| Topic: |
Science > Physics |
| User: |
"Ilja Schmelzer" |
| Date: |
17 May 2006 12:41:31 AM |
| Object: |
Cellular ether theory (large TeX) |
%% The current working version of my cellular ether theory
%% Ilja Schmelzer
\documentclass{amsart} % article
\begin{document}
\newtheorem{theorem}{Theorem}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\newcommand{\g}{\gamma}
\newcommand{\D}{\Delta}
\renewcommand{\d}{\delta}
\newcommand{\e}{\varepsilon}
\renewcommand{\i}{\iota}
\newcommand{\w}{\mbox{$\omega$}}
\newcommand{\s}{\mbox{$\sigma$}}
\newcommand{\alg}[1]{\mathtt{#1}}
\renewcommand{\k}{\kappa}
\renewcommand{\l}{\lambda}
\newcommand{\m}{\mu}
\renewcommand{\qed}{\mbox{$\square$}} %%
\newcommand{\pd}{\mbox{$\partial$}} %%
\newcommand{\Z}{\mbox{$\mathbb{Z}$}}
\newcommand{\R}{\mbox{$\mathbb{R}$}}
\newcommand{\C}{\mbox{$\mathbb{C}$}}
\renewcommand{\H}{\mbox{$\mathbb{H}$}}
\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{$U(1)_{\g}$ }} %% the electromagnetic field
\newcommand{\Uem}{\mbox{$U(1)_{\tilde{\g}}$}} %% charge I_3-1/2
\newcommand{\SUc}{\mbox{$SU(3)_c$}} %% the SM color group
\newcommand{\SUL}{\mbox{$SU(2)_L$}} %% SM weak bosons
\newcommand{\A}{\mbox{$A(3)$}} %% 3-dim. affine group
\newcommand{\E}{\mbox{$E(3)$}} %% 3-dim. Euclidean group
\newcommand{\T}{\mbox{T}} %% The lattice (distorted)
\newcommand{\U}{\mbox{$\Z^3$}} %% The lattice (undistorted)
\newcommand{\n}{\mbox{$\vec{n}$}} %% a lattice node
\newcommand{\CA }{\mbox{$(\C \otimes A(3))(\T)$}}
\newcommand{\CAU}{\mbox{$(\C \otimes A(3))(\U)$}}
\newcommand{\Cons}{\mbox{C}} %% constant pointwise operators on \O
\newcommand{\Cl}{\mbox{$Cl(3,3,\R)$}} %% the Clifford algebra
\newcommand{\p}{\mbox{$p$}} %% momentum variables
\newcommand{\q}{\mbox{$q$}} %% configuration variables
\newcommand{\z}{\mbox{$\psi$}} %% q + i p
\renewcommand{\v}{\mbox{$\varphi$}} %% octet variables
\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{$\hat{0}$}}
\newcommand{\lll}{\mbox{$\hat{1}$}}
\newcommand{\Tc}{\mbox{$\T_c$}} %% The coarse sublattice
\newcommand{\f}{\theta} %% lattice node factor
\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
\renewcommand{\P}{\mbox{P}} %% pointwise, [.,D]=0 \cal{P}
\renewcommand{\S}[1]{\mbox{$\Sigma_{#1}$}} %% staggered sublattice
\newcommand{\Sl}{\S{\l}} %% ... with index \l
%%\renewcommand{\;}{\hfill} %% {\hspace{1cm}}
\renewcommand{\;}{\hspace{1cm}}
\newcommand{\follows}{\hspace{0.6cm}\Longrightarrow\hspace{0.6cm}}
\title{Cellular ether theory}
\author{I. Schmelzer}
\maketitle
\begin{abstract}
The ether theory proposed here is based on two observations: If we
discretize only in space, leaving time continuous, naive fermion doubling
gives an octet and staggered fermions a doublet of fermions, which allows a
physical interpretation. Then, SM gauge fields preserve a global 3D
Euclidean symmetry (rotations between generations, additive shifts on
right-handed neutrinos as translations).
Because of fermion doubling, a lattice Dirac equation on the phase space of
a 3D lattice of cells, whose configuration is described by a 3D affine
deformation, gives all twenty four SM fermions. Gauge fields describe
defects of this lattice. The two simplest types of defects, restricted by
Euclidean symmetry, give the gauge group $U(3)_c \oplus U(2)_L \oplus
U(1)_{em}$.
The gravitational field may be defined, in a variant of ADM decomposition,
by density, velocity and stress tensor of this ether. The postulate that
continuity and Euler equations appear as energy and momentum conservation
laws in a variant of Noethers theorem gives the GR Lagrangian with a
harmonic gauge breaking term.
\end{abstract}
\section{Introduction}
We propose here a lattice theory which promises to give, in the large
distance limit, fermions and gauge fields of the standard model of particle
physics (SM), as well as an effective metric theory of gravity with GR
limit.
To name such a theory ``ether theory'' seems to be a very unfortunate
choice.
Last not least, it refers to a class of theories which have been abandoned a
century ago, long before the development of the standard model, and has been
compromised later by a lot of junk science, characterized by elementary
misunderstandings of special relativity, often (for example in nazistic
``German physics'') connected with antisemitism.
Having nothing in common with this type of crank science, our theory has
very much in common with the old scientific program to find a mechanical
ether theory --- a program which may be reformulated in modern language as a
special, condensed-matter-like, regularization of field theories with
special geometric interpretation and well-defined symmetry properties:
\begin{itemize}
\item The regularization has a preferred frame. Relativistic symmetry is not
fundamental, but appears only in the large distance limit.
\item The regularization gives discrete objects changing their positions
continuously in time. Thus, we have a discretization in space, but not in
time.
\item The degrees of freedom in each node have a three-dimensional geometric
interpretation, as positions or velocities of some fundamental entities
(atoms, elementary cells) in the three-dimensional Euclidean space. This
defines a global action of the three-dimensional Euclidean group \E, on the
lattice as well as in the continuous limit, which may be only spontaneously
broken.
\end{itemize}
For a theory with these properties it would be deception not to name it an
ether theory. On the other hand, despite the bad name of ether theory, there
is nothing wrong with theories of this type. Instead, there is something to
be proud of: The mechanical ether defines a high standard of simplicity and
explanatory power --- all observable fields have to be explained in a very
special geometric way, without introducing additional dimensions or inner
degrees of freedom. To reject this standard remembers soar grapes.
Let's consider now the content of our ether theory. The global action of
\E\/ can be easily identified in the standard model: The group of rotations
$O(3)$ acts on the three generations of the standard model. Translations act
as additive shifts on right-handed neutrinos. This global symmetry is broken
only by the mass terms, which are considered to be a consequence of
spontaneous symmetry breaking already in the SM. That the SM gauge fields
preserve this \E\/ symmetry explains a lot of characteristic properties of
the SM: The symmetry between the three generations, the difference between
quarks and leptons ($SU(3)_c$ instead of $SU(4)_c$ action with lepton
charge as a fourth color), the chiral character of weak interactions
($SU(2)_L$ instead of the $SU(2)_L\oplus SU(2)_R$ left-right symmetric
Pati-Salam model \cite{PatiSalam}), and the difference between leptons and
neutrinos ($Q$ instead of, say, $I_3$ as the EM charge).
A completely independent hint in favour of an ether theory is related with
the fermion doubling problem(cf. \cite{lattice}): A naive discretization of
the Dirac equation on $\C^4(\R^4)$ gives a lattice Dirac equation on
$\C^4(\Z^4)$.
But it's continuous limit is not the original equation on $\C^4(\R^4)$, but
on $(\C^4)^{16}(\R^4)$ --- the equation for sixteen Dirac fermions called
doublers. We obtain a doubling factor two in each direction of spacetime.
Now, if we discretize in space only, we obtain a lattice evolution equation
on $\C^4(\Z^3\times\R)$. The same doubling effect gives now only eight
fermions, as much as we need for an SM octet.
There is an interesting variant known as ``staggered fermions''
\cite{Kogut},
where we have a decomposition of lattice equation into four equations on
four ``staggered'' sublattices. In the continuous limit, this gives a
decomposition $(\C\otimes\Lambda)^4(\R^4)$ into four groups. Each group is
described by the Dirac-K\"ahler equation \cite{Kaehler} on
$\C\otimes\Lambda(\R^4)$, which describes four Dirac fermions.
In our approach, we have a similar decomposition of the octet into four
groups $(\C\otimes\Lambda)^4(\R^3)$, based on an evolution equation on the
three-dimensional bundle $\C\otimes\Lambda(\R^3)$ equivalent to the Dirac
equation for two fermions. Thus, we have only two Dirac fermions in each
group, similar to the decomposition of the SM octets into four electroweak
doublets.
\footnote{Three-dimensional geometric representations of the Dirac equation
have been considered independently by Daviau \cite{Daviau}. The idea to
connect geometric fermions with electroweak doublets has been proposed by
Hestenes \cite{Hestenes}.}
This observation allows to use fermion doubling as a tool instead of a bug:
To define the Dirac equation for a whole octet, we need only one ``naive''
Dirac fermion on $\C^4(\Z^3\times\R)$, and $(\C^4)^3(\Z^3\times\R)$ for all
standard model fermions.
Now we can identify $(\C^4)^3$ with $\C\otimes A(3)$, which defines a
natural left action of \E\/ on it. Thus, we can describe all SM fermions
with a first order Hamiltonian evolution equation on \CAU. But \CAU,
together with the \E\/ action on it, is the phase space for a very simple
ether model: A simple lattice $\Z^3$ of cells. Indeed, it is natural to
describe the configuratin of a cell by it's deformation from a standard
reference cell, which is an element of \A. Thus, we can describe the SM
fermions using a simple 3D lattice of cells.
In condensed matter theories gauge fields are used to describe lattice
defects. The related gauge groups have to preserve the global \E\/ symmetry.
In our cellular ether theory, lattice defects may be decomposed into two
types of defects: Defects of the basic lattice (the lattice of points
defined by the cell centers), and defects of the cell configurations which
leave the basic lattice unchanged.
The latter may be described by Wilson lattice gauge fields. This allows to
define an $U(12)$ vector action with exact gauge invariance on the lattice.
The part of this $U(12)$ action which preserves \E\/ symmetry is $U(3)$. Its
subgroup $SU(3)\subset U(3)$ has all properties of $SU(3)_c$: It is a vector
action, particles with the same charge appear in doublets, it acts trivially
on three doublets (the leptons) and preserves generations. The leptons may
be identified with the degrees of freedom which describe the cell centers.
The other type of defects acts on cells as if they were points, and will be
defined by its action on the leptonic sector, which has to be preserved.
This gives the maximal gauge group $U_L(6)\oplus U_R(6)$. The maximal
subgroup which may be preserved by an \E\/ action is $\UL \oplus \Uem$.
Here, with \Uem\/ we denote the gauge group with charge $I_3-1/2$, which
coinsides with the EM field on the leptonic sector. It's difference to \Ue\/
is the baryon charge. It is nice to have, together with $SU(3)_c$, also the
diagonal of \Uc, which has this charge, because we obtain now
\be
\SUc \oplus \SUL \oplus \Ue \subset \Uc \oplus \UL \oplus \Uem,
\ee
thus, all standard model gauge field may be described by these two types of
gauge fields. The additional two abelian gauge fields are diagonal and do
not lead to additional particle decays.
To incorporate gravity is also possible. A three-dimensional geometric
interpretation for metric theories of gravity is already known, as ADM
decomposition \cite{ADM}: For a fixed choice of a foliation of spacetime,
the gravitational field decomposes into a scalar field, a three-vector
field, and the a three-metric. We interpret them as density, velocity, and
stress tensor of the ether. A hint that this is a very natural
identification is that continuity and Euler equations translate into the
harmonic conditions.
For a Lagrange formalism it seems natural to postulate that continuity and
Euler equations should appear as Noether conservation laws. This allows to
define the general form of the Lagrangian. The next surprise is that this
general Lagrangian appears to be the Lagrangian of general relativity in
harmonic gauge. Thus, we obtain a metric theory of gravity with exact
Einstein equivalence principle and with the Einstein equations of general
relativity in a natural limit.
Thus, in the classical domain we have, at the kinematic level, a
surprisingly simple ether theory --- a simple cellular lattice, with
defects. It allows to describe all observed until now fields --- fermions
and gauge fields of the SM and metric gravity. From point of view of the
ether theory presented here, the failure of old ether theory is easy to
explain: At that time, nothing except the EM field was known from the
standard model, and nothing except the Newtonian limit about the
gravitational field. With this incomplete information, last century ether
theory has not had a chance. Moreover, an ether theory at that time would
have had much less explanatory power: The ether would have allowed to
explain the fundamental nature of four field components $A_i$ of the EM
field and, possibly, one potential $V$ of gravity. The ether theory
presented here does this for $24 \times 8 = 192$ components of fermion
fields, $12 \times 4 = 48$ components of gauge fields, and $10$ components
of the gravitational field, thus, $250$ different field degrees of freedom.
Some questions have to be left to future research. We need a different
understanding of fermion quantization rules, because we need classical
instead of Grassmann variables as the classical limit of fermions. We need a
theory of spontaneous symmetry breaking, which gives mass to fermions and
gauge fields. We need a general Lagrangian for a lattice with defects, which
allows to derive the dynamics of gauge fields and gravity in the continuous
limit.
On the other hand, lot's of serious quantization problems in the standard
relativistic approach, like the problem of time in quantum gravity,
disappear in our approach.
\section{Free fermions}
We use latin indices $1\le i,j,k,\ldots\le 3$ for spatial indices, the first
greek indices $0\le\a,\b\,\ldots\le 3$ for four-indices, and the greek
indices $\k,\l,\m,\ldots$ for indices in eight-dimensional spaces.
\subsection{Kinematics of cellular ether theory}
The ``cellular ether'' is a spatial lattice with a small cell in each
lattice node. The lattice $\T$ is a three-dimensional lattice in Euclidean
space. Below we consider lattice defects, but in the regular case $\T = \U$,
the lattice may be characterized in the following way:
\be
\T = \{x \in \R^3 | \f_i(x)= e^{i\pi(h_i,x)} = \pm 1 \},
\ee
where the three vectors \hi (which are not necessarily orthogonal) are the
basic lattice vectors. We can distinguish the ``coarse lattice'' as the
sublattice defined by
\be
\Tc = \{x \in \R^3 | \f_i(x)= e^{i\pi(h_i,x)} = 1 \}.
\ee
Shift which preserve the coarse sublattice will be named even shifts. The
lattice function
\be
\f(\n)=\f_1(\n)\f_2(\n)\f_3(\n)
\ee
subdivides the lattice into even and odd sublattices. With $\ti=\ti^+$,
$\ti^-$ we denote the shift operators:
\be
(\ti^\pm\v)_\k(\n) = \v_\k(\n\pm\hi).
\ee
The configuration $\q$ of each cell is described by a single
three-dimensional affine transformation $\q\in\A$ from some standard
reference cell. The deformation from the reference cell may be described by
a $3\times 4$ matrix $\q^i_\a$. The configuration space of the theory is
therefore $\A(\T)$, described by lattice functions $\q^i_\a(\n)\in\R$,
$\n\in\T$. The phase space also contains the corresponding momentum
variables $\p^i_\a(\n)\in\R$. Taken together, they form a $3\times 8$ matrix
of real lattice functions $\v^i_\k(\n)\in\R$.
We have a natural action of the Euclidean group on this lattice space. Its
representation depends on the physical meaning of the components $\q^i_\a$.
If $\q^i_0$ defines the translation of the center of the reference cell, and
$\q^i_j$ for $j>0$ a linear deformation around the center, we have the
following action:
\be
\q^i_\a \to e^i_k \q^k_\a + \delta_{0\a}e^i_0,\; \p^i_\a \to e^i_k \p^k_\a.
\ee
But it may be preferable to describe the state of the cell by the positions
of four points $\q_\a$. In this case, translations act in a slightly
different way:
\be
\q^i_\a \to e^i_k \q^k_\a + e^i_0, \; \p^i_\a \to e^i_k \p^k_\a.
\ee
What these possibilities (and other imaginable variants) have in common is
that we have three ``directions of translation'' $\c^i$ transformed by
Euclidean rotations $e^i_k$ into each other so that
\be
\v^i_\k \to e^i_j \v^j_\k + e^i_0 c^i_\k.
\ee
Time is a continuous evolution parameter. The general Lagrangian of the
theory is of the form
\be
L(t) = \sum_{\n\in\T} \p^i_\a(\n) \dot{\q}^i_\a(\n) - H
\ee
where the Hamiltonian $H$ does not depend on time. We assume that the phase
space has also a standard complex structure so that the variables
$(\q^i_\a(\n),\p^i_\a(\n))$ define a complex plane. The phase space of our
cellular ether theory is therefore \CA.
We have a decomposition $\CA\cong \O^3$ into subspaces \O\/ described by
eight real lattice functions $\v_\k(\n) \in \R$. The subspaces \O\/ will be
identified with standard model generations. Because the particles in
different generations have different masses, Euclidean symmetry is broken.
We assume that the mass terms appear only after some spontaneous symmetry
breaking, and omit them. This gives exact rotational symmetry between the
three generations. Inside a single generation, Euclidean symmetry reduces to
translational symmetry in one direction of translation \c:
\be
\v_\k \to \v_\k + e_0 c_\k.
\ee
\subsection{Clifford algebra representation}
The real coordinates are useful to describe the algebra \Cons\/ of
node-preserving constant real operators on \O. We have $\Cons\cong
M_8(\R)\cong\Cl$. For the generators $\a^i,\b^i$:
\be
\{\a^i,\a^j\} = 2\d^{ij}, \;
\{\a^i,\b^j\} = 0, \;
\{\b^i,\b^j\} = -2\d^{ij},
\ee
we use the following (standard) representation:
\be
a_i\a^i\v = \\
\left(\begin{array}{cccccccc}
& a_3 & a_2 & & a_1 & & & \\
a_3 & & & a_2 & & a_1 & & \\
a_2 & & & -a_3 & & & a_1 & \\
& a_2 & -a_3 & & & & & a_1 \\
a_1 & & & & & -a_3 & -a_2 & \\
& a_1 & & & -a_3 & & & -a_2 \\
& & a_1 & & -a_2 & & & a_3 \\
& & & a_1 & & -a_2 & a_3 &
\end{array}\right)
\left(\begin{array}{c}
\v_{000}\\
\v_{001}\\
\v_{010}\\
\v_{011}\\
\v_{100}\\
\v_{101}\\
\v_{110}\\
\v_{111}
\end{array}\right),
\ee
\be
b_\mu\b^\mu\v= \left(\begin{array}{cccccccc}
b_0 & -b_3 & -b_2 & & -b_1 & & & \\
b_3 & -b_0 & & -b_2 & & -b_1 & & \\
b_2 & & -b_0 & b_3 & & & -b_1 & \\
& b_2 & -b_3 & b_0 & & & & -b_1 \\
b_1 & & & & -b_0 & b_3 & b_2 & \\
& b_1 & & & -b_3 & b_0 & & b_2 \\
& & b_1 & & -b_2 & & b_0 & -b_3 \\
& & & b_1 & & -b_2 & b_3 & -b_0
\end{array}\right)
\left(\begin{array}{c}
\v_{000}\\
\v_{001}\\
\v_{010}\\
\v_{011}\\
\v_{100}\\
\v_{101}\\
\v_{110}\\
\v_{111}
\end{array}\right).
\ee
The diagonal operator $\b^0=(-1)^{\deg{\k}}$ may be used to define a dual
set of generators:
\be
\a_i=\b^0\b^i,\; \b_i=\b^0\a^i,
\ee
\be
\{\a_i,\a_j\} = 2\d_{ij}, \;
\{\a_i,\b_j\} = 0, \;
\{\b_i,\b_j\} = -2\d_{ij}.
\ee
Note that
\be
\b^0 = \a_1\b_1\a_2\b_2\a_3\b_3 = \a^1\b^1\a^2\b^2\a^3\b^3.
\ee
\subsection{The Dirac equation on the exterior bundle}
\newcommand{\CAL}{\mbox{$(\C \otimes A(3) \otimes \Lambda)(\R^3)$ } }
\newcommand{\CL }{\mbox{$(\C \otimes \Lambda)(\R^3)$ } }
\renewcommand{\L }{\mbox{$\Lambda(\R^3)$ } }
\newcommand{\di }{\mbox{$\partial_i$}}
\newcommand{\co }{\mbox{$e^{(000)}$} }
The space \O\/ may be interpreted as the result of the naive discretization
of the three-dimensional exterior bundle \L\/ on the lattice \T:
\be
\v(x) = \sum_{\k} \v_{\k}(x) e^\k; \;
e^\k = (dx^1)^{\k_1} \wedge (dx^2)^{\k_2} \wedge (dx^3)^{\k_3}
\ee
We have in \L\/ a natural candidate for the neutral direction: $\c = \co$.
The exterior derivative $d$ is defined by
\be
d = (\a^i+\b^i)\di,\; d^2=0.
\ee
Its adjoint operator $d^*=-*d*$, where $*$ is the Hodge star operator, is
defined by
\be
d^* = (\a^i-\b^i)\di,\; (d^*)^2=0.
\ee
These operators define the harmonic operator $\Delta$ and the geometric
Dirac operator $D$ as
\be \label{Dc}
D = d + d^*, \; \Delta = dd^*+d^*d = D^2
\ee
The Dirac operator may be used to define an evolution equation:
\be \label{Dceq}
\partial_t \v = \pm D \v; \; \partial_t^2 \v = \D \v.
\ee
\subsection{The naive lattice Dirac equation}
Let's define now a ``Dirac equation'' on our lattice space \O as a naive
discretization of the continuous Dirac equation (\ref{Dceq}). We simply use
a central differences discretization of the continuous Dirac operator
(\ref{Dc}):
\be
\label{Di} \di \to \D_i = \frac{1}{2} (\ti^+ - \ti^-).
\ee
This gives (on flat background) the following lattice Dirac operator:
\be \label{Dop}
D = \a^i \D_i; \; D^2 = \D = \sum_i (\D_i)^2.
\ee
Time is left continuous. This gives an evolution equation on our cellular
lattice:
\be \label{Deq}
\partial_t \v = \pm D \v; \; \partial_t^2 \v = \D \v.
\ee
\subsection{Decomposition into staggered sublattices}
The operators
\be
\a_i = \b^0 \b^i; \; \g_i=\b^0\a^i\t_i.
\ee
commute with the Dirac operator
\be
[\a_i,\a^j\D_j] = [\g_i,\a^j\D_j] = 0.
\ee
and form a Clifford algebra $\P\cong \Cl$:
\be
\{\a_i,\a_j\} = 2\d_{ij}, \;
\{\a_i,\g_j\} = 0, \;
\{\g_i,\g_j\} = -2\d_{ij}.
\ee
We can define now ``charge operators''
\be
q_i=\a_i\g_i: \; [q_i,q_j]=0, \; (q_i)^2 = 1,
\ee
which allow to define a decomposition into eight common eigenspaces \Sl
\be
\O = \bigoplus_{\l=(\l_1,\l_2,\l_3)\in\{0,1\}^3} \Sl; \;
\Sl = \{\v\in\O | q_i \v = (-1)^{\l_i} \v\}
\ee
The subspaces \Sl\/ appear to be located on the staggered sublattices
defined by the condition $\n+\k+\l=0\bmod 2$:
\be
\Sl = \{ \v \in \O | \v_k(\n)\neq 0 \follows \forall i: n_i+\k_i+\l_i = 0
\bmod 2 \}
\ee
The algebra of lattice operators which preserve the \Sl\/ only approximates,
modulo even shifts, the Clifford algebra \Cl. The approximate generators are
\be
\a^i_\pm=\a^i\ti^\pm; \; \b^i_\pm=\a^i_\pm\f_i.
\ee
Indeed,
\be
\{\a^i_+,\a^j_+\} = 2\d_{ij} (\ti^+)^2, \;
\{\a^i_+,\b^j_+\} = 0, \;
\{\b^i_+,\b^j_+\} = -2\d_{ij} (\ti^+)^2.
\ee
Because they commute with all elements of $\P$:
\be
[\a_i,\a^j_\pm]=[\g_i,\a^j_\pm]=[\a_i,\b^j_\pm]=[\g_i,\b^j_\pm]=0,
\ee
they also preserve the common eigenspaces \Sl\/ of the $\q_i\in\P$. Note
also the decomposition of the Dirac operator
\be
D = \a^i D_i = \a^i_+ D_i^-, \; D_i^- = \frac{1}{2} (1 - (\ti^-)^2),
\ee
into parts $\a^i_+$ and $D_i^-$ which preserve the \Sl. We conclude that the
lattice Dirac equation on each \Sl\/ gives, in the continuous limit, a
complete Dirac equation (\ref{Dceq}) on \L.
As a consequence, the lattice Dirac equation on \CAU\/ gives, in the
continuous limit, the Dirac equation on \CAL: Each real degree of freedom on
a node of the lattice \CAU\/ corresponds to a whole bundle \L.
This effect is known as ``fermion doubling''. In the standard,
four-dimensional, approach, it leads to sixteen doublers. For these sixteen
doublers we have no natural physical interpretation.
\subsection{Doublets}
\newcommand{\Db}[1]{\mbox{$\Theta_{#1}$}} %% doublet
\newcommand{\Da}{\Db{\a}} %% ... with index \a
\newcommand{\Dn}{\Db{\ooo}} %% leptonic doublet
The space \Sl\/ does not have a natural complex structure. Instead, we have
such a natural complex structure on \CAU, defined by the following
operators:
\be
i = \a_1\a_2\a_3 \in \P; \;
C = \a_1\g_1\a_2\g_2\a_3\g_3 \in \P;
\ee
\be
i^2 = -1; \; C^2 = 1; \; \{i,C\} = 0.
\ee
They acts pointwise, but do not preserve the \Sl. Instead, pairs $\Da = \Sl
\oplus \S{\dual{\l}}$ are preserved. In the continous limit we obtain the
bundle \CL, which describes two geometric Dirac fermions. The pairs \Da\/
will be interpreted as electroweak doublets.
This identification is not possible in the four-dimensional approach. The
four-dimensional analogon of the bundle \CL\/ is $\C\otimes\Lambda(\R^4)$.
The Dirac equation on this bundle is known as the ``Dirac-K\"ahler
equation''. Its staggered discretization, the analogon of \Da, is known as
``staggered fermions''. But it describes four Dirac spinors. Therefore an
interpretation in terms of electroweak doublets is not possible.
Thus, in the four-dimensional approach a ``naive'' (non-staggered) lattice
Dirac equation gives sixteen fermions, which decompose into four groups of
four ``staggered'' fermions. No physical interpretation seems possible. If
we leave time continuous, we obtain eight fermions, which decompose into
four groups of two ``staggered'' fermions. The situation cries for a
physical interpretation in terms of octets and doublets of the standard
model.
Note that the physical interpretation of the doublers leads to an essential
simplification of the theory. Instead of ninety six complex degrees of
freedom, which seem to be necessary to describe all fermions of the standard
model, we need only twelf of them on each node. It is this simplification
which allows an identification with the phase space of a simple
three-dimensional cellular lattice.
\subsection{Geometric fermions}
The connection between our Dirac equation (\ref{Dceq}) on \L\/ and the
standard Dirac equation needs some more detailed consideration. Especially
\L\/ is a geometric bundle, while Dirac particles define spinor bundles
which are not geometric. This seems to be a contradiction. Let's, therefore,
consider the symmetries in more detail.
First, remember that our complex structure does not preserve \Sl. Therefore
\L\/ defines only the real part of a doublet. Now, geometric rotations on
\L\/ are generated by the following operators:
\be
\w_{ij} = \a^i\a^j - \b^i\b^j
\ee
They form, indeed, a representation of the commutation relations of
$\alg{so}(3) \cong \alg{su}(2)$, and preserve the degree of a form.
Therefore they define three-dimensional representations on one- and
two-forms. Instead, their parts
\be
\s_{ij} = \a^i\a^j, \; \varepsilon_{ijk} \i_k = \b^i\b^j
\ee
define spinor representations of $SU(2)$. The representation defined by the
$\s_{ij}$ is the standard spinor representation related with the Dirac
equation. The additional part defined by the $\i_k$ commutes with the Dirac
equation and has to be identified with the isospin:
\be
\i_k=2iI_k.
\ee
Now, isospin symmetry is obviously broken. We have a preferred isospin
direction $I_3$. In our geometric interpretation that means that isotropy of
space is also broken. But this follows already from the definition of
Euclidean symmetry: The mass terms give different masses to particles in
different generation, thus, do not preserve Euclidean rotations.
The spinor symmetry $\s_{ij}$ is, therefore, only what remains from
rotational symmetry after spontaneous symmetry breaking. Note that on the
lattice we have only approximations of the $\i_i$:
\be \label{ii}
\i^\pm_1 = \b^2_\pm\b^3_\pm, \;
\i^\pm_2 = \b^3_\pm\b^1_\pm, \;
\i^\pm_3 = \b^1_\pm\b^2_\pm.
\ee
The same holds for the spinor representation $\s_{ij}$, $\s_{0i}=\a^i$: It
defines a representation of the Lorentz group only modulo irrelevant shifts.
\subsection{Geometric inversion}
\newcommand{\gp}{\mbox{$\gamma^5_+$} }
The operator $\g^5=-i\i$ becomes a simple diagonal shift operator:
\be\label{g5p}
\gp \v(\n) = -i\a^1_+\a^2_+\a^3_+ \v(\n) = \v(\n+h_1+h_2+h_3),
\ee
It seems strange that \gp has nothing to do with spatial inversion. To
understand this, we have to consider the complex structure on the doublets
\Da in more detail. It differs from the naive $\C\otimes \Sl$ complex
structure which would be
\be\label{tensor}
i e^\k_\Re = e^\k_\Im, \; i e^\k_\Im = - e^\k_\Re.
\ee
Instead, we have
\be\label{skew}
i e^\k_\Re = \i e^\k_\Im, \; i e^\k_\Im = \i e^\k_\Re.
\ee
with $\i=\a^1_+\a^2_+\a^3_+$ modulo even shifts. Above representations are
algebraically equivalent, their difference is a choice of the basis in the
imaginary part. Nonetheless, the related geometric interpretation differs.
For the geometric inversion we have:
\be
P_g dx^i = - dx^i \follows
P_g \v_\k(x) e^\k = \b^0 \v_\k(-x) e^\k.
\ee
The standard algebraic definition of $P$
\be
P \v_\k(x) e^\k = \g^0 \v_\k(-x) e^\k
\ee
requires the definition of $\g^0$. With $\b^0$, $C\b^0$ we have two nice
candidates, and to choose between them we can use the condition
$[i,\g^0]=0$:
\bea
\mbox{ equ. }(\ref{tensor}) \follows& i\b^0= \b^0 i
&\follows \g^0 = \b^0, \quad P_g = P \\
\mbox{ equ. }(\ref{skew}) \follows& i\b^0=-\b^0 i
&\follows \g^0 = C\b^0, \quad P_g = CP.
\eea
Thus, in our representation we have $P_g=CP$, and \gp\/ is not connected
with spatial inversion, but a lattice shift operator.
\section{Gauge fields}
\subsection{Wilson lattice gauge fields}
\newcommand{\om}{\omega} %% gauge transformation
In Wilson lattice gauge theory, the gauge field is represented by elements
$U(\n,i)\in G$ of the gauge group located on the edges $(\n,\n+\hi)$ of the
lattice. Gauge transformations $\om(\n,t) \in G$ act as
\be
U(\n,i) \to \om(\n)U(\n,i)\om^{-1}(\n+\hi)
\ee
To define the action of a Wilson lattice gauge field with gauge group $G$ on
our lattice space $\CAL$ we need a representation $T$ of $G$ on the lattice
space so that the gauge transformation $\om(n,t) \in G$ acts on $\O^3$ as
\be
\v(\n) \to T(\om(\n)) \v(\n).
\ee
To make the theory gauge-invariant, we have to replace usual lattice shifts
$\t^\pm_i$ by ``covariant lattice shifts'' $\tilde{\t}^\pm_i$:
\bea
(\tilde{\t}^+_i \v)(\n) &=& (T(U (\n, i))\v(\n+h_i), \\
(\tilde{\t}^-_i \v)(\n) &=& (T(U^{-1}(\n-\hi,i))\v(\n-h_i).
\eea
The difference operators $\D_i, \D_i^-$ have to be replaced by their
covariant versions following the same scheme as in (\ref{Di}):
\bea
\tilde{\D}_i &=& \frac{1}{2} (\tilde{\t}_i^+ - \tilde{\t}_i^-)\\
\tilde{\D}^-_i &=& \frac{1}{2} (1 - (\tilde{\t}_i^-)^2)
\eea
and, last not least,
\be
\tilde{D} = -i\a^i\tilde{\D_i}
\ee
The Wilson lattice gauge fields which preserve Euclidean symmetry may be
characterized by the following theorem:
\begin{theorem} \label{U3c}
The maximal Wilson gauge action which preserves Euclidean symmetry and
multiplication with $i$ is $U(3)$.
\end{theorem}
On \O, we have found for every constant operator an operator in \P\/ which
commutes with $D$. This may be extended to $\A C=\O^3$. We obtain the group
$O(24)$ for Wilson gauge fields commuting with $D$. The requirement of
commutation with Euclidean rotations reduces $O(24)$ to $O(8)$ acting from
the right on \O. Commutation with $i$ reduces $O(8)$ to $U(4)$.
Commutation with Euclidean translations means that the neutral direction
\c\/ has to be preserved. Together with \c\/ also $i\c$ will be preserved.
This reduces $U(4)$ to $U(3)$. \qed
We identify this group of Wilson gauge fields \Uc\/ with the unitary
extension of \SUc. It contains, together with \SUc, also its diagonal, the
abelian gauge field \UB\/ related with baryon charge. Note that this
additional gauge field is only allowed, not required, by our conditions.
The subspace preserved by \Uc\/ we identify with the leptonic sector, the
remaining part the quark sector. The degrees of freedom related with the
leptonic sector are three pairs of dual variables $(\p^i,\q^i)$, with
translations $e^i_0$ acting on them as
\be
(\p^i,\q^i) \to (\p^i,\q^i+e^i_0).
\ee
But these are simply degrees of freedom for a single point in each lattice
node. It seems natural to identify this point with the center of the cell.
Instead, translations leave the quark variables unchanged. Thus, the quark
fields describe the deformation of the cell relative to its center.
\subsection{Restrictions from Euclidean symmetry for weak fields}
Let's consider now another class of gauge fields of the standard model which
we name ``weak fields''. They are characterized by the property that they
preserve leptons and act on leptons and quarks in exactly the same way. This
includes, of course, the weak gauge field \SUL. But it does not include the
EM field in its standard form, because the EM charge is different for
leptons and quarks. Fortunately, there exists a decomposition
\be
Q = 2 I_B + (I_3-\frac{1}{2})
\ee
The first term (the baryon charge $I_B$ which is $1/3$ on quarks) is part of
the Lie algebra of \Uc, namely its diagonal \UB. Thus, it may be described
as a Wilson gauge field. The second part is the same for leptons and quarks
and therefore fits into our definition of ``weak fields''. The gauge group
generated by this part we denote with \Uem.
\begin{theorem} The maximal possible group of weak fields which is
compatible with Euclidean symmetry is $\UL \oplus \Uem$.
\end{theorem}
Again, rotational symmetry restricts these gauge groups to gauge groups
which preserve the generations and act on all generations in the same way.
Thus, the weak fields have to preserve electroweak doublets and act on all
doublets in the same way. What remains from Euclidean symmetry in one
generation \O\/ is the preservation of some translational direction \c. This
requires a nontrivial common eigenspace with eigenvalue $1$ for the gauge
group, that means, $0$ for the corresponding Lie algebra.
Let's consider the algebra of operators which commute with $i$ and $D$ on
doublets. It is generated by $i$, $\g^5$, and the isospin operators
$\i_i=2iI_i$. There are various but equivalent maximal subalgebras of operat
ors which share a nontrivial $0$ eigenspace. One of these subalgebras is
generated by the operators $I_3-1/2$, $1-\g^5$, $(1-\g^5)I_i$. The shared
$0$ eigenspace is in this case the right-handed neutrino. \qed.
\subsection{How to put chiral gauge fields on the lattice?}
To describe weak gauge fields on our lattice we cannot use Wilson gauge
fields. Even worse, in our approach a gauge-invariant way to describe these
gauge fields seems impossible. Indeed, we have already identified all
lattice operators for the generators of the weak gauge groups:
\newcommand{\op}{A}
\be
\op \in \{1,\g^5,I_i,\g^5I_i\} \to
\op^- \in \{1,\g^5_-,\frac{-i}{2}\i_i^-,\frac{-i}{2}\g^5_-\i_i^+\},
\ee
and these lattice operators define a representation of the Lie algebra only
modulo irrelevant shifts. We propose to accept this as a fact: Strong and
weak gauge fields are fields of different fundamental nature, and while
strong gauge fields have, as Wilson lattice gauge fields, gauge invariance
on the lattice, this does not hold for weak gauge fields. Without the
requirement of gauge invariance it is not difficult to introduce weak gauge
fields on the lattice:
\be
\a^i A^\k_i(x) \op_\k \to \a^i_+ A^\k_i(\n) \op_\k^-, \;
A^\k_0(x) \op_\k \to A^\k_0(\n) \op_\k^-.
\ee
How to quantize gauge fields without gauge invariance will be discussed
below.
\section{Gravity} \label{Gravity}
For metric theories of gravity there is a simple way to obtain an ether
interpretation. 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 propose to identify these fields as density $\rho$,
velocity $v^i$ and stress tensor $\s^{ij}$ of the ether in the following
way:
\bea
\nonumber g^{00}\sqrt{-g} &=& \rho, \\
\label{ADM} g^{0i}\sqrt{-g} &=& \rho v^i, \\
\nonumber g^{ij}\sqrt{-g} &=& \rho v^i v^j - \s^{ij}.
\eea
For these condensed matter fields, we would like to have continuity and
Euler equations:
\bea
\label{continuity} \pd_t \rho + \pd_i (\rho v^i) &=& 0 \\
\label{Euler} \pd_t (\rho v^i) + \pd_i(\rho v^i v^j - \s^{ij}) &=& 0.
\eea
They coincide with the harmonic conditions for the metric (\ref{ADM}):
\be
\label{harmonic} \pd_\a (g^{\a\b}\sqrt{-g}) = 0.
\ee
With this ether interpretation, we need a metric theory of gravity which
includes the harmonic condition as a physical equation. A simple theory with
this property is general relativity in harmonic gauge. We have to add a
non-covariant term to the GR Lagrangian which enforces harmonic coordinates:
\be \label{L}
L=\Xi_\a g^{\a\a}\sqrt{-g}+ L_{GR}(g^{\a\b},\psi^{matter})
\ee
For some constants $\Xi_{\a}$. Its dependence on the preferred coordinates
$X^\a(x)$ can be made explicit (that means, if we would forget about their
geometric nature and consider the $X^\a(x)$ simply as four scalar fields,
the expression would be covariant):
\be \label{Lfull}
L= \frac{-1}{2} \Xi_{\g} g^{\a\b}X^\g_{,\a}X^\g_{,\b}\sqrt{-g}+
L_{GR}(g^{\a\b},\psi^{matter})
\ee
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:
\be\label{EL}
\frac{\delta S}{\delta X^\g} =
\Xi_\g \pd_{\b} (g^{\a\b}\sqrt{-g} \pd_a X^{\g}),
\ee
thus, the preferred coordinates $X^\a$ are harmonic. The Lagrangian
(\ref{L}) 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 an ether theory?
\begin{theorem} The Lagrangian (\ref{Lfull}) follows from the following two
conditions:
\bea
\label{No}
\frac{\delta S}{\delta X^0} &=& \Xi_0 (\pd_t\rho+\pd_i(\rho v^i))\\
\label{Ni}
\frac{\delta S}{\delta X^i} &=& \Xi_i (\pd_t(\rho v^i)+\pd_j(\rho v^i v^j-
\s^{ij}))
\eea
\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
\be
\frac{\delta S}{\delta X^\a} = 0,
\ee
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 Noethers 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:
\be \label{Noether}
\frac{\delta S}{\delta X^\a} = -\pd_{\b}\left(\frac{\pd L}{\pd_{\b}X^\a} +
\ldots\right)
\ee
Thus, the left hand side of (\ref{No}), (\ref{Ni}) defines the Noether
conservation law 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 an ether theory.
\section{Quantization}
The theory is, yet, purely classical. Once we have a preferred frame, the
straightforward way to quantize the theory is canonical quantization, that
means, the definition of one-time commutation or anticommutation relations
for the operators $\v^i_\k(\n)$. A simple choice would be to take their
complex linear combinations
\be
\z^i_\k(\n) = \v^i_\k(\n) + i \v^i_{\dual{\k}}(\n),\; \k \mbox{ even }
\ee
and to postulate anticommutation relations for them:
\be
\{\z^i_\a(\n),\z_\b^{j\dagger}(\n')\} = f_{\a\b}\delta^{ij}\delta(\n-\n')
\ee
This seems to be a way to define a consistent quantum theory. On the other
hand, with these standard anticommutation relations, the classical limit is
problematic --- we obtain Grassmann variables instead of the classical
real-valued fields which describe our ether theory. On the other hand, we
have natural anticommutation relations in the classical limit too --- for
the forms of odd degree in \L. These relations differ from the
anticommutation relations we need in the straightforward approach --- only
forms of odd degree anticommute. Despite this, the author believes that they
have to play a role in quantization, and, therefore, that the
straightforward anticommutation relations are not the way to procede. We
have to leave this question to future research.
Despite this, we can discuss here some qualitative aspects of the resulting
quantization.
\subsection{Quantization of non-gauge-invariant gauge fields}
\sloppy
According to the standard approach to gauge field quantization, exact gauge
invariance is considered to be necessary. Without exact gauge invariance,
the standard quantization scheme leads to a theory which is not unitary and
therefore physically meaningless.
\fussy
Is it possible to modify the quantization scheme in such a way that we
obtain a physically meaningful theory even without exact gauge invariance?
It is. Even more, the original quantization of the electromagnetic field,
proposed by Fermi \cite{Fermi} and Dirac \cite{Dirac}, has had these
properties. Only more recent manifestly covariant approaches to gauge field
quantization, following Gupta \cite{Gupta} and Bleuer \cite{Bleuer}, lead,
if gauge invariance is not exact, to a physically meaningless, non-unitary
theory.
In above approaches, we start with a ``big'' space, where operators
$A_\mu(x)$ are used to describe the gauge fields. That means, different but
gauge equivalent gauge field configurations are represented, in the big
space, as different states.
The definition of the big space is not unproblematic, because in a
manifestly covariant approach we obtain for the time-like components
commutation relations with the ``wrong sign'':
\be
[c_\mu^{\dagger}(k),c_\nu(k')] = \eta_{\mu\nu}\delta(k-k')
\ee
In the Gupta-Bleuer approach we do not care about this and use a manifest
covariant definition of the vacuum state:
\be
c_\mu(k) |0\rangle = 0.
\ee
The resulting space contains states with negative norm and is, therefore,
physically meaningless. But this is not important, because this space is
only an intermediate tool. The physical Hilbert space is obtained from this
unphysical big space by factorization --- a factorization which requires
exact gauge invariance.
In the Fermi-Dirac approach this problem was solved in another,
non-covariant, way --- the Hilbert space was defined in such a way that for
the vacuum state
\be
c_i(k) |0\rangle = 0, \; c_0^{\dagger}(k) |0\rangle = 0,
\ee
thus, $c_0(k)$ and $c_0^\dagger(k)$ change their roles. This Hilbert space
does not contain states with negative norm. The theory is, therefore,
unitary and meaningful already for the big space. But this approach violates
manifest covariance. This is, again, not problematic, because effective
Lorentz covariance is recovered for the ``physical states'', which are
defined as the invariant subspace
\be
\partial_\mu A_\mu |0\rangle = 0.
\ee
In the case of the EM field these two approaches are equivalent. But this
equivalence no longer holds if we have only approximate gauge invariance. In
this case, the Gupta-Bleuer approach obviously fails, we obtain a physically
meaningless, non-unitary theory with negative probabilities --- the theory
we have used for the big space. If gauge invariance fails in the Fermi-Dirac
approach, we also obtain the theory we have used in the big space. But it is
a well-defined, physically meaningful, unitary theory. What goes wrong is
only exact relativistic invariance.
In our approach, we quantize the basic operators in such a way that the
resulting theory is, from the start, unitary. We have, anyway, no chance to
obtain exact relativistic invariance. First, manifest relativistic
invariance is already violated by the decision to discretize only in spatial
directions. Moreover, the spinor representation of the Lorentz group is
defined on \Sl only approximately, by the generators
\be
\s_{ij}^+ = \a^i_+\a^j_+, \; \s_{0i}^+ = \a^i_+.
\ee
Thus, quantization in our approach leads to a theory of the Fermi-Dirac
type, with manifest unitarity but without manifest relativistic covariance.
We should not be afraid of negative probabilities.
As a consequence of these considerations, other arguments related with gauge
field quantization should be reconsidered too. Especially the case of
anomalous gauge theories should be reevaluated.
One characteristic property of a quantization scheme without exact gauge
invariance is worth to be mentioned: The gauge degrees of freedom become
physical. Different gauge-equivalent gauge field configurations are, in this
scheme, different physical states. Thus, we obtain additional particles
which interact with usual matter only is a very weak way --- in the
classical continous limit we have no interaction at all. Thus, we obtain
some new candidates for dark matter.
\subsection{Quantization of gravity}
An ether theory of gravity, as proposed in section \ref{Gravity}, solves the
known problems of quantization of general relativity. Especially we have no
problem of time \cite{Isham}: We have a background geometry with absolute
time, thus, everything we need for canonical quantization is available. Many
related problems disappear too: We have no diffeomorphism or Hamiltonian
constraints, but a well-defined, non-degenerated energy-momentum tensor,
which is simply $g^{\a\b}\sqrt{-g}$, and local as well as global
conservation laws related with translational symmetry.
Quantization itself follows the same scheme as quantization of condensed
matter theories: We have to quantize the discrete (atomic) theory and, then,
derive a reasonable field theory limit, following, for example,
\cite{Landau}. This limit is not unproblematic \cite{Wagner}, but this is
already a mathematical problem --- the field-theoretic limit of some
well-defined physical theory --- and no longer a physical one.
Similar to the situation with gauge fields, different but equivalent
effective metrics $g_{\a\b}(x)$ describe different states of the
gravitational field. We obtain additional degrees of freedom which do not
interact with matter fields in the classical continuous limit. Instead, they
appear only in the equations for the gravitational field, as additional
terms, that means, as dark matter or dark energy terms.
But, different from the situation with gauge fields, the related additional
terms are already known:
\be
G^\a_\b = 8\pi G (T_{matter})^\a_\b + (\Lambda + \Xi_{\g}
g^{\g\g})\delta^\a_\b
- 2 g^{\a\g} \Xi_{\g} \delta^{\g}_{\b}
\ee
These additional terms prevent, for $\Xi_0<0$, black hole and big bang
singularities \cite{GLET}. As a consequence, problems related with such
singularities also disappear. Especially without black holes there is no
black hole information loss problem (see for example \cite{Preskill} about
this problem).
The whole class of GR quantization problems related with nontrivial topology
and closed causal loops also disappears. We have trivial topology, and the
condition $\rho>0$ requires a global time-like function $X^0$. Note that
initial conditions with $\rho>0$ can lead, in principle, to solutions with
$\rho\le 0$ later --- but in this case it is the condensed matter
approximation which fails in the region with $\rho\le 0$: For sufficiently
small $\rho$ we have to go back to the fundamental theory.
\section{Conclusion}
A lot of interesting questions has to be left to future research. First of
all, the consideration of electroweak symmetry breaking. Then, the
appropriate generalization of the lattice Dirac equation to more general
configurations, especially for other lattices and lattices with defects. For
a true unification of the SM with gravity we need a general lattice
Lagrangian which, on one hand, gives the Dirac equation defined here, for a
regular lattice, and, on the other hand, in the continuous limit a variant
of the Lagrangian derived here for gravity.
And, last not least, we have to quantize the resulting lattice theory, which
requires a better understanding of the classical limit of fermion fields.
Despite these open problems, the ether theory presented here defines a
reasonable candidate for a quantum theory of everything. We have obtained,
even if only on a classical and kinematic level, all observed fields, the
standard model fermion and gauge fields as well as the metric for gravity.
Moreover, we have a classical framework of absolute space and time, which
allows a canonical approach to quantization. The major problems of
quantization of general relativity, especially the ``problem of time'' and
derivative problems, are not present in our approach.
Last not least, let's mention here another, completely independent hint in
favour of a preferred frame --- the violation of Bell's inequality
\cite{Bell}, which defines a contradiction between classical realism and
Einstein causality. As a consequence, realistic hidden variable theories for
quantum theory --- like Bohmian mechanics \cite{Bohm} or Nelsonian
stochastics \cite{Nelson} --- need a preferred frame.
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\bibitem{Wagner} H.-J. Wagner, Das inverse Problem der Lagrangeschen
Feldtheorie in Hydrodynamik, Plasmaphysik und hydrodynamisches Bild der
Quantenmechanik, Univ. Paderborn, 1997
\end{thebibliography}
\end{document}
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| User: "Ilja Schmelzer" |
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| Title: Re: Cellular ether theory (large TeX) |
07 Jun 2006 08:56:27 AM |
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"Ilja Schmelzer" <q6867901@mailstore.fernuni-hagen.de> schrieb
%% The current working version of my cellular ether theory
%% Ilja Schmelzer
Only a typo but nonetheless an important correction:
The \t_i in the following formula
--------------------------
\subsection{Decomposition into staggered sublattices}
The operators
\be
\a_i = \b^0 \b^i; \; \g_i=\b^0\a^i\t_i.
\ee
--------------------------
(formula (25) in pdf) should be replaced by \theta_i, so that we obtain
--------------------------
\subsection{Decomposition into staggered sublattices}
The operators
\be
\a_i = \b^0 \b^i; \; \g_i=\b^0\a^i\theta_i.
\ee
--------------------------
Ilja
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| User: "FrediFizzx" |
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| Title: Re: Cellular ether theory (large TeX) |
07 Jun 2006 12:39:16 PM |
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"Ilja Schmelzer" <q6867901@mailstore.fernuni-hagen.de> wrote in message
news:e66lvm$62i$1@sycamore.fernuni-hagen.de...
"Ilja Schmelzer" <q6867901@mailstore.fernuni-hagen.de> schrieb
%% The current working version of my cellular ether theory
%% Ilja Schmelzer
Only a typo but nonetheless an important correction:
The \t_i in the following formula
--------------------------
\subsection{Decomposition into staggered sublattices}
The operators
\be
\a_i = \b^0 \b^i; \; \g_i=\b^0\a^i\t_i.
\ee
--------------------------
(formula (25) in pdf) should be replaced by \theta_i, so that we
obtain
--------------------------
What PDF? Did you put this up on your website in PDF? If not, you
should. I don't have the time to convert your TeX.
FrediFizzx
http://www.vacuum-physics.com
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| User: "Ilja Schmelzer" |
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| Title: Re: Cellular ether theory (large TeX) |
08 Jun 2006 01:24:03 AM |
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"FrediFizzx" <fredifizzx@hotmail.com> schrieb
What PDF? Did you put this up on your website in PDF?
Not yet.
If not, you should.
I'm in the process of rewriting the whole site. But it will
take some time.
I don't have the time to convert your TeX.
Feel free to ignore my theory until I have solved the
remaining open problems too.
Ilja
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| User: "The Ghost In The Machine" |
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| Title: Re: Cellular ether theory (large TeX) |
20 May 2006 11:00:05 AM |
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On Wed, 17 May 2006 07:41:31 +0200, Ilja Schmelzer wrote:
Running this through LaTeX generated 18 pages of something that looks
generally readable but which had a number of problems with references and
suggested that Abstract should precede \maketitle on line 93.
I'll leave it to others to critique the actual theory; I for one can't
make heads or tails out of it. :-)
Followups.
%% The current working version of my cellular ether theory
%% Ilja Schmelzer
\documentclass{amsart} % article
\begin{document}
\newtheorem{theorem}{Theorem}
[snippage]
\end{thebibliography}
\end{document}
--
#191,
It's still legal to go .sigless.
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| User: "Andreas Most" |
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| Title: Re: Cellular ether theory (large TeX) |
22 May 2006 07:51:26 AM |
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The Ghost In The Machine wrote:
On Wed, 17 May 2006 07:41:31 +0200, Ilja Schmelzer wrote:
Running this through LaTeX generated 18 pages of something that looks
generally readable but which had a number of problems with references and
suggested that Abstract should precede \maketitle on line 93.
You have to run Latex twice to let it figure out the references correctly.
The second time Latex runs without any warnings. RTFM.
I'll leave it to others to critique the actual theory; I for one can't
make heads or tails out of it. :-)
To put it in my unskilled words:
Ilja has developed a quantized ether theory which he calls
GLET (General Lorentz Ether Theory).
The GLET requires a so called preferred frame and it
contains SRT and ART as an approximation.
Predictions concern elementary particles as well as cosmological
models (no black holes; flat universe; ...)
Andreas.
Followups.
%% The current working version of my cellular ether theory
%% Ilja Schmelzer
\documentclass{amsart} % article
\begin{document}
\newtheorem{theorem}{Theorem}
[snippage]
\end{thebibliography}
\end{document}
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