Smooth Quantum Mechanics provides a way to avoid the discontinuity in the wavefunction collapse. This post presents the wave-like nature of the particles in Smooth QM, as compared to the standard QM. The probabilities also have a different character, being rooted in the initial conditions, instead of discontinuities.
In Smooth Quantum Mechanics, there are only (entangled) waves.
In standard QM, the particle behavior is manifest when the observable is a position operator acting on the Hilbert space. In this case, the system is found in an eigenstate of the position operator, which is a Dirac distribution (a generalized function which is zero everywhere, except in one point, where is infinite, such that its integral is 1). The wave and particles are, in standard QM, all vectors in the Hilbert space, being therefore wave functions or distributions. When expressed as wavefunction (in a basis of positions), the “pure waves” are eigenstates of momenta, and the “pure Dirac distributions” are eigenstates of positions - these are extreme situations, in which infinities occur.
One interpretation of the waves in QM is that they underlie probabilities of finding the particles at a given point in space. Of course, this is true, grace to the Born rule, but this does not necessarily mean that the point particles are the fundamental ones, and the wave represents only probabilities. Yet, many like to think at particles as being fundamentally point-like, therefore the waves need to be interpreted as underlying probabilities.
In Smooth QM, it is considered that the only real physical states of the system are smooth: it is employed a space of smooth wavefunctions of finite norm. By completing the space, we obtain a rigged Hilbert space, which contains state vectors that are abstract, being distributions or having infinite norm. By observing the position, we get only a smooth wave concentrated around that position, but not a Dirac distribution. The position is never determined to be in a point, but in an open subset of the space. I think that the two extremes: eigenstates of pure momenta and pure positions, are non-physical abstractions, because they bring in infinities, but they are useful to express and explain the wavefunction duality. In Smooth QM, by not allowing the distributions, by eliminating the infinities, the unity between the two complementary aspects is more manifest and more physical. All particles are waves, in various shapes, depending on the observable. Of course, the Hilbert space is very useful, and it is easy to solve the equations in this space, and many operations become simpler. But in Smooth QM it is considered abstract. The wave is physical, not a probability wave.
Therefore, we return to the old idea of particles as wave packets. One main problem of this view was the dispersion of the wave packet. In Smooth QM, the laws are the same, so the dispersion is not eliminated, but the mechanism is such that the dispersion is no longer a problem. A measurement of position, finding the wave-like particle localized very well, will imply dispersion both in the future, and in the past. The waves converge, the wave packet is “con-persed” until the small sized packet is obtained and determined by measurement, and then it disperses again. The “delayed initial conditions” mechanism allows this.
The probabilistic behavior occurs only when we measure an observable which does not contain the observed state among its eigenstates. In this case, the Born rule expresses the probabilistic character, but the probability in Smooth QM resides only in the undetermined initial conditions. Here “determined” has two meanings, active and passive: to cause, and to measure. The probabilistic behavior occurs when a collapse occurs, but in this approach the collapse is smooth and deterministic, and the probabilities reside in the initial conditions. Even the Heisenberg relations, which are often thought as representing irreducible probabilities, are not necessarily such. They can be obtained from Fourier analysis, and apply for deterministic waves. The probabilistic character occurs because of measurements, which entails the (smooth) collapse, which brings the probabilities of the unknown initial conditions.
Although I said that the waves are physical, and real, we should not forget that they are entangled. For a particle, the wave can be a field depending on space and position, but for more particles, we need to consider the tensor products, therefore the entanglement.