sci.physics.particle

The theory can be found at ilja-schmelzer.de/clm, with the paper at
ilja-schmelzer.de/clm/paper.pdf.

For reasons of priority protection I post here the tex text of this
paper.
------------------------------------------------
\documentclass{amsart} % article
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\usepackage{hyperref,graphicx,color}

\begin{document}
\sloppypar \sloppy
\newcommand{\B}{\mbox{$\mathbb{Z}_2$}} %% binary group Z_2
\newcommand{\Z}{\mbox{$\mathbb{Z}$}}
\newcommand{\R}{\mbox{$\mathbb{R}$}}
\newcommand{\C}{\mbox{$\mathbb{C}$}}
%\renewcommand{\H}{\mbox{$\mathbb{H}$}}

\newcommand{\A}{\mbox{\textrm{Aff}(3)}} %% 3-dim. affine group
\newcommand{\E}{\mbox{$E(3)$}} %% 3-dim. Euclidean group
\newcommand{\CAZ}{\mbox{$(\C \otimes \A)(\Z^3)$}}
\newcommand{\AZ}{\mbox{$\A(\Z^3)$}}
\newcommand{\CAL}{\mbox{$(\A \otimes \C \otimes \Lambda)(\R^3)$}}
\newcommand{\CL }{\mbox{$(\C \otimes \Lambda)(\R^3)$}}
\newcommand{\CZ }{\mbox{$\C(\Z^3)$}}

%\newcommand{\T}{\mbox{T}} %% The lattice (distorted)
%\newcommand{\U}{\mbox{$\Z^3$}} %% The lattice (undistorted)
%\newcommand{\n}{\mbox{$\vec{n}$}} %% a lattice node

%\newcommand{\CAD }{\mbox{$(\C \otimes \A)(\T)$}}
%\newcommand{\Cons}{\mbox{C}} %% constant pointwise operators on \O
\newcommand{\Cl}{\mbox{$Cl(3,3,\R)$}} %% the Clifford algebra

\newtheorem{theorem}{Theorem}
\newtheorem{axiom}{Axiom}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{postulate}[axiom]{Postulate}
%\newtheorem{acknowledgement}[theorem]{Acknowledgement}
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%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{criterion}[theorem]{Criterion}
%\newtheorem{example}[theorem]{Example}
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{0.5em}}
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\renewcommand{\b}{\beta}
\newcommand{\g}{\gamma}
\renewcommand{\d}{\delta}
\newcommand{\D}{\Delta}
\newcommand{\e}{\varepsilon}
\renewcommand{\i}{\iota}
\renewcommand{\k}{\kappa}
\renewcommand{\l}{\lambda}
\renewcommand{\L}{\Lambda}
%\newcommand{\m}{\mu}
\newcommand{\s}{\mbox{$\sigma$}}
\newcommand{\w}{\mbox{$\omega$}}
%\newcommand{\f}{\theta} %% lattice node factor

\newcommand{\pd}{\mbox{$\partial$}} %%

\newcommand{\alg}[1]{\mathfrak{#1}}

%\newcommand{\qed}{\mbox{$\square$}} %%

%\newcommand{\Uc }{\mbox{$U(3)_c$}} %% the color group
%\newcommand{\UB }{\mbox{$U(1)_B$}} %% the baryon charge group
%\newcommand{\UL }{\mbox{$U(2)_L$}} %% weak bosons
%\newcommand{\ULd}{\mbox{$U(1)_L$}} %% the diagonal of the weak bosons
\newcommand{\Ue }{\mbox{$U(1)_{\tilde{\g}}$ }} %% with charge \Ie
\newcommand{\Ie }{\mbox{$I_{\tilde{\g}}$}} %% =3D I_3-1/2
\newcommand{\Uem}{\mbox{$U(1)_{em}$}} %% the electromagnetic field
%\newcommand{\SUc}{\mbox{$SU(3)_c$}} %% the SM color group
%\newcommand{\SUL}{\mbox{$SU(2)_L$}} %% SM weak bosons

\renewcommand{\c}{\mbox{$\vec{c}$}} %% the neutral direction

%\renewcommand{\O }{\mbox{$\Omega$}} %% a single generation
%\renewcommand{\deg}[1]{\mbox{deg$(#1)$}}
%\newcommand{\dual}[1]{\mbox{$\bar{#1}$}}
%\newcommand{\ooo}{\mbox{$\hat{0}$}}
%\renewcommand{\lll}{\mbox{$\hat{1}$}}

%\newcommand{\Tc}{\mbox{$\T_c$}} %% The coarse sublattice
\renewcommand{\t}{\mbox{$\tau$}} %% lattice shift
\newcommand{\ti}{\mbox{$\tau_i$}} %% basic lattice shift in direction
i
\newcommand{\hi}{\mbox{$\vec{h}_i$}} %% basic lattice vector in
direction i
%\newcommand{\h}{\mbox{$\vec{h}$}} %% basic lattice vector in
direction i
\newcommand{\h}{{\xi}} %% basic lattice vector in direction i

%\newcommand{\follows}{\hspace{0.6cm}\Longrightarrow\hspace{0.6cm}}

\title{Yet Another Exceptionally Simple Theory of Everything}
%\title{Geometric and condensed matter interpretation of SM fermions
and gauge fields}
\author{I. Schmelzer}

\begin{abstract}

We present the bundle \CAL, with a geometric Dirac equation on it, as
a three-dimensional geometric interpretation of the SM fermions. Each
\CL\/ describes two Dirac particles. It has a doubler-free staggered
spatial discretization on the lattice space \CAZ. This space has a
simple physical interpretation as a phase space of a lattice of cells.

We find the SM $SU(3)_c\times SU(2)_L\times U(1)_{Y}$ action on \CAL\/
to be a maximal gauge action preserving \E\/ symmetry, symplectic
structure, and anomaly freedom, and which can be constructed using two
simple types of gauge-like lattice fields: Wilson gauge fields and
correction terms for lattice deformations.

The lattice fermion fields we propose to quantize as low energy states
of a canonical quantum theory with \B-degenerated vacuum state. We
construct anticommuting fermion operators for the resulting \B-valued
(spin) field theory.

A metric theory of gravity compatible with this model is presented
too.

\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}

After the success of relativity, the interest of modern physics has
been centered on four-dimensional spacetime. If a concept requires a
preferred frame, this is, for many physicists, sufficient to reject
it. Of course, to be acceptable, a theory with preferred frame has to
explain the observable relativistic symmetry. But this is possible: In
\cite{GLET}, GR in harmonic gauge -- and, especially, the Einstein
equivalence principle -- is derived from principles of condensed
matter theory. The basic ideas of this derivation we present here in
appendix \ref{Gravity}. Once this basic problem is solved, there seems
to be nodecisive argument against a preferred frame.

One of the assumptions of this theory is, that matter fields describe
material properties of the condensed matter. As a consequence, we need
a condensed matter model for the SM fields too. The aim of this paper
is to present such a model. We have found a three-dimensional
geometric interpretation of SM fermions as \CAL, together with a
doubler-free discretization on the lattice space \CAZ, which allows a
condensed matter interpretation, as the phase space of a lattice of
cells. Moreover, this model allows, essentially, to compute the SM
gauge group and its action on the fermions. Thus, all SM fields
observed until now can be described in this way.

Let's start with the bundle \CAL, which we propose as a three-
dimensional geometric interpretation of the SM fermions. The bundle \CL
\/ describes an electroweak doublet.\footnote{Independent of this
paper, three-dimensional geometric fermions have been proposed by
Daviau \cite{Daviau}. The idea that geometric fermions may be used to
describe electroweak doublets has been proposed by Hestenes
\cite{Hestenes}.}
Each of the $3\cdot(3+1)$ components $(a^i_\mu)\in\A$ of an affine
transformation we associate with such an electroweak doublet: The
upper index $i$ denotes the generation, $\mu=3D0$ the leptonic sector, $
\mu>0$ the quark sector, and the three positive values $\mu\in\{1,2,3\}
$ define the three quark colors.

On the bundle \CL\/ exists a three-dimensional geometric Dirac
operator -- an analogon of the Dirac-K\"{a}hler operator on $(\C
\otimes \Lambda)(\R^4)$. This operator is sufficient to define the
Dirac matrices $\a^i$. We find also natural operators $I_i$ as well as
$\b=3D\g^0$ on \CL. The Dirac equation we define in its original Dirac
form $i\partial_t\psi =3D H \psi$, as an evolution equation on \CL. This
equation contains eight complex fields and describes a doublet of
Dirac particles.

In analogy with the staggered discretization of the four-dimensional
bundle $(\C \otimes \Lambda)(\R^4)$, we have also a staggered
discretization of the three-dimensional Dirac operator. It lives on a
three-dimensional spatial lattice $\Z^3$. It is a staggered
discretization, with only one complex component on each lattice node,
and eight different types of lattice nodes. Similar to the four-
dimensional staggered discretization of $\Lambda(\R^4)$ on $\Z^4$ (see
\cite{Susskind},\cite{Becher}), it is a doubler-free discretization of
\CL. In other words, we obtain a lattice evolution equation on a three-
dimensional lattice \CZ, which gives, in the continuous limit, two
Dirac fermions.

For all SM fermions (the bundle \CAL) we obtain a first order lattice
equation on \CAZ. This lattice space has a physical interpretation as
the phase space for a three-dimensional lattice of elementary cells,
where the state of each cell is described by a single affine
transformation from a standard reference cell (see figure
\ref{fig:lattice}).

\begin{figure}
\includegraphics[angle=3D270,width=3D0.8\textwidth]{lattice}
\label{fig:lattice}\caption{The space \CAZ\/ of the lattice model
suggests an interpretation as the phase space (with configuration
space \AZ) of a three-dimensional lattice of deformable three-
dimensional cells. The configuration of each cell is described by an
affine transformation from a standard reference cell.}
\end{figure}

This physical interpretation gives us two important structures: First,
a symplectic structure of the phase space, second, a natural action of
the Euclidean group \E. These structures may be used to restrict the
gauge groups. For a compact gauge group we can always construct a
preserved Euclidean metric, which, together with a preserved
symplectic structure, allows to construct a preserved complex
structure. Thus, preservation of the symplectic structure requires
unitarity of the gauge groups.

The left action of \E\/ on \AZ\/ transforms the lattice as a whole.
The requirement of preserving this symmetry for the gauge groups
consists of two parts:

\begin{itemize}
\item To commute with rotations, gauge groups have to preserve
generations and to act on all three generations in the same way. This
holds for all SM gauge fields.
\item To commute with translations, one direction in the leptonic
sector has to be preserved. All SM gauge fields leave right-handed
neutrinos and their antiparticles invariant, thus, a common invariant
direction exists in the SM.
\end{itemize}

Thus, for an appropriate identification of the invariant direction,
all SM gauge fields preserve \E\/ symmetry.

The lattice theory also leads to another important restriction for the
gauge fields: We have to define an appropriate lattice model for the
gauge fields. A well-known way to put gauge fields on the lattice are
Wilson gauge fields. Their modification to a three-dimensional lattice
with continuous time is trivial. But Wilson gauge fields cannot act in
a nontrivial way inside the doublets \CL, because these are
represented on the lattice as \CZ, which leaves only $U(1)$. Thus,
Wilson gauge fields have to have the same charge on all parts of a
doublet. The maximal group of Wilson gauge fields compatible with \E\/
symmetry and symplectic structure is $U(3)\cong SU(3)_{QCD}\times
U(1)_B$.

We find another modification of the lattice equations which, in the
large distance limit, leads to a gauge-like interaction term for
fermions. It describes correction terms for lattice deformations. As
a consequence, the coefficients depend only on the geometry of the
lattice, thus, has to preserve doublets \CL, and act identically on
all doublets. The maximal group of this type compatible with \E\/
symmetry and symplectic structure is generated by chiral $U(2)_L\cong
SU(2)_L\times U(1)_L$ and a vector field \Ue with charge $\Ie =3D I_3-
\frac{1}{2}$.

The EM field does not fit into any of the two types. But it can be
constructed as a combination of them, by the simple formula $Q=3D2I_B+\Ie
$. Thus, our two types of gauge-like lattice fields are already
sufficient to construct all SM gauge fields.

Last not least, we have to look at the additional fields -- The field
$U(1)_B$ with baryon charge $I_B$, and the diagonal $U(1)_L$ of the
weak group $U(2)_L$. Above are, in the presence of the other SM
fields, anomal. If we, as a last condition, add anomaly freedom, we
can get rid of them too. Thus, we have, essentially, computed the SM
gauge group.

Of course, there are a lot of things left to future research. We have
not considered yet the mass terms and the Higgs sector. They break \E
\/ symmetry, thus, to describe them, we need some spontaneous \E-
symmetry breaking. Once the broken symmetry is \E, it is not clear if
we need a separate Higgs sector at all. This has to be left to future
research.

What about quantization? The first problem is fermion quantization. We
use classical, commuting c-number fields in the lattice theory \CAZ,
not Grassmann variables as in the Berezin approach to fermion
quantization. Thus, we need a completely different quantization scheme
for fermions.

We propose such an alternative in section \ref{fermionQuantization}.
We consider canonical quantization of a field with \B-symmetric
degenerated vacuum state. The lowest energy states of such a field
define a \B-valued (spin) field, yet with commuting operators on
different lattice nodes. Then we define anticommuting fermion
operators on this space. The transformation is nonlocal and depends on
an order between the lattice nodes. We fix such an order and motivate
this choice. As a result, the staggered lattice Dirac operator in
fermion operators may be obtained from a simpler, non-staggered,
symmetric operator in terms of the spin field operators.

A new approach is required for gauge field quantization too. The
reason is that the gauge-like lattice fields, which describe lattice
deformations, do not have exact gauge invariance on the lattice. This
has to be left to future research. Nonetheless, we can already suggest
an approach which does not lead to unitarity violations. Quantization
of gravity has to follow the scheme of quantization of condensed
matter theories. The details have been left to future research too.

\section{Geometric interpretation of SM fermions}

Let's consider now the geometric interpretation of the SM fermions
\CAL. Throughout this paper, we do not consider mass terms. Without
the mass terms, the three generations of SM fermions appear completely
identical, simply as three identical copies of the same representation
of the SM gauge group $SU(3)_c\times SU(2)_L\times U(1)$.

The group \A\/ is the group of three-dimensional affine
transformations $y^i=3Da^i_j x^j + a^i_0$ on $\R^3$. Each $a^i_\mu$ we
can identify with an electroweak doublet of the SM according to the
following simple rules: The upper index $i, 1\le i \le 3$ defines the
generation. The translational components $a^i_0$ we identify with the
leptonic sector. The linear part $a^i_j$, $j>0$ we identify with the
quark sector. The lower index $j$, $1\le j\le 3$ denotes the color of
the quark doublet.

This identification of the $3\times(3+1)$ SM doublets with a
$3\times(3+1)$ affine matrix may be considered, up to now, as pure
numerology. But it defines a natural action of the Euclidean group \E,
by multiplication from the left. This action commutes with all gauge
fields and plays an important part in the computation of the SM gauge
action.

Each electroweak doublet is defined by the bundle \CL. It is assumed
here that right-handed neutrinos exists, so that neutrinos form usual
Dirac particles. Thus, qualitatively there is no difference between
electroweak quark doublets and electroweak lepton doublets. Above
contain two Dirac particles. The bundle \CL consists of three-
dimensional complex inhomogeneous differential forms
\begin{eqnarray} \Psi &=3D& \sum_{\k_i\in\{0,1\}} \psi_{\k_1\k_2\k_3}(x)
e^{\k_1\k_2\k_3}\\
\nonumber &=3D& \psi_{000}(x) + \psi_{100}(x)\, dx^1 + \psi_{010}(x)
\,dx^2+\psi_{001}(x)\,dx^3 \\
\nonumber &+& \psi_{110}(x)\, dx^1 \wedge dx^2 + \psi_{011}(x)
\,dx^2\wedge dx^3 + \psi_{101}(x)\, dx^1 \wedge dx^3 \\
\nonumber &+& \psi_{111}(x)\, dx^1 \wedge dx^2 \wedge dx^3.
\end{eqnarray}

Thus, we have $1+3+3+1=3D8$ complex functions, which gives two Dirac
fermions. This allows a physical interpretation in terms of a standard
model electroweak doublet. The use of a three-dimensional bundle is
essential. In spacetime, we have only the bundle $\C\times
\Lambda(\R^4)$, with the Dirac-K\"{a}hler equation \cite{Kaehler},
which describes four Dirac fermions.

On the external bundle $\Lambda(\R^d)$ exists a natural geometric
Dirac operator $D$ as a square root of the Laplace operator $\Delta =3D
D^2$. For a general metric, the definition is given in appendix
\ref{DiracCurved}. In the Euclidean case $g_{\mu\nu}=3D\delta_{\mu\nu}$,
this Dirac operator has the form
\begin{equation}
D =3D d + d^* =3D -i\a^i\pd_i.
\end{equation}
with operators $\a^i$ which fulfill the anticommutation relations $\
{\a^i,\a^j\} =3D 2\delta^{ij}$. Now, together with the skew-symmetric $
\a^i$, it is useful to consider also corresponding symmetric operators
$\b^i$. They may be defined by
\begin{equation}
d - d^* =3D -i\b^i\pd_i.
\end{equation}

Together, they define a set of generators of $M_{2^d}(\R)\cong
\textit{Cl}^{d,d}(\R)$:
\begin{equation}
\{\a^i,\a^j\} =3D 2\d^{ij}, \;
\{\a^i,\b^j\} =3D 0, \;
\{\b^i,\b^j\} =3D -2\d^{ij}.
\label{eq:aibi}
\end{equation}

For $d=3D3$, the explicit representation of the matrices $\a^i,\b^i$ is:
\begin{eqnarray*}
-i\a^i\pd_i\Psi =3D& \left(\begin{array}{cccccccc}
0 & -\pd_3 & -\pd_2 & 0 & -\pd_1 & 0 & 0 & 0
\\
+\pd_3 & 0 & 0 & -\pd_2 & 0 & -\pd_1 & 0 & 0
\\
+\pd_2 & 0 & 0 & +\pd_3 & 0 & 0 & -\pd_1 & 0
\\
0 & +\pd_2 & -\pd_3 & 0 & 0 & 0 & 0 & -\pd_1
\\
+\pd_1 & 0 & 0 & 0 & 0 & +\pd_3 & +\pd_2 & 0
\\
0 & +\pd_1 & 0 & 0 & -\pd_3 & 0 & 0 & +\pd_2
\\
0 & 0 & +\pd_1 & 0 & -\pd_2 & 0 & 0 & -\pd_3
\\
0 & 0 & 0 & +\pd_1 & 0 & -\pd_2 & +\pd_3 & 0
\end{array}\right)&
\left(\begin{array}{c}
\psi_{000}\\
\psi_{001}\\
\psi_{010}\\
\psi_{011}\\
\psi_{100}\\
\psi_{101}\\
\psi_{110}\\
\psi_{111}
\end{array}\right)\\
-i\b^i\pd_i\Psi =3D& \left(\begin{array}{cccccccc}
0 & +\pd_3 & +\pd_2 & 0 & +\pd_1 & 0 & 0 & 0
\\
+\pd_3 & 0 & 0 & +\pd_2 & 0 & +\pd_1 & 0 & 0
\\
+\pd_2 & 0 & 0 & -\pd_3 & 0 & 0 & +\pd_1 & 0
\\
0 & +\pd_2 & -\pd_3 & 0 & 0 & 0 & 0 & +\pd_1
\\
+\pd_1 & 0 & 0 & 0 & 0 & -\pd_3 & -\pd_2 & 0
\\
0 & +\pd_1 & 0 & 0 & -\pd_3 & 0 & 0 & -\pd_2
\\
0 & 0 & +\pd_1 & 0 & -\pd_2 & 0 & 0 & +\pd_3
\\
0 & 0 & 0 & +\pd_1 & 0 & -\pd_2 & +\pd_3 & 0
\end{array}\right)&
\left(\begin{array}{c}
\psi_{000}\\
\psi_{001}\\
\psi_{010}\\
\psi_{011}\\
\psi_{100}\\
\psi_{101}\\
\psi_{110}\\
\psi_{111}
\end{array}\right)
\end{eqnarray*}

The last Dirac operator $\g^0$ can be obtained now as
\begin{equation}
\b =3D \g^0 =3D \b^1\b^2\b^3\a^1\a^2\a^3 =3D \a^1\b^1\a^2\b^2\a^3\b^3,
\label{eq:betafromaibi}
\end{equation}
and appears to be a diagonal operator, which measures the \B-
graduation of $\Lambda(\R^3)$. The matrices $\a^i,\b$ define a
representation of the standard Dirac algebra
\begin{equation}
\{\a^i,\a^j\}=3D2\d^{ij};\qquad\{\a^i,\b\}=3D0;\qquad (\a^i)^2=3D\b^2=3D1.
\label{eq:DiracAlgebra}
\end{equation}

For the (massless) Dirac equation we prefer to use the original form,
as proposed by Dirac, with the operators $\a^i$:
\begin{equation}
i\pd_t \Psi =3D -i\a^i\pd_i\Psi.
\end{equation}

The operators $I_i$ defined by
\begin{equation}
2i\varepsilon^{ijk}I_i =3D \b^j\b^k,
\end{equation}
define a vector representation of the isospin algebra $\mathfrak{su}
(2)$. We identify them with the (weak) vector isospin $I_i =3D \tau^i_L
+ \tau^i_R$. The $I_i$ commute, as they should, with the Dirac
equation as well as with $\g^0$. Thus, the operator $I_3$ may be used
to split the bundle \CL\/ into two parts with eigenvalues $I_3=3D\pm
\frac{1}{2}$, so that each of the parts contains a full representation
of the Dirac algebra.

An interesting question is how the spinor representation $\sigma^{ij}=3D
\a^i\a^j$ on the Dirac particles is connected with the representation $
\mathfrak{so}(3)$ of geometric rotations of the bundle \CL. The answer
is that geometric rotations are generated by the operators $\omega^{ij}
$ defined by
\begin{equation}
\omega^{ij} =3D \a^i\a^j - \b^i\b^j =3D \sigma^{ij} - 2i
\varepsilon^{ijk}I_i.
\end{equation}
Thus, the true, geometric rotations of our geometric interpretation
are a combination of spinor rotations and isospin rotations.

The operator $\g^5=3D-i\a^3\a^2\a^1$ turns out to be the (modified)
geometric Hodge $\ast$ operator (\ref{astdef}).

\subsection{Symplectic structure}\label{complex}

We have a complex structure in our geometric interpretation. Now,
every complex structure defines a natural symplectic structure $
\omega=3Ddz\wedge d\bar{z}$. We know that all the SM gauge groups are
unitary groups, thus, they preserve the complex structure. As a
consequence, they also preserve the symplectic structure. Therefore,
we can postulate the following:

\begin{postulate}\label{postulate:symplectic}
All gauge fields preserve the symplectic structure derived from the
complex structure of \CL.
\end{postulate}

The question we want to consider here is if we really need the complex
structure. May be the symplectic structure is already sufficient? Or
do we obtain, in this way, some additional gauge fields? No, at least
as long as we consider only compact gauge groups. For compact gauge
groups, we have the invariant Haar measure, and it has a finite norm.
This allows to construct, for a given action of a compact group, an
invariant Euclidean norm $\langle.,.\rangle$. All we have to do is to
start with an arbitrary norm $\langle.,.\rangle_0$ and to compute the
average of the Haar measure:
\begin{equation}
\langle a,b \rangle =3D \int \langle ga,gb \rangle_0 dg.
\end{equation}
The resulting Euclidean distance $\langle.,.\rangle$ is already
preserved by the gauge group action. Once we have a preserved
Euclidean metric together with a preserved symplectic structure, we
can already construct a preserved complex structure by the rule
\begin{equation}
\omega(a,ib) =3D \langle a,b \rangle.
\end{equation}

As a consequence, our second postulate is sufficient to restrict the
gauge group to an unitary group. Thus, in the geometric interpretation
we can forget about the complex structure and restrict ourself to the
symplectic structure. Thus, we can interpret the space \CAL as a phase
space.

\subsection{Euclidean symmetry}

On \A, we have a well-defined left action of the Euclidean symmetry
group $\E\subset\A$.

The action of the rotation group $O(3)\subset\E$ extends immediately
to \CAL as
\begin{equation}
\omega: \Psi^i_\mu \to \omega^i_j\Psi^j_\mu.
\end{equation}
In terms of our interpretation, these rotations rotate the three
generations of the SM. Now, all SM gauge groups preserve generations.
(Remember that we consider here the massless case, thus, define
generations in such a way that they contain electroweak doublets
completely.) Moreover, they act on the different generations in
exactly the same way. As a consequence, they commute with the action
of our group of rotations $O(3)$.

Let's extend now the action of the subgroup of translation $T^3\subset
\E$ on \CAL. For this purpose, we have to define a shift operator
\begin{equation}
t: \Psi^i_0 \to \tau(t^i)\Psi^i_0,
\end{equation}
where $\tau(t): \Psi\to\Psi'$ defines a scalar shift operator on \CL.
This is an action of $\R$ on $\CL$ and should not depend on $x$.
Therefore, it is uniquely defined by a single shift vector $\c=3D(c_\k)
\in\C^8$ as
\begin{equation}
\c =3D \tau(1)\Psi-\Psi,
\end{equation}
which we name the ``direction of translation''. After this,
translations are defined as $\tau(t)\psi_\k\to\psi_\k+tc_\k$ for all $
\k$, and we have extended the definition of translations from \A\/ to
\CAL.

In our interpretation, translations act, by shifts, only on the
leptonic doublets. Once we already have found that rotations commute
with all gauge groups, it would be nice to have a similar property for
translations too. So, what does it mean for the gauge groups to
commute with translations? The answer is simple --- the gauge groups
have to leave the translational direction \c\/ of the leptonic sector
invariant. Now, the leptonic sector contains a part which is left
invariant by all SM gauge fields --- the right-handed neutrinos and
their antiparticles. Thus, if we identify the direction of translation
\c\/ in such a way that it is inside the right-handed neutrino sector,
then all SM gauge fields preserve translational symmetry too.

Thus, for an appropriate definition of the direction of translation
\c, all SM gauge fields preserve the complete \E\/ symmetry. This
property of the SM gauge fields we use in the following as a
postulate:

\begin{postulate} \label{postulate:Euclidean} All gauge fields
preserve the \E\/ symmetry defined by the left action of \E\/ on \A.
\end{postulate}

Note that this observation gives our \E\/ symmetry large explanatory
power. It explains why all SM gauge fields preserve generations and
act in the same way on the three generations. Moreover, it excludes a
lot of very interesting natural and symmetric extensions of the SM:

\begin{itemize}
\item The extension of $SU(3)_c$ to $SU(4)_c$ with lepton charge as a
forth color, which is part of the Pati-Salam extension of the SM
\cite{PatiSalam},
\item the left-right-symmetric extension of $SU(2)_L\times U(1)_Y$ to
$U(1)_{B-L}\times SU(2)_L\times SU(2)_R$, which is also part of the
Pati-Salam extension of the SM \cite{PatiSalam},
\item and all GUTs which use at least one of these extensions as a
subgroup, especially $SO(10)$ GUT.
\end{itemize}

Indeed, all these extensions of the SM act on right-handed neutrinos
in a nontrivial way, and, therefore, do not leave any direction
invariant. As a consequence, they cannot commute with any
implementation of the translations.

Nonetheless, these principles are not yet sufficient to compute the SM
gauge group. There remain interesting nontrivial extensions like
$SU(5)$ GUT \cite{su5} or chiral color with $SU(3)_L\times SU(3)_R$
instead of $SU(3)_c$ \cite{chiralColor}.

\section{The lattice Dirac operator}\label{doubling}

Let's consider now a discretization of our Dirac equation in space,
leaving time continuous. Using naive central differences, we obtain
the following lattice equation:
\begin{equation}
i\pd_t \psi_\k(n) \ =3D\
\sum_i -i(\alpha^i)_\k^{\k'} (\psi_{\k'}({n+h_i})-\psi_{\k'}({n-
h_i}))
\label{eq:DiracLattice}
\end{equation}
on the lattice space $\Omega=3D\C^8(\Z^3)$.

It is easy to see that this lattice equation contains eight doublers.
Indeed, let's defined eight so-called ``staggered'' sublattices,
labelled by $\l=3D(\l_1,\l_2,\l_3)\in\{0,1\}^3$, defined by the
condition
\begin{equation}
\Omega^\l =3D \{\psi_\k(n)| n =3D \k+\l \;\;\textrm{mod}\;\; 2\}
\end{equation}
so that $\Omega=3D\sum_\l \Omega^\l$. It is easy to see that the naive
lattice Dirac equation preserved the decomposition into the staggered
sublattices. As a consequence, it is easy to get rid of the doublers,
and sufficient to preserve only one of the eight sublattices $
\Omega^{000}$, with $\l=3D\{0,0,0\}$. Thus, our staggered sublattice is
defined by the condition
\begin{equation}\label{eq:DiracStaggered}
n =3D \k \;\;\textrm{mod}\;\; 2.
\end{equation}

This doubler-free lattice equation (\ref{eq:DiracLattice}),
(\ref{eq:DiracStaggered}) can be obtained from a much more genereal,
geometric construction, which is presented in appendix
\ref{DiracLatticeCurved}. It is the same geometric construction which
gives, in the case of the four-dimensional Dirac-K\"{a}hler equation
\cite{Kaehler} on the spacetime bundle $\Lambda(\R^4)$, the staggered
fermions \cite{Susskind} in lattice gauge theory (see \cite{Becher}).

Now, it is interesting to see what happens with the other operators we
have defined in the continuous limit. On $\Omega$, the operators $\a^i,
\b^i,I_i,\g^5$ and the shift operators $\tau_i: \Psi(n)\to\Psi(n+h_i)$
are well-defined. Unfortunately, they do not preserve the
decomposition into staggered subspaces. Fortunately, there are natural
replacements for these operators which already preserve $
\Omega^{000}$. For the generators $\a^i,\b^i$ of $\textit{Cl}^{d,d}(\R)
$ we obtain:
\begin{equation}
\tilde{\a}^i =3D \a^i\tau_i, \qquad \tilde{\b}^i =3D \b^i\tau_i.
\end{equation}
For the other operators $I_i,\g^5$ we can use the same formulas we
have used in the continuous limit to compute them:
\begin{equation}
\tilde{\g}^5=3D-i\tilde{\a}^3\tilde{\a}^2\tilde{\a}^1 =3D
\g^5\tau_1\tau_2\tau_3
\end{equation}
\begin{equation}
2i\varepsilon^{ijk}\tilde{I}_i =3D \tilde{\b}^j\tilde{\b}^k =3D \b^j\b^k
\tau^j\tau^k
\end{equation}

Now, the operators $\tilde{\g}^5$ and $\tilde{I}_i$ generate an
interesting group $\mathcal{A}$ of operators associated with lattice
shifts:

\begin{theorem}\label{th:shiftAlgebra} The group $\mathcal{A}$ of
operators generated by $\tilde{\g}^5$ and $2\tilde{I}_i$ has the
following properties:
\begin{itemize}
\item It preserves the staggered subspaces $\Omega^\l$.
\item It preserves the lattice Dirac equation.
\item There exists an epimorphism $\pi: \mathcal{A} \to \Z^3$ named
``underlying shift operator''.
\item $\mathrm{Ker} \pi \cong\B$ and acts by pointwise
multiplication.
\end{itemize}
\end{theorem}

Note one advantage of using the original form $D=3D\a^i\pd_i$ of the
Dirac equation here: $\g^5$ does not anticommute, but commute with the
Dirac equation. For a shift operator $\tau\in\Z^3$, the equation $
\pi(\tilde{tau})=3D\tau$ defines the operator $\tilde{\tau}$ modulo its
sign.

\subsection{The cellular lattice model}

Let's forget, for some time, about the staggered character of the
lattice Dirac equation. Then, the lattice space of the discretization $
\Omega^{000}$ is simply \CZ, with a single complex number on each
lattice node. For all SM fermions, we obtain the lattice space \CAZ.

Note also that we have a first oder lattice equation on it. This
suggests an interpretation of \CAZ\/ as a phase space of some physical
system:
\begin{equation}
z^i_\mu(n) =3D a^i_\mu(n) + i \pi^i_\mu(n).
\end{equation}
with configuration variables $a^i_\mu(n): \Z^3\to\A$ and momentum
variables $\pi^i_\mu(n)$. On the phase space \CAZ we have the standard
symplectic structure
\begin{equation}\label{symplectic}
\omega =3D \sum_{i,\mu,n} d a^i_\mu(n) \wedge d \pi^i_\mu(n).
\end{equation}

Then, the configuration space \AZ\/ appears in a natural way if we
have a regular lattice of deformable cells (see figure
\ref{fig:lattice}). Here, each cell is described by an affine
transformation from some standard reference cell. This reference cell
is assumed to be located in the origin.

Now, to have such a simple model is, of course, nice and beautiful.
But is it only an otherwise useless toy, or is it helpful to explain
the physics of the SM? We want to show here that this model has
physical importance.

First, of course, this model gives the symplectic structure, which we
have used in section \ref{complex} to derive the unitarity. Thus, the
model allows to explain our postulate \ref{postulate:symplectic}.

But it seems helpful to explain Euclidean symmetry too. Of course,
Euclidean symmetry is not a property of the full SM, the mass matrices
break this symmetry. Thus, we need some spontaneous symmetry breaking
to explain the SM masses. Nonetheless, the lattice model allows to
answer the following simple question: Why do we have to use the left
action of \E\/ on \A\/, instead of the right or adjoint action? For
this purpose, let's see what happens with a lattice of deformed cells
if we apply the different actions of \E:

We consider an almost regular lattice. Then we have approximately

\begin{equation}
a^i_j(n) \approx \delta^i_j, \qquad a^i_0(n) \approx n_ih,
\label{eq:locationInSpace}
\end{equation}

Now, we see that the left action of a rotation rotates the lattice as
a whole, including the shifts $a^i_0(n)$. Instead, the right action of
a rotation leaves the cells on their places $n_ih$ and rotates them
around these places. This, obviously, changes the connection between
neighbour cells. Instead, the left action rotates the lattice as a
whole, leaving the local geometry unchanged. Thus, the left action
seems much more likely to be a symmetry of the theory. In this sense,
our cellular model is useful to explain our postulate
\ref{postulate:Euclidean} as well.

But the most important consequence of the cellular lattice model is
that we can apply now condensed matter theory. Especially we can, in
the large distance limit, define density, velocity, and a stress
tensor, and postulate continuity and Euler equations. But this is what
we need to incorporate gravity into the model. A metric theory of
gravity with GR limit, based on such an ``ether concept'', has been
proposed in \cite{GLET}. We give a short introduction in appendix
\ref{Gravity}.

\section{Lattice gauge fields}

While our postulates \ref{postulate:symplectic} and
\ref{postulate:Euclidean} impose strong restrictions for the gauge
group of the SM, we are yet far away from computing the SM gauge
group. There are gauge groups much larger than $SU(3)_c\times SU(2)_L
\times U(1)_Y$ compatible with these postulates.

But the consideration of the lattice theory allows to impose another
type of restrictions: It should be possible to ``put the gauge action
on the lattice''. We will see that this gives the additional
restrictions we need to compute the SM gauge group almost exactly. The
remaining possibilities for additional gauge fields will be killed by
the standard condition of anomaly freedom.

\subsection{Strong fields as Wilson gauge fields}

The classical way to incorporate gauge fields into a lattice theory
are Wilson gauge fields. The classical formalism of Wilson gauge
fields, even if it was developed for spacetime lattices $\Z^4$ instead
of our lattice of cells $\Z^3$, needs only a sufficiently obvious,
minor modification. This is caused by the fact that we have no
discrete structure in time direction. Formally, it looks like time
remaining continuous. This requires a mixed form for the definition of
the gauge field: The temporal component $A_0(n,t) \in \mathfrak{g}$
is, like in the continuous case, a function with values in the Lie
algebra, but defined on the lattice nodes. Instead, the spatial
(vector potential) part $A_i$ is described, as usual for Wilson gauge
fields, by Lie group valued functions $U(n,i,t)\in G$ located on the
edges $n,n+h_i$ of the lattice. The most important, defining property
of the Wilson gauge field remains unchanged too: The lattice gauge
symmetry is defined by a gauge-group-valued lattice function $g(.) \in
\Z^3\to G$ which acts pointwise on the lattice \CAZ\/ and is uniquely
defined by a gauge action $G\times\C\times\A\to\C\times\A$. The gauge
transformation acts in the following way:
\begin{eqnarray}
\Psi^i_\mu(n,t) &\to& (g(n,t)\Psi)^i_\mu(n,t) \\
U(n,i,t) &\to& g(n,t)U(n,i)g^{-1}(n+\hi,t), \\
A_0(n,t) &\to& g(n,t)A_0(n,t)g^{-1}(n,t) - (\pd_tg(n,t))g^{-1}(n,t).
\label{def:WilsonAction}
\end{eqnarray}

This definition of the gauge action (\ref{def:WilsonAction}) shows
that not all imaginable gauge actions may be defined in this way.
Indeed, the gauge action can act only on the generation and color
indices. Inside a doublet \CL, which is represented on the lattice as
\CZ, it cannot act in a nontrivial way. For a fixed doublet of
generation $i$ and color $\mu$, there is only a single complex number
$z^i_\mu(n)$ in each lattice node. The only possible Wilson gauge
action on the lattice \CZ\/ is an action of $U(1)$ with the same
charge on all parts of the doublet.

Now, we can compute the maximal possible Wilson gauge action which is
compatible with our postulates \ref{postulate:symplectic} and
\ref{postulate:Euclidean}. It should be an unitary group, which acts
on all generations in the same way, preserves the generations, thus,
does not act on the generation index $i$. Then, it acts with the same
charge on all parts of electroweak doublets, thus, cannot act on the
doublet indices $\k$. Thus, it can act only on the remaining index $\mu
$. This gives $U(4)$ as the maximal gauge group. Moreover, to commute
with translations, it has to leave the translational direction \c\/ in
the leptonic sector invariant. But then it has to act trivially on the
leptonic sector $\mu=3D0$. What remains is the group $U(3)$ acting on
the color index $\mu>0$. Its special subgroup $SU(3)$ is, obviously,
the color group $SU(3)_c$ of the SM. The other field is the diagonal
$U(1)_B$ with the baryon charge $I_B$.

Thus, the consideration of Wilson lattice gauge fields has given us an
important part of the SM gauge group --- the strong interactions.

\subsection{Correction terms for lattice deformations}\label{weak}

Assume our lattice $\Z^3$ is not exactly regular but slightly
deformed. This requires also a modification of the lattice Dirac
equation. What can be said about the general form of the corresponding
correction terms?

One way to find a lattice equation for a deformed lattice is to
consider the general form of the Dirac equation on a curved background
$g_{ij}(x)$, described in appendix \ref{DiracCurved}. Then, we can put
the modified continuous equation on the lattice following the scheme
presented in appendix \ref{DiracLatticeCurved}. But this much more
general lattice equation has, nonetheless, a special structure. The
lattice nodes belong to classes of different dimension --- points,
lines, faces, and cells --- and interact only with nodes of neighbour
codimension.

For our cellular lattice model, there are now two possibilities. Or
the cells of the lattice have some additional properties, like a
``dimension'', so that they really interact only with cells of
neighbour dimension. In this case, the lattice equation of the
deformed lattice would be a discretization of the Dirac equation on \CL
\/ with curved background $g_{ij}(x)$. Or there is no such additional
structure. In this case, the special structure of the Dirac equation
on the regular lattice was only accidental. The general equation for a
deformed lattice will destroy some qualitative properties of the
continuous Dirac equation. We will find, in the continuous limit,
additional terms which do not fit into the form of a Dirac equation on
curved background.

On the other hand, however deformed the lattice, the correction
coefficients will have geometric nature. They depend on the deformed
lattice. We obtain a deformed lattice equation on the same bundle \CL.
The deformation of the lattice is certainly no reason to use different
lattice equations for the different components $a^i_\mu\in\A$. This
leads to the following

\begin{postulate}\label{postulate:weak}
Correction terms for lattice deformations preserve doublets \CL\/ and
act on all doublets in the same way.
\end{postulate}

But this is a signature of weak interactions. Thus, we propose the
hypothesis that weak interactions describe correction terms for
lattice deformations.

Let's consider now a more general correction term for lattice
deformations, one which does not preserve the decomposition of the
staggered lattice into the different components $\psi_\k$. Thus, we
consider the lattice \CZ, with a function $\psi(n)$ on it, and an
undistorted lattice Dirac equation. In this equation, we replace now
every occurrence of $\psi(n)$ by a sum over values $\psi(n+\h)$ on
neighbour nodes:
\begin{equation}
\psi(n) \to \psi(n) + \sum_\h g^\h_p(n)\psi(n+\h).
\end{equation}

The geometric coefficients depend on the nodes $n$ themself as well as
on the direction of the neighbour $\h\in\Z^3$ and on the occurrence $p
$ of the term in the undistorted Dirac equation. Using the lattice
shift operator $\tau_\h: \psi(n)\to\psi(n+\h)$, we can rewrite the
expression as
\begin{equation}
\psi(n) \to (1 + \sum_\h g^\h_p(n)\tau_\h) \psi(n).
\end{equation}

Now, the lattice shift operator $\tau_\h$ is not a beautiful choice.
Especially, for the But, given the fact that it stands together with a
complex set of coefficients $g^\h_p(n)$, we can replace it with
different, more beautiful operators, as long as the difference leads
only to a redefinition of the $g^\h_p(n)$. That means, we have the
freedom to replace $\tau_\h$ by operators of type $o^\h(n)\tau_\h$
with $o^\h(n)\in\C$. We will use this freedom to replace the shift
operators $\tau_\h$ by the operators $\tilde{\tau}_\h\in\mathcal{A}$
of theorem \ref{th:shiftAlgebra}. This gives
\begin{equation}
\psi(n) \to (1 + \sum_\h \tilde{g}^\h_p(n)\tilde{\tau}_\h) \psi(n).
\end{equation}

This expression can be interpreted as describing interaction terms of $
\psi$ with some other field described by various lattice fields $
\tilde{g}^\h_p(n)$, with the operators $\tilde{\tau}_\h$ as defining
the interaction.

Now, we propose here the hypothesis that \emph{the large distance
limit of these lattice fields $\tilde{g}^\h_p(n)$ will be a gauge
field $A^\h_i(x)$}. The proof of this hypothesis has to be left to
future research. It would require a more concrete model of the
geometric coefficients related with a lattice deformation and, then, a
consideration of the large distance limit. But it seems useful, at
this point, to remember a major lecture of the Wilson approach to
renormalization, namely, large distance universality. The details of
the microscopic model may be very different, even qualitatively
completely different, nonetheless, the mathematics of the large
distance limit may become identical, defining the same universality
class. This certainly does not replace a proof of the hypothesis, but,
nonetheless, makes it plausible.

On the other hand, we also have to note an essential difference
between these correction coefficients for lattice deformations and
Wilson gauge fields. Wilson gauge fields have an exact gauge symmetry
on the lattice. There is obviously no such thing for these correction
coefficients. The consequences of the missing lattice gauge invariance
for quantization we will discuss below.

Thus, to name them lattice gauge fields seems misleading and
unjustified. Instead, we will name them ``gauge-like lattice fields''.

Anyway, even without having a proof of our hypothesis about the
classical limit of the lattice fields $\tilde{g}^\h_p(n)$ as being
gauge fields $A^\h_i(x)$, we know all we need about the interaction
terms $\tilde{\tau}_\h$. They give, in the large distance limit, the
eight operators $\{1,\g^5,2I_i,2I_i\g^5\}$. Thus, they give all the
operators we need to define weak gauge fields. Thus, if our hypothesis
is correct, we have found a way to put the weak gauge group $SU(2)_L$
on the lattice.

Let's now look at this possibility to construct gauge fields from the
other side. What would be the maximal gauge group, which can be
obtained in this way?

We have already found, that this gauge group has to preserve doublets
and to act on all doublets in the the same way. This is, obviously, in
correspondence with the desription of the gauge group as generated by
linear combinations of the operators $\{1,\g^5,2I_i,2I_i\g^5\}$.
Indeed, all of them preserve doublets and act on them in the same way.
Preservation of the symplectic structure and rotational symmetry do
not give any additional restrictions. But translational symmetry gives
such an additional restriction: On the leptonic sector, there has to
be a preserved direction.

The maximal gauge group $U(2)_L\times U(2)_R$ generated by the
generators $\{1,\g^5,2I_i,2I_i\g^5\}$ does not have any invariant
direction. Thus, any group which fulfills our conditions has to be
smaller than this group. The preferred direction may be left-handed,
right-handed, or a linear combination of above. The last case gives a
more rigorous restriction of the group, because as the left-handed
part, as the right-handed part would have to be preserved. Thus, we
can ignore the last case. Without restriction of generality we,
therefore, assume, that the preserved translational direction is right-
handed. This gives the group $U(2)_L$ as preserving this direction.

Now, let's see which part of $U(2)_R$ preserves one translational
direction. This will be a subgroup $U(1)\subset U(2)_R$ with charge
$0$ on the translational direction and charge $\pm1$ on the orthogonal
direction. Without restriction of generality, we can use here $(1+
\g^5)/2(I_3-1/2)$ as its charge. This choice locates the translational
direction inside the right-handed neutrino sector. Instead of the
right-handed charge $\frac{1+\g^5}{2}(I_3-\frac{1}{2})$, we can as
well use the corresponding vector charge $\Ie=3D I_3-\frac{1}{2}$.
Thus, we obtain the following

\begin{theorem}
The maximal gauge group, obtained from correction terms for lattice
deformations, and compatible with \E\/ symmetry and symplectic
structure, is the group generated by the chiral $U(2)_L$ and the
vector \Ue with charge $\Ie=3D I_3-\frac{1}{2}$.
\end{theorem}

\subsection{The EM field and anomaly freedom}

At a first look, we see the EM field does not fit into any of the two
classes of gauge-like lattice fields. It acts nontrivially inside
doublets, thus, is not a Wilson field. On the other hand, it has
different charges on leptons and quarks, thus, does not fit into the
weak fields too.

Nonetheless, we have already obtained it. It appears as a linear
combination of a strong field and a weak field. Indeed, we have
\begin{equation}
U(1)_{em} \cong S(U(1)_B\times\Ue)
\end{equation}
with the charge
\begin{equation}
Q =3D 2I_B + \Ie
\end{equation}

Thus, we have obtained now all SM gauge fields, and the two types of
lattice gauge fields are already sufficient. Thus, we can add the next
postulate for the gauge fields:

\begin{postulate} \label{postulate:lattice}
All gauge fields have to be constructed using the following two types
of gauge-like lattice fields:
\begin{itemize}
\item Wilson gauge fields,
\item correction terms for lattice deformations.
\end{itemize}

\end{postulate}

With this postulate, we are already very close to the computation of
the gauge group of the SM. Indeed, the maximal gauge group which can
be obtained with our two types of gauge fields is the group
\begin{equation}
G_0 \cong U(3)_c \times U(2)_L \times U(1)_Y \supset SU(3)_c \times
SU(2)_L \times U(1)_Y
\end{equation}
The difference between the maximal possible gauge group $G_0$ and the
SM gauge group is minimal.

To get rid of the remaining diagonal fields, it is sufficient to
remember about the gauge anomaly. The additional diagonal fields are
anomalous. Thus, it remains to add

\begin{postulate} The gauge fields have to be anomaly-free.
\end{postulate}

and we have finished the computation of the SM gauge group. We have
the following theorem:

\begin{theorem}
The SM gauge group $SU(3)_c \times SU(2)_L \times U(1)_Y$ defines a
maximal gauge action on our lattice model \CAZ\/ which preserves \E\/
symmetry and the symplectic structure, gives anomaly freedom, and can
be constructed using only Wilson gauge fields and correction terms for
lattice deformations.
\end{theorem}

Of course, the result of the computation is not unique. There are
other maximal gauge groups which fulfill these postulates. For
example, we could start with $U(1)_B$ or $\Ue$ and extend them as much
as possible. As well, we could use $SU(2)_R$ instead of $SU(2)_L$. It
seems reasonable to hope that future research, especially the
consideration of the renormalization equations, allows to obtain even
better results.

\section{Fermion quantization} \label{fermionQuantization}

Following Berezin \cite{Berezin}, the classical limit of fermion
fields are Grassmann-valued fields. This is, obviously, incompatible
with our geometric interpretation of fermion doublets as \CL, nor with
the lattice model \CZ, which are classical, commuting, fields, with a
standard symplectic structure. The appropriate way to quantize them
would be canonical quantization.

Here we present a way to obtain anticommuting fermion fields via
canonical quantization. It consists of two parts, with a canonically
quantized \B-valued field (spin field) as the intermediate step. To
obtain a \B-valued field from an \R-valued field, all we need is a \B-
degenerated potential $V(\varphi)$. The lowest energy states, then,
already define a \B-valued field theory. This potential $V(\varphi)$
already requires the breaking of \E\/ symmetry, at least if $\varphi$
is the direction of translation \c. But this is necessary, because we
cannot define a translation on \B-valued fields, moreover, on fermion
fields.

The more non-trivial step is from spin fields to fermion fields, or
from commuting to anticommuting operators at different nodes. The
lattice operator algebras appear to be isomorph, but the isomorphism
is, first, nonlocal, and, second, not natural, depends on some order
between different lattice nodes. This leads to a nontrivial
transformation and approximation of the Hamilton operator

We consider here canonical quantization of lattice theories with
configuration space $Q=3D\R(\Z^3)$ resp. $Q=3D\B(\Z^3)$. Our
considerations here do not depend on the dimension $d=3D3$, so we
consider here the more general case $Q=3D\R(\Z^d)$ resp. $Q=3D\B(\Z^d)$.
Canonical quantization consists of the definition of operators on the
Hilbert space $\mathscr{L}^2(Q)$, and a Schr\"{o}dinger equation
\begin{equation}
i\pd_t \Psi(q,t)=3DH\Psi(q,t), \qquad q \in Q=3D\mathscr{F}(\Z^3,Y),
\qquad t \in \R.
\label{eq:Schroedinger}
\end{equation}
Thus, we always have continuous time. Note that in the condensed
matter interpretation the lattice $\Z^3$ is not a ``discretization of
space'' $\R^3$ itself. Instead, it enumerates elementary cells located
in a continuous $\R^3$, where the state of the cells is described by
some affine transformation $\A\subset \R^{12}$ of $\R^3$. Nonetheless,
for the purpose of this section, the lattice $\Z^3$ may be considered
like a ``discretization of space'' $\Z^3\subset \R^3$, and our
geometric interpretation of $\R^{12}$ plays no role here.

\subsection{From fermion fields to spin fields}\label{fermion2spin}

Spin fields have the configuration space $\B(\Z^d)$. On each lattice
node $n\in \Z^d$ we have the Pauli matrices $\sigma^i_n$ as operators:
\begin{equation}
\sigma_n^i\sigma_n^j =3D \delta_{ij}+i\varepsilon_{ijk}\sigma_n^k.\;
\label{eq:sigma:product}
\end{equation}
Spin field operators on different nodes commute:
\begin{equation}
\left[\sigma^i_m, \sigma^j_n\right] =3D 2i\delta_{mn}
\varepsilon_{ijk}\sigma^k_n.
\label{eq:sigmaCommutation}
\end{equation}

Instead, following Berezin\cite{Berezin}, the fermion field operators $
\psi_n, \psi^*_n$ are usually considered to be of qualitatively
different nature. They do not fit into the canonical scheme.
Especially there is no configuration space $Q$. Indeed, operators
related to different nodes do not commute. Instead, they anticommute:
\begin{equation}
\{\psi_m,\psi^*_n\} =3D \delta_{mn},\;\{\psi^*_m,\psi^*_n\}=3D\{\psi_m,
\psi_n\}=3D0.
\label{eq:psiAnticommutation}
\end{equation}
This difference seems to forbid any identification of fermions with
spin fields.

Despite this, the two operator algebras appear to be isomorph. The
isomorphism is well-known in the theory of Clifford algebras and
allows to establish the isomorphism
\begin{equation}
\textit{Cl}^{N,N}(\R)\cong M_2(\textit{Cl}^{N-1,N-1}(\R))\cong M_{2^N}
(\R).
\label{eq:Clifford}
\end{equation}

To see this, let's at first transform the operator algebras in each
node into an equivalent form, by defining operators $\psi^i_n$:
\begin{equation}
\psi_n^1 =3D \psi_n + \psi_n^*,\; \psi_n^2 =3D -i(\psi_n - \psi_n^*),\;
\psi_n^3 =3D -i\psi_n^1\psi_n^2.
\label{eq:def:psi^i}
\end{equation}
This gives
\begin{equation}
\psi_n^i\psi_n^j =3D \delta_{ij}+i\varepsilon_{ijk}\psi_n^k,\;
\label{eq:psi:product}
\end{equation}
similar to (\ref{eq:sigma:product}).
In these variables, the operators $\psi^1_n$ and $i\psi^2_n$ generate
(for a finite lattice with N nodes) the Clifford algebra $
\textit{Cl}^{N,N}(\R)$. On the other hand, $M_{2^N}(\R)$ is the
operator algebra on the $2^N$-dimensional space of \B-valued functions
on the same lattice.

But the isomorphism between these two operator algebras is not
natural. It depends on the choice of some order $>$ between the
lattice nodes. For a given order, the isomorphism is defined by:
\begin{eqnarray}
\label{eq:psi2sigma}
\psi^{1/2}_n =3D \sigma^{1/2}_n \prod_{m>n}{\sigma^3_m},&&\;\psi^3_n=3D
\sigma^3_n,\\
\label{eq:sigma2psi}
\sigma^{1/2}_n =3D \psi^{1/2}_n \prod_{m>n}{\psi^3_m},&&\;\sigma^3_n=3D
\psi^3_n.
\end{eqnarray}
Note also that (different from the $\sigma^i_n$) the operators $
\psi_n^i$ do not act as local operators on the lattice. Instead, they
act like $\sigma^3_m$ on other nodes $m>n$. This is a necessary
property of such an isomorphism, because, obviously, any local
combination of the commuting local operators $\sigma^i_n$ leads only
to another set of commuting local operators.

As a consequence, a Hamilton operator which ``looks local'' in terms
of the $\psi^i_n$ may appear nonlocal in terms of the $\sigma^i_n$
(which we consider to be ``truly local'' operators) and reverse.
Fortunately, there are important examples of operators where this does
not happen. First, there is the operator
\begin{equation}
\label{eq:def:H0}
H_0\ =3D - \frac{1}{2}\sum_n{\sigma_n^3} =3D\ \frac{1}{2}\sum_n
\psi^*_n\psi_n-\psi_n\psi^*_n
\end{equation}

Let's consider now operators with interactions between neighbour
nodes. We are (for reasons which become obvious later) especially
interested in the following linear combination:
\begin{equation}
\label{eq:def:HDd}
H_D\ =3D \frac{1}{2}\sum_{n,i}\sigma^1_n\sigma^1_{n+h_i}-\sigma^2_n
\sigma^2_{n+h_i}
% \label{eq:def:H1d}
% H_1\ &=3D& \frac{1}{2}\sum_{n,i}\sigma^1_n\sigma^1_{n+h_i}+\sigma^2_n
\sigma^2_{n+h_i}
\end{equation}
where $h_i$ are the $d$ basic lattice shifts in the d-dimensional
lattice $\Z^d$.

Now, in the one-dimensional case, we have a natural (up to the sign)
order $>$. For this order, we obtain:
\begin{equation}
\label{eq:def:HD1}
H^{(1)}_D\ =3D i\frac{1}{2}\sum_n(\psi^1_n\psi^2_{n+1}+\psi^2_n
\psi^1_{n+1})\ =3D
\ \sum_n \psi_n\psi_{n+1}-\psi^*_n\psi^*_{n+1}
% \label{eq:def:H11}
% H_1^{(1)}\ &=3D& i\frac{1}{2}\sum_n(\psi^1_n\psi^2_{n+1}-\psi^2_n
\psi^1_{n+1})\ =3D
% \ \sum_n\psi^*_n\psi_{n+1}-\psi_n\psi^*_{n+1}.
\end{equation}

Note that our operator is symmetric for spatial inversion $n\to -n$,
but the representation in the asymmetric (in terms of the $\sigma^i_n
$) operators $\psi^i_n$ hides this symmetry.

\subsection{The case of higher dimensions}

Unfortunately, the transformation of the Hamilton operator in higher
dimensions is not that simple. What we can obtain is only an
approximation
\begin{equation}
\label{eq:def:tHDd}
H^{(d)}_D\ \approx\ \tilde{H}^{(d)}_D\ =3D%&=3D&
\sum_{n,i} \alpha^n_{n+h_i} (\psi_n\psi_{n+h_i}-\psi^*_n\psi^*_{n
+h_i})%\\
% \label{eq:def:tH1d}
% H^{(d)}_1\ \approx\ \tilde{H}_1^{(d)}\ &=3D&
% \sum_{n,i} \alpha^n_i (\psi^*_n\psi_{n+h_i}-\psi_n\psi^*_{n+h_i})
\end{equation}%narray}
where
\begin{equation}
\label{eq:alphadef}
\alpha^n_{n+h_i} =3D \left\{ \begin{array}{cl}
1 & \textrm{if}\ n < n+h_i\\
-1 & \textrm{else}\end{array} \right. \
\end{equation}

The accuracy of this approximation obviously depends on the order $>
$. Indeed, the error
\begin{equation}
\sigma^1_n \sigma^1_{n'} \approx \psi^1_n \psi^2_{n'} =3D \sigma^1_n
\sigma^1_{n'}\prod_{n \label{eq:prodnonlocal}
\end{equation}
resp. for $\sigma^2_n \sigma^2_{n'}$, depends on the number and
location of the nodes $m$ located ``between'' (according to the chosen
ordering) the ``neighbour'' (according to the lattice $\Z^d$) nodes
$n,n'$. Now, instead of the simple lexicographic order (which gives $
\alpha^n_{n+h_i}=3D1$) we propose to use another, more sophisticated
order we name ``alternating lexicographic order''.

It has to be acknowledged that this order has been designed to give
the result below. Fortunately, we can justify this choice of an order
in another way: It gives a better approximation of the original
Hamiltonian operator, in the sense, that some algebraic properties of
the original terms may be preserved exactly.

Note that our interaction terms can be represented as a function of
the differences of the operator $\sigma^1_n$ and its shift:
\begin{equation}
\sigma^1_n\sigma^1_{n+h_i} =3D 1 - \frac{1}{2}((1-\tau_i)\sigma^1_n)^2,
\label{eq:differenceproperty}
\end{equation}
where $\tau_i$ is the shift operator on the lattice. This follows from
$(\sigma^1_n)^2=3D1$ and the commutation relation $[\sigma^1_n,\tau_i
\sigma^1_n]=3D0$. Now, we propose to use an order which allows to
preserve these properties exactly. That means, we want to replace the $
\sigma^1_n$ by some $\tilde{\sigma}^1_n$ with exactly the same
properties:
\begin{equation}
(\tilde{\sigma}^1_n)^2=3D1,\; [\tilde{\sigma}^1_n,\tau_{i}
\tilde{\sigma}^1_{n}]=3D0,
\label{eq:approximationalproperties}
\end{equation}
so that
\begin{equation}
\sigma^1_n\sigma^1_{n+h_i} \approx \tilde{\sigma}^1_n
\tilde{\sigma}^1_{n+h_i}=3D 1 -
\frac{1}{2}((1-\tau_i)\tilde{\sigma}^1_n)^2.
\end{equation}

\begin{figure}
\includegraphics[angle=3D90,width=3D0.8\textwidth]{order}
\label{fig:order}\caption{The alternating lexicographic order}
\end{figure}

For the simple lexicographic order, we have no way to define such $
\tilde{\sigma}^i_{n}$. But it is possible for the alternating
lexicogrpahic order. We define it by induction. Let $>_k$ be the order
defined for a k-dimensional lattice $\Z^k$, and $\pi_k$ the projection
on this lattice defined by the first $k$ coordinates. Then we define
$>_{k+1}$ by the following properties:

\begin{itemize}
\item if $n_{k+1} \lessgtr m_{k+1}$ then $n\lessgtr_{k+1}m$;
\item else if $n_{k+1} (=3D m_{k+1})$ is even and $\pi_k n \lessgtr_{k}
\pi_k m$ then $ n \lessgtr_{k+1} m$;
\item else if $n_{k+1} (=3D m_{k+1})$ is odd and $\pi_k n \lessgtr_{k}
\pi_k m$ then $ n \gtrless_{k+1} m$.
\end{itemize}

Thus, we use the inverse order inside the odd planes. Now the
interaction term can be splitted in the following way:
\begin{equation}
\sigma^{1/2}_n \sigma^{1/2}_{n+h_i}\prod_{n \tilde{\sigma}^{1/2}_n \tilde{\sigma}^{1/2}_{n+h_i}
% (\sigma^{1/2}_n \prod_{\begin{array}{c}n{\sigma^3_m}) \cdot
% (\sigma^{1/2}_{n+h_i}\prod_{\begin{array}{c}m+1\end{array}}{\sigma^3_m})
\label{eq:proddecomposition}
\end{equation}
with
\begin{equation}
\tilde{\sigma}^{1/2}_n=3D\sigma^{1/2}_n
\prod_{\begin{array}{c}n \tilde{\sigma}^{1/2}_{n+h_i}=3D\sigma^{1/2}_{n+h_i}
\prod_{\begin{array}{c}m\end{equation}
we obtain the properties (\ref{eq:approximationalproperties}). The key
is that for each node $m$ with $n\tau_i m$ fulfils $m'term we find a corresponding $\sigma^3_{m'}$ in the second term.

For our choice of $>$, the coefficients $\alpha^n_{n'}$ fulfill the
following relations:
\begin{equation}
\alpha^n_m =3D \alpha^{n+2h_i}_{m+2h_i};\ \ \hfill
\alpha^n_{n+h_i} \alpha^{n+h_i}_{n+2h_i} =3D 1;\ \ \hfill
\alpha^n_{n+h_i} \alpha^{n+h_i}_{n+h_i+h_j} =3D - \alpha^n_{n+h_j}
\alpha^{n+h_j}_{n+h_i+h_j}.
\label{eq:alpharelations}
\end{equation}

\subsection{Transformation of the lattice Dirac operator into
staggered form}\label{staggering}

Now, the operator $H=3D\tilde{H}_D+mH_0$ appears to be a lattice Dirac
operator.
Indeed, let's consider the evolution equation defined by $H$:
\begin{eqnarray}
\label{eq:DiracInPsi}
i\pd_t \psi_n \ =3D\ [H,\psi_n ]&=3D& \phantom{-}
\sum_i \alpha^n_{n+h_i} (\psi_{n+h_i}^*-\psi_{n-h_i}^*)-m\psi_{n},\\
\label{eq:DiracInPsiAdjoint}
i\pd_t \psi_n^* \ =3D\ [H,\psi_n^*]&=3D& -
\sum_i \alpha^n_{n+h_i} (\psi_{n+h_i} -\psi_{n-h_i} )+m\psi^*_{n}.
\end{eqnarray}

As a consequence of the relations (\ref{eq:alpharelations}), the
evolution equations (\ref{eq:DiracInPsi}),(\ref{eq:DiracInPsiAdjoint})
give
\begin{equation}
\pd^2_t\psi_n =3D \sum_i (\psi_{n+2h_i}-2\psi_n+\psi_{n-2h_i})-
m^2\psi_{n}
=3D ((\Delta_{2h}+m^2) \psi)_n,
\label{eq:KleinGordonLattice}
\end{equation}
where $\Delta_{2h}$ is the lattice Laplace operator with doubled
distance $2h_i$ --- a Laplace operator on a coarse lattice.

The lattice Laplace operator $\Delta_{2h}$ acts independently on $2^d$
different sublattices. Let's distinguish these sublattices by
introduction of $2^d$ different lattice functions enumerated by
elements of $\k=3D(\k_1,\ldots,\k_d)\in\left\{0,1\right\}^d$. Using the
denotation $*\psi_n=3D\psi^*_n$, we define
\begin{equation}
\psi_\k(n) =3D *^{\k_1+\ldots+\k_d}\psi_n\qquad\textrm{on}\qquad n
\equiv \k \;\;\textrm{mod}\;\; 2.
\label{eq:def:psikappa}
\end{equation}

Each of the $2^d$ lattice functions $\psi_\k(n)$ is defined on a
``coarse lattice'' containing the nodes of type $n_i =3D 2\tilde{n}_i+
\k_i$ and lattice spacing $2h_i$. Now, the lattice opeartor $
\Delta_{2h}$ acts as the simple Laplace operator on each of the $2^d$
functions $\psi_\k(n)$. In the continuous limit, each $\psi_\k(n)$
gives a function $\psi^\k(x)$ which fulfills the Klein-Gordon equation
\begin{equation}
\pd^2_t\psi_\k(x,t) =3D(\sum_i \pd^2_i - m^2) \psi_\k(x,t) =3D 0.
\label{eq:KleinGordonContinuous}
\end{equation}

The lattice Dirac equations (\ref{eq:DiracInPsi}),
(\ref{eq:DiracInPsiAdjoint}) now establish a connection between these
$2^d$ lattice fields. We can define now $2^d\times 2^d$ matrices $
(\a^i)_\k^{\k'},\b_\k^{\k'}$ so that the original lattice equations
(\ref{eq:DiracInPsi}),(\ref{eq:DiracInPsiAdjoint}) transform into
\begin{equation}
i\pd_t \psi_\k(n) \ =3D\ [H,\psi_\k(n) ]\ =3D\
\sum_i -i(\alpha^i)_\k^{\k'} (\psi_{\k'}({n+h_i})-\psi_{\k'}({n-
h_i})) + m\b_\k^{\k'}\psi_{\k'}(n)
\label{eq:DiracLatticeDerived}
\end{equation}
on $n =3D \k \;\;\textrm{mod}\;\; 2$. Because of the factor $*^{\k_1+
\ldots+\k_d}$ in (\ref{eq:def:psikappa}), equation
(\ref{eq:DiracLattice}) connects only the fields $\psi_\k(n)$, and
it's adjoint only the $(\psi_\k(n))^*$.

This equation is our lattice Dirac equation (\ref{eq:DiracLattice}) on
the staggered lattice (\ref{eq:DiracStaggered}), but already in its
quantized form, with anticommuting fermion operators $\psi_\k(n)$.

\subsection{From spin fields to scalar fields}\label{spin2scalar}

Spin fields are already a much more classical object in comparison
with the original fermion fields. But we need even more classical
objects, namely real-valued fields.

But this is not problematic at all. We can embed the spin field as an
effective description of the lowest energy states of a scalar field
with a \B-symmetric potential with two different vacuum states. For
example, we can consider $\varphi^4$ theory in $\R^d$ with negative
mass parameter $\mu^2$:

\begin{equation}
\mathscr{L} =3D \frac{1}{2}((\pd_t\varphi)^2 - (\pd_i\varphi)^2)-
V(\varphi) \;\textrm{with}\;
V(\varphi)=3D-\frac{\mu^2}{2}\varphi^2+\frac{\lambda}{4!}\varphi^4.
\label{eq:L:phi4}
\end{equation}
The two minima of the potential are $\varphi(x) =3D \pm \varphi_0$ with $
\varphi_0 =3D \sqrt{\frac{6\mu^2}{\lambda}}$.

If the system is near $\varphi_0$, it is convenient to use the $\sigma
$-variable $\sigma(x)=3D\varphi(x)-\varphi_0$ so that
\begin{equation}
V(\sigma) =3D \frac{1}{2}(2\mu^2)\sigma^2+\sqrt{\frac{\lambda}{6}}\mu
\sigma^3+\frac{\lambda}{4!}\sigma^4
\label{eq:V:sigma}.
\end{equation}
This describes a scalar field with mass $\sqrt{2}\mu$ and some
interactions.

Instead, we are interested only in the lowest energy states of this
theory. Let's consider at first the simple case of dimension $d=3D0$,
where QFT reduces to ordinary quantum theory. If we have energies much
below $\mu$, only the two vacuum states $\Psi_\pm(\varphi)$ with $
\langle \Psi_\pm|\varphi|\Psi_\pm\rangle \approx \pm\varphi_0$ are
important. But the true eigenstates of energy are
\begin{equation}
\Psi_{0/1}(\varphi)=3D\frac{1}{\sqrt{2}}(\Psi_+(\varphi) \pm \Psi_-
(\varphi)).
\label{eq:Psi}
\end{equation}
Between them, we have an energy gap of order
\begin{equation}
\Delta =3D E_1-E_0 \sim \exp(-\int_0^{\varphi_0}{\sqrt{V(\varphi)-E_0}d
\varphi}) \sim
\exp(-\frac{\mu^3}{\lambda}).
\label{eq:gap}
\end{equation}
With increasing $\mu$ the mass of the $\sigma$ field increases, but
the energy gap $\Delta$ decreases exponentially. Without any
conspiracy, this leads to two different domains: a high energy domain,
with energies of order $\mu$, where the tunneling may be ignored, and
a low energy domain, with energies of order $\Delta$, where the whole
theory reduces to the two-dimensional space spanned by $\Psi_{0/1}$.
Reduction to this subspace gives
\begin{eqnarray*}
\varphi\qquad\to &\varphi_0\sigma^1&\textrm{ with }\;\varphi_0=3D\int
\overline{\Psi}_0\cdot\varphi\Psi_1 d\varphi\\
\pi=3D\frac{\delta L}{\delta\dot{\varphi}}\qquad \to &\pi_0\sigma^2&
\textrm{ with }\; \pi_0=3D\int\overline{\Psi}_0\cdot\pd_{\varphi}\Psi_1 d
\varphi\\
H\qquad\to & H_0-\frac{1}{2}\Delta \sigma^3&\textrm{ with }\; H_0=3D
\frac{E_0+E_1}{2}.
\label{eq:reductionToPauli}
\end{eqnarray*}
For dimension $d>0$, at least as long as the momentum $k$ is
sufficiently small, we have a similar situation for each of the modes $
\varphi(x)=3D\exp(ikx)\varphi$. For sufficiently large $\mu$, and
suffiently low energies under consideration, the theory reduces to an
effective theory where we have only two degrees of freedom for each
mode. Effectively, the configuration space reduces from $\mathcal{F}
(\Z^d,\R)$ to $\mathcal{F}(\Z^d,\B)$.

Last not least, let's consider typical lattice theory interaction
terms which may appear in the reduction for a Lagrangian of type
(\ref{eq:L:phi4}). We consider lattice approximations where only
neighbour nodes have nontrivial interaction terms. Let $n,n'=3Dn+h_i$ be
these neighbour nodes, $d=3D1$. One possibility is to use $\frac{1}{2}
(\pi_n+\pi_{n'})$ to approximate $\pi(x)$ and $\frac{1}{h} (\varphi_n-
\varphi_{n'})$ to approximate $\pd_i\varphi(x)$. Then, the reduction
gives an effective Hamiltonian
\begin{equation}
H =3D \frac{1}{2}(\pi^2 + c^2 (\pd_x\varphi)^2 +V(\varphi)) \to c_0+
c_1 \sigma^1_n\sigma^1_{n'} +
c_2 \sigma^2_n\sigma^2_{n'} +
c_3 \sigma^3_n
\label{eq:HRtoHZ2}
\end{equation}
for some constants $c_i$. The lattice Dirac operator corresponds to
$c_1 =3D -c_2 =3D 1$, thus, can be obtained in this scheme.

As a consequence of this quantization method for fermions, we obtain
some analogon of a ``supersymmetric partner'' of the fermions. This
partner can be very heavy without any conspiracy. At the current state
of research, no indications about their masses can be given.

\subsection{Generalization of Bohmian mechanics and Nelsonian
stochastics}

A consequence of our approach to fermion quantization is that we
obtain, in this way, a new route for the generalization of hidden
variable theories like Bohmian mechanics \cite{Bohm} or Nelsonian
stochastics \cite{Nelson}. Indeed, these theories have been defined
for multi-particle Schr\"{o}dinger theory. Now, our model, and, even
more, the quantization scheme we have used here, is already very close
to classical multi-particle Schr\"{o}dinger theory.

Indeed, we can identify the state $a^i_\mu(n)$ of the cell $n$ with
the position of four of its points $a_\mu(n)$. In this sense, the
configuration space already coinsides with the configuration space of
a classical multi-particle theory. Thus, we have already the same
configuration space as in Bohmian mechanics as well as in Nelsonian
stochastics.

Moreover, we have also used classical canonical quantization. Thus,
all the quantum operators are also the same as in Bohmian mechanics
or Nelsonian stochastics.

The only difference is that the Schr\"{o}dinger operator is not
exactly of the same form as in multi-particle Schr\"{o}dinger theory.
But the difference is not very big. Indeed, the Hamilton operator
(\ref{eq:HRtoHZ2}) is already of the form
\begin{equation}
H =3D Q(p_i) + V(Q)
\end{equation}
with some non-degenerated quadratic form $Q(p_i)\ge 0$ of the momentum
variables and some potential $V(Q)$ on the configuration space.

Thus, one way to obtain a generalization of Bohmian mechanics and
Nelsonian stochastics is to diagonalize the quadratic form $Q(p_i)$ so
that, for some new, diagonalized, momentum variables $p'_i(p_j)$, we
have
\begin{equation}
Q(p_i) =3D \sum_j \frac{p'_j}{2m_j}
\end{equation}
and we have recovered the standard form of multi-particle Schr
\"{o}dinger theory, so that we can use the standard versions of
Bohmian mechanics and Nelsonian stochastics.

\section{About gauge field quantization}

A first objection against our construction of weak gauge fields in
section \ref{weak} is that it presents a lattice regularization for
chiral gauge field theory. But to obtain such a regularization is a
famous problem of chiral lattice gauge theory \cite{lattice}, and
there are various no-go theorems for such regularizations.

But the regularization problem of chiral gauge theory is the problem
to find a \emph{gauge-invariant} regularization. Our regularization
has no exact gauge invariance on the lattice. Instead, we have only
approximate gauge invariance, modulo even lattice shifts. Thus, our
regularization is not in contradiction with the various no-go theorems
for regularizations with exact gauge invariance.

This answer leads, in a natural way, to a second objection. Last not
least, people have tried to find regularization with exact gauge
invariance not just for fun, but for a good reason --- to quantize
chiral gauge fields. The problem is that the standard procedure to
quantize gauge fields --- BRST quantization --- depends essentially on
exact gauge invariance of the theory. Without exact gauge invariance,
it fails miserably. What remains is a non-unitary theory.

But this failure is a special problem of the manifestly Lorentz-
covariant Gupta-Bleuer approach to gauge field quantization, which
starts with an indefinite Hilbert space structure. Following Gupta
\cite{Gupta} and Bleuer \cite{Bleuer}, in the BRST approach, manifest
relativistic invariance is reached using an unphysical ``big space''
with indefinite Hilbert metric. A physical interpretation of this big
Hilbert space would lead to negative probabilities, which is
nonsensical. To get rid of the states with negative probability,
restriction to an invariant subspace and factorization is used. But
these operations depend on exact gauge invariance. If gauge invariance
fails, the result is fatal for the whole approach.

But there is an alternative --- the earlier approach of Fermi
\cite{Fermi} and Dirac \cite{Dirac}, where the Hilbert space is
definite, but Lorentz covariance is not explicit. Whatever may go
wrong, the Hilbert space remains definite, and at least a probability
interpretation of the results is possible.

In our approach, weak gauge fields appear, in the large distance
limit, as effective fields.
\footnote{Note that, instead, Wilson gauge fields are located on the
links between lattice nodes, their related degrees of freedom are not
part of the basic, fermionic theory. Instead, the weak gauge fields,
described by geometric coefficients of the lattice itself, are already
fixed if the lattice itself is fixed. Thus, the related theory appears
as an effective field theory.}
The fundamental theory is not obliged to contain them explicitly. As
far as we have developed it until now, is a well-defined and unitary
quantum theory, a variant of multi-particle Schr\"{o}dinger theory.
Whatever the complications connected with symmetry breaking, we would
not switch to an indefinite Hilbert space, because the main advantage
of this choice --- manifest Lorentz invariance --- cannot be reached
in our approach anyway.

This answer leads to the next objection: The manifestly Lorentz-
covariant approach has not been introduced without reason too.
Manifest Lorentz covariance, on one hand, simplifies computations.
This is, obviously, not a decisive argument. More serious is that, in
our approach, we do not have relativistic invariance. Indeed, our
construction from the start violates Lorentz covariance, and in many
different ways: First, we handle time and space in different ways,
having a lattice only in space. Then, even a spacetime lattice would
violate the symmetry of the continuous limit. Moreover, the operators $
\sigma_{ij}$, for spatial spinor rotations on our staggered lattice,
are nonlocal, and, therefore, do not define an exact representation of
the algebra $\mathfrak{su}(2)$. An approach, which violates Lorentz
invariance on the fundamental level, has to explain, how it will be
recovered in the large distance limit.

Fortunately, this question has been, at least partially, addressed by
the derivation of the Einstein equvalence principle (which includes
local Lorentz covariance) in our theory of gravity \ref{Gravity}.
Because of the importance of this question we give an introduction
into this theory in appendix \ref{Gravity}.

\section{Discussion}

Many questions have to be left to future research. This includes:

\begin{itemize}

\item Symmetry breaking;

\item The search for a Hamilton operator for a general configuration
of cells, which would allow the derivation for other regular
crystallographic lattices as well as for lattices with deformations
and defects;

\item The large distance limit, especially renormalization group
equations;

\item The connection between the SM lattice model and the theory of
gravity, which are metaphysically compatible, but mathematically yet
unrelated theories; In the large distance limit of our cellular
lattice, we will obviously have notions like density $\rho$, average
velocity $v^i$ and some stress tensor $\sigma^{ij}$.

\end{itemize}

Especially symmetry breaking promises to be interesting. First, we
need it. The \E\/ action does not define a symmetry of the SM. The SM
mass terms clearly violate the rotational symmetry between the three
generations. Moreover, the whole construction of section
\ref{fermionQuantization}, which creates effective \B-fields from the
original \R-fields, violates translational symmetry: We cannot add
constants to \B-valued fields. Thus, to obtain the Hamiltonian of the
SM, even to obtain fermion fields at all, we have to break \E\/
symmetry. Euclidean symmetry is also broken by the EM field, which
prefers the direction associated with $I_3$, which is also associated
with a direction in space.

On the other hand, some arguments for the standard SM theory of
symmetry breaking seem to fail. Especially, it is not clear, if we
need some Higgs sector for symmetry breaking. One major argument ---
that spatial symmetry is not broken --- fails in our geometric
interpretation of the SM fields. Then, we do not need a fundamental
theory, with unbroken gauge symmetry, for electroweak gauge fields.
Our theory has no lattice gauge symmetry for the weak gauge fields
from the start. Note that Wilson gauge fields, with their exact
lattice gauge symmetry, correspond to massless gluons, while the weak,
chiral fields, which do not allow exact lattice gauge invariance,
appear to be massive. Thus, we don't need symmetry breaking to break
an exact fundamental gauge symmetry.

Thus, we need symmetry breaking, but for very different reasons, and a
different symmetry. Thus, the symmetry breaking may be expected to be
very different from the SM symmetry breaking approach.

Despite these open questions, our simple cellular lattice model
already allows to describe kinematically all SM particles observed so
far, and it is compatible with a metric theory of gravity with GR
limit.

\begin{appendix}

\section{Gravity} \label{Gravity}

For metric theories of gravity there is a simple way to obtain a
condensed matter interpretation, closely related to the ADM
decomposition \cite{ADM} or the geometrodynamic interpretation
\cite{Wheeler}. The preferred frame defines an ADM decomposition of
the four-metric $g_{\a\b}$ into a scalar field, a three-vector and a
definite three-metric. We identify these fields with density $\rho$,
velocity $v^i$ and stress tensor $\s^{ij}$ of some form of condensed
matter in the following way:

\begin{eqnarray}
\nonumber g^{00}\sqrt{-g} &=3D& \rho, \\
\label{ADM} g^{0i}\sqrt{-g} &=3D& \rho v^i, \\
\nonumber g^{ij}\sqrt{-g} &=3D& \rho v^i v^j - \s^{ij}.
\end{eqnarray}

For these condensed matter fields, we would like to have continuity
and Euler equations:

\begin{eqnarray}
\label{continuity} \pd_t \rho + \pd_i (\rho v^i) &=3D& 0 \\
\label{Euler} \pd_t (\rho v^i) + \pd_i(\rho v^i v^j - \s^{ij})
&=3D& 0.
\end{eqnarray}

They coincide with the harmonic conditions for the metric (\ref{ADM}):

\begin{equation}
\label{harmonic} \pd_\a (g^{\a\b}\sqrt{-g}) =3D 0.
\end{equation}

This condensed matter interpretation is, therefore, possible for all
metric theories of gravity, which include the harmonic condition as a
physical equation. A simple theory with this property is general
relativity in harmonic gauge. One variant of GR in harmonic gauge is
to add a non-covariant term to the GR Lagrangian which enforces the
harmonic conditions:

\begin{equation} \label{L}
L=3D\Xi_\a g^{\a\a}\sqrt{-g}+ L_{GR}(g^{\a\b},\psi^{matter})
\end{equation}

For some constants $\Xi_{\a}$. Its dependence on the preferred
coordinates $X^\a(x)$ can be made explicit
\footnote{The dependence of some expression on the preferred
coordinates $X^\a(x)$ is, by definition, explicit, if, after a formal
replacement of occurrence of $X^\a(x)$ by four scalar fields $Y^\a(x)
$, the resulting expression is covariant. So, in $a^0$ a replacement
$X^\a(x)\to Y^\a(x)$ changes nothing, the resulting expression $a^0$
is not covariant, thus, the dependence on the preferred coordinates is
implicit. Instead, in the form $a^\mu X^0_{,\mu}$, the replacement
gives the expression $a^\mu Y^0_{,\mu}$, which is covariant, because
$Y^0$ is considered as a scalar field.}

\begin{equation} \label{Lfull}
L=3D \frac{-1}{2} \Xi_{\g} g^{\a\b}X^\g_{,\a}X^\g_{,\b}\sqrt{-g}+
L_{GR}(g^{\a\b},\psi^{matter})
\end{equation}

This explicit form is useful because it allows variation over the
preferred coordinates. We have to take care --- the four functions $X^
\a(x)+\delta X^\a(x)$ have to define a valid system of coordinates ---
but nonetheless variation is possible and gives Euler-Lagrange
equations for the $\Xi^\a$ of the same form as for usual fields. We
obtain:

\begin{equation}\label{EL}
\frac{\delta S}{\delta X^\g} =3D
\Xi_\g \pd_{\b} (g^{\a\b}\sqrt{-g} \pd_a X^{\g}),
\end{equation}

thus, the preferred coordinates $X^\a$ are harmonic. The Lagrangian
(\ref{L}) obviously defines a metric theory of gravity with Einstein
equivalence principle. In the limit $\Xi_\a\to 0$ we obtain the
Einstein equations. The terms $g^{\a\a}\sqrt{-g}$ do not depend on
partial derivatives of the metric, therefore the limit $\Xi_\a\to 0$
is natural for small distances and weak fields.

But, as long as we simply postulate the Lagrangian (\ref{Lfull}), we
have no explanation for these properties. The classical argument
against the Lorentz ether may be raised: It needs some conspiracy,
does not give and explanation for relativistic symmetry. Is it
possible to derive this Lagrangian from some postulates which are more
natural for a theory with preferred frame?

\begin{theorem} The Lagrangian (\ref{Lfull}) follows from the
following two conditions:

\begin{eqnarray}
\label{No}
\frac{\delta S}{\delta X^0} &=3D& \Xi_0 (\pd_t\rho+\pd_i(\rho v^i))\\
\label{Ni}
\frac{\delta S}{\delta X^i} &=3D& \Xi_i (\pd_t(\rho v^i)+\pd_j(\rho v^i
v^j-
\s^{ij}))
\end{eqnarray}

\end{theorem}

Indeed, given (\ref{ADM}), the equations (\ref{No}), (\ref{Ni}) are
equivalent to (\ref{EL}). The general solution of (\ref{EL}) is
defined by a particular solution (given by the first, non-covariant
term of (\ref{Lfull})) and the general solution of the homogeneous
problem

\begin{equation}
\frac{\delta S}{\delta X^\a} =3D 0,
\end{equation}

thus, modulo a covariant Lagrangian. The covariance of the Lagrangian
is what we take here as the definition of the Lagrangian $L_{GR}$ of
general relativity\footnote{Note that we use here the most general
understanding of general relativity, where the Einstein-Hilbert
Lagrangian is only the lowest order term, and higher order terms, or
terms with higher order derivatives in the metric, are, in principle,
allowed, as long as they are covariant. This understanding is standard
for effective field theories --- in the large distance limit, only the
lowest order terms survive.} in (\ref{Lfull}) \qed.

Now, to postulate the equations (\ref{No}), (\ref{Ni}) does not
require much conspiracy. Instead, they can be seen as a combination of
the Noether theorem with the standard interpretation of continuity and
Euler equations, as conservation laws for energy and momentum in
condensed matter theories. Indeed, if the Lagrangian, in its explicit
form, has a symmetry $X^\a\to X^\a+c$, the Euler-Lagrange equation for
the preferred coordinates does not depend on the $X^\a$ themself, but
only on its partial derivatives. In this case, the Euler-Lagrange
equation automatically obtains the form of a conservation law:

\begin{equation} \label{Noether}
\frac{\delta S}{\delta X^\a} =3D -\pd_{\b}\left(\frac{\pd L}{\pd_{\b}X^
\a} +
\ldots\right)
\end{equation}

Thus, the left hand side of (\ref{No}), (\ref{Ni}) define the Noether
conservation laws related with translation in time and space. On the
right hand side we have the continuity and Euler equations --- the
conservation laws for energy and momentum in condensed matter theory.
To identify left and right hand sides is a very natural postulate for
a condensed matter theory.

More details and consequences of this theory of gravity can be found
in \cite{GLET}. Especially, the gauge-breaking term stops (for the
correct sign of the constants) the black hole collapse and prevents
the big bang singularity. The condensed matter approach to gravity
solves many quantization problems of GR quantization: The notorious
``problem of time'' \cite{Isham} simply disappears. Together with the
black hole collapse the related information loss problem
\cite{Preskill} disappears too.

\section{The Dirac operator on $\Lambda(\R^d)$}\label{DiracCurved}

Let's remember the basic formulas for the Dirac operator in the
exterior bundle (see, for example, \cite{Pete}). The exterior bundle
or de Rham complex $\Lambda =3D \sum_{k=3D0}^d \Lambda^k$ consists skew-
symmetric tensor fields of type $(0,k), 0\le k \le d$ which are
usually written as differential forms
\begin{equation}
\psi =3D \psi_{i_1\ldots i_k} dx^{i_1}\wedge \cdots \wedge dx^{i_k} \in
\Lambda^k
\end{equation}
The exterior bundle $\Lambda$ has dimension $2^d$ in the d-dimensional
space. The most important operation on $\Lambda$ is the external
derivative $d:\Lambda^k\to\Lambda^{k+1}$ defined by
\begin{equation}
(d\psi)_{i_1\ldots i_{k+1}}=3D\sum_{q=3D1}^{k+1}\frac{\partial}{\partial
x^{i_q}}
(-1)^q \psi_{i_1\ldots \hat{i}_q\ldots i_{k+1}}
\end{equation}
where $\hat{i}_q$ denotes that the index $i_q$ has been omitted. It's
main property is $d^2=3D0$. In the presence of a metric, we have also
the important $\star$-operator $\Lambda^k\to\Lambda^{d-k}$:
\begin{equation}
(\star\psi)_{i_{k+1}\ldots i_d} =3D \frac{1}{k!} \varepsilon_{i_1\ldots
i_d}
g^{i_1j_1} \cdots g^{i_kj_k}\psi_{j_1\ldots j_k}
\end{equation}
with $\star^2 =3D (-1)^{k(d-k)}\mbox{sgn}(g)$. This allows to define a
global inner product by
\begin{equation}
(\phi,\psi) =3D \int \phi \wedge (\star\psi) =3D \int \psi \wedge (\star
\phi)
\end{equation}
It turns out that the adjoint operator of $d^*: \Lambda^k\to
\Lambda^{k-1}$ of $d$ is
\begin{equation}
d^* =3D (-1)^{kd+d+1} \star d \star
\end{equation}

Note that the expressions for $\star^2$ and $d^*$ depend on the order
of the form $k$, which is not nice. But a minor redefinition of the $
\star$ operator allows to solve this problem. For the operator
\begin{equation}\label{astdef}
\ast =3D i^{k(d-k)}\star
\end{equation}
the resulting expressions no longer depend on $k$:
\begin{equation}
\ast^2 =3D \mbox{sgn}(g), \qquad d^* =3D (-i)^{d+1} \ast d \ast
\end{equation}

In this general context we can define the Laplace operator as
\begin{equation}
\Delta =3D d d^* + d^* d.
\end{equation}
Then, the Dirac operator (as it's square root) can be defined as
\begin{equation}
D =3D d+d^*,
\end{equation}
so that $\Delta=3DD^2$. Indeed, we have $d^2=3D0$ as well as $(d^*)^2=3D0$.

\subsection{Discretization of the Dirac operator}
\label{DiracLatticeCurved}

The special geometric nature of the exterior bundle allows to define a
nice doubler-free discretization of the Dirac equation on a general
cell complex. Such a cell complex consists of cells $c_i$ of dimension
$k=3Ddim(c_i)$, which are embeddings of the $k$-dimensional unit cube
$I^k$ into the manifold so that the image of the boundary is part of
the image of lower-dimensional cells of the complex, and the image of
all cells of the cell complex covers the whole manifold.

On such a cell complex, $k$-dimensional differential forms are
represented on the lattice by their integrals over the $k$-dimensional
cells $c_i$ of the cell complex:

\begin{equation}
\Psi \to \{\psi_i\}, \psi_i =3D \int_{c_i} \Psi
\end{equation}

The external derivative defines in a similar natural way a derivative
for functions on the mesh, with the same most important exact property
$d^2=3D0$.

For the definition of the $\star$-operator we need a metric and a dual
mesh. A metric $g_{\mu\nu}$ on the manifold defines in a natural way
for every cell $c_i$ it's area $a_i=3Da(c_i)> 0$. For a triangulation
on Euclidean background, the values $\a_i$ depend on each other. But
in the general case they may be considered as independent variables,
which approximate the metric on the cell complex. In the following we
consider them as given and defining the metric.

A dual mesh is a mesh with cells $\hat{c}_i$ so that for each cell $c_i
$ of the original mesh with dimension $k$ we have a corresponding
``dual cell'' $\hat{c}_i$ of dimension $d-k$ which intersects only the
cell $c_i$, in a single point, orthogonally and with positive
intersection index. The metric defines the areas $\hat{a}_i$ of the
dual cells in a similar way. Now, the lattice Hodge $\star$-operator
may be defined as

\begin{equation}
\star \psi_i =3D \frac{\hat{a}_i}{a_i} \hat{\psi}_i
\end{equation}

and maps functions on the mesh to functions on the dual mesh. Note
that the dual of the dual mesh has the same cells as the original
mesh, but possibly with different orientations. Therefore, for $
\star^2$ we obtain an additional factor $(-1)^{k(d-k)}$ as in the
continuous case.

Thus, we can define the exterior derivative as well as the Hodge $\star
$ operator on the lattice preserving their algebraic properties
$d^2=3D0$, $\star^2=3D(-1)^{k(d-k)}$. As a consequence, the remaining part
of the theory can be transferred on the lattice too.

It is a general consequence of the geometric character of the
continuous Dirac equation as well as its lattice discretization that
the lattice discretization does not have doublers. See, for example,
\cite{Becher}.

\end{appendix}

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\end{document}