The wavefunction isn't an invariant quantity, it's a means to the calculation of invariant quantities, which are the probability distributions for the field configurations of particles.

The probability for a field configuration is exp(-S[Φ(x)])/Z, where S(Φ(x)] is the Euclidian action (including quantum corrections) and Z is the partition function. Particle number isn't conserved, in the presence of interactions, but it's possible from the probability density to compute the probability of finding a state with some number of particles and so on.

In the absence of interactions the probability density is a Gaussian.

The probability density isn't discrete, it just can be recovered from the correlation functions. It's the correlation functions that really matter.

Great question, thanks for asking.

Stam Nicolis

What you state is what in your view of quantum physics the wave function can be. I know that the square of the modulus of the wave function is a location density distribution. A stochastic process generated the footprint of the particle that conforms to this distribution. The chance is large that the stochastic process generates a normal distribution. If the generated collection is discrete and if that is done step by step, then a hopping path will be generated. In some way that hopping path must generate a hop landing location swarm that is described by the location density distribution. The stochastic generation is an ongoing mechanism. The particle resides on a platform that can archive the hopping path as a cord of quaternions that become eigenvalues in the eigenspace of a dedicated operator. A quaternionic separable Hilbert space can easily archive the complete hopping path during the full lifetime of the particle. Some other mechanism must embed the ongoing hopping path in a dynamic field that represents our universe. This dynamic field can be archived in the eigenspace of a dedicated operator that resides in a non-separable Hilbert space. The model contains a system of Hilbert spaces that all apply the same underlying vector space.

The form of the wave function depends on what the particle is doing.

Is the particle just moving freely in a straight line?

Is the particle trapped in a box?

Is the particle bound to an atom?

Particle subject to a magnetic field?

Particle in a thin layered material?

Different in each case. Those who handle quantum problems know this immediately. Thinking about hopping around not productive for this.

You have to learn to solve the Q problem.

Juan Weisz

If an elementary particle is represented by its own private Hilbert space and its hopping path is archived in the eigenspace of a dedicated operator, then this archive does not depend on the embedding of the representing Hilbert space into a background platform that archives the dynamic field that represents our universe.

The system of Hilbert spaces that apply the same underlying vector space supports a location density distribution that is constant and describes the recurrently regenerated hop landing location swarm that the ongoing hopping path produces.

The system of Hilbert spaces is a purely mathematical model of physical reality that differs from what is taught as the mainstream model of particles. The mainstream model derives QFT, QED, and QCD from a Lagrangian and that is derived from the stationary action principle. I do not consider that as a suitable fundament of a model of physical reality.

During my studies at the TUE I was taught the vision that resulted after the introduction of the Schroedinger equation and Heisenberg's Matrix-based approach that Dirac combined in a single theory. During my career, I raised more and more doubts about the approach that mainstream physics took. I started to investigate the approach that Hilbert and others developed. I think that they took a better inroad on the foundation of physical reality. I have put my vision in Preprint How to design a universe

Hans

There is no particle path to start with in QM.

You are right, you probably are far off the beaten path.

The beaten path has no proper foundation. Without foundation, the building may rest on quicksand. I find it safer to start at a foundation that operates at the level of the most basic space. As far as I know, the most basic space is a vector space.

I decorated the vector space until it worked as a system of Hilbert spaces that all apply the same underlying vector space. Each Hilbert space corresponds to a different coverage of the vector space. That coverage is navigatable by a coordinate system that places coordinate markers that are identified by members of a number system. This construct appears to possess properties that show a surprising resemblance with the Standard Model of the elementary fermions.

The Standard Model of the elementary fermions is a part of the Standard Model of the experimental particle physicists. The coverage is uncountable and that makes the coverage a continuum in which all converging series of locations end in a limit that is a member of the coverage. The existence of the limit makes the continuum differentiable and differential calculus describes the behavior of the continuum.

Each member Hilbert space owns a state vector that is taken from the underlying vector space. It is also the vector that defines the footprint of the represented elementary fermion. Its expected value in terms of the eigenvalues of the corresponding footprint operator equals the geometric center of the private parameter space of the Hilbert space.

A physical sitation cannot be so mathematized as you describe. Even by a Hilbert himself.

You describe like a minority of math people have the right answer.

They should spend more time in labs.

The foundation is mostly empirical. Experiment after eperiment, like they do in good physics programs.

Juan Weisz

I visited the LHC and was astonished at what they can measure and even more by what they cannot measure. The advantage of the LHC is its enormous luminosity. That enables them to measure very precisely. The signal-to-noise ratio is huge. On the other hand, many measurements are indirect and require a mathematical theory to translate the observations into the final measuring results. During my career, I was an experimental physicist and I built part of my measuring equipment. Behind every measuring apparatus serves a mathematical theory that tells how the measuring results follow from the raw observations. I helped develop image intensifiers. These imaging devices turn every detected quant into pixels consisting of a huge number of quanta. Consequently, the device can show how individual quanta behave. Image intensifiers are applied in night vision equipment and in X-ray imaging equipment. In those applications, low dose imaging is a rule. The signal-to-noise ratio is very low. I learned that the quantum theory that was lectured at the university differs essentially from what occurred in practice. That is why I started looking at alternative approaches for quantum physical theories. In my opinion, David Hilbert and his surround found a far better approach than mainstream physics was following.

I disagree with your view that experiments constitute the foundation. In fact, most fundamental facts are not accessible to measurements. First, you need a well-founded mathematical model. Then you can check whether the model conforms enough with the observations. If that is not the case then the model must be rejected and the investigation must be restarted by a better agreeing mathematical model. Most physicists make the fault that they start by tackling much too advanced concepts. The Lagrangian is a notorious example.

My model of the universe starts with empty space. That space is a container, but empty space contains nothing that you can refer to. It has no measures. It does not have a center and it certainly contains no coordinates that could help to navigate the contents of space. A container has the capability to contain objects. These objects must be simple. As soon as more than one object is present, then relations between objects become possible. A vector space contains vectors that obey vector arithmetic. A vector combines two points and the direction line that connects them. One point is the base point and the other point is the pointer. The vector has a length, which is a scalar. Via the vector arithmetic, all locations in the vector space can be reached. Thus, at the location, the pointer of the vector and the location coincide. A location is a point-like object. These locations are the more interesting objects. Set theory tells what happens when the number of locations supersedes accountability. In that case, the locations form a continuum and all convergent series of locations end in a limit that is again a location. The consequence is that the continuum is differentiable. Being differentiable means that the continuum can change. Differential calculus tells how the continuum can change. The cause of the change is a curious subject. These events are not expected and are completely counterintuitive. Also, the reaction of the continuum is unexpected. The reaction follows from differential equations. These phenomena are corollaries of the coverage of space with point-like objects.

No current physical theory explains these phenomena. Instead, physical theories apply differential calculus and consider it as a handy tool.

Even more curious is the fact that a small extension of the vector space transfers it into a Hilbert space. Each Hilbert space owns a private parameter space that turns the Hilbert space into a function space. This is the consequence of the fact that Hilbert spaces can archive members of a selected version of a number system in the eigenspace of a corresponding operator. It is possible to define a system of Hilbert spaces that all share the same underlying vector space.

This introduces interesting possibilities for the coverage of space by point-like objects. The system of Hilbert spaces also covers the vector space and each Hilbert space does that with its own symmetry, which derives from the symmetry of its parameter space. This coverage is archived in the combined storage capacity of the Hilbert spaces. One Hilbert space plays a special role. It embeds the state vectors of the members of the system of Hilbert spaces into the continuum eigenspace of a dedicated operator that manages this continuum as a dynamic field. Physicists call this dynamic field our universe.

I have not yet found any inconsistencies in this coverage of space. It is in concordance with set theory, current number theory, vector space theory, and Hilbert space technology.