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\def\yeardone{\number1990}
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\def\sec#1{{{\partial^2}\over{\partial {#1}^2}}}                                      

\begin{document}
\input{notecomm.tex}

\newcommand{\ram}{{\sc ram}}
\newcommand{\rom}{{\sc rom}}
\newcommand{\eprom}{{\sc eprom}}
\newcommand{\eeprom}{{\sc eeprom}}
\newcommand{\sci}{{\sc sci}}
\hyphenation{one-di-men-sion-al 
di-men-sion-al % 
wave-pack-et wave-pack-ets %
Klein-Gor-don}

\title{ Discrete representations of
the $n$-dimension\-al wave and Klein-Gordon equations}
\author{Hrvoje Hrgov\v ci\'c}
\address{{\em   Department of Physics}\\%
{\em   Massachusetts Institute of Technology}\\%
{\em Cambridge, Massachusetts 02139}}
\date{}
\def\yeardone{\number1990}

\maketitle

%\beginabstract}
{\begin{center}
%	{\sl Abstract}
\rule{.0ex}{0ex}
\end{center}}
The decomposition
of a one-di\-men\-sion\-al solution
of the  wave equation into
traveling waves is extended to
higher dimensions.
Developing this procedure provides a novel way of obtaining
the $n$-di\-men\-sion\-al discrete wave equation
and Klein-Gordon equation as
the continuum limit of a
discrete, ran\-dom-walk-like process, 
much the same way as the diffusion
equation is obtained by considering the
random walks of particles on a lattice.
These results are applied to the construction
of a strictly local model of multi-particle
quantum mechanics, in ordinary 
three-di\-men\-sion\-al configuration space.
Moreover, time is nowhere Euclidianized.

%MAKE CLEAR THAT WHAT YOU'RE DOING HERE IS
%AN ANALOGY WITH THE RANDOM-WALK--DIFFUSION-EQ
%CONNECTION. ALSO, NOTE THAT THE LATTICE POINTS MUST BE
%ABLE TO ACCOMMODATE AN ARBITRARILY LARGE NUMBER OF PARTICLES,
%ALTHOUGH IN PRINCIPLE, HOW LARGE DEPENDING ON
%HOW LONG WE WANT THE RESULTS TO BE VALID

%We present solutions of the $n$-di\-men\-sion\-al dis\-crete wave equa\-tion
%in terms of traveling waves, allowing the wave equation to be
%represented as a network flow of a conserved quantities whose relationship
%to the wave function is easily seen. {\tt{note: explain to Tom}}  
%We next show
%how this may be used to give a random walk interpretation
%of wave phenomena amenable to simulation on a CAM, 
%and how we may generalize our results to the Klein-Gordon Equation.
%\end{abstract}

\vspace{.6\baselineskip}
\noindent{PACS numbers: 03.65.Bz, 11.10.Qr}
\vspace{.5\baselineskip}

The general solution of the
one-di\-men\-sion\-al wave equation
may be written as a sum of
two functions, or traveling waves. The equations of motion
of these traveling waves, first-order in time and space, 
are simple translations in opposite directions with a constant speed.

%At one time or another, almost everyone who
%studies the one-di\-men\-sion\-al wave equation
%learns how to decompose it into traveling
%waves that merely translate in opposite 
%directions.
%Being a very physical, flow-like interpretation of
%the wave equation, this decomposition provides
%a very tangible way of
%picturing and computing the evolution of the wave.

The first section of the paper  extends the benefits of such an approach
to higher dimensions, so that in general, the $n$-di\-men\-sion\-al 
wave equation is also represented as the sum of $2n$ functions,
first order in time,  each of which implies a motion
along the positive or negative directions of the $n$ axes of the space.
The domain of the wave functions are a discrete spacetime lattice;
nevertheless,
the resultant wave equations and their solutions are relativistically covariant, in
the continuum limit.

The second section shows that this traveling wave decomposition lends itself to
a statistical implementation, involving summations of random walks on a lattice.  
Recall that the kernel
of the diffusion equation may be expressed in terms of the continuum limit of
random walks on a lattice, so that the time evolution of the equation may in principle
be computed by ensembles of particles executing random walks, if  the
initial distributions of the particles correspond to the initial conditions of the
desired solutions.  By allowing particles
to carry an extra bit of information, that may be likened to
a sign,  it is possible to simulate
wave equations by the same random walk methods (e.g., Monte Carlo and
cellular automata$^1$)
%\footnote{
%T. Toffoli and N. Margolus,
%{\sl Cellular Automata Machines, A New Environment for
%Modeling\/}, MIT Press, Cambridge, Massachusetts, 1987.
%D. Farmer, T. Toffoli, S. Wolfram, {\sl Cellular Automata\/}),
%(North-Holland, Amsterdam, 1984.}) 
as those employed for the diffusion equation.

The third section shows that since this statistical rendering of the 
wave equation 
allows discrete particles to collectively exhibit wavelike 
properties, it can be used to construct a model,
not just a computation, of multi-particle quantum mechanics,
in ordinary three-di\-men\-sion\-al configuration space.
In general, a single particle state is represented, or simulated,
by many particles propagating throughout the lattice
at once. Nevertheless,  it is still possible to obtain the 
correct single and mul\-ti-particle
expectation values by use of proper counting methods, and
by using large ensembles of such simulations.

In the fourth section, the traveling wave decomposition and
is applied to the
Klein-Gordon equation. Subsequently, it is shown that the presence of
the mass term in the equation may be seen as the result of an
interaction
of the ran\-dom-walk particles with yet another 
distribution of particles.


\section{The Discrete Wave Equation}
{\obeylines
\noindent{{\it Lattice waves}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
The 
discrete analogue of the wave equation is defined on an
orthonormal,
$(n + 1)$-di\-men\-sion\-al spacetime lat\-tice 
of points whose spacing is unity.  
The second order partial derivatives
of time and space appearing in the
D'Alembertian are replaced by their finite-difference analogues
$$
{{\partial^2}\over{\partial x_{i}^2}}\psi(\ldots,x_i,\ldots) \equiv 
     \psi(\ldots,x_i + 1,\ldots)\rule{10ex}{0mm}$$
\begin{equation}
\label{eq:op-def} 
 - 2\psi(\ldots,x_i,\ldots)
 + \psi(\ldots,x_i - 1,\ldots)
\end{equation}
where
$$   x_i = x_1, x_2, ..., x_n, t\ \ . $$
Thus, the wave equation retains its usual form,
\begin{equation}
{1\over c^2}{{\partial^2}\over{\partial t^2}}\psi  = %
\left({{\partial^2}\over{\partial x_{1}^2}} + 
{{\partial^2}\over{\partial x_{2}^2}} +\cdots+
{{\partial^2}\over{\partial x_{n}^2}}\right)\psi\ {\rm .} \label{eq:wave}
\end{equation}

Setting
the constant $c$, which will be called the speed of light, to $\sqrt{1/n}$ confers several
useful properties on the equations;$^2$
%\footnote{
%R. Courant, K. Lewy, and O. Friedrichs,
%``On the Partial Difference Equations of Mathematical Physics'', originally
%published in {\em Mathematische Annalen}, 1928, and translated into
%English in the {\em IBM Journal}, March, 1967. Quantities such as the
%momentum and energy, that are conserved and that converge in the
%continuum limit to their usual definitions, also exist.} 
this is done throughout.
%(setf sum-phi (+ sum-phi (aref tp i j))
%	      sum-sq-phi (+ sum-sq-phi (* (aref tp i j) (conjugate (aref tp i j))))
%	      f-mom-x-2 (+ f-mom-x-2 (* (^ -1 (mod i 2)) (- (aref tp i j) (aref tb (mod (+ i 1) n) j))))
%	      b-mom-x-2 (+ b-mom-x-2 (* (^ -1 (mod i 2)) (- (aref tp (mod (+ i 1) n) j) (aref tb i j))))
%	      f-mom-y-2 (+ f-mom-y-2 (* (^ -1 (mod j 2)) (- (aref tp i j) (aref tb i (mod (+ j 1) m)))))
%	      b-mom-y-2 (+ b-mom-y-2 (* (^ -1 (mod j 2)) (- (aref tp i (mod (+ j 1) m)) (aref tb i j))))
%	      f-energy-x-2 (+ f-energy-x-2 (* (- (aref tp i j) (aref tb (mod (+ i 1) n) j))
%					      (conjugate (- (aref tp i j) (aref tb (mod (+ i 1) n) j)))))
%	      b-energy-x-2 (+ b-energy-x-2 (* (- (aref tp  (mod (+ i 1) n) j) (aref tb i j))
%					      (conjugate (- (aref tp  (mod (+ i 1) n) j) (aref tb i j)))))				%  
%	      f-energy-y-2 (+ f-energy-y-2 (* (- (aref tp i j) (aref tb i (mod (+ j 1) m)))
%					      (conjugate (- (aref tp i j) (aref tb i (mod (+ j 1) m))))))
%	      b-energy-y-2 (+ b-energy-y-2 (* (- (aref tp i (mod (+ j 1) m)) (aref tb i j))
%					      (conjugate (- (aref tp i (mod (+ j 1) m)) (aref tb i j))))))))))
In particular, because  of the cancellation of
all terms in the above equation 
corresponding to the central term
of the right-hand side of (\ref{eq:op-def}),
the spacetime is split into two 
causally separate lattices, even and odd
(Fig.~{\ref{fi:checkered-space}}).
A spacetime point $({\bf x}, t)$ in the even (odd) sublattice
has the property that
$$ t+ \sum_{i=1}^{n}x_i$$
is even (odd).
\begin{figure}
\psfig{file=hx/twfig/twf-check-sp-2.ps,height=1.93in,width=3.0in}
\vspace{1.3\baselineskip}
%box is 7 1/4" by 3 7/8", lower left corner 1/8" 6 7/8"
%                         upper right corner 7 3/8" 10 3/4"
%Have a checkered lattice of points connected
%by dashed lines. 
\caption{Two-di\-men\-sion\-al spacetime is split into
 two causally separate lattices. Diagonal lines
connect causally related points.}
\label{fi:checkered-space}
\end{figure}
Without loss of generality, it will be assumed, 
unless stated otherwise, that the solutions considered henceforth
are nonzero on only one of these lattices.

It can be seen from the two preceding
equations
that solutions of the
discrete wave equations form a
vector space whose dimension\-ality
is equal to twice the number of
points in the lat\-tice (i.e., the sublattice).  An
element of this space of solutions
is com\-plete\-ly determined by
arbitrarily assigning values
on all points of
the lat\-tice at two successive
times.  

Any solution of the $n$-di\-men\-sion\-al discrete 
wave equation for a lat\-tice of $N^n$ points
may also be given in terms of
harmonic solutions via Fourier analysis,
e.g., for $n = 2$,
\begin{equation}
\psi(x,y,t) = \sum_{{\kappa}_{x}, {\kappa}_{y}}  
        {\cal A}_{\kappa_{x}, \kappa_{y}} 
        e^{{{2{\pi}i}\over {N}} %
\left(\kappa_{x} x + {\kappa}_{y} y %
+ {\omega} t\right)}\ {\rm ,} \label{eq:fourier}
\end{equation}
where $\kappa_{x}$ and $\kappa_y$ each
range  from 0 to $N-1$, and                               
where the ${\cal A}_{{\kappa}_{x}, \kappa_{y}}$ depend on
the initial conditions.
The frequency $\omega$, whose
functional dependence on the
$\kappa_x$ and $\kappa_y$ 
has been suppressed for clarity,
is given by the
dispersion relation
\begin{equation}
\label{eq:dis-dis}
 \sin^2{\omega \over2} = {1\over2}\left(
        \sin^2{{\kappa_x}\over2} +
        \sin^2{{\kappa_y}\over2}\right)\ {\rm .}
\end{equation}
This passes in the continuum limit
 to
the familiar
%
\begin{equation}
 \omega^2 = {1\over2}\left(k_x^2 +
                           k_y^2\right)\ {\rm .}
\end{equation}
%
where the $k_x$ and $k_y$ are the respective
conjugate variables of the Fourier integral
appearing in the continuum generalization of (\ref{eq:fourier}).
Note that if $c^{2} = 1/n$, the
numerical integration of
(\ref{eq:wave})
is 
stable.  If one were to make the
{\sl a\,priori\/} simpler
choice of setting $c^2 = 1$, or indeed to any
value greater than $\sqrt{1/n}$,
then for all except the one-di\-men\-sion\-al
case, there would be
spatial frequencies for which
the modulus of the right-hand side
of (\ref{eq:dis-dis}) would be greater than one.
The time frequency $\omega$ would 
then contain
an imaginary component, leading
in general
to an exponential growth of 
the solution.  As it is,
the solutions
have the property that
all the 
frequencies are real, so that
even if they do not obey
the continuum dispersion
relations, they retain
their characteristic undamped periodicity.

How well this discrete system
mimics the continuum case is determined
by the spectral range of the
initial conditions. Those solutions
containing only spatial frequencies  for
which the  dispersion equations
yield numerically similar oscillation frequencies,
i.e., those whose associated wavelengths are much
larger than the lattice spacing,
are well approximated by the
discrete analogue.  In principle, the
correspondence may be made as close as desired.


\subsection{The one-di\-men\-sion\-al case}

{\obeylines
\noindent{{\it Traveling waves and links}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
As is well known, 
the general solution of the one-di\-men\-sion\-al discrete wave equation
may be written as the sum of two traveling waves,
%
\begin{equation}
\label{eq:one-dim-wave}
\psi(x,t) = f_+(u) + f_-(v)\ {\rm ,} 
\end{equation}
where
these two traveling waves
are respectively functions of the single variables
$u = x - t$, and $v = x + t$, so that
they translate unchanged
in opposite directions as time passes. 
Indeed, this propagation from one
spacetime point to its neighbors
suggests that these traveling
waves are functions more naturally defined
on the the two kinds of {\sl links\/}, so to speak, connecting
neighboring spacetime points (Fig.~\ref{fi:checkered-space}),
although the utility of such a
view becomes fully manifest
only in higher dimensions.  

Let $f_{+}^{\rm{in}}(x,t)$ and $f_{-}^{\rm{in}}(x,t)$ denote
functions, which will
be referred to as {\sl flows\/}, defined on the links
connecting spacetime points of
the form $(x \mp 1,t-1)$. 
%
%The sum $f_{+}^{\rm{in}}(x,t) + f_{-}^{\rm{in}}(x,t)$
%will then be called the incoming flow
%--at the point $(x,t)$.
%
Similarly, at the risk of redundancy,
let  $f_{\pm}^{\rm{out}}(x,t)$ denote
functions defined on links 
connecting spacetime points
of the form $(x\pm 1,t+1)$.
%,
%whose sum we shall call the
%outgoing flow at $(x,t)$.
%
Obviously,  
\begin{equation}
\label{eq:equ-rel}
f_{\pm}^{\rm{in}}(x,t) \equiv f_{\pm }^{\rm{out}}(x\mp 1,t-1)\ {\rm .}
\end{equation}
The decomposition of
(\ref{eq:one-dim-wave}) then
implies  
\begin{equation}
\label{eq:one-dim-mix}
f_{\pm}^{\rm{in}}(x,t) = f_{\pm}^{\rm{out}}(x,t)\ {\rm .}
\end{equation}
Any solution of (\ref{eq:wave})
may be cast in this way into a system of first-order
partial difference equations.

The set of link-so\-lu\-tions also forms a
vector space, and the traveling wave
decomposition implies that there exists
a homomorphism from the link-solutions
to the point-solutions (Fig.~\ref{fi:hom-tim}).  Moreover, this
homomorphism is preserved over time, under the
respective time evolution of each system. By
the implicit use of this homomorphism, 
\begin{figure}
$$
\begin{array}{ccccc}
%{c@{\hspace{.4ex}}c@{\hspace{.4ex}}c@{\hspace{.4ex}}c@{\hspace{.1ex}}c}
                 &(\ref{eq:one-dim-mix})      &                   &(\ref{eq:one-dim-mix}) \\
\vspace{-1.6em} \\
\{ \sum f(x,t)\} & \longrightarrow      & \{ \sum f(x,t+1)\}  & \longrightarrow & \cdots \\
\vspace{0em}\\
\mid             &   & \mid   \\
\vspace{-1em} \\
{\cal H} &   & {\cal H} \\
\vspace{-1em} \\
%\mid             &   & \mid   \\
\downarrow          &   & \downarrow \\
\vspace{.2em}                 &(\ref{eq:wave})&                   & (\ref{eq:wave})\\
\vspace{-1.6em} \\
\{ \psi(x,t) \}  & \longrightarrow      & \{ \psi(x,t+1)\}    & \longrightarrow  & \cdots     
\end{array}$$
\caption{There exists a homomorphism $ \cal H $ from the
space of link-so\-lu\-tions to the space
of point-so\-lu\-tions that is preserved
under their respective time evolutions.}
%given by (\ref{eq:wave}) and (\ref{eq:one-dim-mix}).}
\label{fi:hom-tim}
\end{figure}
(\ref{eq:one-dim-wave}) may be restated 
as
\begin{equation}
\psi(x,t)= f_{+}^{\rm{in}}(x,t) + f_{-}^{\rm{in}}(x,t) =%
	   f_{+}^{\rm{out}}(x,t) + f_{-}^{\rm{out}}(x,t)\ {\rm .} 
\end{equation}
The homomorphism is obviously
not one-to-one, since by  everywhere setting
$$f_{+}^{\rm{out}}(x,t) = -f_{-}^{\rm{out}}(x,t)= \alpha\ ,$$ 
where $\alpha $ is an arbitrary constant,
one obtains an infinity of link-so\-lu\-tions 
that under the homomorphism
are mapped onto the trivial 
(i.e., always and everywhere zero) point-so\-lu\-tion.

Note that by a proper choice of the basis
functions,
the expression of a given solution
in terms of the flows can
(in any number of dimensions)
be made unique, so
that the relation between the
solutions defined on points and 
those defined on links
becomes an isomorphism.

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Completeness}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
%
In considering the notion of 
completeness, regardless  of the
number of 
dimensions, the space in question is assumed for convenience to
be a torus of length $4N$ in all spatial dimensions.
Moreover, let all solutions have the property that
\begin{equation}
 \psi(\ldots,x_i,\ldots) =
        - \psi(\ldots,x_i + 2N,\ldots)\ {\rm ,}
\end{equation}
where the `$+$' sign in the expression
$x_i + 2N $ implies addition modulo $2N$.
Therefore, a solution is completely determined
by its boundary conditions on the subspace of points
for which the $x_i$ range from 0 to $2N-1$.
By making $N$ as large as necessary, 
the desired
generality is retained.

Let $h_{\pm}(x-x_0)$
stand for the 
traveling wave solutions 
that at $t = t_0$ 
are respectively zero everywhere except on the links
going out from the point $x_0$
to one of the  points $x_0\pm 1$, where they are 1.
Such solutions, which will be called {\sl hodotic\/}
solutions (from the Greek word for ``path''), have many unusual 
properties, especially in higher dimensional systems.  
Some of these properties will be discussed in the appendix.
For purposes of clarity, 
their time dependence has been suppressed.

A moment's thought will show
that under the homomorphism, these link solutions
correspond to the point solutions
that at $t=0$ are
equal to $\delta_{x,x_0}$
and at $t=1$ are respectively equal to $\delta_{x,x_0\pm1}$,
where $\delta_{x,x'}$
is the Kronecker delta.
 %(Fig.~\ref{fi:mak-hod}). 
%\begin{figure}
%\rule{.2pt}{13em}
%\caption{The boundary conditions corresponding to
%a hodotic solution.}\label{fi:mak-hod}
%\end{figure}

Next,
let $G_{0}(x-x_{0})$ and $G_{1}(x-x_{0})$,  
be the point solutions of the wave equation
whose initial conditions (at times $t_0$ and $t_1=t_0+1$),
are
\begin{eqnarray}
\label{eq:gre-fun-a}
  G_{0}(x-x_{0}) = & \delta_{x,x_0}\delta_{t,t_0}\nonumber\\
\\
  G_{1}(x-x_{0}) = & \delta_{x,x_0}\delta_{t,t_1}\nonumber
\end{eqnarray}
$$x, x_0=0,1,\ldots 2N-1\ {\rm .}$$
Again,  their dependence on time has been suppressed.

It is clear that any solution of the 
wave equation 
may be expressed as a linear combination
of the $G_0$ and $G_1$.
On the subspace of length $2N$ mentioned above,  
The function  $G_{0}(x-x_{0})$  and $G_{1}(x-x_{0})$ may be 
expressed in terms of the link functions, $h_{+}$ and $h_{-}$, as 
%
\begin{eqnarray}
\label{eq:gre-fun}
  G_{0}(x-x_{0}) ={1\over{2}}\sum_{x' = 0,2,\ldots}^{2N-2}
	\left\{ h_{+}(x - x_{0} - x'){\rm Sgn}(x-x_0)\right.\nonumber \\
\phantom{ G_{0}(x-x_{0}) ={1\over{2}}a} - %
\left.h_{-}(x- x_{0} - x'){\rm Sgn}(x-x_0-1)\right\} \nonumber \\
\\
  G_{1}(x-x_{0}) ={1\over{2}}\sum_{x' = 0,2,\ldots}^{2N-2}
	\left\{-h_{+}(x - x_{0} - x'){\rm Sgn}(x'-x_0)\right.\nonumber \\
\phantom{ G_{0}(x-x_{0}) ={1\over{2}}a} + %
\left. h_{-}(x- x_{0} - x'){\rm Sgn}(x'-x_0)\right\}\ {\rm ,} \nonumber
\end{eqnarray}
where
$${\rm Sgn}(x)=\left\{\begin{array}{ll}
                       \phantom{-} 1& \mbox{if $x\geq 0$}\\
                       -1& \mbox{otherwise} {\rm .} 
\end{array}\right. $$
Again, note that if $t_0$ is even (odd), then the solutions in question are assumed to be nonzero  on
the even (odd) sublattice. Therefore, $x_0$ is even for $G_{0}(x-x_{0})$,  and odd for
$G_{1}(x-x_{0})$.
\begin{figure}
%\rule{.2pt}{13em}
\psfig{file=hx/twfig/twf-1d-gr-2.ps,height=.72in,width=3.0in}
$$\phantom{some  {\large empty} chatter fill}$$
%box is 7 1/4" by 3 1/4", lower left corner 1/4" 7 1/4"
%                         upper right corner 7 1/2" 10 1/2"
%Shows an arrangement of positive and negative flows that
%collectively yields a Green's function.
\caption{An arrangement of hodotic solutions that yields a
Green's function $G_{1}(x)$ for the discrete wave equation.}
\label{fi:one-dim-greens-function}
\end{figure}

As the equations show, these Green's functions for the discrete
wave function may be constructed by interlocking positive and negative
hodotic solutions of opposite sign  
(Fig.~\ref{fi:one-dim-greens-function}).
Thus the traveling waves completely suffice to
specify the solutions of the wave equation.

Note that the Green's
functions require nonzero flows
across the entire space. Therefore, even
if the solution one wishes to simulate is initially
nonzero only in some localized proper subset
of the space, say, one that
can be covered by a line segment, or
in $n$ dimensions, an $n$-cube,  its expression 
in terms of the Green's functions
involves nonzero flows in a region
extending across the entire space. However,  
if this solution has the additional property
that at 
two successive times, the respective integrals of the amplitude over
the support is the same, then one can find an
arrangement of hodotic solutions that is also
nonzero only in a localized subset.

\subsection{The two-di\-men\-sion\-al case}

In passing to the two-di\-men\-sion\-al case via the present analysis,
solutions of
the wave equation are again to be expressed as a linear sum of 
components. Each of these components will likewise be
associated with flows along the
links connecting nearest neighbors.
However, since in higher dimensions even a localized wave packet
spreads and deforms, it is to be expected that there will be
mixing among these modes of
propagation, instead of the
trivial translation found in the one-di\-men\-sion\-al case.

At any time, each lattice point can be seen
as a kind of black box, 
into which enter and
from which exit four flows. 
The latter, 
are to be determined solely in 
terms of the former, 
in a linear fashion. If, as
in the one-di\-men\-sion\-al case,  the 
flow out from the point $(x,y,t)$
and in to the point $(x+1,y,t+1)$  is denoted as
$f_{x_{+}}^{\rm{out}}(x,y,t)$, or
equivalently as $f_{x_{+}}^{\rm{in}}(x+1,y,t+1)$ and 
the 
flows to other points are analogously denoted,
the above considerations may be restated as
\begin{equation}
\label{eq:two-dim-ass-1}
\psi(x,y,t)= \sum_\sigma f_{\sigma}^{\rm{in}}(x,y,t) %
           = \sum_\sigma f_{\sigma}^{\rm{out}}(x,y,t)\ , 
\end{equation}
\begin{equation}
\label{eq:two-dim-ass-2}
 f_\sigma^{\rm{out}} = \sum_{\sigma'}c^{\rm{in}}_{\sigma,\sigma'}%
     f_{\sigma'}^{\rm{in}}\ {\rm ,}
\end{equation}
where $   \sigma,\sigma' \in \left\{x_{+}, x_{-}, %k9
        y_{+}, y_{-} \right\} $, and
where the coefficients of the matrix $c^{\rm{in}}_{\sigma,\sigma'}$
are to be determined.  

Consider the solution of the wave equation
corresponding to a nonzero flow in only
one link, say, $f_{x_{+}}^{\rm{in}}(0,0,0) = 1$,
with all other incoming flows to all
other points being zero; i.e.,  
a two-di\-men\-sion\-al hodotic solution.
As in the one-di\-men\-sion\-al case, 
the boundary conditions 
corresponding to this
solution 
are $\psi(x,y,0) = \delta_{x,0}\delta_{y,0}$, and
$\psi(x,y,1) = \delta_{x,1}\delta_{y,0}$.

Since  the wave amplitude is thus specified on the
entire space at two successive times, one can iterate
these boundary conditions according to
the wave equation  (Fig.~\ref{fi:x+}),%
\begin{figure}
%\rule{.2pt}{13em}
\psfig{file=hx/twfig/twf-xplus-2.ps,height=2.7in,width=3.0in}
%height=2.83in,width=3.0in}
\vspace{.6\baselineskip}
%box is 5 1/8" by 6 3/8", lower left corner 1/4" 7 1/4"
%                         upper right corner 7 1/2" 10 1/2"
%Have a diamond, with the coefficients ${1\over2}$ on 
%home plate, first and second base, with $-{1\over2}$
%on third base.  If clearer, make the drawing three-di\-men\-sion\-al
%with time along the vertical axis.
\caption{The $x_{+}$ hodotic solution, at $t=0,1$ and 2.}
\label{fi:x+}
\end{figure}
and thus uniquely determine the first column of the 
matrix $c^{\rm{in}}_{\sigma,\sigma'}$ . 
Repeating this analysis for the hodotic solutions of
the remaining three directions completes the
specification of the matrix,
so that
\begin{equation}
\label{eq:two-mat}
\left(\rule{-.5ex}{0ex}  %
\begin{array}{c} f_{x_{+}} \\ 
                 f_{x_{-}} \\
                 f_{y_{+}} \\
                 f_{y_{-}} 
\end{array}\rule{-.5ex}{0ex}  % 
\right)^{\rule{-.8ex}{0ex}\rm{out}}%
\rule{-2.5ex}{0ex} ={1\over2}
\left(\rule{-1.ex}{0ex}
\begin{array}{r@{\hspace{.7ex}}r@{\hspace{.7ex}}%
r@{\hspace{.7ex}}r} 
         1 & -1 &  1 &  1 \\  
        -1 &  1 &  1 &  1 \\
         1 &  1 &  1 & -1 \\ 
         1 &  1 & -1 &  1 
\end{array}\rule{-.5ex}{0ex} 
\right)
\left(\rule{-.5ex}{0ex}%
\begin{array}{c} 
f_{x_{+}} \\
f_{x_{-}} \\
f_{y_{+}} \\
f_{y_{-}} 
\end{array}\rule{-.5ex}{0ex}%
\right)^{\rule{-.8ex}{0ex}\rm{in}}\ {\rm .}
\end{equation}

By using the time invariance properties of the
wave equation, one also could have just 
as easily determined the coefficients of the
inverse of
the above matrix,
$c^{\rm{out}}_{\sigma,\sigma'}$, and would have found them
to be identical. It is then  a matter of simple algebra to
show that the sum of the ingoing or outgoing flows obeys 
the discrete wave equation.

Note that the sum of the coefficients
along any column of the matrix is one, so
that the sum of flows entering a point
at any time is equal to the sum of
the exiting flows. This is also 
obvious from (\ref{eq:two-dim-ass-1}).  
Furthermore,
the matrix is unitary.  Therefore, the sum of
the squares of the entering (and thus the exiting) flows is conserved.
Thus we can  picture the evolution of
such a system as   a network flow of some conserved quantity.

The considerations of completeness may be
directly generalized to higher dimensions
merely by imbedding the one-di\-men\-sion\-al
expressions for the Green's functions (\ref{eq:gre-fun}) along
any of the axes. This is arguably the
simplest way to express the Green's functions,
although in higher dimensions more elaborate
expressions are possible 
(Fig.~\ref{fi:greens-function-crazy}), since
the overcompleteness of the hodotic solutions is
more extensive.
%
\begin{figure}
%\rule{.2pt}{13em}
\psfig{file=hx/twfig/twf-cr-gr-2.ps,height=1.65in,width=3.0in}
\vspace{1.8\baselineskip}
%box is 4 3/4" by 6 1/2", lower left corner 1/2" 5 5/8"
%                         upper right corner 7" 10 1/2"
%Shows a two-di\-men\-sion\-al spacetime graph of a Green's function.
\caption{A two-di\-men\-sion\-al non-collinear arrangement of hodotic solutions that yields a
Green's function for the discrete wave equation. 
Nearest neighbor points are displaced by one unit of time. The circle
marks the support of the point solution.}
\label{fi:greens-function-crazy}
\end{figure}

\subsection{Higher-di\-men\-sion\-al cases}

This method of considering
the behavior of hodotic solutions, in order
to determine the coefficients of the transition matrix,
can be readily generalized to higher dimensions.
For example, in three dimensions, the transition
matrix is determined to be
\begin{equation}
\left(\rule{-.5ex}{0ex}
\begin{array}{c} 
f_{x_{+}} \\
f_{x_{-}} \\                      
f_{y_{+}} \\ 
f_{y_{-}} \\
f_{z_{+}} \\ 
f_{z_{-}}%
\end{array}\rule{-.5ex}{0ex} %
\right)^{\rule{-.8ex}{0ex}\rm{out}}
\rule{-2.5ex}{0ex} ={1\over3}
\left(\rule{-1.2ex}{0ex}
\begin{array}{r@{\hspace{.7ex}}r@{\hspace{.7ex}}%
r@{\hspace{.7ex}}r@{\hspace{.7ex}}r@{\hspace{.7ex}}r} 
       1 & -2 &  1 &  1 &  1 &   1 \\
      -2 &  1 &  1 &  1 &  1 &   1 \\
       1 &  1 &  1 & -2 &  1 &   1 \\
       1 &  1 & -2 &  1 &  1 &   1 \\
       1 &  1 &  1 &  1 &  1 &  -2 \\
       1 &  1 &  1 &  1 & -2 &   1 %
\end{array}\rule{-.6ex}{0ex}
\right) 
\left(\rule{-.5ex}{0ex}
\begin{array}{c}
f_{x_{+}}\\
f_{x_{-}}\\
f_{y_{+}}\\
f_{y_{-}}\\
f_{z_{+}} \\ 
f_{z_{-}}%
\end{array}\rule{-.5ex}{0ex}
\right)^{\rule{-.8ex}{0ex}\rm{in}}\ {\rm .}
\rule{-1.2ex}{0ex} 
\end{equation}
In the general $n$-di\-men\-sion\-al case,
the coefficients of the matrix in any row or
column will be $1/n$, except for the coefficient
connecting oppositely directed flows, (e.g., $c_{x_{+},x_{-}}$),
which will be $-(n-1)/n$.

In each case, the number of traveling waves 
is determined by the number of adjacent neighbors of the
points of the lattice, and the evolution of
a given wave may be determined by the study of
the corresponding hodotic solutions.

It is possible to formally extend
the notions of traveling waves
to non-or\-tho\-nor\-mal 
lattices, generalizing the discrete wave equation
to hexagonal, tetragonal, indeed
to any set of points where each point $j$ has
some privileged subset ${\cal N} (j)$
of generalized nearest neighbors;
the ``wave equation''
then becomes
\begin{equation}
\psi(j,t+1) + \psi(j,t-1)  = {2\over{\left| {\cal N}(j)\right|}}\sum_{j' \in N(j)}\psi(j',t)\ {\rm ,}
\end{equation}
where $\left| {\cal N}(j) \right|$ is the number of
nearest neighbors corresponding to point $j$, and
which may vary throughout the space.
Again, each pair of neighbors will
induce a flow, with
an interaction among flows that may
be determined by studying the behavior
of the
associated hodotic solutions.  As before,
there will be a
conservation of the sum of the 
flows, as well as the sum of their squares.
Of course, how well, if at all, the resultant
system mimics  the 
continuum wave equation in some limit depends on
the particular lattice employed and,
if definable, the
associated dispersion relation.
%kill this if you don't need this, or
%point out that all your stuff was derived independently.
In the course of his research on discrete
quantum probability spaces, S.~ Gudder
has studied even more general systems
of flows on  lattices and graphs, subject only
to the condition that the transition matrix
at each lattice point or graph node
satisfies a unitarity condition..$^3$
%\footnote{S.~Gudder, {\it Quantum Probability,} Boston, 1988.}


\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Effective dimensionality}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
As an aside, note that
in a discrete system, the number of
dimensions in certain
situations is arguable. Consider 
an orthonormal toroidal lattice, as before,
and suppose the number of dimensions $n$, is
very large, but  that all except $m$ dimensions
are exactly one unit long.  
This means that for all but the first $m$ dimensions,
the links and flows leading out of a point will 
lead back in to the very same point, representing
stationary flows.
This also means that the even and odd parity
sublattices are now one and the same.

In effect,
such a system is $m$-di\-men\-sion\-al;
however,
the speed of light
is still $\sqrt{n}$ rather than $\sqrt{m}$, and
the number of types of flows is
likewise equal to twice the
actual dimension $n$, rather than $m$.

The utilization of such systems
with ``thin'' dimensions is a
very useful trick, and will appear again in the
study of the Klein-Gordon equation.
Note that the addition
of these spurious  dimensions does
not change the fact that
the associated difference equations
are numerically stable, and
that
the Fourier frequencies are always real.
Trivial
modifications of the above equations also allow one
to extend the above results to spaces containing
perfectly reflecting or absorbing obstacles,
as well as situations  in which the speed of light,
or alternately, the index of refraction, varies
throughout the space.

\section{Random walks and summations of paths}

The previous section describes a decomposition
of the discrete wave equations into a linear system
of $2n$ first order equations, or traveling waves. 
One benefit
of this decomposition is that
it allows us to relate traveling waves, and thus the wave equation itself,
to ensemble averages of
particle motions on a lattice.  Therefore, this calls to
mind the analogous relation between the
diffusion equation and ensemble averages
of particles executing  random walks.

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Random walks of the diffusion equation}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
The emergence of the $n$-di\-men\-sion\-al diffusion
equation as the continuum realization of
a random walk process, is well known.
The analysis gives the $n$-di\-men\-sion\-al Gaussian
distribution as the large N limit of a summation 
of terms, each of which can be seen as describing a
distinct walk by a particle on an $n$-di\-men\-sion\-al lattice,
in which the particle  starts from the origin and continues
for some  number $T$ time steps.
To wit, consider
the multinomial
\begin{equation}
\left(\tilde{p}_{1_{+}}+\tilde{p}_{1_{-}}+\tilde{p}_{2_{+}}+\ldots+
\tilde{p}_{n_{+}}+
\tilde{p}_{n_{-}}\right)^T\ {\rm ,}
\label{eq:gen-fun}
\end{equation}
where the $\tilde{p}_{i_{+}}$ and $\tilde{p}_{i_{-}}$ are to be
treated as the generators of the free group with
$2n$ dimensions.  
When multiplied out, each of
the $(2n)^T$ terms is a string (i.e., a product) of
$T$ factors. 
These strings can be
bijectively mapped to the
set composed of
the $(2n)^T$ paths of
length $T$ (on an $n$-di\-men\-sion\-al lattice) 
that begin at, say, the origin.
%representable as a string of $T$ choices
%from a set of $2n$ elements.
This is done simply by letting
each successive element of a string signify
the positive or negative
direction, along one of the $n$ axes of the space,
of each successive step of the
random walk.

Consider the probability $P({\bf x},T;{\bf 0},0)$ of a particle starting from
the origin at time $t=0$ and making $T$ independent
choices of steps in order to arrive at a point 
$(x_1,x_2,\ldots,x_n)$.
This may be obtained from the above multinomial
by substituting the  group generators
with the probabilities $\{p_{1_{+}},p_{1_{-}},p_{2_{+}},\ldots,p_{n_{-}}\}$
of choosing a particular direction for a particle step,
(so that $\sum_\sigma p_\sigma = 1$).
The result is
\begin{equation}
P({\bf x},T;{\bf 0},0)= {\sum}' \prod_{j=1}^{T}{\cal S}_{j}\ {\rm ,}\\
\label{eq:pat-pro}
\end{equation}
where 
$${\cal S}_{j}\in\{p_{1_{+}},p_{1_{-}},p_{2_{+}},\ldots,p_{n_{-}}\}\ .$$
The summation is over all distinct assignments of
the ${\cal S}_{j}$, with
the restriction,
denoted by the prime, that the number of
$p_{i_{+}}$ factors in any summation term
minus the number of $p_{i_{-}}$ factors
must equal $x_i$. Thus it  is a summation over paths.

%The probability of a particle starting from
%the origin and making $T$ independent
%choices of steps in order to arrive at a point 
%$(x_1,x_2,\ldots,x_n)$ may then
%be written as
%\begin{equation}
%P({\bf x},T;{\bf 0},0) = {\sum}' \prod_{j=1}^{T}{\cal S}_{j}\ {\rm ,}
%\label{eq:pat-pro}
%\end{equation}
%where
%the ${\cal S}_j$ are each members of the
%set \break $\{p_{1_{+}},p_{1_{-}},p_{2_{+}},\ldots,p_{n_{-}}\}$,
%which we now see as real numbers, each representing the statistical
%weight associated with 
%choosing a particular direction for the particle's step,
%%in the simplest case, where the choices of a direction are equally likely, these would
%%all be equal to $(2n)^{-1}$.
%so that  and $0\leq p_\sigma \leq 1$.


For future reference, note that  (\ref{eq:pat-pro}) can
be separated into
the sum of $2n$ different summations,
\begin{eqnarray}
P({\bf x},T;{\bf 0},0)= %
\sum_{\sigma}P_{\sigma}({\bf x},T;{\bf 0},0)\\
\sigma \in \{x_{1_{+}}, x_{1_{-}},\ldots,x_{n_{+}}, %
x_{n_{-}}\}\ {\rm .}\nonumber
\end{eqnarray}
where each of the summands of the above equation
is itself a summation of the form
(\ref{eq:pat-pro}), but
only contains paths whose first step
is along the direction denoted by 
the $\sigma$ index.
It is these subsummations that will
be related to the hodotic
solutions of the wave equation.

%By the use of the multinomial series,
One can obtain a closed form
for (\ref{eq:pat-pro}).
%Then, by using Stirling expansions to
%express the factorials, 
By taking appropriate limits of the
resultant multinomial coefficients,
the result is seen to be
kernel of the $n$-di\-men\-sion\-al
diffusion equation.$^4$
%\footnote{
%W. Feller, {\sl An Introduction 
%to Probability Theory and Its Applications\/}, Vol.~I,  
%John Wiley \& Sons, Inc., New York, 1968.}
Computation schemes and computer architectures
have been devised that exploit this connection
between random walks and the associated 
differential equation.$^5$
%\footnote{
%T. Toffoli and N. Margolus, {\it op.~cit.}}
Diffusion phenomena
can today be simulated by actually generating
large numbers of particles executing random walks, thereby
recapitulating the physical processes that
underly the diffusion.

Many of the equations
of fundamental physics, including those involving 
wavelike phenomena
are derivable from
processes similar to random walks.  Often, such
calculations are done by
analytically continuing the real variable
representing time 
onto the imaginary axis, thereby
converting a hyperbolic equation into
a more tractable elliptic analogue.
Although the advantages of this
somewhat unphysical technique are undeniable,
it may be bypassed altogether using
the results  of the previous section.

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Random walks of the wave equation}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
The hodotic equations can, in any dimension, also be brought
to a summation of the form (\ref{eq:pat-pro}).
For example, consider the
two-di\-men\-sion\-al hodotic
solution $h_{x_{+}}(x,y, t)$.
As stated previously, this is a solution
initially having a nonzero flow only along the positive
$x$ axis emanating from the 
%point $(-1,0)$. 
origin.

Observe
what happens to this solution in the first time step
(Fig.~\ref{fi:x+}).
The amplitudes at $t=0$ and $t=1$ suggest that
if one were to implement this system of network flows 
by the actual motion of some physical fluid composed of particles,
all the particles of the fluid  would 
initially execute a step 
from the origin to the point $(1,0)$.
%from  the point $(-1,0)$ to the origin.

By observing what happens in
the subsequent time interval ending at $t=2$,
one may, in a purely
formal way, give the following random-walk 
interpretation of the situation:
\begin{quote}
a particle starting from the
origin in
the positive $x$ direction, may
in the subsequent time step travel in any of 
the four possible directions, with the
`probability' being $-1/2$
for a step that represents a reversal
of its previous motion,
and $1/2$ for steps in
the remaining directions.
\end{quote}

Since the sum of the squares of the
above weights  equals one, and since
the sum of the squares of the flows
along the links is a conserved quantity,
it might seem more natural to
treat the squares of the amplitude
as probabilities. However,
%
%it turns out to be easier to work with
%the linear amplitudes. This is because 
%
it is the sum of the linear wave solutions,
not the sum of their squares, for which
the principle of superposition holds.
Therefore, it is preferable to use the linear
amplitudes as probabilities and to
suspend for the moment,
any consideration of what
a negative probability might mean.

Continuing in the same way,
and noting that the system obeys
{\sl first\/}-order equations
of motion,
one sees that at time $t = T$,
the amplitude 
of the wave may be expressed as
\begin{equation}
\label{eq:pat-amp}
h_{x_{+}}(x,y,T)=
{\sum}' \prod_{j=1}^{T}{\cal A}_j\ {\rm .}
\end{equation}
Again, each summation term
corresponds to a path on
the lattice
consisting of $T$ choices,
symbolically represented by the ${\cal A}_j$
from a set of $2n = 4$ steps.

This time however,
the numerical
values represented by the ${\cal A}_j$
are generated according to the following
recipe: 
\begin{quote}
${\cal A}_{1}$ always has the value 
$+1/2$ and represents a step in the $x+$ direction.
(If we were studying say, $h_{y_{-}}$, it
would always be step in the $y-$ direction, and so on.) For all
other $j$,
${\cal A}_{j}$ is $+1/2$, unless the associated
step is taken in the reverse direction to that
of the previous step ${\cal A}_{j-1}$, whereupon it is $-1/2$.
\end{quote}
Note there must be a reversal of directions in order
for the weight to change---right angle kinks in the path are given
the same weight as those that continue along the same direction.
The prime on the summation sign again indicates 
a restriction to choices of steps such that
at time $T$ the path ends at the point $(x,y)$.

Therefore,  each path, i.e., each summation term,
has a contribution of $({1\over2})^{T-R}(-{1\over2})^{R}$, where $R$ is the
number of reversals within the path.
In the general $n$-di\-men\-sion\-al cases, the contribution would
likewise be $({1\over n})^{T-R}({{1-n}\over n})^{R}$.
In other words, suppose $\phi_{\sigma}$  is some indexing of
all the paths of length $T$ that begin at the origin with
a step along the $\sigma $ direction, and arrive at
the point ${\bf x}$. Then
\begin{equation}
h_{\sigma}({\bf x}, T)= ({1\over n})^T \sum_{\{\phi_\sigma\}} (1-n)^{R(\phi_{\sigma})}\ ,
\end{equation}
where $R(\phi_{\sigma})$ is the number of reversals for the path $\phi_{\sigma}$.
(It is assumed that $n>1$.)

Note that the sum of the absolute values
of the ``probabilities'' associated with steps in the four possible
directions is greater than one, and one of
them is in fact negative.  Even so, it is desirable 
to retain as much
of the path summation formalism as is possible.
One reason for this is that the retention of
this formalism  would allows us to take the abovementioned computation schemes 
that have proved successful with diffusion phenomena, and then
directly apply them to the wave equation.
That is, we could simulate wave phenomena by performing ensemble averages
over large numbers of particles executing random walks.

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Branching processes and phases}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
The path summation formula can indeed be retained, by
augmenting the
random walk into a branching process by endowing the
particles with phases.
In defining a branching process, note that a particle 
undergoing a random walk on a lattice may be described
in the following terms.  Consider, as before, each of the
lattice points to be ``black boxes'' into which
particles with unit speed enter.  
Suppose that the entering particle annihilates as soon
as it reaches the black box while at the same time
it causes it to emit an outgoing particle
into any of the $2n$ possible directions, with this
choice of direction being made randomly.

Suppose next that for every
particle that comes in, the black box is
allowed to emit more than one particle.
The emission probabilities
will be defined below.
Such a process will be referred to as a
{\it branching process;} obviously, its
introduction implies that the particle number is
no longer a constant.  

%ensemble average with phase

 
Next, assume that the lattice particles can be
in one of four states, or phases, so that
a particle can be positive, negative,
positive imaginary, and negative imaginary. 
For the sake of convenience,  let it be said
that a particle in one of the four above states
has a phase factor of 1, $+i$, $-1$, or $-i$.

%	In taking an ensemble average, 
The wave
amplitude can be obtained from
the lattice 
By simulating a large number of
identically prepared lattices and then
taking the the  number
of positive particles {\it minus} the number of
negative particles, plus $i$ times the
mean number of positive imaginary particles, and so on,
one can stochastically obtain the wave amplitude
(times a normalization factor equal to the number
of lattices involved in the computation).

The use of complex phases, although superfluous
in the case of the real wave equations, will
be necessary later on.

%rather
%than the corresponding multiple of $\pi/4$, even though
%this is an abuse of the usual language.

Now consider again
the above discussion of the two-dimensional
hodotic solution.  The
number $1/2$ can be interpreted to mean that, with probability one half,
the presence of a particle entering any lattice point
along the $x_{+}$ direction will cause the lattice point  to produce a particle
of the same phase exiting that point along the $x_{+}$ direction.
Likewise, it will with probability one half produce
a particle of the same phase exiting along the $y_{+}$ and $y_{-}$ directions.
The particles exiting along the $x_{-}$ direction will be generated
with the same probability, but will have a phase that is
the negative of that of the incoming particle.
Thus, a particle entering a lattice point may produce
more than one exiting particle, or none at all.


Assume next that whenever particles of opposite phase 
are found within a link, they
annihilate, so that at every
link only particles of at most two phases remain, with one phase
being $\pm i$ times the other.
Otherwise, the outputs of any particles coming
into a point along different links will be independent of one another.
That is, when two or more particles enter a lattice point,
their outputs will be generated independently of one another.
Therefore,  there may in principle
be an arbitrary number of particles traveling along any link. 
Moreover, assume that even for a single incoming particle,
the ``output'' particles produced
in one outgoing link will be generated independently of
the output particles in another outgoing link.

Such a prescription of the dynamics, and their
$n$-di\-men\-sion\-al analogues, will be preferred to as the {\it binomial dynamics},
since input particles produce or do not produce output particles
according to a binomial distribution.

Even though the binomial dynamics is perhaps
the easiest to consider, 
the wavefunctions can also be simulated with a particle
dynamics for which there are more general
distributions of the outgoing particles. Since it is only
the {\it net} number of particles which will be used to
determine the wave function,
an input particle may produce a variety of outputs in an output link.
For example, denote by $p(j,\phi)$ the probability that
an incoming particle produces $j$ particles in some output link, all of which
have a phase that is equal to $\phi$ times that of the input link,
where
$$\sum_{j,\phi}p(j,\phi)=1$$
then all that is  required is that the weighted sum 
$$\sum_{j,\phi}p(j,\phi)j\phi$$
is  equal to $+1/2$ for all but the negatively directed
output links, for which  the weighted sum must equal $-1/2$.
Even more general schemes will suffice, although the rest of
the paper assumes that the binomial dynamics are in effect.
%work, but for convenience, we
%shall continue to assume that the output particles from one incoming
%link are independent of those in another link, and
%that the particles in each output link are also generated
%independently of one another.


\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Ensembles}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
Suppose there is  a large, possibly infinite ensemble of 
identical, initially blank lattices, with which one
wishes to simulate a hodotic solution.
Any spacetime point $({\bf x},t)$ in one of the lattices
has a corresponding member in another lattice.
Suppose that at time $t=0$ one places at each lattice a positive particle
at the $x_{+}$ link leading {\it to} the origin. 
The statistics of the particles involved are
independent, so that knowing 
a certain particle has produced a given output in one lattice
tells  nothing about what any other particle will produce.

Again, the above initial
distribution of particles per lattice need not  have been  so uniform
as to have  one particle per lattice. There could
have been an arbitrary varying number of positive, negative, or imaginary, particles, so long as
the net mean number of particles is one.
%at the
%links in question, so long as if whenever we take a
%count of all the particles found in the corresponding links
%of say, $M$ lattices, the count we obtain as $M$ becomes large, approaches $M$.)


%NO COUNTER YET?????
%In subsequent sections, it shall be practical to consider all
%the members of the ensemble of lattices as regions of one
%large lattice.  Moreover for the purposes of
%making an ensemble average, it is best to 
%posit a {\it counter}. This counter is nothing more than
%a 

%Let us postulate a 
%{\it counter}, with which we go from spacetime point to
%spacetime point and increment the count according to the
%phase and number of the particles we find there,
%so that the count we obtain a complex integer.


Suppose that at $t=2$, one takes a count at the lattice point $(x,y)=(0,1)$, of
a large selection of $M$ of the above lattices.
The number on
the counter will, for large $M$, be close to $M/2$. Likewise,
at the point $(-1,0)$, the count will be close to $-M/2$.
In $n$-di\-men\-sion\-al lattices in general, the corresponding 
counts would be
$M(1-n)/n$ for the point corresponding to a reverse path step, and
$M/n$ otherwise.

Proceeding further, the net count at 
time $t=T$ at some point $(x,y)$, when divided by $M$, 
will be  close to
$h_{x_{+}}(x+1, y; T)$. Given the elementary rules
for the probability of multiple, independent events,  
when one calculates the expected
count of particles arriving
at a certain point,  the expression is, as above,
a sum of terms  similar to (\ref{eq:pat-amp}), each of which can
be related to a path.  
%Though the support of a hodotic
%solution at time $T$ is, strictly speaking an $n$-di\-men\-sion\-al
%cube,  the amplitude is negligible outside a spherical
%region of radius $cT$.

The above means of simulating
the hodotic solutions can be extended to any sum of hodotic solutions, and
thus any solution of the discrete wave equation.
To do so one  needs a large number $M$, of
lattices, having the following property: if one initially takes
a count along the points of each lattice that
correspond to ${\bf x}$, the net count divided by $M$
must be close to $\psi({\bf x};0)$, i.e., the initial
value of the wave solution that is to be  computed.

Henceforth, the {\it sample set} of  lattices
corresponding to a given wave solution,
will mean a (possibly infinite) set of lattices 
having the property that any random selection among them
of size $M$, where $M$ is a large number, has
the property that the initial normalized count is 
close to the initial wave solution.

If one is interested in computing phenomena
that last for $T$ time steps, then in general,
assuming that
at least $M_0$ lattices are required to
get the normalized initial count as close as desired to
the initial solution (and assuming that the
a binomial dynamics are in effect),
the results  obtained by normalized counting will be accurate if
$M$ is much greater than $T^{n} M_0$, where
$n$ is the dimension of the lattice.


Thus the preceding section  gives a statistical representation of the flows
and traveling waves of the wave equation.  The path summation formalism 
is retained, except that the  paths are converted into chains of causality.
The Huygens picture of wave propagation, whereby
any portion of a wave itself emits a wavelet, is
here given a statistical rendering, with the result
that motions of discrete particles can be used
to generate wavelike phenomena.

\section{Quantum mechanics}


Insofar as motions of discrete particles produce wavelike
phenomena, the above formalism is reminiscent of quantum mechanics.
This section  will further develop this similarity.
In quantum mechanics the quantity that
is experimentally observed is $\psi^{*}({\bf x}, t)\psi({\bf x}, t)$.
Later on, the Klein-Gordon equation and multi-particle systems will be considered;
for now,  assume that $\psi({\bf x}, t)$ satisfies the lattice wave equation,
and corresponds to the quantum mechanical 
wave function of some massless, spinless particle.  The question of normalization
will also be deferred until later.

%TO SUMMARIZE WHAT FOLLOWS.  ONCE PROBABILITY MEASURES
%HAVE BEEN GENERALIZED TO THE NONPOSITIVE REALM, TH STANDARD
%QM FORMALISM ALONG WITH THE REDUCTION OF THE WAVE FUNCTION
%INTO A PATH SUMMATION
%ALLOWS THE PROBABILITY OF OBSERVING A CERTAIN
%EVENT TO BE A SUM OF TERMS, EACH OF WHICH MAY BE RELATED
%TO A PAIR OF PATHS, WHICH IN KEEPING WITH USUAL
%NOTATION, SHALL BE REFERRED TO AS THE BRA PATHS AND KET PATHS.

%PUT THIS IN PAPERTW4 AND THEN SEND IT INSTEAD OF PAPERTW3, BUT
%KEEP THE METHOD OF INDISTINGUISHABLE PARTICLES?

%THUS IN A TWO-PARTIC

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Bilattices}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
%
Consider again the sample set of lattices corresponding to
some wave function $\psi({\bf x}, 0)$. Consider next 
the set formed by all {\it ordered pairs} of elements
of this set.  
The particles on
each of the two lattices may be superimposed onto one of these
{\it bilat\-tices,} so long as
the  particles belonging to each member of the pair are distinguished.
Particles belonging to the first member of the ordered pair
will be called {\it bra\/} particles. Particles belonging to
the second pair will be called {\it ket\/} particles.
The motion of the bra particles is totally independent
of that of the ket particles.


\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Event counting}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
At any given time during the evolution of a large set of bilat\-tices,
one can make a count at the points corresponding to ${\bf x}$ on
each 
of the bilat\-tices, in the following fashion:
finding $j$ bra particles in the $\phi$ phase, and
$k$ ket particles in the $\phi'$ phase at any of the points,  increments
the count by the number $jk \phi^{*}\phi'$. 
Note that the bra and ket particles need not have come to the point  on
the same link. 

Again, $\phi$  in this context means the number $1, +i, -1$, or $ -i$, according
to whether the phase of the particles is respectively positive, positive imaginary, negative, or
negative imaginary.  For the same respective listing of phases, the  conjugated phase
$\phi^{*}$, will mean the number $1, -i, -1$, or $+i$.  The increment will be 
linear in the number of ket particles and antilinear
in the number of bra prticles, so that
if  the bra or the ket particles
are of more than one phase, the increment will be the one obtained by the corresponding
complex multiplication.

Because the assignment of particles in
any pair of lattices is statistically independent,
the above count divided by $M$ will 
approach $\psi^{*}({\bf x}, t)\psi({\bf x}, t)$, for large $M$.
By way of analogy, this counting of the simultaneous
presence of bra and ket particles will be referred to as single particle {\it event counting},
even though the integer increments can be complex, and with a modulus greater than one.
Note however, that the expected net event count will be a real, nonnegative number.
Later on, this counting will be extended to multi-particle event counting, where
at  multiple points ${\bf x}_1,\ldots,{\bf x}_N$,  the
count will be incremented according to the product of the single particle increment at ${\bf x}_1$
times the single particle increment at ${\bf x}_2$, and so  on. This count will be taken
in such a way as to obtain correct results
for $N$-particle states of fermions.

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Phase invariance}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
Consider the operation of multiplying
all the ket particles in a bilattice
by $+i$.  This  means the operation of changing
all positive particles into positive imaginary particles,
all positive imaginary particles into negative particles, and so on.
One may similarly speak of multiplying the particles  by $-1$, or $-i$.

Note that in considering a large set of
bilat\-tices, if one chooses to multiply
all the bra particles in any bilat\-tice by a random choice
of $1, +i, -1$, or $-i$, 
the event count will be unchanged if
 the
phases of the ket particles are correspondingly
multiplied by $1, -i, -1$, or $+i$, respectively.
Henceforth,  assume this has been done
to all the bilat\-tices. 

Therefore, because
of the resultant incoherence of phases,
when one counts only the number of bra particles
at the points corresponding to ${\bf x}$ in any selection of sample bilat\-tices of length $M$,
the count approaches zero as $M$ becomes large,
and likewise for the ket particles.
Also, any multi-point cross correlation of bra particles
alone, or ket particles alone  also vanishes.
However, multiparticle counts for which 
one  increments the  count by the product of the bra and 
(the conjugated) ket particles there,
will remain unchanged.

Note that one may count events at more than one point {\bf x} in
a bilat\-tice.  That is, one can increment the count at several points
per lattice, or in the case of $N$-particle correlations, at several
sets of distinct lattice points. One can also consider situations in
which all of the bilat\-tices corresponding to a selection from a
sample set are regions within one large bilat\-tice, and in which the
count is taken along a set of points which can be connected by a
spacetime path on the lattice.


Such counting procedures are reminiscent of physical experiments of
the kind in which information about quantum mechanical states is
obtained by having a beam of particles sweep past a detector, or
equivalently, experiments in which a detector sweeps through regions
corresponding to the wavefunctions of many stationary particles.

\subsection{Single particle wave functions}

{\obeylines
\noindent{{\it Initialization and normalization}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
Consider next the problem of initializing the bilat\-tices
corresponding to a given set of initial conditions.

The
discrete integral of
the absolute squares of the flows along the links in the lattice
is constant, and nonnegative. It is this quantity which
will be treated as a probability density. Henceforth,
assume that all the wave functions are normalized so 
as to make the latter integral equal to one.
Obviously, this quantity
is not a true probability; nevertheless, it will be true
that for a large number $M$  of properly initialized
bilat\-tices, the  event count at
the points corresponding to $({\bf x};t)$ will
be close to $M|\psi({\bf x}, t)|^2$.

It is trivial to initialize a bilat\-tice corresponding
to the two-di\-men\-sion\-al hodotic solution considered above.
In any such lattice, one must with probability one (i.e., with
certainty) place a single bra and a single ket in the $x_{+}$ 
link leading into the origin. The phase will be arbitrary, but
will be the same for both the bra and the ket particles at
any point. For convenience,
assume it is positive. At the next time step, if
one were to perform an event count at the point $(-1,0)$ (implying a reverse step), 
then with
probability $1/4$, the  event count will be incremented by one, since
with probability $1/2$, there will be a negative bra particle
and with probability  $1/2$ there will be a negative 
ket particle coming into that point. 

Consider next how to initialize bilat\-tices corresponding to 
a sum of two hodotic solutions
\begin{equation}
\psi({\bf x}, t) = {1\over{\sqrt2}}
 \left(h_{\sigma_1}({\bf x} - {\bf x}_1, t) + 
  e^{i \beta}h_{\sigma_2}({\bf x} - {\bf x}_2, t)\right)\ {\rm ,}
\end{equation}
where ${\bf x}_1$ and ${\bf x}_2$ are distinct.
Assume first that $\beta= 0$.

First, one randomly chooses one of the four phases;  denote the choice by $\phi$. 
For the purposes of such initial assignments,
assume there exists a random bit generator with a parameter $p$,
that
with probability $p$ outputs a `1' and with probability $(1-p)$ outputs
a `0', where the probability $p$ may be chosen at will to be any value
between zero and one.
Multiple outputs from this generator are  statistically independent.

Then, setting the parameter to $1/\sqrt2$,  a bra particle
of the phase $\phi$ will be placed into the $\sigma_1$ link
leading into the point ${\bf x}_1$ if the output is a `1'.
If the output is `0', no particle will be placed there. 
Likewise, if the next bit is a `1',
a ket particle of the identical phase will be placed there.
The identical procedure is repeated at the point ${\bf x}_2$.
Thus, at the end of the assignment, there may be anywhere from zero to two
bra particles on the lattice, and zero to two ket particles, all
of the same phase.

Suppose next that one wishes to initialize a bilattice
corresponding to a wavefunction for which $\beta=\pi$.  Then at the
point ${\bf x}_2$, if the output from the
bit generator is a `1', one places a bra particle with a phase
of $-\phi$, and then repeats this procedure to determine whether
or not to place a ket particle of phase $-\phi$.

For $0<\beta<\pi$, the procedure at the link at ${\bf x}_2$ would
be to first set the bit generator's
parameter to $\cos(\beta)/\sqrt2$, and then place a
bra particle of phase $\phi$ if the output is a `1'.
This procedure would be repeated for the ket particle.
Then, the bit generator's parameter would then be set to
$\sin(\beta)/\sqrt2$, and if the next bit were a `1',
a bra particle of phase $i\phi$ would be placed in the
same link. With the parameter still set to $\sin(\beta)/\sqrt2$,
the next bit would determine  the placing of a ket particle 
of phase $i\phi$ at ${\bf x}_2$.

The general procedure should now be clear.
Suppose  $\psi({\bf x},t)$ is expressed as
as a sum of hodotic solutions
\begin{equation}
\psi({\bf x},t)= \sum_{{\bf x'},\sigma}{\cal A}({\bf x'},\sigma)h_{\sigma}({\bf x} - {\bf x'},t)\ {\rm .}
\end{equation}
Assume also that an initial phase
$\phi$ has been randomly chosen. At each link on the lattice, one must independently
determine whether or not to place a bra particle there.
At say, the $\sigma $ link leading into the point ${\bf x}$,
one must first set the bit generator's parameter to the
absolute value of the real part of ${\cal A}({\bf x},\sigma)$,
$|{\rm Re}\{{\cal A}({\bf x},\sigma)\}|$.
This quantity will not be greater than one if, as assumed,
the normalization discussed above has been performed.
If the output is a `1', a bra particle is placed there, of phase $\phi$ if
${\rm Re}\{{\cal A}({\bf x},\sigma)\}$ is positive and of phase $-\phi$ otherwise.
Next  the bit generator is set to $|{\rm Im}\{{\cal A}({\bf x},\sigma)\}|$, and
if the output is `1', a bra particle is placed there, of phase $i\phi$ if
${\rm Im}\{{\cal A}({\bf x},\sigma)\}$ is positive and of phase $-i\phi$ otherwise.
The above procedure is then repeated at this link to determine 
what kind of ket particles, if any, to place there.  Bra and ket
particles are assigned to all the other links of the bilattice in the same way.

If one repeats the above procedure for a large number $M$
of bilat\-tices,
%(we assume that for each bilat\-tice, we randomly choose a new phase,) 
then by  letting each one
evolve,  one can at any  time take
a count at the spacetime points corresponding to $({\bf x};t)$.  Then by dividing
this count by $M$, one can obtain a computation of $|\psi({\bf x},t)|^2$, with
the  accuracy of this result depending on the size of $M$.
One can go on to construct a series of bilat\-tices corresponding to wave pack\-ets of
some quantum particle traveling along some direction, whose momentum
along that direction
is given by the reciprocal of the average wavelength 
within the wave pack\-et.
%\footnote{See, for example, C.~Cohen-Tannoudji, B.~Diu, F. Lalo\"e,
%{\it Quantum Mechanics}, Vol.~I, Chap. 1, John Wiley \& Sons, New York, 1977.}
It is necessary to  assume, of course,  that any such wavelengths will be much larger than
the lattice spacings.

%KILL THIS NEXT PARAGRAPH.
%Imagine now combining  the configurations of $M$ bilat\-tices into one.
%First let us assume that we are trying to simulate a two-di\-men\-sion\-al wave function
%that corresponds to a wave packet that is appreciably non-zero in some region
%much smaller than the length of an entire bilat\-tice, and which is initially located about, say the
%point $({\bf x}_0)$.  After using the above procedure for constructing a large number $M$ of
%such initial bilat\-tices, each of which has its own randomly selected
%initial phase, let us imagine translating the origin of each lattice by
%some random vector.  (We suppose for convenience that the bilat\-tices are toroidal.)
%Let us now superimpose all the bilat\-tices onto the first one. That is,
%let us at each point $({\bf x})$ of the first lattice add the
%particles in the second bilat\-tice corresponding to that point
%({\it after} the random translation), then the particles of
%the third lattice, and so on, allowing the particles of
%opposite phase to annihilate.  
%We suppose that the support of the wave packet is so small, that
%the probability that any two supports overlap after their
%random translations is very small.  Thus we will be
%left with one lattice covered with small regions in
%which there is a probability of finding some particles.
%Assuming that  the wave function in question is normalized
%to 1, then if we imagine making an event-count at every point
%in the space, the expected value of the counter would be $M$.
%%%%%%


\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Double-slit diffraction}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
As mentioned previously, the above results for single particle wave functions
may be used to produce an ensemble of similar wave functions, all on one
bilat\-tice.
Consider
the {\sl ge\-dan\-ken\-ex\-pe\-ri\-ment\/}
in which a beam of microscopic particles  is made to exhibit
diffraction behavior, such as is described in
introductory quantum mechanics texts.$^6$
%\footnote{
%For example, R.~P.~Feynman, 
%R.~B.~Leighton, and M. Sands, {\sl The Feynman
%Lectures on Physics}, Vol.~III, Chap. 1, Addison-Wesley, Reading, Massachusetts, 1966.}
In such an experiment, there is a beam of $M$ particles qua wave packets, all of which
have the same momenta along, say, the $z$ axis, ${\bf p} = k {\bf \hat z}$.
These wave packets impinge on a screen, oriented orthogonally to
the $z$ axis (Fig.~\ref{fi:ele-dif}).  

Suppose that
the concentration of wave packets in the beam is
so dilute as to make the overlap between
them negligible. 
On the screen are two thin, parallel
slits, I and II (oriented along the $x$ axis),
separated by a distance on the order of the 
de Broglie  wavelength of the electrons.  At some
point, a distance of many wavelengths away along
the $z$ axis, is another screen.  It is at one point, ${\bf x}_c$, of
this far screen at which the  counting will be performed.
(The screens and other classical objects may 
be taken to be 
perfect absorbers of the incident wave packets.)

Assume that the above initialization method has been applied to
every one of the regions in the particle beam
for which the wave function is nonzero, so that in effect, an $M$-particle wavefunction
is being simulated.
That is, at each point in space where the hodotic expansion of
one of the wave pack\-ets is nonzero,
 bra and ket particles are assigned to each link in the region,
according to the amplitude there.  The choice of initial  phase will be
the same throughout the region corresponding to 
any single wave packet, but will  differ
from region to region.  
(The discussion of how to initialize regions where
wave pack\-ets do overlap will be deferred until the
discussion of multi-particle simulations.)
Thus, given that each wave pack\-et is
initially normalized to one,
if one were to make an event count at
every point of the space, the expected value of the count would
be $M$.

As far as $M$-event simulations are concerned, the single
bilat\-tice in question is useless for determining
$M$-particle correlations.  For such a purpose, 
one needs to initialize a large
number of such bilat\-tices, each corresponding to
the same set of single-particle wave functions.
Yet, just as  an ensemble of similar wave pack\-ets allows
the physicist to make accurate determinations
regarding single particle behavior, the above
bilattice will serve  as 
$M$ separate single-particle simulations.

%Let us suppose that we perform an event
%count at a point on the screen at
%which the probability
%
\begin{figure}
%\rule{.2pt}{13em}
\psfig{file=hx/twfig/twf-gedank-2.ps,height=2.162in,width=3.0in}
\vspace{2.0\baselineskip}
%box is 3" by 5 1/8", lower left corner 1" 7 1/2"
%                         upper right corner 6 1/8" 10 1/2"
%Steal a pic from Feynman's book.
\caption{The two-slit
particle diffraction
{\sl ge\-dan\-ken\-ex\-pe\-ri\-ment}. A beam of
particles impinges on a screen with two parallel slits, which results
in a diffraction pattern at a farther screen. }
\label{fi:ele-dif}
\end{figure}
%



\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Coherence}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
Quantum mechanics tells us that for
the {\sl ge\-dan\-ken\-ex\-pe\-ri\-ment\/} which the
bilat\-tice simulates, the
rate of events recorded, which
in the real world come in discrete (and positive) pulses, is 
proportional to
$$\left|\psi_{\rm I}({\bf x}_c) + %
\psi_{\rm II}({\bf x}_c)\right|^2$$
where $\psi_{\rm I}$ is the amplitude
whose square would determine the rate
of particles on the screen if only slit I were open, and so on.
%of a wave pack\-et that passes through slit I, and so on.

Suppose that the place where
event counting is to be performed corresponds to
a region on the observation screen where 
there is complete cancellation, i.e.,
where the amplitude associated with
one slit totally cancels the amplitude associated with
the other.  

Instead of there being no events 
whatsoever,
the event count of the simulation  would  be the result of many events,
but as many positive as negative, so that
after a long period of time,
the net number in the  count would be vanishingly small
in comparison with the event count obtained
in some region where cancellation does not occur.

The absolute square of wave function at the point ${\bf x}_c$ where the
event count is performed can be written as
%
\begin{equation}
\psi_{\rm I}^{*}\psi_{\rm I}^{\vphantom{*}} + 
\psi_{\rm II}^{*}\psi_{\rm II}^{\vphantom{*}} +
\psi_{\rm I}^{*}\psi_{\rm II}^{\vphantom{*}} +
\psi_{\rm II}^{*}\psi_{\rm I}^{\vphantom{*}}\ {\rm ,} 
\label{eq:two-slt}
\end{equation}
where the dependence on ${\bf x}_c$ has been  suppressed.
%
Note that the events, i.e., the simultaneous arrivals of
bra and ket particles, can be conceptually placed into
four classes, corresponding to each of the above four terms.
A particle from the first class represents an event for
which both the bra and ket particle came through slit I;
the second, an event for which both came through slit II;
the third, an event for which the bra particle came through
slit I and the ket came through slit II; and
the fourth, an event for which the bra particle came through
slit II and the ket came through slit I.
(Saying  that the  bra particle of a certain event came through
a certain slit means that the causality chain
of interactions that led to the
event count increment can be related by a
spacetime path that leading through that particular slit.)
Because the contribution coming from slit I is the exact
negative of that coming from slit II, the net result
of the latter two classes exactly cancels (on the average)
the contribution from the first two classes.


If  slit II is closed,
the outcome in both the corresponding physical situation
and in the simulation  is such that the
event count  count is proportional 
to $\psi_{\rm I}^{*}\psi_{\rm I}^{\vphantom{*}}$;
i.e., only the events for which both the bra and
the ket come through slit I are present.

In general, there are also terms in
the event counts representing situations for which a
bra particle originating from one wave pack\-et arrives at
${\bf x}_c$ at the same time as a ket particle originating
from some other wave packet. These terms  have been ignored in (\ref{eq:two-slt}). Now
because of the random assignments of phase
to each wave packet, the probability that
the relative phase
of the bra and ket particles is, say $\phi$,   is equal
to the probability that it is $-\phi$. Therefore,
for a large number of initial wave pack\-ets,
the net contribution from such occurrences  vanishes. But note
that if 
all the initial phases had been the same,
the event counts  would have been proportional
to $M^2$, a situation similar in some respects to the physical phenomenon known 
as  superradiance.

Next, suppose slit II is again opened  and
some phase-shifting device is inserted there,
whose effect is to negate the phase of all
particles traveling through that slit. Then
the last two terms in (\ref{eq:two-slt}) would
be negated, so that the expected event count would be four times what
it would have been with slit II closed. 

Next, suppose  one of the slits is perturbed in a random, time-va\-ry\-ing
fashion so that the phase of particles entering there
would likewise be randomly shifted. Then the latter two terms
in (\ref{eq:two-slt}) would be zero, i.e., there would
no longer be  evidence of interference.
The event count would then correspond to classical phenomena
for which the number of events coming from the two
slits is a sum of the ones obtained by first closing only one
slit, and then only closing the other.
The {\sl ge\-dan\-ken\-ex\-pe\-ri\-ments}   which are
used to exhibit the Heisenberg uncertainty principle
can also be illustrated in the context of the lattice model.  There,  it is 
the perturbations from a physical
detector which destroy the interference effects, i.e., the
wavelike behavior.

In general then, a set of unpolarized events, corresponding
to some statistical mix of states, that is typically described
by a density matrix, is to be seen, in terms of the  model,
 as resulting from several subsets for which
the relative phase responsible for the interference effect
varies randomly from one subset to the next. This destroys the 
net contribution of events responsible
for exhibiting the polarization. 
Note  that these subsets could have been rendered unpolarized 
at the initial assignment of particles;
the incoherence need not have been induced through perturbing the slits.
%
%
%For example, we could  initially have had  wave packets corresponding
%to two
%differing momenta, such that the net event count contributions
%corresponding to the last two terms in (\ref{eq:two-slt}) are
%respectively positive and negative for the two types of wave pack\-ets.

One can go on to reproduce other physical  experiments.
Note that the dynamics of the particles involve no  momentum. In fact, the particle motions are only slightly removed from  random walks.
However, suppose that one were to replace the double-slit screen in
the above example with an arrangement of slits  corresponding to
a diffraction grating. Then the net event count for all points
on the distant screen would be zero except for one region
(or, depending on the grating, several equidistant regions)
where there would be a large peak in the counting rate. The shape and location of this
peak would depend on the Fourier transform
of the associated wave packets, as well as the slit spacing.  For incoherent ensembles of
particles, corresponding
to wave packets having differing momenta, there
would  be several such peaks.  In other words, any
ensemble of these randomly meandering particles, properly prepared,
can indeed be said to be in a quantum state with a certain momentum, or in a superposition of
such states.

\subsection{Multi-particle phenomena}

Consider next the simulation of multi-particle phenomena.
Two methods will be presented here: in the first, all the lattice particles
associated with a single particle wave function carry an
identical label of some sort. Thus, they can be distinguished
from the particles associated with some other one-par\-ti\-cle wavefunction.
In the second method, all particles are indistinguishable.
The first method is particularly useful as an introduction to the
second,
which is easier to implement, but otherwise more difficult to explain.
Assume for the purposes of the presentation that all the particles 
are fermions.  The former method can, with trivial modifications in
the event counting, also be applied to bosons as well.
If one wishes to simulate boson states with the latter method,
the bosons must be represented as mul\-ti-par\-ti\-cle
states of some even number of fermions, just as
pions are in fact composed of two fermionic quarks.



%\vspace{.6\baselineskip}
%\noindent{{\large The Method of Distinguishable Particles}}\nopagebreak
%\vspace{.3\baselineskip}\nopagebreak
%
\subsubsection{The method of distinguishable particles}


Proceeding to the first method, suppose that the
lattice particles have an extra degree of freedom.
The constancy of this degree of freedom is such that
it will be best to think of it as merely a label.
%The specifications of these
%identifying labels, which serve only to distinguish each wavefunction's particles,
%are few. 
For now, it is only necessary to assume that these labels are
members of a possibly infinite set of distinct tags, 
denoted here by  letters.  In other words, it is not necessary for the
set 
to possess an ordering relation.
In fact, the number of possible tags does not in  practice
 need to be infinite either. It 
must only be as large as the number of single-particle states in
the physical situation one  wishes to simulate. 

Also, because the particles involved are fermions,
 it will be shown below how it is possible to alter the dynamics in such a way
that at any link in the lattice,
 bra particles of only one tag are present. Likewise, the ket particles
at any link are also all of the same tag, resulting in a great  
simplification of the implementation of the model.

The tags can be seen as a kind of rudimentary momentum, in
that they represent an extra degree of freedom
that a given particle can possess. However,
they do not constrain the behavior
of a given particle in any way, but merely serve to 
distinguish one particle from another.


\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Two-particle states}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
Consider first the simulation of a two-particle universe
where $\phi({\bf x}; t)$  and $\chi({\bf x}; t)$
correspond to two hodotic solutions, that initially are
nonzero at
links leading from the distinct points ${\bf z}_1$ 
and ${\bf z}_2$, respectively.
The directions of these links is unimportant; assume
they are both in the $x_{+}$ direction.
The quantum wave function is 
\begin{equation}
\psi({\bf x}_1, {\bf x}_2) = 
{1\over\sqrt2}\left(\phi({\bf x}_1)\chi({\bf x}_2) -  
\chi({\bf x}_1)\phi({\bf x}_2)\right)\ {\rm ,}
\end{equation}
where the dependence on time has been suppressed.


This is in fact a Slater determinant of states, of order two.
For present purposes, it is only necessary  to consider  one of the
terms in the determinant, say,
$$\phi({\bf x}_1)\chi({\bf x}_2)\ \ .$$
To simulate the above system, one places, with probability
one (i.e.,~with certainty), a bra particle with some tag, call it $A$, in
the $x_{+}$ link emanating from the point ${\bf z}_1$. The same is done
for a ket particle with the same tag. 
The bra and the ket particle are of the same
phase, which has been chosen at random.  At the $x_{+}$ link leading from
${\bf z}_2$, one must also place a bra and a ket particle of some
other tag, call it $B$.  Again, the bra and ket particles will have
the same phase, but one that is 
totally random with respect to the $A$ particles.

Assume just for the present that the dynamics of the $A$ and the $B$ particles are totally
independent of one another, so that at any point in space there may in principle be
two kinds of bra particles.  The $A$ particles may be in at most two phases (otherwise
annihilation occurs), and likewise for the $B$ particles.  

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Two-particle event counting}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
%
Consider the simultaneous arrival of particles at two observation points,
${\bf x}_1$ and ${\bf x}_2$. A two-par\-ti\-cle event refers to 
the simultaneous arrival of two distinctly tagged bra particles
at the respective observation points, along with the simultaneous arrival
of two distinctly tagged ket particles at those same points, such that each bra particle's
tag is the same as the tag of one of the ket particles. 

The sign of the event  depends on the the relative location of the
tags. Assume first that the bra and
the ket particles at   ${\bf x}_1$ both have the same tag, (and thus
the bra and
ket particle at ${\bf x}_2$ both have some other tag).  For such an event,
the event count is incremented
by the product of the
phases of the ket particles, times the conjugated product of the phases
of the ket particles.  The count is again
linear in the number of ket particles and antilinear in the number of bra particles at each point. 
Thus, if there are simultaneously two ket particles of the same phase at ${\bf x}_1$
the count is double what it would be if only one were present, and so on.

This type of event will be denoted by a diagram
of the form
$$
{\setlength{\unitlength}{12pt}
\begin{picture}(1,1)
\put(0,1){\line(0,-1){1}}
\put(1,1){\line(0,-1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\end{picture}}\ \ \ .
$$
The top circles represent the ket particles at ${\bf x}_1$ and ${\bf x}_2$,
respectively. Likewise, the bottom circles represent the bra particles
at the two respective points.  The line connecting the first ket circle with the
first bra circle indicates that the  particles at the first point have the same tags,
and likewise for the line connecting the second bra and ket.

Next, assume the two-particle event is such that a ket particle  with a
given tag arrives at the point  ${\bf x}_1$ while 
simultaneously  a bra particle with the same tag arrives at ${\bf x}_2$,
(and thus simultaneously a ket particle of another tag arrives at ${\bf x}_2$ 
while a bra particle with that same  other tag arrives at ${\bf x}_1$).
In this case the event count is incremented by the {\it negative} of the product
of the ket phases times the conjugate of the product of the bra particle phases.
The corresponding diagram for this type of event is
$$
{\setlength{\unitlength}{12pt}
\begin{picture}(1,1)
\put(0,0){\line(1,1){1}}
\put(0,1){\line(1,-1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\end{picture}}\ \ \ .
$$
Note that if one exchanges the lower circles
of the  second diagram (keeping the identically tagged particles
connected by lines), the result will be a diagram like
the first. Also note that although
particles connected by lines must have the same tags,
they do not in general have the same phases.

Consider now the event count in terms of
the original wave functions.  Because of
the independence of the branching process at
each point in space, the expected value
of the positive counts, call 
it ${\cal C}({\bf x}_1,{\bf x}_2)$  can be written in terms of
the sum
\begin{eqnarray}
{\cal C}({\bf x}_1,{\bf x}_2)=  %
\phantom{{\cal C}({\bf x}_1,{\bf x}_2){\cal C}%
({\bf x}_1{\cal C}({\bf x}_1,{\bf x}_2){\cal C}%
({\cal C}({\bf x}_1{\cal C}({\bf x}_1,{\bf x}_2){\cal C}({\bf x}_2)} 
\nonumber \\
\phi^{*}({\bf x}_1)\phi({\bf x}_1)\chi^{*}({\bf x}_2)\chi({\bf x}_2) - %
\phi^{*}({\bf x}_1)\chi({\bf x}_1)\chi^{*}({\bf x}_2)\phi({\bf x}_2)\phantom{\ .} \nonumber \\
%\phantom{{\cal C}({\bf x}_1,{\bf x}_2)}%
- \chi^{*}({\bf x}_1)\phi({\bf x}_1)\phi^{*}({\bf x}_2)\chi({\bf x}_2) + %
  \chi^{*}({\bf x}_1)\chi({\bf x}_1)\phi^{*}({\bf x}_2)\phi({\bf x}_2)\ {\rm .} \nonumber
\end{eqnarray}
Assume the assignment of $A$ and $B$ with respect to the two
single par\-ti\-cle states is as above. That is, the $A$ particles
are initially at ${\bf z}_1$ and the $B$ particles are initially 
at ${\bf z}_2$.
Then the first term in the previous expression is belongs to the class of events for which
an $A$ bra and ket particle arrive at ${\bf x}_1$ (from  ${\bf z}_1$) and
 a $B$ bra and ket particle arrive at ${\bf x}_2$  (from  ${\bf z}_2$).
Likewise, the second term belongs to the class of events for which
the $A$ bra and ket particles arrive respectively at ${\bf x}_1$ and  ${\bf x}_2$,
while the  $B$ bra and ket particles arrive respectively at  ${\bf x}_2$ and  ${\bf x}_1$,
and so on.

%Conversely, the negative counts can be written as
%\begin{equation}
%\phi^{*}({\bf x}_1\chi({\bf x}_1\chi^{*}({\bf x}_2)\phi({\bf x}_2) +
%\chi^{*}({\bf x}_1\phi({\bf x}_1\phi^{*}({\bf x}_2)\chi({\bf x}_2) 
%\end{equation}
%and we can likewise partition any event according
%to whether the bra particle at ${\bf x}_1$ is
%has an $A$ tag or a $B$ tag.

Again, note that in  the formalism presented here, there is
no difference between the phase space point corresponding
to $\psi({\bf x}_1,{\bf x}_2)$ and the one
corresponding to $\psi({\bf x}_2,{\bf x}_1)$.
It is the {\it sum} of the probabilities associated
with these phase space points that is being determined, hence the absence
of any factors of $1/\sqrt2$, (or in the
general case, $1/\sqrt{N!}$) in the
initial assignment of bra and ket particles.
Alternately,  it does not matter which tag
is called $A$ and which one is called $B$, and
there is a probability of $1/2$ of either
assignment of tags.  One can therefore account for the
discrepancy of the factor of $1/\sqrt{N!}$ by considering it
as the result (in the general, $N$ particle case) of selecting one of the $N!$ specific terms
in the
Slater determinant.

Also note that while it is possible to include in the event count 
those arrivals of bra and ket particles corresponding to the terms
\begin{equation}
\phi^{*}({\bf x}_1)\chi({\bf x}_1)\phi^{*}({\bf x}_2)\chi({\bf x}_2) 
\end{equation}
and their like, these would on the average vanish anyway. 
This is because of the initial randomness of the relative phase
between the particles assigned to the $\phi({\bf x})$ state and the particles
assigned to the $\chi({\bf x})$ state. (This is one benefit
of introducing complex phases.)
For the same reason, if one were to perform a single particle event count
at any point, one  could also include events for which the bra and ket particle
were not of the same tag; they too would vanish, for the same reason.

%Note that each particle arriving at a given point can be related to a path
%leading either from ${\bf z}_1$ or ${\bf z}_2$ to that given point.
%Likewise,  a term of the form $\phi({\bf x}_1)\chi({\bf x}_2)$ can
%be related to a sum over pairs of such paths, for which
%a ket particle of one tag, say $A$, initially at the point ${\bf z}_1$  arrives at ${\bf x}_1$,
%simultaneously to a ket particle of another tag, say $B$, arriving at ${\bf x}_2$, having
%initially been at  ${\bf z}_2$.

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Coalescing particle tags}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
Because the wavefunction at any point corresponding to either state
can be written as a sum over particle paths, the
product of  the two  wave functions at two points may be written as a sum over pairs
of particle paths.
Consider all such pairs of paths for which the two distinctly tagged
ket particles are at some time within the same link.  Thus, the $A$ and $B$
particles may be said to travel (via branching processes)
into the same link, whereupon exiting, the $A$ particle
eventually gets to, say, the observation point ${\bf x}_1$ in the same instant as the $B$ particle
gets to ${\bf x}_2$ (cf.~Fig.~\ref{fi:swi-pat}).


Note that because the two particles were initially
in the same link, the  probability of
one of them subsequently arriving at a given point with some phase  is
the same for both particles  (up to a factor representing
their   phase differences  at the time when they were both in the same link).
In other words, for any pair of paths in which the $A$ particle arrives at ${\bf x}_1$
and the $B$ particle arrives at ${\bf x}_2$, there is another equally probable
pair of paths, call it the switched pair, for which the $A$ particle arrives at ${\bf x}_2$ 
while the $B$ particle arrives at ${\bf x}_1$, i.e., for which
the particles exchange their paths subsequent to entering the common link.

The first pair of paths  would appear in the path summation expression
of  $\phi({\bf x}_1)\chi({\bf x}_2)$,
while the switched path  would appear in the
expression  of $\chi({\bf x}_1)\phi({\bf x}_2)$.

\begin{figure}
%\rule{.2pt}{13em}
\psfig{file=hx/twfig/twf-wf-prod-2.ps,height=1.925in,width=3.0in}
\vspace{2.0\baselineskip}
%box is 7" by 3", lower left corner 1/2" 7 1/2"
%                         upper right corner 6 1/2" 10 1/2"
\caption{Products of hodotic wavefunctions
%which were initially nonzero at ${\bf z}_1$ and ${\bf z}_2$ respectively
%and which lead to the simultaneous incidence of particles
%at the observation points ${\bf x}_1$ and ${\bf x}_2$
can be related to a sum over pairs of
paths.  Any pair of paths which coincide at some link 
can be matched
with a pair  whose steps subsequent to entering the link
are interchanged.  The net contribution of such pairs is zero.}
\label{fi:swi-pat}
\end{figure}

Now any two such particle events for which the two ket paths
have a path step in common is just as  likely as  
an event for which the bra paths are identical, but for
which the the ket paths are the switched paths.  But 
because of the counting scheme, the two events will
have opposite signs,  and will therefore cancel. 
%(since they
%reverse the parity of
%the ket permutation of tags with respect to the
%fixed bra permutation of tags).
This calls to mind  the  Pauli exclusion principle,
which says that two indistinguishable particles cannot
be in the same state. In the context of this model,
the manifestation of this principle is due to the fact that
events which imply the simultaneous presence of
distinct particles have a vanishing net contribution to
the event count.

By taking advantage of this property, it is possible
to greatly  simplify
the particle dynamics.  One cannot simply have
coincident particles of opposite phase annihilate;
that would distort the event counts involving  any one of the two kinds of tagged particles.
Rather, whenever two particles of differing tags are to be placed
into an outgoing link, one can instead place two particles having  {\sl only one tag}.
That is, one can have either of the particles take on the tag of the other.
If there are only two particles present, then the surviving tag has
an equal probability of having either of the two initial tags.
Rather than having an $A$ particle and a $B$ particle exiting a link with probability one,
there will instead be two $A$ particles or two $B$ particles, each with a probability of
$1/2$. Therefore, the statistical weight attributable
to each of the path steps in question remains unaltered.  

In order for this dynamical refinement to produce the
proper sign, a particle that assumes another particle's
tag must also assume the phase of the other particle.
As an example, suppose a positive $A$ particle and a negative
$B$ particle arrive at a given link.  In that case, the exiting  
particles will both be
positive, and of tag $A$, or
both of them negative, with tag $B$; the probability
of either outcome is $1/2$.

It is possible to generalize this procedure
to other configurations of incoming particles.
Suppose that $j$
bra particles of tag $A$, and phase $\phi_A$, and $k$
bra particles of tag $B$ particles of phase $\phi_B$
are found in a single link.  (It is assumed  that
all oppositely phased particle with identical tags
have already been annihilated.)
Then with probability $j/(j+k)$ all the
particles will be given the tag $A$ and the  phase $\phi_A$, or with
probability $k/(j+k)$ they are all given  the
tag $B$ and the phase $\phi_B$. 


In general, the particles with a given tag are of two
phases, with a relative phase difference of $\pm i$.
One can simply choose to ignore the
particles corresponding to one of the phases for
the purpose of the tag reassignment. However,
this method of coalescing properties can 
also be used to combine particles of one tag having two distinct phases, thereby
reducing the number of possible phases at any given link from two to
one. For example, instead of having in one link 3 positive and also 5 negative
imaginary particles, all of one tag, all eight particles can instead
be set
to one phase, by making them all positive, with a probability of $3/8$, or
all negative imaginary, with a probability $5/8$. Therefore the statistical
weight of the paths leading into the link with either phase is unaltered. 

The ability to coalesce the identities of coincident particles
greatly simplifies the implementation of the model. Instead
of having to keep track of a large number of tags at each link in the lattice,
one only needs to consider a single tag (for a total of two per bilat\-tice). 
Thus, at any link of a lattice, there needs to be a record only of the number of
particles there, and if that number is nonzero, the (single) phase and tag of
those particles.


\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Changes of basis}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
Returning to the two-particle hodotic solution considered above,
consider, as an aside, the following similar two-particle solution.
Instead of having $\phi({\bf x})$ be the hodotic solution $h({\bf x} -{\bf z}_1)$,
and having $\chi({\bf x})$ be the hodotic solution $h({\bf x} - {\bf z}_2)$
(the link specifications have again been suppressed), one may 
instead set
\begin{eqnarray}
      \phi({\bf x}) = {1\over\sqrt2}\left(h({\bf x} -{\bf z}_1)+h({\bf
x} -{\bf z}_2)\right)\phantom{.}  \nonumber \\
\\ 
      \chi({\bf x}) = {1\over\sqrt2}\left(h({\bf x} -{\bf z}_1)-h({\bf
x} -{\bf z}_2)\right). \nonumber
\end{eqnarray}
As before, the two solutions are normalized to one and are orthogonal.
(Two solutions having the hodotic expansions
\begin{eqnarray}
\phi({\bf x}) = \sum_{{\bf x}, \sigma}A({\bf x'},\sigma)
h_{\sigma}({\bf x} - {\bf x'})\phantom{\ {\rm ,}} \nonumber \\ 
\\
\chi({\bf x}) = \sum_{{\bf x}, \sigma}B({\bf x'},\sigma) h_{\sigma}({\bf x} - {\bf x'})\ {\rm ,} \nonumber 
\end{eqnarray}
are said to be orthogonal if
\begin{equation}
\sum_{{\bf x}, \sigma}A({\bf x},\sigma)B({\bf x},\sigma) = 0\  \ {\rm .)}
\end{equation}

The bilat\-tices corresponding to this solution are to be initialized
in the following way. First, in order to initialize $\phi({\bf x})$, 
the random bit generator is again used in order to place, with probability $1/\sqrt2$,
a ket particle of a given tag, say $A$, and randomly chosen phase at
${\bf z}_1$. Next, 
a new random bit with the same probability is used
to determine whether or not to place a ket particle of the same phase and tag at ${\bf z}_2$.
This procedure is repeated, with the same phases and tags,
for the ket particles. 

Next, in order to initialize the  $\chi({\bf x})$ solution
corresponding to the second particle,  the assignment procedure is 
carried out again,
for both bra and ket particles. This is done
with a new tag and new randomly chosen phase, and with the
following modification:  because $\chi({\bf x})$ has at ${\bf z}_1$ a phase
opposite that of its phase at  ${\bf z}_2$, the particles, if any, that
are to be placed at the latter point's link will
have a phase opposite that of the initial randomly chosen phase
(cf.~the initialization examples for single particle solutions,
with $\beta=\pi$).

Finally, at the links in which there are bra (or ket) particles
of two distinct tags,
%(in this case, this means there is one  bra (or ket) particle of each
%tag), 
one particle's randomly chosen 
tag is changed to that of its partner. That is, the tags are coalesced, as described above. 
Note that the choice of which tag is to be coalesced is made independently at
each point in the space.

Suppose that one now performs an event count at the two points next to
${\bf z}_1$ and ${\bf z}_1$, toward   which the particles
 initially propagate.  Let the term
$$ A^{*}A(-B)^{*}(-B) $$
signify an event for which one finds an $A$ bra and an 
$A$ ket particle at ${\bf z}_1$,  while finding  a $B$ bra particle and
a $B$ ket particle at ${\bf z}_2$. To be more precise,
each successive two labels in such a term would be
subscripted in such a way as to indicate the  point at which their location is measured,
but for notational clarity, this denotion is suppressed. The reason the
$B$ factors have a negative sign preceding them is
that the $B$ factors at ${\bf z}_2$ have a phase
that is opposite that of the initially chosen phase for that solution.
(Finding  more than one particle of a given tag does not
alter the above classification of the  events; instead,  it  merely 
causes the count to be incremented  
by an amount twice as great as when only one such particle
is found there.)

There are four possible non-zero classifications for   events
that cannot {\sl a priori} be shown to vanish,
which will be denoted as
$$\begin{array}{ccc}
 A^{*}A(-B)^{*}(-B) &  \quad & B^{*}BA^{*}A\phantom{\ \ .}\\
 B^{*}AA^{*}(-B) &  \quad & A^{*}B(-B)^{*}A\ \ .
\end{array}$$
Again, note that even if one had chosen to
count events for which there is say, one  bra particle having the tag $A$,
with all the other bra and ket particles 
having the tag $B$, the expected 
contribution of such
events is automatically seen to vanish, because of the
random phase difference between $A$  and $B$ particles.
%Likewise, if in taking say a single particle event
%count at  ${\bf z}_1$, the events for which
%a bra particle of one tag arrives simultaneously
%%with a ket particle of another tag are automatically seen to cancel
%for the same reason.

The expected event count from each of
the above four cases of events is positive.  This is because those
on the bottom row are negated
once by virtue of the fact that the
ket particles written alone are an odd 
permutation of the bra particles (i.e., they
correspond to the second diagram introduced above).  They are
negated once again by the presence of
the negative sign preceding the $B$ particle at  ${\bf z}_2$
(i.e., the relative phase difference between the $B$ particles
at  ${\bf z}_1$ and  ${\bf z}_2$.)

Thus, even though the initialization of the two-par\-ti\-cle
wave function in this example differs from the
two-par\-ti\-cle solution considered first,
%we have considered, 
%reflecting
%the different initial choices of the single particle
%wave functions, 
the expectation of the two-par\-ti\-cle event count
is the same for both cases.


The extension to arbitrary two-par\-ti\-cle functions for which the
constituent single particle wave functions are orthonormal, is a 
direct generalization of the two-par\-ti\-cle initialization. In each case, 
one assigns a tag to
each of the single particle states independently, and
then (if so desired) coalesces the tags, again, independently at each point.
That the expected two-par\-ti\-cle event count at any
two observation points gives the proper quantum mechanical
value may  be verified as in the single particle case, 
by writing all the participant amplitudes
in terms of paths.



\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Multi-particle initialization}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
The generalization to $N$-particle wavefunctions is similarly done.
The assignment of distinct tags to each of the $N$ orthonormal
wavefunctions amounts to selecting one of the 
terms of the corresponding Slater determinant.  Each particle's
wavefunction is initialized independently, although for each one, the
choice of a tag and an initial phase (for both its bra and its ket particles together)
are the same throughout
the support  of the wave function.

Once the initial particles have been assigned,
one may again coalesce the tags. At a point in space at which
there are $j_1$ particles of tag $A_1$, $j_2$ particles of tag $A_2$ and so on,
up to  $j_N$ particles of tag $A_N$, all particles will assume one tag,
this choice being made independently at each point in space and separately
for the bra and the ket particles.
The probability of tag $A_m$ being 
chosen for say the ket particles is equal to $j_m/(j_1+j_2\cdots+j_N)$.

Suppose that $j_{m,\phi}$ of the $A_m$ particles are of one phase $\phi$,
and $j_{m,\phi'}$ particles are of a phase $\phi'$ (equal to $\pm i$ times $\phi$)  so that
$j_{m,\phi} +  j_{m,\phi'}$ equals  $j_m$.  Then the  probability  of
the phase $\phi$ being chosen for all the particles is $j_{m,\phi}/j_m$ while the probability of
the phase $\phi'$ being chosen is   $j_{m,\phi'}/j_m$.  As mentioned previously, these
two remaining phases may be coalesced into one. The probability
of  $\phi$ or the  $\phi'$ phases  being chosen is  $j_{m,\phi}/j_m$
and  $j_{m,\phi'}/j_m$, respectively.


\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Multi-particle event counts}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
An  $\tilde{N}$ particle event (where  $\tilde{N}$ can be less than  $N$)
at the the observation points ${\bf x}_1, {\bf x}_2,\ldots,{\bf x}_{\tilde{N}}$, 
is the simultaneous arrival of $\tilde{N}$ distinctly tagged bra particles at those
respective points and  $\tilde{N}$ distinctly tagged ket particles.  This 
arrival must be such
that the respective tags on the bra particles found at the
observation points are a permutation of the tags on the ket particles
found at those points. For non-distinct observation points,
the event count is  zero by fiat.

The sign of the overall count is given
by the product of the phases of the ket particles times the conjugate
of the product of the phases of the bra particles, times an additional
factor of negative one if the abovementioned permutation is odd.
The magnitude of an event count is linear (antilinear) in
the number of ket (bra) particles arriving at each point. 
%Thus,
%an event for which there are $k$ ket particles of a given tag
%at a given point  increments the event count $k$ times as
%much as if only one such particle arrived there.

Diagrams can again be used to classify such  events.
As an example, for $N=6$, an event corresponding to
the diagram
$${\setlength{\unitlength}{12pt}
\begin{picture}(5,1)
\put(0,0){\line(2,1){2}}
\put(1,0){\line(0,1){1}}
\put(2,0){\line(1,1){1}}
\put(3,0){\line(2,1){2}}
\put(0,1){\line(4,-1){4}}
\put(4,1){\line(1,-1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(2,1){\circle*{.24}}
\put(3,1){\circle*{.24}}
\put(4,1){\circle*{.24}}
\put(5,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\put(2,0){\circle{.22}}
\put(3,0){\circle{.22}}
\put(4,0){\circle{.22}}
\put(5,0){\circle{.22}}
\end{picture}}$$
implies an event for which
a bra particle of a given tag
arrives at the first observation
point while a ket particle
of
the same tag (but not necessarily the same phase)
arrives at the fifth observation point, and
a bra and ket of the same tag arrive at the
second observation point, and so on.

Note that the parity of the permutation of the
ket particles with respect to the bra particles
may be determined by the number of line crossings
in the diagram, and will be even if the number
of such crossings is even, and odd otherwise. 
This may be proven
by showing that exchanging two adjacent circles
in the diagram while keeping them connected to
their original lines changes the number of crossings by plus or
minus one. (Where $j$ lines come together at a point, it  will be assumed
that there are actually $(j!)/2$ separate crossings, as there would
be if the lines were slightly distorted, so that they did not
all intersect at one point.) 

For the purposes of the model, any other
method of determining the parity of the relative permutation
of the bra and ket particles would also suffice. However,
at some point it might be necessary to model
physical phenomena whose outcome depends on
some
%interactions producing 
specific multi-particle correlations. For
example, one might wish a physical quantum mechanical model of an {\sc AND} gate that
outputs (with the proper phase) a single particle event every time there is
a corresponding $N$ particle event at some given $N$ points.
For such a purpose, some
methods of determining permutation parity will be easier  to
incorporate into the dynamics than others.

%Another important point is

\vspace{.6\baselineskip}{\obeylines
\noindent{{\it Contextual dependence of phase}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
% 
Also note
that a single particle event of one phase may
be a constituent of a multi-particle event 
of a different phase.  For example, a
positive bra and a positive ket particle with
a given tag
arriving at a  point ${\bf z}_1$ constitute a
positive single event.  Yet if simultaneously,
a negative bra particle and
a negative imaginary ket particle of some other tag
arrive at some other point 
${\bf z}_2$, then the corresponding two-par\-ti\-cle 
event is positive imaginary. Thus, the
value associated with the arrival of a bra and
ket particle at some point(s) depends on the
context of the 
%kind of event 
kind of event being measured.
Even so, the expectation of the corresponding
%single and double particle 
event counts, when considered separately, are each in accord
with the quantum mechanical values.

Since there are multiple directions
along which a particle may enter a lattice point, it is possible, even
when one coalesces phases, 
for there to be an arrival of bra and ket particles  
such that more than one diagram may be
drawn. For example, consider an event in which there is a bra particle 
of tag $A$ {\sl and\/} one of tag $B$
coming into each of two lattice points, say ${\bf x}_1$ and ${\bf x}_2$,
as well as ket particles of the same type at both of those points.
Thus,  one may draw the first two columns of the diagram as
$$
{\setlength{\unitlength}{12pt}
\begin{picture}(1,1)
\put(0,0){\line(0,1){1}}
\put(1,0){\line(0,1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\put(1.6,0.48){$\ldots$\phantom{.}}
\end{picture}}
$$
{\it or} as
$$
{\setlength{\unitlength}{12pt}
\begin{picture}(1,1)
\put(0,0){\line(1,1){1}}
\put(0,1){\line(1,-1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\put(1.6,0.48){$\ldots.$}
\end{picture}}
$$
Such arrivals will not be considered valid events,
and therefore do not increment the event count.
Note that the two diagrams that can be associated with such events
denote permutations of opposite parity, so that even
if one were to use the linearity property of event counting to
separate out the two valid events and count each of them individually,
the second
count increment would cancel the first.

In spite of the fact that the two (or in general, $N$)
observation points are separated in space, if one were to
actually implement a counting device within the lattice,
one could keep the event count a local process, by 
making sure that
%physically connecting
the device that performs the counting could in principle
be connected to the observation
points (i.e.,~the  probes of the device) by physical wires or channels of some sort.
Then the
event counting could be said to occur only when the data from
all the observation points travel to their common destination.
This point will be  discussed further below.




%\vspace{\baselineskip}
%\noindent{{\large The Method of Indistinguishable Particles}}\nopagebreak
%\vspace{.6\baselineskip}\nopagebreak
\subsubsection{The method of indistinguishable particles}
%
%
%
There is an alternate method of event counting,
for which there are no tags.  For the purposes of this
method, all bra particles are, apart
from their locations on the lattice and their phases, are for
the purposes of this method, truly
indistinguishable.  In the previous method,
one obtains  results by
simulating a large number  of  bilat\-tices, (or
equivalently, a large number of similar initial configurations
in one large bilat\-tice).  All these bilattices
are themselves indistinguishable.  For the purposes of this next method, whenever
one wishes to use the ensemble of bilat\-tices to obtain an $N$-particle event
count, one must give each of the {\it bilat\-tices} a tag of, say,  $A_1$ 
through $A_N$.  For this method, it will be assumed that
the tags are a well ordered set.



\vspace{.6\baselineskip}{\obeylines
\noindent{{\it The two-particle case}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%To see how this is done, 
Consider first a two-particle event count,
even though the bilat\-tices  can be initialized to correspond to
any $N$ particle wave function, where $N$ may be larger than two.
In this method, it is each bilattice  in the ensemble to be used
in the simulation that will be given a  tag of $A$ or $B$.


Note that for purposes of
event counting, it has already been implicitly assumed in the
previous sections 
that the bilat\-tices have been given some ordering,
namely  the order in which bilattices are observed as the event
count proceeds.
%(That is, the successor of any bilat\-tice
%in this ordering will be that lattice 
%at which we next determine an event count.)
Therefore, one can label the first 
bilat\-tice at which events are observed as $A$; the second
bilattice is labeled $B$; the third as $A$, the fourth $B$, and so on.

To initialize the bilat\-tices, one repeats the procedure described in
the above method of distinguishable particles. However, when this
procedure is completed,
one then removes, or chooses to ignore, the particle tags. Note
that if the individual single particle wavefunctions
that one wishes to simulate are initially 
nonzero only on mutually nonoverlapping regions,
one can remove the tags even before the
initialization procedure.

Let the bilat\-tices evolve for any 
desired number $T$ of time steps, and denote
the observation points 
as $({\bf x}_1; T)$ and $({\bf x}_2; T)$.

The simplest way to explain the event counting for this
new method is again through
the use of diagrams, or equivalently, through the 
permutations of bra particles with respect to the
ket particles.  All events will be classified
in accordance with some such permutation.  
Assume that
at any point in performing the event count, one is aware of the particles arriving
at the observation points of {\it  two\/} independently initialized
bilat\-tices. One can do this by, say, recording what was found
in the $A$ bilattice  and waiting until the successive $B$ lattice
is observed before making the event count, and it may be assumed
that this storing, unlike the previous memories considered here,
is absolute.
A two-par\-ti\-cle event corresponding to the
diagram
$$
{\setlength{\unitlength}{12pt}
\begin{picture}(1,1)
\put(0,1){\line(0,-1){1}}
\put(1,1){\line(0,-1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\end{picture}}
$$
is one for which there is a bra and ket
particle at ${\bf x}_1$ in bilat\-tice $A$
and a bra and a ket particle at ${\bf x}_2$ in bilat\-tice $B$.
The sign of the event is given by the product of the
signs of the ket particles times the conjugate of the product
of the bra particles. The event count is, as always, linear (antilinear)
in the number of ket (bra) particles at each point.

An event corresponding to the diagram
$$
{\setlength{\unitlength}{12pt}
\begin{picture}(1,1)
\put(0,0){\line(1,1){1}}
\put(0,1){\line(1,-1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\end{picture}}
$$
is likewise one for which there is a ket
particle at ${\bf x}_1$ and a bra particle at ${\bf x}_2$ in bilat\-tice $A$
while in  bilat\-tice $B$,  there is a ket
particle at ${\bf x}_2$ and a bra particle at  ${\bf x}_1$.
Thus, each  line on a diagram corresponds to a single
one of the bilattices involved in the event.

Writing out all amplitudes in terms of paths, one can again 
classify the events corresponding to the first diagram into the
classes denoted as
$$\phi^{*}_A\phi^{\vphantom{*}}_A\phi^{*}_B\phi^{\vphantom{*}}_B\quad%
\phi^{*}_A\phi^{\vphantom{*}}_A\chi^{*}_B\chi^{\vphantom{*}}_B\phantom{\ .}$$
$$\phi^{*}_A\phi^{\vphantom{*}}_A\chi^{*}_B\chi^{\vphantom{*}}_B\quad%
\chi^{*}_A\chi^{\vphantom{*}}_A\phi^{*}_B\phi^{\vphantom{*}}_B \ .$$
Again the first two members of a term refer to the bra and
ket particles found at the first observation point ${\bf x}_1$, and the second
two symbols to the particles at the second observation point ${\bf x}_2$.
The first term denotes events for which all the arriving particles
can be traced back to the initialization of the $\phi$ state, the
one  to its right denotes events for which the bra and ket particle
in the $A$ bilat\-tice come from the $\phi$ states (and arrive at
${\bf x}_1$) and the ones in the $B$ bilat\-tice come from the $\chi$ states (and arrive at
${\bf x}_2$), and so on.  There is another set
of such terms that are obtained by interchanging $A$ and $B$.

Of course, one cannot, just by looking at what arrives
at the observation points, know that a given event belongs to
any of these classes, for the particle tags have been removed. But
in principle, one knows that the nonzero contribution to the event count
will arise from events corresponding to the above classes.

There are, as before, other classes as well which  have not been written down,
because their contributions can automatically be seen to vanish. An example
would be a class denoted by a term in which three of the four
factors were due to a single initial state, e.g.,
$$\phi^{*}_A\phi^{\vphantom{*}}_A\phi^{*}_B\chi^{\vphantom{*}}_B\ .$$
Again, these automatically vanish, on the average, because of the
random phase difference between the $\phi$ and $\chi$ states.

Note in particular that terms of the form
$$\phi^{*}_A\chi^{\vphantom{*}}_A\chi^{*}_B\phi^{\vphantom{*}}_B%
\quad\chi^{*}_A\phi^{\vphantom{*}}_A\phi^{*}_B\chi^{\vphantom{*}}_B\ {\rm ,}$$
represent  vanishing contributions as well, although
this would not be true if one had taken event counts on one lattice
at a time, instead of separating the lattices and getting a portion of
a given event from each bilat\-tice.  Had the lattices not been separated, 
then there would have been no phase incoherence in
such events and they would have
made a net positive contribution to the event count; 
since a bra and a ket from a given initial state
both constitute part of an event, there is no overall random
phase factor to make them vanish on the average.  
However, given
that two lattices are
employed, then in the class corresponding
to the first term listed above, the  $\chi$ ket particle (i.e.,
the ket particle attributable to the initialization of the
$\chi$ state)
that arrives at ${\bf x}_1$ in bilat\-tice $A$ has no fixed phase relative to
the bra $\chi$ particle that arrives  at ${\bf x}_2$ in bilat\-tice $B$, and
likewise for the $\phi$ particles. Thus, on the average, the contribution
for such events  vanishes.

Consider now the nonvanishing terms corresponding to the second diagram,
denoted by
$$\phi^{*}_A\phi^{\vphantom{*}}_B\phi^{*}_B\phi^{\vphantom{*}}_A\quad%
\phi^{*}_A\chi^{\vphantom{*}}_B\chi^{*}_B\phi^{\vphantom{*}}_A\phantom{\ .}$$
$$\chi^{*}_A\phi^{\vphantom{*}}_B\phi^{*}_B\chi^{\vphantom{*}}_A\quad%
  \chi^{*}_A\chi^{\vphantom{*}}_B\chi^{*}_B\chi^{\vphantom{*}}_A\ .$$

For example, the second term in the first row implies
an event for which, in lattice $A$, a $\phi$ bra particle arrives at  ${\bf x}_1$
and a  $\phi$ ket particle arrives at  ${\bf x}_2$, while at the same corresponding
time in lattice $B$, a $\chi$ ket particle arrives at  ${\bf x}_1$
and a  $\chi$ bra particle arrives at  ${\bf x}_2$.  Again, there
is a corresponding set of terms with $A$ and $B$ reversed, as well as
a set of terms representing contributions which are immediately seen to cancel
due to phase incoherence.

Note in particular the absence of the classes represented by the terms
$$\phi^{*}_A\phi^{\vphantom{*}}_B\chi^{*}_B\chi^{\vphantom{*}}_A\quad%
\chi^{*}_A\chi^{\vphantom{*}}_B\phi^{*}_B\phi^{\vphantom{*}}_A\ .$$
Again, had one performed an event count on each lattice individually,
the events corresponding to these terms would make a positive contribution.
As it is, in considering the first term, one can see that the $\phi$ bra particle
that arrives at ${\bf x}_1$ in bilat\-tice $A$, has no fixed phase relative to
the bra $\phi$ particle that arrives  at that same point in bilat\-tice $B$,
since the two are initialized with different phases. Therefore, such
events are seen to cancel.

If one agrees to subtract the event count contributions corresponding to
the second diagram from that of the first, then the unphysical, ``bi-location'' events,
i.e., those events given by the terms
$$\phi^{*}_A\phi^{\vphantom{*}}_A\phi^{*}_B\phi^{\vphantom{*}}_B\quad%
\chi^{*}_A\chi^{\vphantom{*}}_A\chi^{*}_B\chi^{\vphantom{*}}_B\ {\rm ,}$$ 
which imply that the single particle of a given single particle state
arrives in two places at once, and which appear in counts corresponding to both diagrams, also vanish.

Note that even though  two bilat\-tices are required to perform an event 
count, there also two ways of obtaining events (corresponding
to terms such as those written above, but for which the $A$ and
$B$ labels are switched).  Therefore, the expected number of events
per bilat\-tice is the same as before. 




\vspace{.6\baselineskip}{\obeylines
\noindent{{\it The multi-particle case}}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
%
%
The general case of event counts involving a higher 
number $N$ of particles is handled in 
the same way.  In order to make such event counts, 
one must partition all the bilat\-tices used 
in obtaining an event count into groups of $N$, giving 
a label to each bilat\-tice in a group, 
$A_1$, $A_2$, up to $A_N$.
The initialization procedure for the 
bilat\-tices is a straightforward 
generalization of the two-par\-ti\-cle case.
As before, the number of single particle states
per bilattice may be larger than $N$.  

Again, events are most simply described with the help of diagrams.
A simultaneous arrival of bra and ket 
particles at the $N$ distinct observation
points of the $N$ bilat\-tices within a 
group qualifies as an event if one can
associate with it an $N$-fold diagram, such that the top member
of any column is connected by a line to the 
bottom member of one of the columns.
The $N$ columns in a diagram signify the 
observation points in the respective order
${\bf x}_1,{\bf x}_2,\ldots,{\bf x}_N$. 
Also, the top $N$ circles are to
be associated with the $N$ bilat\-tices 
in the order  $A_1$, $A_2$, up to $A_N$, 
{\it or any cyclic permutation thereof.} 
For the moment, however,  assume that the
lattices have been renamed so that the 
first circle is in fact associated with
lattice  $A_1$.  In that case, a line leads from the 
$j^{th}$ top circle to the  $k^{th}$
bottom circle  if, for the arrival of 
bra and ket particles in question,
there is in lattice $A_j$ a ket particle 
at the point ${\bf x}_j$ and also a bra 
particle at the 
point ${\bf x}_k$. 

As mentioned before, each diagram implies a permutation of  $N$ objects, such that a line
leading from the  $j^{th}$'s column top circle  to the  $k^{th}$'s column bottom circle
implies that, under the permutation, the  $j^{th}$ object becomes the  $k^{th}$.
The phase of an event equals  the product  of the ket particle phases times
the complex conjugate of the product of the bra particle phases, times an
additional factor of negative one if the permutation implied by the associated
diagram is odd.

As in the previous method,  there may be multiple ways to construct such a diagram.
For example in a two-par\-ti\-cle event, it may happen that there are
bra and ket particles at both observation points, in both of the lattices.
Again, such occurrences will not be considered valid events. Even if one were to
use the
linearity property to separate out such events and count each of them individually,
the permutations involved in each of the events are of opposite parity, so that
the net contribution is zero anyway.

As in the two-par\-ti\-cle case, one can verify that by using
$N$ independently initialized bilat\-tices for each
$N$-particle event,  phase incoherence will cancel
away all  events that cannot be associated with valid diagrams. Moreover,
the fact that one subtracts the events corresponding to
odd permutations from those corresponding to even ones, cancels
away events which imply that a single particle
may be found in more than one location at any given time.
In other words, despite the fact that the branching
processes prescribed by the dynamics imply that even the one-particle
in general involve many lattice particles,
the counting procedures  ensure that the 
evidence of this multiplicity is cancelled away.

An example will make the above discussion clearer.
Suppose that for a four-particle event count, at the
time $t=T$ in each bilattice there is:
in the bilattice $A_1$, a ket particle at ${\bf x}_1$
and a bra particle at ${\bf x}_3$; in the bilattice
$A_2$, there is a ket
particle at ${\bf x}_2$ and a bra particle at ${\bf x}_1$; in the bilattice $A_3$, there
is a ket particle
at ${\bf x}_3$ and a bra particle at ${\bf x}_4$; and in bilattice $A_4$, there is
a ket particle at ${\bf x}_4$ and a bra particle at ${\bf x}_2$.
This event would correspond to the diagram
$${\setlength{\unitlength}{12pt}
\begin{picture}(3,1)
\put(0,0){\line(1,1){1}}
\put(1,0){\line(2,1){2}}
\put(0,1){\line(2,-1){2}}
\put(2,1){\line(1,-1){1}}
\put(0,1){\circle*{.24}}
\put(1,1){\circle*{.24}}
\put(2,1){\circle*{.24}}
\put(3,1){\circle*{.24}}
\put(0,0){\circle{.22}}
\put(1,0){\circle{.22}}
\put(2,0){\circle{.22}}
\put(3,0){\circle{.22}}
\end{picture}}$$
and would have a phase that is the negative of the
phases of the ket particles multiplied by
the conjugate of the phases of the bra particles.
For an event identical to the one above, except with the labels of
any two bilattices switched, the event would not be considered a valid one, i.e.,~would not
increment the count,
whereas, as mentioned previously, for any cyclic permutations of the bilattice labels $A_1$,$\ldots$,$A_N$,
the event would indeed be valid.
It is because these cyclic permutations of the bilattices are allowed
that the expected number of events per bilattice is the same as in the previous method.

%there will in fact be $N$ different ways to obtain an event from
%any grouping of $N$ bilat\-tices, so that the number
%of events per bilat\-tice is the same as in the
%first method we considered.
Thus the information obtained by being able
to distinguish one particle from another can also
be obtained instead distinguishing the members of  the ensemble of bilat\-tices used
to determine a multi-particle event count.
%There are similar, alternate  ways in which we 
%the latter method could have been executed,
%For example, by superposing $N$ independent bilat\-tices
%into one, as we superposed two lattices into one bilat\-tice,
%we can make the second means of multi-particle event counting
%more similar to the 
%first, but otherwise, the counting procedure will be
%identical.
Note that if one chooses to use the
method of indistinguishable particles, one will in general need a larger
ensemble of bilat\-tices to obtain results to within
a degree of accuracy, in order that the terms
which are required to vanish because of phase incoherence actually do.
A similar disadvantage is encountered in making the transition
from continuous, infinitely divisible flows on a lattice, to the random
motions of discrete particles. What is lost in complexity in
going from one model to another 
must be compensated in the brute size of the computation.

%NOTE THAT SINCE IN ANY EVENT COUNT WE REFER TO DISCONTINUOUS
%POINTS, THE MODEL IS NOT LOCAL YET. HOWEVER, WE MAY MAKE THE
%EVENT COUNTER TRAVERSE PHYSICAL PATHS, AND PERHAPS HAVE A SHUTTER.

%In either method, it is the case that wave phenomena are
%obtained by a gas of particles on a lattice.
%At any point (or more precisely, link) in the
%lattice, there will be some number
%of particles of up to two phases (and in the
%first multi-particle model considered, 
%of one tag, that will differ from link to link.)
%The event counts will always 
%be integral (albeit complex). 
%By considering
%a large number of properly prepared 
%replicas of initial states,  one  may
%obtain evidence of the underlying wave equations.
%Note that one may in principle make the entire system
%deterministic by using a suitable background
%heat bath of lattice particles to act as a
%deterministic pseudorandom number generator.
%%Again we note that in order to use the second method
%%to simulate bosonic wavefunctions, one will
%%have to suppose that the bosonic (i.e., integral-spin) particles
%%are composites of an even number of fermi constituents.




%
%
In closing this section, note that
if it seems a puzzle as to why fermi particles
exist in nature, it is more obvious why they are
useful in the context of the model presented here.
As mentioned before, they
serve to erase away the observability of  the
branching process of the particle dynamics, in which
even single particle states imply a large number of
bra and ket particles.  Therefore one is left, in the macroscopic
limit,  with phenomena suggesting the
simpler notion of single particle paths.

%\vspace{.6\baselineskip}{\obeylines
%\noindent{{\it Extensions}}\nopagebreak
%\vspace{.3\baselineskip}\nopagebreak
%
\subsubsection{Extensions of event counting}
%
In addition to the coincidence counting that
was used in determining information about
multi-par\-ti\-cle events,  it will in general be necessary
to combine additively the information
from many different event count sites, at each of which  
information is being collected simultaneously. Just as multi-par\-ti\-cle  event counting
provides the physical manifestation of
the {\sc AND} construction in logic, there is
likewise a manifestation of the {\sc OR}
construction, allowing one to obtain the
event count whose
value is equal to the sum of the event counts
at several other locations. 
%In Principle, the disparate
%event counts need not be of the same phenomenon.
%One can record
%sing\-le-par\-ti\-cle encounter of one kind of fermion,
%say an electron,
%while another can record the mul\-ti-par\-ti\-cle
%events pertaining to some other kind of fermion, say
%a certain type of neutrino.
Such an operation is useful, in that it allows one to simulate the
results obtained by physical measuring devices whose collection
windows are of macroscopic proportions.  

Note again,  in order for the event counts to
correspond to the information obtained
any real measuring device, 
it is necessary to constrain the devices by
requiring that event count increments are combined only from collection points
that could in principle be connected by wires or some
other information pipes,
along which signals  propagate at some
subluminal velocity.



There are other  unphysical combinations
of coincidence and combination counting
one must be careful to avoid, in addition to those
that disobey relativistic constraints.  
The event counting procedure
as presented here
is something that occurs independently
of the dynamics; however, in the real
world, the event counting
is done by physical devices that necessarily
interact and therefore interfere with the
phenomena they measure.
This must be continually kept in mind when one
is determining a suitable event counting
procedure. 

As an example, it is possible, within the context of the model, to 
determine the location
of a wave packet at one time and place,
and then do a correlation experiment with that
same (unperturbed) packet at some
later stage of its evolution.
Such a procedure has no counterpart in
the real world, where the first
measurement  unpredictably
distort the wave packet. In order
to do a realistic simulation
of such a coincidence experiment,
one has to include in some other
way the random perturbations that physical measuring
devices produce.

%There is thus a sort of mind-body duality
%between the theoretical counting procedure
%and the physical one. One can
%speak of acts of perception without  specifying
%the nature of the perceiver.  However, in
%quantum mechanics, one must keep in mind that in the real world
%there are always physical objects corresponding to
%any measurement process.



The event counting procedure is thus reminiscent  of the Gaussian surfaces
encountered in the study of electromagnetism; they too allow one to gather information from a
physical system without interacting with that system in any way. Note
that even in the study of electromagnetism, 
when the Gaussian surface is to correspond to some physical object---such
as the surface of some conducting material whose shape is altered
by the physical processes taking place---then
presumably the Gaussian surfaces must also reflect the dynamics. 
%In the case
%of event counting
%quantum measurements however, there is less choice in the matter, for
%physical measuring devices  inevitably interact with
%the phenomena they observe.





The model of the wave equation presented here calls
to mind the similar random-walk-type model of the 
one-di\-men\-sion\-al Klein-Gordon equation introduced by
R. P. Feynman, in the 
exposition of his path integral
approach to quantum 
physics,$^7$
%\footnote{
%R. P. Feynman and A. R. Hibbs, 
%{\sl Quantum Mechanics and Path Integrals\/}
%Prob.~2-6,
%McGraw-Hill, New York, 1965.}
although this author is unaware of
high\-er-di\-men\-sion\-al extensions
of his model.  Such methods, which
relate particle motions to
wave phenomena,
can be used to give quantum mechanics
a completely local interpretation.
%\footnote{
%H. Hrgov\v ci\'c,
%``The Localist Interpretation of Quantum Mechanics''. }

This is done by ascribing a physical reality to  negative 
events.  The  measurement process, in turn,  becomes a direct
generalization of the event counting procedure, and
all physical information is obtained by way
of counting the number of some preferred particles (say electrons) encountered
along some spacetime path.
Therefore, in this context, negative events merely serve to decrement
the event count.
The reason that these negative events, or the implied erasures of
past events, are not observable
at the macroscopic level by beings that are themselves
constructions on the lattice is that quantum mechanics, along with the law of
large numbers,
assures that event counts involving large numbers of events
will be positive, since they are are positively proportional
to the absolute square of the associated wavefunctions.
Even if the experiments are sensitive to quantum details,
e.g. ones in which the recording devices are 
photoelectric plates, SQUIDS, etc.,
  assessing the outcome of the
experiments still involves observing (i.e. making measurements on)
the macroscopic measuring devices.
The elaboration of this interpretation,
as well as its application to such macroscopic
phenomena such as the EPR experiment and the like is discussed elsewhere.$^8$

%	, the act by
%	which physical situations on a lattice are 
%	transformed into numbers by measuring devices internal
%	to those lattices




\section{The Klein-Gordon equation}

%{\tt Add in arbitrary potentials?}\hfil\break
%It is possible to generalize 
The above results for the wave equation can be
extended to the 
computation of the
Klein-Gordon equation.
On an $m$-di\-men\-sion\-al Euclidean
space, The Klein-Gor\-don equation takes the form
\begin{equation}
{1\over c^2}{{\partial^2}\over{\partial t^2}}\psi  = %
\left({{\partial^2}\over{\partial x_{1}^2}} + 
{{\partial^2}\over{\partial x_{2}^2}} 
+\cdots+
{{\partial^2}\over{\partial x_{m}^2}}\right)\psi 
+ \eta^2\psi\label{eq:K-G}\ {\rm ,}
\end{equation}
where $\eta$ is a real parameter.
(Note again that the above
equation is in Cartesian
space, so that the partial second derivatives
have their usual definitions.)
The Fourier frequencies of this equation
obey a dispersion relation of the form
$$\omega^2c^2 = k_1^2+k_2^2+\cdots k_m^2 + \eta^2$$.

%\vspace{.6\baselineskip}
%\noindent{{\large Manipulating Dimensions}}\nopagebreak
%\vspace{.3\baselineskip}\nopagebreak
%
\subsection{ Manipulating dimensions}
Recall the discussion of the wave equation
of a toroidal space of $n$ dimensions,
all except $m$ of which
are ``thin'', i.e., are of length one.
Suppose now that these thin
dimensions were of length two
instead of one. (If the length of all
the dimensions were even, then it 
would again make sense to speak of even and odd
sublattices.)  The lattice
would still be effectively
$m$-di\-men\-sion\-al, but there would
also be extra degrees of freedom
to consider. 

First suppose that $n-m=1$.
A thin dimension of length two
implies that 
one more set of spatial frequencies,
$\kappa_{x_n}$,
would be required to specify the system,
corresponding to motions along the
thin dimension.
However, since the index $x_n$
only takes the values `0' or `1'
(or by an appropriate
change of variables `$\pm1/2$'),
it could
be viewed as merely a specifier between
two different ``components''
of the solutions.

The dispersion relation
of this system would
then be of the form
\begin{equation}
\sin^2{\omega \over2} = {1\over n}\left( %
\sin^2{{\kappa_{x_1}}\over2} +\cdots %
\sin^2{{\kappa_{x_m}}\over2}+%
\sin^2\left({{\pm\pi}\over2}\right)\right)%
\end{equation}
(the last term of course being equal to one).
For extremely small lattice spacings,
this expression would become,
in analogy with the discrete wave equation 
dispersion relation described above,
\begin{equation}
\omega^2c^2 = k_1^2+\cdots+k_m^2 +%
\left(\alpha\over d\right)^2\ {\rm ,}
\end{equation}
where $\alpha$ is a constant on the order of one,
and $d$ is the
length of the lattice spacing.  
This is precisely of the
form of the Klein-Gordon
dispersion relation, with ${d/\alpha}$
acting as the Compton wavelength
of the particle whose wave function
obeys the Klein-Gordon equation.

This approach therefore is loosely reminiscent
of  Kaluza-Klein models.
Here, the concept of mass is the
manifestation of the topology of
the space corresponding to
motion along a ``matter'' dimension.
Of course, this system is
only useful for situations
involving spatial frequencies,
i.e., momenta,
much smaller than $d^{-1}$, so
that only nonrelativistic
phenomena can be accurately
simulated.

Suppose, however, that $m$ is the
typical 2 or 3, corresponding
to two and three dimension\-al
cases, and that $n$ is large,
even astronomically so.
Such a system has an evolution insufferably 
hard to compute
(the speed of light is unbearably ``slow'', i.e.,
a relatively large amount of computation 
is necessary to execute 
a given time interval of the simulation). 
Even so, the
above considerations for the case
$n=m+1$ 
still hold, except that
the mass term in the
dispersion relation, $\alpha$, 
would be multiplied by a factor 
of $n$, thereby increasing
the spectral range in which
the approximation to the continuous
case is good.  Also,
the approximately $n$
momenta corresponding
to motion along the thin dimensions
can likewise  be further manipulated,
depending on the internal substructures and
symmetries of the particles one wishes to simulate.

%Also, by choosing dimensions of
%length 1 or 2 in this way,
%we can simulate 
%(depending on the mutual
%interactions)  several coupled Klein-Gordon
%systems, each having different (possibly vanishing)
%masses, depending merely on the structure
%of the associated thin dimensions.


%\vspace{\baselineskip}
%\noindent{{\large The Modulation Method }}\nopagebreak
%\vspace{.6\baselineskip}\nopagebreak
%
\subsection{The modulation method }
Another
way of obtaining the
Klein-Gordon equation
from the wave equation is to first  double the number
of neighboring links. For example, in two dimensions, a
point then has eight neighbors instead of the previous four: two
neighbors, say `top' and `bottom',
in the $x_{+}$ direction,
two in the $y_{+}$ direction and
so on.  Alternatively, one may suppose
the associated particles have yet another
degree of freedom, and can be in either
an `top' or `bottom' state. 
%	The relationship of
%	this degree of freedom to spin will be postponed
%	until the addition of an interacting electromagnetic field.
%	For now, keep in mind that the equation under consideration is a
%	scalar one.
Each one of the doubled number of links, will 
just as before, lead to separate flows,
which will be respectively denoted by $f_{x_{+}}$ and
$g_{x_{+}}$, etc. The equation of
motion for the traveling
waves are then be given by the system
of equations
\begin{equation}
\left(\rule{-.5ex}{0ex}
\begin{array}{c} 
f_{x_{+}} \\
f_{x_{-}} \\
f_{y_{+}} \\ 
f_{y_{-}} \\
g_{x_{+}} \\
g_{x_{-}} \\
g_{y_{+}} \\ 
g_{y_{-}}%
\end{array}\rule{-.5ex}{0ex} %
\right)^{\rule{-.8ex}{0ex}\rm{out}}
\rule{-2.5ex}{0ex} ={1\over4}
\left(\rule{-1.2ex}{0ex}
\begin{array}{r@{\hspace{.5ex}}r@{\hspace{.5ex}}%
r@{\hspace{.5ex}}r@{\hspace{.5ex}}r@{\hspace{.5ex}}
r@{\hspace{.5ex}}r@{\hspace{.5ex}}r} 
       1 & -3 &  1 &  1 &  1 &   1 &  1 &  1 \\
      -3 &  1 &  1 &  1 &  1 &   1 &  1 &  1 \\
       1 &  1 &  1 & -3 &  1 &   1 &  1 &  1 \\
       1 &  1 & -3 &  1 &  1 &   1 &  1 &  1 \\
       1 &  1 &  1 &  1 &  1 &  -3 &  1 &  1 \\
       1 &  1 &  1 &  1 & -3 &   1 &  1 &  1 \\
       1 &  1 &  1 &  1 &  1 &   1 &  1 & -3 \\
       1 &  1 &  1 &  1 &  1 &   1 & -3 &  1 %
\end{array}\rule{-.6ex}{0ex}
\right) 
\left(\rule{-.5ex}{0ex}
\begin{array}{c}
f_{x_{+}} \\
f_{x_{-}} \\
f_{y_{+}} \\ 
f_{y_{-}} \\
g_{x_{+}} \\
g_{x_{-}} \\
g_{y_{+}} \\ 
g_{y_{-}}%
\end{array}\rule{-.5ex}{0ex}
\right)^{\rule{-.8ex}{0ex}\rm{in}}
\rule{-1.2ex}{0ex}\ {\rm .} 
\end{equation}

The system is still, in effect,
two-di\-men\-sion\-al. Indeed,
one sees that by
adding the respective
`top' and `bottom' coefficients
of the matrix
and treating the quantity $f_\sigma + g_\sigma $
as one flow,
the original
two-di\-men\-sion\-al system of 
equations (\ref{eq:two-mat}) is recovered.

Next multiply every even numbered
column by $e^{i\alpha}$ and
every odd numbered column by $e^{-i\alpha}$.
%	  but suppose, however, that
%	after dividing the above matrix into
%	four equal quadrants, the entries
%	in the two diagonal quadrants are multiplied by
%	$e^{i\alpha}$
%	 and and the ones
%	in the off-diagonal quadrants by $e^{-i\alpha}$, where $\alpha$ is some
%	small (real) constant.
%	This implies that each input flow modulates
%	its contributions to an output flow by a
%	positive or negative angle, according to whether
%	the output is of the same group (i.e., top or bottom)
%	as the input.
How this  modulation can be implemented
by way of a discrete particle-like process
is discussed below.
%as each top flow
%left a node it was multiplied
%by $e^{i\alpha}$ and
%each bottom flow was
%multiplied by $e^{-i\alpha}$,
%where $\alpha$ is some
%small (real) constant.
%The sum of the coefficients
%along any column of the matrix would then no
%longer equal to one; however, the interaction 
%matrix would still be unitary.
%(Since we have  introduced
%imaginary components, we 
%have also complicated the
%random-walk simulation schemes
%proposed above; nevertheless,
%they are still applicable
%if in addition to having
%particles in the plus and
%minus state we also postulate
%an appropriate $+i$ and
%$-i$ state and
%make the interactions such that
%probablistically, the amplitudes
%are appropriately modulated.)
 
If one computes
the resultant dispersion relation for this system,
one obtains
%\begin{equation}
%\cos\omega = {1\over4} \left( \cos(\kappa_{x} + \alpha)\right. +%
%\cos(\kappa_{x} - \alpha) 
%\end{equation}
%$$\left. + \cos(\kappa_{y} + \alpha) +%
%\cos(\kappa_{y} - \alpha) \right)$$
%which when expanding the cosine terms becomes
\begin{eqnarray}
\cos\omega = {1\over4} \left( \cos(\kappa_{x} + \alpha) +%
 \cos(\kappa_{x} - \alpha)\right. \nonumber\\
%\phantom{\cos\omega}
\left.+ \cos(\kappa_{y} + \alpha) +%
 \cos(\kappa_{y} + \alpha)\right)\phantom{\cos\omega} \\
={{\cos\alpha}\over2}\left( \cos\kappa_{x} +%
\cos\kappa_{y} \right)\ {\rm .}\phantom{\cos\omega} \nonumber
\end{eqnarray}
Note again that, as in the case of the
wave equations, 
the frequencies are always real.
Expanding the
above equation in a power series shows that
for a small lattice spacing,
this system satisfies (for sufficiently small $\alpha$)
a dispersion relation of the form
\begin{equation}
\omega^2 = {1\over2} \left(k_x^2 + k_y^2 + \left(\alpha\over d\right)^2 \right)\ {\rm ,}
\end{equation}
where the $k_x$ and $k_y$ have their
usual continuum definitions, and
where $d$ is now a length on the order of
the lattice spacing.
This again, is precisely the
form of the dispersion relation of
the Klein-Gordon
equation.
This approach
can immediately be extended to any number of
dimensions and 
has the advantage over the 
previous one in that
the extra components that have been
introduced do not change
the value of the speed of light
from $1/\sqrt{n}$, where $n$ is the corresponding effective dimension.

%???Also, by modifying the modulation
%factor $\alpha$ into a
%function of the space and time, $\alpha + V({\bf x},t)$,
%we can simulate a system subject to a
%potential $V({\bf x,t})$??? 



\vspace{.6\baselineskip}{\obeylines
\noindent {\it Modulation and Moderators}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
The lattice interaction
whereby an incoming flow is
modulated by a factor of $e^{i\alpha}$ from what
it would have been in the case of the wave equation, may be
implemented via a particle approach, just as the
flows themselves may be interpreted in terms of the statistical
motions of discrete particles.

%For the sake of simplicity consider the two-di\-men\-sion\-al case, where
%every lattice point is connected to each of its
%four nearest neighbors a top and a bottom link.
%In the case of the wave equation, a particle of phase $\phi$ coming
%into a lattice point along the top (bottom) link in say the $\sigma$ direction, will
%with probability $1/4$ yield a particle of phase $\phi$
%in each links leading out of that lattice point, except for the
%up (down) link leading in the direction opposite to $\sigma$, where
%it will with probability $3/4$ yield a particle of phase $-\phi$.



%\vspace{.6\baselineskip}
%\noindent {\it Moderators}\nopagebreak
%\vspace{.3\baselineskip}\nopagebreak
%
Consider again the lattice in which the Klein-Gordon
equation is to be implemented.
Aside
from the particles executing wavelike motion, let there
also be a number of particles comprising what
will be called the {\it moderator\/} gas.  The dynamics of
these particles as they move from point to
point along the links is arbitrary, except that
it is  assumed that the particles
have at all times a Poisson distribution.  That is,
at each link, there is a probability 
%
\begin{equation}
P(\alpha; k) = {\alpha^k \over {k!}}e^{-\alpha}\ {\rm }
\end{equation}
%
of finding exactly $k$ particles.  Therefore, if the
number of points in the lattice is $N^n$, the
expected number of moderator particles, assuming
$2n$ links per point, is $2n\alpha N^n$.

Suppose next that the presence of one moderator particle
in an incoming link multiplies the outgoing
distribution by $(1+i\epsilon)$, where $\epsilon$ is a positive number.  That is, whereas in the
absence of any moderator particles,
the presence of a wave particle of phase $\phi$ produces outgoing
wave particles of the same phase with a probability of $1/4$, now
there will also be produced, with a probability of
$\epsilon/4$, a wave  particle of the phase $i\phi$.
For the reverse link, where the probability associated
with the outgoing particles is  $3/4$, the extra particles
are produced with a probability of $3\epsilon/4$ and a phase of $-i\phi$.
(Once the particles of this alternate phase are produced,
they may, if desired, be coalesced with the particles of the other phase,
as described above, and as always, the resultant particles
of opposite phases annihilate each other.)
In the absence of any moderator particles,
there is no extra $\epsilon$ interaction.

Thus the moderator particles act to multiply the number
of wave particles. In a given link, in the presence of one
moderator particle, 
the input of one wave particle leads
to an expected number of $(1+i\epsilon)$ particles.
An additional moderator particle will act
on the output particles of the first one in
the same way, resulting in an
expected number of $(1+i\epsilon)^2$ particles.

Therefore, the net expected number of output
particles is
$$e^{-\alpha} + (1 + i\epsilon){\alpha\over{1!}}e^{-\alpha} +  
(1 + i\epsilon)^2{\alpha^2\over{2!}}e^{-\alpha}\cdots   = e^{i\epsilon\alpha}\ .$$
Thus,  a discrete process can indeed act to modulate
the statistical flows of particles by an arbitrary
real parameter.  The moderator distribution
thus serves here a function similar to that of the Higgs fields of
quantum field theories,
in that it induces a mass.
Note  that if the extra particle 
which the moderators produced are
of a phase $-i$ times the phase of the initial particle, making $\epsilon$, in effect, a negative number, 
then the corresponding modulation
factor likewise has a negative phase. 

Suppose next that there are two types of moderator particles,
each with a Poisson distribution parametrized by $\alpha$ and $\alpha'$, 
respectively, and
both of which  operate on the wave particles of phase $\phi$ to produce 
an additional particle of phase $i\phi$, with the probabilities
$\epsilon$ and $\epsilon'$. The modulation is then 
$$e^{i\epsilon\alpha+i\epsilon'\alpha'}\ .$$

Next consider a situation in which the
two types of moderator particles, each initially
with a Poisson distribution parametrized by 
$\alpha_+$ and $\alpha_-$, at each link of
the lattice, are given the respective opposite phases of $+1$ and $-1$.
That is, the moderator particles are themselves given phases
and while all $\alpha_+$ particles  have a positive phase, 
the $\alpha_-$ particles  have a negative one. 
(Note that imaginary phases are not included.)
Each positive and negative  particles multiplies the
outgoing distribution of wave function particles by a factor
of $(1+i\epsilon)$ and $(1-i\epsilon)$, respectively.
Where moderator particles of both types are found in any link of the lattice,
they again will be made to annihilate, so that only one type of particle remains.

Thus, in  considering both types of particles at once, one can
say that at any link of the lattice, there is
a certain probability of finding any number $k$ of particles,
where $k$ may be negative as well as positive. Finding a negative number of
particles simply means finding a positive number of particles of
the second type.

The probability of finding any number is then given by
\begin{equation}
P(\alpha_+, \alpha_-; k)=\left\{%
\begin{array}{ll} \sum_{m=0}^\infty P(\alpha_+; k+m)P(\alpha_-; m) & k\ge0 \\
       \sum_{m=0}^\infty P(\alpha_+; m)P(\alpha_-; m-k) & k<0 
\end{array} \right. {\rm .}
\end{equation}
Writing out the individual Poisson distributions, one obtains
\begin{equation}
P(\alpha_+, \alpha_-; k)=%
 e^{-\alpha_+ - \alpha_-}\sum_{m=0}^\infty%
{ {{\alpha_+}^{|k|+m}{\alpha_-}^m}\over{(|k|+m)\,! m\,! }}
\end{equation}
$$ \phantom{P(\alpha_+, \alpha_-; k)}=  e^{-\alpha_+ - \alpha_-}%
\left({\alpha_+\over\alpha_-}\right)^{k/2}I_{|k|}\left(\sqrt{4\alpha_+ \alpha_-}\right)$$
$$ \phantom{P(\alpha_+, \alpha_-; k)} k = \ldots, -1,0,1,2,\ldots\ {\rm ,}$$
where $I_k(x)$ is the modified  Bessel function of the first kind, of
order $k$, which satisfies the differential equation
$$x^2 I''(x) + x I'(x) = (x^2 + k^2)I(x)\ .$$
%If the $\alpha_{\pm}$ particles act on each lattice particle
%in such a way to produce on the average $(1\pm i\epsilon)$ particles, then 
The 
modulation factor arising from such a distribution of particles
is proportional to $(\alpha_+ - \alpha_-)$.



In the limit of infinitely long times, the
same kind of modulation can be produced by
a moderator with a binomial instead of a
Poisson distribution, because of the
similarity of the two when the appropriate limits are taken.
(By a binomial distribution, what is meant is that there is
a probability  $p$ of finding one moderator
particle at any link, and a probability  $(1-p)$ 
of finding none.) 




\vspace{.6\baselineskip}{\obeylines
\noindent {\it Potentials}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
Such results are useful not only
for the simulation of the Klein-Gordon equation, but
for the analogous equation for particles subject to
electromagnetic or other potentials that themselves satisfy
wave equations. The potential amplitude can,
just as the quantum wave functions
and the moderator, be given statistically with discrete particles,
which will  operate on the quantum wave functions just as the moderator particles do.
Perhaps such an approach will be less troubled by
the infinities that are so problematic in standard field theories.



It is hoped that
the  method presented, of
treating the
wave and other related equation
will lead to
new insights regarding
all wave phenomena.
%and the construction of field theories
%free from the usual infinities.
%MAKE CLEAR THE CONNECTION BETWEEN DIFFUSIONEQUATION AND RANDOM WALK
In particular, just as the realization that the study of particles executing a random
walk on a lattice in some sense 
explain the diffusion equation,   
% and is its logical source,
the particle approach to the wave equation ma also
be seen as 
a way of explaining the
wave-par\-ticle duality of microscopic physics.
I wish to thank
Tommaso Toffoli
and all the members
of the Information Mechanics
Group at MIT, whose help 
and guidance made this
work possible. I also wish to thank Mark Smith especially
for his  help.
Support was provided in part by the National
Science Foundation, grant no. 8618002-IRI,
and in part by the Defense Advanced Research
Projects Agency, grant no. N00014-89-J-1988.


\vspace{.6\baselineskip}{\obeylines
\noindent {\large \bf Appendix. Hodotic Solutions}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
Some unusual properties of
the hodotic solutions of
the wave equation are
easily established by
resorting to summations over paths.
As before, let $\sum_{\bf x} f({\bf x},t)$, be the
sum over all space of some function $f({\bf x},t)$ 
at some instant of time.  That is, it is a sum over
the {\it point values\/} of a given solution.

It is easy to show from the equations
of motion that the hodotic solutions have the property
that $\sum_{\bf x} h_{\sigma}({\bf x};t)$ is equal to one for all time,
where $$\sigma \in \{ x_{1+},x_{1-},x_{2+},\ldots,x_{n-}\}\ \ .$$ 
What is much more remarkable is
that on any $n$-di\-men\-sion\-al torus whose
lengths along any of the coordinate axes are the same,
the quantity $\sum_{\bf x} h^2_{\sigma}({\bf x};t)$ is also equal to one for
all time.  That is, without regard to the traveling wave {\it link\/}-solutions,
the integral of the square of the {\it point\/}-solution is conserved.  
This property does not hold, in general, for an arbitrary
sum of the hodotic solutions.
The proof of this is somewhat tedious, but it will be outlined 
briefly for the two-di\-men\-sion\-al case.  

Since  $h_{x_{+}}(x,y,T)$, for example,
may be written as a sum over all walks of length $T$, leading from
the point $(1,0)$ to the point $(x,y)$, one can see that $h^2_{x_{+}}(x,y,T)$
may be written as a sum of terms, each of which represents a
{\sl pair\/} of such paths.  Let $h^{x_{+}}_{\sigma}(x,y,T)$ represent
the contribution to $h_{\sigma}(x,y,T)$ of paths
whose  $T^{th}$ (i.e., whose final) step is along the $x_{+}$ direction,
so that 
\begin{equation}h_{x_{+}}(x,y,T)=\sum_{\sigma} h_{x_{+}}^{\sigma}(x,y,T)\ {\rm .}\end{equation}
Then
\begin{equation}
\sum_{\bf x} h_{x_{+}}^{\sigma}= \sum_{\bf x}\sum_{\sigma} \left(h_{x_{+}}^{\sigma}\right)^2
+ 2\sum_{\bf x}\sum_{\sigma \neq \sigma'}h_{x_{+}}^{\sigma}h_{x+}^{\sigma'}\ {\rm ,}
\end{equation}
where the functional dependence on $x$, $y$, and $T$, has
been suppressed.
Now the first (double) sum on the right is equal to one.
This is because
it represents the discrete integral of the sum of the squares of the
traveling wave components for this solution of the
wave equation. This quantity, due to the unitarity of
the transition matrix, is always conserved (and therefore
equal to its initial value of one).  Therefore, it remains only to
show that the second sum on the right vanishes.

First suppose that the two-di\-men\-sion\-al torus under
consideration is of
infinite length along either axis.
Unlike the first sum, the second contains
no terms representing pairs of 
walks that are identical. This is because  if the two paths in such a term were identical,
they would obviously have their $T^{th}$ step along
the same direction and therefore would belong in
the first sum.  It is easy to see graphically, 
that any such pair of nonidentical paths may
be related in a one-to-one way to another
pair of paths (belonging to the
same summation) whose cumulative number
of reversals differs by one.
\begin{figure}
\psfig{file=hx/twfig/twf-pat-op-2.ps,height=1.75in,width=3.0in}
%box is 7" by 3 3/8", lower left corner 1/4" 7 1/8"
%                         upper right corner 7 1/8" 10 1/2"
%show two side by side photos; on the left, a
%pair of paths of length T starting at origin,
%being identical for the first two or three steps
%then making a loop at the point (x,y)
%(each end of the path terminates at this point).
%on the right, the manipulated graph with
%one additional reversal, and therefore
%a  contribution that negates the first.
$\phantom{two paths some{\rm{\large{(a)}}}}%
\phantom{two paths that are paths{\rm{\large{(b)}}}}%
$
\caption{Two paths (a) that are identical 
except for their final $m$ steps, where
$m\ne0$, but arrive at the same place at
$t=T$, may by a symmetry operation
% rotation of the
%nonidentical portions of the path 
%(in 
%this case, a rotation of $\pi/4$ about 
%the point $P$) 
be related to another pair 
of paths (b) such that the total number 
of reversals in the latter pair differs 
by one, so that the net contribution of 
these two pairs  is zero.}
\label{fi:pat-ope}
\end{figure}
Since
the value of the contribution of any pair of paths
depends only on the number of reversals involved,
the 
contributions to the summation of these two sets of paths
therefore cancels (Fig.~\ref{fi:pat-ope}).
The same result can of course be obtained without resorting to
graphical methods by representing each
path as a string of $T$ choices from the
set of possible directions, and then showing
that the contributions from certain
classes of strings cancel.

To extend these results to a torus of finite length, it is 
also  necessary to consider terms in the
sum representing pairs of paths that, when connected and
considered as one path, 
describe a loop that
is not homotopic to a point.
(The reference here is of course to the continuous paths
that can be obtained from the discrete ones by
imbedding the lattice in a Euclidean space, and
by likewise transforming any steps between two
lattice points into
a continuous path along the straight line
segment joining those points.)

If the two axes of the torus are of equal length,
then the contributions from those pairs of paths
that wrap around the torus along one direction,
may be seen to cancel the contributions of
those that wrap in the perpendicular direction.  However,
suppose the lengths of the lattice along the two axes
are unequal.  Then at some time, there will be
a pair of paths describing a loop around the shorter
length of the torus that is
not cancelled by a pair of paths along the other axis, since
only pairs of paths that compose closed loops enter into
the summation (Fig.~\ref{fi:axi-neq}).
\begin{figure}
%Show a torus that looks exactly like an
%inner tube resting stably on the floor.
%Show the two paths as arrows that,
%when taken together (they will
%emanate from the same point in
%opposite directions and meet at
%the point 180 degrees north or south,)
%define a sub-great circle of the torus,
%i.e., the path of shortest length
%that cannot be homeomorphically 
%shrunk to a point.
%\rule{.2pt}{13em}
\psfig{file=hx/twfig/twf-torus-2.ps,height=2.286in,width=3.0in}
%box is 2 3/4" by 6 3/8", lower left corner 5/8" 6 3/4"
%                         upper right corner 6 7/8" 10 1/2"
\vspace{2\baselineskip}
\caption{If the two axes of the %
torus are of unequal  length, % 
say, $L_y > L_x$, then at %
$t = L_{x}/2$,  a pair of paths
forming a loop along the $x$-axis
will not be cancelled by a
corresponding
loop along the $y$-direction, since
such loops could only be made of paths
of length greater than $t$.}
\vspace{2.0\baselineskip}
\label{fi:axi-neq}
\end{figure}

In $n>2$
dimensions, the proof is more tedious, because
the coefficients in the transition matrix no longer have the same modulus. A 
pair of nonidentical
paths, that diverge with one of them making a reversal,
must be added to the contributions of $n-1$
pairs of nonidentical paths that diverge
without either member of the pair making a reversal
\begin{figure}
%Show on the left two paths that terminate at same point
%but diverge with one of them making a reversal;
%show on the right two paths that diverge without either
%making a reversal.  In each case
%distinguish the two paths involved by marking
%them with different lines, say, dotted and dashed.
%\rule{.2pt}{13em}
\psfig{file=hx/twfig/twf-div-pat-2.ps,height=1.23in,width=3.0in}
\vspace{2\baselineskip}
%box is 2 3/4" by 6 3/8", lower left corner 1/2" 6 3/4"
%                         upper right corner 6 7/8" 9 1/2"
\caption{The pair of paths on the left diverges with 
one path making a reversal; the pair on the right
diverges with neither of the paths making a reversal.
(The solid line indicates
the portions of the two paths that are identical.)}
\vspace{2.0\baselineskip}
\label{fi:div-pat}
\end{figure} 
(Fig.~\ref{fi:div-pat}) in 
order to obtain a cancellation.
(Adding pairs of paths together means of course,
adding the terms in the summation that may be related to
the path pairs in question.)
Similarly, in extending the result
to an $n$-di\-men\-sion\-al torus,
a given number of 
pairs representing traversals
in one direction will in general
cancel a contribution from
some unequal number of
terms representing traversals
along some other direction,
but only if the axes of the torus are of equal length.
Otherwise, the proof is similar
to the two-di\-men\-sion\-al case.

%Note again that the proof holds only
%for a single hodotic solution.
%In general, any combination of
%two or more hodotic solutions			
%produces a wave solutions such that the square of the
%amplitude  summed over the whole 
%space is not conserved.

We turn next to another curious
property of the ho\-do\-tic solutions, that
becomes useful in decomposing solutions of the
wave equation into components other than the
$2n$ traveling waves discussed above.
Let {\it promotion} of an $m$-di\-men\-sion\-al
solution of (\ref{eq:wave}) to an $n$-di\-men\-sion\-al
space, with $n > m$,  denote
the following procedure:
\begin{quote}
take the ($m$-di\-men\-sion\-al) hodotic
solution at any two successive
time values and imbed it in a space of
dimension $n$. That is, use the solution as the boundary
conditions on a hyperplane of dimension $m$
in an $n$-di\-men\-sion\-al space, with
the solution being 0 everywhere outside
the hyperplane. Afterwards, iterate according
to (\ref{eq:wave}), (the speed of light
becoming $\sqrt{1/n}$).
\end{quote}

%As a concrete example, 
%suppose that in a four-di\-men\-sion\-al
%space at time $t = 0$ and 1,
%we assign the value of 0 to all points
%outside of a two-di\-men\-sion\-al
%plane intersecting the origin.
%On this plane, the function at
%time 0 will be identical to
%the two-di\-men\-sion\-al hodotic solution
%described above at say $t = 14$,
%while the function at $t = 1$
%will be identical to the 
%two-di\-men\-sion\-al hodotic solution
%at $t = 15$.  
%We have thus promoted
%the two-di\-men\-sion\-al hodotic
%solution (at time 15) to
%a four-di\-men\-sion\-al space.

The hodotic solutions
have the interesting property
that they may be iterated
an arbitrary number of times,
promoted to a higher dimension,
iterated again for some
other arbitrary number of times,
promoted to a still 
high\-er\-di\-men\-sion\-al subspace, and
so on, all the while producing a
solution whose discrete integrals
over the corresponding space
of the point-so\-lu\-tion, as well as its square,
are constant. The proof is
similar to the one outlined previously;
classes of pairs of nonidentical
paths can again be shown to cancel,
although for the purposes of
the proof, the number of classes
to which a pair of paths may belong
varies according to the time at which
the two members of the pair diverged
(more precisely, according to
whether they diverged before or after the given imbedding
in the higher-di\-men\-sion\-al space).
Note
that in general, this
result does not hold for
a torus of any finite 
 length.

Finally, note that
the $n$-di\-men\-sion\-al
hodotic solutions,
are simply the promotions
of one-di\-men\-sion\-al hodotic solutions 
to the appropriate higher dimension.




\vspace{.6\baselineskip}{\obeylines
\noindent {\large \bf References}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
\noindent$^1$D.~Farmer, T.~Toffoli, S.~Wolfram, {\sl Cellular Automata\/},
(North-Holland, Am\-ster\-dam, 1984). 
T.~Toffoli and N.~Margolus,
{\sl Cellular Automata Machines, A New Environment for
Modeling\/}, MIT Press, Cambridge, Massachusetts, 1987.


\medskip

\noindent$^2$R.~Courant, K.~Lewy, and O.~Friedrichs,
``On the Partial Difference Equations of Mathematical Physics'', originally
published in {\em Mathematische Annalen}, 1928, and translated into
English in the {\em IBM Journal}, March, 1967. Quantities such as the
momentum and energy, that are conserved and that converge in the
continuum limit to their usual definitions, also exist.

\medskip

\noindent$^3$S.~Gudder, {\it Quantum Probability,} Boston, 1988.

\medskip

\noindent$^4$W.~Feller, {\sl An Introduction 
to Probability Theory and Its Applications\/}, Vol.~I,  
(John Wiley \& Sons, Inc., New York, 1968).

\medskip

\noindent$^5$T. Toffoli and N. Margolus, {\it op.~cit.}

\medskip

\noindent$^6$For example, R.~P.~Feynman, 
R.~B.~Leighton, and M. Sands, {\sl The Feynman
Lectures on Physics}, Vol.~III, 
(Addison-Wesley, Reading, Massachusetts, 1966), Chap.~1.

\medskip

\noindent$^7$R. P. Feynman and A. R. Hibbs, 
{\sl Quantum Mechanics and Path Integrals\/}
(McGraw-Hill, New York, 1965), Prob.~2-6.

\medskip

\noindent$^8$H. Hrgov\v ci\'c,
``A negative probability approach to quantum mechanics'', 
MIT Technical Report 441.



\end{document}





PUT THE (FINISHED VERSION OF THESE) NEXT MATHY PARAGRAPH DOWN AWAYS?; OTHERWISE, KILL?
We thus have arrived at a statistical version of a flow.
In summary, Let us notate all this:  ${\cal C}({\bf x};t)$ is the counter value
of the particles encountered at the point ${\bf x}$ REDO THIS LAST SENTENCE.
of a large, possibly infinite set of identical lattices, or correspondingly, a large set
of corresponding subregions in one lattice.  We will refer to it as
the sample space of lattices, and for any given spacetime point
in one lattice, there will be a corresponding point in all the other lattices.
 Suppose that we wish to simulate
some wave function, which may be written as
a linear combination of hodotic solutions
$$\psi({\bf x};t) = \sum{{\bf x},\sigma} A({\bf x'}h_{\sigma}({\bf x} - {\bf x'};t}) $$

Let $$E\left({\cal C}({\bf x})\right)$$ is the expected value of the counter.
We require that when we select any $M$-member subset of sample space lattices
that the quantity $E\left({\cal C}({\bf x};t)\right/M)$ approach $\psi({\bf x}$
as $M$ becomes large.

Then $$E\left({\cal C}((x,y);t)\right)=E\left({\cal C}((x+1,y);t)\right)\ellipsis$$
$$E\left(\product_{i=1}^{M}{\cal C}({\bf x_i};t)\right)%
=\product_{i=1}^{N} E\left({\cal C}({\bf x_i};t)\right)$$
$$\sum_{\rm input} E\left({\rm inputs};t)\right) %
= \sum_{\rm outputs}E\left({\rm outputs};t)\right)$$

??For the future, we shall require that the lattices are statistically independent.??















Define $\delta(x-x_0,y-y_0)$ to be the function whose value
at the point ${\bf{x}}_0$ is 1, and 0 everywhere else, and
let the boundary conditions of the wave equation
be the following:
$$\psi(\bf{x}, -1) = \delta(x+1,y)\quad \psi(\bf{x},0) = \delta(x,y)\eqno(3)$$
so that we have a situation as if all the particles of
some gas, which were originally localized at the point $(-1,0)$,
have moved to the origin at time $t=0$ by 
traveling in the positive-$x$ direction. 
This solution and the ones corresponding to
motions in the positive or negative directions 
along any axis, which we shall refer to as
the {\it{hodotic}} solutions, have several interesting properties;
the discrete integral of $\psi$ and $\psi^2$ over all space
is equal to 1 at all times, and this is true
regardless of the dimension of the space.
(All the properties listed here, except
where otherwise noted, also
hold for the $n$-di\-men\-sion\-al torus as well.)
This may be proved by induction and
we leave the details as an exercise.
{\tt{See Appendix}} We are therefore encouraged
in viewing these solutions as amenable to a 
fluid interpretation.


{\tt{delete the following paragraph in
any condensed version of this topic.}}
Let us examine the two-di\-men\-sion\-al solution
described above more closely 
at $t = 2$ (Fig. 1)
Note that if we were to consider the
squares of the solutions as
a probability distribution
of particles that immediately
prior to $t=0$ were moving
in the direction along the
positive-$x$ axis, we
might suggest the following
as a description of the
dynamics of the system:
a fluid particle moving in
a given direction immediately
prior to a given time has
a probability of $1/4$
of moving in any direction
immediately after this
time with the proviso
that if it moves in
the direction exactly
opposite to its
subsequently prior direction of
motion, it becomes
``negative''.  That is,
a particle carries
with itself
an extra bit of information
which may be likened to a sign.







	      
{\tt  The remainder discusses how
to view these results in terms of
summations of paths and other
discrete motions, and ends with
a generalization of Feynman's
discrete 1-D version of
the Dirac equation to $n$ dimensions.}
\end{document} 
{\tt Should I men\-tion Pen\-rose lat\-tices, 
and how they may or may not be used to 
si\-mu\-late the wave equa\-tion, or should 
I rather save it for ano\-ther pa\-per?
Is there any\-thing else I should add, 
aside from my ac\-know\-ledge\-ments to you 
and the group?}

Explain why, if we want to represent the wave equation
by a set of solutions that obey a continuity equation
we have a puzzle.  Although there is a positive-definite
quantity which is conserved, it is difficult to interpret it
as a density because of the relations it must satisfy,
{\tt Delete these next two paragraphs at first.}
Write a paragraph or two giving your motivation
in writing this stuff. say: We in the
CAM world are interested in simulating
the fundamental equations of physics via
lattice method.  Furthermore, since
CAM's do relativistic hydrodynamics
in terms  of lattice gases,
and since indeed, quantum mechanics
makes nature itself seem like
a computation of wave like phenomena
using particles, we desire to
implement such equations in terms of
statistical averages of ensemble of particles although there
are admittedly phenomena in
physics, such as the EPR experiment,
that suggest that such attempts 
are at best doomed to produce no more than
a sometimes useful approximation.

Other problems are immediately apparent.
It is much easier to work with
dynamical laws that conserve
the numbers of particles involved,
so that we would like to have
access to conserved quantities
which may be simulated as densities.
In the case of the wave equation,
we are confronted with a problem
in that the wave equation readily admits
solutions whose integrated square
is not conserved.  Although conserved,
energy-like, densities exist, it
is difficult, on the lattice level
to interpret these quantities
of densities of particles,
as they involve gradients and
time derivatives.
??????The best that we can do is
to compute the wave in some
other way and have the particle float along
(pilot wave)???????

The two-di\-men\-sion\-al case.  Keep the same motivation i.e., note that each of
the component wave equations may be obtained by considering the $2n$
solutions of the wave equation in which we have a moving delta function.

Mark's commutation diagram? 

We immediately have an equation that satisfies the Klein-Gordon equation.
(Reference Feynman, Bohm)

(Note you can still do higher dimensions as densities in terms of $1/(n^2)$
if the back scattered waves split into components each of which
adds up to (n-1).

\section{Appendix}
Define $\psi({\bf x}, t)$ to be
$\sum_\sigma f_\sigma$, i.e., the sum
of the hodotic solutions at any given point.
\begin{itemize}
\item[{\bf Theorem 1}] $\psi({\bf x},t)$ obeys 
the wave equation.

\item[{\bf Proof 1}] We give the proof for
the two-di\-men\-sion\-al case. $\psi({\bf x},t+1)$
may be obtained from the outgoing
traveling waves at neighboring points
at the previous time step as
$$ \psi(x,y,t+1) = {1\over2} f^{\rm{out}}_{x+}(x-1) %
                     - {1\over2} f^{\rm{out}}_{x-}(x+1) %
                     + {1\over2} f^{\rm{out}}_{y+}(y-1) %
                     + {1\over2} f^{\rm{out}}_{y-}(y+1)
        \eqno(?)$$
Similarly, $\psi({\bf x},t-1)$ may be written as terms
of waves at the successive time step as
$$ \psi(x,y,t-1) = - {1\over2} f^{\rm{in}}_{x+}(x+1) %
                     + {1\over2} f^{\rm{in}}_{x-}(x-1) %
                     + {1\over2} f^{\rm{in}}_{y+}(y+1) %
                     + {1\over2} f^{\rm{in}}_{y-}(y-1)
        \eqno(?)$$
By using Eqs. 4 to eliminate either the ingoing or outgoing
terms and adding the above two equations together,
we arrive at the discrete wave equation. \bullet
\end{itemize}

Let ${\displaystyle \sum_{\bf x}}f$, be the
sum over all space of some function 
at some instant of time.

\begin{itemize}
\item[{\bf Theorem 2}] For all non-negative times and for all
solutions of the wave equation expressed at $t=0$
in terms of the hodotic solutions,
$I_1(t)\equiv\displaystyle{ \sum_{\bf x}}f$ and 
$I_2(t)\equiv=\displaystyle{ \sum_{{\bf x}, \sigma}}\psi_{\sigma}$
are respectively equal to their initial values.
\item[{\bf Proof 2}]
By summing (\ref{wave}) over all space, we have
$$ I_1(t+1) + I_1(t-1) = I(t) \eqno(?)$$
Eqs.



Finally, we prove some results for solutions that
may be written as a single hodotic solution.
For the solution $\psi_\sigma$ of
the wave equation which coincides at times $t=0,1$ with a hodotic solution,
$\displaystyle{ \sum_{\bf x}}\psi_{\sigma}^2$, i.e., the point-wise sime of the square of
the solution without considering its traveling wave components, is equal to 1 for all
non-negative times.
\item[{\bf Proof 2}]
\end{itemize}

We shall not discuss the time-reversal properties of these solutions, except
to remark that they may be defined so as to make the above theorem hold for
all times, negative as well as positive.
\vfill\
\end{document}

%some other conserved quantities, in various dimensions
%and names.


% Some of these are conserved, for any solution of eq 2, some only for hodotic solutions.
%(list sum-phi (float sum-sq-phi)
%				       (list f-mom-y-2 b-mom-y-2)  ; f and b imply forward and backward
%				       (list f-mom-x-2 b-mom-x-2)
%				       (list (/ (+ f-energy-x-2 f-energy-y-2) 2))
%				       (list (/ (+ b-energy-x-2 b-energy-y-2) 2))
%				       (list (/ (+ f-energy-x-2 b-energy-x-2
%						   f-energy-y-2 b-energy-y-2) 2))))
%(setf sum-phi (+ sum-phi (aref tp i j))
%	      sum-sq-phi (+ sum-sq-phi (* (aref tp i j) (conjugate (aref tp i j))))
%	      f-mom-x-2 (+ f-mom-x-2 (* (^ -1 (mod i 2)) (- (aref tp i j) (aref tb (mod (+ i 1) n) j))))
%	      b-mom-x-2 (+ b-mom-x-2 (* (^ -1 (mod i 2)) (- (aref tp (mod (+ i 1) n) j) (aref tb i j))))
%	      f-mom-y-2 (+ f-mom-y-2 (* (^ -1 (mod j 2)) (- (aref tp i j) (aref tb i (mod (+ j 1) m)))))
%	      b-mom-y-2 (+ b-mom-y-2 (* (^ -1 (mod j 2)) (- (aref tp i (mod (+ j 1) m)) (aref tb i j))))
%	      f-energy-x-2 (+ f-energy-x-2 (* (- (aref tp i j) (aref tb (mod (+ i 1) n) j))
%					      (conjugate (- (aref tp i j) (aref tb (mod (+ i 1) n) j)))))
%	      b-energy-x-2 (+ b-energy-x-2 (* (- (aref tp  (mod (+ i 1) n) j) (aref tb i j))
%					      (conjugate (- (aref tp  (mod (+ i 1) n) j) (aref tb i j)))))				%  
%	      f-energy-y-2 (+ f-energy-y-2 (* (- (aref tp i j) (aref tb i (mod (+ j 1) m)))
%					      (conjugate (- (aref tp i j) (aref tb i (mod (+ j 1) m))))))
%	      b-energy-y-2 (+ b-energy-y-2 (* (- (aref tp i (mod (+ j 1) m)) (aref tb i j))
%					      (conjugate (- (aref tp i (mod (+ j 1) m)) (aref tb i j))))))))))

% 
%		sum-cross-sq-phi (+ sum-cross-sq-phi
%				(* (conjugate (- (aref xp i j) (aref xn i j)))
%				   (- (aref xp i j) (aref xn i j)))
%				(* (conjugate (- (aref yp i j) (aref yn i j)))
%				   (- (aref yp i j) (aref yn i j))))
%		error-sq (+ error-sq
%			    (* (conjugate (- temp-sum (aref tp i j)))
%			       (- temp-sum (aref tp i j))))))))))

% sum-across-sq-phi (+ sum-across-sq-phi
%				      (* (- (aref tp (mod (+ i 1) n) j k l)
%					    (aref tp (mod (- i 1) n) j k l))
%					 (conjugate (- (aref tp (mod (+ i 1) n) j k l)
%						       (aref tp (mod (- i 1) n) j k l))))
%				      (* (- (aref tp i (mod (+ j 1) m) k l)
%					    (aref tp i (mod (- j 1) m) k l))
%					 (conjugate (- (aref tp i (mod (+ j 1) m) k l)
%						       (aref tp i (mod (- j 1) m) k l))))
%				      (* (- (aref tp i j (mod (+ k 1) o) l)
%					    (aref tp i j (mod (- k 1) o) l))
%					 (conjugate (- (aref tp i j (mod (+ k 1) o) l)
%						       (aref tp i j (mod (- k 1) o) l))))
%				      (* (- (aref tp i j k (mod (+ l 1) p))
%					    (aref tp i j k (mod (- l 1) p)))
%					 (conjugate (- (aref tp i j k (mod (+ l 1) p))
%						       (aref tp i j k (mod (- l 1) p)))))))))))))
\end{document}    
