# The cell lattice model

The model presented here is not mainstream, but an alternative approach. But, different from most "alternative physics theories" you can find on the net, my article has been published in the peer-reviewed journal Foundations of Physics:

Schmelzer, I.: A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Foundations of Physics, vol. 39, 1, p. 73 – 107 (2009),
DOI: 10.1007/s10701-008-9262-9
,
with unrestricted access at arXiv:0908.0591;

The actual version of a beamer presentation and some further ideas in the article
Schmelzer, I.: Neutrinos as pseudo-acoustic ether phonons, arXiv:0912.3892v1 may be interesting for professionals too (it contains some first ideas for the explanation of mass terms).

## Introduction

The aim of this lattice model is to obtain all fermions and gauge fields of the standard model of particle physics (SM), together with a Lorentz metric for gravity.

Here we give some introduction into the model for interested laymen. If the text below is, nonetheless, too much math for you, try this.

This picture is not simply some symbolic representation of something which cannot be represented in the usual three-dimensional space, as we know it from illustrations for general relativity and, especially, string theory. The model really is that simple — a three-dimensional lattice, consisting of three-dimensional elementary cells. These cells oscillate around their average positions in our usual three-dimensional space. A cell may moved, rotated, and stretched in different directions.

Intuitively, this model seems far too simple to describe such a complex set of fields as the SM. Nonetheless, all observable fields have a place in it: Fermions appear as oscillations of the lattice, the gravitational field combines density, average velocity and stress tensor, strong and weak gauge fields appear as different types of deformations: of the material between the cells, and the lattice itself. Last not last, the EM field is a combination of above types of gauge fields.

### Fermions

Let's start with the fermions. The state of deformation of each cell we describe by an affine transformation $$a^i_\mu$$, which is a $$3\times (3+1)$$ matrix, and an an element of the affine group Aff(3). On the whole lattice $$\mathbb{Z}^3$$ this gives a configuration space $$\mbox{Aff}(3)(\mathbb{Z}^3)$$ of lattice funtions $a^i_\mu(n_1,n_2,n_3): \mathbb{Z}^3 \to \mbox{Aff}(3).$

One needs also the corresponding momentum variables $$\pi^i_\mu(n_1,n_2,n_3)$$, so that we obtain the phase space $$\mbox{Aff}(3)\otimes\mathbb{C}(\mathbb{Z}^3)$$. On the phase space, one can introduce some complex structure.

Then, the continuous limit gives an interesting doubling effect — together with almost continuous solutions, we have to consider also heavily oscillating solutions in different oscillating modes. In the limit, these oscillating doublers may be identified with differential forms, so that each lattice function gives the whole bundle of inhomogeneous differential forms $$\Lambda(\mathbb{R}^3)$$. Taken together, our lattice model leads, in the continuous limit, to the following phase space: $\mathbf{\mbox{Aff}(3)\otimes\mathbb{C}\otimes\Lambda(\mathbb{R}^3)}$

This defines our geometric interpretation of the SM fermions. The number of fields fits $$\mathbb{C}\otimes\Lambda(\mathbb{R}^3)$$ contains eight complex fields, as many as two Dirac fermions, and the twelf electroweak doublets appear in a similar $$3\times (3+1)$$ structure as elements of Aff(3).

## Dirac equation

The "exterior bundle" $$\Lambda(\mathbb{R}^n)$$ is especially interesting because mathematicians know very well that there exists a version of the Dirac operator (a square-root of the harmonic operator) on this bundle. Especially there is a well-known Dirac-Kähler equation on $$\mathbb{C}\otimes\Lambda(\mathbb{R}^4)$$, which describes four Dirac fermions in Minkowski spacetime.

We obtain only a three-dimensional Dirac operator on $$\mathbb{C}\otimes\Lambda(\mathbb{R}^3)$$, but this operator is sufficient to write down the Dirac equation in the original form as proposed by Dirac: $i \partial_t \psi(x,t) = i \alpha^i \partial_i \psi(x,t) - \beta m \psi(x,t)$

Nonetheless, the reduction from $$\mathbb{C}\otimes\Lambda(\mathbb{R}^4)$$ on spacetime to $$\mathbb{C}\otimes\Lambda(\mathbb{R}^3)$$ on space has one important effect: Instead of four Dirac particles, we have only two of them. Together with the Dirac operators $$\alpha^i$$ we find also an operator $$\beta$$ as well as isospin operators $$I_i$$, thus, all the operators we need.

The Dirac equation on the bundle $$\mathbb{C}\otimes\Lambda(\mathbb{R}^4)$$ is also famous for its correspondence to a lattice Dirac equation on $$\mathbb{C}(\mathbb{Z}^4)$$ known as "staggered fermions". A similar "staggered fermions" lattice equation can be found for our Dirac equation too. It lives on $$\mathbb{C}(\mathbb{Z}^3)$$, that means, on a lattice in space, with $$t$$ as a continuous evolution parameter.

### Gauge fields

On the space of 24 Dirac fermions of the SM, which could be considered as well as 48 Weyl fermions, it would be possible to define U(48), with $$48^2$$ different gauge fields, as the maximal possible gauge group. If we would not care about the complex structure, even O(96) with $$48\cdot 95$$ gauge fields would be allowed. Instead, only 12 of them, the group $$SU(3)_c\times SU(2)_L \times U(1)_Y$$, are observed, and their action shows very strange and unexplained regularities.

Now, our approach allows to explain all these regularities. What we obtain as the maximal possible gauge group is the same $$SU(3)_c\times SU(2)_L \times U(1)_Y$$.

The principles which we use to obtain this reductions are the following ones:

• Preservation of the complex structure, which follows from preservation of the symplectic structure on the phase space.
• Preservation of Euclidean symmetry.
• The gauge fields should live in a natural way on the lattice. We find two such natural types of lattice gauge fields:
• Strong fields may be described with a straightforward generalization of so-called Wilson lattice gauge fields, a known way approximate gauge fields in lattice theories. This gives the maximal group U(3)c, containing the color group SU(3)c of the standard model and, as an additional field, a field $$U(1)_B$$ with the baryon charge as its charge.
• Weak fields describe effects related with deformations of the lattice itself. This gives the group $$U(2)_L \times U(1)_R$$ as the maximal possible gauge group, which contains the standard model group $$SU(2)_L$$ together with two additional fields.
• The EM field can be obtained as a combination of the three additional fields.
• Restriction to the special subgroup (the determinant should be 1). This can be justified by the requirement of neutrality of the vacuum state (the Dirac sea). This condition allows to get rid of the $$U(1)_B$$ field.

The remaining group $$S(U(3)_c\times U(2)_L\times U(1)_R)$$ contains the whole SM gauge group and only one additional gauge field, which I have named upper axial boson.

We have also an association between exact gauge symmety in our model and masslessness, thus, some qualitative explanation of the mass terms of the gauge bosons.

### Gravity

Last but not least, gravity describes density, velocity and stress tensor in the continuous (condensed matter) limit of the model.

This is a separate theory. It does not depend on the details of the lattice model, but is a kind of general theory for condensed matter which fulfills some sufficiently simple and natural axioms. As a consequence of these axioms, we can derive the general Lagrangian — which is very close to GR, with only two additional terms. These terms enforce harmonic coordinates, and do not contain matter terms. As a consequence, the matter Lagrangian is the same as in GR, thus, we have the Einstein equivalence principle. Moreover, in some natural limit the additional terms vanish, and we obtain the Einstein equations of GR.

### Summary

Thus, all particles and fields observed until now may be described in this surprisingly simple model. And it explains a lot of properties of the SM - the number and structure of the fermions, the gauge group, and its action. The theory of gravity explains the Einstein equivalence principle and the Einstein equations of GR.

Moreover, at least some part of the theory is already quantized.

On the other hand, there are a lot of open problems and things yet to do. The most important thing is symmetry breaking. Our lattice model has Euclidean symmetry. The SM, in our identification of the fields, has not. Especially the fermion masses destroy Euclidean symmetry.

Then, we have to consider the large distance limit of the whole construction. It may be, that we need, to obtain the observable values of the observable large distance parameters, some extreme fine-tuning of the parameters of the lattice model. The SM contains a lot of such fine-tuning problems, and the question is if our lattice model allows to solve them.

Last but not least, the quantization of the gauge fields will be very different from the method used today to quantize gauge fields. Here, the details have to be worked out.