# Cellular lattice model

Abstract:

I propose a simple cellular lattice model with configuration space $Aff(3)(Z^3)$.

The model allows to obtain, in the continuous limit, all fermions of the SM, the gauge group of the SM, it's action on the fermionic sector, and an effective Lorentz metric.

1 a) Importance of a unification of all observed particles and fields does not need further comment. b) The novelty of the approach is obvious. c) The SM itself is a phenomenological theory, which does not explain its particle and fields content at all. d) This gap is filled by the new model, which allows to explain the particle content of the fermionic and gauge sector of the SM completely. Unification of the SM with gravity and the quantization of gravity are other important problems which may be solved in the new approach: The gravitational field is included too. Moreover, the theory is, essentially, a condensed matter theory on a Newtonian background, and, therefore, a theory which can be canonically quantized.

2.) For details and references see the attached article. Further information can be found at ilja-schmelzer.de/clm and ilja-schmelzer.de/glet.

Basic Ideas: We consider a lattice of simple cells in R^3. The state of each cell is described by an affine deformation from a standard reference cell. Therefore, the configuration space of the lattice model is $Aff(3)(Z^3)$, resp. the phase space $Aff(3)\otimes C(Z^3)$.

We have a lattice Dirac equation on this lattice. Because of an analogon of the fermion doubling effect, this equation gives, in the continuous limit, a Dirac equation on $Aff(3)\otimes C\otimes \Lambda(R^3)$. We identify the eight complex fields $C\otimes \Lambda(R^3)$ with electroweak doublets. The twelf components of the 3x(3+1) affine matrix can be easily identified with twelf electroweak doublets (3 generations x (3 quark colors + leptons)) of the SM.

The lattice model gives two natural things to be preserved by gauge fields: Euclidean symmetry and the symplectic structure of the phase space. We show that these requirements explain the following properties of the SM gauge group:

1.) They preserve generations and act on the three generations in the same way
2.) There exists some direction in each generation preserved by all gauge fields (right-handed neutrinos)
3.) The gauge groups are unitary

We consider the way gauge actions can be defined on the lattice. We find that Wilson gauge fields cannot have different charges inside electroweak doublets. As a consequence, the maximal possible Wilson gauge group is $U(3) \cong SU(3)_c \times U(1)_B$.

We show that deformations of the underlying lattice Z^3 require correction terms which, in the continuous limit, lead to another set of effective gauge fields. These gauge fields have to preserve electroweak doublets and are generated by the operators I_i and gamma^5. Combined with the previously found requirements, this gives electroweak $U(2)_L \times U(1)_R$, so that right-handed neutrinos have charge 0. In combination with U(1)_B we can obtain hypercharge Y and, therefore, all SM gauge fields.

As a last requirement, we use anomaly freedom. All additional fields (the diagonals of $SU(3)_c$ and $SU(2)_L$) are anomalous, thus, the SM gauge group appears to be a maximal anomaly-free subgroup of the group $U(3) \times U(2)_L \times U(1)_R$ we have obtained.

The gravitational field can be incorporated too. We construct an effective Lorentz metric using density, three-velocity and stress tensor. This decomposition is, essentially, the ADM decomposition in the preferred frame of the Newtonian background. We have continuity and Euler equations, which correspond to the harmonic condition for the effective metric. We require that they appear as Noether conservation laws. This allows to derive the general Lagrangian of the theory: It is the sum of a term which enforces the harmonic gauge and the most general Lagrangian of general relativity. As a consequence, in the resulting theory we has an exact Einstein equivalence principle, and obtain, in a natural limit, the Einstein equations of GR.

As far, the considerations have been classical. But an important step of the quantization is also done: We have found a way to obtain anticommuting fermion fields starting from canonical quatization of the lattice theory. This method requires the introduction of some additional heavy bosons.

Some questions have been left to future research: The mass terms, which require symmetry breaking, dynamics of the gauge fields, quantization of gauge fields and gravity, and renormalization.

#### Attachments

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There are a number of cellular automata quantum models one might consider. The type of model here has some analogues with atoms in a laser trap.

Lawrence B. Crowell

I find it strange that nobody comments anything - no criticism, nothing else, only a single answer with no direct relation to my theory.

BTW, it is published now in Foundations of Physics. See http://ilja-schmelzer.de/clm" [Broken] for more.

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I find it strange that nobody comments anything - no criticism, nothing else, only a single answer with no direct relation to my theory.
I think, the aether concept is correct, if you take spacetime of GR as this aether. And it has to have three imaginary 'dimensions' and time in encoded into rotation.
The problem is the same as in my own idea (what is influenced by yours btw.), that you can't keep the idea of real particles. If particles are nodes, vortices (or as in your model deformations of a lattice), than there is no need for real, lasting particles anymore.
That is quite a thing for all kinds of subsequent sciences (e.g. chemistry, biology, ...) and you will not get very much support from this direction.

I think, the aether concept is correct, if you take spacetime of GR as this aether. And it has to have three imaginary 'dimensions' and time in encoded into rotation.
The problem is the same as in my own idea (what is influenced by yours btw.), that you can't keep the idea of real particles. If particles are nodes, vortices (or as in your model deformations of a lattice), than there is no need for real, lasting particles anymore.
That is quite a thing for all kinds of subsequent sciences (e.g. chemistry, biology, ...) and you will not get very much support from this direction.

Spacetime of GR is something I derive, not something I take as fundamental.

Then I don't think that there is much connection with chemistry and biology. They don't care much about fundamental physics. And in quantum field theory, at least in semiclassical gravity, the particle concept (with real, lasting particles) is already dead. The fundamental things in field theory are fields.

Ilja

Spacetime of GR is something I derive, not something I take as fundamental.
I think spacetime could be a 'real' foundation to our observations, because intervals are observer invariant. Maybe there is more behind spacetime, but that I don't know.
Then I don't think that there is much connection with chemistry and biology. They don't care much about fundamental physics. And in quantum field theory, at least in semiclassical gravity, the particle concept (with real, lasting particles) is already dead. The fundamental things in field theory are fields.
With the same construct I would put fields into daubt: if there are connections within a continuum, that could build stable structures, than only these connections are 'real' and fields describe their behaviour.
This is the same relation as between air and a tornado: the tornado we call a thing, but it's only air.

I think spacetime could be a 'real' foundation to our observations, because intervals are observer invariant. Maybe there is more behind spacetime, but that I don't know.

In my theory the background is simply flat Euclidean space. It does the job as well.

With the same construct I would put fields into daubt: if there are connections within a continuum, that could build stable structures, than only these connections are 'real' and fields describe their behaviour.
This is the same relation as between air and a tornado: the tornado we call a thing, but it's only air.

The comparison of particles with a tornado is nice.

In this analogy, my theory is about the atoms which air is made of.

In my theory the background is simply flat Euclidean space. It does the job as well.
I use a 'flat spacetime' model, that is build from 'inside-> observation' and is based on quaternions.
The ideas are very close to aether ideas, but different.
Actually it's more philosophy than physics. The world is assumned as the behavior of things, that interact with their direct vincinity and create systems, that we observe and call: space, time, fields or things. The relations are sorted into categories of our experience and called e.g. an atom, that wiggles due to heat. My aim is to find a system, that could behave in such a way, that we would experience this way because we as humans are part of that.
This is in fact possible if one assumnes only local interactions (that I call 'influence') and sum that over to the world we observe.