The doublers and external forms

We have various functions on the lattice (living in the phase space). But the distance between the lattice cells \(\Delta\) is much lower than what we can measure. Then, the question is what do we measure.

The straightforward answer is that we measure some continuous function \(f_0(x)\), which changes only slowly, thus, essentially does not change between x and \(x+\Delta\), so that the lattice function f(n) is defined by \[ f(n) = f_0(n\cdot \Delta).\]

A wave of this type is, therefore, almost constant on the lattice. But, because of the doubling effect, to consider only almost constant solutions is not enough. We have to care, as well, about some types of heavily oscillating solutions. Fortunately, these heavily oscillating solutions may be described with the help of some continuous functions too: The oscillations are of a very special type, where values on odd nodes have approximately the other sign as the values on even nodes. Therefore, we can describe them, in the one-dimensional case, by another continuous function \(f_1(x)\). The corresponding oscillating solution is defined by \[ f(n) = (-1)^n f_1(n\cdot\Delta)\]

That means, to describe the solutions which are important for large distance considerations, we need two functions \(f_0(x)\), \(f_1(x)\), so that \[ f(n) = \sum_{k\in\{0,1\}} (-1)^{kn} f_k(n\cdot\Delta)\]

For higher lattice dimensions, the same effect appears in every direction independently. Thus, we obtain a factor two for each lattice dimension. In the three-dimensional case, this leads to eight different functions \(f_{000}(x,y,z), \ldots , f_{111}(x,y,z)\) so that \[ f(n_1,n_2,n_3) = \sum_{k_i\in\{0,1\}} (-1)^{\sum k_i n_i} f_{k_1 k_2 k_3} (n_1\Delta, n_2\Delta, n_3\Delta)\]

Differential forms

It is possible to identify the so-called "doublers" with inhomogeneous differential forms. In the one-dimensional case, such a form looks like \[ f = f_0(x) = f_1(x) dx\]

In the three-dimensional case, it looks in the following way: \[ f = f_{000} + f_{100} dx + f_{010}dy + f_{001}dz + f_{110}dx \land dy + f_{011}dy \land dz+ f_{101}dz \land dx + f_{111}dx \land dy\land dz\]

This space of three-dimensional inhomogeneous forms is denoted by \(\Lambda(\mathbb{R}^3)\). The space of inhomogeneous differential forms on a space is also known as its external bundle.

Why is it interesting to have such a representation in terms of differential forms? The reason is that this set of functions has been studied by mathematicians, and there are a lot of things known about these forms. Especially, it is known that there exists a nice, beautiful version of the Dirac equation on this space.

Especially, it is known how to write this geometric Dirac equation for a different, nontrivial metric \(g_{ij}(x)\) on the space, and how to put this equation on the lattice.

The geometric interpretation of the SM fermions

Taking together the affine group Aff(3), the complex structure on the phase space C, and the external bundle, we obtain the three-dimensional geometric interpretation of the SM as \[\mbox{Aff}(3) \otimes \mathbb{C} \otimes \Lambda(\mathbb{R}^3).\]