**5. Where are the fast variables and the slow variables in the standard model?**

At first sight we may seem to be a long way from describing quantum field theories such as the Standard Model. In principle, one may expect something resembling a cellular automaton, where we may be able to project the various field variables as data on a cellular lattice. However, as described in Section 3, we have to deal with the question how continuous and discrete symmetry patterns, essential for the Standard Model to work, can be introduced. As is well-known, once we have all local and global symmetries in place, the entire Standard Model is almost fixed, with only a few dozen interaction parameters to be determined. We make a gentile attempt at finding some sign posts that could indicate to us where to start.

In a very important paper [17], F. Jegerlehner describes the Standard model as a minimalistic outcome of an algebraic structure whose basic interaction properties

<sup>5</sup> This then would be an example of the 'butterfly effect'. It is not as crazy as it sounds. As soon as we include the fast variables in the discussion, the dynamics becomes invariant under time reversal, and the statement that a later photon is correlated with settings chosen earlier is then not strange at all.

are essentially natural near its ultimate cut-off scale, the Planck length, except that the Higgs field self coupling happens to vanish, or almost vanish, at that scale. It seems as if the universe is metastable, or perhaps just at the edge of stability. When we scale towards the TeV scale, using the renormalization group equations, one discovers that the Higgs self-coupling slowly grows towards its present value, and this appears to explain the recently observed Higgs mass remarkably well.

There are important new questions that may be raised in connection with the present work. One is where in the cellular automaton this copious algebra is generated; and of course we want to know how any kind of fast oscillating variables can arise. Previously, this author was just thinking of very heavy virtual particles such as the vector bosons that represent the remaining grand unification symmetry, but there is a problem with that: as described in Section 2, **Figure 2**, the dynamical fast variable must have the geometry of a multi-dimensional torus, whereas fields have a more trivial topological structure if they indeed form vector representations of the unifying algebra, see Section 2.

This perhaps can be done better [18]. The general philosophy that might be useful here starts from a fundamental observation. Fields that describe data at the Planck scale, can only propagate as fields at much more conventional scales (from milli-electronVolts to nearly a TeV), if there is a mechanism that prevents them from obtaining effective mass terms. To be precise, the dynamical field equations must allow them to be shifted by a constant with only minor effects on the energy of the state. At our scale of physics (to be referred to as the SM scale), fields can be shifted in any way, depending on space and time, such that energies also change within the energy domain of our SM scale. This means that the effective mass term must be at the SM scale. When we move towards the Planck scale, this mass term must rapidly approach to zero. Physically, the only mechanism that can do this is the Goldstone mechanism:

Only if a field effectively describes a symmetry transformation, and if, at the Planck scale, our world is invariant under this symmetry transformation, then we can understand how this field can propagate all the way to the SM scale.

Since the Standard Model has a rich spectrum of possible fields (fermionic and bosonic), this would force us to suspect that each of these fields must represent a symmetry transformation under which the Planck-scale theory is either exactly invariant (when the mass term vanishes) or invariant in a very good approximation (when the mass is of SM scale or smaller). Indeed, this should also hold for the fermionic fields, and this points towards supersymmetry at the Planck scale.

In short, every field component in the SM represents a generator for an almost exact symmetry of the Planck scale model. If we would be dealing with only scalar fields in the Standard Model, this would give us all the symmetry transformations, including estimates on how well the system is invariant.

Unfortunately, the real situation will be a lot more complicated. We have fermionic fields that transform as spinors under rotation, and we have vector fields that themselves again obey local gauge symmetries. How do we deal with that? It would be a great assignment for a team of PhD students to design and elaborate a logically coherent mathematical scheme.

This scheme might eventually produce logical guidelines for setting up cellular automaton models in such a way that their behavior at SM scales indeed reproduce the SM. But this is not all. The resulting automaton will still be a *quantum* automaton. What we now need is a set of variables that can play the role of fast variables. These are fields, but they cannot live on a flat field-space, they must form toruses as in Section 2. Now it would be tempting to consider the gauge groups. All group parameters of the local gauge groups *SU*ð Þ2 , *SU*ð Þ3 , and *U*ð Þ1 , form toroidal spaces

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

or spheres. What's more, we know that the physical quantum states are invariant under these group transformations, so regardless their time-dependence, our world should be in the invariant state, just like the energy ground state. This could be an alley towards understanding how quantum behaviour could follow from a classical cellular automaton.
