@Remi Cornwall:

Let me make some remarks concerning the origin of this question:  

(A)

If one considers an object (particle) having an indefinite charge density rho(x) restricted to a compact region of R^3 then it is straight foreward to decompose this charge density into two parts, i.e. a pos. part rho_+(x) and a neg. part rho_-(x)

and to define total pos. charge Q_+ and total neg. charge Q_- and a total charge Q and a center of pos. charge X_cc+ and a center of neg. charge X_cc- via integrals and to define a 'total' center of charge X_cc as a weighted sum of X_cc+ and X_cc-. Then, by defining the complex function

psi(x) := ( sqrt( rho_+(x) ) / sqrt(|Q|) , i sqrt( rho_-(x) ) / sqrt(|Q|) ) , Q := total charge

one obtains

integral |psi(x)|**2 dx = 1

which means that psi(x) is normalized to 1

and

integral x |psi(x)|**2 dx = X_cc

which means that the 'expectation value' ( or mean value ) of 'position operator' of non-rel. QM ( or of Schroedinger wave mechanics ) equals center of charge of the object !

Therefore, the complex psi(x) can be viewed as a convenient means for a formal description of the definition of the center of charge of an object having indefinite charge density.

(B)

Now, as a model, consider a kind of gas of fluctuating tiny dipoles,

i.e. dipoles which extend from zero-length to some non-zero length and back again to zero length. In the limit !!! of many, many dipoles this system can be considered as a 'field', which has an indefinite charge density, and a center of charge could be defined for that field according to (A).

(C)

Now, consider the 'gas bubble' described in (B) to be attached to a free electron,

i.e. consider that the tiny dipoles embody the presumed 'vacuum fluctuations'  surrounding the electron, and consider the free electron in its rest system.

This is an object which has a fluctuating charge structure and which can be described by a complex function psi(x) according to (A), but now, integral x |psi(x)|**2 dx is the position of the center of charge of these fluctuating dipoles with respect to the center-of-mass position of the electron. If such an electron is bound to an atom, then it is quite natural to assume that the e.m. field of the nucleus influences the fluctuations

and thereby the structure of the field which is characterized by psi(x).

What Schroedinger wave mechanics or QM actually does for electrons bound 'inside'  an atom is to set up a differential equation which determines psi(x) as an eigenvalue problem of a s.a. operator (Hamiltonian). This part of non-rel. QM (for a bound electron) can be understood even in the case where psi(x) has an ontology as specified in (A).

(D)

The above arguements can be extended to the case where the tiny dipols have a spin ('up' and 'down' with respect to the dipole axis).

One arrives at two functions psi_1(x) and psi_2(x), same ontology as above,

and (following Pauli) one obtains a diff. eq. (Pauli eq.) for ( psi_1(x) , psi_2(x) ) which transforms as a 2-spinor.

(E)

Finally, if one consideres one of the tiny dipoles to be splitted up into a minus-object (electron) and a plus-object (positron) and if one considers the bubble of reminder dipoles as a compound object of two bubbles one arrives at ( psi_1(x) , psi_2(x) ; psi_3(x) , psi_4(x) ), same ontology as above, and diff. eq. for 4-spinors. 

Dirac eq. can be obtained by group theoretical arguement (unitary repr. of inhomog. Lorentz group, cf. Straumann's excellent review article published in arXiv.org).

Proceeding this way one has started from the very beginning with a well defined ONTOLOGY which is free from the Born-Bohr mystery of 'complex-valued probability amplitudes'. 

Indeed, the Dirac eq. has solutions which characterize a trembling change of a 'position' ('Zitterbewegung'). This supports an interpretation of psi(x) as a 4-tupel of complex-valued 'charge amplitudes' which have their origins in a chaotic or (quasi)stochastic dynamics of fluctuating dipoles which are described in the sense of an approximation (many, many dipoles!) as a tiny 'field' in the space-time continuum.

By the way:

The concept of fluctuating dipoles surrounding a particle yields a surprising result even in the case of a photon:

The Mach-Zehnder phenomenon ("quantum rubber") can be understood qualitatively on an almost pure classical level, i.e. simply by means of Lorentz force law of classical electrodynamics and some considerations concerning a splitting of the dipole bubble and spin and magn. field of the dipole bubbles and momentum transfer to the photon when it arrives at the second half-mirror and interacts with one of the bubbles.

But this would be just another lengthy story.

@Remi Cornwall :

The question of the physical meaning of the 'Zitterbewegung' feature is involved with the question of the physical meaning of psi-spinor in the Dirac eq.

To my point of view, and in contrast to what is written commonly in most of the textbooks on relativistic QM, the Dirac field could make sense for describing a free electron or a free positron, provided ! that the electron or positron is considered as an object with charge that has a flucuating dynamical structure which behaves like a gas bubble consisting out of many, many fluc. (massless) dipoles plus one (or two) single 'point' charge objects with mass>0.

Considering the Dirac eq. in some inertial frame of ref. it turns out that solutions with both, pos. AND neg. frequency parts show up the feature that the expectation value of an operator v, which is claimed to represent a 'velocity', is highly oscillating near the speed of light. Therefore, it makes no sense to consider      as the 'velocity' of the free electron.

The only way out of this unphysical feature seems to be to consider psi(x) as a quantity that characterizes an object which has a charge distribution caused by some presumed (massless) dipole fluctuations. In this case it is the position of the center of charge which is "trembling" with respect to the center of mass (or center of energy) of the field.

But even this is questionable, because the Dirac field is linear, i.e. the diff.eq. is linear. It cannot be expected that the diff.eq. has a single wave packet solution   representing some definite structure. To accomplish this a non-linear Dirac field can be considered, and to achieve a local gauge symmetry of the solutions of the field eqs. by  'adding' gauge field amplitudes A_mue to the Lagrangian, and to see how the nonlinearities can produce solutions which correspond to an energy density which looks like a wave packet and to a charge density which is (slightly) fluctuating and which causes the "trembeling" of the center of charge with respect to the center of energy (center of mass in the rest system) of the 'extended' particle. In doing so, one leaves the scope of 'ordinary' QED or QFT and arrives at non-linear field theories.

The notion of space-time position co-ordinates (and space-time fields) is a construction of the observer. Particles have precise quantum numbers. The "smearing" into a space-time density arises when the observer interprets those quantum numbers in a space-time co-ordinate frame.

@Mike York:   That's how I see it, too.

Furthermore, space-time as a 'continuum' (which is tacitly presumed by the 'observer') should be considered as a smooth approximation of a more general discrete structure which is 'filled' with some 'fundamental' 'pulsating objects' which are characterized by some 'fundamental'  'states' and 'state transitions' and 'histories' (sequences of states) and which (at least some of them) can be considered as 'fundamental clocks' and 'fundmenatal rods'. For each such an object it would be desirable to construct some discrete countable structure which can be made 'dense' by some approximation processes, and thereby yields a Lorentzian (1+3)-structure, and, by considering a huge ensemble of such (1+3)-structures as an ensemble of tangent spaces, it would be desirable to end up at an ontology of space-time that is used in 'ordinary' physics, i.e. 'flat space-time' of SR or 'curved space-time' of GR.

Concerning 'particles' the approximation process which yields space-time as a continuum must then be 'recompensated' by a 'smearing' process of those particle quantities which are involved with space-time coordinates. 

My conjecture is that 'vacuum fluctuations', considered as a kind of 'fluctuating dipoles' for which some discrete dynamics is postulated, might be good candidates for the 'pulsating objects' mentioned above.

My worry about all this is the issue of the description of two (distant, generally noninteracting) particles. In QM, you ill have a wave function of two variables, giving probabilities of 2-particle configurations. This allows for all the remarkable features of entanglement. It does not seem to me that this will readily lend itself to the kind of modelling you suggest.

@F. Leyvraz:

To my point of view the interpretations of experiments which are involved with 'entanglement' rely heavily on some of the questionable axioms or credos of QM.

QM-CREDO-1:

The allowed (i.e. supposed to be allowed) values of characteristical quantities of a system in an INITIAL situation (preparation of the system) before it interacts with some other system can only be specified by an 'observer' up to some uncertainty, and therefore, the possibility of specifying an initial situation or 'state' with respect to 'certainty' is limited.

REMARK-1:

The most critical point of 'Copenhagen'-QM seems to be that this 'uncertainty' is attached to the system by describing the system by a 'wave function' (or mathematical more elegantly by a 'state-vector') and to connect this kind of 'uncertainty' with some 'randomness' of FINAL situations after some interaction with some other system. Thereby, the 'uncertainty', which stems from the limited capabilities of the oberserver in tackling the system in a preparation process to get a 'well-defined' initial 'state', is identified with the 'randomness' as a property of the system. But the dynamical nature of the system itself might be intrinsically 'random' or stochastic or highly chaotic. If this is the case, then by QM-CREDO-1 the 'uncertainty' due to the observer is mixed !!! with the 'intrinsic randomness' of the system from the very beginning of the theory.

QM-CREDO-2:

The Hilbert space of state-vectors of a system is given by the tensor product of the Hilbert spaces of the subsystems, in any case, whatever the system is.

REMARK-2:

The convincing significance of QM-CREDO-2 becomes clear even on a (semi-)classical level when the electromagnetic field is 'quantized' by Fock space methods.

QM-CREDO-3A:

The state-vector shortly before a measuring process of some of the characteristic quantities ('observables') of the system (i.e. an interaction of the system with some other system) in general is NOT an eigenvector of the selfadjoint linear operator which is associated to the observable to be 'measured'.

QM-CREDO-3B:

But the state-vector shortly after a measuring process is an eigenvector of the selfadjoint linear operator which is associated to the observable which has been 'measured' ("collapse of the wave function").

REMARK-3:

Applying this QM-CREDO-3A/3B to the EPR-like experiments leads to the puzzling arguements which are stated in the context of 'entanglement'. Following the way QED can be constructed via Fock space methods (applied to Dirac field and e.m. field) one obtains a theory (QED) which in a limit contains (non-relativistic) QM of charged objects (electrons) interacting with e.m. field. In such a theory-set-up it seems to be not necessary the postulate QM-CREDO-3A/3B. Then, whenever 'probabilites' are calculated from 'probability amplitudes' and expressions like ||**2, one can ask the question of what is the 'nature of the wave function' and wether there are some other reasons, for example some 'intrinsic' random or quasi-stochastic or highly chaotic unknown properties of the systems under consideration, which might explain why "||**2" leads to reasonable results for the statistics observed in experiments.

QM-CREDO-4:

Allowed values of physical (dynamical) quantities are eigenvalues of s.a. operators in the Hilbert space of state-vectors.

REMARK-4:

Usually in axiomatic QM QM-CREDO-3 is used to 'prove' that the state-vector shortly after measurement must be an eigenvector of the operator corresponding to the observable to be measured. The relevance of eigenvalues and eigenvectors of s.a. operators with respect to the discrete energy levels of excited atomes or molecules and the discrete energy spectrum of emitted photons can be founded mathematically by considering properties of the square-variance -**2 of s.a. operators A and vectors v in a Hilbert space. A more general QM-CREDO would be to postulate a minimum principle: The system tries to get into a state where, during and-?-or after the interaction, the square-variance of the energy (represented by a s.a. operator in a Hilbert space) and some other (commuting/anticommuting) s.a. operators is minimized (and thereby, even opening a way to handle constraints in numerical calculations of approximations to eigenvalues and eigenvectors).

FINALLY:

To my point of view EPR-like experiments indicate that QM-CREDO-3A/3B should be more questioned and QM-CREDO-1 should be stated much more carefully. Up to now it cannot be excluded that the hard facts of EPR-like experiments can be understood by means of still unknown conservation laws which influence the behaviour of photons.

The crucial point, as far as I can see, in EPR experiments, is that they have very little to do with quantum mechanics: two particles, or whatever, go towards two detectors. These detectors simultaneously are randomly set, say to any one of three positions A1, B1 and C1 for the first detector, A2, B2 and C2 for the second. The detectors then give answers 1 or 0. This is repeated millions of times. The instrument settings are recorded, and so are the results. One finds that

1) Whenever the settings were the same, the instruments gave the same answer (either both 0 or both 1) all the time.

2) Whenever the settings were different, the results were the same 25% of the time.

The fact that this can be done is experimental fact. The question is: can one in any way explain it in terms of the kind of classical picture you have in mind? Quantum mechanics, using a completely straightforward calculation, does predict that this is indeed possible, but it does not say how it should be explained in classical terms. Indeed, it says, if anything, that such behaviour cannot be explained classically.

@ F.Leyvraz:

Right now I do not have a comprehensive answer to your question. All I can do is to tell you what I have done so far to get some insight what kind of physical processes might be the reason for the strange results of the typical experiments which are usually considered to favour QM.

The most astonishing result is that the Mach-Zehnder experiment (quantum ereaser experiment) can be understood qualitatively on pure classical level using Maxwell electrodynamics, if one makes some assumptions about the nature of the photon.

The key is to consider the photon as an 'extended' object (neglecting for the moment some difficulties which are related to the fact that the photon has no mass and no rest system) which has energy and momentum PLUS a fluctuating charge distribution (imagine a tiny 'cloud' of massless fluctuating dipoles) and to assume that a very tiny amount of that 'cloud' can be separated in interactions of the photon with atomes or molecules, forming a 'ghost'-cloud, and that this 'ghost'-cloud obtains a tiny spin and a tiny magn. moment and a tiny polarized e.m. field, and moves just as the photon at speed c.

The difference between the entanglement/non-locality experiments and a classical two gloves in boxes experiment is that in the glove case, someone puts the gloves in the boxes and so the information is available in the universe and therefore any future measurement must be consistent with that information if nature does not conspire against us -- even if the glove receivers don't have that information before they receive the gloves. In the quantum case, the information as to which particle is in which state is not available anywhere in the universe and each particle state is therefore a superposition to a space-time observer.

The presence of the Bohr radius suggests a centre of charge with a maximum probability radius of charge