Wilson gauge fields describe one type of irregularities of the cellular lattice. The other type describes deformations of the lattice itself.
The cells in our model do not live in empty space. There is something between and around them which transmits forces between them. In the regular cases, the influence of the medium between the cells is described by the regular lattice equation for the cells.
But assume now that something in the medium becomes irregular. The cells themself can oscillate around their positions as usual, but between them something goes wrong. How can this "something" influence the motion of the cells?
An interesting point is that the particular mechanism of influence is not important at all. At least, if the distance between the cells is small enough, we will be unable to distinguish most of the details of the influence, simply because the observable effects of different mechanisms coinside for large distances. This effect is called "large distance universality".
Because of large distance universality, it makes not much sense to speculate about the particular mechanism of influence. What counts is another thing: how the unknown irregularity modifies the equations of the cells.
Let's consider two neighbour cells. Assume something between them has changed. Let's try to compensate this distorted influence of the medium between them on the left cell by modifying the state of the right cell. If this is possible, that means, if the influence of the irregularity can be completely (or at least approximately) compensated by a modification of the state of the neighbour, we can use this modification of the state of the neighbour cell to describe the irregularity. After this, a description of the properties of the medium between the cells is no longer necessary: All the information we need for the equation is contained in the compensating modification of the neighbour cell — an operation which can be handled, mathematically, even without knowing anything about the medium between the cells, because it is an operation which is defined on the cells.
Let's consider the mathematical part of this description. First, we have modifications of the state of the cell. These modifications define a group. Thus, we have some group G of transformations of the state of a single cell. The configuration of a cell we have already identified with the group Aff(3), which is a dense subset of the twelf-dimensional space R12. But the state of the cell is more than the configuration. We have to include also the velocities or momentum variables, thus, we have to consider the phase space, which gives R24.
In principle, the group could be allowed to act on the phase space in an arbitrary way. We consider only the simplest case of linear transformations. Thus, we assume the group of modifications acts as a linear group on R24.
Once we have fixed the group G and its action on the phase space R24, we can define the influence of the material between two neighbour cells with by an element g(n,n') of the group G for each pair of neighbour nodes (n,n'). Now, the relative position between two cells is the same if we apply g(n,n') to the cell in n', or if we apply g(n,n')-1 to the cell in n. Therefore, we also assume that g(n,n') and g(n',n) are inverse to each other:
g(n,n') g(n',n) = 1.
Given the g(n,n'), we would like to know how they modify the equations of motion of the cells. Now, let's consider the equation of motion for the cell n. This equation depends on the state of the neighbour nodes n'. To compensate the influence of the medium between the nodes, we have to replace the occurrences of the state φ(n') in the equation for n by g(n,n')φ(n'):
φ(n') on n ⇒ g(n,n')φ(n').
The typical term where neighbour nodes appear are difference operators like φ(n)-φ(n'). On n, they have to be replaced by
φ(n)-φ(n') on n ⇒ φ(n)-g(n,n')φ(n').
It is also interesting to compare this with the modification on n':
φ(n)-φ(n') on n' ⇒ g(n',n)φ(n)-φ(n').
Here our compatibility condition g(n,n') g(n',n) = 1 becomes useful: If φ(n)-g(n,n')φ(n') =0, then it follows that g(n',n)φ(n)-φ(n') = 0 too.
Fields on a lattice, which are defined on edges (n,n'), with values g(n,n') in a group G, so that g(n,n') g(n',n) = 1, are well-known in lattice gauge theory as "Wilson gauge fields". Technically, the only difference between our construction and the Wilson gauge fields of lattice gauge theory is that the Wilson gauge fields are defined on a four-dimensional lattice in spacetime, while we have a lattice only in space. We ignore here the resulting minor differences and name the fields we obtain in this way "Wilson gauge fields".
As follows from our considerations, the action of Wilson gauge fields is defined on the degrees of freedom of a single cell. The action of the gauge group G will be the same on all cells.
Remember now that we have defined electroweak doublets based on the doubling effect, using only a single, scalar cell function for each doublet. The different parts of the electroweak doublet, distinguished by isospin and parity, are represented as different types of oscillating modes for the same function. But the Wilson gauge field acts only on a single cell — it cannot "see" the mode of oscillations, because this requires the consideration of several neighbour cells.
As a consequence, the interactions described by Wilson gauge fields cannot depend on parity or isospin of the particles. In other words, the charge of the Wilson gauge field is the same for all parts of an electroweak doublet, it cannot depend on parity or isospin. Therefore, the electroweak gauge fields cannot be represented by Wilson gauge fields.
On the other hand, this property of the Wilson fields is also a characteristic property of the action of SU(3)c in QCD, the gauge action which describes strong interactions in the standard model. Their action does not depend nor on the isospin, nor on parity of the particles.
The most interesting property of the gauge fields of the SM is that they preserve Euclidean symmetry. Let's consider what follows from the preservation of Euclidean symmetry for Wilson gauge fields.
The phase space for each cell n is given by the twenty four parameters aiμ, πiμ. On this space, we have to define the action of the Wilson gauge group.
This action is not completely arbitrary, because it is obligued to preserve the mathematical structure related with the phase space, which is name "symplectic structure". A save way to guarantee this is to use a complex structure by introducing complex variables
ziμ = aiμ + i πiμ
and to consider only actions which commute with the complex multiplication with i.
Next, we have to consider the restrictions following from Euclidean symmetry:
Rotations rotate the three generations, which are described by the upper index i. To commute with rotations, the action should preserve generations and act on all three generations in the same way. That means, it is sufficient to define the gauge action on one generation only. Thus, we have to care only about the possible gauge actions on one generation, taken alone, with parameters
zμ = aμ + i πμ
Translations (ti) act as shifts on the components ai0. Having to consider only one generation, we have to consider only one type of shift t:
a0 -> a0 + t.
A gauge action can commute with such a translation only in one case: if the direction a0 remains invariant. The requirement of preservation of the complex structure (commutation with multiplication with i) leads to the next restriction: π0 should be left invariant too. Thus, what remains are actions on the three complex variables
zj = aj + i πj, j > 0.
The largest possible compact gauge group with these properties is the group U(3). It's subgroup SU(3) coinsides with the color group SU(3)c of the standard model exactly, in all its properties: The color charge does not depend on the generation, as well as on isospin and parity, and leaves the leptonic sector untouched.
The only additional Wilson gauge field which is allowed is a the diagonal of U(3) — a single diagonal field, with the baryon charge IB as its charge, which I denote U(1)B. We will need it to construct the EM field.
An important type of irregularities of our lattice model — inconsistent influences of the material between the cells on the motion of the cells — can be described technically by athree-dimensional variant of Wilson gauge fields. The characteristic property of the resulting gauge field action in our model is that it cannot depend on parity and isospin of the fermions. This forbids to use them for the description of electroweak gauge fields.
On the other hand, it fits with the action of SU(3), which also does not depend on parity and isospin.
Preservation of the symplectic structure and Euclidean symmetry are the other two important restriction. With these restrictions, the maximal possible gauge group of this type is U(3). This group contains the color group SU(3)c of strong interactions of the standard model. This fit between the computed maximal possible Wilson gauge group U(3) and the observed color gauge group SU(3)c is almost ideal.