No Individual Particles

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.

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?

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

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 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?

"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!!!