**Previously in series**: Classical Configuration Spaces

Yesterday, we looked at configuration spaces in classical physics. In classical physics, configuration spaces are a useful, but optional, point of view.

Today we look at quantum physics, which *inherently* takes place inside a configuration space, and *cannot be taken out*.

For a start, as you might guess, in quantum physics we deal with distributions of complex amplitudes, rather than probability distributions made up of positive real numbers. At left, I've used up 3 dimensions drawing a complex distribution over the position of *one* particle, A.

You may recall that yesterday, 3 dimensions let us display the position of two 1-dimensional particles plus the system evolution over time. 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. Which is why we did classical configuration spaces first.

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

The up-and-down direction, and the invisible third dimension that leaps out of the paper, are devoted to the complex amplitudes. Since a complex amplitude has a real and imaginary part, they use up 2 of our 3 dimensions.

Richard Feynman said to just imagine the complex amplitudes as little 2-dimensional arrows. This is as good a representation as any; little 2D arrows behave just the same way complex numbers do. (You add little arrows by starting at the origin, and moving along each arrow in sequence. You multiply little arrows by adding the angles and multiplying the lengths. This is isomorphic to the complex field.) So we can think of each position of the A particle as having a little arrow associated to it.

As you can see, the position of A bulges in two places—a big bulge to the left, and a smaller bulge at right. Way up at the level of classical observation, there would be a large probability (integrating over the squared modulus) of finding A somewhere to the left, and a smaller probability of finding it at the small bulge to the right.

Drawing a neat little graph of the A+B system would involve having a complex amplitude for each joint position of the A and B particles, which you could visualize as a hypersurface in 4 dimensions. I'd draw it for you, but I left my 4-dimensional pencil in the pocket of the 3rd leg of my other pants.

You may recall from yesterday that a plaid rectangular probability distribution factorizes into the product of two independent probability distributions.

This kind of independence-structure is one of several keys to recovering the illusion of individual particles from quantum amplitude distributions. If the amplitude distribution roughly factorizes, has subsystems A and B with Amplitude(X,Y) ~ Amplitude(X) * Amplitude(Y), then X and Y will seem to evolve roughly independently of each other.

But maintaining the illusion of individuality is harder in quantum configuration spaces, because of the identity of particles. This identity cuts down the size of a 2-particle configuration space by 1/2, cuts down the size of a 3-particle configuration space by 1/6, and so on. Here, the diminished configuration space is shown for the 2-particle case:

The quantum configuration space is over joint possibilities like "a particle here, a particle there", not "this particle here, that particle there". What would have been a neat little plaid pattern gets folded in on itself.

You might think that you could recover the structure by figuring out which particle is "really which"—i.e. if you see a "particle far forward, particle in middle", you can guess that the first particle is A, and the second particle is B, because only A can be far forward; B just stays in the middle. (This configuration would lie in at the top of the original plaid pattern, the part that got folded over).

The problem with this is the little triangular region, where the folded plaid intersects itself. In this region, the folded-over amplitude distribution gets superposed, added together. Which makes an experimental difference, because the squared modulus of the sum is not the sum of the squared moduli.

In that little triangular region of quantum configuration space, there is simply no fact of the matter as to "which particle is which". Actually, there *never was* any such fact; but there was an illusion of individuality, which in this case has broken down.

But even *that* isn't the ultimate reason why you can't take quantum physics out of configuration space.

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 distributions develop into new amplitude distributions. They do not describe, *even in principle,* how one configuration develops into another configuration.

(I pause to observe that physics books make it way, way, way too hard to figure out this extremely important fact. 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*.*" When I first saw the Schrödinger Equation it confused the hell out of me, because I thought the equation was supposed to apply to single configurations.)

If I've understood the laws of physics correctly, quantum mechanics still has an extremely important property of *locality:* You can determine the instantaneous change in the amplitude of a single configuration using only the infinitesimal neighborhood. If you forget that the space is continuous and think of it as a mesh of computer processors, each processor would only have to talk to its immediate neighbors to figure out what to do next. You do have to talk to your neighbors—but *only* your next-door neighbors, no telephone calls across town. (Technical term: "Markov neighborhood.")

Conway's Game of Life has the discrete version of this property; the future state of each cell depends only on its own state and the state of neighboring cells.

The second derivative—Laplacian, actually—is not a *point* property. But it is a *local* property, where knowing the immediate neighborhood tells you everything, regardless of what the rest of the distribution looks like. Potential energy, which also plays a role in the evolution of the amplitude, can be computed at a *single* positional configuration (if I've understood correctly).

There are mathematical transformations physicists use for their convenience, like viewing the system as an amplitude distribution over momenta rather than positions, which throw away this neighborhood structure (e.g. by making potential energy a non-locally-computable property). Well, mathematical convenience is a fine thing. But I *strongly suspect* that the physically real wavefunction has local dynamics. This kind of locality seems like an *extremely* important property, a candidate for something hardwired into the nature of reality and the structure of causation. Imposing locality is part of the jump from Newtonian mechanics to Special Relativity.

The temporal behavior of each amplitude in configuration space depends only on the amplitude at neighboring points. But you cannot figure out what happens to the amplitude of a point in quantum configuration space, by looking *only* at that *one* point. The *future* amplitude depends on the *present* second derivative of the amplitude distribution.

So you can't say, as you can in classical physics, "If I had infinite knowledge about the system, all the particles would be in one definite position, and then I could figure out the exact future state of the system."

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's like asking, in Conway's Game of Life: "What is the future state of this one cell, regardless of the cells around it?" The immediate future of the cell depends on its immediate neighbors; its distant future may depend on distant neighbors.

Imagine trying to say, in a classical universe, "Well, we've got this probability distribution over this classical configuration space... but to find out where the system evolves, where the probability flows from each point, we've got to twice differentiate the probability distribution to figure out the dynamics."

In classical physics, the position of a particle is a separate fact from its momentum. You can know exactly where a particle is, but not know exactly how fast it is moving.

In Conway's Game of Life, the velocity of a glider is not a separate, additional fact about the board. Cells are only "alive" or "dead", and the *apparent* motion of a glider arises from a configuration that repeats itself as the cell rules are applied. If you know the life/death state of all the cells in a glider, you know the glider's velocity; they are not separate facts.

In quantum physics, there's an amplitude distribution over a configuration space of particle positions. Quantum dynamics specify how that amplitude distribution evolves over time. Maybe you start with a blob of amplitude centered over position X, and then a time T later, the amplitude distribution has evolved to have a similarly-shaped blob of amplitude at position X+D. Way up at the level of human researchers, this looks like a particle with velocity D/T. But at the quantum level this behavior arises *purely* out of the amplitude distribution over *positions,* and the laws for how amplitude distributions evolve over time.

In quantum physics, if you know the exact current amplitude distribution over particle positions, you know the exact future behavior of the amplitude distribution. Ergo, you know how blobs of amplitude appear to propagate through the configuration space. Ergo, you know how fast the "particles" are "moving". Full knowledge of the amplitude distribution over positions implies full knowledge of momenta.

Imagine trying to say, in a classical universe, "I twice differentiate the probability distribution over these particles' positions, to *physically determine* how fast they're going. So if I learned new information about where the particles were, they might end up moving at different speeds. If I got very precise information about where the particles were, this would physically cause the particles to start moving very fast, because the second derivative of probability would be very large." Doesn't sound all that sensible, does it? Don't try to interpret this nonsense—it's not even analogously correct. We'll look at the horribly misnamed "Heisenberg Uncertainty Principle" later.

But that's why you can't take quantum physics out of configuration space. Individual configurations don't *have* physics. Amplitude distributions have physics.

(Though you can regard the *entire* state of a quantum system—the whole amplitude distribution—as a single point in a space of infinite dimensionality: "Hilbert space." But this is just a convenience of visualization. You imagine it in N dimensions, then let N go to infinity.)

Part of *The Quantum Physics Sequence*

Next post: "Feynman Paths"

Previous post: "Classical Configuration Spaces"

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)

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

isa configuration, and it obeys a certain law of motion whi... (read more)Psy-Kosh: In Quantum Field Theory, the fields (the analog of wavefunctions in non-relativistic Quantum Mechanics) evolve locally on the spacetime. This is given a precise, observer-independant (i.e. covariant) meaning. This property reduces to the spatially-local evolution of the wavefunction in QM which Eliezer is describing. Further, this indeed identifies position-space as "special", compared to momentum-space or any other decomposition of the Hilbert space.

Eliezer: The wavefunctions in QM (and the fields in QFT) evolve locally under norma... (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!

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

positionortime intervalseparatedly 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

whatthe "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.

Komponisto, I looked up your references and found that the Hilbert space of QM is generally believed to have a countable basis, though there are occasional theories which make the basis uncountable.

I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable. But in QM it's countable... because everything comes in multiples of Planck's constant, perhaps? Though I haven't seen the actual reason stated, and perhaps it's something beyond my current grasp.

In any case, I've edited the text to read sim... (read more)

Ahh... here's something I can help with. To see why Hilbert space has a countable basis, let's first define Hilbert space. So let

= the set of all functions such that the integral of is finite, and let

= the set of all functions such that the integral of is zero. This includes for example the Dirichlet function which is one on rational numbers but zero on irrational numbers. So it's actually a pretty big space.

Hilbert space is defined to be the quotient space . To see that it has a countable basis, it suffices to show that it contains a countable dense set. Then the Gram-Schmidt orthogonalization process can turn that set into a basis. What does it mean to say that a set is dense? Well, the metric on Hilbert space is given by the formula

%20=%20\sqrt{\int%20|f%20-%20g|%5E2}),

so a sequence is dense if for every element of Hilbert space, you can find a sequence such... (read more)

(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 uncountableWell, 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, map

makingin 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 basisIt 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 freedomRight -- 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

quantizedharmonic 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:

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:

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)

This series is great. But, I'm having a little trouble understanding the fourth diagram, the one with the folded configuration space.

I sort of get it: the original configuration space distinguished between two particles, which is wrong, so in reality only half of the configuration space's area matters when it comes to information. But I don't get how that means you delete the probability from half of the space. Why is it wrong to make the space symmetrical across the diagonal line? It seems a little arbitrary to me; is there a physical reason, or is this a... (read more)

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

Is it contradictory to be an infinite set atheist, and be a realist about a continuous configuration space?

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)

So the parts have no physical presence, but the whole does?

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.