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I've been reading an article that describes what seems to be a classical particle as a regularity in the global wavefunction over a quantum configuration space:

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.

...which looks sensible enough, but I cannot figure out what is actually meant by this configuration space; specifically, what is the identity of an individual point in that space?

A simplified picture - mentioned in the above article - tells us that an individual point corresponds to something like "a photon here, a photon there (etc., for every photon), an electron here, an electron there, etc. for every species of elementary particles". But this can't be quite right: if an electron corresponds to a blob of amplitude in the configuration space, a single point in that space can't correspond to a position of this blob... and besides, what would a dimension of that space even be, since the number of particles in the universe isn't fixed.

An "explanation" offered in the article is as follows:

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.

But what does this actually mean?

As I'm given to understand, in quantum field theory one has a wavefunction(al) defined over all configurations of a specific field, so if one jumps over to the real world where space-time is filled with many fields, is it possible to speak of the quantum configuration space over which the universal wavefunction is defined as a space where every point corresponds to some configuration of every quantum field, and if so, can this picture still be used to see an electron as a roughly independent part/blob of this giant amplitude distribution or not?

I've been gnawing at this for two days now and I'd really appreciate some help here...

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1 Answer 1

In terms of quantum field theory, an electron is an elementary excitation of the electron field, in a similar way as a water wavelet is an excitation of a water surface. There is a slight difference, though, as excitations of a quantum field are quantized, hence come in discrete bunches (1,2,3,... electrons) describing their size, while water wavelets are (in the usual classical setting) unquantized, and hence can be of any size. The quantization is the reason why an electron is indivisible (hence an elementary particle), while water is infinitely divisible (in the classical setting).

There are also differences in the more detailed description: The form of a water wavelet is determined by a real-valued function of space, giving the amplitude of the wavelet at any particular point. The form of an electron wavelet is given by its wave function $\psi(x)$, which is a spinor-valued function of space. (A spinor is a complex vector with two components.)

The position of an electron with wave function $\psi$ is not a single point but the region of space where $|\psi(x)|^2$ is significantly different from zero. (Just as the position of a city is not a point, but a whole region.)

In an electron trap, this is a tiny region; in an electron beam, this is a moving needle-shaped region in the direction of the beam. So you describe the electron by a few dof characterizing the shape of $\psi$ inside this region, and declare everything else to be environment.

In a quantum dot, a few lowlying excitations are usually enough to get the essence, and modeling it more accurately is a waste of resources unless the measurement accuracy is very high.

When there are multiple electrons, the difference between water wavelets and electron wavelets becomes conspicuous. While two water wavelets are modelled by a superposition $\rho_1(x) + \rho_2(x)$ of the individual wavelets $\rho_j(x)$ depending on the same 3D position variable $x$, two electron wavelets are modelled by a so-called Slater determinant $\psi_1(x_1)\psi_2(x_2)-\psi_2(x_1)\psi_1(x_2)$ depending on two sets $x_1$, and $x_2$ of 3D coordinates, one for each particle.

And things get worse with more particles. While $N$ water wavelets are described by a superposition of individual wavelets sharing the 3D position coordinate vector, $N$ electron wavelets form a Slater determinant with $N!$ terms in $3N$ dimensions. In the most general situation, the situation is even worse, as we may have a superposition of Slater determinants. This causes headaches (or guarantees their job, if you see it from a more positive perspective) to quantum chemists, as molecules usually have lots if electrons. It also makes a visualization of what happens vbery difficult, beyond what can be explained here in a few lines.

Thus, if you want to have more details, you shood look at Chapter A6: The structure of physical objects and Chapter B2: Photons and Electrons of my theoretical physics FAQ.

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Perhaps it isn't clear what I'm asking, since your answer pressuposes that an electron already exists as a fundamental object (and the wavefunction describes it's "posssible" positions), but what I'm actually interested in is how can an appearance of individual electron be described as some kind of regularity in a quantum configuration space of all quantum fields (or, for starters, just an electron field). –  Rok Mar 16 '12 at 10:17
    
I edited my answer to account for your more precise specifications. –  Arnold Neumaier Mar 16 '12 at 10:44
    
Well, it does get closer to what I'm interested in, so thank you, but I have a few followup questions. :) –  Rok Mar 16 '12 at 20:34
    
You said that an electron wavelet can be described by a spinor-valued wavefunction of space, but since we have more than one electron in the real world - and moreover, their number isn't fixed, and in general, they can all be entangled with each other - how do we model this kind of situation? –  Rok Mar 16 '12 at 20:46
    
I edited my answer to account for your followup questions. Roughly, quantum superposition = entanglement, but this is not the full answer. See ''Indistinguishable particles and entanglement'' in Chapter B3: Basics on quantum fields of my FAQ cited above. –  Arnold Neumaier Mar 16 '12 at 21:19

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