncountable dimension, because there are continuum-many degrees of freedom.

Given any cardinal number, there exists a Hilbert space with that orthogonal dimension. Note, however, that even if the dimension is uncountable, individual elements are still given by linear combinations with countably many terms. In othe... (read more)

I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable.

A mere typo. I meant, of course, that I'd thought the quantum Hilbert space was uncountably dimensional, because a point in that Hilbert space corresponded to an amplitude distribution over points in a real space, and a function of a real space has uncountable degrees of freedom.

a function of a real space has uncountable degrees of freedom

Right -- that's exactly the misunderstanding I was addressing in my earlier comment.

An arbitrary function does indeed have uncountable degrees of freedom, but in that context you're notconsidering it as an element of a Hilbert space. (Those degrees of freedom do not correspond to basis vectors.)

I want to throw in a simple way to think about quantum field theory, for people who understand the quantization of the simple harmonic oscillator.

You think of the field's Fourier modes as independent harmonic oscillators. The quantum field is therefore a tensor product of uncountably many quantized harmonic oscillators. Call the energy levels of a single oscillator |0>, |1>, |2>, etc. In QFT 101, you say that one increment of energy level in one mode corresponds to one particle with momentum p = hbar.k, where k is the wave vector of the Fourier mo... (read more)

Roland asked:

A) In the real universe there is a particle located in one specific position in that diagram. (In that case it is not clear to me what the amplitude distribution represents.) OR B) No particle exists in the real universe but all you have is an amplitude distribution over several positions giving the illusion that there is a particle in one position.

B.

Thanks for answering Eliezer,

No particle exists in the real universe but all you have is an amplitude distribution over several positions giving the illusion that there is a particle in one position.

Is this amplitude distribution defined only in a limited space or is it defined on all infinity(bounded by the limits of the universe) in all directions(either in the 1D example or the 3D real world)?

Eliezer,

another question relating to this: how does the amplitude distribution change over time? Is there an axis of symmetry somewhere?

On April 17, 2008 at 02:07 PM, Eliezer_Yudkowsky said:

No particle exists in the real universe but all you have is an amplitude distribution over several positions giving the illusion that there is a particle in one position.

Is this building up to the conclusion that evolution(s) has(ve) led our brains/minds to, tell us that objects are located at the areas of high concentration of amplitude distribution, as a useful model of the world?

Just in case it's not clear from the above: there are uncountably many degrees of freedom to an arbitrary complex function on the real line, since you can specify its value at each point independently.

A continuous function, however, has only countably many degrees of freedom: it is uniquely determined by its values on the rational numbers (or any dense set).

Eliezer: I thought your analogy to Conway's Game of Life and the glider were brilliant. And in fact, you don't take it far enough.

It's a very intuitive example of position implying momentum, just as happens in QM. So you should do more examples of that.

For example: Let's say there's just a single photon, in 1D-land. It has some amplitude distribution for its position. Let's say it's mostly concentrated at some particular point, X. What does that amplitude distribution look like? Most people would probably naively guess some kind of bell-shaped curve... (read more)

"You'd think they'd tell you up front, "Hey, the evolution of a quantum system depends on stuff like the second derivative of the amplitude distribution, so you can't possibly break it down into the evolution of individual configurations." It's worse than that; they wait until the 2nd semester to even start talking about time-evolution. They spend the first semester trying to find, for a given Hamiltonian, a set of wavefunctions for which the value of a particular observable, energy (or in a few cases, momentum), is unchanging in time. Time-... (read more)

If you had a point mass of amplitude, an infinitely sharp spike in the quantum arena, the amplitude distribution would not be twice differentiable and the future evolution of the system would be undefined. The known laws of physics would crumple up like tinfoil. Individual configurations don't have quantum dynamics; amplitude distributions do.

A point mass of amplitude, concentrated into a single exact position in configuration space, does not correspond to a precisely known state of the universe. It is physical nonsense.

It might be helpful here to re... (read more)

If you had a point mass of amplitude, an infinitely sharp spike in the quantum arena, the amplitude distribution would not be twice differentiable and the future evolution of the system would be undefined.

This just caught my eye, and it's not clear to me what the actual mathematics behind it is. An "infinitely sharp spike" is an intuitive description of something that can be formalised, but not as a function mapping points of configuration space to amplitudes (because "infinity" in this context is not a number). The concept of an inf... (read more)

Here is a great simulation of two electrons in a wire that looks just like your drawing of a two particle configuration space, and is quite helpful for showing how it moves and what it means about the particles.

I'm confused. If future configurations are determined as a result of calculating amplitudes, if you calculate the future amplitudes why haven't you calculated the future configuration(s)? What's the significant difference between configurations and amplitudes?

Here, the issue starts to clear out, just becoming non-physics in my mind (as it is trained to understand the world). Now it seems worhty to have read all the previous parts. Thanks.