Followup toCan You Prove Two Particles Are Identical?, Feynman Paths

Even babies think that objects have individual identities.  If you show an infant a ball rolling behind a screen, and then a moment later, two balls roll out, the infant looks longer at the expectation-violating event.  Long before we're old enough to talk, we have a parietal cortex that does spatial modeling: that models individual animals running or rocks flying through 3D space.

And this is just not the way the universe works.  The difference is experimentally knowable, and known.  Grasping this fact, being able to see it at a glance, is one of the fundamental bridges to cross in understanding quantum mechanics.

If you shouldn't start off by talking to your students about wave/particle duality, where should a quantum explanation start?  I would suggest taking, as your first goal in teaching, explaining how quantum physics implies that a simple experimental test can show that two electrons are entirely indistinguishable —not just indistinguishable according to known measurements of mass and electrical charge.

To grasp on a gut level how this is possible, it is necessary to move from thinking in billiard balls to thinking in configuration spaces; and then you have truly entered into the true and quantum realm.

In previous posts such as Joint Configurations and The Quantum Arena, we've seen that the physics of our universe takes place in a multi-particle configuration space.

Conf6_2The illusion of individual particles arises from approximate factorizability of a multi-particle distribution, as shown at left for a classical configuration space.

If the probability distribution over this 2D configuration space of two classical 1D particles, looks like a rectangular plaid pattern, then it will factorize into a distribution over A times a distribution over B.

In classical physics, the particles A and B are the fundamental things, and the configuration space is just an isomorphic way of looking at them.

In quantum physics, the configuration space is the fundamental thing, and you get the appearance of an individual particle when the amplitude distribution factorizes enough to let you look at a subspace of the configuration space, and see a factor of the amplitude distribution—a factor that might look something like this:


This isn't an amplitude distribution, mind you.  It's a factor in an amplitude distribution, which you'd have to multiply by the subspace for all the other particles in the universe, to approximate the physically real amplitude distribution.

Most mathematically possible amplitude distributions won't factor this way.  Quantum entanglement is not some extra, special, additional bond between two particles.  "Quantum entanglement" is the general case.  The special and unusual case is quantum independence.

Reluctant tourists in a quantum universe talk about the bizarre phenomenon of quantum entanglement.  Natives of a quantum universe talk about the special case of quantum independence.  Try to think like a native, because you are one.

I've previously described a configuration as a mathematical object whose identity is "A photon here, a photon there; an electron here, an electron there."  But this is not quite correct.  Whenever you see a real-world electron, caught in a little electron trap or whatever, you are looking at a blob of amplitude, not a point mass.  In fact, what you're looking at is a blob of amplitude-factor in a subspace of a global distribution that happens to factorize.

Clearly, then, an individual point in the configuration space does not have an identity of "blob of amplitude-factor here, blob of amplitude-factor there"; so it doesn't make sense to say that a configuration has the identity "A photon here, a photon there."

But what is an individual point in the configuration space, then?

Well, it's physics, and physics is math, and you've got to come to terms with thinking in pure mathematical objects.  A single point in quantum configuration space, is the product of multiple point positions per quantum field; multiple point positions in the electron field, in the photon field, in the quark field, etc.

When you actually see an electron trapped in a little electron trap, what's really going on, is that the cloud of amplitude distribution that includes you and your observed universe, can at least roughly factorize into a subspace that corresponds to that little electron, and a subspace that corresponds to everything else in the universe.  So that the physically real amplitude distribution is roughly the product of a little blob of amplitude-factor in the subspace for that electron, and the amplitude-factor for everything else in the universe.  Got it?

One commenter reports attaining enlightenment on reading in Wikipedia:

'From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.'

Okay, but that doesn't make the basic jump into a quantum configuration space that is inherently over multiple particles.  It just sounds like you're talking about individual disturbances in the aether, or something.  As I understand it, an electron isn't an excitation of a quantum electron field, like a wave in the aether; the electron is a blob of amplitude-factor in a subspace of a configuration space whose points correspond to multiple point positions in quantum fields, etc.

The difficult jump from classical to quantum is not thinking of an electron as an excitation of a field.  Then you could just think of a universe made up of "Excitation A in electron field over here" + "Excitation B in electron field over there" + etc.  You could factorize the universe into individual excitations of a field.  Your parietal cortex would have no trouble with that one—it doesn't care whether you call the little billiard balls "excitations of an electron field" so long as they still behave like little billiard balls.

The difficult jump is thinking of a configuration space that is the product of many positions in many fields, without individual identities for the positions.  A configuration space whose points are "a position here in this field, a position there in this field, a position here in that field, and a position there in that field".  Not, "A positioned here in this field, B positioned there in this field, C positioned here in that field" etc.

You have to reduce the appearance of individual particles to a regularity in something that is different from the appearance of particles, something that is not itself a little billiard ball.

Oh, sure, thinking of photons as individual objects will seem to work out, as long as the amplitude distribution happens t factorize.  But what happens when you've got your "individual" photon A and your "individual" photon B, and you're in a situation where, a la Feynman paths, it's possible for photon A to end up in position 1 and photon B to end up in position 2, or for A to end up in 2 and B to end up in 1?  Then the illusion of classicality breaks down, because the amplitude flows overlap:

In that triangular region where the distribution overlaps itself, no fact exists as to which particle is which, even in principle—and in the real world, we often get a lot more overlap than that.

I mean, imagine that I take a balloon full of photons, and shake it up.

Amplitude's gonna go all over the place.  If you label all the original apparent-photons, there's gonna be Feynman paths for photons A, B, C ending up at positions 1, 2, 3 via a zillion different paths and permutations.

The amplitude-factor that corresponds to the "balloon full of photons" subspace, which contains bulges of amplitude-subfactor at various different locations in the photon field, will undergo a continuously branching evolution that involves each of the original bulges ending up in many different places by all sorts of paths, and the final configuration will have amplitude contributed from many different permutations.

It's not that you don't know which photon went where.  It's that no fact of the matter exists. The illusion of individuality, the classical hallucination, has simply broken down.

And the same would hold true of a balloon full of quarks or a balloon full of electrons.  Or even a balloon full of helium. Helium atoms can end up in the same places, via different permutations, and have their amplitudes add just like photons.

Don't be tempted to look at the balloon, and think, "Well, helium atom A could have gone to 1, or it could have gone to 2; and helium atom B could have gone to 1 or 2; quantum physics says the atoms both sort of split, and each went both ways; and now the final helium atoms at 1 and 2 are a mixture of the identities of A and B."  Don't torture your poor parietal cortex so.  It wasn't built for such usage.

Just stop thinking in terms of little billiard balls, with or without confused identities.  Start thinking in terms of amplitude flows in configuration space.  That's all there ever is.

And then it will seem completely intuitive that a simple experiment can tell you whether two blobs of amplitude-factor are over the same quantum field.

Just perform any experiment where the two blobs end up in the same positions, via different permutations, and see if the amplitudes add.


Part of The Quantum Physics Sequence

Next post: "Identity Isn't In Specific Atoms"

Previous post: "Feynman Paths"

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24 comments, sorted by Click to highlight new comments since:

I don't see excitations as billiard balls, but the smallest type of wave possible. The whole point of QFT is so that you can see "particles" as annihilating and changing which isn't very billiard ball like. Start with a bag of positrons and electrons, shake it up and what do get?

Quantum without all the physics (fermions/bosons/anti-particles/vacuum energy) is horribly dry and un-illuminating for me.

"As I understand it, an electron isn't an excitation of a quantum electron field, like a wave in the aether; the electron is a blob of amplitude-factor in a subspace of a configuration space whose points correspond to multiple point positions in quantum fields, etc."

It is hard to tell from the brief description, but it seems to me that you are talking about localized electrons and Wikipedia is talking about delocalized electrons. To describe particles in quantum field theory you have some field in spacetime. In the simplest case of a scalar field it is described by some function f(x,y,z,t). Note that f(x,y,z,t) is not a quantum wavefunction, it is just a classical field. Quantum mechanically, there is an amplitude corresponding to each possible configuration of this field. (Thus the wavefunction is technically a "functional"). Different configurations have different energies. The Langrangian tells you what energy corresponds to what configuration. The Lagrangian for a single field not interacting with anything looks sort of like the Lagrangian for material that can vibrate. (This is just an analogy, it has nothing to do with the aether.)

By a change of basis, we can write the Lagrangian in terms of normal modes, which each behave like harmonic oscillators, and which are decoupled from each other. As a one dimensional example, the normal modes for a violin string are the sine waves whose wavelengths are the length of the string, half the length of the string, 1/3 the length of the string, etc. These modes thus correspond to sinusoidal variation of the field. (This has nothing to do with string theory. The violin string is just a handy example of a vibrating system.) We know how to "quantize" the Harmonic oscillator. It turns out that the allowed energies are (n+1/2)h*omega, where n=1,2,3,..., and omega is the resonant frequency of the mode and h is planck's constant. If the mode of frequency omega is excited to n=1 and the mode of frequency omega' is excited to n=5 that corresponds to a six electron state with one electron of frequency omega and five electrons of frequency omega'. (Similarly for photons, or any other particles. For photons these frequencies correspond to colors.)

We can have superpositions of different such states. For example we could have quantum amplitude 1/sqrt(2) for mode omega to have n=1 and quantum amplitude 1/sqrt(2) for mode omega to have n=2. If we just have quantum amplitude 1 for a given mode omega to be in the n=1 state, and amplitude zero for all other configurations of the field, then this is a one electron state, where the electron is completely delocalized. What state corresponds to an electron in a particular region? A localized electron does not correspond to the field being nonzero in only a small region (e.g. the violin string has a localized bump in it like this ---^---). That would be a multi-electron state, because it decomposes into a classical superposition of many different sine waves, so we would have n>0 in multiple modes. Instead we can build a localized state of an electron by making a quantum superposition over different modes being occupied. It is important not to get the wavefunction confused with the field f(x,y,z,t). (If you have heard about the Dirac and Klein-Gordon equations, the solutions are analogous to f(x,y,z,t), not analogous to Schrodinger wavefunctions. Historically, there was some confusion on this point.)

Everything I have described so far is the quantum field theory of non-interacting particles. Although I may not have explained that well, it is actually not too complicated. However, if the particles interact, then the normal modes are coupled. Nobody knows how to treat this directly, so you need to use perturbation theory. This is where the complicated stuff about Feynman diagrams and so forth comes in.

I hope this is helpful.

Technical caveat: I should have said it's actually the Hamiltonian, not the Lagrangian that directly tells you the energy of a configuration. (Its easy to convert between Hamiltonians and Lagrangians though, and it turns out Lagrangians are handier for QFT.)

Something I'm unclear about is what's the difference between multiple particles, and one particle with multiple degrees of freedom?

Sorry, I should clarify that better: What is the nature of the physical and mathematical differences that cause one factorization into, say, 6 factors to "look like" 1 particle free to move in 6 dimensions vs one particle free to move in two dimensions and another particle free to move in three dimensions?

And so on. ie, What makes one sort of factorization look like one, vs a different factorization look like the other?

Psy-Kosh: I think that is a great question. Here is my take on it:

The wavefunction for six particles will be a function of six variables, x1,y1,z1,x2,y2,z2. You could of course think of these as just six variables without thinking in terms of two particles with three coordinates apiece. However, from this point of view, the system would have certain strange properties that appear coincidental. For example, suppose the two particles are bosons. Then, if we exchange them, nothing happens to the wavefunction. This seems fairly natural. However, from the 6D point of view we have the strange property that if we swap three particular pairs of variables (x1 swapped with x2, y1 swapped with y2, and z1 swapped with z2) the wavefunction is unchanged, whereas in general if we pair the variables in any other way and swap them the wavefunction is changed. Similarly, the potential term in the Hamiltonian will often depend on the distance between the two particles (such as if they repel coulombically). This again seems natural. However, from the 6D point of view this is a mysterious property that the potential depends only on (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2, where we have subtracted variables in pairs in some particular way, rather than in any of the many other ways we could pair them.

Stephen: Thank you very much.

Would you say the swapping properties of the degrees of freedom could be viewed as the fundamental "cause" of the differences between the situations, or merely one property, among others, that shows up differently in the different situations?

That is, would you say the swapping properties is the essense of "illusion of seperate particles" vs "illusion of one particle with multiple degrees of freedom"?

Psy-Kosh: I don't know. Certainly in practice it seems to be useful to focus a lot on the group of symmetries of a system. In the example we discussed the swapping properties were basically the group of permutations of labels leaving the wavefunction invariant. (Or the group of permutations leaving the Hamiltonian invariant in the other example.) I think special relativity can be stated as "the Lagrangian of the universe is invariant under the Lorentz group." So, although I don't know whether swapping properties and so forth are the essence of things, they certainly seem to be important and useful to analyze.

"Well, it's physics, and physics is math, and you've got to come to terms with thinking in pure mathematical objects."

  • Physics is modeled as math - physics and math are not the same thing. Math is a functionally complete descriptive language - but it is not a definitive one. Newton's laws were mathematically perfect, it was the physical things that they incorrectly represented which ultimately broke their backs. Newton's laws also had perfect predictive power over everything in their realm - the macroscopic - for more than a century, before we developed instruments finely tuned enough to detect their inaccuracies.

If you're trying to convince anybody here, you're going to fail, because you start by assuming the very mathematical models which they challenge - asserting repeatedly that particles have no definition, and therefore particles have no definition, is an advancement towards nowhere. If you're trying to enlighten people, you do so from the perspective of one biased in favour of a particular mathematical model.

I can't prove my position, but I generally favour a variant of multiverse theory in which the uncertainty principle is the result of consciousness. That is, the human mind as a conscious entity is a functional quantum computer, and the uncertainty principle is a result of that, rather than a fundamental property of the universe. (The uncertainty is not about what state the particle is in, but what spectrum of probability space the mind inhabits, and thus what spectrum of particle states the mind observes.)

You'll notice that this is a functionally equivalent interpretation. Which is the problem with quantum mechanics - the mathematics describe something, but interpretation is, for now, still up in the air.

You'll also notice that this interpretation suggests that a 'slice' of probability space produces a universe of zombies. But a slice of probability space as an independent structure is no less ridiculous in this model than a slice of 2D space taken out of our "normal" three dimensions when treated independently.

"Well, it's physics, and physics is math, and you've got to come to terms with thinking in pure mathematical objects."

Physics is modeled as math - physics and math are not the same thing.

As Korzybski would say, whatever you say Physics is, it is not.

In short, these posts are not going to convince anyone of anything, except possibly people who already accept these models that the author is familiar with the models.

When is this blog going to address bias and how to overcome it?

Hey Eliezer, your explanations are incredible but I don't understand why we are folding the amplitude distribution over that diagonal line. I do understand that the folded distribution is unfactorable, but what is the motivation behind the folding operation?

I believe the idea here is that because particle A and B are indistinguishable, the probability assigned to the case where particle A is "before" B can be equivalently assigned to the opposite case.

In the same manner that a train schedule with cities across the top and side needs only one entry per cityA / cityB pairing.

I see

I was also having a problem with this, but I think it can be resolved by saying that every operator is symmetric in A and B. Therefore, whenever we take a measurement, we will fail to distinguish between a blob at (1,0) and a blob at (0,1). More fundamentally, any interaction with a third particle will be the same whether the blob is at (1,0) or (0,1).

Even more important is that a blob at (0,1) plus a blob with the opposite phase at (1,0) can have a subtractive effect, even cancelling out entirely if both have the same amplitude.

In a sense, then, factoring the topology of configuration-space by the identity A==B causes the symmetry of operators to be an unavoidable consequence of the topology, rather than some freakish coincidence.

Wait, but how did they fire photons one at a time in the first experiment?


Apparently I shouldn't ask questions when I don't understand something because I get thumbed down for not already knowing the answer.

They fired quanta of energy.

As I understand it, an electron isn't an excitation of a quantum electron field, like a wave in the aether; the electron is a blob of amplitude-factor in a subspace of a configuration space whose points correspond to multiple point positions in quantum fields, etc.

It all depends what you mean by quantum field, doesn't it? If you mean '4-dimensional vector field', then yes. But that's not what's meant by Quantum Field - rather, it'd be a 'Fock space', a special subspace of Hilbert space. Using that, the first is more accurate. An electron IS an excitation in this quantum field of electrons in the Fock space. This excitation only exists in the subspaces of the Fock space corresponding to that electron's existence (referring to 'that' electron by there being an electron in that region of spacetime).

This is the first thing I've found that I'd categorize as not even technically right (I skipped this page on my first time through).

In quantum physics, the configuration space is the fundamental thing, and you get the appearance of an individual particle when the amplitude distribution factorizes...

Um, if individual particles are derived from the amplitude distribution on the configuration space, and the dimension of that space is related to the number of those particles, how do we then know how many dimensions should a particular configuration space have?

Concretely, how did we know that we have to draw a 2-d diagram (+2 dimensions for the amplitudes) up there? One spatial dimension for each photon? Ok, but supposedly we don't know how many photons are there - actually, individual photons don't even exist fundamentaly - since this should be a derived fact from the amplitude distribution, right? Do we just guess and see which picture comes out as the most natural?

It all seems kind of circular... in a confused corner of my mind at least - help, please? :-/

A full diagram in that style would be 1 dimension of graph for each dimension of space for each particle that happens to exist, and you have a separate diagram for each different combination of different numbers of particles.

Obviously, he's simplified it, so the answer to your question is just, "That's how he chose to reduce it".

A single point in quantum configuration space, is the product of multiple point positions per quantum field ...

An attempt at translation: each point in the quantum configuration space corresponds to a particular configuration of every field (e.g. electromagnetic, gluon, electron etc.), and the regularities in these fields tell us how many particles of each species do we have and where they are. Is this even approximately correct?

So... how many dimensions does the real quantum configuration space then have? Uncountably infinite?

Yes, that is correct. The fields that physicists actually use (Fock spaces or Hilbert spaces) are considerably more general than the configuration spaces Eliezer has been using. In Eliezer's case, each position in configuration space corresponds to a particular number of point excitations in each field, and you need to integrate these to reach a full quantum configuration.

A single point in the Hilbert/Fock spaces that physicists use correspond to a complete quantum state; Eliezer's configuration spaces are each just a choice of basis vectors in that space. When you add them up to get the full quantum state, it's adding up the basis vectors to get the full vector. That's how you get from a full distribution in Eliezer's configuration spaces corresponding to a single point in a Hilbert/Fock space.


"amplitude distribution happens t factorize." -> "amplitude distribution happens to factorize."

Not the best first comment, but I've spent too much time fixing inconsequential typos to feel comfortable skipping over it. Excellent sequence, I think I'm starting to get a toehold in a.. very different view of how things are.

I find polynomials to be a helpful analogy:

p(x) = x^2 - 5x + 6

This factors into (x - 3) and (x - 2).

...But what if we SWAPPED the factors??

Clearly this is nonsensical, and this article captures that very well. :) Thank you for elucidating materials!!!