**3. Symmetries and superpositions**

The interpretation of the Schrödinger equation that we obtained is that it merely describes the evolution of the probability distribution for the slow variables, after averaging over the positions of the fast variables. The fast variables dictate the evolution, but they act too fast for us to observe this directly. The new thing in our procedure is that we have the choice to also describe the fast variables using quantum mechanics as a tool: fast and slow variables together go into a vector representation of what happens.<sup>2</sup>

In statistical treatments of moving variables, with well-determined evolution laws, it should be completely clear that the probability distributions of the final state are the result of our choices for the probability distributions of the initial states.

At first sight, the group of rotations can also be regarded as pure permutations, and, although the lattice structure of our local coordinate space *x* ! will be severely affected, one might suspect that our present understanding of physics comes from smearing the lattice back into a continuum; this may be a reasonable approach towards understanding rotational symmetry.

However, more severe problems arise if we consider the notion of spin in a particle. We need to take spin into account when analysing Bell's theorem. In the treatment displayed in the previous section, the spin variable of a particle would be a discretised variable *s*, with integer spacings ranging from �*S* to *S*, where *S* is the

<sup>2</sup> Do keep in mind that the distinction between fast variables and slow variables is a feature of our simplified models, but possibly unnecessary in the real world. All variables are real, evolving classically according to the same or similar classical laws.

total spin quantum number, being integer or half-odd-integer. These would be promoted to the status of classical variables, and then we can set up exactly the right Schrödinger equation for particles like the Dirac particle. What happens with its ontological interpretation if we rotate that?

It is clear that, in this case, rotation transformations transform the 'real states' ∣*s*i into superpositions. In doing so, the rotation group can serve as the prototype of many symmetry considerations in quantum mechanics. How do we analyse the Stern-Gerlach experiment?

The superimposed states obey exactly the same Schrödinger equations as the basis elements do, and we had chosen the latter at will. So one possible answer could be: it does not matter which of the states we call real; there is no experiment to help us make the distinction. But this is debatable. The Stern-Gerlach experiment in its vertical orientation distinguishes particles with spin up from particles with spin down, these have different orbits. Remember that, in this chapter, we focus on going beyond the usual statistical interpretation of quantum mechanics, aiming at a description of pure, real states. The only accepted probabilities are 1 and 0.

The real physical states we work with form a basis of Hilbert space, and the equations we work with ensure that any state that starts off being real, occurring either with probability 1 or with probability 0, continues to be a real state forever. This must also hold for any Stern-Gerlach set-up or any of the other paradoxical contraptions that have been proposed over the years. Real state in ¼ real state out. This was called the 'law of ontology conservation' [14].

At first sight it seems that a Stern-Gerlach experiment, after a rotation over an arbitrary angle, turns into a superposition of several real states. This is true in the mathematical sense. It is the easiest way to visualise what the rotation group stands for. However, if we physically rotate a Stern-Gerlach experiment, by undoing and re-arranging nuts and bolts, We do something else. The new experiment again goes into one of the realistic states; the nuts and bolts also go into new physical states, so this is not quite the same kind of rotation.

Notice however, that if a particle with spin leaves one Stern-Gerlach instrument and continues its way in an other, rotated, device, then, as we know from standard quantum mechanics, it emerges in a superposition, or more accurately, in a probabilistic distribution. Where does this stochastic behavior come from? What happens if we do interference experiments with the various emerging beams of particles?

**Figure 2.**

*The periodic orbits of the fast variables. Points where interactions take place are indicated. If these occur in the orbit of a single fast variable (a), they will be difficult to miss, but in the case of two or more (b), the special points will be hit much less frequently, so that the interactions become slow. The orbit takes the shape of a (multidimensional) torus (c).*

#### *Ontology in Quantum Mechanics DOI: http://dx.doi.org/10.5772/intechopen.99852*

Apparently, there are other variables that play a role. We can blame the fast variables for this. The fast variables for the rotated device do not coincide with the previous fast variables. In specifying the state of the particle in the first device, we forgot to observe where exactly the fast variable was. We couldn't observe this, as it was moving too fast. The transformation (in this case that is the rotation), formally involved the excited energy modes of the fast variables. In practice, we know that the energies of the quantum particles in both devices are too low to detect the excited modes, but in formulating the interactions, using the special points in **Figure 2b** and **c**, the excited modes do play a role because the interaction points are localised.

From these considerations, we claim that whatever is left of the various paradoxes should be nothing to worry about.
